From 01d490ad696ec81629f74ef1bcf4d390b379d34f Mon Sep 17 00:00:00 2001 From: nicholasjclark Date: Tue, 14 Jun 2022 11:01:47 +1000 Subject: [PATCH] Remove strict jags dependencies --- .DS_Store | Bin 0 -> 18436 bytes DESCRIPTION | 4 +- NEON_manuscript/.DS_Store | Bin 0 -> 14340 bytes NEON_manuscript/Case studies/.DS_Store | Bin 0 -> 10244 bytes .../mvgam_case_study2_files/.DS_Store | Bin 0 -> 8196 bytes .../Case studies/mvgam_case_study3.html | 4776 ++++++++--------- NEON_manuscript/Case studies/rodents_data.csv | 1002 ++-- .../Case studies/rsconnect/.DS_Store | Bin 0 -> 8196 bytes .../rsconnect/documents/.DS_Store | Bin 0 -> 10244 bytes .../documents/mvgam_case_study1.Rmd/.DS_Store | Bin 0 -> 8196 bytes .../mvgam_case_study1.Rmd/rpubs.com/.DS_Store | Bin 0 -> 8196 bytes .../documents/mvgam_case_study2.Rmd/.DS_Store | Bin 0 -> 8196 bytes .../mvgam_case_study2.Rmd/rpubs.com/.DS_Store | Bin 0 -> 8196 bytes .../rpubs.com/rpubs/Document.dcf | 20 +- .../rpubs.com/rpubs/Publish Document.dcf | 22 +- NEON_manuscript/Figures/.DS_Store | Bin 0 -> 10244 bytes NEON_manuscript/Manuscript/.DS_Store | Bin 0 -> 10244 bytes 17 files changed, 2913 insertions(+), 2911 deletions(-) create mode 100644 .DS_Store create mode 100644 NEON_manuscript/.DS_Store create mode 100644 NEON_manuscript/Case studies/.DS_Store create mode 100644 NEON_manuscript/Case studies/mvgam_case_study2_files/.DS_Store create mode 100644 NEON_manuscript/Case studies/rsconnect/.DS_Store create mode 100644 NEON_manuscript/Case studies/rsconnect/documents/.DS_Store create mode 100644 NEON_manuscript/Case studies/rsconnect/documents/mvgam_case_study1.Rmd/.DS_Store create mode 100644 NEON_manuscript/Case studies/rsconnect/documents/mvgam_case_study1.Rmd/rpubs.com/.DS_Store create mode 100644 NEON_manuscript/Case studies/rsconnect/documents/mvgam_case_study2.Rmd/.DS_Store create mode 100644 NEON_manuscript/Case studies/rsconnect/documents/mvgam_case_study2.Rmd/rpubs.com/.DS_Store create mode 100644 NEON_manuscript/Figures/.DS_Store create mode 100644 NEON_manuscript/Manuscript/.DS_Store diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..fa9811a29ce4f32e1d1cbb319dd96e7b3b9a1f94 GIT binary patch literal 18436 zcmeHO33Oal8UFutd++tNdFeVc%_Ng9Y14)_X_h8!L#J8VrVA}iS6WCXnV}t+%!JG& z-6$53qaurn>~i2Bs8AFIiXb9F0Rge7)WaqmQGuhJBc3CI$fDo9_sw+XCFy3hIh=Xt zyzMS;x%2)1_kaJpLquHP9V{pE6Oo2h?5*KMF04M2s0?dz5%ps2wG2WDV=clu82yRb zDISUUG_*xxZM|A&3$i1S9f9l!WJe%70@)E52qPf48;b_c&o0>!$c{jE1TscI_y-?W zvG?2iy=pIA2R1o474-=J)q-Q_09od<_kMf7SM6n2^pRb$S6$hb7_7RZ+$Wl2@3;4R z)n0WcR^5s1JF~4&Sl%7|e6l&QU+uHo>p!Wz2MNuxDK~1z0+g9qJ07a>r;?VM4 z6sAr}^t5zEBeCdKv=qk2w7NMGYqvE%%#Ht!((C^%7aYkJZQJ9Wc8h1)IwRqDqO9yi z&KWd#$k1V$W@yF;8+M)_wMqlvQl47?VeKK?((wg z%2MB)vdTSsbZz{U^19|N9XogJzHrY)H^_%4wml4nC3Cq^z1rafN6v%ES9>M-YPvDp z9N}5gykx1lyww`jgQs5EnFzI#Ya_3g1pcSkhdYUQ;D zBJJIgKq9m-5@`wV3ZX1F-rg09L{`P(VR4y_8xo-%2~pRpYQtIaSXZJ!UaZx!y5phM zAzUlm6Kcd2;~T=brRZp}zHj9C{Gut-XI0jo-f$?_${RI$jA@>QL5n8B=Y~UFYqy7k ziESGL@%B(O7>;gT-xH39+ajSg(LhHiFDFmeWiOZ8L}U&3GD%lR6ENB>|TNltlX>fP6x}*DPEMD28_QXBpNo0Xj{# zE}B?aXbH5b)8WO1Cd@&Xb<>Mcmt)t=oQ0Y}K)$N0s5A?$App7RS5s@6)?iqHs;xU6 zwL^h-?a&0b$7ME{O}a2ei%t??pQRV*4*=^&Ogt1o9Sx8c@l-yI=kNkv$c@~@i+ClU z$t}Ew*8-wjIKXXuzJzIvcknL0koWLK{6+pUe+5vz0#LmkP`!!2$#?K~_)fly_woI_ zpC9Ch_+kDDKgkFARsNj=uFD;8eN(~J!9$sivS-IX0wRPKs|cY7Q;=er`TLxQ`*sST zx24O>6)8eFWt7&ZTS|H7-YjR1F~=pT%xTDJ?5F4PHM%l>!DoH5oVUPZW_T27;-dY; 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- - - - - - - -

Here we will use the mvgam package, which fits dynamic -GAMs using MCMC sampling via either the JAGS software -(installation links are found here) or -via the Stan software (installation links are found here), -to estimate paramaters of a Bayesian distributed lag model. These models -are used to describe simultaneously non-linear and delayed functional -relationships between a covariate and a response, and are sometimes -referred to as exposure-lag-response models. If we assume \(\tilde{\boldsymbol{y}}_{t}\) is the -conditional expectation of a discrete response variable \(\boldsymbol{y}\) at time \(\boldsymbol{t}\), the linear predictor for -a dynamic distributed lag GAM with one lagged covariate is written -as:

-

\[log(\tilde{\boldsymbol{y}}_{t})={\boldsymbol{B}}_0+\sum\limits_{k=1}^K{f}(\boldsymbol{b}_{k,t}\boldsymbol{x}_{k,t})+\boldsymbol{z}_{t}\,,\] -where \(\boldsymbol{B}_{0}\) is the -unknown intercept, the \(\boldsymbol{b}\)’s are unknown spline -coefficients estimating how the functional effect of covariate (\(\boldsymbol{x}\)) on \(log(\tilde{\boldsymbol{y}}_{t})\) changes -over increasing lags (up to a maximum lag of (\(\boldsymbol{K}\))) and \(\boldsymbol{z}\) is a dynamic latent trend -component.

-

To demonstrate how these models are estimated in mvgam, -first we load the Portal rodents capture data, which are available from -the portalr package

-
# devtools::install_github('nicholasjclark/mvgam')
-library(mvgam)
-library(dplyr)
-portal_dat <- read.csv("https://raw.githubusercontent.com/nicholasjclark/mvgam/master/NEON_manuscript/Case studies/rodents_data.csv", 
-    as.is = T)
-

We’ll keep data from the year 2004 onwards to make the model quicker -to estimate for this simple example

-
portal_dat_all <- portal_dat %>% dplyr::filter(year >= 
-    2004) %>% dplyr::group_by(year, month) %>% 
-    dplyr::slice_head(n = 1)
-

Below is an exact reproduction of Simon Wood’s lag matrix function -(which he uses in his distributed lag example from his book Generalized -Additive Models - An Introduction with R 2nd edition). Here we -supply a vector and specify the maximum lag that we want, and it will -return a matrix of dimension length(x) * lag. Note that -NAs are used for the missing lag values at the beginning of -the matrix. In essence, the matrix objects represent exposure histories, -where each row represents the lagged values of the predictor that -correspond to each observation in y

-
lagard <- function(x, n.lag = 6) {
-    n <- length(x)
-    X <- matrix(NA, n, n.lag)
-    for (i in 1:n.lag) X[i:n, i] <- x[i:n - 
-        i + 1]
-    X
-}
-

Organise all data needed for modelling into a list. We will focus -only on the species Chaetodipus penicillatus (labelled as -PP), which shows reasonable seasonality in its captures -over time

-
data_all <- list(lag = matrix(0:5, nrow(portal_dat_all), 
-    6, byrow = TRUE), y = portal_dat_all$PP, 
-    season = portal_dat_all$month, year = portal_dat_all$year, 
-    series = rep(as.factor("series1"), NROW(portal_dat_all)), 
-    time = 1:NROW(portal_dat_all))
-data_all$precip <- lagard(portal_dat_all$precipitation)
-data_all$mintemp <- lagard(portal_dat_all$mintemp)
-

The exposure history matrix elements of the data list look as -follows:

-
head(data_all$lag, 5)
-
##      [,1] [,2] [,3] [,4] [,5] [,6]
-## [1,]    0    1    2    3    4    5
-## [2,]    0    1    2    3    4    5
-## [3,]    0    1    2    3    4    5
-## [4,]    0    1    2    3    4    5
-## [5,]    0    1    2    3    4    5
-
head(data_all$precip, 5)
-
##      [,1] [,2] [,3] [,4] [,5] [,6]
-## [1,] 37.8   NA   NA   NA   NA   NA
-## [2,]  8.7 37.8   NA   NA   NA   NA
-## [3,] 43.5  8.7 37.8   NA   NA   NA
-## [4,] 23.9 43.5  8.7 37.8   NA   NA
-## [5,]  0.9 23.9 43.5  8.7 37.8   NA
-
head(data_all$mintemp, 5)
-
##        [,1]   [,2]   [,3]   [,4]  [,5] [,6]
-## [1,] -9.710     NA     NA     NA    NA   NA
-## [2,] -5.924 -9.710     NA     NA    NA   NA
-## [3,] -0.220 -5.924 -9.710     NA    NA   NA
-## [4,]  1.931 -0.220 -5.924 -9.710    NA   NA
-## [5,]  6.568  1.931 -0.220 -5.924 -9.71   NA
-

All other elements of the data list are in the usual vector -format

-
head(data_all$y, 5)
-
## [1]  0  1  2 NA 10
-
head(data_all$series, 5)
-
## [1] series1 series1 series1 series1 series1
-## Levels: series1
-
head(data_all$year, 5)
-
## [1] 2004 2004 2004 2004 2004
-
head(data_all$time, 5)
-
## [1] 1 2 3 4 5
-

View the raw series. There is a clear seasonal pattern to the data, -and there are missing values scattered throughout

-
plot(ts(data_all$y, frequency = 12), ylab = "Captures for PP", 
-    xlab = "", lwd = 2.5, col = "#8F2727")
-

-

Create training and testing sets; start at observation 7 so that the -NA values at the beginning of the covariate lag matrices -are not included. Currently there is no option for on-the-fly imputation -of missing covariate values in mvgam models, though this -can easily be done in JAGS by specifying prior -distributions over these missing entries

-
data_train <- list(lag = data_all$lag[7:174, 
-    ], y = data_all$y[7:174], series = data_all$series[7:174], 
-    season = data_all$season[7:174], year = data_all$year[7:174], 
-    time = 7:174, precip = data_all$precip[7:174, 
-        ], mintemp = data_all$mintemp[7:174, 
-        ])
-data_test <- list(lag = data_all$lag[175:length(data_all$y), 
-    ], y = data_all$y[175:length(data_all$y)], 
-    series = data_all$series[175:length(data_all$y)], 
-    season = data_all$season[175:length(data_all$y)], 
-    year = data_all$year[175:length(data_all$y)], 
-    time = 175:length(data_all$y), precip = data_all$precip[175:length(data_all$y), 
-        ], mintemp = data_all$mintemp[175:length(data_all$y), 
-        ])
-

Now we can fit a Bayesian GAM with distributed lag terms for -precipitation and minimum temperature. The distributed lags are set up -as tensor product smooth functions (see help(te) for an -explanation of tensor product smooth constructions in the -mgcv package) between lag and each covariate. -We will start simply by assuming our data follow a Poisson -observation process

-
mod1 <- mvgam(formula = y ~ te(mintemp, lag, 
-    k = c(8, 4)) + te(precip, lag, k = c(8, 
-    4)), data_train = data_train, data_test = data_test, 
-    family = "poisson", chains = 4, burnin = 10000, 
-    trend_model = "None")
-

Posterior predictive rootograms are a useful way to explore whether a -discrete model is able to capture relevant dispersion in the observed -data. This plot compares the frequencies of observed vs predicted values -for each bin, which can help to identify aspects of poor model fit. For -example, if the gray bars (representing observed frequencies) tend to -stretch below zero, this suggests the model’s simulations predict the -values in that particular bin less frequently than they are observed in -the data. A well-fitting model that can generate realistic simulated -data will provide a rootogram in which the lower boundaries of the grey -bars are generally near zero

-
ppc(mod1, type = "rootogram")
-

-

The Poisson model is not doing a great job of capturing -dispersion, underpredicting the zeros in the data and overpredicting -some of the medium-range values (counts of ~5-30). The -residual Q-Q plot confirms that the Poisson is -not an appropriate distribution for these data

-
plot(mod1, type = "residuals")
-

-

Given the overdispersion present in the data, we will now assume a Geometric-Poisson -observation model, which can be more -flexible than the Negative binomial for modelling -overdispersed count data. In mvgam the -Geometric-Poisson is estimated as a -Tweedie-Poisson model with the power parameter -p fixed at 1.5

-
mod2 <- mvgam(formula = y ~ te(mintemp, lag, 
-    k = c(8, 4)) + te(precip, lag, k = c(8, 
-    4)), data_train = data_train, data_test = data_test, 
-    family = "tw", chains = 4, burnin = 10000, 
-    trend_model = "None")
-

The rootogram for this model looks better, though of course there is -still some overprediction of medium-range values

-
ppc(mod2, type = "rootogram")
-

-

However, the residual plot looks much better for this model

-
plot(mod2, type = "residuals")
-

-

The summary of the model provides useful information on convergence -for unobserved parameters. Notice how strongly positive the -overdispersion parameter is estimated to be, providing further evidence -that this overdispersion is important to capture for these data

-
summary(mod2)
-
## GAM formula:
-
## y ~ te(mintemp, lag, k = c(8, 4)) + te(precip, lag, k = c(8, 
-##     4))
-
##
-
## Family:
-
## Tweedie
-
##
-
## Link function:
-
## log
-
##
-
## Trend model:
-
## None
-
##
-
## N series:
-
## 1
-
##
-
## N observations:
-
## 168
-
##
-
## Status:
-
## Fitted using runjags::run.jags()
-
##
-
## Dispersion parameter estimates:
-
##        2.5%      50%    97.5% Rhat n.eff
-## twdis 1.388 1.932959 2.895216 1.14   331
-
##
-
## GAM smooth term estimated degrees of freedom:
-
##                   edf df
-## te(mintemp,lag) 95.67 31
-## te(precip,lag)  82.94 28
-
##
-
## GAM coefficient (beta) estimates:
-
##                             2.5%          50%       97.5% Rhat n.eff
-## (Intercept)         2.212693e+00  2.403258707  2.56885901 1.09   290
-## te(mintemp,lag).1  -1.054775e+00 -0.097381019  0.35528542 1.75    26
-## te(mintemp,lag).2  -1.106697e+00 -0.368732775  0.12915239 1.33    44
-## te(mintemp,lag).3  -1.474308e+00 -0.734838360  0.09358272 1.38    52
-## te(mintemp,lag).4  -9.798236e-01 -0.605709195 -0.21506225 1.37    49
-## te(mintemp,lag).5  -4.040562e-01 -0.092727382  0.42243535 1.79    43
-## te(mintemp,lag).6   8.517891e-05  0.310276127  0.72603789 1.44    34
-## te(mintemp,lag).7   7.187907e-02  0.491456804  0.77836084 1.23    38
-## te(mintemp,lag).8  -8.162581e-01 -0.515957422 -0.20271474 1.15    43
-## te(mintemp,lag).9  -3.707472e-01 -0.175962485  0.04951066 1.40    65
-## te(mintemp,lag).10 -6.358472e-02  0.123077437  0.31621992 1.10    50
-## te(mintemp,lag).11  1.195955e-01  0.392272177  0.64054687 1.07    46
-## te(mintemp,lag).12 -5.981071e-01 -0.224082393  0.16808352 1.44    35
-## te(mintemp,lag).13 -3.862997e-01 -0.139209569  0.19495414 1.80    66
-## te(mintemp,lag).14 -2.441130e-01  0.025814566  0.33439909 1.46    45
-## te(mintemp,lag).15  3.241991e-02  0.320965039  0.62328350 1.08    41
-## te(mintemp,lag).16 -3.286109e-01 -0.010954410  0.33498460 1.35    41
-## te(mintemp,lag).17 -2.844375e-01 -0.041016059  0.26075539 1.89    64
-## te(mintemp,lag).18 -2.656616e-01 -0.002339434  0.33316216 1.55    56
-## te(mintemp,lag).19 -6.744452e-02  0.203056321  0.52179149 1.09    60
-## te(mintemp,lag).20 -2.222059e-01  0.071034888  0.55035077 1.27    43
-## te(mintemp,lag).21 -2.417919e-01  0.052624654  0.38676519 1.69    47
-## te(mintemp,lag).22 -3.463257e-01 -0.018821781  0.32024342 1.32    32
-## te(mintemp,lag).23 -2.567171e-01  0.023001627  0.41206856 1.09    46
-## te(mintemp,lag).24  1.457483e-02  0.354576859  0.85032897 1.42    48
-## te(mintemp,lag).25 -1.949911e-01  0.108520078  0.54652848 1.92    45
-## te(mintemp,lag).26 -3.749921e-01 -0.130021494  0.21053719 1.52    41
-## te(mintemp,lag).27 -6.237620e-01 -0.351657535 -0.00898590 1.05    34
-## te(mintemp,lag).28  9.617270e-02  0.492034669  0.92753487 1.12    43
-## te(mintemp,lag).29 -3.045994e-01  0.151414518  0.52004441 1.17    29
-## te(mintemp,lag).30 -5.743251e-01 -0.210852728  0.11881993 1.02    35
-## te(mintemp,lag).31 -9.933905e-01 -0.558453233 -0.07249095 1.16    47
-## te(precip,lag).1   -1.822929e-01  0.029293583  0.16142576 1.11    44
-## te(precip,lag).2   -2.422555e-01 -0.053000798  0.11107324 1.32    57
-## te(precip,lag).3   -4.278990e-01 -0.092901543  0.15044128 1.71    51
-## te(precip,lag).4    1.468352e-02  0.167812898  0.38051116 1.16    51
-## te(precip,lag).5   -1.523433e-01  0.007996773  0.08109699 1.17    70
-## te(precip,lag).6   -2.451465e-01 -0.080542477  0.05006037 1.26    59
-## te(precip,lag).7   -4.125140e-01 -0.106601339  0.08656329 1.72    59
-## te(precip,lag).8   -1.110711e-01  0.066748222  0.31861998 1.27    52
-## te(precip,lag).9   -1.741086e-01 -0.038851485  0.08810518 1.10    63
-## te(precip,lag).10  -3.021204e-01 -0.101017687  0.06064512 1.08    38
-## te(precip,lag).11  -4.583086e-01 -0.103610861  0.07695710 1.36    52
-## te(precip,lag).12  -3.216636e-01 -0.104570006  0.19237728 1.44    50
-## te(precip,lag).13  -2.430229e-01 -0.099218023  0.02424224 1.09    63
-## te(precip,lag).14  -2.806793e-01 -0.113844520  0.03463855 1.07    45
-## te(precip,lag).15  -4.500282e-01 -0.081944527  0.10618411 1.17    46
-## te(precip,lag).16  -4.201331e-01 -0.133284892  0.12198342 1.66    56
-## te(precip,lag).17  -3.327693e-01 -0.135333807  0.02824623 1.19    74
-## te(precip,lag).18  -3.679845e-01 -0.129255428  0.06712730 1.06    34
-## te(precip,lag).19  -4.973665e-01 -0.125078896  0.14759108 1.14    38
-## te(precip,lag).20  -1.085044e-01  0.158442743  0.36943597 1.03    45
-## te(precip,lag).21  -1.609977e-01 -0.004588906  0.10242298 1.11    93
-## te(precip,lag).22  -2.865369e-01 -0.130260412 -0.01385738 1.07    82
-## te(precip,lag).23  -3.639076e-01 -0.176572752  0.04987009 1.06    61
-## te(precip,lag).24  -1.790802e-01  0.021304676  0.22629910 1.15    72
-## te(precip,lag).25  -6.432391e-01 -0.094210307  0.32520239 1.56    43
-## te(precip,lag).26  -4.278256e-01 -0.077048068  0.21366462 1.35    41
-## te(precip,lag).27  -2.883587e-01  0.012994431  0.28647043 1.15    44
-## te(precip,lag).28  -2.094542e-01  0.094615123  0.41248432 1.31    60
-
##
-
## GAM smoothing parameter (rho) estimates:
-
##                         2.5%      50%    97.5% Rhat n.eff
-## te(mintemp,lag)   1.83893338 3.192341 4.309709 1.01    71
-## te(mintemp,lag)2  0.28404659 2.421797 4.056106 1.09    53
-## te(mintemp,lag)3 -0.68282687 1.729060 4.122901 1.50   108
-## te(precip,lag)    2.14411956 3.629856 4.698440 1.01   211
-## te(precip,lag)2   0.09545339 2.704613 4.271508 1.25    79
-## te(precip,lag)3   0.20771730 2.566491 4.186006 1.24    82
-
##
-

As this is a timeseries and the residual plot hints at some -autocorrelation remaining in the short-term lags, lets check if an -AR latent trend process improves forecasts compared to the -no-trend model. An important note here is the choice of prior for the -overdispersion parameter twdis. This parameter and the -latent trend variance can interact strongly, particularly when -overdispersion is in the data high. This is because at high values of -the dispersion parameter, there is less need for a latent trend to be -able to capture any outliers and so the latent trend precision can go up -toward infinity, approaching a space of very diffuse likelihood that -forces the Gibbs samplers to take on frustratingly small step sizes. -Likewise when there is not much need for overdispersion, the dispersion -parameter can approach zero and move around in an equally uninformative -parameter space. The latent trend operates on the log scale, so really -we should not expect autocorrelated jumps in trappings of more than -6-8 from timepoint to timepoint (any larger and the trend -will compete strongly with the overdispersion parameter, making it -difficult for us to model the inherent overdispersion process and -instead assuming it is all autocorrelation). A containment prior on the -latent trend sigma will help achieve this

-
mod3 <- mvgam(formula = y ~ te(mintemp, lag, 
-    k = c(8, 4)) + te(precip, lag, k = c(8, 
-    4)), data_train = data_train, data_test = data_test, 
-    family = "tw", sigma_prior = "dexp(2.5)T(0.15, 2)", 
-    chains = 4, burnin = 10000, trend_model = "AR3")
-summary(mod3)
-
## y ~ te(mintemp, lag, k = c(8, 4)) + te(precip, lag, k = c(8, 
-##     4))
-## Tweedie
-## log
-## AR3
-## 1 
-## 168 
-## Fitted using runjags::run.jags() 
-##            2.5%       50%    97.5% Rhat n.eff
-## twdis 0.1442308 0.3038398 0.533641 1.58    35
-##                   edf df
-## te(mintemp,lag) 55.01 31
-## te(precip,lag)  79.24 28
-##                           2.5%          50%        97.5% Rhat n.eff
-## (Intercept)         1.95451829  2.294369149  2.687638507 1.05    79
-## te(mintemp,lag).1  -0.34440489  0.001118830  0.259412270 1.59    34
-## te(mintemp,lag).2  -0.58230901 -0.114278574  0.245323772 2.32    28
-## te(mintemp,lag).3  -1.05666653 -0.380131705  0.112717290 1.81    26
-## te(mintemp,lag).4  -0.90716043 -0.666943517 -0.374398808 1.44    61
-## te(mintemp,lag).5  -0.40434523 -0.193728154  0.042342982 1.23    32
-## te(mintemp,lag).6   0.02152938  0.203985336  0.456034236 1.56    42
-## te(mintemp,lag).7   0.25813250  0.514930957  0.773757069 1.46    74
-## te(mintemp,lag).8  -0.67614540 -0.442897113 -0.202233028 1.33    57
-## te(mintemp,lag).9  -0.27762442 -0.138611196 -0.011473547 1.10    56
-## te(mintemp,lag).10  0.02284141  0.168421464  0.353924370 1.50    60
-## te(mintemp,lag).11  0.26878013  0.486133830  0.749333381 1.21    44
-## te(mintemp,lag).12 -0.34006723 -0.149228275  0.121707811 1.17    33
-## te(mintemp,lag).13 -0.25641787 -0.078535420  0.124577834 1.44    40
-## te(mintemp,lag).14 -0.07347474  0.104024205  0.364380385 1.73    54
-## te(mintemp,lag).15  0.20099240  0.417414813  0.650764551 1.17    46
-## te(mintemp,lag).16 -0.24087569 -0.042252034  0.217570497 1.21    34
-## te(mintemp,lag).17 -0.23693743 -0.039656865  0.140520521 1.45    37
-## te(mintemp,lag).18 -0.16064686  0.059071449  0.260616038 1.57    45
-## te(mintemp,lag).19  0.06594100  0.279170574  0.570433710 1.42    44
-## te(mintemp,lag).20 -0.24701161  0.050345586  0.335014718 1.26    29
-## te(mintemp,lag).21 -0.22732085 -0.016441948  0.219734982 1.41    39
-## te(mintemp,lag).22 -0.29591233 -0.039567014  0.188568090 1.64    36
-## te(mintemp,lag).23 -0.19552442  0.018925677  0.354815056 1.59    40
-## te(mintemp,lag).24  0.00712481  0.257539466  0.534982135 1.26    39
-## te(mintemp,lag).25 -0.14986366  0.028610310  0.240532737 1.27    35
-## te(mintemp,lag).26 -0.35516212 -0.203205407  0.132626064 1.92    43
-## te(mintemp,lag).27 -0.66173660 -0.361493765 -0.139140221 2.02    53
-## te(mintemp,lag).28 -0.10654805  0.315825287  0.718339605 1.35    32
-## te(mintemp,lag).29 -0.19502605  0.057821835  0.351960793 1.19    22
-## te(mintemp,lag).30 -0.49213602 -0.254132913 -0.004726919 1.73    45
-## te(mintemp,lag).31 -0.98269158 -0.601603108 -0.293992943 1.55    36
-## te(precip,lag).1   -0.18071579 -0.007770073  0.129141735 1.23    44
-## te(precip,lag).2   -0.26109947 -0.053181668  0.109517176 1.26    36
-## te(precip,lag).3   -0.29134027 -0.067594472  0.192333185 1.50    34
-## te(precip,lag).4   -0.03682622  0.091643893  0.283454745 1.40    40
-## te(precip,lag).5   -0.16933377 -0.017988505  0.090918611 1.48    65
-## te(precip,lag).6   -0.26882772 -0.057775209  0.074503584 1.66    48
-## te(precip,lag).7   -0.28597650 -0.067388939  0.134412111 1.67    35
-## te(precip,lag).8   -0.14848668  0.011187431  0.178197414 1.03    40
-## te(precip,lag).9   -0.20361931 -0.027252625  0.080875803 1.83    78
-## te(precip,lag).10  -0.25544688 -0.044531181  0.074521842 1.73    53
-## te(precip,lag).11  -0.31157290 -0.051729464  0.124553750 1.54    36
-## te(precip,lag).12  -0.26780541 -0.068596545  0.129182178 1.24    39
-## te(precip,lag).13  -0.15294675 -0.041633221  0.103724179 1.49    69
-## te(precip,lag).14  -0.17124532 -0.031463088  0.105549391 1.19    50
-## te(precip,lag).15  -0.26011511 -0.016574210  0.191909247 1.20    41
-## te(precip,lag).16  -0.29057183 -0.136137976  0.138202557 1.43    48
-## te(precip,lag).17  -0.26211414 -0.089851782  0.102359183 1.39    40
-## te(precip,lag).18  -0.27921814 -0.056206824  0.114079040 1.14    33
-## te(precip,lag).19  -0.32498905 -0.035923817  0.183391255 1.12    33
-## te(precip,lag).20  -0.06375377  0.092346100  0.228629904 1.22    58
-## te(precip,lag).21  -0.17045386 -0.026071316  0.117831174 1.13    47
-## te(precip,lag).22  -0.24797078 -0.086540727  0.062987289 1.27    57
-## te(precip,lag).23  -0.24537849 -0.059480138  0.128256777 1.23    55
-## te(precip,lag).24  -0.23766458 -0.031720898  0.119043396 1.09    54
-## te(precip,lag).25  -0.41009464  0.018395088  0.234195699 1.24    23
-## te(precip,lag).26  -0.26921000 -0.018316715  0.191046564 1.09    33
-## te(precip,lag).27  -0.22769731 -0.043288937  0.193162536 1.22    40
-## te(precip,lag).28  -0.39840734 -0.095952392  0.347629109 1.18    25
-##                         2.5%      50%    97.5% Rhat n.eff
-## te(mintemp,lag)   2.32902372 3.493322 4.464910 1.12   167
-## te(mintemp,lag)2  1.17748107 2.697980 4.056678 1.12   122
-## te(mintemp,lag)3 -0.07146531 1.930241 3.650584 1.15   166
-## te(precip,lag)    2.65403979 3.894396 4.845707 1.01   289
-## te(precip,lag)2   0.86082577 2.670244 4.238180 1.17   100
-## te(precip,lag)3   1.06013195 2.973949 4.354556 1.03   440
-##             2.5%         50%     97.5% Rhat n.eff
-## ar1    0.5131377  0.82615370 1.1411773 1.02   540
-## ar2   -0.2822316  0.11651989 0.5243282 1.02   445
-## ar3   -0.3530288 -0.05626851 0.2378173 1.00   599
-## sigma  0.2385893  0.35052909 0.4733824 1.06   509
-

A pairs plot of the logged versions of the latent trend precision and -the overdispersion parameter suggest there is no strange behaviour in -the joint posterior

-
plot(log(MCMCvis::MCMCchains(mod3$model_output, 
-    "twdis")), log(MCMCvis::MCMCchains(mod3$model_output, 
-    "tau")), ylab = "log(tau)", xlab = "log(twdis)", 
-    pch = 16, col = "#8F272740")
-

-

Our rootogram has not improved much with the addition of the latent -trend

-
ppc(mod3, type = "rootogram")
-

-

But there is no more evidence of autocorrelation in the residuals

-
plot(mod3, type = "residuals")
-

-We can also demonstrate another feature of mvgam, which is -the ability to use Hamiltonian Monte Carlo for parameter estimation via -the software Stan (using the rstan interface). -Note that rstan is currently required for this option to -work, though support for other Stan interfaces will be -added in future. Also note that currently there is no support for -fitting Tweedie responses or dynamic factor models in -Stan, though again these will be added in future. Because -of these current limitations, we will stick with a -Negative Binomial observation process for the -Stan version. However there are great advantages when using -Stan, which includes the option to estimate smooth latent -trends via Hilbert space -approximate Gaussian Processes. This often makes sense for -ecological series, which we expect to change smoothly over time. As -expected when compared to the Gibbs sampler in JAGS, the -Stan version converges very nicely

-
mod4 <- mvgam(formula = y ~ te(mintemp, lag, 
-    k = c(8, 4)) + te(precip, lag, k = c(8, 
-    4)), data_train = data_train, data_test = data_test, 
-    family = "nb", chains = 4, burnin = 2000, 
-    trend_model = "GP", use_stan = T)
-
## [1] "n_eff / iter looks reasonable for all parameters"
-## [1] "Rhat looks reasonable for all parameters"
-## [1] "0 of 6000 iterations ended with a divergence (0%)"
-## [1] "0 of 6000 iterations saturated the maximum tree depth of 10 (0%)"
-## [1] "E-FMI indicated no pathological behavior"
-
summary(mod4)
-
## y ~ te(mintemp, lag, k = c(8, 4)) + te(precip, lag, k = c(8, 
-##     4))
-## Negative Binomial
-## log
-## GP
-## 1 
-## 168 
-## Fitted using rstan::stan() 
-##          2.5%      50%    97.5% Rhat n.eff
-## r[1] 6.387461 12.92774 33.03298 1.05  1923
-##                   edf df
-## te(mintemp,lag) 82.22 31
-## te(precip,lag)  69.35 28
-##                           2.5%          50%       97.5% Rhat n.eff
-## (Intercept)         1.63043329  2.025783349  2.40985714 1.00  4219
-## te(mintemp,lag).1  -1.84873342 -0.572497711  0.30358913 1.01   753
-## te(mintemp,lag).2  -1.66148138 -0.566264805  0.21959178 1.01  1309
-## te(mintemp,lag).3  -1.95995954 -0.729764063  0.48337282 1.01  2200
-## te(mintemp,lag).4  -1.04749937 -0.464180897  0.43679788 1.01   593
-## te(mintemp,lag).5  -0.44704419  0.107171629  0.77444933 1.00   641
-## te(mintemp,lag).6  -0.04357357  0.459788221  1.19749704 1.01   681
-## te(mintemp,lag).7  -0.15864767  0.553884957  1.40885143 1.01   684
-## te(mintemp,lag).8  -0.76184102 -0.272727228  0.56474992 1.02   607
-## te(mintemp,lag).9  -0.27955380  0.008169622  0.39884186 1.01   672
-## te(mintemp,lag).10 -0.05535882  0.240727901  0.66193487 1.01   711
-## te(mintemp,lag).11 -0.30769023  0.449104473  1.08695119 1.01   679
-## te(mintemp,lag).12 -0.38651485  0.105738986  0.96434531 1.02   581
-## te(mintemp,lag).13 -0.30422060  0.138102796  0.67008821 1.01   623
-## te(mintemp,lag).14 -0.16747282  0.241386023  0.82860785 1.01   726
-## te(mintemp,lag).15 -0.29382751  0.431648865  1.19047993 1.01   687
-## te(mintemp,lag).16 -0.31726120  0.155790307  0.98590270 1.02   574
-## te(mintemp,lag).17 -0.23276347  0.131140845  0.59170978 1.01   714
-## te(mintemp,lag).18 -0.21802671  0.166659085  0.64291665 1.01   834
-## te(mintemp,lag).19 -0.39213948  0.325270744  1.01606455 1.01   678
-## te(mintemp,lag).20 -0.27292088  0.211176553  1.07941624 1.02   560
-## te(mintemp,lag).21 -0.18679767  0.193652901  0.66046778 1.01   776
-## te(mintemp,lag).22 -0.30616406  0.093817129  0.59701745 1.01   923
-## te(mintemp,lag).23 -0.56980776  0.121020786  0.82989099 1.01   704
-## te(mintemp,lag).24  0.01290298  0.512449509  1.43650480 1.02   578
-## te(mintemp,lag).25 -0.17722706  0.282828700  0.83553041 1.00   583
-## te(mintemp,lag).26 -0.49579737 -0.072009236  0.53156865 1.01   745
-## te(mintemp,lag).27 -1.16071007 -0.451773534  0.34472747 1.01   711
-## te(mintemp,lag).28 -0.06636782  0.493075446  1.38277533 1.02   701
-## te(mintemp,lag).29 -0.23023057  0.160529302  0.60900733 1.00  1024
-## te(mintemp,lag).30 -0.70273436 -0.274952138  0.14069635 1.01  1101
-## te(mintemp,lag).31 -1.62337075 -0.803700028 -0.13401490 1.01   906
-## te(precip,lag).1   -0.22440154  0.012728382  0.18308133 1.00  1390
-## te(precip,lag).2   -0.23065020 -0.029626388  0.15134365 1.00  2167
-## te(precip,lag).3   -0.26629699 -0.013218309  0.23375989 1.00  2222
-## te(precip,lag).4   -0.02793277  0.129430462  0.30941307 1.00  1323
-## te(precip,lag).5   -0.17502112  0.003256255  0.11530145 1.00  1436
-## te(precip,lag).6   -0.19687565 -0.039007464  0.09853130 1.01  1842
-## te(precip,lag).7   -0.26435419 -0.039320689  0.16622140 1.00  1450
-## te(precip,lag).8   -0.13845187  0.023768094  0.19850417 1.00  2393
-## te(precip,lag).9   -0.17124793 -0.023636438  0.10340766 1.00  2777
-## te(precip,lag).10  -0.19768569 -0.043054191  0.11426486 1.00  1703
-## te(precip,lag).11  -0.28533839 -0.061829023  0.14531645 1.00  1377
-## te(precip,lag).12  -0.27608860 -0.084868783  0.11291892 1.01  2841
-## te(precip,lag).13  -0.19492321 -0.048000359  0.10313426 1.00  2378
-## te(precip,lag).14  -0.21072843 -0.051399264  0.12055919 1.00  1573
-## te(precip,lag).15  -0.30401520 -0.069295651  0.14506673 1.00  1609
-## te(precip,lag).16  -0.40737674 -0.175921781  0.05109252 1.00  3002
-## te(precip,lag).17  -0.32763302 -0.132050298  0.06552842 1.00  2007
-## te(precip,lag).18  -0.32644884 -0.106155695  0.10644345 1.00  1599
-## te(precip,lag).19  -0.37081575 -0.103568705  0.14428941 1.00  1793
-## te(precip,lag).20  -0.20106843  0.008123892  0.21702686 1.00  2683
-## te(precip,lag).21  -0.19946162 -0.048711137  0.10260081 1.00  3983
-## te(precip,lag).22  -0.25990550 -0.098490418  0.06386584 1.00  3309
-## te(precip,lag).23  -0.24564619 -0.042672777  0.15656400 1.00  2795
-## te(precip,lag).24  -0.32775675 -0.064065969  0.12710948 1.00  2499
-## te(precip,lag).25  -1.07300974 -0.115524700  0.34081401 1.02  1224
-## te(precip,lag).26  -0.70919763 -0.056366195  0.28343424 1.02   987
-## te(precip,lag).27  -0.50883218 -0.011973930  0.37685721 1.01  1333
-## te(precip,lag).28  -0.62005663 -0.030384012  0.48489471 1.01  2350
-##                         2.5%       50%    97.5% Rhat n.eff
-## te(mintemp,lag)   1.99749372 3.3323382 4.420704 1.00  2492
-## te(mintemp,lag)2  0.02058432 2.3409064 3.925603 1.00  1100
-## te(mintemp,lag)3 -2.06214616 0.4842372 3.566277 1.00   613
-## te(precip,lag)    2.58635809 3.9113363 4.882824 1.00  2824
-## te(precip,lag)2  -0.18388517 2.3094257 4.163047 1.01   861
-## te(precip,lag)3   0.16522326 2.7762004 4.293537 1.01  1346
-##                  2.5%       50%    97.5% Rhat n.eff
-## alpha_gp[1] 0.6004228 0.8128277 1.112649 1.00  2417
-## rho_gp[1]   2.4203052 3.3710166 4.870697 1.01  1390
-

As with all other mvgam objects, we can create plots of -the estimated forecast distribution

-
plot_mvgam_fc(mod4, series = 1, data_test = data_test, 
-    ylim = c(0, 100))
-

-The trend now evolves smoothly via an infinite dimensional Gaussian -Process

-
plot_mvgam_trend(mod4, series = 1)
-

-

Traceplots of smooth penalties indicate good mixing and convergence -of the four MCMC chains

-
plot_mvgam_trace(mod4, "rho")
-

-

We can also create quick plots of the estimated smooth tensor product -interactions for the distributed lag terms, which basically follow -mgcv’s two-dimensional plotting utility but uses the -mvgam’s estimated coefficients

-
plot_mvgam_smooth(mod4, series = 1, smooth = 1)
-

-
plot_mvgam_smooth(mod4, series = 1, smooth = 2)
-

-

If you are like me then you’ll find these plots rather difficult to -interpret! The more intense yellow/white colours indicate higher -predicted values, with the deeper red colours representing lower -predicted values, but actually making sense of how the functional -response is expected to change over different lags is not easy from -these plots. HOwever, we can use the predict_mvgam function -to generate much more interpretable plots. First we will focus on the -effect of mintemp and generate a series of predictions to -visualise how the estimated function changes over different lags. Set up -prediction data by zeroing out all covariates apart from the covariate -of interest

-
newdata <- data_test
-newdata$year <- rep(0, length(newdata$year))
-newdata$season <- rep(0, length(newdata$season))
-newdata$precip <- matrix(0, ncol = ncol(newdata$precip), 
-    nrow = nrow(newdata$precip))
-

Set up viridis plot colours and initiate the plot window -to be centred around zero. We will then keep all mintemp -values at zero apart from the particular lag being predicted so that we -can visualise how the predicted function changes over lags of -mintemp. Predictions are generated on the link scale in -this case, though you could also use the response scale. Note that we -need to first generate predictions with all covariates (including the -mintemp covariate) zeroed out to find the ‘baseline’ -prediction so that we can shift by this baseline for generating a -zero-centred plot. That way our resulting plot will roughly follow the -traditional mgcv partial effect plots

-
cols <- viridis::inferno(6)
-plot(1, type = "n", xlab = "Mintemp", ylab = "Predicted response function", 
-    xlim = c(min(data_train$mintemp), max(data_train$mintemp)), 
-    ylim = c(-1.6, 1.6))
-
-# Calculate predictions for when mintemp
-# is all zeros to find the baseline value
-# for centring the plot
-newdata$mintemp <- matrix(0, ncol = ncol(newdata$mintemp), 
-    nrow = nrow(newdata$mintemp))
-preds <- predict(mod4, series = 1, newdata = newdata, 
-    type = "link")
-offset <- mean(preds)
-
-for (i in 1:6) {
-    # Set up prediction matrix for mintemp
-    # with lag i as the prediction sequence;
-    # use a sequence of mintemp values across
-    # the full range of observed values in
-    # the training data
-    newdata$mintemp <- matrix(0, ncol = ncol(newdata$precip), 
-        nrow = nrow(newdata$precip))
-    newdata$mintemp[, i] <- seq(min(data_train$mintemp), 
-        max(data_train$mintemp), length.out = length(newdata$year))
-    
-    # Predict on the link scale and shift by
-    # the offset so that values are roughly
-    # centred at zero
-    preds <- predict(mod4, series = 1, newdata = newdata, 
-        type = "link") - offset
-    
-    # Calculate empirical prediction
-    # quantiles
-    probs = c(0.05, 0.2, 0.3, 0.4, 0.5, 0.6, 
-        0.7, 0.8, 0.95)
-    cred <- sapply(1:NCOL(preds), function(n) quantile(preds[, 
-        n], probs = probs))
-    
-    # Plot expected function posterior
-    # intervals (40-60%) and medians in
-    # varying colours per lag
-    pred_upper <- cred[4, ]
-    pred_lower <- cred[6, ]
-    pred_vals <- seq(min(data_train$mintemp), 
-        max(data_train$mintemp), length.out = length(newdata$year))
-    polygon(c(pred_vals, rev(pred_vals)), 
-        c(pred_upper, rev(pred_lower)), col = scales::alpha(cols[i], 
-            0.6), border = scales::alpha(cols[i], 
-            0.7))
-    lines(pred_vals, cred[5, ], col = scales::alpha(cols[i], 
-        0.8), lwd = 2.5)
-}
-abline(h = 0, lty = "dashed")
-legend("topleft", legend = paste0("lag", 
-    seq(0, 5)), bg = "white", bty = "n", 
-    col = cols, lty = 1, lwd = 6)
-

-

This plot demonstrates how the effect of mintemp is -expected to change over different exposure lags, with the 3 - 5 month -lags showing more of a cyclic seasonal pattern (catches expected to -increase in the summer and autumn, roughly 3 - 5 months following cold -minimum winter temperatures) while the recent lags (lags 0 and 1) -demonstrate a more linear response function (catches broadly increasing -as minimum temperature increases). This is hopefully a useful example -for developing a better understanding of how a distributed lag model is -attempting to recreate the data generating process. And here is the same -plot for precipitation, which demonstrates how a u-shaped functional -relationship diminishes toward a flat function at lags 2 - 5 (though -this effect is clearly less important in the model than the mintemp * -lag effect above)

-
newdata <- data_test
-newdata$year <- rep(0, length(newdata$year))
-newdata$season <- rep(0, length(newdata$season))
-newdata$mintemp <- matrix(0, ncol = ncol(newdata$mintemp), 
-    nrow = nrow(newdata$mintemp))
-newdata$precip <- matrix(0, ncol = ncol(newdata$precip), 
-    nrow = nrow(newdata$precip))
-preds <- predict(mod4, series = 1, newdata = newdata, 
-    type = "link")
-offset <- mean(preds)
-plot(1, type = "n", xlab = "Precipitation", 
-    ylab = "Predicted response function", 
-    xlim = c(min(data_train$precip), max(data_train$precip)), 
-    ylim = c(-1.6, 1.6))
-
-for (i in 1:6) {
-    newdata$precip <- matrix(0, ncol = ncol(newdata$precip), 
-        nrow = nrow(newdata$precip))
-    newdata$precip[, i] <- seq(min(data_train$precip), 
-        max(data_train$precip), length.out = length(newdata$year))
-    preds <- predict(mod4, series = 1, newdata = newdata, 
-        type = "link") - offset
-    probs = c(0.05, 0.2, 0.3, 0.4, 0.5, 0.6, 
-        0.7, 0.8, 0.95)
-    cred <- sapply(1:NCOL(preds), function(n) quantile(preds[, 
-        n], probs = probs))
-    pred_upper <- cred[4, ]
-    pred_lower <- cred[6, ]
-    pred_vals <- seq(min(data_train$precip), 
-        max(data_train$precip), length.out = length(newdata$year))
-    polygon(c(pred_vals, rev(pred_vals)), 
-        c(pred_upper, rev(pred_lower)), col = scales::alpha(cols[i], 
-            0.6), border = scales::alpha(cols[i], 
-            0.7))
-    lines(pred_vals, cred[5, ], col = scales::alpha(cols[i], 
-        0.8), lwd = 2.5)
-}
-abline(h = 0, lty = "dashed")
-legend("topleft", legend = paste0("lag", 
-    seq(0, 5)), bg = "white", bty = "n", 
-    col = cols, lty = 1, lwd = 6)
-

-

All of the usual functions in mvgam can also be used for -list data objects and for models fitted with Stan, so long -as they contain the necessary fields series, -season and year. For example, posterior -retrodictive checks for the in-sample training period:

-
ppc(mod4, series = 1, type = "cdf")
-

-

and predictive checks for the out of sample forecast period (which -demonstrates how the model tends to overpredict for the forecast period -in this particular example):

-
ppc(mod4, data_test = data_test, series = 1, 
-    type = "cdf")
-

-

Logical next steps for interrogating this model would be to trial -different trend types (i.e. random walk), replace the distributed lag -function for precip with a standard smooth function (that -does not include lag interactions, as clearly the model above indicates -that these are not supported) and inspect whether different covariates -(such as ndvi or maxtemp) might play a role in -modulating catches of PP. Finally, once we are satisfied -that we have a well-performing model that we can understand and -interrogate, we could expand up to a multivariate model by including -other species as response variables. This would allow us to capture any -possible unobserved dependencies in the catches of multiple co-occurring -species in a single unified modelling framework

- - - - -
- - - - - - - - - - - - - - - + + + + + + + + + + + + + + +mvgam case study 3: distributed lag models + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + +

Here we will use the mvgam package, which fits dynamic +GAMs using MCMC sampling via either the JAGS software +(installation links are found here) or +via the Stan software (installation links are found here), +to estimate paramaters of a Bayesian distributed lag model. These models +are used to describe simultaneously non-linear and delayed functional +relationships between a covariate and a response, and are sometimes +referred to as exposure-lag-response models. If we assume \(\tilde{\boldsymbol{y}}_{t}\) is the +conditional expectation of a discrete response variable \(\boldsymbol{y}\) at time \(\boldsymbol{t}\), the linear predictor for +a dynamic distributed lag GAM with one lagged covariate is written +as:

+

\[log(\tilde{\boldsymbol{y}}_{t})={\boldsymbol{B}}_0+\sum\limits_{k=1}^K{f}(\boldsymbol{b}_{k,t}\boldsymbol{x}_{k,t})+\boldsymbol{z}_{t}\,,\] +where \(\boldsymbol{B}_{0}\) is the +unknown intercept, the \(\boldsymbol{b}\)’s are unknown spline +coefficients estimating how the functional effect of covariate (\(\boldsymbol{x}\)) on \(log(\tilde{\boldsymbol{y}}_{t})\) changes +over increasing lags (up to a maximum lag of (\(\boldsymbol{K}\))) and \(\boldsymbol{z}\) is a dynamic latent trend +component.

+

To demonstrate how these models are estimated in mvgam, +first we load the Portal rodents capture data, which are available from +the portalr package

+
# devtools::install_github('nicholasjclark/mvgam')
+library(mvgam)
+library(dplyr)
+portal_dat <- read.csv("https://raw.githubusercontent.com/nicholasjclark/mvgam/master/NEON_manuscript/Case studies/rodents_data.csv", 
+    as.is = T)
+

We’ll keep data from the year 2004 onwards to make the model quicker +to estimate for this simple example

+
portal_dat_all <- portal_dat %>% dplyr::filter(year >= 
+    2004) %>% dplyr::group_by(year, month) %>% 
+    dplyr::slice_head(n = 1)
+

Below is an exact reproduction of Simon Wood’s lag matrix function +(which he uses in his distributed lag example from his book Generalized +Additive Models - An Introduction with R 2nd edition). Here we +supply a vector and specify the maximum lag that we want, and it will +return a matrix of dimension length(x) * lag. Note that +NAs are used for the missing lag values at the beginning of +the matrix. In essence, the matrix objects represent exposure histories, +where each row represents the lagged values of the predictor that +correspond to each observation in y

+
lagard <- function(x, n.lag = 6) {
+    n <- length(x)
+    X <- matrix(NA, n, n.lag)
+    for (i in 1:n.lag) X[i:n, i] <- x[i:n - 
+        i + 1]
+    X
+}
+

Organise all data needed for modelling into a list. We will focus +only on the species Chaetodipus penicillatus (labelled as +PP), which shows reasonable seasonality in its captures +over time

+
data_all <- list(lag = matrix(0:5, nrow(portal_dat_all), 
+    6, byrow = TRUE), y = portal_dat_all$PP, 
+    season = portal_dat_all$month, year = portal_dat_all$year, 
+    series = rep(as.factor("series1"), NROW(portal_dat_all)), 
+    time = 1:NROW(portal_dat_all))
+data_all$precip <- lagard(portal_dat_all$precipitation)
+data_all$mintemp <- lagard(portal_dat_all$mintemp)
+

The exposure history matrix elements of the data list look as +follows:

+
head(data_all$lag, 5)
+
##      [,1] [,2] [,3] [,4] [,5] [,6]
+## [1,]    0    1    2    3    4    5
+## [2,]    0    1    2    3    4    5
+## [3,]    0    1    2    3    4    5
+## [4,]    0    1    2    3    4    5
+## [5,]    0    1    2    3    4    5
+
head(data_all$precip, 5)
+
##      [,1] [,2] [,3] [,4] [,5] [,6]
+## [1,] 37.8   NA   NA   NA   NA   NA
+## [2,]  8.7 37.8   NA   NA   NA   NA
+## [3,] 43.5  8.7 37.8   NA   NA   NA
+## [4,] 23.9 43.5  8.7 37.8   NA   NA
+## [5,]  0.9 23.9 43.5  8.7 37.8   NA
+
head(data_all$mintemp, 5)
+
##        [,1]   [,2]   [,3]   [,4]  [,5] [,6]
+## [1,] -9.710     NA     NA     NA    NA   NA
+## [2,] -5.924 -9.710     NA     NA    NA   NA
+## [3,] -0.220 -5.924 -9.710     NA    NA   NA
+## [4,]  1.931 -0.220 -5.924 -9.710    NA   NA
+## [5,]  6.568  1.931 -0.220 -5.924 -9.71   NA
+

All other elements of the data list are in the usual vector +format

+
head(data_all$y, 5)
+
## [1]  0  1  2 NA 10
+
head(data_all$series, 5)
+
## [1] series1 series1 series1 series1 series1
+## Levels: series1
+
head(data_all$year, 5)
+
## [1] 2004 2004 2004 2004 2004
+
head(data_all$time, 5)
+
## [1] 1 2 3 4 5
+

View the raw series. There is a clear seasonal pattern to the data, +and there are missing values scattered throughout

+
plot(ts(data_all$y, frequency = 12), ylab = "Captures for PP", 
+    xlab = "", lwd = 2.5, col = "#8F2727")
+

+

Create training and testing sets; start at observation 7 so that the +NA values at the beginning of the covariate lag matrices +are not included. Currently there is no option for on-the-fly imputation +of missing covariate values in mvgam models, though this +can easily be done in JAGS by specifying prior +distributions over these missing entries

+
data_train <- list(lag = data_all$lag[7:174, 
+    ], y = data_all$y[7:174], series = data_all$series[7:174], 
+    season = data_all$season[7:174], year = data_all$year[7:174], 
+    time = 7:174, precip = data_all$precip[7:174, 
+        ], mintemp = data_all$mintemp[7:174, 
+        ])
+data_test <- list(lag = data_all$lag[175:length(data_all$y), 
+    ], y = data_all$y[175:length(data_all$y)], 
+    series = data_all$series[175:length(data_all$y)], 
+    season = data_all$season[175:length(data_all$y)], 
+    year = data_all$year[175:length(data_all$y)], 
+    time = 175:length(data_all$y), precip = data_all$precip[175:length(data_all$y), 
+        ], mintemp = data_all$mintemp[175:length(data_all$y), 
+        ])
+

Now we can fit a Bayesian GAM with distributed lag terms for +precipitation and minimum temperature. The distributed lags are set up +as tensor product smooth functions (see help(te) for an +explanation of tensor product smooth constructions in the +mgcv package) between lag and each covariate. +We will start simply by assuming our data follow a Poisson +observation process

+
mod1 <- mvgam(formula = y ~ te(mintemp, lag, 
+    k = c(8, 4)) + te(precip, lag, k = c(8, 
+    4)), data_train = data_train, data_test = data_test, 
+    family = "poisson", chains = 4, burnin = 10000, 
+    trend_model = "None")
+

Posterior predictive rootograms are a useful way to explore whether a +discrete model is able to capture relevant dispersion in the observed +data. This plot compares the frequencies of observed vs predicted values +for each bin, which can help to identify aspects of poor model fit. For +example, if the gray bars (representing observed frequencies) tend to +stretch below zero, this suggests the model’s simulations predict the +values in that particular bin less frequently than they are observed in +the data. A well-fitting model that can generate realistic simulated +data will provide a rootogram in which the lower boundaries of the grey +bars are generally near zero

+
ppc(mod1, type = "rootogram")
+

+

The Poisson model is not doing a great job of capturing +dispersion, underpredicting the zeros in the data and overpredicting +some of the medium-range values (counts of ~5-30). The +residual Q-Q plot confirms that the Poisson is +not an appropriate distribution for these data

+
plot(mod1, type = "residuals")
+

+

Given the overdispersion present in the data, we will now assume a Geometric-Poisson +observation model, which can be more +flexible than the Negative binomial for modelling +overdispersed count data. In mvgam the +Geometric-Poisson is estimated as a +Tweedie-Poisson model with the power parameter +p fixed at 1.5

+
mod2 <- mvgam(formula = y ~ te(mintemp, lag, 
+    k = c(8, 4)) + te(precip, lag, k = c(8, 
+    4)), data_train = data_train, data_test = data_test, 
+    family = "tw", chains = 4, burnin = 10000, 
+    trend_model = "None")
+

The rootogram for this model looks better, though of course there is +still some overprediction of medium-range values

+
ppc(mod2, type = "rootogram")
+

+

However, the residual plot looks much better for this model

+
plot(mod2, type = "residuals")
+

+

The summary of the model provides useful information on convergence +for unobserved parameters. Notice how strongly positive the +overdispersion parameter is estimated to be, providing further evidence +that this overdispersion is important to capture for these data

+
summary(mod2)
+
## GAM formula:
+
## y ~ te(mintemp, lag, k = c(8, 4)) + te(precip, lag, k = c(8, 
+##     4))
+
##
+
## Family:
+
## Tweedie
+
##
+
## Link function:
+
## log
+
##
+
## Trend model:
+
## None
+
##
+
## N series:
+
## 1
+
##
+
## N observations:
+
## 168
+
##
+
## Status:
+
## Fitted using runjags::run.jags()
+
##
+
## Dispersion parameter estimates:
+
##        2.5%      50%    97.5% Rhat n.eff
+## twdis 1.388 1.932959 2.895216 1.14   331
+
##
+
## GAM smooth term estimated degrees of freedom:
+
##                   edf df
+## te(mintemp,lag) 95.67 31
+## te(precip,lag)  82.94 28
+
##
+
## GAM coefficient (beta) estimates:
+
##                             2.5%          50%       97.5% Rhat n.eff
+## (Intercept)         2.212693e+00  2.403258707  2.56885901 1.09   290
+## te(mintemp,lag).1  -1.054775e+00 -0.097381019  0.35528542 1.75    26
+## te(mintemp,lag).2  -1.106697e+00 -0.368732775  0.12915239 1.33    44
+## te(mintemp,lag).3  -1.474308e+00 -0.734838360  0.09358272 1.38    52
+## te(mintemp,lag).4  -9.798236e-01 -0.605709195 -0.21506225 1.37    49
+## te(mintemp,lag).5  -4.040562e-01 -0.092727382  0.42243535 1.79    43
+## te(mintemp,lag).6   8.517891e-05  0.310276127  0.72603789 1.44    34
+## te(mintemp,lag).7   7.187907e-02  0.491456804  0.77836084 1.23    38
+## te(mintemp,lag).8  -8.162581e-01 -0.515957422 -0.20271474 1.15    43
+## te(mintemp,lag).9  -3.707472e-01 -0.175962485  0.04951066 1.40    65
+## te(mintemp,lag).10 -6.358472e-02  0.123077437  0.31621992 1.10    50
+## te(mintemp,lag).11  1.195955e-01  0.392272177  0.64054687 1.07    46
+## te(mintemp,lag).12 -5.981071e-01 -0.224082393  0.16808352 1.44    35
+## te(mintemp,lag).13 -3.862997e-01 -0.139209569  0.19495414 1.80    66
+## te(mintemp,lag).14 -2.441130e-01  0.025814566  0.33439909 1.46    45
+## te(mintemp,lag).15  3.241991e-02  0.320965039  0.62328350 1.08    41
+## te(mintemp,lag).16 -3.286109e-01 -0.010954410  0.33498460 1.35    41
+## te(mintemp,lag).17 -2.844375e-01 -0.041016059  0.26075539 1.89    64
+## te(mintemp,lag).18 -2.656616e-01 -0.002339434  0.33316216 1.55    56
+## te(mintemp,lag).19 -6.744452e-02  0.203056321  0.52179149 1.09    60
+## te(mintemp,lag).20 -2.222059e-01  0.071034888  0.55035077 1.27    43
+## te(mintemp,lag).21 -2.417919e-01  0.052624654  0.38676519 1.69    47
+## te(mintemp,lag).22 -3.463257e-01 -0.018821781  0.32024342 1.32    32
+## te(mintemp,lag).23 -2.567171e-01  0.023001627  0.41206856 1.09    46
+## te(mintemp,lag).24  1.457483e-02  0.354576859  0.85032897 1.42    48
+## te(mintemp,lag).25 -1.949911e-01  0.108520078  0.54652848 1.92    45
+## te(mintemp,lag).26 -3.749921e-01 -0.130021494  0.21053719 1.52    41
+## te(mintemp,lag).27 -6.237620e-01 -0.351657535 -0.00898590 1.05    34
+## te(mintemp,lag).28  9.617270e-02  0.492034669  0.92753487 1.12    43
+## te(mintemp,lag).29 -3.045994e-01  0.151414518  0.52004441 1.17    29
+## te(mintemp,lag).30 -5.743251e-01 -0.210852728  0.11881993 1.02    35
+## te(mintemp,lag).31 -9.933905e-01 -0.558453233 -0.07249095 1.16    47
+## te(precip,lag).1   -1.822929e-01  0.029293583  0.16142576 1.11    44
+## te(precip,lag).2   -2.422555e-01 -0.053000798  0.11107324 1.32    57
+## te(precip,lag).3   -4.278990e-01 -0.092901543  0.15044128 1.71    51
+## te(precip,lag).4    1.468352e-02  0.167812898  0.38051116 1.16    51
+## te(precip,lag).5   -1.523433e-01  0.007996773  0.08109699 1.17    70
+## te(precip,lag).6   -2.451465e-01 -0.080542477  0.05006037 1.26    59
+## te(precip,lag).7   -4.125140e-01 -0.106601339  0.08656329 1.72    59
+## te(precip,lag).8   -1.110711e-01  0.066748222  0.31861998 1.27    52
+## te(precip,lag).9   -1.741086e-01 -0.038851485  0.08810518 1.10    63
+## te(precip,lag).10  -3.021204e-01 -0.101017687  0.06064512 1.08    38
+## te(precip,lag).11  -4.583086e-01 -0.103610861  0.07695710 1.36    52
+## te(precip,lag).12  -3.216636e-01 -0.104570006  0.19237728 1.44    50
+## te(precip,lag).13  -2.430229e-01 -0.099218023  0.02424224 1.09    63
+## te(precip,lag).14  -2.806793e-01 -0.113844520  0.03463855 1.07    45
+## te(precip,lag).15  -4.500282e-01 -0.081944527  0.10618411 1.17    46
+## te(precip,lag).16  -4.201331e-01 -0.133284892  0.12198342 1.66    56
+## te(precip,lag).17  -3.327693e-01 -0.135333807  0.02824623 1.19    74
+## te(precip,lag).18  -3.679845e-01 -0.129255428  0.06712730 1.06    34
+## te(precip,lag).19  -4.973665e-01 -0.125078896  0.14759108 1.14    38
+## te(precip,lag).20  -1.085044e-01  0.158442743  0.36943597 1.03    45
+## te(precip,lag).21  -1.609977e-01 -0.004588906  0.10242298 1.11    93
+## te(precip,lag).22  -2.865369e-01 -0.130260412 -0.01385738 1.07    82
+## te(precip,lag).23  -3.639076e-01 -0.176572752  0.04987009 1.06    61
+## te(precip,lag).24  -1.790802e-01  0.021304676  0.22629910 1.15    72
+## te(precip,lag).25  -6.432391e-01 -0.094210307  0.32520239 1.56    43
+## te(precip,lag).26  -4.278256e-01 -0.077048068  0.21366462 1.35    41
+## te(precip,lag).27  -2.883587e-01  0.012994431  0.28647043 1.15    44
+## te(precip,lag).28  -2.094542e-01  0.094615123  0.41248432 1.31    60
+
##
+
## GAM smoothing parameter (rho) estimates:
+
##                         2.5%      50%    97.5% Rhat n.eff
+## te(mintemp,lag)   1.83893338 3.192341 4.309709 1.01    71
+## te(mintemp,lag)2  0.28404659 2.421797 4.056106 1.09    53
+## te(mintemp,lag)3 -0.68282687 1.729060 4.122901 1.50   108
+## te(precip,lag)    2.14411956 3.629856 4.698440 1.01   211
+## te(precip,lag)2   0.09545339 2.704613 4.271508 1.25    79
+## te(precip,lag)3   0.20771730 2.566491 4.186006 1.24    82
+
##
+

As this is a timeseries and the residual plot hints at some +autocorrelation remaining in the short-term lags, lets check if an +AR latent trend process improves forecasts compared to the +no-trend model. An important note here is the choice of prior for the +overdispersion parameter twdis. This parameter and the +latent trend variance can interact strongly, particularly when +overdispersion is in the data high. This is because at high values of +the dispersion parameter, there is less need for a latent trend to be +able to capture any outliers and so the latent trend precision can go up +toward infinity, approaching a space of very diffuse likelihood that +forces the Gibbs samplers to take on frustratingly small step sizes. +Likewise when there is not much need for overdispersion, the dispersion +parameter can approach zero and move around in an equally uninformative +parameter space. The latent trend operates on the log scale, so really +we should not expect autocorrelated jumps in trappings of more than +6-8 from timepoint to timepoint (any larger and the trend +will compete strongly with the overdispersion parameter, making it +difficult for us to model the inherent overdispersion process and +instead assuming it is all autocorrelation). A containment prior on the +latent trend sigma will help achieve this

+
mod3 <- mvgam(formula = y ~ te(mintemp, lag, 
+    k = c(8, 4)) + te(precip, lag, k = c(8, 
+    4)), data_train = data_train, data_test = data_test, 
+    family = "tw", sigma_prior = "dexp(2.5)T(0.15, 2)", 
+    chains = 4, burnin = 10000, trend_model = "AR3")
+summary(mod3)
+
## y ~ te(mintemp, lag, k = c(8, 4)) + te(precip, lag, k = c(8, 
+##     4))
+## Tweedie
+## log
+## AR3
+## 1 
+## 168 
+## Fitted using runjags::run.jags() 
+##            2.5%       50%    97.5% Rhat n.eff
+## twdis 0.1442308 0.3038398 0.533641 1.58    35
+##                   edf df
+## te(mintemp,lag) 55.01 31
+## te(precip,lag)  79.24 28
+##                           2.5%          50%        97.5% Rhat n.eff
+## (Intercept)         1.95451829  2.294369149  2.687638507 1.05    79
+## te(mintemp,lag).1  -0.34440489  0.001118830  0.259412270 1.59    34
+## te(mintemp,lag).2  -0.58230901 -0.114278574  0.245323772 2.32    28
+## te(mintemp,lag).3  -1.05666653 -0.380131705  0.112717290 1.81    26
+## te(mintemp,lag).4  -0.90716043 -0.666943517 -0.374398808 1.44    61
+## te(mintemp,lag).5  -0.40434523 -0.193728154  0.042342982 1.23    32
+## te(mintemp,lag).6   0.02152938  0.203985336  0.456034236 1.56    42
+## te(mintemp,lag).7   0.25813250  0.514930957  0.773757069 1.46    74
+## te(mintemp,lag).8  -0.67614540 -0.442897113 -0.202233028 1.33    57
+## te(mintemp,lag).9  -0.27762442 -0.138611196 -0.011473547 1.10    56
+## te(mintemp,lag).10  0.02284141  0.168421464  0.353924370 1.50    60
+## te(mintemp,lag).11  0.26878013  0.486133830  0.749333381 1.21    44
+## te(mintemp,lag).12 -0.34006723 -0.149228275  0.121707811 1.17    33
+## te(mintemp,lag).13 -0.25641787 -0.078535420  0.124577834 1.44    40
+## te(mintemp,lag).14 -0.07347474  0.104024205  0.364380385 1.73    54
+## te(mintemp,lag).15  0.20099240  0.417414813  0.650764551 1.17    46
+## te(mintemp,lag).16 -0.24087569 -0.042252034  0.217570497 1.21    34
+## te(mintemp,lag).17 -0.23693743 -0.039656865  0.140520521 1.45    37
+## te(mintemp,lag).18 -0.16064686  0.059071449  0.260616038 1.57    45
+## te(mintemp,lag).19  0.06594100  0.279170574  0.570433710 1.42    44
+## te(mintemp,lag).20 -0.24701161  0.050345586  0.335014718 1.26    29
+## te(mintemp,lag).21 -0.22732085 -0.016441948  0.219734982 1.41    39
+## te(mintemp,lag).22 -0.29591233 -0.039567014  0.188568090 1.64    36
+## te(mintemp,lag).23 -0.19552442  0.018925677  0.354815056 1.59    40
+## te(mintemp,lag).24  0.00712481  0.257539466  0.534982135 1.26    39
+## te(mintemp,lag).25 -0.14986366  0.028610310  0.240532737 1.27    35
+## te(mintemp,lag).26 -0.35516212 -0.203205407  0.132626064 1.92    43
+## te(mintemp,lag).27 -0.66173660 -0.361493765 -0.139140221 2.02    53
+## te(mintemp,lag).28 -0.10654805  0.315825287  0.718339605 1.35    32
+## te(mintemp,lag).29 -0.19502605  0.057821835  0.351960793 1.19    22
+## te(mintemp,lag).30 -0.49213602 -0.254132913 -0.004726919 1.73    45
+## te(mintemp,lag).31 -0.98269158 -0.601603108 -0.293992943 1.55    36
+## te(precip,lag).1   -0.18071579 -0.007770073  0.129141735 1.23    44
+## te(precip,lag).2   -0.26109947 -0.053181668  0.109517176 1.26    36
+## te(precip,lag).3   -0.29134027 -0.067594472  0.192333185 1.50    34
+## te(precip,lag).4   -0.03682622  0.091643893  0.283454745 1.40    40
+## te(precip,lag).5   -0.16933377 -0.017988505  0.090918611 1.48    65
+## te(precip,lag).6   -0.26882772 -0.057775209  0.074503584 1.66    48
+## te(precip,lag).7   -0.28597650 -0.067388939  0.134412111 1.67    35
+## te(precip,lag).8   -0.14848668  0.011187431  0.178197414 1.03    40
+## te(precip,lag).9   -0.20361931 -0.027252625  0.080875803 1.83    78
+## te(precip,lag).10  -0.25544688 -0.044531181  0.074521842 1.73    53
+## te(precip,lag).11  -0.31157290 -0.051729464  0.124553750 1.54    36
+## te(precip,lag).12  -0.26780541 -0.068596545  0.129182178 1.24    39
+## te(precip,lag).13  -0.15294675 -0.041633221  0.103724179 1.49    69
+## te(precip,lag).14  -0.17124532 -0.031463088  0.105549391 1.19    50
+## te(precip,lag).15  -0.26011511 -0.016574210  0.191909247 1.20    41
+## te(precip,lag).16  -0.29057183 -0.136137976  0.138202557 1.43    48
+## te(precip,lag).17  -0.26211414 -0.089851782  0.102359183 1.39    40
+## te(precip,lag).18  -0.27921814 -0.056206824  0.114079040 1.14    33
+## te(precip,lag).19  -0.32498905 -0.035923817  0.183391255 1.12    33
+## te(precip,lag).20  -0.06375377  0.092346100  0.228629904 1.22    58
+## te(precip,lag).21  -0.17045386 -0.026071316  0.117831174 1.13    47
+## te(precip,lag).22  -0.24797078 -0.086540727  0.062987289 1.27    57
+## te(precip,lag).23  -0.24537849 -0.059480138  0.128256777 1.23    55
+## te(precip,lag).24  -0.23766458 -0.031720898  0.119043396 1.09    54
+## te(precip,lag).25  -0.41009464  0.018395088  0.234195699 1.24    23
+## te(precip,lag).26  -0.26921000 -0.018316715  0.191046564 1.09    33
+## te(precip,lag).27  -0.22769731 -0.043288937  0.193162536 1.22    40
+## te(precip,lag).28  -0.39840734 -0.095952392  0.347629109 1.18    25
+##                         2.5%      50%    97.5% Rhat n.eff
+## te(mintemp,lag)   2.32902372 3.493322 4.464910 1.12   167
+## te(mintemp,lag)2  1.17748107 2.697980 4.056678 1.12   122
+## te(mintemp,lag)3 -0.07146531 1.930241 3.650584 1.15   166
+## te(precip,lag)    2.65403979 3.894396 4.845707 1.01   289
+## te(precip,lag)2   0.86082577 2.670244 4.238180 1.17   100
+## te(precip,lag)3   1.06013195 2.973949 4.354556 1.03   440
+##             2.5%         50%     97.5% Rhat n.eff
+## ar1    0.5131377  0.82615370 1.1411773 1.02   540
+## ar2   -0.2822316  0.11651989 0.5243282 1.02   445
+## ar3   -0.3530288 -0.05626851 0.2378173 1.00   599
+## sigma  0.2385893  0.35052909 0.4733824 1.06   509
+

A pairs plot of the logged versions of the latent trend precision and +the overdispersion parameter suggest there is no strange behaviour in +the joint posterior

+
plot(log(MCMCvis::MCMCchains(mod3$model_output, 
+    "twdis")), log(MCMCvis::MCMCchains(mod3$model_output, 
+    "tau")), ylab = "log(tau)", xlab = "log(twdis)", 
+    pch = 16, col = "#8F272740")
+

+

Our rootogram has not improved much with the addition of the latent +trend

+
ppc(mod3, type = "rootogram")
+

+

But there is no more evidence of autocorrelation in the residuals

+
plot(mod3, type = "residuals")
+

+We can also demonstrate another feature of mvgam, which is +the ability to use Hamiltonian Monte Carlo for parameter estimation via +the software Stan (using the rstan interface). +Note that rstan is currently required for this option to +work, though support for other Stan interfaces will be +added in future. Also note that currently there is no support for +fitting Tweedie responses or dynamic factor models in +Stan, though again these will be added in future. Because +of these current limitations, we will stick with a +Negative Binomial observation process for the +Stan version. However there are great advantages when using +Stan, which includes the option to estimate smooth latent +trends via Hilbert space +approximate Gaussian Processes. This often makes sense for +ecological series, which we expect to change smoothly over time. As +expected when compared to the Gibbs sampler in JAGS, the +Stan version converges very nicely

+
mod4 <- mvgam(formula = y ~ te(mintemp, lag, 
+    k = c(8, 4)) + te(precip, lag, k = c(8, 
+    4)), data_train = data_train, data_test = data_test, 
+    family = "nb", chains = 4, burnin = 2000, 
+    trend_model = "GP", use_stan = T)
+
## [1] "n_eff / iter looks reasonable for all parameters"
+## [1] "Rhat looks reasonable for all parameters"
+## [1] "0 of 6000 iterations ended with a divergence (0%)"
+## [1] "0 of 6000 iterations saturated the maximum tree depth of 10 (0%)"
+## [1] "E-FMI indicated no pathological behavior"
+
summary(mod4)
+
## y ~ te(mintemp, lag, k = c(8, 4)) + te(precip, lag, k = c(8, 
+##     4))
+## Negative Binomial
+## log
+## GP
+## 1 
+## 168 
+## Fitted using rstan::stan() 
+##          2.5%      50%    97.5% Rhat n.eff
+## r[1] 6.387461 12.92774 33.03298 1.05  1923
+##                   edf df
+## te(mintemp,lag) 82.22 31
+## te(precip,lag)  69.35 28
+##                           2.5%          50%       97.5% Rhat n.eff
+## (Intercept)         1.63043329  2.025783349  2.40985714 1.00  4219
+## te(mintemp,lag).1  -1.84873342 -0.572497711  0.30358913 1.01   753
+## te(mintemp,lag).2  -1.66148138 -0.566264805  0.21959178 1.01  1309
+## te(mintemp,lag).3  -1.95995954 -0.729764063  0.48337282 1.01  2200
+## te(mintemp,lag).4  -1.04749937 -0.464180897  0.43679788 1.01   593
+## te(mintemp,lag).5  -0.44704419  0.107171629  0.77444933 1.00   641
+## te(mintemp,lag).6  -0.04357357  0.459788221  1.19749704 1.01   681
+## te(mintemp,lag).7  -0.15864767  0.553884957  1.40885143 1.01   684
+## te(mintemp,lag).8  -0.76184102 -0.272727228  0.56474992 1.02   607
+## te(mintemp,lag).9  -0.27955380  0.008169622  0.39884186 1.01   672
+## te(mintemp,lag).10 -0.05535882  0.240727901  0.66193487 1.01   711
+## te(mintemp,lag).11 -0.30769023  0.449104473  1.08695119 1.01   679
+## te(mintemp,lag).12 -0.38651485  0.105738986  0.96434531 1.02   581
+## te(mintemp,lag).13 -0.30422060  0.138102796  0.67008821 1.01   623
+## te(mintemp,lag).14 -0.16747282  0.241386023  0.82860785 1.01   726
+## te(mintemp,lag).15 -0.29382751  0.431648865  1.19047993 1.01   687
+## te(mintemp,lag).16 -0.31726120  0.155790307  0.98590270 1.02   574
+## te(mintemp,lag).17 -0.23276347  0.131140845  0.59170978 1.01   714
+## te(mintemp,lag).18 -0.21802671  0.166659085  0.64291665 1.01   834
+## te(mintemp,lag).19 -0.39213948  0.325270744  1.01606455 1.01   678
+## te(mintemp,lag).20 -0.27292088  0.211176553  1.07941624 1.02   560
+## te(mintemp,lag).21 -0.18679767  0.193652901  0.66046778 1.01   776
+## te(mintemp,lag).22 -0.30616406  0.093817129  0.59701745 1.01   923
+## te(mintemp,lag).23 -0.56980776  0.121020786  0.82989099 1.01   704
+## te(mintemp,lag).24  0.01290298  0.512449509  1.43650480 1.02   578
+## te(mintemp,lag).25 -0.17722706  0.282828700  0.83553041 1.00   583
+## te(mintemp,lag).26 -0.49579737 -0.072009236  0.53156865 1.01   745
+## te(mintemp,lag).27 -1.16071007 -0.451773534  0.34472747 1.01   711
+## te(mintemp,lag).28 -0.06636782  0.493075446  1.38277533 1.02   701
+## te(mintemp,lag).29 -0.23023057  0.160529302  0.60900733 1.00  1024
+## te(mintemp,lag).30 -0.70273436 -0.274952138  0.14069635 1.01  1101
+## te(mintemp,lag).31 -1.62337075 -0.803700028 -0.13401490 1.01   906
+## te(precip,lag).1   -0.22440154  0.012728382  0.18308133 1.00  1390
+## te(precip,lag).2   -0.23065020 -0.029626388  0.15134365 1.00  2167
+## te(precip,lag).3   -0.26629699 -0.013218309  0.23375989 1.00  2222
+## te(precip,lag).4   -0.02793277  0.129430462  0.30941307 1.00  1323
+## te(precip,lag).5   -0.17502112  0.003256255  0.11530145 1.00  1436
+## te(precip,lag).6   -0.19687565 -0.039007464  0.09853130 1.01  1842
+## te(precip,lag).7   -0.26435419 -0.039320689  0.16622140 1.00  1450
+## te(precip,lag).8   -0.13845187  0.023768094  0.19850417 1.00  2393
+## te(precip,lag).9   -0.17124793 -0.023636438  0.10340766 1.00  2777
+## te(precip,lag).10  -0.19768569 -0.043054191  0.11426486 1.00  1703
+## te(precip,lag).11  -0.28533839 -0.061829023  0.14531645 1.00  1377
+## te(precip,lag).12  -0.27608860 -0.084868783  0.11291892 1.01  2841
+## te(precip,lag).13  -0.19492321 -0.048000359  0.10313426 1.00  2378
+## te(precip,lag).14  -0.21072843 -0.051399264  0.12055919 1.00  1573
+## te(precip,lag).15  -0.30401520 -0.069295651  0.14506673 1.00  1609
+## te(precip,lag).16  -0.40737674 -0.175921781  0.05109252 1.00  3002
+## te(precip,lag).17  -0.32763302 -0.132050298  0.06552842 1.00  2007
+## te(precip,lag).18  -0.32644884 -0.106155695  0.10644345 1.00  1599
+## te(precip,lag).19  -0.37081575 -0.103568705  0.14428941 1.00  1793
+## te(precip,lag).20  -0.20106843  0.008123892  0.21702686 1.00  2683
+## te(precip,lag).21  -0.19946162 -0.048711137  0.10260081 1.00  3983
+## te(precip,lag).22  -0.25990550 -0.098490418  0.06386584 1.00  3309
+## te(precip,lag).23  -0.24564619 -0.042672777  0.15656400 1.00  2795
+## te(precip,lag).24  -0.32775675 -0.064065969  0.12710948 1.00  2499
+## te(precip,lag).25  -1.07300974 -0.115524700  0.34081401 1.02  1224
+## te(precip,lag).26  -0.70919763 -0.056366195  0.28343424 1.02   987
+## te(precip,lag).27  -0.50883218 -0.011973930  0.37685721 1.01  1333
+## te(precip,lag).28  -0.62005663 -0.030384012  0.48489471 1.01  2350
+##                         2.5%       50%    97.5% Rhat n.eff
+## te(mintemp,lag)   1.99749372 3.3323382 4.420704 1.00  2492
+## te(mintemp,lag)2  0.02058432 2.3409064 3.925603 1.00  1100
+## te(mintemp,lag)3 -2.06214616 0.4842372 3.566277 1.00   613
+## te(precip,lag)    2.58635809 3.9113363 4.882824 1.00  2824
+## te(precip,lag)2  -0.18388517 2.3094257 4.163047 1.01   861
+## te(precip,lag)3   0.16522326 2.7762004 4.293537 1.01  1346
+##                  2.5%       50%    97.5% Rhat n.eff
+## alpha_gp[1] 0.6004228 0.8128277 1.112649 1.00  2417
+## rho_gp[1]   2.4203052 3.3710166 4.870697 1.01  1390
+

As with all other mvgam objects, we can create plots of +the estimated forecast distribution

+
plot_mvgam_fc(mod4, series = 1, data_test = data_test, 
+    ylim = c(0, 100))
+

+The trend now evolves smoothly via an infinite dimensional Gaussian +Process

+
plot_mvgam_trend(mod4, series = 1)
+

+

Traceplots of smooth penalties indicate good mixing and convergence +of the four MCMC chains

+
plot_mvgam_trace(mod4, "rho")
+

+

We can also create quick plots of the estimated smooth tensor product +interactions for the distributed lag terms, which basically follow +mgcv’s two-dimensional plotting utility but uses the +mvgam’s estimated coefficients

+
plot_mvgam_smooth(mod4, series = 1, smooth = 1)
+

+
plot_mvgam_smooth(mod4, series = 1, smooth = 2)
+

+

If you are like me then you’ll find these plots rather difficult to +interpret! The more intense yellow/white colours indicate higher +predicted values, with the deeper red colours representing lower +predicted values, but actually making sense of how the functional +response is expected to change over different lags is not easy from +these plots. HOwever, we can use the predict_mvgam function +to generate much more interpretable plots. First we will focus on the +effect of mintemp and generate a series of predictions to +visualise how the estimated function changes over different lags. Set up +prediction data by zeroing out all covariates apart from the covariate +of interest

+
newdata <- data_test
+newdata$year <- rep(0, length(newdata$year))
+newdata$season <- rep(0, length(newdata$season))
+newdata$precip <- matrix(0, ncol = ncol(newdata$precip), 
+    nrow = nrow(newdata$precip))
+

Set up viridis plot colours and initiate the plot window +to be centred around zero. We will then keep all mintemp +values at zero apart from the particular lag being predicted so that we +can visualise how the predicted function changes over lags of +mintemp. Predictions are generated on the link scale in +this case, though you could also use the response scale. Note that we +need to first generate predictions with all covariates (including the +mintemp covariate) zeroed out to find the ‘baseline’ +prediction so that we can shift by this baseline for generating a +zero-centred plot. That way our resulting plot will roughly follow the +traditional mgcv partial effect plots

+
cols <- viridis::inferno(6)
+plot(1, type = "n", xlab = "Mintemp", ylab = "Predicted response function", 
+    xlim = c(min(data_train$mintemp), max(data_train$mintemp)), 
+    ylim = c(-1.6, 1.6))
+
+# Calculate predictions for when mintemp
+# is all zeros to find the baseline value
+# for centring the plot
+newdata$mintemp <- matrix(0, ncol = ncol(newdata$mintemp), 
+    nrow = nrow(newdata$mintemp))
+preds <- predict(mod4, series = 1, newdata = newdata, 
+    type = "link")
+offset <- mean(preds)
+
+for (i in 1:6) {
+    # Set up prediction matrix for mintemp
+    # with lag i as the prediction sequence;
+    # use a sequence of mintemp values across
+    # the full range of observed values in
+    # the training data
+    newdata$mintemp <- matrix(0, ncol = ncol(newdata$precip), 
+        nrow = nrow(newdata$precip))
+    newdata$mintemp[, i] <- seq(min(data_train$mintemp), 
+        max(data_train$mintemp), length.out = length(newdata$year))
+    
+    # Predict on the link scale and shift by
+    # the offset so that values are roughly
+    # centred at zero
+    preds <- predict(mod4, series = 1, newdata = newdata, 
+        type = "link") - offset
+    
+    # Calculate empirical prediction
+    # quantiles
+    probs = c(0.05, 0.2, 0.3, 0.4, 0.5, 0.6, 
+        0.7, 0.8, 0.95)
+    cred <- sapply(1:NCOL(preds), function(n) quantile(preds[, 
+        n], probs = probs))
+    
+    # Plot expected function posterior
+    # intervals (40-60%) and medians in
+    # varying colours per lag
+    pred_upper <- cred[4, ]
+    pred_lower <- cred[6, ]
+    pred_vals <- seq(min(data_train$mintemp), 
+        max(data_train$mintemp), length.out = length(newdata$year))
+    polygon(c(pred_vals, rev(pred_vals)), 
+        c(pred_upper, rev(pred_lower)), col = scales::alpha(cols[i], 
+            0.6), border = scales::alpha(cols[i], 
+            0.7))
+    lines(pred_vals, cred[5, ], col = scales::alpha(cols[i], 
+        0.8), lwd = 2.5)
+}
+abline(h = 0, lty = "dashed")
+legend("topleft", legend = paste0("lag", 
+    seq(0, 5)), bg = "white", bty = "n", 
+    col = cols, lty = 1, lwd = 6)
+

+

This plot demonstrates how the effect of mintemp is +expected to change over different exposure lags, with the 3 - 5 month +lags showing more of a cyclic seasonal pattern (catches expected to +increase in the summer and autumn, roughly 3 - 5 months following cold +minimum winter temperatures) while the recent lags (lags 0 and 1) +demonstrate a more linear response function (catches broadly increasing +as minimum temperature increases). This is hopefully a useful example +for developing a better understanding of how a distributed lag model is +attempting to recreate the data generating process. And here is the same +plot for precipitation, which demonstrates how a u-shaped functional +relationship diminishes toward a flat function at lags 2 - 5 (though +this effect is clearly less important in the model than the mintemp * +lag effect above)

+
newdata <- data_test
+newdata$year <- rep(0, length(newdata$year))
+newdata$season <- rep(0, length(newdata$season))
+newdata$mintemp <- matrix(0, ncol = ncol(newdata$mintemp), 
+    nrow = nrow(newdata$mintemp))
+newdata$precip <- matrix(0, ncol = ncol(newdata$precip), 
+    nrow = nrow(newdata$precip))
+preds <- predict(mod4, series = 1, newdata = newdata, 
+    type = "link")
+offset <- mean(preds)
+plot(1, type = "n", xlab = "Precipitation", 
+    ylab = "Predicted response function", 
+    xlim = c(min(data_train$precip), max(data_train$precip)), 
+    ylim = c(-1.6, 1.6))
+
+for (i in 1:6) {
+    newdata$precip <- matrix(0, ncol = ncol(newdata$precip), 
+        nrow = nrow(newdata$precip))
+    newdata$precip[, i] <- seq(min(data_train$precip), 
+        max(data_train$precip), length.out = length(newdata$year))
+    preds <- predict(mod4, series = 1, newdata = newdata, 
+        type = "link") - offset
+    probs = c(0.05, 0.2, 0.3, 0.4, 0.5, 0.6, 
+        0.7, 0.8, 0.95)
+    cred <- sapply(1:NCOL(preds), function(n) quantile(preds[, 
+        n], probs = probs))
+    pred_upper <- cred[4, ]
+    pred_lower <- cred[6, ]
+    pred_vals <- seq(min(data_train$precip), 
+        max(data_train$precip), length.out = length(newdata$year))
+    polygon(c(pred_vals, rev(pred_vals)), 
+        c(pred_upper, rev(pred_lower)), col = scales::alpha(cols[i], 
+            0.6), border = scales::alpha(cols[i], 
+            0.7))
+    lines(pred_vals, cred[5, ], col = scales::alpha(cols[i], 
+        0.8), lwd = 2.5)
+}
+abline(h = 0, lty = "dashed")
+legend("topleft", legend = paste0("lag", 
+    seq(0, 5)), bg = "white", bty = "n", 
+    col = cols, lty = 1, lwd = 6)
+

+

All of the usual functions in mvgam can also be used for +list data objects and for models fitted with Stan, so long +as they contain the necessary fields series, +season and year. For example, posterior +retrodictive checks for the in-sample training period:

+
ppc(mod4, series = 1, type = "cdf")
+

+

and predictive checks for the out of sample forecast period (which +demonstrates how the model tends to overpredict for the forecast period +in this particular example):

+
ppc(mod4, data_test = data_test, series = 1, 
+    type = "cdf")
+

+

Logical next steps for interrogating this model would be to trial +different trend types (i.e. random walk), replace the distributed lag +function for precip with a standard smooth function (that +does not include lag interactions, as clearly the model above indicates +that these are not supported) and inspect whether different covariates +(such as ndvi or maxtemp) might play a role in +modulating catches of PP. Finally, once we are satisfied +that we have a well-performing model that we can understand and +interrogate, we could expand up to a multivariate model by including +other species as response variables. This would allow us to capture any +possible unobserved dependencies in the catches of multiple co-occurring +species in a single unified modelling framework

+ + + + +
+ + + + + + + + + + + + + + + diff --git a/NEON_manuscript/Case studies/rodents_data.csv b/NEON_manuscript/Case studies/rodents_data.csv index 25f7340e..6583c2d5 100644 --- a/NEON_manuscript/Case studies/rodents_data.csv +++ b/NEON_manuscript/Case studies/rodents_data.csv @@ -1,501 +1,501 @@ -"moon","ntraps","nplots","BA","DM","DO","DS","NA.","OL","OT","PB","PE","PF","PH","PL","PM","PP","RF","RM","RO","SF","SH","SO","total","moondate","period","censusdate","year","month","mintemp","maxtemp","meantemp","precipitation","ndvi","source" -28,196,4,0,6,0,3,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,"1979-09-21",27,"1979-09-22",1979,9,NA,NA,NA,NA,NA,NA -29,196,4,0,6,1,6,1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,18,"1979-10-20",28,"1979-10-24",1979,10,NA,NA,NA,NA,NA,NA -30,196,4,0,9,1,4,2,0,3,0,0,0,0,0,0,1,0,2,0,0,0,0,22,"1979-11-19",29,"1979-11-17",1979,11,NA,NA,NA,NA,NA,NA -31,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,"1979-12-19",NA,NA,1979,12,NA,NA,NA,NA,NA,NA -32,196,4,0,16,1,4,2,5,2,0,0,2,0,0,0,0,0,1,0,0,0,0,33,"1980-01-17",30,"1980-01-15",1980,1,-9.4,21.5,5.38235294117647,3.81,NA,"hist" -33,196,4,0,4,2,2,1,3,0,0,0,2,0,0,0,0,0,1,0,0,0,0,15,"1980-02-16",31,"1980-02-24",1980,2,-10,23,5.16666666666667,38.862,NA,"hist" -34,196,4,0,7,1,7,0,2,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,"1980-03-16",32,"1980-03-21",1980,3,1,24.5,10.651724137931,7.874,NA,"hist" -35,196,4,0,4,1,7,0,3,1,0,0,1,0,0,0,0,0,0,0,0,0,0,17,"1980-04-14",33,"1980-04-17",1980,4,-2,29,11.6896551724138,0,NA,"hist" -36,196,4,0,4,2,10,2,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,20,"1980-05-14",34,"1980-05-17",1980,5,3,31,18.67,1.016,NA,"hist" -37,196,4,0,8,4,11,0,1,2,0,0,1,0,0,0,0,0,0,0,0,0,0,27,"1980-06-12",35,"1980-06-22",1980,6,2,39,25.6344827586207,8.382,NA,"hist" -38,196,4,0,7,4,10,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,24,"1980-07-11",36,"1980-07-21",1980,7,18,41,28.3551724137931,13.462,NA,"hist" 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