diff --git a/docs/articles/mvgam_overview.html b/docs/articles/mvgam_overview.html index b5008e2a..480b2fbf 100644 --- a/docs/articles/mvgam_overview.html +++ b/docs/articles/mvgam_overview.html @@ -77,7 +77,7 @@

Nicholas J Clark

-

2023-11-22

+

2023-11-23

Source: vignettes/mvgam_overview.Rmd
mvgam_overview.Rmd
@@ -384,7 +384,16 @@

Example time series datadplyr::glimpse(portal_data) 
## Rows: 199
 ## Columns: 10
-## $ moon          <int> 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 3…## $ DM            <int> 10, 14, 9, NA, 15, NA, NA, 9, 5, 8, NA, 14, 7, NA, NA, 9…## $ DO            <int> 6, 8, 1, NA, 8, NA, NA, 3, 3, 4, NA, 3, 8, NA, NA, 3, NA…## $ PP            <int> 0, 1, 2, NA, 10, NA, NA, 16, 18, 12, NA, 3, 2, NA, NA, 1…## $ OT            <int> 2, 0, 1, NA, 1, NA, NA, 1, 0, 0, NA, 2, 1, NA, NA, 1, NA…## $ year          <int> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 20…## $ month         <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6,…## $ mintemp       <dbl> -9.710, -5.924, -0.220, 1.931, 6.568, 11.590, 14.370, 16…## $ precipitation <dbl> 37.8, 8.7, 43.5, 23.9, 0.9, 1.4, 20.3, 91.0, 60.5, 25.2,…## $ ndvi          <dbl> 1.4658889, 1.5585069, 1.3378172, 1.6589129, 1.8536561, 1…
+## $ moon <int> 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 3… +## $ DM <int> 10, 14, 9, NA, 15, NA, NA, 9, 5, 8, NA, 14, 7, NA, NA, 9… +## $ DO <int> 6, 8, 1, NA, 8, NA, NA, 3, 3, 4, NA, 3, 8, NA, NA, 3, NA… +## $ PP <int> 0, 1, 2, NA, 10, NA, NA, 16, 18, 12, NA, 3, 2, NA, NA, 1… +## $ OT <int> 2, 0, 1, NA, 1, NA, NA, 1, 0, 0, NA, 2, 1, NA, NA, 1, NA… +## $ year <int> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 20… +## $ month <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6,… +## $ mintemp <dbl> -9.710, -5.924, -0.220, 1.931, 6.568, 11.590, 14.370, 16… +## $ precipitation <dbl> 37.8, 8.7, 43.5, 23.9, 0.9, 1.4, 20.3, 91.0, 60.5, 25.2,… +## $ ndvi <dbl> 1.4658889, 1.5585069, 1.3378172, 1.6589129, 1.8536561, 1…

We will focus analyses on the time series of captures for one specific rodent species, the Desert Pocket Mouse Chaetodipus penicillatus. This species is interesting in that it goes into a @@ -406,17 +415,17 @@

Manipulating data for modellinghead(data$data_train, 12) 
##    y season year   series time
 ## 1  0      1    1 series_1    1
-## 2  1      1    1 series_2    1
-## 3  1      1    1 series_3    1
-## 4  1      1    1 series_4    1
-## 5  1      2    1 series_1    2
+## 2  0      1    1 series_2    1
+## 3  2      1    1 series_3    1
+## 4  2      1    1 series_4    1
+## 5  2      2    1 series_1    2
 ## 6  1      2    1 series_2    2
-## 7  1      2    1 series_3    2
+## 7  2      2    1 series_3    2
 ## 8  0      2    1 series_4    2
-## 9  1      3    1 series_1    3
-## 10 1      3    1 series_2    3
-## 11 1      3    1 series_3    3
-## 12 0      3    1 series_4    3
+## 9 2 3 1 series_1 3 +## 10 7 3 1 series_2 3 +## 11 8 3 1 series_3 3 +## 12 4 3 1 series_4 3

Notice how we have four different time series in these simulated data, but we do not spread the outcome values into different columns. Rather, there is only a single column for the outcome variable, labelled @@ -471,7 +480,12 @@

Manipulating data for modellingdplyr::glimpse(model_data) 
## Rows: 199
 ## Columns: 6
-## $ series  <fct> PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP…## $ year    <int> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 20…## $ time    <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,…## $ count   <int> 0, 1, 2, NA, 10, NA, NA, 16, 18, 12, NA, 3, 2, NA, NA, 13, NA,…## $ mintemp <dbl> -9.710, -5.924, -0.220, 1.931, 6.568, 11.590, 14.370, 16.520, …## $ ndvi    <dbl> 1.4658889, 1.5585069, 1.3378172, 1.6589129, 1.8536561, 1.76132…
+## $ series <fct> PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP… +## $ year <int> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 20… +## $ time <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,… +## $ count <int> 0, 1, 2, NA, 10, NA, NA, 16, 18, 12, NA, 3, 2, NA, NA, 13, NA,… +## $ mintemp <dbl> -9.710, -5.924, -0.220, 1.931, 6.568, 11.590, 14.370, 16.520, … +## $ ndvi <dbl> 1.4658889, 1.5585069, 1.3378172, 1.6589129, 1.8536561, 1.76132…

You can also summarize multiple variables, which is helpful to search for data ranges and identify missing values

@@ -547,7 +561,13 @@ 
GLMs with temporal random effectsdplyr::glimpse(model_data)
## Rows: 199
 ## Columns: 7
-## $ series   <fct> PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, P…## $ year     <int> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2…## $ time     <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18…## $ count    <int> 0, 1, 2, NA, 10, NA, NA, 16, 18, 12, NA, 3, 2, NA, NA, 13, NA…## $ mintemp  <dbl> -9.710, -5.924, -0.220, 1.931, 6.568, 11.590, 14.370, 16.520,…## $ ndvi     <dbl> 1.4658889, 1.5585069, 1.3378172, 1.6589129, 1.8536561, 1.7613…## $ year_fac <fct> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2…
+## $ series <fct> PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, PP, P… +## $ year <int> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2… +## $ time <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18… +## $ count <int> 0, 1, 2, NA, 10, NA, NA, 16, 18, 12, NA, 3, 2, NA, NA, 13, NA… +## $ mintemp <dbl> -9.710, -5.924, -0.220, 1.931, 6.568, 11.590, 14.370, 16.520,… +## $ ndvi <dbl> 1.4658889, 1.5585069, 1.3378172, 1.6589129, 1.8536561, 1.7613… +## $ year_fac <fct> 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2004, 2… 
 levels(model_data$year_fac)
##  [1] "2004" "2005" "2006" "2007" "2008" "2009" "2010" "2011" "2012" "2013"
@@ -598,8 +618,8 @@ 
GLMs with temporal random effects## 1             vector[1] mu_raw;            1 s(year_fac) pop mean
 ## 2 vector<lower=0>[1] sigma_raw;            1   s(year_fac) pop sd
 ##                               prior                 example_change
-## 1            mu_raw ~ std_normal();   mu_raw ~ normal(0.77, 0.71);
-## 2 sigma_raw ~ student_t(3, 0, 2.5); sigma_raw ~ exponential(0.95);
+## 1            mu_raw ~ std_normal();   mu_raw ~ normal(0.38, 0.41);
+## 2 sigma_raw ~ student_t(3, 0, 2.5); sigma_raw ~ exponential(0.36);
 ##   new_lowerbound new_upperbound
 ## 1             NA             NA
 ## 2             NA             NA
@@ -635,33 +655,33 @@

GLMs with temporal random effects## ## ## GAM coefficient (beta) estimates: -## 2.5% 50% 97.5% Rhat n_eff -## s(year_fac).1 1.80 2.1 2.3 1.00 3249 -## s(year_fac).2 2.50 2.7 2.8 1.00 3312 -## s(year_fac).3 3.00 3.1 3.2 1.00 2722 -## s(year_fac).4 3.10 3.3 3.4 1.00 3079 -## s(year_fac).5 1.90 2.1 2.3 1.00 3407 -## s(year_fac).6 1.50 1.8 2.0 1.00 2531 -## s(year_fac).7 1.80 2.0 2.3 1.00 2370 -## s(year_fac).8 2.80 3.0 3.1 1.00 2926 -## s(year_fac).9 3.10 3.2 3.4 1.00 2730 -## s(year_fac).10 2.60 2.8 2.9 1.00 3159 -## s(year_fac).11 3.00 3.1 3.2 1.00 2881 -## s(year_fac).12 3.10 3.2 3.3 1.00 2495 -## s(year_fac).13 2.00 2.3 2.5 1.00 2640 -## s(year_fac).14 2.50 2.6 2.8 1.00 2965 -## s(year_fac).15 1.90 2.2 2.4 1.00 2659 -## s(year_fac).16 1.90 2.1 2.3 1.00 3067 -## s(year_fac).17 -0.32 1.1 1.9 1.01 394 +## 2.5% 50% 97.5% Rhat n_eff +## s(year_fac).1 1.8 2.1 2.3 1.00 2341 +## s(year_fac).2 2.5 2.7 2.8 1.00 3001 +## s(year_fac).3 3.0 3.1 3.2 1.00 2975 +## s(year_fac).4 3.1 3.3 3.4 1.00 3032 +## s(year_fac).5 1.9 2.1 2.3 1.00 3122 +## s(year_fac).6 1.5 1.8 2.0 1.00 2763 +## s(year_fac).7 1.8 2.0 2.3 1.00 2168 +## s(year_fac).8 2.8 3.0 3.1 1.00 3152 +## s(year_fac).9 3.1 3.2 3.4 1.00 2549 +## s(year_fac).10 2.6 2.8 2.9 1.00 2710 +## s(year_fac).11 3.0 3.1 3.2 1.00 3196 +## s(year_fac).12 3.1 3.2 3.3 1.00 2348 +## s(year_fac).13 2.0 2.2 2.4 1.00 2729 +## s(year_fac).14 2.5 2.6 2.8 1.00 3219 +## s(year_fac).15 1.9 2.2 2.4 1.00 2484 +## s(year_fac).16 1.9 2.1 2.3 1.00 2909 +## s(year_fac).17 -0.3 1.0 1.9 1.01 560 ## ## GAM group-level estimates: -## 2.5% 50% 97.5% Rhat n_eff -## mean(s(year_fac)) 2.00 2.50 2.8 1.05 141 -## sd(s(year_fac)) 0.46 0.68 1.2 1.03 116 +## 2.5% 50% 97.5% Rhat n_eff +## mean(s(year_fac)) 2.10 2.4 2.8 1.02 170 +## sd(s(year_fac)) 0.47 0.7 1.2 1.02 241 ## ## Approximate significance of GAM observation smooths: ## edf Chi.sq p-value -## s(year_fac) 13.6 23000 <2e-16 *** +## s(year_fac) 13.8 24510 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## @@ -672,7 +692,7 @@

GLMs with temporal random effects## 0 of 2000 iterations saturated the maximum tree depth of 12 (0%) ## E-FMI indicated no pathological behavior ## -## Samples were drawn using NUTS(diag_e) at Wed Nov 22 12:42:10 PM 2023. +## Samples were drawn using NUTS(diag_e) at Thu Nov 23 8:34:58 AM 2023. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split MCMC chains ## (at convergence, Rhat = 1) @@ -689,7 +709,23 @@

GLMs with temporal random effectsdplyr::glimpse(beta_post) 
## Rows: 2,000
 ## Columns: 17
-## $ `s(year_fac).1`  <dbl> 2.08031, 2.14139, 2.08884, 1.99740, 2.26875, 1.90646,…## $ `s(year_fac).2`  <dbl> 2.60488, 2.52888, 2.74309, 2.59697, 2.77037, 2.57275,…## $ `s(year_fac).3`  <dbl> 3.08624, 3.18546, 2.96344, 3.07401, 3.12205, 2.95258,…## $ `s(year_fac).4`  <dbl> 3.26653, 3.26699, 3.35323, 3.24137, 3.20626, 3.33566,…## $ `s(year_fac).5`  <dbl> 2.15022, 2.24475, 2.06944, 2.04412, 2.26502, 2.18564,…## $ `s(year_fac).6`  <dbl> 1.67474, 1.97130, 1.56335, 1.72251, 1.77394, 1.62831,…## $ `s(year_fac).7`  <dbl> 1.90693, 2.18487, 1.98117, 2.01677, 2.12961, 1.99228,…## $ `s(year_fac).8`  <dbl> 2.94260, 2.95542, 2.97893, 3.04033, 2.90723, 3.09810,…## $ `s(year_fac).9`  <dbl> 3.24186, 3.22605, 3.21179, 3.29233, 3.23729, 3.28691,…## $ `s(year_fac).10` <dbl> 2.79245, 2.72758, 2.81575, 2.72156, 2.81095, 2.79049,…## $ `s(year_fac).11` <dbl> 3.15032, 3.10892, 3.10850, 3.11659, 3.12006, 3.05922,…## $ `s(year_fac).12` <dbl> 3.23851, 3.21248, 3.16899, 3.25140, 3.28649, 3.14034,…## $ `s(year_fac).13` <dbl> 2.39143, 2.24833, 2.44651, 2.32984, 2.18376, 2.41337,…## $ `s(year_fac).14` <dbl> 2.74540, 2.56089, 2.75172, 2.71387, 2.61187, 2.52972,…## $ `s(year_fac).15` <dbl> 2.21686, 2.17691, 2.37968, 2.21372, 2.12467, 2.09551,…## $ `s(year_fac).16` <dbl> 2.08770, 2.04529, 2.10352, 2.11526, 2.00120, 2.08496,…## $ `s(year_fac).17` <dbl> 1.515040, 1.211260, 1.241590, 1.527410, 0.669471, 0.5…
+## $ `s(year_fac).1` <dbl> 2.08212, 2.13563, 2.09978, 1.96775, 2.10025, 2.00987,… +## $ `s(year_fac).2` <dbl> 2.71049, 2.76985, 2.78095, 2.64928, 2.71481, 2.71621,… +## $ `s(year_fac).3` <dbl> 2.99601, 3.19365, 3.13145, 3.07174, 3.07684, 3.09534,… +## $ `s(year_fac).4` <dbl> 3.17865, 3.34493, 3.26995, 3.28116, 3.15369, 3.35802,… +## $ `s(year_fac).5` <dbl> 2.20924, 2.07454, 2.19393, 2.10433, 2.20351, 2.14543,… +## $ `s(year_fac).6` <dbl> 1.75460, 1.86416, 1.84508, 1.72438, 1.73111, 1.92814,… +## $ `s(year_fac).7` <dbl> 1.88514, 2.06802, 1.95780, 1.82762, 2.08620, 2.19353,… +## $ `s(year_fac).8` <dbl> 2.96932, 2.98186, 2.82675, 2.82874, 2.86850, 3.05983,… +## $ `s(year_fac).9` <dbl> 3.30681, 3.24499, 3.21878, 3.21961, 3.16487, 3.34114,… +## $ `s(year_fac).10` <dbl> 2.79073, 2.62311, 2.83885, 2.70922, 2.73095, 2.66744,… +## $ `s(year_fac).11` <dbl> 3.10926, 3.03815, 2.98585, 2.97678, 3.08263, 3.04552,… +## $ `s(year_fac).12` <dbl> 3.13376, 3.28079, 3.18349, 3.20083, 3.13156, 3.30719,… +## $ `s(year_fac).13` <dbl> 2.13703, 2.50151, 2.21896, 2.07110, 2.27643, 2.23768,… +## $ `s(year_fac).14` <dbl> 2.58957, 2.79552, 2.63736, 2.65114, 2.62641, 2.75834,… +## $ `s(year_fac).15` <dbl> 2.16848, 2.24516, 2.33078, 2.17398, 2.17238, 2.16534,… +## $ `s(year_fac).16` <dbl> 2.10648, 2.07510, 2.07197, 1.92124, 1.93862, 2.05727,… +## $ `s(year_fac).17` <dbl> 1.251520, 0.866643, 0.770535, 0.742428, 1.040220, 1.4…

With any model fitted in mvgam, the underlying Stan code can be viewed using the code function:

@@ -810,7 +846,7 @@

## $ test_observations : NULL ## $ test_times : NULL ## $ hindcasts :List of 1 -## ..$ PP: num [1:2000, 1:199] 12 11 10 10 12 7 6 11 7 7 ... +## ..$ PP: num [1:2000, 1:199] 8 11 7 6 13 5 8 11 10 9 ... ## .. ..- attr(*, "dimnames")=List of 2 ## .. .. ..$ : NULL ## .. .. ..$ : chr [1:199] "ypred[1,1]" "ypred[2,1]" "ypred[3,1]" "ypred[4,1]" ... @@ -823,7 +859,7 @@ 

 hc <- hindcast(model1, type = 'link')
 range(hc$hindcasts$PP)
-
## [1] -2.37133  3.48166
+
## [1] -2.40801  3.46010

Objects of class mvgam_forecast have an associated plot function as well:

@@ -875,14 +911,14 @@ 
Automatic forecasting for new dataplot(model1b, type = 'forecast')

## Out of sample DRPS:
-## [1] 182.8079
+## [1] 180.5517

We can also view the test data in the forecast plot to see that the predictions do not capture the temporal variation in the test set

 plot(model1b, type = 'forecast', newdata = data_test)

## Out of sample DRPS:
-## [1] 182.8079
+## [1] 180.5517

As with the hindcast function, we can use the forecast function to automatically extract the posterior distributions for these predictions. This also returns an object of @@ -910,12 +946,12 @@

Automatic forecasting for new data## ..$ PP: int [1:39] NA 0 0 10 3 14 18 NA 28 46 ... ## $ test_times : num [1:39] 161 162 163 164 165 166 167 168 169 170 ... ## $ hindcasts :List of 1 -## ..$ PP: num [1:2000, 1:160] 10 2 10 15 10 5 8 10 10 10 ... +## ..$ PP: num [1:2000, 1:160] 3 10 10 2 5 11 5 5 11 16 ... ## .. ..- attr(*, "dimnames")=List of 2 ## .. .. ..$ : NULL ## .. .. ..$ : chr [1:160] "ypred[1,1]" "ypred[2,1]" "ypred[3,1]" "ypred[4,1]" ... ## $ forecasts :List of 1 -## ..$ PP: num [1:2000, 1:39] 6 5 11 6 7 10 16 8 14 9 ... +## ..$ PP: num [1:2000, 1:39] 9 13 10 14 10 12 9 16 13 7 ... ## .. ..- attr(*, "dimnames")=List of 2 ## .. .. ..$ : NULL ## .. .. ..$ : chr [1:39] "ypred[161,1]" "ypred[162,1]" "ypred[163,1]" "ypred[164,1]" ... @@ -972,33 +1008,33 @@

Adding predictors as “fixed” eff ## ## GAM coefficient (beta) estimates: ## 2.5% 50% 97.5% Rhat n_eff -## ndvi 0.32 0.39 0.46 1 1661 -## s(year_fac).1 1.10 1.40 1.70 1 2466 -## s(year_fac).2 1.80 2.00 2.20 1 2410 -## s(year_fac).3 2.20 2.40 2.60 1 2149 -## s(year_fac).4 2.30 2.50 2.70 1 1994 -## s(year_fac).5 1.20 1.40 1.60 1 2250 -## s(year_fac).6 1.00 1.30 1.50 1 2659 -## s(year_fac).7 1.10 1.40 1.70 1 2015 -## s(year_fac).8 2.10 2.30 2.50 1 2224 -## s(year_fac).9 2.70 2.90 3.00 1 2627 -## s(year_fac).10 2.00 2.20 2.40 1 2601 -## s(year_fac).11 2.30 2.40 2.60 1 2050 -## s(year_fac).12 2.50 2.70 2.80 1 2144 -## s(year_fac).13 1.40 1.60 1.90 1 2548 -## s(year_fac).14 0.57 2.00 3.30 1 1089 -## s(year_fac).15 0.64 2.00 3.40 1 1517 -## s(year_fac).16 0.56 2.00 3.30 1 1509 -## s(year_fac).17 0.63 2.00 3.30 1 1021 +## ndvi 0.33 0.39 0.45 1 1774 +## s(year_fac).1 1.10 1.40 1.70 1 2528 +## s(year_fac).2 1.80 2.00 2.20 1 2279 +## s(year_fac).3 2.20 2.40 2.60 1 2055 +## s(year_fac).4 2.30 2.50 2.70 1 1911 +## s(year_fac).5 1.20 1.40 1.60 1 2351 +## s(year_fac).6 1.00 1.30 1.50 1 2919 +## s(year_fac).7 1.10 1.40 1.70 1 2459 +## s(year_fac).8 2.10 2.30 2.40 1 2153 +## s(year_fac).9 2.70 2.90 3.00 1 2025 +## s(year_fac).10 2.00 2.20 2.40 1 2439 +## s(year_fac).11 2.30 2.40 2.60 1 2031 +## s(year_fac).12 2.50 2.70 2.80 1 2214 +## s(year_fac).13 1.40 1.60 1.80 1 2122 +## s(year_fac).14 0.65 2.00 3.30 1 1055 +## s(year_fac).15 0.54 2.00 3.30 1 1622 +## s(year_fac).16 0.69 2.00 3.20 1 1566 +## s(year_fac).17 0.60 2.00 3.30 1 1186 ## ## GAM group-level estimates: -## 2.5% 50% 97.5% Rhat n_eff -## mean(s(year_fac)) 1.60 2.00 2.3 1.01 307 -## sd(s(year_fac)) 0.41 0.59 1.0 1.00 377 +## 2.5% 50% 97.5% Rhat n_eff +## mean(s(year_fac)) 1.60 2.0 2.3 1.01 332 +## sd(s(year_fac)) 0.41 0.6 1.0 1.01 463 ## ## Approximate significance of GAM observation smooths: ## edf Chi.sq p-value -## s(year_fac) 10.4 2952 <2e-16 *** +## s(year_fac) 10.6 2792 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## @@ -1009,7 +1045,7 @@

Adding predictors as “fixed” eff ## 0 of 2000 iterations saturated the maximum tree depth of 12 (0%) ## E-FMI indicated no pathological behavior ## -## Samples were drawn using NUTS(diag_e) at Wed Nov 22 12:43:14 PM 2023. +## Samples were drawn using NUTS(diag_e) at Thu Nov 23 8:36:12 AM 2023. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split MCMC chains ## (at convergence, Rhat = 1) @@ -1020,24 +1056,24 @@

Adding predictors as “fixed” eff 
 coef(model2)
##                     2.5%      50%     97.5% Rhat n_eff
-## ndvi           0.3228257 0.391166 0.4593855    1  1661
-## s(year_fac).1  1.1404162 1.400395 1.6634465    1  2466
-## s(year_fac).2  1.7973598 1.995695 2.1988837    1  2410
-## s(year_fac).3  2.1774477 2.376875 2.5715922    1  2149
-## s(year_fac).4  2.3233060 2.502500 2.6789605    1  1994
-## s(year_fac).5  1.1939365 1.422650 1.6418732    1  2250
-## s(year_fac).6  1.0095840 1.274575 1.5165723    1  2659
-## s(year_fac).7  1.1392430 1.411065 1.6643688    1  2015
-## s(year_fac).8  2.0748615 2.265685 2.4505097    1  2224
-## s(year_fac).9  2.7139455 2.856145 2.9870925    1  2627
-## s(year_fac).10 1.9799230 2.187460 2.3882100    1  2601
-## s(year_fac).11 2.2644308 2.433500 2.6032955    1  2050
-## s(year_fac).12 2.5326728 2.687635 2.8342965    1  2144
-## s(year_fac).13 1.3667295 1.612645 1.8541005    1  2548
-## s(year_fac).14 0.5727090 1.957480 3.2950085    1  1089
-## s(year_fac).15 0.6379996 1.997195 3.3552087    1  1517
-## s(year_fac).16 0.5632498 1.956545 3.2594935    1  1509
-## s(year_fac).17 0.6325674 1.982975 3.2627203    1  1021
+## ndvi 0.3254700 0.388940 0.4545993 1 1774 +## s(year_fac).1 1.1324400 1.404140 1.6680297 1 2528 +## s(year_fac).2 1.8013830 2.001210 2.1912828 1 2279 +## s(year_fac).3 2.1858860 2.384775 2.5708725 1 2055 +## s(year_fac).4 2.3185475 2.508285 2.6786000 1 1911 +## s(year_fac).5 1.2007655 1.425115 1.6407507 1 2351 +## s(year_fac).6 1.0184522 1.277490 1.5145570 1 2919 +## s(year_fac).7 1.1476518 1.415705 1.6663517 1 2459 +## s(year_fac).8 2.0888960 2.272105 2.4487602 1 2153 +## s(year_fac).9 2.7135970 2.856790 2.9818367 1 2025 +## s(year_fac).10 1.9795910 2.183405 2.3811530 1 2439 +## s(year_fac).11 2.2688975 2.434785 2.6082938 1 2031 +## s(year_fac).12 2.5372467 2.695205 2.8332955 1 2214 +## s(year_fac).13 1.3754218 1.615400 1.8429202 1 2122 +## s(year_fac).14 0.6519133 1.971190 3.3202952 1 1055 +## s(year_fac).15 0.5373446 1.969180 3.2645155 1 1622 +## s(year_fac).16 0.6892467 1.960490 3.2228937 1 1566 +## s(year_fac).17 0.5975932 2.015125 3.3076672 1 1186

Look at the estimated effect of ndvi using plot.mvgam with type = 'pterms'

@@ -1053,7 +1089,24 @@ 
Adding predictors as “fixed” eff
 dplyr::glimpse(beta_post)
## Rows: 2,000
 ## Columns: 18
-## $ ndvi             <dbl> 0.387436, 0.409739, 0.394457, 0.431873, 0.396274, 0.3…## $ `s(year_fac).1`  <dbl> 1.41734, 1.49543, 1.54026, 1.24608, 1.33257, 1.41159,…## $ `s(year_fac).2`  <dbl> 1.95613, 1.97027, 2.00336, 1.94539, 1.92974, 1.98538,…## $ `s(year_fac).3`  <dbl> 2.41307, 2.38483, 2.39814, 2.27193, 2.37262, 2.37026,…## $ `s(year_fac).4`  <dbl> 2.39442, 2.33973, 2.50900, 2.39682, 2.50362, 2.51380,…## $ `s(year_fac).5`  <dbl> 1.38814, 1.35694, 1.33459, 1.48374, 1.35462, 1.59126,…## $ `s(year_fac).6`  <dbl> 1.24112, 1.39580, 1.21348, 1.32198, 1.23243, 1.33269,…## $ `s(year_fac).7`  <dbl> 1.31151, 1.29583, 1.36247, 1.32749, 1.22643, 1.30892,…## $ `s(year_fac).8`  <dbl> 2.28249, 2.24587, 2.33273, 2.13411, 2.34726, 2.30157,…## $ `s(year_fac).9`  <dbl> 2.75439, 2.77804, 2.92003, 2.70158, 2.96359, 2.73798,…## $ `s(year_fac).10` <dbl> 2.12669, 2.18653, 2.20835, 2.16660, 2.24748, 2.26067,…## $ `s(year_fac).11` <dbl> 2.50111, 2.43007, 2.41899, 2.39531, 2.33690, 2.52664,…## $ `s(year_fac).12` <dbl> 2.65021, 2.70536, 2.59644, 2.72438, 2.62080, 2.75586,…## $ `s(year_fac).13` <dbl> 1.63676, 1.66940, 1.69642, 1.44458, 1.68236, 1.73516,…## $ `s(year_fac).14` <dbl> 2.00488, 1.86698, 1.72716, 1.33882, 1.50478, 2.20462,…## $ `s(year_fac).15` <dbl> 1.62293, 1.73438, 1.90062, 1.82146, 1.66189, 2.03626,…## $ `s(year_fac).16` <dbl> 1.472810, 1.407540, 1.365660, 1.079730, 1.394120, 2.3…## $ `s(year_fac).17` <dbl> 1.93064, 2.10818, 1.44301, 2.19165, 1.74017, 2.22774,…
+## $ ndvi <dbl> 0.424042, 0.415251, 0.376440, 0.366467, 0.361792, 0.3… +## $ `s(year_fac).1` <dbl> 1.22925, 1.24749, 1.47788, 1.20037, 1.47978, 1.35059,… +## $ `s(year_fac).2` <dbl> 1.98318, 1.87325, 1.98441, 2.06521, 2.04639, 2.00794,… +## $ `s(year_fac).3` <dbl> 2.18793, 2.36020, 2.29305, 2.50610, 2.49024, 2.41114,… +## $ `s(year_fac).4` <dbl> 2.39819, 2.41437, 2.52095, 2.59109, 2.47269, 2.66913,… +## $ `s(year_fac).5` <dbl> 1.29312, 1.19872, 1.51329, 1.36905, 1.62790, 1.55644,… +## $ `s(year_fac).6` <dbl> 1.20561, 1.01872, 1.25503, 1.25678, 1.10984, 1.13588,… +## $ `s(year_fac).7` <dbl> 1.18002, 1.47339, 1.21908, 1.51392, 1.60791, 1.47573,… +## $ `s(year_fac).8` <dbl> 1.98866, 2.28440, 2.18800, 2.33697, 2.22476, 2.32740,… +## $ `s(year_fac).9` <dbl> 2.84229, 2.75877, 2.87801, 2.93136, 2.88772, 2.86256,… +## $ `s(year_fac).10` <dbl> 2.10992, 2.04506, 2.23292, 2.26566, 2.30330, 2.27192,… +## $ `s(year_fac).11` <dbl> 2.33570, 2.32394, 2.45942, 2.50878, 2.43429, 2.46903,… +## $ `s(year_fac).12` <dbl> 2.71171, 2.54388, 2.76026, 2.72311, 2.83329, 2.70795,… +## $ `s(year_fac).13` <dbl> 1.33811, 1.55696, 1.59988, 1.67098, 1.76702, 1.54182,… +## $ `s(year_fac).14` <dbl> 1.429890, 2.359990, 2.555390, 2.251150, 2.325550, 2.6… +## $ `s(year_fac).15` <dbl> 1.1201100, 1.7265900, 2.2085200, 1.2997700, 1.4786000… +## $ `s(year_fac).16` <dbl> 1.447770, 1.581720, 2.045710, 0.965276, 0.979640, 1.8… +## $ `s(year_fac).17` <dbl> 1.560560, 2.003960, 1.462440, 2.417490, 2.196010, 2.1…

The posterior distribution for the effect of ndvi is stored in the ndvi column. A quick histogram confirms our inference that log(counts) respond positively to increases @@ -1190,26 +1243,26 @@

Adding predictors as smooths## ## GAM coefficient (beta) estimates: ## 2.5% 50% 97.5% Rhat n_eff -## (Intercept) 2.00 2.10 2.200 1.01 932 -## ndvi 0.26 0.33 0.400 1.01 968 -## s(time).1 -2.10 -1.10 -0.033 1.00 504 -## s(time).2 0.41 1.30 2.400 1.01 357 -## s(time).3 -0.54 0.43 1.600 1.01 354 -## s(time).4 1.50 2.40 3.700 1.01 340 -## s(time).5 -1.20 -0.23 0.910 1.01 347 -## s(time).6 -0.65 0.33 1.600 1.01 346 -## s(time).7 -1.50 -0.55 0.630 1.01 387 -## s(time).8 0.54 1.40 2.600 1.01 326 -## s(time).9 1.10 2.00 3.300 1.01 340 -## s(time).10 -0.41 0.50 1.700 1.01 355 -## s(time).11 0.77 1.70 2.900 1.01 333 -## s(time).12 0.59 1.50 2.600 1.01 344 -## s(time).13 -1.20 -0.36 0.690 1.01 407 -## s(time).14 -7.90 -4.10 -0.970 1.01 435 +## (Intercept) 2.00 2.10 2.200 1.00 888 +## ndvi 0.26 0.33 0.390 1.00 868 +## s(time).1 -2.10 -1.10 0.026 1.01 476 +## s(time).2 0.41 1.30 2.300 1.01 410 +## s(time).3 -0.54 0.47 1.500 1.01 357 +## s(time).4 1.50 2.50 3.500 1.01 351 +## s(time).5 -1.20 -0.18 0.850 1.01 384 +## s(time).6 -0.60 0.37 1.500 1.01 383 +## s(time).7 -1.50 -0.50 0.570 1.01 392 +## s(time).8 0.60 1.50 2.500 1.01 346 +## s(time).9 1.20 2.10 3.100 1.01 353 +## s(time).10 -0.40 0.56 1.500 1.01 369 +## s(time).11 0.83 1.80 2.800 1.01 344 +## s(time).12 0.65 1.50 2.400 1.01 359 +## s(time).13 -1.20 -0.31 0.620 1.00 523 +## s(time).14 -7.40 -4.10 -1.100 1.01 451 ## ## Approximate significance of GAM observation smooths: ## edf Chi.sq p-value -## s(time) 9.32 640 <2e-16 *** +## s(time) 9.76 722 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## @@ -1220,7 +1273,7 @@

Adding predictors as smooths## 0 of 2000 iterations saturated the maximum tree depth of 12 (0%) ## E-FMI indicated no pathological behavior ## -## Samples were drawn using NUTS(diag_e) at Wed Nov 22 12:44:04 PM 2023. +## Samples were drawn using NUTS(diag_e) at Thu Nov 23 8:37:11 AM 2023. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split MCMC chains ## (at convergence, Rhat = 1) @@ -1347,7 +1400,7 @@

Latent dynamics in mvgamplot(model3, type = 'forecast', newdata = data_test) 

## Out of sample DRPS:
-## [1] 286.0611
+## [1] 287.1214

Why is this happening? The forecasts are driven almost entirely by variation in the temporal spline, which is extrapolating linearly forever beyond the edge of the training data. Any slight @@ -1430,33 +1483,33 @@

Latent dynamics in mvgam## ## GAM coefficient (beta) estimates: ## 2.5% 50% 97.5% Rhat n_eff -## (Intercept) 1.200 2.0000 2.800 1.12 39 -## s(ndvi).1 -0.160 -0.0086 0.083 1.00 476 -## s(ndvi).2 -0.190 0.0170 0.340 1.00 349 -## s(ndvi).3 -0.056 -0.0016 0.052 1.00 738 -## s(ndvi).4 -0.290 0.1200 1.400 1.01 255 -## s(ndvi).5 -0.074 0.1500 0.340 1.00 395 +## (Intercept) 1.100 1.8000 2.400 1.20 18 +## s(ndvi).1 -0.160 -0.0090 0.079 1.01 444 +## s(ndvi).2 -0.130 0.0160 0.240 1.00 518 +## s(ndvi).3 -0.043 -0.0011 0.049 1.00 850 +## s(ndvi).4 -0.230 0.1200 1.100 1.02 288 +## s(ndvi).5 -0.073 0.1500 0.360 1.01 421 ## ## Approximate significance of GAM observation smooths: ## edf Chi.sq p-value -## s(ndvi) 1.25 83.8 0.087 . +## s(ndvi) 1.09 85.9 0.065 . ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Latent trend parameter AR estimates: ## 2.5% 50% 97.5% Rhat n_eff -## ar1[1] 0.70 0.81 0.91 1.01 345 -## sigma[1] 0.67 0.80 0.96 1.00 424 +## ar1[1] 0.71 0.82 0.92 1.03 98 +## sigma[1] 0.67 0.80 0.97 1.02 330 ## ## Stan MCMC diagnostics: ## n_eff / iter looks reasonable for all parameters -## Rhats above 1.05 found for 84 parameters +## Rhats above 1.05 found for 114 parameters ## *Diagnose further to investigate why the chains have not mixed ## 0 of 2000 iterations ended with a divergence (0%) ## 0 of 2000 iterations saturated the maximum tree depth of 12 (0%) ## E-FMI indicated no pathological behavior ## -## Samples were drawn using NUTS(diag_e) at Wed Nov 22 12:45:21 PM 2023. +## Samples were drawn using NUTS(diag_e) at Thu Nov 23 8:38:40 AM 2023. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split MCMC chains ## (at convergence, Rhat = 1) @@ -1472,7 +1525,7 @@

Latent dynamics in mvgamplot(model4, type = 'forecast', newdata = data_test) 

## Out of sample DRPS:
-## [1] 154.5162
+## [1] 149.1606

The trend is evolving as an AR1 process, which we can also view:

 plot(model4, type = 'trend', newdata = data_test)
@@ -1487,7 +1540,7 @@

Latent dynamics in mvgam## Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details. 
##        elpd_diff se_diff
 ## model4    0.0       0.0 
-## model3 -563.9      66.3
+## model3 -559.8 66.5

The higher estimated log predictive density (ELPD) value for the dynamic model suggests it provides a better fit to the in-sample data.

@@ -1502,7 +1555,7 @@

Latent dynamics in mvgamscore_mod3 <- score(fc_mod3, score = 'drps') score_mod4 <- score(fc_mod4, score = 'drps') sum(score_mod4$PP$score, na.rm = TRUE) - sum(score_mod3$PP$score, na.rm = TRUE) -
## [1] -131.5449
+
## [1] -137.9608

A strongly negative value here suggests the score for the dynamic model (model 4) is much smaller than the score for the model with a smooth function of time (model 3)

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