lab1b_sdm-regress.Rmd

---
title: 'Lab 1b. Species Distribution Modeling - Logistic Regression'
editor_options: 
  chunk_output_type: console
---

```{r setup, include=F}
knitr::opts_chunk$set(echo = T)
```

# Learning Objectives {-}

- **Exploratory Data Analysis** (cont'd): 
  - Pairs plot to show correlation between variables and avoid **multicollinearity** (see [8.2 Many predictors in a model](https://openintro-ims.netlify.app/model-mlr.html#many-predictors-in-a-model))
- **Logistic Regression** seen as an evolution of techniques
  - **Linear Model** to show simplest multivariate regression, but predictions can be outside the binary values.
  - **Generalized Linear Model** uses a logit transformation to constrain the outputs to being within two values.
  - **Generalized Additive Model** allows for "wiggle" in predictor terms.
  - **Maxent** (Maximum Entropy) is a presence-only modeling technique that allows for a more complex set of shapes between predictor and response.

# Explore (cont'd)

Let's load R packages and data from previous [Explore](https://bbest.github.io/eds232-ml/lab1_sdm-explore.html) session last time for your species.

```{r load pkgs data}
librarian::shelf(
  DT, dplyr, dismo, GGally, here, readr, tidyr)
select <- dplyr::select # overwrite raster::select
options(readr.show_col_types = F)

dir_data    <- here("data/sdm")
pts_env_csv <- file.path(dir_data, "pts_env.csv")

pts_env <- read_csv(pts_env_csv)
nrow(pts_env)

datatable(pts_env, rownames = F)
```

Let's look at a pairs plot (using [GGally::ggpairs()](http://ggobi.github.io/ggally/articles/ggpairs.html)) to show correlations between variables. 

```{r ggpairs, fig.cap="Pairs plot with `present` color coded.", fig.width=11, fig.height=11, eval=T}
GGally::ggpairs(
  select(pts_env, -ID),
  aes(color = factor(present)))
```

# Logistic Regression

We'll work up to the modeling workflow using multiple regression methods along the way.

```{r fig.cap="Full model workflow with split, fit, predict and evaluate process steps.", echo=F}
# [model-workflow - Google Drawings](https://docs.google.com/drawings/d/1bnevzcNpkRtopo3jagpwBy0IyNN-uh_mVJXvQg4WRtU/edit)
knitr::include_graphics("https://docs.google.com/drawings/d/e/2PACX-1vSab7DR3sDS1aoRWNDCfb8NUDE5a1P701NIZrb5zb5ss3RAhvfLJxuRLhHpJpKAW8krtGRkwm-jKa1Y/pub?w=932&h=344")
```

## Setup Data

Let's setup a data frame with only the data we want to model by:

- Dropping rows with any NAs. Later we'll learn how to ["impute"](https://bbest.github.io/eds232-ml/glossary.html#imputation) values with guesses so as to not throw away data.
- Removing terms we don't want to model. We can then use a simplified formula $present \sim .$ to predict $present$ based on all other fields in the data frame (i.e. the $X$`s in $y \sim x_1 + x_2 + ... x_n$).

```{r setup d}
# setup model data
d <- pts_env %>% 
  select(-ID) %>%  # remove terms we don't want to model
  tidyr::drop_na() # drop the rows with NA values
nrow(d)
```

## Linear Model

Let's start as simply as possible with a linear model `lm()` on multiple predictors `X` to predict presence `y` using a simpler workflow.

```{r fig.cap="Simpler workflow with only fit and predict of all data, i.e. no splitting.", echo=F}
# [model-workflow_fit-all-predict-all - Google Drawings](https://docs.google.com/drawings/d/1JtWF-4dltWR0hSAZBGaqQdLShReLVjLhKZG8iRpVY2A/edit)
knitr::include_graphics("https://docs.google.com/drawings/d/e/2PACX-1vRRvQdrDELT9QRPJxj8ooPvABgjQ-qqjzC0Ri9Hhv6n1O7N31eb4o6c7ZhZd6QtQ7XgZybNhUBESK9L/pub?w=572&h=170")
```


```{r fit lm}
# fit a linear model
mdl <- lm(present ~ ., data = d)
summary(mdl)

y_predict <- predict(mdl, d, type="response")
y_true    <- d$present

range(y_predict)
range(y_true)
```

The problem with these predictions is that it ranges outside the possible values of present `1` and absent `0`. (Later we'll deal with converting values within this range to either `1` or `0` by applying a cutoff value; i.e. any values > 0.5 become 1 and below become 0.)

## Generalized Linear Model

To solve this problem of constraining the response term to being between the two possible values, i.e. the **probability** $p$ of being one or the other possible $y$ values, we'll apply the logistic transformation on the response term.

$$
logit(p_i) = \log_{e}\left( \frac{p_i}{1-p_i} \right)
$$
We can expand the expansion of the predicted term, i.e. the probability $p$ of being either $y$, with all possible predictors $X$ whereby each coeefficient $b$ gets multiplied by the value of $x$:

$$
\log_{e}\left( \frac{p_i}{1-p_i} \right) = b_0 + b_1 x_{1,i} + b_2 x_{2,i} + \cdots + b_k x_{k,i}
$$

```{r fit glm}
# fit a generalized linear model with a binomial logit link function
mdl <- glm(present ~ ., family = binomial(link="logit"), data = d)
summary(mdl)

y_predict <- predict(mdl, d, type="response")

range(y_predict)
```

Excellent, our response is now constrained between 0 and 1. Next, let's look at the term plots to see the relationship between predictor and response.

```{r termplots glm}
# show term plots
termplot(mdl, partial.resid = TRUE, se = TRUE, main = F, ylim="free")
```

## Generalized Additive Model

With a generalized additive model we can add "wiggle" to the relationship between predictor and response by introducing smooth `s()` terms.

```{r fit gam}
librarian::shelf(mgcv)

# fit a generalized additive model with smooth predictors
mdl <- mgcv::gam(
  formula = present ~ s(WC_alt) + s(WC_bio1) + 
    s(WC_bio2) + s(ER_tri) + s(ER_topoWet) + s(lon) + s(lat), 
  family = binomial, data = d)
summary(mdl)

# show term plots
plot(mdl, scale=0)
```

```{r gam-todo, eval=F, echo=F}
# TODO: model selection ?mgcv::gam
AIC(b0,bt) ## interaction worse than additive
bt <- gam(y~s(x0)+s(x1)+s(x2)+s(x3)+ti(x1,x2,k=6),
            data=dat,method="REML")
bs <- gam(y~s(x0,x1,k=40)+s(x2)+s(x3),data=dat,
          method="REML")
AIC(b0,bt,bs) ## additive still better.
## Now do automatic terms selection as well
b1 <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),data=dat,
       method="REML",select=TRUE)

# TODO: ?predict.gam
pred <- predict.gam(mdl, newd)
```

## Maxent (Maximum Entropy)

Maxent is probably the most commonly used species distribution model ([Elith 2011](http://dx.doi.org/10.1111/j.1472-4642.2010.00725.x)) since it performs well with few input data points, only requires presence points (and samples background for comparison) and is easy to use with a Java graphical user interface (GUI).

```{r fit maxent}
# load extra packages
librarian::shelf(
  maptools, sf)

mdl_maxent_rds <- file.path(dir_data, "mdl_maxent.rds")

# show version of maxent
if (!interactive())
  maxent()

# get environmental rasters
# NOTE: the first part of Lab 1. SDM - Explore got updated to write this clipped environmental raster stack
env_stack_grd <- file.path(dir_data, "env_stack.grd")
env_stack <- stack(env_stack_grd)
plot(env_stack, nc=2)

# get presence-only observation points (maxent extracts raster values for you)
obs_geo <- file.path(dir_data, "obs.geojson")
obs_sp <- read_sf(obs_geo) %>% 
  sf::as_Spatial() # maxent prefers sp::SpatialPoints over newer sf::sf class

# fit a maximum entropy model
if (!file.exists(mdl_maxent_rds)){
  mdl <- maxent(env_stack, obs_sp)
  readr::write_rds(mdl, mdl_maxent_rds)
}
mdl <- read_rds(mdl_maxent_rds)

# plot variable contributions per predictor
plot(mdl)

# plot term plots
response(mdl)

# predict
y_predict <- predict(env_stack, mdl) #, ext=ext, progress='')

plot(y_predict, main='Maxent, raw prediction')
data(wrld_simpl, package="maptools")
plot(wrld_simpl, add=TRUE, border='dark grey')
```

Notice how the `plot()` function produces different outputs depending on the class of the input object. You can view help for each of these with R Console commands: `?plot.lm`, `?plot.gam` and <code  class="r">`plot,DistModel,numeric-method`</code>.

Next time we'll split the data into test and train data and **evaluate** model performance while learning about tree-based methods.

```{r split rspatial, eval=F, echo=F}
samp <- sample(nrow(sdmdata), round(0.75 * nrow(sdmdata)))
traindata <- sdmdata[samp,]
traindata <- traindata[traindata[,1] == 1, 2:9]
testdata <- sdmdata[-samp,]
```