MixedEffects-copper.Rmd

---
title: "Copper Regression"
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# 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*