forked from stormwaterheatmap/s8Regressions
-
Notifications
You must be signed in to change notification settings - Fork 0
/
MixedEffects-copper.Rmd
752 lines (565 loc) · 19.6 KB
/
MixedEffects-copper.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
---
title: "Copper Regression"
output:
html_notebook: default
---
<!-- --- -->
<!-- title: 'Watershed Regression - Part 1: Linear Mixed Models (adapted to GLS Survival -->
<!-- Analysis) ' -->
<!-- author: "Christian Nilsen" -->
<!-- output: -->
<!-- html_notebook: -->
<!-- toc: yes -->
<!-- highlight: zenburn -->
<!-- df_print: paged -->
<!-- fig_caption: yes -->
<!-- html_document: -->
<!-- toc: yes -->
<!-- highlight: zenburn -->
<!-- df_print: paged -->
<!-- word_document: -->
<!-- toc: yes -->
<!-- --- -->
# License
Copyright (c) 2020 Geosyntec Consultants, Inc.
[Mozilla Public License Version 2.0](https://choosealicense.com/licenses/mpl-2.0/)
This software is provided "as is", without warranty of any kind, express or implied, including but not limited to the warranties of merchantability, fitness for a particular purpose and noninfringement. In no event shall the authors or copyright holders be liable for any claim, damages or other liability, whether in an action of contract, tort or otherwise, arising from, out of or in connection with the software or the use or other dealings in the software.
# Introduction
This notebook uses spatial data, rainfall data, and outfall monitoring data to develop predictive linear regression relationships related to concentrations of total suspended solids (TSS) in urban stormwater.
Although land use is a commonly used predictor for runoff water quality, recent investigations have found that other landscape and meteorologic variables are more correlated with pollutant concentrations.
Here, we compare predictive models using remote sensed landscape data, precipitation, and other factors to models relying on land use.
<!-- clear workspace and load libraries. (code not shown in html file ) -->
```{r knitr_init, cache=FALSE, include=FALSE}
# load packages
#
#code for word export-------------
library(officedown)
library(officer)
fp <- fp_par(
text.align = "center",
padding.bottom = 20, padding.top = 120,
border.bottom = fp_border()
)
ft <- fp_text(shading.color = "#EFEFEF", bold = TRUE)
#------------
#---libraries
library(car)
library(caret)
library(fitdistrplus)
library(gamlss)
library(glmmLasso)
library(ggrepel)
library(ggthemr)
library(knitr)
library(lattice)
library(lubridate)
library(magrittr)
library(MASS)
library(Metrics)
library(NADA)
library(readr)
library(PerformanceAnalytics)
library(psych)
library(scam)
library(sjPlot)
library(survival)
library(texreg)
library(tidyverse)
library(visreg)
#options --------------
opts_chunk$set(
prompt = FALSE,
message = FALSE,
warning = FALSE
)
options(scipen = 1, digits = 3)
set.seed(50)
ggthemr("fresh")
```
# Methods
## Outfall Monitoring Data
The primary source of measured stormwater data is the S8.D Municipal Stormwater Permit Outfall Data (referred to as the S8 Data in this document) provided by the Washington Department of Ecology (William Hobbs et al. 2015). Special Condition S8.D of the 2007-2012 Phase I Municipal Stormwater Permit required permittees to collect and analyze data to evaluate pollutant loadings of stormwater discharged from different land uses: high density (HD) residential, low density (LD) residential, commercial, and industrial. Phase I Permittees5 collected water quality and flow data, sediment data, and toxicity information from stormwater discharges during storm events.
## Data import and cleaning
The stormwater outfall data is available from Ecology via an open-data api at: https://data.wa.gov/Natural-Resources-Environment/Municipal-Stormwater-Permit-Outfall-Data/d958-q2ci.
A .csv file is saved in ```WatershedRegression/data/S8_data.csv``` Here we import the cleaned dataset.
```{r}
#load data
load("./data/s8_cleaned.rda")
#load helper functions
source("./R/helpers.R")
#Filter for coc
coc <- 'Copper - Water - Total'
units <- 'μg/L'
df.coc <- dplyr::filter(s8data.wPredictors,parameter == coc)
#get rid of NA columns
df.coc <- Filter(function(x)!all(is.na(x)), df.coc)
#add a log transformed response column
df.coc$logConcentration <- log(df.coc$concentration)
```
Most stormwater studies use land use as the main predictor of pollutant concentrations.
The data include four land use categories:
* Industrial (IND)
* Low density residential (LDR)
* High density residential (HDR)
* Commercial (COM)
Collected by 6 Agencies:
* City of Seattle
* City of Tacoma
* King County
* Pierce County
* Snohohmish County
* Port of Tacoma
(Clark County was not included in this analysis. Port of Seattle, likewise was removed due to the small watershed area used for monitoring).
## Fit a distribution for parametric models.
Produce Q-Q plots to aid in distribution selection.
```{r fig.height=5, fig.width=7}
#TODO: Add Survival Object for censored data.
plot_distributions(coc,df.coc)
```
From the above, We will choose the lognormal distribution for fitting. Below in the ```gamlss``` function we will select ```family = LOGNO```.
# Parameter Selection
## Landscape Data
For each watershed contained in the S8 dataset, potentially relevant landscape data was extracted from the following sources below:
| Layer | ID | Source |
|-------------------------|-------------|---------|
| Particulate Matter 2.5μm | pm25 | [van Donkelaar et al. 2018.](https://doi.org/10.7927/H4ZK5DQS)
| Imperviousness | impervious | TNC Puget Sound land cover
|Logarithm of Population Density |logPopulation |[CIESIN - Columbia University Gridded Population of the World, Version 4](https://doi.org/10.7927/H49C6VHW)
|Logarithm of Average Daily Traffic Volume |logTraffic|[INRIX Traffic](https://inrix.com/products/volume)
|Percent Tree Cover | percent_tree_cover | NLCD Database
First center and scale the landscape predictors.
```{r}
# Use the preProcess function from the caret package to generate preprocess elements
predictors <-
c("impervious",
"logTraffic",
"percent_tree_cover",
"pm25",
"population_density"
)
scaled_centered <- scale_and_center(df.coc, predictors)
df.scaled <- scaled_centered$scaled
df.transformed <- scaled_centered$transformed
```
## View Correlations
Generate a correlation matrix chart to visualize correlations of the response (log concentration) to predictors.
```{r}
#Select data for charting
df.toChart <- dplyr::select(df.transformed,
c(logConcentration,
impervious,
population_density,
logTraffic ,percent_tree_cover,
pm25,
)
)
lab = coc
#show kendall correlation coefficients
chart.Correlation(df.toChart,method="kendall",title=(lab))
```
Below we plot the predictors against concentration to investigate predictive relationships.We use p-splines here to show fits.
```{r}
# pivot for plotting
landscape_cols <- pivot_longer(
df.transformed,
c(
impervious,
population_density,
logTraffic,
pm25,
percent_tree_cover
),
names_to = "Predictor",
names_repair = "check_unique",
values_to = "lsvalue"
)
# plot data with p splines
ggplot(landscape_cols, aes(x = lsvalue, y = concentration)) +
geom_smooth(method = "scam", formula = y ~ s(x, k = 4, bs = "ps")) +
facet_wrap(~Predictor, scales = "free_x") +
scale_y_log10() +
theme(
axis.line = element_line(
size = 0.3,
linetype = "solid"
), axis.ticks = element_line(size = 0.7),
panel.background = element_rect(
fill = NA,
colour = "black", size = 0, linetype = "solid"
)
) +
geom_point(aes(color = study_name), fill = NA, alpha = 0.4) +
ggtitle(lab)
```
## Collinearity
To address multicolinearity, we calculate the variance inflation factor (VIF) and remove parameters with with high VIF.
```{r}
vif_results <- sort(vif(lm(log(concentration) ~
impervious+
population_density+
logTraffic+
pm25+
percent_tree_cover,
data=df.transformed)))
(vif_results)
```
All vif values are below 10. We do not need to remove any.
## Categorical Predictors
Other predictors to be investigated are:
* Study ID - Which jurisdiction conducted the study (factor)
* Season - Season of the year data were collected (1-Winter, 2-Spring, 3-Summer, 4-Fall)
* Type - Self reported land use type
* AMC - Antecedent moisture condition (AMC) - Three day antecedent rainfall depth *(Gray, et al 1982)*. (1-Dry, 2-Avg, 3 - Wet).
```{r}
# Coerce data to factors.
factor_cols <- c("season", "study_name", "Location", "type", "AMC")
df.transformed[factor_cols] <- lapply(df.transformed[factor_cols], factor)
factor_cols <- pivot_longer(
df.transformed,
c(season, AMC, type),
names_to = "Predictor",
values_to = "factor_value",
names_repair = "minimal"
)
# plot the data
ggplot(factor_cols, aes(x = factor_value, y = concentration)) +
geom_boxplot() +
facet_wrap(~Predictor, scales = "free_x") +
scale_y_log10() +
geom_jitter(aes(color = study_name), alpha = 0.4) +
ggtitle(lab)
```
## Penalized parameter selection
We employ LASSO (Least Absolute Shrinkage and Selection Operator) regression to select parameters that minimize model complexity. It uses a loss function and penalty parameter (lambda) to estimate regression coefficients.
We start with all potential predictors and interactions.
```{r message=FALSE, warning=FALSE}
#make a formula with all predictors
formula.1 <-
as.formula(
logConcentration ~ logTraffic + impervious + percent_tree_cover + pm25 + population_density
+ logTraffic:impervious + logTraffic:percent_tree_cover + logTraffic:population_density +
logTraffic:pm25 + impervious:percent_tree_cover + impervious:pm25 + impervious:population_density
+ percent_tree_cover:pm25 + percent_tree_cover:population_density + pm25:population_density
)
# Run the function (can take a while for a lot of parameters)
param.select.1 <- selectLasso(formula.1, df.transformed)
# plot the results
ggplot(param.select.1) +
geom_smooth(aes(x = lambda, y = val, group = predictor), color = "light grey", se = FALSE, size = 0.1) +
geom_text_repel(
data = param.select.1 %>%
filter(lambda == last(lambda)),
aes(label = predictor, x = lambda * 0.9, y = val * 1.2), label.size = 0.01
) +
theme(legend.position = "none") +
geom_hline(yintercept = 0)
# display the last lambda
kable(param.select.1 %>%
filter(lambda == last(lambda)))
```
This chart is kind of a mess, but it shows that many predictors quickly go to zero as the loss function increases in penalty. We will remove these and show the predictors that seem to hold up.
Make this clearer:
```{r}
formula.2 <- (as.formula(logConcentration ~ logTraffic + percent_tree_cover + pm25 + impervious))
param.select.2 <- selectLasso(formula.2, df.transformed)
ggplot(param.select.2) +
geom_smooth(aes(x = lambda, y = val, color = predictor), se = FALSE) +
geom_text_repel(
data = param.select.2 %>%
filter(lambda == last(lambda)),
aes(label = predictor, x = lambda * 0.9, y = val * 1.2, color = predictor)
) +
geom_hline(yintercept = 0) +
theme(legend.position = "none")
kable(param.select.2 %>%
filter(lambda == last(lambda)))
```
# Begin Fitting a model
Now it is time to begin with model fitting. We use the ```gamlss``` package, an additive univariate regression model that can include non-linear relationships, random effects, and various parametric distributions.
## Step 1 - Model with no random effects
We begin with our fitting our null hypotheses. We actually have to null hypotheses.
* H.null.a: There is not a relationship between predictors.
* H.null.b: Land use is the best predictor.
```{r echo=TRUE, message=FALSE, warning=FALSE}
# gamlss does automatic centering and scaling. Abandon the transformed data.
rm(df.transformed)
# get our non-transformed data frame.
df.coc <-
dplyr::select(
df.coc,
c(
season,
impervious,
pm25,
percent_tree_cover,
concentration,
logTraffic,
nondetect_flag,
population_density,
study_id,
Location,
study_name,
AMC,
type
)
)
# coerce factors
factor_cols <-
c(
"season",
"study_id",
"study_name",
"Location",
"type",
"AMC"
)
df.coc[factor_cols] <- lapply(df.coc[factor_cols], factor)
# Fit Null Models
model.nulla <- gamlss(concentration ~ 1, data = df.coc, family = LOGNO)
model.nullb <- gamlss(concentration ~ type, data = df.coc, family = LOGNO)
# fit linear models
lm.1 <- gamlss(concentration ~
logTraffic + pm25 + impervious + population_density + percent_tree_cover, data = df.coc, family = LOGNO)
lm.2 <- gamlss(concentration ~
logTraffic + pm25 + percent_tree_cover, data = df.coc, family = LOGNO)
mods.step1 <- list(model.nulla, model.nullb, lm.1, lm.2)
```
### Step 1 Results
```{r echo=TRUE}
##huxtablereg(mods.step1,single.row = T)
screenreg(mods.step1,single.row=T)
```
Based on AIC, model 4 seems to be the best model. It outperforms the null models.
```{r}
best.step1 <- mods.step1[[4]]
```
## Step 2 - Adjust scale parameter
GAMLSS includes a scale parameter `sigma`, which can be used to adjust the distribution standard deviation. Here we try different sigma formulas.
```{r}
step1.formula <- formula(best.step1)
mod.step2.1 <- gamlss(step1.formula,
data = df.coc, family = LOGNO,
trace = F,
sigma.formula = ~study_id
)
mod.step2.2 <- gamlss(step1.formula,
data = df.coc, family = LOGNO,
trace = F,
sigma.formula = ~AMC
)
mod.step2.3 <- gamlss(step1.formula,
data = df.coc, family = LOGNO,
trace = F,
sigma.formula = ~season
)
# create list of models. Include the best from the previous step.
mods.step2 <- list(best.step1, mod.step2.1, mod.step2.2, mod.step2.3)
```
### Step 2 Results
```{r}
# display results
screenreg(mods.step2,single.row=T)
```
Based on AIC, Model 4 from step 2 seems to be the best model.
```{r}
best.step2 <- mods.step2[[4]]
```
## Step 3 - Explore non-linear relationships
```{r}
#get the formula
step2.formula <- formula(best.step2)
# make new formulas by updating previous steps
step3.1.formula <- update(step2.formula, . ~ . - logTraffic - pm25 + pbm(logTraffic) + pbm(pm25))
step3.2.formula <- update(step2.formula, . ~ . - logTraffic + pbm(logTraffic))
step3.3.formula <- update(step2.formula, . ~ . - pm25 + pbm(pm25))
step3.4.formula <- update(step2.formula, . ~ . - logTraffic - pm25 + pbm(logTraffic))
# fit models
mod.step3.1 <- gamlss(
step3.1.formula,
data = df.coc,
family = LOGNO,
trace = F,
sigma.formula = ~season
)
mod.step3.2 <- gamlss(
step3.2.formula,
data = df.coc,
family = LOGNO,
trace = F,
sigma.formula = ~season
)
mod.step3.3 <- gamlss(
step3.3.formula,
data = df.coc,
family = LOGNO,
trace = F,
sigma.formula = ~season
)
mod.step3.4 <- gamlss(
step3.4.formula,
data = df.coc,
family = LOGNO,
trace = F,
sigma.formula = ~season
)
# make a list of models
mods.step3 <- list(best.step2, mod.step3.1, mod.step3.2, mod.step3.3, mod.step3.4)
```
### Step 3 Results
```{r}
# display results
screenreg(mods.step3, single.row = T)
```
Based on AIC, Model 2 from step 3 seems to be the best model.
```{r}
best.step3 <- mods.step3[[2]]
step3.formula <- formula(best.step3)
```
## Step 4 - Select random effects
Finally, we look at different random effects. To avoid over fitting, we do not use the sigma formulas from step 2 (meaning the fitted standard deviation is the same for all categories)
```{r}
formula.step4.1 <- update(step3.formula, . ~ . + random(AMC))
formula.step4.2 <- update(step3.formula, . ~ . + random(study_id))
formula.step4.3 <- update(step3.formula, . ~ . + random(season))
mod.step4.1 <-
gamlss(
formula.step4.1,
# sigma.formula = ~season,
family = LOGNO,
data = df.coc,
trace = FALSE
)
mod.step4.2 <-
gamlss(formula.step4.2,
# sigma.formula = ~season,
trace = T,
data = df.coc,
family = LOGNO
)
mod.step4.3 <- gamlss(formula.step4.3,
family = LOGNO,
# sigma.formula = ~season,
trace = T,
data = df.coc, method = mixed()
)
# make a list
mods.step4 <- list(best.step3, mod.step4.1, mod.step4.1, mod.step4.2, mod.step4.3)
```
### Step 4 results
```{r}
#display results
screenreg(mods.step4, single.row = T)
```
Model 1 seems the best (same as from step 3). This means we have no random effects. We revert to Model 1.
```{r}
best.step4 <- mods.step4[[1]]
```
## Summary of best bodel from steps 1-4
```{r}
#display a summary
summary(best.step4)
```
# Grouped K-Fold Cross validation
We use cross-validation with grouped folds to find the best model, and then test that model on a subset of the original data. Since our data is grouped by Location, we assign a fold for each location.
Split the data into folds by Location.
```{r}
folds <-(as.numeric(df.coc$Location))
(max(folds)) #check number of folds
```
We have 14 folds, one for each location.
Make a function for cross-validation using the ```gamlssCV``` package.
```{r message=FALSE, warning=FALSE}
```
Make a list of models to validate, run through the cross-validation function and show the results.
```{r message=FALSE, warning=FALSE}
models.to.validate <-
list(
"nulla" = model.nulla,
"nullb" = model.nullb,
"step1" = best.step1,
"step2" = best.step2,
"step3" = best.step3,
"step4" = best.step4
)
cv.results <- cv.function(models.to.validate)
```
## Cross validation results
Show the resulting average AIC from the cross-validation results.
```{r}
print(cv.results)
```
Seems like our more complicated models may be overfilling the data.
The best model is the step 2 resulting model.
# Results
The best validated model is step 2. Plot the model diagnostics.
```{r}
selected_model <- best.step2
plot(selected_model,summary=T)
```
Plot the predictions.
```{r fig.height=5, fig.width=6}
term.plot(selected_model,what="mu",rug=F,pages=1,ask=F,partial.resid=T,ylim="free",main=coc)
```
# Hypotheses test results
1. Is the land use (nullb) model better than using average concentration (nulla) (i.e. single intercept; slope = 0)?
```{r}
#use Vuong and Clarke tests
VC.test(model.nulla,model.nullb)
```
2. Is the selected model better than the land use only model?
```{r}
VC.test(model.nullb,selected_model)
```
Plot results from the land use model vs. the selected model
```{r fig.height=5.5, fig.width=8}
diag.df <- add.fit(selected_model, df.coc)
diag.nullb <- add.fit(model.nullb, df.coc)
ggthemr('fresh', "clean", spacing = 0.8)
selected_model.fit <- diag.df %>%
group_by(study_name, type) %>%
summarize(Int = mean(.fit)) %>%
add_column(model = "this study")
#
null_model.fit <- diag.nullb %>%
group_by(study_name, type) %>%
summarize(Int = mean(.fit)) %>%
add_column(model = "null model")
#
model.fits <- rbind(selected_model.fit, null_model.fit)
#
summary.plot <- ggplot(diag.df, aes(x = logConcentration)) +
geom_density(color = "#757575",
fill = "#e0e0e0",
alpha = 0.8) +
geom_vline(
data = model.fits,
aes(
xintercept = Int,
group = model,
color = model,
linetype = (model)
),
size = 1,
#linetype = "solid",
alpha = 0.9
) + legend_bottom() +
scale_linetype_manual(
name = "Model",
breaks = c("null model", "this study"),
labels = c("null model", "this study"),
values = c('dashed', 'solid')
) + scale_color_manual(
values = c("#E84646", "#109B37") ,
name = "Model",
breaks = c("null model", "this study"),
labels = c("null model", "this study")
) +
labs(title = coc, subtitle = "Average predicted concentration vs. density plots of observered data") +
scale_y_continuous(labels = scales::number_format(accuracy = .1,
decimal.mark = '.'))
summary.plot +facet_wrap(study_name~type,
drop = T,
nrow = 4,
scales = 'free_y',labeller=labeller(.multi_line = F),dir='v') #,scales="free")+
```
*fin*