diff --git a/jupyter/radon/LICENSE b/jupyter/radon/LICENSE new file mode 100644 index 00000000..8b782ee2 --- /dev/null +++ b/jupyter/radon/LICENSE @@ -0,0 +1,354 @@ +The code is distributed under the BSD 3-clause license. + +The text and images are distributed under the CC BY-ND 4.0 license. + +====================================================================== + +BSD 3-Clause License + +Copyright (c) 2018--2022, Stan Development Team and their Assignees All +rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + +* Redistributions of source code must retain the above copyright +notice, this list of conditions and the following disclaimer. + +* Redistributions in binary form must reproduce the above copyright +notice, this list of conditions and the following disclaimer in the +documentation and/or other materials provided with the distribution. + +* Neither the name of the copyright holder nor the names of its + contributors may be used to endorse or promote products derived from + this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. 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If the provision + cannot be reformed, it shall be severed from this Public License + without affecting the enforceability of the remaining terms and + conditions. + + c. No term or condition of this Public License will be waived and no + failure to comply consented to unless expressly agreed to by the + Licensor. + + d. Nothing in this Public License constitutes or may be interpreted + as a limitation upon, or waiver of, any privileges and immunities + that apply to the Licensor or You, including from the legal + processes of any jurisdiction or authority. + +====================================================================== \ No newline at end of file diff --git a/jupyter/radon/README.md b/jupyter/radon/README.md index e69de29b..90d9acaf 100644 --- a/jupyter/radon/README.md +++ b/jupyter/radon/README.md @@ -0,0 +1,6 @@ +Notebook based on Gelman and Hill 2007, Radon case study. +Introduces hierarchical models and Python packages `cmdstanpy`, and `plotnine`. + +Author: Mitzi Morris + + diff --git a/jupyter/sum-to-zero/LICENSE b/jupyter/sum-to-zero/LICENSE new file mode 100644 index 00000000..8b782ee2 --- /dev/null +++ b/jupyter/sum-to-zero/LICENSE @@ -0,0 +1,354 @@ +The code is distributed under the BSD 3-clause license. + +The text and images are distributed under the CC BY-ND 4.0 license. + +====================================================================== + +BSD 3-Clause License + +Copyright (c) 2018--2022, Stan Development Team and their Assignees All +rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + +* Redistributions of source code must retain the above copyright +notice, this list of conditions and the following disclaimer. + +* Redistributions in binary form must reproduce the above copyright +notice, this list of conditions and the following disclaimer in the +documentation and/or other materials provided with the distribution. + +* Neither the name of the copyright holder nor the names of its + contributors may be used to endorse or promote products derived from + this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +====================================================================== + +Creative Commons Attribution-NoDerivatives 4.0 International Public +License + +By exercising the Licensed Rights (defined below), You accept and agree +to be bound by the terms and conditions of this Creative Commons +Attribution-NoDerivatives 4.0 International Public License ("Public +License"). To the extent this Public License may be interpreted as a +contract, You are granted the Licensed Rights in consideration of Your +acceptance of these terms and conditions, and the Licensor grants You +such rights in consideration of benefits the Licensor receives from +making the Licensed Material available under these terms and +conditions. + + +Section 1 -- Definitions. + + a. Adapted Material means material subject to Copyright and Similar + Rights that is derived from or based upon the Licensed Material + and in which the Licensed Material is translated, altered, + arranged, transformed, or otherwise modified in a manner requiring + permission under the Copyright and Similar Rights held by the + Licensor. For purposes of this Public License, where the Licensed + Material is a musical work, performance, or sound recording, + Adapted Material is always produced where the Licensed Material is + synched in timed relation with a moving image. + + b. Copyright and Similar Rights means copyright and/or similar rights + closely related to copyright including, without limitation, + performance, broadcast, sound recording, and Sui Generis Database + Rights, without regard to how the rights are labeled or + categorized. For purposes of this Public License, the rights + specified in Section 2(b)(1)-(2) are not Copyright and Similar + Rights. + + c. Effective Technological Measures means those measures that, in the + absence of proper authority, may not be circumvented under laws + fulfilling obligations under Article 11 of the WIPO Copyright + Treaty adopted on December 20, 1996, and/or similar international + agreements. + + d. Exceptions and Limitations means fair use, fair dealing, and/or + any other exception or limitation to Copyright and Similar Rights + that applies to Your use of the Licensed Material. + + e. Licensed Material means the artistic or literary work, database, + or other material to which the Licensor applied this Public + License. + + f. Licensed Rights means the rights granted to You subject to the + terms and conditions of this Public License, which are limited to + all Copyright and Similar Rights that apply to Your use of the + Licensed Material and that the Licensor has authority to license. + + g. Licensor means the individual(s) or entity(ies) granting rights + under this Public License. + + h. Share means to provide material to the public by any means or + process that requires permission under the Licensed Rights, such + as reproduction, public display, public performance, distribution, + dissemination, communication, or importation, and to make material + available to the public including in ways that members of the + public may access the material from a place and at a time + individually chosen by them. + + i. Sui Generis Database Rights means rights other than copyright + resulting from Directive 96/9/EC of the European Parliament and of + the Council of 11 March 1996 on the legal protection of databases, + as amended and/or succeeded, as well as other essentially + equivalent rights anywhere in the world. + + j. You means the individual or entity exercising the Licensed Rights + under this Public License. Your has a corresponding meaning. + + +Section 2 -- Scope. + + a. License grant. + + 1. Subject to the terms and conditions of this Public License, + the Licensor hereby grants You a worldwide, royalty-free, + non-sublicensable, non-exclusive, irrevocable license to + exercise the Licensed Rights in the Licensed Material to: + + a. reproduce and Share the Licensed Material, in whole or + in part; and + + b. produce and reproduce, but not Share, Adapted Material. + + 2. Exceptions and Limitations. For the avoidance of doubt, where + Exceptions and Limitations apply to Your use, this Public + License does not apply, and You do not need to comply with + its terms and conditions. + + 3. Term. The term of this Public License is specified in Section + 6(a). + + 4. Media and formats; technical modifications allowed. The + Licensor authorizes You to exercise the Licensed Rights in + all media and formats whether now known or hereafter created, + and to make technical modifications necessary to do so. The + Licensor waives and/or agrees not to assert any right or + authority to forbid You from making technical modifications + necessary to exercise the Licensed Rights, including + technical modifications necessary to circumvent Effective + Technological Measures. For purposes of this Public License, + simply making modifications authorized by this Section 2(a) + (4) never produces Adapted Material. + + 5. Downstream recipients. + + a. Offer from the Licensor -- Licensed Material. Every + recipient of the Licensed Material automatically + receives an offer from the Licensor to exercise the + Licensed Rights under the terms and conditions of this + Public License. + + b. No downstream restrictions. You may not offer or impose + any additional or different terms or conditions on, or + apply any Effective Technological Measures to, the + Licensed Material if doing so restricts exercise of the + Licensed Rights by any recipient of the Licensed + Material. + + 6. No endorsement. Nothing in this Public License constitutes or + may be construed as permission to assert or imply that You + are, or that Your use of the Licensed Material is, connected + with, or sponsored, endorsed, or granted official status by, + the Licensor or others designated to receive attribution as + provided in Section 3(a)(1)(A)(i). + + b. Other rights. + + 1. Moral rights, such as the right of integrity, are not + licensed under this Public License, nor are publicity, + privacy, and/or other similar personality rights; however, to + the extent possible, the Licensor waives and/or agrees not to + assert any such rights held by the Licensor to the limited + extent necessary to allow You to exercise the Licensed + Rights, but not otherwise. + + 2. Patent and trademark rights are not licensed under this + Public License. + + 3. To the extent possible, the Licensor waives any right to + collect royalties from You for the exercise of the Licensed + Rights, whether directly or through a collecting society + under any voluntary or waivable statutory or compulsory + licensing scheme. In all other cases the Licensor expressly + reserves any right to collect such royalties. + + +Section 3 -- License Conditions. + +Your exercise of the Licensed Rights is expressly made subject to the +following conditions. + + a. Attribution. + + 1. If You Share the Licensed Material, You must: + + a. retain the following if it is supplied by the Licensor + with the Licensed Material: + + i. identification of the creator(s) of the Licensed + Material and any others designated to receive + attribution, in any reasonable manner requested by + the Licensor (including by pseudonym if + designated); + + ii. a copyright notice; + + iii. a notice that refers to this Public License; + + iv. a notice that refers to the disclaimer of + warranties; + + v. a URI or hyperlink to the Licensed Material to the + extent reasonably practicable; + + b. indicate if You modified the Licensed Material and + retain an indication of any previous modifications; and + + c. indicate the Licensed Material is licensed under this + Public License, and include the text of, or the URI or + hyperlink to, this Public License. + + For the avoidance of doubt, You do not have permission under + this Public License to Share Adapted Material. + + 2. You may satisfy the conditions in Section 3(a)(1) in any + reasonable manner based on the medium, means, and context in + which You Share the Licensed Material. For example, it may be + reasonable to satisfy the conditions by providing a URI or + hyperlink to a resource that includes the required + information. + + 3. If requested by the Licensor, You must remove any of the + information required by Section 3(a)(1)(A) to the extent + reasonably practicable. + + +Section 4 -- Sui Generis Database Rights. + +Where the Licensed Rights include Sui Generis Database Rights that +apply to Your use of the Licensed Material: + + a. for the avoidance of doubt, Section 2(a)(1) grants You the right + to extract, reuse, reproduce, and Share all or a substantial + portion of the contents of the database, provided You do not Share + Adapted Material; + b. if You include all or a substantial portion of the database + contents in a database in which You have Sui Generis Database + Rights, then the database in which You have Sui Generis Database + Rights (but not its individual contents) is Adapted Material; and + c. You must comply with the conditions in Section 3(a) if You Share + all or a substantial portion of the contents of the database. + +For the avoidance of doubt, this Section 4 supplements and does not +replace Your obligations under this Public License where the Licensed +Rights include other Copyright and Similar Rights. + + +Section 5 -- Disclaimer of Warranties and Limitation of Liability. + + a. UNLESS OTHERWISE SEPARATELY UNDERTAKEN BY THE LICENSOR, TO THE + EXTENT POSSIBLE, THE LICENSOR OFFERS THE LICENSED MATERIAL AS-IS + AND AS-AVAILABLE, AND MAKES NO REPRESENTATIONS OR WARRANTIES OF + ANY KIND CONCERNING THE LICENSED MATERIAL, WHETHER EXPRESS, + IMPLIED, STATUTORY, OR OTHER. THIS INCLUDES, WITHOUT LIMITATION, + WARRANTIES OF TITLE, MERCHANTABILITY, FITNESS FOR A PARTICULAR + PURPOSE, NON-INFRINGEMENT, ABSENCE OF LATENT OR OTHER DEFECTS, + ACCURACY, OR THE PRESENCE OR ABSENCE OF ERRORS, WHETHER OR NOT + KNOWN OR DISCOVERABLE. WHERE DISCLAIMERS OF WARRANTIES ARE NOT + ALLOWED IN FULL OR IN PART, THIS DISCLAIMER MAY NOT APPLY TO YOU. + + b. TO THE EXTENT POSSIBLE, IN NO EVENT WILL THE LICENSOR BE LIABLE + TO YOU ON ANY LEGAL THEORY (INCLUDING, WITHOUT LIMITATION, + NEGLIGENCE) OR OTHERWISE FOR ANY DIRECT, SPECIAL, INDIRECT, + INCIDENTAL, CONSEQUENTIAL, PUNITIVE, EXEMPLARY, OR OTHER LOSSES, + COSTS, EXPENSES, OR DAMAGES ARISING OUT OF THIS PUBLIC LICENSE OR + USE OF THE LICENSED MATERIAL, EVEN IF THE LICENSOR HAS BEEN + ADVISED OF THE POSSIBILITY OF SUCH LOSSES, COSTS, EXPENSES, OR + DAMAGES. WHERE A LIMITATION OF LIABILITY IS NOT ALLOWED IN FULL OR + IN PART, THIS LIMITATION MAY NOT APPLY TO YOU. + + c. The disclaimer of warranties and limitation of liability provided + above shall be interpreted in a manner that, to the extent + possible, most closely approximates an absolute disclaimer and + waiver of all liability. + + +Section 6 -- Term and Termination. + + a. This Public License applies for the term of the Copyright and + Similar Rights licensed here. However, if You fail to comply with + this Public License, then Your rights under this Public License + terminate automatically. + + b. Where Your right to use the Licensed Material has terminated under + Section 6(a), it reinstates: + + 1. automatically as of the date the violation is cured, provided + it is cured within 30 days of Your discovery of the + violation; or + + 2. upon express reinstatement by the Licensor. + + For the avoidance of doubt, this Section 6(b) does not affect any + right the Licensor may have to seek remedies for Your violations + of this Public License. + + c. For the avoidance of doubt, the Licensor may also offer the + Licensed Material under separate terms or conditions or stop + distributing the Licensed Material at any time; however, doing so + will not terminate this Public License. + + d. Sections 1, 5, 6, 7, and 8 survive termination of this Public + License. + + +Section 7 -- Other Terms and Conditions. + + a. The Licensor shall not be bound by any additional or different + terms or conditions communicated by You unless expressly agreed. + + b. Any arrangements, understandings, or agreements regarding the + Licensed Material not stated herein are separate from and + independent of the terms and conditions of this Public License. + + +Section 8 -- Interpretation. + + a. For the avoidance of doubt, this Public License does not, and + shall not be interpreted to, reduce, limit, restrict, or impose + conditions on any use of the Licensed Material that could lawfully + be made without permission under this Public License. + + b. To the extent possible, if any provision of this Public License is + deemed unenforceable, it shall be automatically reformed to the + minimum extent necessary to make it enforceable. If the provision + cannot be reformed, it shall be severed from this Public License + without affecting the enforceability of the remaining terms and + conditions. + + c. No term or condition of this Public License will be waived and no + failure to comply consented to unless expressly agreed to by the + Licensor. + + d. Nothing in this Public License constitutes or may be interpreted + as a limitation upon, or waiver of, any privileges and immunities + that apply to the Licensor or You, including from the legal + processes of any jurisdiction or authority. + +====================================================================== \ No newline at end of file diff --git a/jupyter/sum-to-zero/README.md b/jupyter/sum-to-zero/README.md new file mode 100644 index 00000000..b575cc82 --- /dev/null +++ b/jupyter/sum-to-zero/README.md @@ -0,0 +1,17 @@ +Notebooks to demonstrate correctness and efficiency of constrained parameter type `sum_to_zero_vector`, +introduced in Stan 2.36. + +HTML page has all necessary resources. To rebuild, add `stan-dev/quarto-config` submodule. + +Included in the GitHub repository for this notebook are several python files of helper functions. + +* eval_efficiencies.py - run models repeatedly, get average performance stats. +* utils.py - simulate data for binomial model. +* utils\_html.py - format Stan summaries for this notebook. +* utils\_bym2.py - compute data inputs to the BYM2 model. +* utils\_nyc\_map.py - munge the New York City census tract map. + +Author: Mitzi Morris + + + diff --git a/jupyter/sum-to-zero/binomial_runtimes_large.json b/jupyter/sum-to-zero/binomial_runtimes_large.json new file mode 100644 index 00000000..6230346e --- /dev/null +++ b/jupyter/sum-to-zero/binomial_runtimes_large.json @@ -0,0 +1 @@ +{"runtime, ave":{"ozs large":2.41,"hard large":3.25,"soft large":17.41},"runtime, std dev":{"ozs large":0.08,"hard large":0.09,"soft large":0.5},"ESS_bulk\/s":{"ozs large":1305.86,"hard large":981.53,"soft large":181.15}} diff --git a/jupyter/sum-to-zero/binomial_runtimes_small.json b/jupyter/sum-to-zero/binomial_runtimes_small.json new file mode 100644 index 00000000..6c390e72 --- /dev/null +++ b/jupyter/sum-to-zero/binomial_runtimes_small.json @@ -0,0 +1 @@ +{"runtime, ave":{"ozs small":2.21,"hard small":3.07,"soft small":54.49},"runtime, std dev":{"ozs small":0.05,"hard small":0.08,"soft small":2.54},"ESS_bulk\/s":{"ozs small":1327.11,"hard small":988.47,"soft small":63.14}} diff --git a/jupyter/sum-to-zero/data/nyc_study.geojson b/jupyter/sum-to-zero/data/nyc_study.geojson new file mode 100644 index 00000000..dd7285d2 --- /dev/null +++ b/jupyter/sum-to-zero/data/nyc_study.geojson @@ -0,0 +1,2102 @@ +{ +"type": "FeatureCollection", +"name": "nyc_poptracts", +"crs": { "type": "name", "properties": { "name": "urn:ogc:def:crs:EPSG::2263" } }, +"features": [ +{ "type": "Feature", "properties": { "CTLabel": "2", "BoroCode": "2", "BoroName": "Bronx", "CT2010": "000200", "BoroCT2010": "2000200", "CDEligibil": "I", 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pd +from typing import Any, Dict, List, Tuple +from cmdstanpy import CmdStanModel, set_cmdstan_path + + +import logging +cmdstanpy_logger = logging.getLogger("cmdstanpy") +cmdstanpy_logger.setLevel(logging.FATAL) + +import warnings +warnings.filterwarnings('ignore') + +set_cmdstan_path(os.path.join('/Users', 'mitzi', 'github', 'stan-dev', 'cmdstan')) + +from utils import simulate_data + +# Fit model and dataset for N iterations. +# For each run, save wall clock time and effective samples / second (N_Eff/sec) +# Return np.ndarray of size (N,2) with timing information. +def time_fits(N: int, model: CmdStanModel, data: dict) -> np.ndarray: + fit_times = np.ndarray(shape=(N, 2), dtype=float) + for i in range(N): + print('Run', i) + fit = model.sample(data=data, parallel_chains=4, + show_progress=False, show_console=False, refresh=10_000) + fit_summary = fit.summary() + total_time = 0 + times = fit.time + for j in range(len(times)): + total_time += times[j]['total'] + + fit_times[i, 0] = total_time + fit_times[i, 1] = fit_summary.loc['lp__', 'ESS_bulk/s'] + return fit_times + + +# Given a list of label, time pairs, populate dataframe +# of means of time and std dev wall clock time, and N_Eff/sec +def summarize_times(data_pairs: List[Tuple[str, np.ndarray]]) -> pd.DataFrame: + result_data = [] + for label, array in data_pairs: + result_data.append({ + 'label': label, + 'mean': np.mean(array, axis=0)[0], + 'std dev': np.std(array, axis=0)[0], + 'ESS_bulk/s': np.mean(array, axis=0)[1] + }) + df = pd.DataFrame(result_data) + return df.set_index('label').round(2) + + +# Create datasets - fix sizes, and seed +N_eth = 3 +N_edu = 5 +N_age = 9 +baseline = -3.5 +sens = 0.75 +spec = 0.9995 +data_small = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 17, seed=45678) +data_large = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 200, seed=45678) + +# sum to zero vector + +binomial_ozs_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_ozs.stan')) +times_ozs_large = time_fits(100, binomial_ozs_mod, data_large) +times_ozs_small = time_fits(100, binomial_ozs_mod, data_small) + +# hard sum-to-zero constraint + +binomial_hard_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_hard.stan')) +times_hard_small = time_fits(100, binomial_hard_mod, data_small) +times_hard_large = time_fits(100, binomial_hard_mod, data_large) + +# soft sum-to-zero constraint + +binomial_soft_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_soft.stan')) +times_soft_small = time_fits(100, binomial_soft_mod, data_small) +times_soft_large = time_fits(100, binomial_soft_mod, data_large) + + +df_small = summarize_times([('ozs small', times_ozs_small), + ('hard small', times_hard_small), + ('soft small', times_soft_small)]) +df_small.to_json("binomial_runtimes_small.json") + +df_large = summarize_times([('ozs large', times_ozs_large), + ('hard large', times_hard_large), + ('soft large', times_soft_large)]) +df_large.to_json("binomial_runtimes_large.json") diff --git a/jupyter/sum-to-zero/stan/binomial_4preds_hard.stan b/jupyter/sum-to-zero/stan/binomial_4preds_hard.stan new file mode 100644 index 00000000..8880155a --- /dev/null +++ b/jupyter/sum-to-zero/stan/binomial_4preds_hard.stan @@ -0,0 +1,53 @@ +// multi-level model for binomial data with 4 categorical predictors. +data { + int N; // number of strata + int N_age; + int N_eth; + int N_edu; + + array[N] int pos_tests; + array[N] int tests; + array[N] int sex; + array[N] int age; + array[N] int eth; + array[N] int edu; + + // hyperparameters + real sens; + real spec; +} +parameters { + real beta_0; + real beta_sex_raw; + real sigma_age, sigma_eth, sigma_edu; + vector[N_age - 1] beta_age_raw; + vector[N_eth - 1] beta_eth_raw; + vector[N_edu - 1] beta_edu_raw; +} +transformed parameters { + vector[2] beta_sex = [beta_sex_raw, -beta_sex_raw]'; + + vector[N_age] beta_age = append_row(beta_age_raw, -sum(beta_age_raw)); + vector[N_eth] beta_eth = append_row(beta_eth_raw, -sum(beta_eth_raw)); + vector[N_edu] beta_edu = append_row(beta_edu_raw, -sum(beta_edu_raw)); + + vector[N] eta = inv_logit(beta_0 + beta_sex[sex] + beta_age[age] + beta_eth[eth] + beta_edu[edu]); + vector[N] prob_pos_test = eta * sens + (1 - eta) * (1 - spec); +} +model { + pos_tests ~ binomial(tests, prob_pos_test); // likelihood + + // priors + beta_0 ~ normal(0, 2.5); + beta_sex ~ std_normal(); + // centered parameterization + beta_age_raw ~ normal(0, sigma_age); + beta_eth_raw ~ normal(0, sigma_eth); + beta_edu_raw ~ normal(0, sigma_edu); + sigma_eth ~ std_normal(); + sigma_age ~ std_normal(); + sigma_edu ~ std_normal(); +} +generated quantities { + array[N] inty_rep = binomial_rng(tests, prob_pos_test); +} diff --git a/jupyter/sum-to-zero/stan/binomial_4preds_hard_ppc.stan b/jupyter/sum-to-zero/stan/binomial_4preds_hard_ppc.stan new file mode 100644 index 00000000..875684ab --- /dev/null +++ b/jupyter/sum-to-zero/stan/binomial_4preds_hard_ppc.stan @@ -0,0 +1,38 @@ +// multi-level model for binomial data with 4 categorical predictors. +data { + int N; // number of strata + int N_age; + int N_eth; + int N_edu; + + // hyperparameters + real sens; + real spec; +} +parameters { + real beta_0; + real beta_sex_raw; + real sigma_age, sigma_eth, sigma_edu; + vector[N_age - 1] beta_age_raw; + vector[N_eth - 1] beta_eth_raw; + vector[N_edu - 1] beta_edu_raw; +} +transformed parameters { + vector[2] beta_sex = [beta_sex_raw, -beta_sex_raw]'; + + vector[N_age] beta_age = append_row(beta_age_raw, -sum(beta_age_raw)); + vector[N_eth] beta_eth = append_row(beta_eth_raw, -sum(beta_eth_raw)); + vector[N_edu] beta_edu = append_row(beta_edu_raw, -sum(beta_edu_raw)); +} +model { + // priors + beta_0 ~ normal(0, 2.5); + beta_sex ~ std_normal(); + // centered parameterization + beta_age_raw ~ normal(0, sigma_age); + beta_eth_raw ~ normal(0, sigma_eth); + beta_edu_raw ~ normal(0, sigma_edu); + sigma_eth ~ std_normal(); + sigma_age ~ std_normal(); + sigma_edu ~ std_normal(); +} diff --git a/jupyter/sum-to-zero/stan/binomial_4preds_ozs.stan b/jupyter/sum-to-zero/stan/binomial_4preds_ozs.stan new file mode 100644 index 00000000..6a84eb3e --- /dev/null +++ b/jupyter/sum-to-zero/stan/binomial_4preds_ozs.stan @@ -0,0 +1,62 @@ +// multi-level model for binomial data with 4 categorical predictors. +data { + int N; // number of strata + int N_age; + int N_eth; + int N_edu; + + array[N] int pos_tests; + array[N] int tests; + array[N] int sex; + array[N] int age; + array[N] int eth; + array[N] int edu; + + // hyperparameters + real sens; + real spec; +} +transformed data { + real mean_sex = mean(sex); + vector[N] sex_c = to_vector(sex) - mean_sex; + // scaling factors for marginal variances of sum_to_zero_vectors + // https://discourse.mc-stan.org/t/zero-sum-vector-and-normal-distribution/38296 + real s_age = sqrt(N_age * inv(N_age - 1)); + real s_eth = sqrt(N_eth * inv(N_eth - 1)); + real s_edu = sqrt(N_edu * inv(N_edu - 1)); +} +parameters { + real beta_0; + real beta_sex; + real sigma_age, sigma_eth, sigma_edu; + sum_to_zero_vector[N_age] beta_age; + sum_to_zero_vector[N_eth] beta_eth; + sum_to_zero_vector[N_edu] beta_edu; +} +transformed parameters { + // true prevalence + vector[N] p = inv_logit(beta_0 + beta_sex * sex_c + beta_age[age] + + beta_eth[eth] + beta_edu[edu]); + // incorporate test sensitivity and specificity. + vector[N] p_sample = p * sens + (1 - p) * (1 - spec); +} +model { + pos_tests ~ binomial(tests, p_sample); // likelihood + + // priors + beta_0 ~ normal(0, 2.5); + beta_sex ~ std_normal(); + sigma_eth ~ std_normal(); + sigma_age ~ std_normal(); + sigma_edu ~ std_normal(); + + // centered parameterization + // scale normal priors on sum_to_zero_vectors + beta_age ~ normal(0, s_age * sigma_age); + beta_eth ~ normal(0, s_eth * sigma_eth); + beta_edu ~ normal(0, s_edu * sigma_edu); +} +generated quantities { + real beta_intercept = beta_0 - mean_sex * beta_sex; + array[N] inty_rep = binomial_rng(tests, p_sample); +} diff --git a/jupyter/sum-to-zero/stan/binomial_4preds_ozs_ppc.stan b/jupyter/sum-to-zero/stan/binomial_4preds_ozs_ppc.stan new file mode 100644 index 00000000..ea733f04 --- /dev/null +++ b/jupyter/sum-to-zero/stan/binomial_4preds_ozs_ppc.stan @@ -0,0 +1,38 @@ +// generate sample from model priors, (before seeing any data) +data { + int N; // number of strata + int N_age; + int N_eth; + int N_edu; + // omit observational data +} +transformed data { + // scaling factors for marginal variances of sum_to_zero_vectors + // https://discourse.mc-stan.org/t/zero-sum-vector-and-normal-distribution/38296 + real s_age = sqrt(N_age * inv(N_age - 1)); + real s_eth = sqrt(N_eth * inv(N_eth - 1)); + real s_edu = sqrt(N_edu * inv(N_edu - 1)); +} +parameters { + real beta_0; + real beta_sex; + real sigma_age, sigma_eth, sigma_edu; + sum_to_zero_vector[N_age] beta_age; + sum_to_zero_vector[N_eth] beta_eth; + sum_to_zero_vector[N_edu] beta_edu; +} +model { + // omit likelihood + // priors + beta_0 ~ normal(0, 2.5); + beta_sex ~ std_normal(); + sigma_eth ~ std_normal(); + sigma_age ~ std_normal(); + sigma_edu ~ std_normal(); + + // centered parameterization + // scale normal priors on sum_to_zero_vectors + beta_age ~ normal(0, s_age * sigma_age); + beta_eth ~ normal(0, s_eth * sigma_eth); + beta_edu ~ normal(0, s_edu * sigma_edu); +} diff --git a/jupyter/sum-to-zero/stan/binomial_4preds_soft.stan b/jupyter/sum-to-zero/stan/binomial_4preds_soft.stan new file mode 100644 index 00000000..0b39d8e3 --- /dev/null +++ b/jupyter/sum-to-zero/stan/binomial_4preds_soft.stan @@ -0,0 +1,58 @@ +// multi-level model for binomial data with 4 categorical predictors. +data { + int N; // number of strata + int N_age; + int N_eth; + int N_edu; + + array[N] int pos_tests; + array[N] int tests; + array[N] int sex; + array[N] int age; + array[N] int eth; + array[N] int edu; + + // hyperparameters + real sens; + real spec; +} +transformed data { + real mean_sex = mean(sex); + vector[N] sex_c = to_vector(sex) - mean_sex; +} +parameters { + real beta_0; + real beta_sex; + real sigma_age, sigma_eth, sigma_edu; + vector[N_age] beta_age; + vector[N_eth] beta_eth; + vector[N_edu] beta_edu; +} +transformed parameters { + // non-standard link function + vector[N] p = inv_logit(beta_0 + beta_sex * sex_c + beta_age[age] + + beta_eth[eth] + beta_edu[edu]); + vector[N] p_sample = p * sens + (1 - p) * (1 - spec); +} +model { + pos_tests ~ binomial(tests, p_sample); // likelihood + + // priors + beta_0 ~ normal(0, 2.5); + beta_sex ~ std_normal(); + // centered parameterization + beta_age ~ normal(0, sigma_age); + beta_eth ~ normal(0, sigma_eth); + beta_edu ~ normal(0, sigma_edu); + sigma_eth ~ std_normal(); + sigma_age ~ std_normal(); + sigma_edu ~ std_normal(); + // soft sum-to-zero constraint + sum(beta_age) ~ normal(0, 0.001 * N_age); + sum(beta_eth) ~ normal(0, 0.001 * N_eth); + sum(beta_edu) ~ normal(0, 0.001 * N_edu); +} +generated quantities { + real beta_intercept = beta_0 - mean_sex * beta_sex; + array[N] inty_rep = binomial_rng(tests, p_sample); +} diff --git a/jupyter/sum-to-zero/stan/bym2_hard.stan b/jupyter/sum-to-zero/stan/bym2_hard.stan new file mode 100644 index 00000000..bc1d8b8b --- /dev/null +++ b/jupyter/sum-to-zero/stan/bym2_hard.stan @@ -0,0 +1,63 @@ + data { + int N; + array[N] int y; // count outcomes + vector[N] E; // exposure + int K; // num covariates + matrix[N, K] xs; // design matrix + + // spatial structure + int N_edges; // number of neighbor pairs + array[2, N_edges] int neighbors; // columnwise adjacent + + real tau; // scaling factor +} +transformed data { + vector[N] log_E = log(E); + // center continuous predictors + vector[K] means_xs; // column means of xs before centering + matrix[N, K] xs_centered; // centered version of xs + for (k in 1:K) { + means_xs[k] = mean(xs[, k]); + xs_centered[, k] = xs[, k] - means_xs[k]; + } +} +parameters { + real beta0; // intercept + vector[K] betas; // covariates + real rho; // proportion of spatial variance + vector[N-1] phi_raw; // spatial random effects + vector[N] theta; // heterogeneous random effects + real sigma; // scale of combined effects +} +transformed parameters { + vector[N] phi = append_row(phi_raw, -sum(phi_raw)); + vector[N] gamma = (sqrt(1 - rho) * theta + sqrt(rho / tau) * phi); // BYM2 +} +model { + y ~ poisson_log(log_E + beta0 + xs_centered * betas + gamma * sigma); + rho ~ beta(0.5, 0.5); + target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]); // ICAR + beta0 ~ std_normal(); + betas ~ std_normal(); + theta ~ std_normal(); + sigma ~ std_normal(); +} +generated quantities { + real beta_intercept = beta0 - dot_product(means_xs, betas); // adjust intercept + array[N] int y_rep; + { + vector[N] eta = log_E + beta0 + xs_centered * betas + gamma * sigma; + if (max(eta) > 26) { + // avoid overflow in poisson_log_rng + print("max eta too big: ", max(eta)); + for (n in 1:N) { + y_rep[n] = -1; + // log_lik[n] = -1; + } + } else { + for (n in 1:N) { + y_rep[n] = poisson_log_rng(eta[n]); + } + } + } +} diff --git a/jupyter/sum-to-zero/stan/bym2_multicomp.stan b/jupyter/sum-to-zero/stan/bym2_multicomp.stan new file mode 100644 index 00000000..d417b9d3 --- /dev/null +++ b/jupyter/sum-to-zero/stan/bym2_multicomp.stan @@ -0,0 +1,128 @@ +functions { + /** + * Component-wise constrain sum-to-zero vectors + * + * @param phi unconstrained vector of zero-sum slices + * @param idxs component start and end indices + * @param sizes component sizes + * @return vector phi_ozs, the vector whose slices sum to zero + */ + vector zero_sum_components(vector phi, array[ , ] int idxs, array[] int sizes) { + vector[sum(sizes)] phi_ozs; + int idx_phi = 1; + int idx_ozs = 1; + for (i in 1:size(sizes)) { + int n = sizes[i]; + phi_ozs[idx_ozs : idx_ozs + n - 1] = + zero_sum_constrain(segment(phi, idx_phi, n - 1)); + idx_phi += n - 1; + idx_ozs += n; + } + return phi_ozs; + } + + /** + * Constrain sum-to-zero vector + * + * @param y unconstrained zero-sum parameters + * @return vector z, the vector whose slices sum to zero + */ + vector zero_sum_constrain(vector y) { + int N = num_elements(y); + vector[N + 1] z = zeros_vector(N + 1); + real sum_w = 0; + for (ii in 1:N) { + int i = N - ii + 1; + real n = i; + real w = y[i] * inv_sqrt(n * (n + 1)); + sum_w += w; + z[i] += sum_w; + z[i + 1] -= w * n; + } + return z; + } +} + +data { + int N; + array[N] int y; // count outcomes + vector[N] E; // exposure + int K; // num covariates + matrix[N, K] xs; // design matrix + + int N_components; + array[N_components] int component_sizes; + vector[N_components] scaling_factors; + + // neighbor graph structure + int N_edges; // number of neighbor pairs + array[2, N_edges] int neighbors; // columnwise adjacent +} + +transformed data { + int N_connected = sum(component_sizes); + int N_singletons = N - N_connected; + if (N_singletons < 0) { + reject("Inconsistent inputs: sum(component_sizes) > N"); + } + vector[N_connected] taus; + array[N_components, 2] int component_idxs; + int idx = 1; + for (n in 1:N_components) { + taus[idx: component_sizes[n] + idx - 1] + = rep_vector(scaling_factors[n], component_sizes[n]); + component_idxs[n, 1] = idx; + component_idxs[n, 2] = component_sizes[n] + idx - 1; + idx += component_sizes[n]; + } + + vector[N] log_E = log(E); + // center continuous predictors + vector[K] means_xs; // column means of xs before centering + matrix[N, K] xs_centered; // centered version of xs + for (k in 1:K) { + means_xs[k] = mean(xs[, k]); + xs_centered[, k] = xs[, k] - means_xs[k]; + } +} + +parameters { + real beta0; // intercept + vector[K] betas; // covariates + + real sigma; // random effects scale + real rho; // proportion unstructured vs. spatially structured variance + + vector[N_connected] theta; // heterogeneous effects + vector[N_singletons] singletons_re; // random effects for areas with no neighbours + + // concatenation of unconstrained vectors, each size N - 1 + vector[N_connected - N_components] phi_raw; +} + +transformed parameters { + vector[N_connected] phi = zero_sum_components(phi_raw, component_idxs, component_sizes); + vector[N] gamma; // BYM2, per Freni-Sterrantino + gamma[1 : N_connected] = sqrt(1 - rho) * theta + sqrt(rho * inv(taus)) .* phi; + gamma[N_connected + 1 : N] = singletons_re; +} + +model { + y ~ poisson_log(log_E + beta0 + xs_centered * betas + gamma * sigma); + beta0 ~ std_normal(); + betas ~ std_normal(); + theta ~ std_normal(); + singletons_re ~ std_normal(); + sigma ~ std_normal(); + rho ~ beta(0.5, 0.5); + target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]); // ICAR +} + +generated quantities { + real beta_intercept = beta0 - dot_product(means_xs, betas); // adjust intercept + array[N] int y_rep; + { + vector[N] eta = log_E + beta0 + xs_centered * betas + gamma * sigma; + y_rep = max(eta) < 26 ? poisson_log_rng(eta) : rep_array(-1, N); + } +} diff --git a/jupyter/sum-to-zero/stan/bym2_multicomp_soft.stan b/jupyter/sum-to-zero/stan/bym2_multicomp_soft.stan new file mode 100644 index 00000000..32b95216 --- /dev/null +++ b/jupyter/sum-to-zero/stan/bym2_multicomp_soft.stan @@ -0,0 +1,82 @@ +data { + int N; + array[N] int y; // count outcomes + vector[N] E; // exposure + int K; // num covariates + matrix[N, K] xs; // design matrix + + // neighbor graph structure + int N_components; + array[N_components] int component_sizes; + int N_edges; // number of neighbor pairs + array[2, N_edges] int neighbors; // columnwise adjacent + vector[N_components] scaling_factors; +} + +transformed data { + int N_connected = sum(component_sizes); + int N_singletons = N - N_connected; + if (N_singletons < 0) { + reject("Inconsistent inputs: sum(component_sizes) > N"); + } + vector[N_connected] taus; + array[N_components, 2] int node_idxs; + array[N_components, 2] int edge_idxs; + int node_idx = 1; + int edge_idx = 1; + for (n in 1:N_components) { + taus[node_idx: component_sizes[n] + node_idx - 1] + = rep_vector(scaling_factors[n], component_sizes[n]); + node_idxs[n, 1] = node_idx; + node_idxs[n, 2] = component_sizes[n] + node_idx - 1; + node_idx += component_sizes[n]; + } + vector[N] log_E = log(E); + // center continuous predictors + vector[K] means_xs; // column means of xs before centering + matrix[N, K] xs_centered; // centered version of xs + for (k in 1:K) { + means_xs[k] = mean(xs[, k]); + xs_centered[, k] = xs[, k] - means_xs[k]; + } +} +parameters { + real beta0; // intercept + vector[K] betas; // covariates + + real sigma; // random effects scale + real rho; // proportion unstructured vs. spatially structured variance + + vector[N_connected] theta; // heterogeneous effects + vector[N_singletons] singletons_re; // random effects for areas with no neighbours + + vector[N_connected] phi; +} +transformed parameters { + vector[N] gamma; // BYM2, per Freni-Sterrantino + gamma[1 : N_connected] = sqrt(1 - rho) * theta + sqrt(rho * inv(taus)) .* phi; + gamma[N_connected + 1 : N] = singletons_re; +} + +model { + y ~ poisson_log(log_E + beta0 + xs_centered * betas + gamma * sigma); + beta0 ~ std_normal(); + betas ~ std_normal(); + theta ~ std_normal(); + singletons_re ~ std_normal(); + sigma ~ std_normal(); + rho ~ beta(0.5, 0.5); + target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]); // ICAR + for (n in 1:N_components) { // component-wise sum-to-zero constraint + sum(phi[node_idxs[n, 1] : node_idxs[n, 2]]) ~ normal(0, + 0.001 * component_sizes[n]); + } +} +generated quantities { + real beta_intercept = beta0 - dot_product(means_xs, betas); // adjust intercept + array[N] int y_rep; + { + vector[N] eta = log_E + beta0 + xs_centered * betas + gamma * sigma; + y_rep = max(eta) < 26 ? poisson_log_rng(eta) : rep_array(-1, N); + } +} diff --git a/jupyter/sum-to-zero/stan/bym2_ozs.stan b/jupyter/sum-to-zero/stan/bym2_ozs.stan new file mode 100644 index 00000000..77a91b91 --- /dev/null +++ b/jupyter/sum-to-zero/stan/bym2_ozs.stan @@ -0,0 +1,51 @@ +data { + int N; + array[N] int y; // count outcomes + vector[N] E; // exposure + int K; // num covariates + matrix[N, K] xs; // design matrix + + // spatial structure + int N_edges; // number of neighbor pairs + array[2, N_edges] int neighbors; // columnwise adjacent + + real tau; // scaling factor +} +transformed data { + vector[N] log_E = log(E); + // center continuous predictors + vector[K] means_xs; // column means of xs before centering + matrix[N, K] xs_centered; // centered version of xs + for (k in 1:K) { + means_xs[k] = mean(xs[, k]); + xs_centered[, k] = xs[, k] - means_xs[k]; + } +} +parameters { + real beta0; // intercept + vector[K] betas; // covariates + real rho; // proportion of spatial variance + sum_to_zero_vector[N] phi; // spatial effects + vector[N] theta; // heterogeneous random effects + real sigma; // scale of combined effects +} +transformed parameters { + vector[N] gamma = sqrt(1 - rho) * theta + sqrt(rho * inv(tau)) * phi; // BYM2 +} +model { + y ~ poisson_log(log_E + beta0 + xs_centered * betas + gamma * sigma); + rho ~ beta(0.5, 0.5); + target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]); // ICAR prior + theta ~ std_normal(); + beta0 ~ std_normal(); + betas ~ std_normal(); + sigma ~ std_normal(); +} +generated quantities { + real beta_intercept = beta0 - dot_product(means_xs, betas); // adjust intercept + array[N] int y_rep; + { + vector[N] eta = log_E + beta0 + xs_centered * betas + gamma * sigma; + y_rep = max(eta) < 26 ? poisson_log_rng(eta) : rep_array(-1, N); + } +} diff --git a/jupyter/sum-to-zero/stan/bym2_soft.stan b/jupyter/sum-to-zero/stan/bym2_soft.stan new file mode 100644 index 00000000..0b160432 --- /dev/null +++ b/jupyter/sum-to-zero/stan/bym2_soft.stan @@ -0,0 +1,62 @@ +data { + int N; + array[N] int y; // count outcomes + vector[N] E; // exposure + int K; // num covariates + matrix[N, K] xs; // design matrix + + // spatial structure + int N_edges; // number of neighbor pairs + array[2, N_edges] int neighbors; // columnwise adjacent + + real tau; // scaling factor +} +transformed data { + vector[N] log_E = log(E); + // center continuous predictors + vector[K] means_xs; // column means of xs before centering + matrix[N, K] xs_centered; // centered version of xs + for (k in 1:K) { + means_xs[k] = mean(xs[, k]); + xs_centered[, k] = xs[, k] - means_xs[k]; + } +} +parameters { + real beta0; // intercept + vector[K] betas; // covariates + real rho; // proportion of spatial variance + vector[N] phi; // spatial random effects + vector[N] theta; // heterogeneous random effects + real sigma; // scale of combined effects +} +transformed parameters { + vector[N] gamma = (sqrt(1 - rho) * theta + sqrt(rho / tau) * phi); // BYM2 +} +model { + y ~ poisson_log(log_E + beta0 + xs_centered * betas + gamma * sigma); + rho ~ beta(0.5, 0.5); + target += (-0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]) + + normal_lupdf(sum(phi) | 0, 0.001 * rows(phi))); + beta0 ~ std_normal(); + betas ~ std_normal(); + theta ~ std_normal(); + sigma ~ std_normal(); +} +generated quantities { + real beta_intercept = beta0 - dot_product(means_xs, betas); // adjust intercept + array[N] int y_rep; + { + vector[N] eta = log_E + beta0 + xs_centered * betas + gamma * sigma; + if (max(eta) > 26) { + // avoid overflow in poisson_log_rng + print("max eta too big: ", max(eta)); + for (n in 1:N) { + y_rep[n] = -1; + } + } else { + for (n in 1:N) { + y_rep[n] = poisson_log_rng(eta[n]); + } + } + } +} diff --git a/jupyter/sum-to-zero/stan/gen_binomial_4_preds.stan b/jupyter/sum-to-zero/stan/gen_binomial_4_preds.stan new file mode 100644 index 00000000..28174fd3 --- /dev/null +++ b/jupyter/sum-to-zero/stan/gen_binomial_4_preds.stan @@ -0,0 +1,63 @@ +data { + int N; // total number of tests + int N_age; + int N_eth; + int N_edu; + real baseline; + real sens; + real spec; +} +transformed data { + int strata = 2 * N_age * N_eth * N_edu; +} +generated quantities { + real beta_0 = baseline; + // some difference by sex, unequal observations + real beta_sex = normal_rng(0, 0.5); + vector[2] pct_sex = [0.4, 0.6]'; + + vector[N_age] pct_age = dirichlet_rng(rep_vector(2, N_age)); + vector[N_age] beta_age; + for (n in 1:N_age) { + beta_age[n] = std_normal_rng(); + } + + vector[N_eth] pct_eth = dirichlet_rng(rep_vector(1, N_eth)); + vector[N_eth] beta_eth; + for (n in 1:N_eth) { + beta_eth[n] = std_normal_rng(); + } + + vector[N_edu] pct_edu = dirichlet_rng(rep_vector(1.5, N_edu)); + vector[N_edu] beta_edu; + for (n in 1:N_edu) { + beta_edu[n] = std_normal_rng(); + } + + array[strata] int sex, age, eth, edu, pos_tests, tests; + array[strata] real p; + array[strata] real p_sample; + + int idx = 1; + for (i_sex in 1:2) { + for (i_age in 1:N_age) { + for (i_eth in 1:N_eth) { + for (i_edu in 1:N_edu) { + + // corresponds to unmodeled data inputs + sex[idx] = i_sex; age[idx] = i_age; eth[idx] = i_eth; edu[idx] = i_edu; + tests[idx] = to_int(pct_sex[i_sex] * pct_age[i_age] * pct_eth[i_eth] * pct_edu[i_edu] * N); + + // corresponds to transformed parameters + p[idx] = inv_logit(beta_0 + beta_sex * (i_sex) + + beta_age[i_age] + beta_eth[i_eth] + beta_edu[i_edu]); + p_sample[idx] = p[idx] * sens + (1 - p[idx]) * (1 - spec); + + // corresponds to likelihood + pos_tests[idx] = binomial_rng(tests[idx], p_sample[idx]); + idx += 1; + } + } + } + } +} diff --git a/jupyter/sum-to-zero/sum_to_zero_evaluation.html b/jupyter/sum-to-zero/sum_to_zero_evaluation.html new file mode 100644 index 00000000..0cf575ec --- /dev/null +++ b/jupyter/sum-to-zero/sum_to_zero_evaluation.html @@ -0,0 +1,7016 @@ + + + + + + + + + + +The Sum-to-Zero Constraint in Stan + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +
+ +
+
+

The Sum-to-Zero Constraint in Stan

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+ + + +
+ +
+
Author
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+

Mitzi Morris

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+ + + +
+ + + +
+ + +

Load libraries and set up notebook.

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+
+Show Code +
# libraries used in this notebook
+import os
+import numpy as np
+import pandas as pd
+import geopandas as gpd
+import plotnine as p9
+import libpysal
+from splot.libpysal import plot_spatial_weights 
+from random import randint
+
+from cmdstanpy import CmdStanModel, cmdstan_path, set_cmdstan_path
+import logging
+cmdstanpy_logger = logging.getLogger("cmdstanpy")
+cmdstanpy_logger.setLevel(logging.FATAL)
+set_cmdstan_path(os.path.join('/Users', 'mitzi', 'github', 'stan-dev', 'cmdstan'))
+
+import warnings
+warnings.filterwarnings('ignore')
+
+import matplotlib
+import matplotlib.pyplot as plt
+%matplotlib inline
+
+from utils_html import *
+
+# notebook display options
+plt.rcParams['figure.figsize'] = (7, 7)
+plt.rcParams['figure.dpi'] = 100
+
+np.set_printoptions(precision=2)
+np.set_printoptions(suppress=True)
+pd.set_option('display.precision', 2)
+pd.options.display.float_format = '{:.2f}'.format
+
+
+
+

Introducing the sum_to_zero_vector Constrained Parameter Type

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The sum_to_zero_vector constrained parameter type was introduced in the Stan 2.36 release.

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The parameter declaration:

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  sum_to_zero_vector[K] beta;
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produces a vector of size K such that sum(beta) = 0. The unconstrained representation requires only K - 1 values because the last is determined by the first K - 1.

+

If you put a normal prior on the zero-sum vector the resulting variance will be less than the intended normal variance. To get the same variance as the intended normal prior do

+
parameters {
+  sum_to_zero_vector[N] z;
+}
+nnmodel {
+  z ~ normal(0, sqrt((N / (N - 1.0)) * sigma)
+}
+

where sigma is the intended standard deviation. (Note: it’s slightly more efficient to pre-calculate the sqrt(N / (N - 1.0)) in transformed_data.)

+

Prior to Stan 2.36, a sum-to-zero constraint could be implemented either

+
    +

  • as a “hard” sum to zero constraint, where the parameter is declared to be an \(N-1\) length vector with a corresponding \(N\)-length transformed parameter whose first \(N-1\) elements are the same as the corresponding parameter vector, and the \(N^{th}\) element is the negative sum of the \(N-1\) elements.

    • +

  • as a “soft” sum to zero constraint, where the parameter is an \(N\)-length vector with a prior on the sum of the elements that keeps it within \(\epsilon\) of \(0\).

    • +
+

The choice between the two was largely an empirical one. As a general rule, for small vectors, the hard sum-to-zero constraint is more efficient; for larger vectors, the soft sum-to-zero constraint is faster, but much depends on the specifics of the model and the data.

+

The hard sum-to-zero is often problematic. For small \(N\) and models with sensible priors, the hard sum-to-zero is usually satisfactory. But as the size of the vector grows, it distorts the marginal variance of the \(N^{th}\). Given a parameter vector: \[ +x_1, x_2, \dots, x_{N-1} \sim \text{i.i.d. } N(0, \sigma^2) +\] by the properties of independent normal variables, each of the free elements \(x_1, \ldots, x_{N-1}\) has variance \(\sigma^2\). However, the \(N^{th}\) element is defined deterministically as: \[ +x_N = -\sum_{i=1}^{N-1} x_i +\] and its variance is inflated by a factor of \(N-1\). \[ +\operatorname{Var}(x_N) = \operatorname{Var}\Bigl(-\sum_{i=1}^{N-1} x_i\Bigr) += \sum_{i=1}^{N-1} \operatorname{Var}(x_i) += (N-1)\sigma^2. +\] For large vectors, MCMC samplers struggle with the hard sum-to-zero constraint, as every change to any of the \(N-1\) elements also requires a corresponding change to the \(N^{th}\) element; balancing these changes introduces potential non-identifiabilities.

+

The soft sum-to-zero constraint is problematic for the following reasons.

+
    +
  • The tolerance \(\epsilon\) (the scale of the penalty) must be chosen by the analyst. Too large, and the result is too far from zero to be effective, too small and the sampler cannot satisfy the constraint.
    • +
  • The soft constraint only penalizes deviations from zero, leading to weaker identifiability of the parameters. This can lead to slow convergence and mixing, as the sampler explores nearly non-identified regions.
    • +
  • The marginal variances may not reflect the intended prior.
    • +
+

The sum_to_zero_vector transform ensures that each element of the resulting constrained vector has the same variance. This improves the sampler performance, providing fast computation and good effective sample size. This becomes increasingly noticeable as models increase in size and complexity. To demonstrate this, in this notebook we consider two different classes of models:

+
    +
  • Multi-level regressions for binomial data with group-level categorical predictors.
    • +
  • Spatial models for areal data.
    • +
+

For these models, we provide three implementations which differ only in the implementation of the sum-to-zero constraint: the built-in sum_to_zero_vector, and the hard and soft sum-to-zero implementations. We fit each model to the same dataset, using the same random seed, and then compare the summary statistics for the constrained parameter values. Since the models are nearly equivalent except in the implied prior distributions on the vector, we expect that all three implementations should produce nearly equivalent estimates; what differs is the speed of computation, as measured by effective samples per second.

+
+
+
+ +
+
+Note +
+
+
+

The spatial models are taken from the a set of notebooks available from GitHub repo https://github.com/mitzimorris/geomed_2024.

+
+
+
+
+

Multi-level Models with Group-level Categorical Predictors

+

In this section we consider a model which estimates per-demographic disease prevalence rates for a population. The model is taken from the Gelman and Carpenter, 2020 Bayesian Analysis of Tests with Unknown Specificity and Sensitivity. It combines a model for multilevel regression and post-stratification with a likelihood that accounts for test sensitivity and specificity.

+

The data consists of:

+
    +

  • A set of per-demographic aggregated outcomes of a diagnostic test procedure with unequal number of tests per demographic.

    • +

  • A corresponding set of demographic descriptors encoded as a vector of categorical values. In this example these are named sex, age, eth, and edu, but there can be any number of demographic predictors with any number of categories.

    • +

  • The specified test sensitivity and specificity

    • +
+

Putting a sum-to-zero constraint on the categorical allows us to model all categories as offset from the mean. We prefer this to the alternative of dropping a category for reference or other contrast coding strategy.

+
+

The Stan model

+

The full model is in file binomial_4_preds_ozs.stan. It provides an estimate of the true prevalence based on binary tests with a given (or unknown) test sensitivity and specificity, and uses a sum_to_zero_vector to constrain the group-level parameters age, eth, and edu.

+

In order to put a standard normal prior on beta_age, beta_eth, and beta_edu, we need to scale the variance, as suggested above. The scaling factors are pre-computed in the transformed data block, and applied as part of the prior.

+

relevant excepts from binomial_4_preds_ozs.stan

+
transformed data {
+  // scaling factors for marginal variances of sum_to_zero_vectors
+  real s_age = sqrt(N_age / (N_age - 1.0));
+  real s_eth = sqrt(N_eth / (N_eth - 1.0));
+  real s_edu = sqrt(N_edu / (N_edu - 1.0));
+}
+parameters {
+  real beta_0;
+  real beta_sex;
+  real<lower=0> sigma_age, sigma_eth, sigma_edu;
+  sum_to_zero_vector[N_age] beta_age;
+  sum_to_zero_vector[N_eth] beta_eth;
+  sum_to_zero_vector[N_edu] beta_edu;
+}
+transformed parameters {
+  // true prevalence
+  vector[N] p = inv_logit(beta_0 + beta_sex * sex_c + beta_age[age]
+              + beta_eth[eth] + beta_edu[edu]);
+  // incorporate test sensitivity and specificity.
+  vector[N] p_sample = p * sens + (1 - p) * (1 - spec);
+}
+model {
+  pos_tests ~ binomial(tests, p_sample);  // likelihood
+  // scale normal priors on sum_to_zero_vectors
+  beta_age ~ normal(0, s_age * sigma_age);
+  beta_eth ~ normal(0, s_eth * sigma_eth);
+  beta_edu ~ normal(0, s_edu * sigma_edu);
+  ...
+}
+
+

Simulating Data

+

To investigate the predictive behavior of this model at different timepoints in a pandemic, we have written a data-generating program to create datasets given the baseline disease prevalence, test specificity and sensitivity, and the desired number of diagnostic tests. In the generated quantities block we use Stan’s PRNG functions to populate vthe true weights for the categorical coefficient vectors, and the relative percentages of per-category observations. Then we use a set of nested loops to generate the data for each demographic, using the PRNG equivalent of the model likelihood.

+

The full data-generating program is in file gen_binomial_4_preds.stan. The helper function simulate_data in file utils.py sets up the data-generating program according to the specified number of demographic categories, approximate average number of observations per category, and baseline disease prevalence, test specificity and sensitivity. This allows us to create datasets for large and small populations and for finer or more coarse-grained sets of categories. The larger the number of strata overall, the more observations are needed to get good coverage. Because the modeled data pos_tests is generated according to the Stan model’s likelihood, the model is a priori well-specified with respect to the data.

+
+
+
+ +
+
+Note +
+
+
+

Because the data-generating parameters and percentage of observations per category are generated at random, some datasets may have very low overall disease rates and/or many unobserved strata. While this is informative for understanding what is consistent with the generating distributions, it results in estimates of varying effect parameters with near-zero values, which are on the boundary of parameter space, and hence very challenging to fit. Because we wish to compare baseline performance, we have cherry-picked a seed which generates datasets that avoid this issue.

+
+
+

Generate several datasets of different sizes.

+
+
+Show Code +
# specify data shapes and test characteristics
+N_eth = 3
+N_edu = 5
+N_age = 9
+baseline = -3.5
+sens = 0.75
+spec = 0.9995
+
+
+# generate datasets
+from utils import simulate_data
+data_tiny = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 4, seed=56789)
+data_small = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 17, seed=data_tiny['seed'])
+data_large = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 200, seed=data_tiny['seed'])
+
+
+

The data generating program creates unequal numbers of tests per demographic. The following tables summarize the resulting distribution.

+
+
+
+
+Show Code +
print("Small Dataset")
+print("tests total " + str(np.sum(data_small['tests'])))
+small_df = pd.DataFrame({'tests per demographic': data_small['tests']}).describe()
+small_df.drop('count', inplace=True)
+small_df
+
+
+
Small Dataset
+tests total 4451
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
tests per demographic
mean16.49
std18.51
min0.00
25%5.00
50%10.00
75%21.75
max119.00
+ +
+
+
+
+
+
+Show Code +
print("Large Dataset")
+print("tests total " + str(np.sum(data_large['tests'])))
+large_df = pd.DataFrame({'tests per demographic': data_large['tests']}).describe()
+large_df.drop('count', inplace=True)
+large_df
+
+
+
Large Dataset
+tests total 53863
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
tests per demographic
mean199.49
std218.00
min4.00
25%61.25
50%123.00
75%258.50
max1401.00
+ +
+
+
+
+
+
+Show Code +
print("Tiny Dataset")
+print("tests total " + str(np.sum(data_tiny['tests'])))
+tiny_df = pd.DataFrame({'tests per demographic': data_tiny['tests']}).describe()
+tiny_df.drop('count', inplace=True)
+tiny_df
+
+
+
Tiny Dataset
+tests total 947
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
tests per demographic
mean3.51
std4.38
min0.00
25%1.00
50%2.00
75%5.00
max28.00
+ +
+
+
+
+
+
+
+
+

Model Fitting

+

We fit each model to the the large and small datasets. We record the seed used for the first run and use it for all subsequent fits; this insures that the models have the same set of initial parameter values for all parameters except the constrained params beta_eth, beta_edu, and beta_age.

+

Model 1: sum_to_zero_vector

+

This model is in file binomial_4_preds_ozs.stan.

+

Compile and fit the model to data.

+
+
+Show Code +
binomial_ozs_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_ozs.stan'))
+print("Model 1:  sum_to_zero_vector, small dataset")
+binomial_ozs_fit = binomial_ozs_mod.sample(data=data_small, parallel_chains=4)
+
+a_seed = binomial_ozs_fit.metadata.cmdstan_config['seed']
+print("Model 1:  sum_to_zero_vector, large dataset")
+binomial_ozs_fit_lg = binomial_ozs_mod.sample(data=data_large, parallel_chains=4, seed=a_seed)
+
+
+

Model 2: Hard sum-to-zero constraint

+

This model is in file binomial_4_preds_hard.stan.

+

Compile and fit the model to data.

+
+
+Show Code +
binomial_hard_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_hard.stan'))
+print("\nModel 2:  hard sum-to-zero constraint, small dataset")
+binomial_hard_fit = binomial_hard_mod.sample(data=data_small, parallel_chains=4, seed=a_seed)
+print("Model 2:  hard sum-to-zero constraint, large dataset")
+binomial_hard_fit_lg = binomial_hard_mod.sample(data=data_large, parallel_chains=4, seed=a_seed)
+
+
+

Model 3: soft sum-to-zero constraint

+

This model is in file binomial_4_preds_soft.stan.

+

Compile and fit the model to data.

+
+
+Show Code +
binomial_soft_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_soft.stan'))
+print("\nModel 3:  soft sum-to-zero constraint, small dataset")
+binomial_soft_fit = binomial_soft_mod.sample(data=data_small, parallel_chains=4, seed=a_seed)
+print("Model 3:  soft sum-to-zero constraint, large dataset")
+binomial_soft_fit_lg = binomial_soft_mod.sample(data=data_large, parallel_chains=4, seed=a_seed)
+
+
+
+
+

Model Checking and Comparison

+
+

Check implementations

+

If all three implementations are correct, we expect that they will produce the same estimates for all parameters. As always, we check the R-hat values and effective sample sizes. To do this, we tabulate the summary statistics from each of the three models, fit to the large and small datasets.

+

Tabulate summary statistics from all fits.

+
+
+Show Code +
# small dataset
+ozs_fit_summary = binomial_ozs_fit.summary(sig_figs=2)
+ozs_fit_summary.index =  ozs_fit_summary.index.astype(str) + "  - ozs"
+
+
+hard_fit_summary = binomial_hard_fit.summary(sig_figs=2)
+hard_fit_summary.index = hard_fit_summary.index.astype(str) + " - hard"
+
+soft_fit_summary = binomial_soft_fit.summary(sig_figs=2)
+soft_fit_summary.index = soft_fit_summary.index.astype(str) + " - soft"
+
+small_data_fits_summary = pd.concat([ozs_fit_summary, hard_fit_summary, soft_fit_summary])
+
+# large dataset
+ozs_fit_lg_summary = binomial_ozs_fit_lg.summary(sig_figs=2)
+ozs_fit_lg_summary.index =  ozs_fit_lg_summary.index.astype(str) + "  - ozs"
+
+hard_fit_lg_summary = binomial_hard_fit_lg.summary(sig_figs=2)
+hard_fit_lg_summary.index = hard_fit_lg_summary.index.astype(str) + " - hard"
+
+soft_fit_lg_summary = binomial_soft_fit_lg.summary(sig_figs=2)
+soft_fit_lg_summary.index = soft_fit_lg_summary.index.astype(str) + " - soft"
+
+large_data_fits_summary = pd.concat([ozs_fit_lg_summary, hard_fit_lg_summary, soft_fit_lg_summary])
+
+
+

Eth

+

Fitted estimates

+
+
+Show Code +
beta_eth_summary = summarize_predictor(small_data_fits_summary, 'beta_eth\[')
+beta_eth_summary_lg = summarize_predictor(large_data_fits_summary, 'beta_eth\[')
+labels = expand_true(data_small['beta_eth'], "Datagen<br>values")
+small_html = style_dataframe(beta_eth_summary, 3).to_html()
+large_html = style_dataframe(beta_eth_summary_lg, 3).to_html()
+display_side_by_side(small_html, large_html, labels)
+
+sigma_eth_summary = summarize_predictor(small_data_fits_summary, 'sigma_eth')
+sigma_eth_summary_lg = summarize_predictor(large_data_fits_summary, 'sigma_eth')
+small_html = style_dataframe(sigma_eth_summary, 4).to_html()
+large_html = style_dataframe(sigma_eth_summary_lg, 4).to_html()
+display_side_by_side(small_html, large_html)
+
+
+ +
+
+   + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Datagen
+values
1.97
 
 
-0.80
 
 
0.75
 
 
+ +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_eth[1] - ozs1.400.102800.001.00
beta_eth[1] - hard1.400.101900.001.00
beta_eth[1] - soft1.400.1098.001.00
beta_eth[2] - ozs-1.500.083100.001.00
beta_eth[2] - hard-1.500.081900.001.00
beta_eth[2] - soft-1.500.08100.001.00
beta_eth[3] - ozs0.100.083200.001.00
beta_eth[3] - hard0.100.085300.001.00
beta_eth[3] - soft0.100.08130.001.00
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_eth[1] - ozs1.300.032700.001.00
beta_eth[1] - hard1.300.031900.001.00
beta_eth[1] - soft1.300.03460.001.00
beta_eth[2] - ozs-1.500.023200.001.00
beta_eth[2] - hard-1.500.021800.001.00
beta_eth[2] - soft-1.500.02530.001.00
beta_eth[3] - ozs0.170.023600.001.00
beta_eth[3] - hard0.180.025100.001.00
beta_eth[3] - soft0.180.02570.001.00
+ + +
+
+ +
+
+ +
+
+   + +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
sigma_eth - ozs1.100.403600.001.00
sigma_eth - hard1.300.443100.001.00
sigma_eth - soft1.200.41180.001.00
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
sigma_eth - ozs1.000.383500.001.00
sigma_eth - hard1.300.453300.001.00
sigma_eth - soft1.200.40880.001.00
+ + +
+
+ +
+
+

Edu

+

Fitted estimates

+
+
+Show Code +
beta_edu_summary = summarize_predictor(small_data_fits_summary, 'beta_edu\[')
+beta_edu_summary_lg = summarize_predictor(large_data_fits_summary, 'beta_edu\[')
+labels = expand_true(data_small['beta_edu'], "Datagen<br>values")
+small_html = style_dataframe(beta_edu_summary, 3).to_html()
+large_html = style_dataframe(beta_edu_summary_lg, 3).to_html()
+display_side_by_side(small_html, large_html, labels)
+
+sigma_edu_summary = summarize_predictor(small_data_fits_summary, 'sigma_edu')
+sigma_edu_summary_lg = summarize_predictor(large_data_fits_summary, 'sigma_edu')
+small_html = style_dataframe(sigma_edu_summary, 4).to_html()
+large_html = style_dataframe(sigma_edu_summary_lg, 4).to_html()
+display_side_by_side(small_html, large_html)
+
+
+ +
+
+   + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Datagen
+values
0.24
 
 
-0.22
 
 
0.87
 
 
-0.85
 
 
1.50
 
 
+ +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_edu[1] - ozs-0.180.123600.001.00
beta_edu[1] - hard-0.190.122700.001.00
beta_edu[1] - soft-0.180.13160.001.00
beta_edu[2] - ozs-0.360.123900.001.00
beta_edu[2] - hard-0.370.122500.001.00
beta_edu[2] - soft-0.360.12150.001.00
beta_edu[3] - ozs0.630.113100.001.00
beta_edu[3] - hard0.620.112000.001.00
beta_edu[3] - soft0.630.10130.001.00
beta_edu[4] - ozs-1.200.212500.001.00
beta_edu[4] - hard-1.200.211600.001.00
beta_edu[4] - soft-1.200.2195.001.00
beta_edu[5] - ozs1.100.122700.001.00
beta_edu[5] - hard1.100.112100.001.00
beta_edu[5] - soft1.100.11120.001.00
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_edu[1] - ozs-0.050.044100.001.00
beta_edu[1] - hard-0.050.042700.001.00
beta_edu[1] - soft-0.050.04680.001.00
beta_edu[2] - ozs-0.530.034000.001.00
beta_edu[2] - hard-0.530.032400.001.00
beta_edu[2] - soft-0.530.03750.001.00
beta_edu[3] - ozs0.600.033400.001.00
beta_edu[3] - hard0.600.032300.001.00
beta_edu[3] - soft0.600.03550.001.00
beta_edu[4] - ozs-1.200.063100.001.00
beta_edu[4] - hard-1.200.061700.001.00
beta_edu[4] - soft-1.200.06410.001.00
beta_edu[5] - ozs1.200.033100.001.00
beta_edu[5] - hard1.200.031900.001.00
beta_edu[5] - soft1.200.03530.001.00
+ + +
+
+ +
+
+ +
+
+   + +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
sigma_edu - ozs0.840.292900.001.00
sigma_edu - hard0.850.322300.001.00
sigma_edu - soft0.920.31130.001.00
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
sigma_edu - ozs0.890.313200.001.00
sigma_edu - hard0.880.312700.001.00
sigma_edu - soft0.960.30860.001.00
+ + +
+
+ +
+
+

Age

+

Fitted estimates

+
+
+Show Code +
beta_age_summary = summarize_predictor(small_data_fits_summary, 'beta_age\[')
+beta_age_summary_lg = summarize_predictor(large_data_fits_summary, 'beta_age\[')
+labels = expand_true(data_small['beta_age'], "Datagen<br>values")
+small_html = style_dataframe(beta_age_summary, 3).to_html()
+large_html = style_dataframe(beta_age_summary_lg, 3).to_html()
+display_side_by_side(small_html, large_html, labels)
+
+sigma_age_summary = summarize_predictor(small_data_fits_summary, 'sigma_age')
+sigma_age_summary_lg = summarize_predictor(large_data_fits_summary, 'sigma_age')
+small_html = style_dataframe(sigma_age_summary, 4).to_html()
+large_html = style_dataframe(sigma_age_summary_lg, 4).to_html()
+display_side_by_side(small_html, large_html)
+
+
+ +
+
+   + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Datagen
+values
-0.74
 
 
-1.43
 
 
-1.67
 
 
1.00
 
 
-0.52
 
 
-0.77
 
 
0.96
 
 
1.31
 
 
0.25
 
 
+ +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_age[1] - ozs-0.780.213900.001.00
beta_age[1] - hard-0.780.213100.001.00
beta_age[1] - soft-0.780.20150.001.00
beta_age[2] - ozs-1.200.561200.001.00
beta_age[2] - hard-1.200.541200.001.00
beta_age[2] - soft-1.100.5378.001.00
beta_age[3] - ozs-1.300.254500.001.00
beta_age[3] - hard-1.300.242400.001.00
beta_age[3] - soft-1.300.25150.001.00
beta_age[4] - ozs1.200.152000.001.00
beta_age[4] - hard1.200.141700.001.00
beta_age[4] - soft1.200.15110.001.00
beta_age[5] - ozs0.040.273400.001.00
beta_age[5] - hard0.030.282600.001.00
beta_age[5] - soft0.030.28160.001.00
beta_age[6] - ozs-0.790.192600.001.00
beta_age[6] - hard-0.790.192300.001.00
beta_age[6] - soft-0.790.19140.001.00
beta_age[7] - ozs0.940.131700.001.00
beta_age[7] - hard0.950.131600.001.00
beta_age[7] - soft0.940.13100.001.00
beta_age[8] - ozs1.200.161900.001.00
beta_age[8] - hard1.200.161900.001.00
beta_age[8] - soft1.200.16110.001.00
beta_age[9] - ozs0.690.182400.001.00
beta_age[9] - hard0.720.192300.001.00
beta_age[9] - soft0.690.19140.001.00
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_age[1] - ozs-0.590.064400.001.00
beta_age[1] - hard-0.590.062500.001.00
beta_age[1] - soft-0.590.06610.001.00
beta_age[2] - ozs-1.300.161200.001.00
beta_age[2] - hard-1.300.161100.001.00
beta_age[2] - soft-1.300.16290.001.00
beta_age[3] - ozs-1.500.084400.001.00
beta_age[3] - hard-1.500.082700.001.00
beta_age[3] - soft-1.500.08730.001.00
beta_age[4] - ozs1.200.042400.001.00
beta_age[4] - hard1.200.041600.001.00
beta_age[4] - soft1.200.04410.001.00
beta_age[5] - ozs-0.280.093200.001.00
beta_age[5] - hard-0.280.092500.001.00
beta_age[5] - soft-0.280.09730.001.00
beta_age[6] - ozs-0.620.052800.001.00
beta_age[6] - hard-0.620.062700.001.00
beta_age[6] - soft-0.620.05620.001.00
beta_age[7] - ozs1.100.041900.001.00
beta_age[7] - hard1.100.041300.001.00
beta_age[7] - soft1.100.04370.001.00
beta_age[8] - ozs1.500.052100.001.00
beta_age[8] - hard1.500.051700.001.00
beta_age[8] - soft1.500.05430.001.00
beta_age[9] - ozs0.450.062400.001.00
beta_age[9] - hard0.450.062200.001.00
beta_age[9] - soft0.450.06580.001.00
+ + +
+
+ +
+
+ +
+
+   + +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
sigma_age - ozs1.000.272400.001.00
sigma_age - hard1.100.292200.001.00
sigma_age - soft1.000.27120.001.00
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
sigma_age - ozs1.100.253400.001.00
sigma_age - hard1.200.283200.001.00
sigma_age - soft1.100.26760.001.00
+ + +
+
+ +
+
+

These comparisons demonstrate that

+
    +

  • All models have R-hat values of 1.00 for all group-level parameters and high effective sample sizes.

    • +

  • The models recover the sign of the parameter, but not the exact value. In the case of lots of observations and only a few categories they do a better job of recovering the true parameters.

    • +

  • In almost all cases, estimates for each parameter are the same across implementations to 2 significant figures. In a few cases they are off by 0.01; where they are off, the percentage of observations for that parameter is correspondingly low.

    • +

  • The sum_to_zero_vector implementation has the highest number of effective samples per second, excepting a few individual parameters for which the hard sum-to-zero performs equally well.

    • +
+
+
+

Model efficiency

+

Model efficiency is measured by iterations per second, however, as the draws from the MCMC sampler may be correlated, we need to compute the number of effective samples across all chains divided by the total sampling time - this is ESS_bulk/s, the effective samples per second. The following table shows the average runtime for 100 runs of each of the three models on large and small datasets. This data was generated by script eval_efficiencies.py.

+

Tabulate summary statistics from all fits.

+
+
+Show Code +
with open(os.path.join('binomial_runtimes_small.json'), 'r') as file:
+     data = json.load(file)
+df_small = pd.DataFrame(data)
+
+with open(os.path.join('binomial_runtimes_large.json'), 'r') as file:
+     data = json.load(file)
+df_large = pd.DataFrame(data)
+
+small_html = style_dataframe(df_small, 3).to_html()
+large_html = style_dataframe(df_large, 3).to_html()
+display_side_by_side(small_html, large_html)
+
+
+ +
+
+   + +
+
+ Small Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 runtime, averuntime, std devESS_bulk/s
ozs small2.210.051327.11
hard small3.070.08988.47
soft small54.492.5463.14
+ + +
+
+ Large Dataset + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 runtime, averuntime, std devESS_bulk/s
ozs large2.410.081305.86
hard large3.250.09981.53
soft large17.410.50181.15
+ + +
+
+ +
+
+

The sum_to_zero_vector is far more efficient than either the hard or soft sum-to-zero constraint.

+
+
+

Prior predictive checks

+

Prior predictive checks simulate data directly from the prior, in the absense of any observed data. Here we use the simulated dataset to examine the prior marginal variances of the elements of the sum-to-zero vector under the hard-sum-to-zero constraint and the built-in sum_to_zero transform. In particular, we are interested in the marginal variances of the elements of the sum-to-zero constrained parameters.

+

We have written two models: binomial_4preds_ozs_ppc.stan and binomial_4preds_hard_ppc.stan. We generate a sample from each, and compare the marginal variances of the elements of sum-to-zero constrained parameter beta_age.

+

Marginal variances of the hard sum-to-zero constraint, sample from the prior

+

Generate sample from the prior of the hard sum-to-zero model.

+
+
+Show Code +
binomial_hard_ppc_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_hard_ppc.stan'))
+binomial_hard_ppc_fit = binomial_hard_ppc_mod.sample(data=data_small, parallel_chains=4)
+
+
+

Get marginal variances of elements of parameter beta_age from the prior.

+
+
+Show Code +
np.var(binomial_hard_ppc_fit.beta_age, axis=0)
+
+
+
array([1.04, 1.19, 1.03, 1.22, 1.23, 1.07, 0.96, 1.02, 9.06])
+
+
+

The above summaries show how the hard sum-to-zero constraint distorts the variance of the \(N^{th}\) element. This is only a problem for very sparse datasets, where the prior swamps the data. With data, this problem goes away; to see this we fit the hard sum-to-zero constraint model on the tiny and small datasets.

+

Marginal variances of the hard sum-to-zero constraint, tiny dataset

+

Fit hard sum-to-zero model to the tiny dataset.

+
+
+Show Code +
binomial_hard_fit_tiny = binomial_hard_mod.sample(data=data_tiny, parallel_chains=4)
+
+
+

Get marginal variances of elements of parameter beta_age from the posterior.

+
+
+Show Code +
np.var(binomial_hard_fit_tiny.beta_age, axis=0)
+
+
+
array([0.18, 0.86, 0.25, 0.09, 0.31, 0.15, 0.07, 0.11, 0.16])
+
+
+

Marginal variances of the hard sum-to-zero constraint, small dataset

+

Fit hard sum-to-zero model to the small dataset.

+
+
+Show Code +
binomial_hard_fit_small = binomial_hard_mod.sample(data=data_small, parallel_chains=4)
+
+
+

Get marginal variances of elements of parameter beta_age from the posterior.

+
+
+Show Code +
np.var(binomial_hard_fit_small.beta_age, axis=0)
+
+
+
array([0.04, 0.31, 0.06, 0.02, 0.08, 0.03, 0.02, 0.03, 0.04])
+
+
+

The marginal variances for the sum_to_zero_vector elements are roughly equal in both the prior and the posterior.

+

Marginal variances of the built-in sum_to_zero_vector, sample from the prior

+

Generate sample from the prior of the sum_to_zero_vector model.

+
+
+Show Code +
binomial_ozs_ppc_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_ozs_ppc.stan'))
+binomial_ozs_ppc_fit = binomial_ozs_ppc_mod.sample(data=data_small, parallel_chains=4)
+
+
+

Get marginal variances of elements of parameter beta_age from the prior.

+
+
+Show Code +
np.var(binomial_ozs_ppc_fit.beta_age, axis=0)
+
+
+
array([0.08, 0.08, 0.17, 0.12, 0.25, 0.11, 0.12, 0.11, 0.12])
+
+
+

Marginal variances of the built-in sum_to_zero_vector, tiny dataset

+

Fit the sum_to_zero_vector model to the tiny dataset.

+
+
+Show Code +
binomial_hard_fit_tiny = binomial_hard_mod.sample(data=data_tiny, parallel_chains=4)
+
+
+

Get marginal variances of elements of parameter beta_age from the posterior.

+
+
+Show Code +
np.var(binomial_hard_fit_tiny.beta_age, axis=0)
+
+
+
array([0.18, 0.81, 0.24, 0.09, 0.33, 0.15, 0.07, 0.1 , 0.16])
+
+
+
+
+
+

Discussion

+

Comparison of these three models with different data regimes demonstrates the following.

+
    +

  • For a multi-level model with group-level categorical predictors the sum_to_zero_vector provides fast results and good effective sample sizes for both datasets.

    • +

  • Model binomial_4preds_ozs.stan. shows how to properly scale the variance of a sum_to_zero_vector constrained parameter in order to put a standard normal prior on it.

    • +

  • Prior predictive checks demonstrate the difference between the marginal variances of the sum_to_zero_vector and hard sum-to-zero implementations.

    • +
+
+
+
+

Spatial Models with an ICAR component

+

Spatial auto-correlation is the tendency for adjacent areas to share similar characteristics. Conditional Auto-Regressive (CAR) and Intrinsic Conditional Auto-Regressive (ICAR) models, first introduced by Besag, 1974, account for this by pooling information from neighboring regions. The BYM model, (Besag, York, Mollié, 1991) extends a lognormal Poisson model plus ICAR component for spatial auto-correlation by adding an ordinary random-effects component for non-spatial heterogeneity. The BYM2 model builds on this model and subsequent refinements.

+

The ICAR, BYM2, and BYM2_multicomp models are more fully explained in a series of notebooks available from GitHub repo: https://github.com/mitzimorris/geomed_2024, see notebooks:

+ +
+

ICAR Models

+

The key element of the BYM2 model is the ICAR component. Its conditional specification is a multivariate normal random vector \(\mathbf{\phi}\) where each \({\phi}_i\) is conditional on the values of its neighbors.

+

The joint specification rewrites to a Pairwise Difference,

+

\[ p(\phi) \propto \exp \left\{ {- \frac{1}{2}} \sum_{i \sim j}{({\phi}_i - {\phi}_j)}^2 \right\} \]

+

Each \({({\phi}_i - {\phi}_j)}^2\) contributes a penalty term based on the distance between the values of neighboring regions. However, \(\phi\) is non-identifiable, constant added to \(\phi\) washes out of \({\phi}_i - {\phi}_j\). Therefore, a sum-to-zero constraint is needed to both identify and center \(\phi\).

+

The Stan implementation of the ICAR component computes the sum of the pairwise distances by representing the spatial adjacency matrix as a array of pairs of neighbor indices.

+
data {
+  ...
+  // spatial structure
+  int<lower = 0> N_edges;  // number of neighbor pairs
+  array[2, N_edges] int<lower = 1, upper = N> neighbors;  // columnwise adjacent
+

The ICAR prior comes into the model as parameter phi.

+
model {
+  ...
+  target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]);  // ICAR prior
+

The ICAR model requires that the neighbor graph is fully connected for two reasons:

+
    +

  • The joint distribution is computed from the pairwise differences between a node and its neighbors; singleton nodes have no neighbors and are therefore undefined.

    • +

  • Even if the graph doesn’t have any singleton nodes, when the graph has multiple connected components a sum-to-zero constraint on the entire vector fails to properly identify the model.

    • +
+

Because the BYM2 model includes an ICAR component, it too requires a fully connected neighbor graph. In order to apply the BYM2 model to the full NYC dataset, it is necessary to extend the BYM2 model to account for disconnected components and singleton nodes.

+

First we use the BYM2 model for fully connected graphs to compare implementations of the sum-to-zero vector on phi, and then we extend the BYM2 model following the INLA implementation developed by Freni-Sterrantino et al. in 2018 and presented in A note on intrinsic Conditional Autoregressive models for disconnected graphs.

+

The dataset we’re using is that used in the analysis published in 2019 Bayesian Hierarchical Spatial Models: Implementing the Besag York Mollié Model in Stan. It data consists of motor vehicle collisions in New York City, as recorded by the NYC Department of Transportation, between the years 2005-2014, restricted to collisions involving school age children 5-18 years of age as pedestrians. Each crash was localized to the US Census tract in which it occurred, using boundaries from the 2010 United States Census, using the 2010 Census block map for New York City. New York City is comprised of several large and small islands, a chunk of the mainland (the Bronx), and the southern tip of Long Island Sound (Brooklyn and Queens). To analyze the full New York City dataset requires the extended BYM2 model, which we do in the second part of this section.

+
+
+

Model 1: The BYM2 model, Riebler et al. 2016

+

We compare three implementations of the BYM2 model which differ only in their implementation of phi, which is constrained to sum to zero.

+

BYM2 v1: sum_to_zero_vector

+

Parameter phi is declared as a sum_to_zero_vector. See file bym2_ozs.stan.

+

BYM2 v2: Hard sum-to-zero constraint

+

Parameter phi_raw is an vector of size N-1 and the N-length vector phi is computed in the transformed parameters block. See file bym2_hard.stan.

+

BYM2 v3: soft sum-to-zero constraint

+

Parameter phi is a vector of size N and its sum is soft-centered on \(0\). See file bym2_soft.stan.

+
+

Data assembly

+

The inputs to the BYM2 model are

+
    +

  • The Poisson regression data

    +
      +
    • int<lower=0> N - number of regions
      • +
    • array[N] int<lower=0> y - per-region count outcome
      • +
    • vector<lower=0>[N] E - the population of each region (a.k.a. “exposure”),
      • +
    • int<lower=1> K - the number of predictors
      • +
    • matrix[N, K] xs - the design matrix
      • +
    • +

  • The spatial structure

    +
      +
    • int<lower = 0> N_edges - the number of neighbor pairs
      • +
    • array[2, N_edges] int<lower = 1, upper = N> neighbors - the graph structure
      • +
    • real tau - the scaling factor, introduced in the BYM2
      • +
    • +
+
+
+
+ +
+
+Note +
+
+
+

The scaling factor tau was introduced by Riebler et al so that the variance of the spatial and ordinary random effects are both approximately equal to 1, thus allowing for a straightforward estimate of the amount of spatial and non-spatial variance. We have written a helper function called get_scaling_factor, in file utils_bym2.py which takes as its argument the neighbor graph and computes the geometric mean of the corresponding adjacency matrix.

+
+
+

To assemble the spatial data from the US Census data maps we use a set of utilities described more fully in notebooks 2 and 5 of the geomed_2024 Spatial Data Processing in Stan workshop. The neighbors graph of New York City consists of several large connected components, notably Brooklyn and Queens, the Bronx, Manhattan, and Staten Island, and a few other smaller components and singletons. The helper function nyc_soft_by_comp_size adds component info to the geodataframe and sorts the data by component size.

+

Get NYC dataset, pre-process as needed for BYM2 model, display neighbors graph.

+
+
+Show Code +
from utils_nyc_map import nyc_sort_by_comp_size
+nyc_geodata = gpd.read_file(os.path.join('data', 'nyc_study.geojson'))
+(nyc_nbs, nyc_gdf, nyc_comp_sizes) = nyc_sort_by_comp_size(nyc_geodata)
+
+from splot.libpysal import plot_spatial_weights 
+print("New York City neighbors graph")
+plot_spatial_weights(nyc_nbs, nyc_gdf)
+
+
+
New York City neighbors graph
+
+
+
+
+

+
+
+
+
+

The BYM2 model requires that the neighbors graph be fully connected - use the largest subcomponent Brooklyn and Queens.

+
+
+Show Code +
from libpysal.weights import Queen
+brklyn_qns_gdf = nyc_gdf[nyc_gdf['comp_id']==0].reset_index(drop=True)
+brklyn_qns_nbs = Queen.from_dataframe(brklyn_qns_gdf , geom_col='geometry')
+from utils_bym2 import get_scaling_factor, nbs_to_adjlist
+brklyn_qns_nbs_adj = nbs_to_adjlist(brklyn_qns_nbs)
+tau = get_scaling_factor(brklyn_qns_nbs)
+
+
+

Put all NYC predictors on the same scale - log transform columns as needed.

+
+
+Show Code +
design_vars = np.array(['pct_pubtransit','med_hh_inc', 'traffic', 'frag_index'])
+design_mat = brklyn_qns_gdf[design_vars].to_numpy()
+design_mat[:, 1] = np.log(design_mat[:, 1])
+design_mat[:, 2] = np.log(design_mat[:, 2])
+
+
+

Assemble all inputs into dictionary brklyn_qns_data.

+
+
+Show Code +
brklyn_qns_data = {"N":brklyn_qns_gdf.shape[0],
+            "y":brklyn_qns_gdf['count'].astype('int'),
+            "E":brklyn_qns_gdf['kid_pop'].astype('int'),
+            "K":4,
+            "xs":design_mat,
+            "N_edges": brklyn_qns_nbs_adj.shape[1],
+            "neighbors": brklyn_qns_nbs_adj,
+            "tau": tau
+}
+
+
+
+
+

Model fitting

+

The BYM2 model requires many warmup iterations in order to reach convergence for all parameters, including hyperparameters rho and sigma. We run all three models using the same seed, in order to make the initial parameters as similar as possible.

+

Compile and fit all three models to the Brooklyn-Queens data.

+
+
+Show Code +
bym2_ozs_mod = CmdStanModel(stan_file=os.path.join('stan', 'bym2_ozs.stan'))
+brklyn_qns_ozs_fit = bym2_ozs_mod.sample(data=brklyn_qns_data, iter_warmup=7000)
+a_seed = brklyn_qns_ozs_fit.metadata.cmdstan_config['seed']
+
+bym2_hard_mod = CmdStanModel(stan_file=os.path.join('stan', 'bym2_hard.stan'))
+brklyn_qns_hard_fit = bym2_hard_mod.sample(data=brklyn_qns_data, iter_warmup=7000, seed=a_seed)
+
+bym2_soft_mod = CmdStanModel(stan_file=os.path.join('stan', 'bym2_soft.stan'))
+brklyn_qns_soft_fit = bym2_soft_mod.sample(data=brklyn_qns_data, iter_warmup=7000, seed=a_seed)
+
+
+
+
+

Check implementations

+

If all three implementations are correct, we expect that they will produce the same estimates for all parameters. As always, we check the R-hat values and effective sample sizes. To do this, we tabulate the summary statistics from each of the three models for the hierarchical parameters beta, sigma, and rho.

+

Tabulate summary statistics from all fits.

+
+
+Show Code +
brklyn_qns_ozs_summary = brklyn_qns_ozs_fit.summary()
+brklyn_qns_ozs_summary.index = brklyn_qns_ozs_summary.index.astype(str) + "  - ozs"
+
+brklyn_qns_hard_summary = brklyn_qns_hard_fit.summary()
+brklyn_qns_hard_summary.index = brklyn_qns_hard_summary.index.astype(str) + " - hard"
+
+brklyn_qns_soft_summary = brklyn_qns_soft_fit.summary()
+brklyn_qns_soft_summary.index = brklyn_qns_soft_summary.index.astype(str) + " - soft"
+
+brklyn_qns_fits_summary = pd.concat([brklyn_qns_ozs_summary, brklyn_qns_hard_summary, brklyn_qns_soft_summary])
+intercept_summary = summarize_predictor(brklyn_qns_fits_summary, 'beta_intercept')
+betas_summary = summarize_predictor(brklyn_qns_fits_summary, 'betas')
+sigma_summary = summarize_predictor(brklyn_qns_fits_summary, 'sigma')
+rho_summary = summarize_predictor(brklyn_qns_fits_summary, 'rho')
+
+brklyn_qns_summary = pd.concat([intercept_summary, betas_summary, sigma_summary, rho_summary])
+display(HTML(style_dataframe(brklyn_qns_summary, 3).to_html()))
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_intercept - ozs-8.030.9624.301.01
beta_intercept - hard-7.950.957.321.01
beta_intercept - soft-8.020.9421.441.01
betas[1] - ozs1.230.2517.261.01
betas[1] - hard1.240.263.071.01
betas[1] - soft1.220.2512.111.01
betas[2] - ozs0.170.0830.081.01
betas[2] - hard0.160.088.011.01
betas[2] - soft0.170.0823.701.01
betas[3] - ozs0.090.0369.321.00
betas[3] - hard0.090.0318.111.00
betas[3] - soft0.100.0352.171.00
betas[4] - ozs0.040.0238.911.00
betas[4] - hard0.030.028.131.00
betas[4] - soft0.040.0233.271.00
sigma - ozs0.750.037.621.03
sigma - hard0.750.032.101.02
sigma - soft0.750.034.461.03
rho - ozs0.400.075.291.04
rho - hard0.390.071.461.02
rho - soft0.390.073.291.04
+
+
+

The sum_to_zero vector is the most efficient. In contrast to the outcomes for the binomial model above, the soft sum-to-zero constraint is more efficient than the hard sum-to-zero constraint.

+
+
+
+

Model 2: The BYM2_multicomp model, Freni-Sterrantino et al, 2018

+

In order to apply the BYM2 model to the full NYC dataset, it is necessary to extend the BYM2 model to account for disconnected components and singleton nodes. (The alternative is to add edges to the neighbor graphs so that the neighbors graph is a single connected component. This is problematic, see Notebook 6 The BYM2_multicomp model in Stan for details.)

+

In A note on intrinsic Conditional Autoregressive models for disconnected graphs Freni-Sterrantino et al. in 2018, the authors show how to extend the BYM2 model to handle disconnected components and singleton nodes. They provide the following recommendations:

+
    +
  • Non-singleton nodes are given the BYM2 prior
    • +
  • Singleton nodes (islands) are given a standard Normal prior
    • +
  • Compute per-connected component scaling factor
    • +
  • Impose a sum-to-zero constraint on each connected component
    • +
+

We have followed these recommendations and implemented this model in Stan. The full model is in file bym2_multicomp.stan

+

For this case study, we compare 2 implementations of the BYM2_multicomp model: one which uses the sum_to_zero_vector and one which implements the soft sum-to-zero constraint.

+

It is necessary to constrain the the elements of the spatial effects vector phi on a component-by-component basis. Stan’s slicing with range indexes, provides a way to efficiently access each component. The helper function nyc_sort_by_comp_size both sorts the study data by component and adds the component index to the geodataframe.

+

In the BYM2 model for a fully connected graph the sum-to-zero constraint on phi is implemented directly by declaring phi to be a sum_to_zero_vector, which is a constrained parameter type. The declaration:

+
  sum_to_zero_vector[N] phi;  // spatial effects
+

creates a constrained variable of length \(N\), with a corresponding unconstrained variable of length \(N-1\).

+

In order to constrain slices of the parameter vector phi, we do the following:

+
    +
  • In the parameters block, we declare the unconstrained parameter phi_raw as an regular vector vector (instead of a sum_to_zero_vector). +
      +
    • For a fully connected graph of size \(N\), the size of the unconstrained sum-to-zero vector is \(N-1\). For a disconnected graph with \(M\) non-singleton nodes, the size of phi_raw is \(M\) minus the number of connected components.
      • +
    • +
+
  vector[N_connected - N_components] phi_raw;  // spatial effects
+
    +

  • In the functions block, we implement the unconstraining transform.

    • +

  • In the transformed parameters block, we apply the constraining transform.

    • +
+
  vector[N_connected] phi = zero_sum_components(phi_raw, component_idxs, component_sizes);
+

The constraining transform is broken into two functions:

+
    +

  • function zero_sum_constrain, the actual constraining transform, which corresponds directly to the built-in zero_sum_vector transform.

    • +

  • function zero_sum_constrain_components, which handles the slicing, and calls zero_sum_constrain on each component.

    • +
+
  /**
+   * Constrain sum-to-zero vector
+   *
+   * @param y unconstrained zero-sum parameters
+   * @return vector z, the vector whose slices sum to zero
+   */
+  vector zero_sum_constrain(vector y) {
+    int N = num_elements(y);
+    vector[N + 1] z = zeros_vector(N + 1);
+    real sum_w = 0;
+    for (ii in 1:N) {
+      int i = N - ii + 1; 
+      real n = i;
+      real w = y[i] * inv_sqrt(n * (n + 1));
+      sum_w += w;
+      z[i] += sum_w;     
+      z[i + 1] -= w * n;    
+    }
+    return z;
+  }
+
    +
  • zero_sum_components: slices vector phi by component, applies constraining transform to each.
    • +
+
  /**
+   * Component-wise constrain sum-to-zero vectors
+   *
+   * @param phi unconstrained vector of zero-sum slices
+   * @param idxs component start and end indices
+   * @param sizes component sizes
+   * @return vector phi_ozs, the vector whose slices sum to zero
+   */
+  vector zero_sum_components(vector phi,
+                                array[ , ] int idxs,
+                                array[] int sizes) {
+    vector[sum(sizes)] phi_ozs;
+    int idx_phi = 1;
+    int idx_ozs = 1;
+    for (i in 1:size(sizes)) {
+      phi_ozs[idx_ozs : idx_ozs + sizes[i] - 1] =
+        zero_sum_constrain(segment(phi, idx_phi, sizes[i] - 1));
+      idx_phi += sizes[i] - 1;
+      idx_ozs += sizes[i];
+    }
+    return phi_ozs;
+  }
+
+
+
+ +
+
+Note +
+
+
+

The constraining transform is a linear operation, leading to a constant Jacobian determinant which is therefore not included. As of Stan 2.36, transforms which include a Jacobian adjustment can do so with the jacobian += statement and must have names ending in _jacobian. See section functions implementing change-of-variable adjustments in the Stan User’s Guide chapter on user-defined functions for details.

+
+
+
+

Data Assembly

+

Get NYC dataset, pre-process as needed for BYM2_multicomp model

+
+
+Show Code +
from utils_nyc_map import nyc_sort_by_comp_size
+from utils_bym2 import nbs_to_adjlist, get_scaling_factors
+nyc_geodata = gpd.read_file(os.path.join('data', 'nyc_study.geojson'))
+(nyc_nbs, nyc_gdf, nyc_comp_sizes) = nyc_sort_by_comp_size(nyc_geodata)
+nyc_nbs_adj = nbs_to_adjlist(nyc_nbs)
+component_sizes = [x for x in nyc_comp_sizes if x > 1]
+scaling_factors = get_scaling_factors(len(component_sizes), nyc_gdf)
+
+
+

Put all NYC predictors on the same scale - log transform columns as needed.

+
+
+Show Code +
design_vars = np.array(['pct_pubtransit','med_hh_inc', 'traffic', 'frag_index'])
+design_mat = nyc_gdf[design_vars].to_numpy()
+design_mat[:, 1] = np.log(design_mat[:, 1])
+design_mat[:, 2] = np.log(design_mat[:, 2])
+
+
+

Assemble all inputs into dictionary bym2_multicomp_data.

+
+
+Show Code +
bym2_multicomp_data = {
+    "N":nyc_gdf.shape[0],
+    "y":nyc_gdf['count'].astype('int'),
+    "E":nyc_gdf['kid_pop'].astype('int'),
+    "K":4,
+    "xs":design_mat,
+    "N_edges": nyc_nbs_adj.shape[1],
+    "neighbors": nyc_nbs_adj,
+    "N_components": len(component_sizes),
+    "component_sizes": component_sizes,
+    "scaling_factors": scaling_factors
+}
+
+
+
+
+

Model Fitting

+

The BYM2_multicomp model requires many warmup iterations in order to reach convergence for all parameters, including hyperparameters rho and sigma, and a larger set of draws from the posterior in order to obtain satisfactory R-hat values.

+

Compile and fit the BYM2_multicomp model using the sum_to_zero_vector constraining transform.

+
+
+Show Code +
bym2_multicomp_ozs_file = os.path.join('stan', 'bym2_multicomp.stan')
+bym2_multicomp_ozs_mod = CmdStanModel(stan_file=bym2_multicomp_ozs_file)
+bym2_multicomp_ozs_fit = bym2_multicomp_ozs_mod.sample(data=bym2_multicomp_data, iter_warmup=9000)
+
+
+

Tabulate summary statistics.

+
+
+Show Code +
bym2_multicomp_ozs_summary = bym2_multicomp_ozs_fit.summary()
+bym2_multicomp_ozs_summary[['Mean', 'StdDev', 'ESS_bulk/s', 'R_hat']].round(2).loc[
+  ['beta_intercept', 'betas[1]', 'betas[2]', 'betas[3]', 'betas[4]', 'sigma', 'rho']]
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
MeanStdDevESS_bulk/sR_hat
beta_intercept-5.780.5510.511.01
betas[1]0.820.169.861.01
betas[2]0.040.059.221.01
betas[3]0.030.0223.191.00
betas[4]0.040.0113.381.01
sigma0.780.023.711.01
rho0.390.062.701.02
+ +
+
+
+
+
+

Model Comparison

+

We can compare the sum_to_zero_vector implementation to the corresponding soft sum-to-zero constraint. The full model is in file bym2_multicomp_soft.stan

+

In model bym2_soft.stan the soft sum-to-zero constraint is combined directly with the ICAR prior:

+
  target += (-0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]])
+         + normal_lupdf(sum(phi) | 0, 0.001 * rows(phi)));
+

For the BYM2_multicomp model, this operation is carried out in two steps. First the ICAR prior is applied to phi, next we iterate through the components, applying the sum-to-zero constraint to each in turn.

+
  target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]);  // ICAR
+  for (n in 1:N_components) {   // component-wise sum-to-zero constraint
+    sum(phi[node_idxs[n, 1] : node_idxs[n, 2]]) ~ normal(0,
+                             0.001 * component_sizes[n]);
+

The data inputs are the same. To ensure (roughly) the same initialization, we reuse the seed from bym2_multicomp_ozs_fit. This model fits very slowly and requires increasing the max_treedepth; at the default setting, all iterations hit this limit.

+

Compile and fit the BYM2_multicomp model using the soft sum-to-zero constraint.

+
+
+Show Code +
a_seed = bym2_multicomp_ozs_fit.metadata.cmdstan_config['seed']
+bym2_multicomp_soft_file = os.path.join('stan', 'bym2_multicomp_soft.stan')
+bym2_multicomp_soft_mod = CmdStanModel(stan_file=bym2_multicomp_soft_file)
+bym2_multicomp_soft_fit = bym2_multicomp_soft_mod.sample(data=bym2_multicomp_data, max_treedepth=13, iter_warmup=9000, seed=a_seed)
+
+
+

The following table compares the results for the hierarchical parameters beta, rho, and sigma.

+

Tabulate summary statistics from both fits.

+
+
+Show Code +
bym2_multicomp_ozs_summary.index =  bym2_multicomp_ozs_summary.index.astype(str) + "  a) ozs"
+bym2_multicomp_soft_summary = bym2_multicomp_soft_fit.summary()
+bym2_multicomp_soft_summary.index =  bym2_multicomp_soft_summary.index.astype(str) + "  b) soft"
+bym2_multicomp_summary = pd.concat([bym2_multicomp_ozs_summary, bym2_multicomp_soft_summary])
+
+intercept_summary = summarize_predictor(bym2_multicomp_summary, 'beta_intercept')
+betas_summary = summarize_predictor(bym2_multicomp_summary, 'betas')
+sigma_summary = summarize_predictor(bym2_multicomp_summary, 'sigma')
+rho_summary = summarize_predictor(bym2_multicomp_summary, 'rho')
+nyc_summary = pd.concat([intercept_summary, betas_summary, sigma_summary, rho_summary])
+
+display(HTML(style_dataframe(nyc_summary, 2).to_html()))
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 MeanStdDevESS_bulk/sR_hat
beta_intercept a) ozs-5.780.5510.511.01
beta_intercept b) soft-5.780.570.601.01
betas[1] a) ozs0.820.169.861.01
betas[1] b) soft0.820.150.481.01
betas[2] a) ozs0.040.059.221.01
betas[2] b) soft0.040.050.461.01
betas[3] a) ozs0.030.0223.191.00
betas[3] b) soft0.030.021.101.00
betas[4] a) ozs0.040.0113.381.01
betas[4] b) soft0.040.010.931.00
sigma a) ozs0.780.023.711.01
sigma b) soft0.780.020.191.01
rho a) ozs0.390.062.701.02
rho b) soft0.380.060.121.01
+
+
+
+
+
+

Discussion

+

The BYM2 model has a relatively complex multilevel structure, and the BYM2_multicomp model is even more complex. Before Stan 2.36, the recommended implementation was to use the soft sum-to-zero constraint for the BYM2 model.

+

For the BYM2_multicomp model, model bym2_multicomp.stan shows how to implement the sum_to_zero_vector constraining transform as a Stan function. Both implementation get exactly the same estimates, confirming that both models are correctly implemented. Fitting the extended BYM2_multicomp model with a soft sum-to-zero is painfully slow. Here we show that the sum_to_zero_vector greatly improves the run time; which is critical because to fit this particular dataset requires a very large number of warmup iterations.

+
+
+
+

Conclusion: the sum_to_zero_vector just works!

+

The more complex the model, the greater the need for the sum_to_zero_vector. When considering the effective sample size, it is important to remember that what is most important is effective samples per second. In these experiments, the sum_to_zero_vector models consistently have the highest ESS/sec.

+

Key takeaways:

+
    +

  • In the first part of this case study, we show how to use the sum_to_zero_vector for categorical predictors in a hierarchical model, and how to adjust the marginal variances for a normal prior.

    • +

  • In the second part of this case study we see how to implement the constraining transform used for the sum_to_zero_vector as a Stan function in order to handle the case where a vector is the concatenation of several sum-to-zero vectors.

    • +
+
+
+

References

+ +
+ +
+ + + +
+ + + + + \ No newline at end of file diff --git a/jupyter/sum-to-zero/sum_to_zero_evaluation.qmd b/jupyter/sum-to-zero/sum_to_zero_evaluation.qmd new file mode 100644 index 00000000..3854f3a6 --- /dev/null +++ b/jupyter/sum-to-zero/sum_to_zero_evaluation.qmd @@ -0,0 +1,1130 @@ +--- +title: "The Sum-to-Zero Constraint in Stan" +author: "Mitzi Morris" +format: + html: + theme: + light: [cosmo, quarto-config/theme.scss] + dark: [cosmo, quarto-config/theme-dark.scss] + syntax-definitions: + - quarto-config/stan.xml + highlight-style: + light: quarto-config/tango.theme + dark: quarto-config/nord.theme + code-copy: true + code-overflow: wrap + code-fold: true + code-summary: "Show Code" + toc: true + css: quarto-config/quarto_styles.css + page-layout: full + toc-location: right + embed-resources: true +execute: + engine: jupyter +include-in-header: + - text: | + +--- + +*Load libraries and set up notebook.* +```{python} +# libraries used in this notebook +import os +import numpy as np +import pandas as pd +import geopandas as gpd +import plotnine as p9 +import libpysal +from splot.libpysal import plot_spatial_weights +from random import randint + +from cmdstanpy import CmdStanModel, cmdstan_path, set_cmdstan_path +import logging +cmdstanpy_logger = logging.getLogger("cmdstanpy") +cmdstanpy_logger.setLevel(logging.FATAL) +set_cmdstan_path(os.path.join('/Users', 'mitzi', 'github', 'stan-dev', 'cmdstan')) + +import warnings +warnings.filterwarnings('ignore') + +import matplotlib +import matplotlib.pyplot as plt +%matplotlib inline + +from utils_html import * + +# notebook display options +plt.rcParams['figure.figsize'] = (7, 7) +plt.rcParams['figure.dpi'] = 100 + +np.set_printoptions(precision=2) +np.set_printoptions(suppress=True) +pd.set_option('display.precision', 2) +pd.options.display.float_format = '{:.2f}'.format +``` + +## Introducing the `sum_to_zero_vector` Constrained Parameter Type + + +The [`sum_to_zero_vector`](https://mc-stan.org/docs/reference-manual/transforms.html#zero-sum-vector) +constrained parameter type was introduced in the [Stan 2.36 release](https://github.com/stan-dev/cmdstan/releases/tag/v2.36.0). + +The parameter declaration: + +```stan + sum_to_zero_vector[K] beta; +``` +produces a vector of size `K` such that `sum(beta) = 0`. +The unconstrained representation requires only `K - 1` values because the +last is determined by the first `K - 1`. + +If you put a normal prior on the zero-sum vector the resulting variance will be less than the intended normal variance. +To get the same variance as the intended normal prior do + +```stan +parameters { + sum_to_zero_vector[N] z; +} +nnmodel { + z ~ normal(0, sqrt((N / (N - 1.0)) * sigma) +} +``` +where sigma is the intended standard deviation. +(Note: it’s slightly more efficient to pre-calculate the `sqrt(N / (N - 1.0))` in transformed_data.) + +Prior to Stan 2.36, a sum-to-zero constraint could be implemented either + +- as a "hard" sum to zero constraint, where the parameter is declared to be an $N-1$ length vector with a corresponding $N$-length transformed parameter +whose first $N-1$ elements are the same as the corresponding parameter vector, and the $N^{th}$ element is the negative sum of the $N-1$ elements. + +- as a "soft" sum to zero constraint, where the parameter is an $N$-length vector with a prior on the sum of the elements that keeps it within $\epsilon$ of $0$. + +The choice between the two was largely an empirical one. +As a general rule, for small vectors, the hard sum-to-zero constraint is more efficient; +for larger vectors, the soft sum-to-zero constraint is faster, +but much depends on the specifics of the model and the data. + +The hard sum-to-zero is often problematic. +For small $N$ and models with sensible priors, the hard sum-to-zero is usually satisfactory. +But as the size of the vector grows, it distorts the marginal variance of the $N^{th}$. +Given a parameter vector: +$$ +x_1, x_2, \dots, x_{N-1} \sim \text{i.i.d. } N(0, \sigma^2) +$$ +by the properties of independent normal variables, each of the free elements $x_1, \ldots, x_{N-1}$ has variance $\sigma^2$. +However, the $N^{th}$ element is defined deterministically as: +$$ +x_N = -\sum_{i=1}^{N-1} x_i +$$ +and its variance is inflated by a factor of $N-1$. +$$ +\operatorname{Var}(x_N) = \operatorname{Var}\Bigl(-\sum_{i=1}^{N-1} x_i\Bigr) += \sum_{i=1}^{N-1} \operatorname{Var}(x_i) += (N-1)\sigma^2. +$$ +For large vectors, MCMC samplers struggle with the hard sum-to-zero constraint, +as every change to any of the $N-1$ elements also requires a corresponding change to +the $N^{th}$ element; balancing these changes introduces potential non-identifiabilities. + +The soft sum-to-zero constraint is problematic for the following reasons. + +* The tolerance $\epsilon$ (the scale of the penalty) must be chosen by the analyst. Too large, +and the result is too far from zero to be effective, too small and the sampler cannot satisfy the +constraint. +* The soft constraint only penalizes deviations from zero, leading to weaker identifiability of the parameters. +This can lead to slow convergence and mixing, as the sampler explores nearly non-identified regions. +* The marginal variances may not reflect the intended prior. + +The `sum_to_zero_vector` transform ensures that each element of the resulting constrained vector has the same variance. +This improves the sampler performance, providing fast computation and good effective sample size. +This becomes increasingly noticeable as models increase in size and complexity. +To demonstrate this, in this notebook we consider two different classes of models: + +- Multi-level regressions for binomial data with group-level categorical predictors. +- Spatial models for areal data. + +For these models, we provide three implementations which differ only in the +implementation of the sum-to-zero constraint: the built-in `sum_to_zero_vector`, +and the hard and soft sum-to-zero implementations. +We fit each model to the same dataset, using the same random seed, and then +compare the summary statistics for the constrained parameter values. +Since the models are nearly equivalent except in the implied prior distributions +on the vector, we expect that all three implementations +should produce nearly equivalent estimates; what differs is the speed of computation, +as measured by effective samples per second. + +::: {.callout-note} +The spatial models are taken from the a set of notebooks available from GitHub repo [https://github.com/mitzimorris/geomed_2024](https://github.com/mitzimorris/geomed_2024). +::: + +## Multi-level Models with Group-level Categorical Predictors + +In this section we consider a model which estimates per-demographic disease prevalence rates for a population. +The model is taken from the Gelman and Carpenter, 2020 +[Bayesian Analysis of Tests with Unknown Specificity and Sensitivity](https://doi.org/10.1111/rssc.12435). +It combines a model for multilevel regression and post-stratification with a likelihood that +accounts for test sensitivity and specificity. + +The data consists of: + +* A set of per-demographic aggregated outcomes of a diagnostic test procedure +with unequal number of tests per demographic. + +* A corresponding set of demographic descriptors encoded as a vector of categorical values. +In this example these are named `sex`, `age`, `eth`, and `edu`, but there can be any number +of demographic predictors with any number of categories. + +* The specified test sensitivity and specificity + +Putting a sum-to-zero constraint on the categorical allows us to model all categories as offset from the mean. +We prefer this to the alternative of dropping a category for reference or other contrast coding strategy. + +### The Stan model + + +The full model is in file [binomial_4_preds_ozs.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4_preds_ozs.stan). +It provides an estimate of the true prevalence based on binary tests with +a given (or unknown) test sensitivity and specificity, and uses a `sum_to_zero_vector` +to constrain the group-level parameters `age`, `eth`, and `edu`. + +In order to put a standard normal prior on `beta_age`, `beta_eth`, and `beta_edu`, +we need to scale the variance, as suggested above. +The scaling factors are pre-computed in the `transformed data` block, +and applied as part of the prior. + +*relevant excepts from* `binomial_4_preds_ozs.stan` +```stan +transformed data { + // scaling factors for marginal variances of sum_to_zero_vectors + real s_age = sqrt(N_age / (N_age - 1.0)); + real s_eth = sqrt(N_eth / (N_eth - 1.0)); + real s_edu = sqrt(N_edu / (N_edu - 1.0)); +} +parameters { + real beta_0; + real beta_sex; + real sigma_age, sigma_eth, sigma_edu; + sum_to_zero_vector[N_age] beta_age; + sum_to_zero_vector[N_eth] beta_eth; + sum_to_zero_vector[N_edu] beta_edu; +} +transformed parameters { + // true prevalence + vector[N] p = inv_logit(beta_0 + beta_sex * sex_c + beta_age[age] + + beta_eth[eth] + beta_edu[edu]); + // incorporate test sensitivity and specificity. + vector[N] p_sample = p * sens + (1 - p) * (1 - spec); +} +model { + pos_tests ~ binomial(tests, p_sample); // likelihood + // scale normal priors on sum_to_zero_vectors + beta_age ~ normal(0, s_age * sigma_age); + beta_eth ~ normal(0, s_eth * sigma_eth); + beta_edu ~ normal(0, s_edu * sigma_edu); + ... +} +``` + +#### Simulating Data + +To investigate the predictive behavior of this model at different timepoints in a pandemic, +we have written a data-generating program to create datasets given the +baseline disease prevalence, test specificity and sensitivity, +and the desired number of diagnostic tests. +In the `generated quantities` block we use Stan's PRNG functions to populate + vthe true weights for the categorical coefficient vectors, and the relative percentages +of per-category observations. +Then we use a set of nested loops to generate the data for each demographic, +using the PRNG equivalent of the model likelihood. + +The full data-generating program is in file [gen_binomial_4_preds.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4_preds_ozs.stan). +The helper function `simulate_data` in file `utils.py` sets up the data-generating program +according to the specified number of demographic categories, approximate average number of +observations per category, and baseline disease prevalence, test specificity and sensitivity. +This allows us to create datasets for large and small populations +and for finer or more coarse-grained sets of categories. +The larger the number of strata overall, the more observations are needed to get good coverage. +Because the modeled data `pos_tests` is generated according to the Stan model's likelihood, +the model is a priori well-specified with respect to the data. + + +::: {.callout-note} +Because the data-generating parameters and percentage of observations per category are generated at random, +some datasets may have very low overall disease rates and/or many unobserved strata. +While this is informative for understanding what is consistent with the generating distributions, +it results in estimates of varying effect parameters with near-zero values, +which are on the boundary of parameter space, and hence very challenging to fit. +Because we wish to compare baseline performance, +we have cherry-picked a seed which generates datasets that avoid this issue. +::: + + +*Generate several datasets of different sizes.* +```{python} +#| output: false + +# specify data shapes and test characteristics +N_eth = 3 +N_edu = 5 +N_age = 9 +baseline = -3.5 +sens = 0.75 +spec = 0.9995 + + +# generate datasets +from utils import simulate_data +data_tiny = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 4, seed=56789) +data_small = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 17, seed=data_tiny['seed']) +data_large = simulate_data(N_eth, N_edu, N_age, baseline, sens, spec, 200, seed=data_tiny['seed']) +``` +The data generating program creates unequal numbers of tests per demographic. +The following tables summarize the resulting distribution. + +:::: columns +::: {.column width="32%"} + +```{python} +print("Small Dataset") +print("tests total " + str(np.sum(data_small['tests']))) +small_df = pd.DataFrame({'tests per demographic': data_small['tests']}).describe() +small_df.drop('count', inplace=True) +small_df +``` +::: +::: {.column width="32%"} +```{python} +print("Large Dataset") +print("tests total " + str(np.sum(data_large['tests']))) +large_df = pd.DataFrame({'tests per demographic': data_large['tests']}).describe() +large_df.drop('count', inplace=True) +large_df +``` +::: +::: {.column width="32%"} +```{python} +print("Tiny Dataset") +print("tests total " + str(np.sum(data_tiny['tests']))) +tiny_df = pd.DataFrame({'tests per demographic': data_tiny['tests']}).describe() +tiny_df.drop('count', inplace=True) +tiny_df +``` +::: +:::: + + + +### Model Fitting + +We fit each model to the the large and small datasets. +We record the seed used for the first run and use it for all subsequent fits; this insures that the models have the same set of initial parameter values +for all parameters except the constrained params `beta_eth`, `beta_edu`, and `beta_age`. + + +**Model 1: `sum_to_zero_vector`** + +This model is in file [binomial_4_preds_ozs.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4_preds_ozs.stan). + +*Compile and fit the model to data.* +```{python} +#| output: false +binomial_ozs_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_ozs.stan')) +print("Model 1: sum_to_zero_vector, small dataset") +binomial_ozs_fit = binomial_ozs_mod.sample(data=data_small, parallel_chains=4) + +a_seed = binomial_ozs_fit.metadata.cmdstan_config['seed'] +print("Model 1: sum_to_zero_vector, large dataset") +binomial_ozs_fit_lg = binomial_ozs_mod.sample(data=data_large, parallel_chains=4, seed=a_seed) +``` +**Model 2: Hard sum-to-zero constraint** + +This model is in file [binomial_4_preds_hard.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4_preds_hard.stan). + +*Compile and fit the model to data.* +```{python} +#| output: false +binomial_hard_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_hard.stan')) +print("\nModel 2: hard sum-to-zero constraint, small dataset") +binomial_hard_fit = binomial_hard_mod.sample(data=data_small, parallel_chains=4, seed=a_seed) +print("Model 2: hard sum-to-zero constraint, large dataset") +binomial_hard_fit_lg = binomial_hard_mod.sample(data=data_large, parallel_chains=4, seed=a_seed) +``` + +**Model 3: soft sum-to-zero constraint** + +This model is in file [binomial_4_preds_soft.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4_preds_soft.stan). + +*Compile and fit the model to data.* +```{python} +#| output: false +binomial_soft_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_soft.stan')) +print("\nModel 3: soft sum-to-zero constraint, small dataset") +binomial_soft_fit = binomial_soft_mod.sample(data=data_small, parallel_chains=4, seed=a_seed) +print("Model 3: soft sum-to-zero constraint, large dataset") +binomial_soft_fit_lg = binomial_soft_mod.sample(data=data_large, parallel_chains=4, seed=a_seed) +``` + + +### Model Checking and Comparison + +#### Check implementations + +If all three implementations are correct, we expect that they will produce the same +estimates for all parameters. As always, we check the R-hat values and effective sample sizes. +To do this, we tabulate the summary statistics from each of the three models, fit to the +large and small datasets. + + +*Tabulate summary statistics from all fits.* +```{python} +# small dataset +ozs_fit_summary = binomial_ozs_fit.summary(sig_figs=2) +ozs_fit_summary.index = ozs_fit_summary.index.astype(str) + " - ozs" + + +hard_fit_summary = binomial_hard_fit.summary(sig_figs=2) +hard_fit_summary.index = hard_fit_summary.index.astype(str) + " - hard" + +soft_fit_summary = binomial_soft_fit.summary(sig_figs=2) +soft_fit_summary.index = soft_fit_summary.index.astype(str) + " - soft" + +small_data_fits_summary = pd.concat([ozs_fit_summary, hard_fit_summary, soft_fit_summary]) + +# large dataset +ozs_fit_lg_summary = binomial_ozs_fit_lg.summary(sig_figs=2) +ozs_fit_lg_summary.index = ozs_fit_lg_summary.index.astype(str) + " - ozs" + +hard_fit_lg_summary = binomial_hard_fit_lg.summary(sig_figs=2) +hard_fit_lg_summary.index = hard_fit_lg_summary.index.astype(str) + " - hard" + +soft_fit_lg_summary = binomial_soft_fit_lg.summary(sig_figs=2) +soft_fit_lg_summary.index = soft_fit_lg_summary.index.astype(str) + " - soft" + +large_data_fits_summary = pd.concat([ozs_fit_lg_summary, hard_fit_lg_summary, soft_fit_lg_summary]) +``` + +**Eth** + +*Fitted estimates* +```{python} +beta_eth_summary = summarize_predictor(small_data_fits_summary, 'beta_eth\[') +beta_eth_summary_lg = summarize_predictor(large_data_fits_summary, 'beta_eth\[') +labels = expand_true(data_small['beta_eth'], "Datagen
values") +small_html = style_dataframe(beta_eth_summary, 3).to_html() +large_html = style_dataframe(beta_eth_summary_lg, 3).to_html() +display_side_by_side(small_html, large_html, labels) + +sigma_eth_summary = summarize_predictor(small_data_fits_summary, 'sigma_eth') +sigma_eth_summary_lg = summarize_predictor(large_data_fits_summary, 'sigma_eth') +small_html = style_dataframe(sigma_eth_summary, 4).to_html() +large_html = style_dataframe(sigma_eth_summary_lg, 4).to_html() +display_side_by_side(small_html, large_html) +``` + + +**Edu** + +*Fitted estimates* +```{python} +beta_edu_summary = summarize_predictor(small_data_fits_summary, 'beta_edu\[') +beta_edu_summary_lg = summarize_predictor(large_data_fits_summary, 'beta_edu\[') +labels = expand_true(data_small['beta_edu'], "Datagen
values") +small_html = style_dataframe(beta_edu_summary, 3).to_html() +large_html = style_dataframe(beta_edu_summary_lg, 3).to_html() +display_side_by_side(small_html, large_html, labels) + +sigma_edu_summary = summarize_predictor(small_data_fits_summary, 'sigma_edu') +sigma_edu_summary_lg = summarize_predictor(large_data_fits_summary, 'sigma_edu') +small_html = style_dataframe(sigma_edu_summary, 4).to_html() +large_html = style_dataframe(sigma_edu_summary_lg, 4).to_html() +display_side_by_side(small_html, large_html) +``` + +**Age** + +*Fitted estimates* +```{python} +beta_age_summary = summarize_predictor(small_data_fits_summary, 'beta_age\[') +beta_age_summary_lg = summarize_predictor(large_data_fits_summary, 'beta_age\[') +labels = expand_true(data_small['beta_age'], "Datagen
values") +small_html = style_dataframe(beta_age_summary, 3).to_html() +large_html = style_dataframe(beta_age_summary_lg, 3).to_html() +display_side_by_side(small_html, large_html, labels) + +sigma_age_summary = summarize_predictor(small_data_fits_summary, 'sigma_age') +sigma_age_summary_lg = summarize_predictor(large_data_fits_summary, 'sigma_age') +small_html = style_dataframe(sigma_age_summary, 4).to_html() +large_html = style_dataframe(sigma_age_summary_lg, 4).to_html() +display_side_by_side(small_html, large_html) +``` + +These comparisons demonstrate that + +* All models have R-hat values of 1.00 for all group-level parameters and high effective sample sizes. + +* The models recover the sign of the parameter, but not the exact value. +In the case of lots of observations and only a few categories they do a better job of recovering the true parameters. + +* In almost all cases, estimates for each parameter are the same across implementations to 2 significant figures. +In a few cases they are off by 0.01; where they are off, the percentage of observations for that parameter is correspondingly low. + +* The `sum_to_zero_vector` implementation has the highest number of effective samples per second, +excepting a few individual parameters for which the hard sum-to-zero performs equally well. + + +#### Model efficiency + +Model efficiency is measured by iterations per second, however, as the draws from the MCMC sampler +may be correlated, we need to compute the number of effective samples across all chains +divided by the total sampling time - this is *ESS_bulk/s*, the effective samples per second. +The following table shows the average runtime for 100 runs +of each of the three models on large and small datasets. +This data was generated by script `eval_efficiencies.py`. + + +*Tabulate summary statistics from all fits.* +```{python} +with open(os.path.join('binomial_runtimes_small.json'), 'r') as file: + data = json.load(file) +df_small = pd.DataFrame(data) + +with open(os.path.join('binomial_runtimes_large.json'), 'r') as file: + data = json.load(file) +df_large = pd.DataFrame(data) + +small_html = style_dataframe(df_small, 3).to_html() +large_html = style_dataframe(df_large, 3).to_html() +display_side_by_side(small_html, large_html) +``` + +The `sum_to_zero_vector` is far more efficient than either the hard or soft sum-to-zero constraint. + +#### Prior predictive checks + +[Prior predictive checks](https://mc-stan.org/docs/stan-users-guide/posterior-predictive-checks.html#prior-predictive-checks) +simulate data directly from the prior, in the absense of any observed data. +Here we use the simulated dataset to examine the prior marginal variances +of the elements of the sum-to-zero vector under the hard-sum-to-zero constraint +and the built-in `sum_to_zero` transform. +In particular, we are interested in the marginal variances of the elements of the sum-to-zero constrained parameters. + +We have written two models: [binomial_4preds_ozs_ppc.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4preds_ozs_ppc.stan) +and [binomial_4preds_hard_ppc.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4preds_hard_ppc.stan). +We generate a sample from each, and compare the marginal variances of the elements of sum-to-zero constrained parameter `beta_age`. + +**Marginal variances of the hard sum-to-zero constraint, sample from the prior** + +*Generate sample from the prior of the hard sum-to-zero model.* +```{python} +#| output: false +binomial_hard_ppc_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_hard_ppc.stan')) +binomial_hard_ppc_fit = binomial_hard_ppc_mod.sample(data=data_small, parallel_chains=4) +``` +*Get marginal variances of elements of parameter `beta_age` from the prior.* +```{python} +np.var(binomial_hard_ppc_fit.beta_age, axis=0) +``` + +The above summaries show how the hard sum-to-zero constraint distorts the variance of the $N^{th}$ element. +This is only a problem for very sparse datasets, where the prior swamps the data. +With data, this problem goes away; to see this we fit the hard sum-to-zero constraint model +on the tiny and small datasets. + +**Marginal variances of the hard sum-to-zero constraint, tiny dataset** + +*Fit hard sum-to-zero model to the tiny dataset.* +```{python} +#| output: false +binomial_hard_fit_tiny = binomial_hard_mod.sample(data=data_tiny, parallel_chains=4) +``` +*Get marginal variances of elements of parameter `beta_age` from the posterior.* +```{python} +np.var(binomial_hard_fit_tiny.beta_age, axis=0) +``` + +**Marginal variances of the hard sum-to-zero constraint, small dataset** + +*Fit hard sum-to-zero model to the small dataset.* +```{python} +#| output: false +binomial_hard_fit_small = binomial_hard_mod.sample(data=data_small, parallel_chains=4) +``` +*Get marginal variances of elements of parameter `beta_age` from the posterior.* +```{python} +np.var(binomial_hard_fit_small.beta_age, axis=0) +``` + +The marginal variances for the `sum_to_zero_vector` elements are roughly equal in both the prior +and the posterior. + +**Marginal variances of the built-in `sum_to_zero_vector`, sample from the prior** + +*Generate sample from the prior of the `sum_to_zero_vector` model.* +```{python} +#| output: false +binomial_ozs_ppc_mod = CmdStanModel(stan_file=os.path.join('stan', 'binomial_4preds_ozs_ppc.stan')) +binomial_ozs_ppc_fit = binomial_ozs_ppc_mod.sample(data=data_small, parallel_chains=4) +``` +*Get marginal variances of elements of parameter `beta_age` from the prior.* +```{python} +np.var(binomial_ozs_ppc_fit.beta_age, axis=0) +``` + +**Marginal variances of the built-in `sum_to_zero_vector`, tiny dataset** + +*Fit the `sum_to_zero_vector` model to the tiny dataset.* +```{python} +#| output: false +binomial_hard_fit_tiny = binomial_hard_mod.sample(data=data_tiny, parallel_chains=4) +``` + +*Get marginal variances of elements of parameter `beta_age` from the posterior.* +```{python} +np.var(binomial_hard_fit_tiny.beta_age, axis=0) +``` + + +### Discussion + +Comparison of these three models with different data regimes demonstrates the following. + +* For a multi-level model with group-level categorical predictors the `sum_to_zero_vector` provides fast results and good effective sample sizes for both datasets. + +* Model [binomial_4preds_ozs.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/binomial_4preds_ozs.stan). +shows how to properly scale the variance of a `sum_to_zero_vector` constrained parameter in order to put a standard normal prior on it. + +* Prior predictive checks demonstrate the difference between the marginal variances of +the `sum_to_zero_vector` and hard sum-to-zero implementations. + + + +## Spatial Models with an ICAR component + +Spatial auto-correlation is the tendency for adjacent areas to share similar characteristics. +Conditional Auto-Regressive (CAR) and Intrinsic Conditional Auto-Regressive (ICAR) models, +first introduced by Besag, 1974, account for this by pooling information from neighboring regions. +The BYM model, (Besag, York, Mollié, 1991) extends a lognormal Poisson model +plus ICAR component for spatial auto-correlation by adding an ordinary +random-effects component for non-spatial heterogeneity. +The BYM2 model builds on this model and subsequent refinements. + +The ICAR, BYM2, and BYM2_multicomp models are more fully explained in a series of notebooks +available from GitHub repo: [https://github.com/mitzimorris/geomed_2024](https://github.com/mitzimorris/geomed_2024), see notebooks: + +* [The ICAR model in Stan](https://github.com/mitzimorris/geomed_2024/blob/main/python-notebooks/h4_icar.ipynb) +* [The BYM2 model in Stan](https://github.com/mitzimorris/geomed_2024/blob/main/python-notebooks/h5_bym2.ipynb) +* [The BYM2_multicomp model in Stan](https://github.com/mitzimorris/geomed_2024/blob/main/python-notebooks/h6_bym2_multicomp.ipynb) + +### ICAR Models + +The key element of the BYM2 model is the ICAR component. +Its conditional specification is a +multivariate normal random vector $\mathbf{\phi}$ +where each ${\phi}_i$ is conditional on the values of its neighbors. + +The joint specification rewrites to a _Pairwise Difference_, + +$$ p(\phi) \propto \exp \left\{ {- \frac{1}{2}} \sum_{i \sim j}{({\phi}_i - {\phi}_j)}^2 \right\} $$ + +Each ${({\phi}_i - {\phi}_j)}^2$ contributes a +penalty term based on the distance between the values of neighboring regions. +However, $\phi$ is non-identifiable, constant added to $\phi$ washes out of ${\phi}_i - {\phi}_j$. +Therefore, **a sum-to-zero constraint is needed to both identify and center $\phi$.** + +The Stan implementation of the ICAR component computes the sum of the pairwise distances +by representing the spatial adjacency matrix as a array of pairs of neighbor indices. + +```stan +data { + ... + // spatial structure + int N_edges; // number of neighbor pairs + array[2, N_edges] int neighbors; // columnwise adjacent +``` + +The ICAR prior comes into the model as parameter `phi`. +```stan +model { + ... + target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]); // ICAR prior +``` + +The ICAR model requires that the neighbor graph is fully connected for two reasons: + +* The joint distribution is computed from the pairwise differences between a node and its neighbors; +singleton nodes have no neighbors and are therefore undefined. + +* Even if the graph doesn't have any singleton nodes, when the graph has multiple connected components +a sum-to-zero constraint on the entire vector fails to properly identify the model. + +Because the BYM2 model includes an ICAR component, it too requires a fully connected neighbor graph. +In order to apply the BYM2 model to the full NYC dataset, it is necessary to +extend the BYM2 model to account for disconnected components and singleton nodes. + +First we use the BYM2 model for fully connected graphs +to compare implementations of the sum-to-zero vector on `phi`, +and then we extend the BYM2 model following the INLA implementation +developed by Freni-Sterrantino et al. in 2018 and presented in +[A note on intrinsic Conditional Autoregressive models for disconnected graphs](https://arxiv.org/abs/1705.04854). + +The dataset we're using is that used in the analysis published in 2019 +[Bayesian Hierarchical Spatial Models: Implementing the Besag York Mollié Model in Stan](https://www.sciencedirect.com/science/article/pii/S1877584518301175). +It data consists of motor vehicle collisions in New York City, +as recorded by the NYC Department of Transportation, between the years 2005-2014, +restricted to collisions involving school age children 5-18 years of age as pedestrians. +Each crash was localized to the US Census tract in which it occurred, using boundaries from the 2010 United States Census, +using the [2010 Census block map for New York City](https://data.cityofnewyork.us/City-Government/2010-Census-Blocks/v2h8-6mxf). +New York City is comprised of several large and small islands, a chunk of the mainland (the Bronx), and the southern tip of Long Island Sound (Brooklyn and Queens). +To analyze the full New York City dataset requires the extended BYM2 model, which we do in the second part of this section. + +### Model 1: The BYM2 model, Riebler et al. 2016 + +We compare three implementations of the BYM2 model which differ only in their implementation of `phi`, +which is constrained to sum to zero. + +**BYM2 v1: `sum_to_zero_vector`** + +Parameter `phi` is declared as a `sum_to_zero_vector`. +See file [bym2_ozs.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_ozs.stan). + +**BYM2 v2: Hard sum-to-zero constraint** + +Parameter `phi_raw` is an vector of size N-1 and the N-length vector `phi` is computed in the `transformed parameters` block. +See file [bym2_hard.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_hard.stan). + +**BYM2 v3: soft sum-to-zero constraint** + +Parameter `phi` is a vector of size N and its sum is soft-centered on $0$. +See file [bym2_soft.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_soft.stan). + + +#### Data assembly + +The inputs to the BYM2 model are + +* The Poisson regression data + + + `int N` - number of regions + + `array[N] int y` - per-region count outcome + + `vector[N] E` - the population of each region (a.k.a. "exposure"), + + `int K` - the number of predictors + + `matrix[N, K] xs` - the design matrix + +* The spatial structure + + + `int N_edges` - the number of neighbor pairs + + `array[2, N_edges] int neighbors` - the graph structure + + `real tau` - the scaling factor, introduced in the BYM2 + +::: {.callout-note} +The scaling factor `tau` was introduced by Riebler et al so that the +variance of the spatial and ordinary random effects are both approximately equal to 1, +thus allowing for a straightforward estimate of the amount of spatial and non-spatial variance. +We have written a helper function called `get_scaling_factor`, in file +[utils_bym2.py](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/utils_bym2.py) +which takes as its argument the neighbor graph and computes the geometric mean of the +corresponding adjacency matrix. +::: + +To assemble the spatial data from the US Census data maps we use a set of utilities +described more fully in notebooks 2 and 5 of the [geomed_2024 Spatial Data Processing in Stan workshop](https://github.com/mitzimorris/geomed_2024). +The neighbors graph of New York City consists of several large connected components, +notably Brooklyn and Queens, the Bronx, Manhattan, and Staten Island, and a few other +smaller components and singletons. +The helper function `nyc_soft_by_comp_size` adds component info to the geodataframe +and sorts the data by component size. + +*Get NYC dataset, pre-process as needed for BYM2 model, display neighbors graph.* +```{python} +from utils_nyc_map import nyc_sort_by_comp_size +nyc_geodata = gpd.read_file(os.path.join('data', 'nyc_study.geojson')) +(nyc_nbs, nyc_gdf, nyc_comp_sizes) = nyc_sort_by_comp_size(nyc_geodata) + +from splot.libpysal import plot_spatial_weights +print("New York City neighbors graph") +plot_spatial_weights(nyc_nbs, nyc_gdf) +``` + +*The BYM2 model requires that the neighbors graph be fully connected - +use the largest subcomponent Brooklyn and Queens.* +```{python} +from libpysal.weights import Queen +brklyn_qns_gdf = nyc_gdf[nyc_gdf['comp_id']==0].reset_index(drop=True) +brklyn_qns_nbs = Queen.from_dataframe(brklyn_qns_gdf , geom_col='geometry') +from utils_bym2 import get_scaling_factor, nbs_to_adjlist +brklyn_qns_nbs_adj = nbs_to_adjlist(brklyn_qns_nbs) +tau = get_scaling_factor(brklyn_qns_nbs) +``` + +*Put all NYC predictors on the same scale - log transform columns as needed.* +```{python} +design_vars = np.array(['pct_pubtransit','med_hh_inc', 'traffic', 'frag_index']) +design_mat = brklyn_qns_gdf[design_vars].to_numpy() +design_mat[:, 1] = np.log(design_mat[:, 1]) +design_mat[:, 2] = np.log(design_mat[:, 2]) +``` + +*Assemble all inputs into dictionary `brklyn_qns_data`.* +```{python} +brklyn_qns_data = {"N":brklyn_qns_gdf.shape[0], + "y":brklyn_qns_gdf['count'].astype('int'), + "E":brklyn_qns_gdf['kid_pop'].astype('int'), + "K":4, + "xs":design_mat, + "N_edges": brklyn_qns_nbs_adj.shape[1], + "neighbors": brklyn_qns_nbs_adj, + "tau": tau +} +``` + +#### Model fitting + +The BYM2 model requires many warmup iterations in order to reach convergence for all parameters, +including hyperparameters `rho` and `sigma`. +We run all three models using the same seed, in order to make the initial parameters as similar +as possible. + + +*Compile and fit all three models to the Brooklyn-Queens data.* +```{python} +#| output: false +bym2_ozs_mod = CmdStanModel(stan_file=os.path.join('stan', 'bym2_ozs.stan')) +brklyn_qns_ozs_fit = bym2_ozs_mod.sample(data=brklyn_qns_data, iter_warmup=7000) +a_seed = brklyn_qns_ozs_fit.metadata.cmdstan_config['seed'] + +bym2_hard_mod = CmdStanModel(stan_file=os.path.join('stan', 'bym2_hard.stan')) +brklyn_qns_hard_fit = bym2_hard_mod.sample(data=brklyn_qns_data, iter_warmup=7000, seed=a_seed) + +bym2_soft_mod = CmdStanModel(stan_file=os.path.join('stan', 'bym2_soft.stan')) +brklyn_qns_soft_fit = bym2_soft_mod.sample(data=brklyn_qns_data, iter_warmup=7000, seed=a_seed) +``` + +#### Check implementations + +If all three implementations are correct, we expect that they will produce the same +estimates for all parameters. As always, we check the R-hat values and effective sample sizes. +To do this, we tabulate the summary statistics from each of the three models +for the hierarchical parameters `beta`, `sigma`, and `rho`. + +*Tabulate summary statistics from all fits.* +```{python} +brklyn_qns_ozs_summary = brklyn_qns_ozs_fit.summary() +brklyn_qns_ozs_summary.index = brklyn_qns_ozs_summary.index.astype(str) + " - ozs" + +brklyn_qns_hard_summary = brklyn_qns_hard_fit.summary() +brklyn_qns_hard_summary.index = brklyn_qns_hard_summary.index.astype(str) + " - hard" + +brklyn_qns_soft_summary = brklyn_qns_soft_fit.summary() +brklyn_qns_soft_summary.index = brklyn_qns_soft_summary.index.astype(str) + " - soft" + +brklyn_qns_fits_summary = pd.concat([brklyn_qns_ozs_summary, brklyn_qns_hard_summary, brklyn_qns_soft_summary]) +intercept_summary = summarize_predictor(brklyn_qns_fits_summary, 'beta_intercept') +betas_summary = summarize_predictor(brklyn_qns_fits_summary, 'betas') +sigma_summary = summarize_predictor(brklyn_qns_fits_summary, 'sigma') +rho_summary = summarize_predictor(brklyn_qns_fits_summary, 'rho') + +brklyn_qns_summary = pd.concat([intercept_summary, betas_summary, sigma_summary, rho_summary]) +display(HTML(style_dataframe(brklyn_qns_summary, 3).to_html())) +``` + +The `sum_to_zero` vector is the most efficient. +In contrast to the outcomes for the binomial model above, +the soft sum-to-zero constraint is more efficient than the hard sum-to-zero constraint. + +### Model 2: The BYM2_multicomp model, Freni-Sterrantino et al, 2018 + +In order to apply the BYM2 model to the full NYC dataset, it is necessary to +extend the BYM2 model to account for disconnected components and singleton nodes. +(The alternative is to add edges to the neighbor graphs so that the neighbors graph +is a single connected component. This is problematic, see Notebook 6 +[The BYM2_multicomp model in Stan](https://github.com/mitzimorris/geomed_2024/blob/main/python-notebooks/h6_bym2_multicomp.ipynb) for details.) + +In [A note on intrinsic Conditional Autoregressive models for disconnected graphs](https://arxiv.org/abs/1705.04854) Freni-Sterrantino et al. in 2018, the authors +show how to extend the BYM2 model to handle disconnected components and singleton nodes. +They provide the following recommendations: + +* Non-singleton nodes are given the BYM2 prior +* Singleton nodes (islands) are given a standard Normal prior +* Compute per-connected component scaling factor +* **Impose a sum-to-zero constraint on each connected component** + +We have followed these recommendations and implemented this model in Stan. +The full model is in file +[bym2_multicomp.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_multicomp.stan) + +For this case study, we compare 2 implementations of the BYM2_multicomp model: +one which uses the `sum_to_zero_vector` and one which implements the soft sum-to-zero constraint. + +It is necessary to constrain the the elements of the spatial effects vector `phi` on a component-by-component basis. +Stan's [slicing with range indexes](https://mc-stan.org/docs/stan-users-guide/multi-indexing.html#slicing-with-range-indexes), +provides a way to efficiently access each component. +The helper function `nyc_sort_by_comp_size` both sorts the study data by component and adds the component index to the geodataframe. + +In the BYM2 model for a fully connected graph the sum-to-zero constraint on `phi` +is implemented directly by declaring `phi` to be a `sum_to_zero_vector`, which is a +[constrained parameter type](https://mc-stan.org/docs/reference-manual/transforms.html#variable-transforms.chapter). +The declaration: +```stan + sum_to_zero_vector[N] phi; // spatial effects +``` +creates a *constrained* variable of length $N$, with a corresponding unconstrained variable of length $N-1$. + +In order to constrain slices of the parameter vector `phi`, we do the following: + +* In the `parameters` block, we declare the *unconstrained* parameter `phi_raw` as an regular vector `vector` (instead of a `sum_to_zero_vector`). + + For a fully connected graph of size $N$, the size of the unconstrained sum-to-zero vector is $N-1$. +For a disconnected graph with $M$ non-singleton nodes, the size of `phi_raw` is $M$ minus the +number of connected components. + +```stan + vector[N_connected - N_components] phi_raw; // spatial effects +``` + +* In the `functions` block, we implement the unconstraining transform. + + +* In the `transformed parameters` block, we apply the constraining transform. + +```stan + vector[N_connected] phi = zero_sum_components(phi_raw, component_idxs, component_sizes); +``` + +The constraining transform is broken into two functions: + +* function `zero_sum_constrain`, the actual constraining transform, which corresponds directly +to the built-in `zero_sum_vector` transform. + +* function `zero_sum_constrain_components`, which handles the slicing, and calls `zero_sum_constrain` on each component. + +```stan + /** + * Constrain sum-to-zero vector + * + * @param y unconstrained zero-sum parameters + * @return vector z, the vector whose slices sum to zero + */ + vector zero_sum_constrain(vector y) { + int N = num_elements(y); + vector[N + 1] z = zeros_vector(N + 1); + real sum_w = 0; + for (ii in 1:N) { + int i = N - ii + 1; + real n = i; + real w = y[i] * inv_sqrt(n * (n + 1)); + sum_w += w; + z[i] += sum_w; + z[i + 1] -= w * n; + } + return z; + } +``` + +* `zero_sum_components`: slices vector `phi` by component, applies constraining transform to each. + +```stan + /** + * Component-wise constrain sum-to-zero vectors + * + * @param phi unconstrained vector of zero-sum slices + * @param idxs component start and end indices + * @param sizes component sizes + * @return vector phi_ozs, the vector whose slices sum to zero + */ + vector zero_sum_components(vector phi, + array[ , ] int idxs, + array[] int sizes) { + vector[sum(sizes)] phi_ozs; + int idx_phi = 1; + int idx_ozs = 1; + for (i in 1:size(sizes)) { + phi_ozs[idx_ozs : idx_ozs + sizes[i] - 1] = + zero_sum_constrain(segment(phi, idx_phi, sizes[i] - 1)); + idx_phi += sizes[i] - 1; + idx_ozs += sizes[i]; + } + return phi_ozs; + } +``` + +::: {.callout-note} +The constraining transform is a linear operation, leading to a constant Jacobian determinant +which is therefore not included. +As of Stan 2.36, transforms which include a Jacobian adjustment can do so with the +`jacobian +=` statement and must have names ending in `_jacobian`. +See section [functions implementing change-of-variable adjustments](https://mc-stan.org/docs/stan-users-guide/user-functions.html#functions-implementing-change-of-variable-adjustments) in the Stan User's Guide +chapter on user-defined functions for details. +::: + + +#### Data Assembly + + +*Get NYC dataset, pre-process as needed for BYM2_multicomp model* +```{python} +from utils_nyc_map import nyc_sort_by_comp_size +from utils_bym2 import nbs_to_adjlist, get_scaling_factors +nyc_geodata = gpd.read_file(os.path.join('data', 'nyc_study.geojson')) +(nyc_nbs, nyc_gdf, nyc_comp_sizes) = nyc_sort_by_comp_size(nyc_geodata) +nyc_nbs_adj = nbs_to_adjlist(nyc_nbs) +component_sizes = [x for x in nyc_comp_sizes if x > 1] +scaling_factors = get_scaling_factors(len(component_sizes), nyc_gdf) +``` + +*Put all NYC predictors on the same scale - log transform columns as needed.* +```{python} +design_vars = np.array(['pct_pubtransit','med_hh_inc', 'traffic', 'frag_index']) +design_mat = nyc_gdf[design_vars].to_numpy() +design_mat[:, 1] = np.log(design_mat[:, 1]) +design_mat[:, 2] = np.log(design_mat[:, 2]) +``` + +*Assemble all inputs into dictionary `bym2_multicomp_data`.* +```{python} +bym2_multicomp_data = { + "N":nyc_gdf.shape[0], + "y":nyc_gdf['count'].astype('int'), + "E":nyc_gdf['kid_pop'].astype('int'), + "K":4, + "xs":design_mat, + "N_edges": nyc_nbs_adj.shape[1], + "neighbors": nyc_nbs_adj, + "N_components": len(component_sizes), + "component_sizes": component_sizes, + "scaling_factors": scaling_factors +} +``` + +#### Model Fitting + +The BYM2_multicomp model requires many warmup iterations in order to reach convergence for all parameters, +including hyperparameters `rho` and `sigma`, and a larger set of draws from the posterior in order +to obtain satisfactory R-hat values. + +*Compile and fit the BYM2_multicomp model using the `sum_to_zero_vector` constraining transform.* +```{python} +#| output: false +bym2_multicomp_ozs_file = os.path.join('stan', 'bym2_multicomp.stan') +bym2_multicomp_ozs_mod = CmdStanModel(stan_file=bym2_multicomp_ozs_file) +bym2_multicomp_ozs_fit = bym2_multicomp_ozs_mod.sample(data=bym2_multicomp_data, iter_warmup=9000) +``` +*Tabulate summary statistics.* +```{python} +bym2_multicomp_ozs_summary = bym2_multicomp_ozs_fit.summary() +bym2_multicomp_ozs_summary[['Mean', 'StdDev', 'ESS_bulk/s', 'R_hat']].round(2).loc[ + ['beta_intercept', 'betas[1]', 'betas[2]', 'betas[3]', 'betas[4]', 'sigma', 'rho']] +``` + +#### Model Comparison + +We can compare the `sum_to_zero_vector` implementation to the corresponding soft sum-to-zero constraint. +The full model is in file [bym2_multicomp_soft.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_multicomp_soft.stan) + +In model [bym2_soft.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_soft.stan) +the soft sum-to-zero constraint is combined directly with the ICAR prior: + +```stan + target += (-0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]) + + normal_lupdf(sum(phi) | 0, 0.001 * rows(phi))); +``` + +For the BYM2_multicomp model, this operation is carried out in two steps. +First the ICAR prior is applied to `phi`, next we iterate through the +components, applying the sum-to-zero constraint to each in turn. + +```stan + target += -0.5 * dot_self(phi[neighbors[1]] - phi[neighbors[2]]); // ICAR + for (n in 1:N_components) { // component-wise sum-to-zero constraint + sum(phi[node_idxs[n, 1] : node_idxs[n, 2]]) ~ normal(0, + 0.001 * component_sizes[n]); +``` + +The data inputs are the same. +To ensure (roughly) the same initialization, we reuse the seed from `bym2_multicomp_ozs_fit`. +This model fits *very* slowly and requires increasing the `max_treedepth`; at the default setting, +all iterations hit this limit. + +*Compile and fit the BYM2_multicomp model using the soft sum-to-zero constraint.* +```{python} +#| output: false +a_seed = bym2_multicomp_ozs_fit.metadata.cmdstan_config['seed'] +bym2_multicomp_soft_file = os.path.join('stan', 'bym2_multicomp_soft.stan') +bym2_multicomp_soft_mod = CmdStanModel(stan_file=bym2_multicomp_soft_file) +bym2_multicomp_soft_fit = bym2_multicomp_soft_mod.sample(data=bym2_multicomp_data, max_treedepth=13, iter_warmup=9000, seed=a_seed) +``` + +The following table compares the results for the hierarchical parameters `beta`, `rho`, and `sigma`. + +*Tabulate summary statistics from both fits.* +```{python} +bym2_multicomp_ozs_summary.index = bym2_multicomp_ozs_summary.index.astype(str) + " a) ozs" +bym2_multicomp_soft_summary = bym2_multicomp_soft_fit.summary() +bym2_multicomp_soft_summary.index = bym2_multicomp_soft_summary.index.astype(str) + " b) soft" +bym2_multicomp_summary = pd.concat([bym2_multicomp_ozs_summary, bym2_multicomp_soft_summary]) + +intercept_summary = summarize_predictor(bym2_multicomp_summary, 'beta_intercept') +betas_summary = summarize_predictor(bym2_multicomp_summary, 'betas') +sigma_summary = summarize_predictor(bym2_multicomp_summary, 'sigma') +rho_summary = summarize_predictor(bym2_multicomp_summary, 'rho') +nyc_summary = pd.concat([intercept_summary, betas_summary, sigma_summary, rho_summary]) + +display(HTML(style_dataframe(nyc_summary, 2).to_html())) +``` + +### Discussion + +The BYM2 model has a relatively complex multilevel structure, +and the BYM2_multicomp model is even more complex. +Before Stan 2.36, the recommended implementation was to use the +soft sum-to-zero constraint for the BYM2 model. + +For the BYM2_multicomp model, +model [bym2_multicomp.stan](https://github.com/stan-dev/example-models/tree/master/jupyter/sum-to-zero/stan/bym2_multicomp.stan) +shows how to implement the `sum_to_zero_vector` constraining transform as a Stan function. +Both implementation get exactly the same estimates, confirming that both models are correctly implemented. +Fitting the extended BYM2_multicomp model with a soft sum-to-zero is painfully slow. +Here we show that the `sum_to_zero_vector` greatly improves the run time; +which is critical because to fit this particular dataset requires a very large +number of warmup iterations. + + +## Conclusion: the `sum_to_zero_vector` just works! + +The more complex the model, the greater the need for the `sum_to_zero_vector`. +When considering the effective sample size, it is important to remember that +what is most important is effective samples **per second**. +In these experiments, the `sum_to_zero_vector` models consistently have the highest ESS/sec. + +Key takeaways: + +* In the first part of this case study, we show how to use the `sum_to_zero_vector` for categorical predictors in a hierarchical model, +and how to adjust the marginal variances for a normal prior. + +* In the second part of this case study we see how to implement the constraining transform +used for the `sum_to_zero_vector` as a Stan function in order to handle the case where a +vector is the concatenation of several sum-to-zero vectors. + + +## References + +* Fraser, 1951: Normal Samples With Linear Constraints and Given Variances. Canadian Journal of Mathematics, 3, 363–366. [doi:10.4153/CJM-1951-041-9](https://doi.org/10.4153/CJM-1951-041-9). + +* Freni-Sterrantino et al.,2018: [A note on intrinsic conditional autoregressive models for disconnected graphs](https://arxiv.org/pdf/1705.04854.pdf) + +* Gelman and Carpenter, 2020: [Bayesian Analysis of Tests with Unknown Specificity and Sensitivity](https://doi.org/10.1111/rssc.12435) + +* Morris et al., 2019: [Bayesian Hierarchical Spatial Models: Implementing the Besag York Mollié Model in Stan](https://www.sciencedirect.com/science/article/abs/pii/S1877584518301175) + +* Pinkney, 2025: [Zero-sum-vector and normal distribution](https://discourse.mc-stan.org/t/zero-sum-vector-and-normal-distribution/38296). + +* Riebler et al., 2016: [An intuitive Bayesian spatial model for disease mapping that accounts for scaling](https://arxiv.org/abs/1601.01180) + +* Seybolt, 2024: [Add ZeroSumNormal distribution](https://github.com/pyro-ppl/numpyro/pull/1751#issuecomment-1980569811) + +* Stan Development Team: [Stan Documentation Suite](https://mc-stan.org/docs/) + +* Vehtari et al, 2021 [Rank-Normalization, Folding, and Localization: An Improved $\widehat{R}$ for Assessing Convergence of MCMC (with Discussion)](https://projecteuclid.org/journals/bayesian-analysis/volume-16/issue-2/Rank-Normalization-Folding-and-Localization--An-Improved-R%cb%86-for/10.1214/20-BA1221.full) diff --git a/jupyter/sum-to-zero/utils.py b/jupyter/sum-to-zero/utils.py new file mode 100644 index 00000000..f965920d --- /dev/null +++ b/jupyter/sum-to-zero/utils.py @@ -0,0 +1,60 @@ +import os +import os.path +import logging +import numpy as np +import pandas as pd + +from typing import Any, Dict +from cmdstanpy import CmdStanModel + +cmdstanpy_logger = logging.getLogger("cmdstanpy") +cmdstanpy_logger.setLevel(logging.ERROR) + +import warnings +warnings.filterwarnings('ignore') + + +# Runs data-generating model and returns a Dict containing simulated data and metadata. +def simulate_data( + N_eth: int, N_edu: int, N_age: int, + baseline: float, sens: float, spec: float, + N_obs_per_stratum: int, *, seed: int = 345678 + ) -> Dict[str, Any]: + N = 2 * N_eth * N_edu * N_age * N_obs_per_stratum + inputs_dict = { + 'N': N, 'N_eth': N_eth, 'N_edu': N_edu, 'N_age': N_age, + 'baseline': baseline, 'sens': sens, 'spec': spec + } + datagen_mod = CmdStanModel(stan_file=os.path.join('stan', 'gen_binomial_4_preds.stan')) + sim_data = datagen_mod.sample(data=inputs_dict, iter_warmup=1, iter_sampling=1, chains=1, + show_progress=False, show_console=False, refresh=10_000, + seed=45678) + gen_data = { + 'N':sim_data.pos_tests.shape[1], + 'N_age':N_age, + 'N_eth':N_eth, + 'N_edu':N_edu, + 'sens': sens, + 'spec': spec, + 'intercept_prior_mean': baseline, + 'intercept_prior_scale': 2.5, + 'pos_tests':sim_data.pos_tests[0].astype(int), + 'tests':sim_data.tests[0].astype(int), + 'sex':sim_data.sex[0].astype(int), + 'age':sim_data.age[0].astype(int), + 'eth':sim_data.eth[0].astype(int), + 'edu':sim_data.edu[0].astype(int), + 'beta_0': sim_data.beta_0[0], + 'pct_sex': sim_data.pct_sex[0], + 'beta_sex': sim_data.beta_sex[0], + 'pct_age': sim_data.pct_age[0], + 'beta_age':sim_data.beta_age[0], + 'pct_eth': sim_data.pct_eth[0], + 'beta_eth':sim_data.beta_eth[0], + 'pct_edu': sim_data.pct_edu[0], + 'beta_edu':sim_data.beta_edu[0], + 'seed':sim_data.metadata.cmdstan_config['seed'], + } + return gen_data + + diff --git a/jupyter/sum-to-zero/utils_bym2.py b/jupyter/sum-to-zero/utils_bym2.py new file mode 100644 index 00000000..852223f2 --- /dev/null +++ b/jupyter/sum-to-zero/utils_bym2.py @@ -0,0 +1,85 @@ +""" +BYM2 model helper functions for computing scaling factors. +""" + +import numpy as np +import pandas as pd +import geopandas as gpd +import scipy.sparse as sp +import scipy.sparse.linalg as splinalg +from libpysal.weights import Queen, W +from scipy import stats +from typing import Optional, Union, List + +def nbs_to_adjlist(nbs: W) -> np.ndarray: + nbs_adj = nbs.to_adjlist(remove_symmetric=True) + j1 = nbs_adj['focal'] + 1 + j2 = nbs_adj['neighbor'] + 1 + edge_pairs = np.vstack([j1, j2]) + return (edge_pairs) + +def compute_jittered_precision_matrix(nbs: W) -> sp.spmatrix: + """ + Compute the precision matrix Q = D - W with added numerical jitter. + - param nbs: Neighborhood structure (W object from libpysal) + - return: Sparse precision matrix with jitter for numerical stability + """ + # Get adjacency matrix in CSR format + W_sparse = sp.csr_matrix(nbs.sparse) + # Compute diagonal matrix D (degree matrix) + D = sp.diags(W_sparse.sum(axis=1).A1) + # Compute precision matrix Q = D - W + Q = D - W_sparse + # Add jitter for numerical stability (similar to R's max(diag(Q)) * sqrt(machine precision)) + jitter = np.max(Q.diagonal()) * np.sqrt(np.finfo(float).eps) + Q_pert = Q + sp.eye(nbs.n) * jitter + return Q_pert + +def q_inv_dense(Q: sp.spmatrix, A: Optional[np.ndarray] = None) -> np.ndarray: + """ + Compute the inverse (or pseudo-inverse) of a precision matrix Q. + - param Q: Sparse precision matrix + - param A: Optional constraint matrix (default: None) + - return: Dense matrix representing Q^(-1) or constrained inverse + """ + Q_dense = Q.todense() + try: + Q_inv = np.linalg.inv(Q_dense) + except np.linalg.LinAlgError: + Q_inv = np.linalg.pinv(Q_dense) # use pseudo-inverse for better stability + if A is None: + return Q_inv + # Apply constraint if provided (ensuring spatial structure is respected) + W = Q_inv @ A.T + Q_inv_const = Q_inv - W @ np.linalg.inv(A @ W) @ W.T + return Q_inv_const + + +def get_scaling_factor(nbs: W) -> np.float64: + """ + Compute the geometric mean of the variances from the spatial covariance matrix. + - param nbs: Neighborhood structure (W object from libpysal) + - return: Geometric mean of the variances + """ + Q = compute_jittered_precision_matrix(nbs) + + # Compute inverse of precision matrix + A = np.ones((1, Q.shape[0])) # Ensure same constraint matrix as in R + Q_inv = q_inv_dense(Q, A) + + # Compute geometric mean of positive variances (diagonal elements of Q_inv) + variances = np.diag(Q_inv) + valid_vars = variances[variances > 0] # Exclude any zero or negative values + scaling = np.exp(np.mean(np.log(valid_vars))) + return scaling + + +def get_scaling_factors(N_components: int, gdf: gpd.GeoDataFrame ) -> List[np.float64]: + scaling_factors = np.ones(N_components) + for i in range(N_components): + comp_gdf = gdf[gdf['comp_id'] == i].reset_index(drop=True) + comp_nbs = Queen.from_dataframe(comp_gdf, geom_col='geometry') + component_w = W(comp_nbs.neighbors, comp_nbs.weights) + scaling_factors[i] = get_scaling_factor(component_w) + + return scaling_factors diff --git a/jupyter/sum-to-zero/utils_html.py b/jupyter/sum-to-zero/utils_html.py new file mode 100644 index 00000000..49a1a58a --- /dev/null +++ b/jupyter/sum-to-zero/utils_html.py @@ -0,0 +1,77 @@ +# json to HTML +import json +import pandas as pd +from IPython.display import HTML +import numpy as np +import pandas as pd +from typing import List + + +def extract_numeric_index(idx: pd.Index) -> pd.Series: + return idx.str.extract(r'[a-z_]*\[(\d+)\]', expand=False).astype(int) + +# add dividers every nth row +from pandas.io.formats.style import Styler +def style_dataframe(df: pd.DataFrame, modulus: int) -> Styler: + def highlight_every_nth_row(row: pd.Series, row_index: int, modulus: int) -> list[str]: + if (row_index + 1) % modulus == 0: # add border + return ['border-bottom: 3px double black'] * len(row) + return [''] * len(row) + return (df.style + .apply(lambda row: highlight_every_nth_row(row, df.index.get_loc(row.name), modulus), axis=1) + .format(precision=2) + ) + + +# Filter and sort predictors +def summarize_predictor(df: pd.DataFrame, name: str) -> pd.DataFrame: + pred_summary = df.filter(regex=name, axis=0).sort_index() + if "[" in name: + pred_summary = pred_summary.sort_index(key=extract_numeric_index) + + return pred_summary[['Mean', 'StdDev', 'ESS_bulk/s', 'R_hat']] + + +# side-by-side tables +from IPython.core.display import display, HTML +def display_side_by_side( + html_left: str, + html_right: str, + labels: str = "", + title_left: str = "Small Dataset", + title_right: str = "Large Dataset" +) -> None: + """ + Displays two HTML tables side by side in a Jupyter Notebook. + """ + html_code = f""" +
+
+   + {labels} +
+
+ {title_left} + {html_left} +
+
+ {title_right} + {html_right} +
+
+ """ + display(HTML(html_code)) + +def expand_true(datagen_values: List[float], header: str) -> str: + # Create HTML for datagen values column + datagen_values_html = f"" + + # Add each datagen value three times (one for each model) + for val in datagen_values: + # Show value for first row of each group + datagen_values_html += f"" + # Empty cells for the other two rows + datagen_values_html += "" + datagen_values_html += "" + datagen_values_html += "
{header}
{val:.2f}
 
 
" + return datagen_values_html diff --git a/jupyter/sum-to-zero/utils_nyc_map.py b/jupyter/sum-to-zero/utils_nyc_map.py new file mode 100644 index 00000000..a50cac03 --- /dev/null +++ b/jupyter/sum-to-zero/utils_nyc_map.py @@ -0,0 +1,75 @@ +import numpy as np +import pandas as pd +import geopandas as gpd +from libpysal.weights import Queen, W +from typing import (Dict, List) + +def disconnect_nbs( + nbs: W, target_indices: List[int], remove_indices: List[int] + ) -> None: + """ + Modify a neighbors graph by removing indices from the list of neighbors + and resetting neighbors, weights lists accordingly. + - param nbs: neighbor graph + - param target_indices: list of node ids to process + - param remove_indices: list of non-neighbor node ids + """ + remove_set = set(remove_indices) # Convert to set for O(1) lookups + for node in target_indices: + # Filter neighbors and weights in a single pass + clean_neighbors = [] + clean_weights = [] + for neighbor, weight in zip(nbs.neighbors[node], nbs.weights[node]): + if neighbor not in remove_set: + clean_neighbors.append(neighbor) + clean_weights.append(weight) + nbs.neighbors[node] = clean_neighbors + nbs.weights[node] = clean_weights + +def nyc_cleanup(nbs: W, gdf: gpd.GeoDataFrame) -> W: + """ + Modify neighbor graph of NYC to remove neighbor pairs between + Manhattan and other boroughs (Brooklyn, Queens). + - param nbs: neighbor graph + - param gdf : geopandas.GeoDataFrame of NYC Census Tract data + """ + # Get indices for each borough + manhattan_indices = gdf[gdf['BoroName'] == 'Manhattan'].index + brooklyn_indices = gdf[gdf['BoroName'] == 'Brooklyn'].index + queens_indices = gdf[gdf['BoroName'] == 'Queens'].index + brooklyn_and_queens = list(brooklyn_indices) + list(queens_indices) + # Disconnect Manhattan from Brooklyn/Queens and vice versa + disconnect_nbs(nbs, manhattan_indices, brooklyn_and_queens) + disconnect_nbs(nbs, brooklyn_indices, manhattan_indices) + disconnect_nbs(nbs, queens_indices, manhattan_indices) + + return W(nbs.neighbors, nbs.weights) + + +def nyc_sort_by_comp_size(nyc_gdf: gpd.GeoDataFrame + ) -> tuple[W, gpd.GeoDataFrame, List[int]]: + """ + Process NYC geodataframe - sort by component size descending + - param nyc_gdf : geopandas.GeoDataFrame + - return: tuple containing: neighbors, sorted gdf, + nodes_per_component, edges_per_component + """ + # Compute initial neighborhood graph + nyc_nbs = Queen.from_dataframe(nyc_gdf, geom_col='geometry') + # Clean borough connections and get components + nyc_nbs_tmp = nyc_cleanup(nyc_nbs, nyc_gdf) + # Add component info to dataframe + nyc_gdf['comp_id'] = nyc_nbs_tmp.component_labels + sizes = nyc_gdf['comp_id'].value_counts() + nyc_gdf['comp_size'] = nyc_gdf['comp_id'].map(sizes) + # Sort by component size and reset index + nyc_gdf_sorted = (nyc_gdf.sort_values(by='comp_size', ascending=False) + .reset_index(drop=True) + .set_geometry('geometry')) + # Recompute neighborhood graph, update gdf, get component sizes + nyc_nbs_sorted = Queen.from_dataframe(nyc_gdf_sorted, geom_col='geometry') + nyc_nbs_clean = nyc_cleanup(nyc_nbs_sorted, nyc_gdf_sorted) + nyc_gdf_sorted['comp_id'] = nyc_nbs_clean.component_labels + component_sizes = list(sizes.sort_values(ascending=False)) + + return nyc_nbs_clean, nyc_gdf_sorted, component_sizes