Favorability in General

\[ \begin{equation} F(\mathbf{s}) = f(w_1X_1(\mathbf{s}), w_2X_2(\mathbf{s}), w_3X_3(\mathbf{s}), ..., w_mX_m(\mathbf{s})) \end{equation} \]

• Logistic regression treats \(f(x)\) as a (generalized) linear function

• Allows for multiple qualitative classes

• Ensures that estimates of \(F(\mathbf{s})\) are [0,1]

Key assumptions of logistic regression

• Dependent variable must be binary

• Observations must be independent (important for spatial analyses)

• Predictors should not be collinear

• Predictors should be linearly related to the log-odds

• Sample Size

Beyond Linearity

• Logistic (and other generalized linear models) are relatively interpretable

• Probability theory allows robust inference of effects

• Predictive power can be low

• Relaxing the linearity assumption can help

Classification Trees

• Use decision rules to segment the predictor space

• Series of consecutive decision rules form a ‘tree’

• Terminal nodes (leaves) are the outcome; internal nodes (branches) the splits

Classification Trees

• Divide the predictor space (\(R\)) into \(J\) non-overlapping regions

• Every observation in \(R_j\) gets the same prediction

• Recursive binary splitting

• Pruning and over-fitting

An Example

Inputs from the dismo package

An Example

The sample data

head(pres.abs)

Simple feature collection with 6 features and 1 field
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -106.75 ymin: 31.25 xmax: -98.75 ymax: 37.75
Geodetic CRS:  GCS_unknown
  y              geometry
1 0  POINT (-99.25 35.25)
2 1  POINT (-98.75 36.25)
3 1 POINT (-106.75 35.25)
4 0 POINT (-100.75 31.25)
5 1  POINT (-99.75 37.75)
6 1 POINT (-104.25 36.75)

An Example

Building our dataframe

pts.df <- terra::extract(pred.stack, vect(pres.abs), df=TRUE)
head(pts.df)

  ID MeanAnnTemp TotalPrecip PrecipWetQuarter PrecipDryQuarter MinTempCold
1  1         155         667              253               71         350
2  2         147         678              266               66         351
3  3         123         261              117               40         329
4  4         181         533              198               69         348
5  5         127         589              257               48         338
6  6          83         438              213               38         278
  TempRange
1       -45
2       -58
3       -64
4        -5
5       -81
6      -107

An Example

Building our dataframe

pts.df[,2:7] <- scale(pts.df[,2:7])
summary(pts.df)

       ID          MeanAnnTemp       TotalPrecip      PrecipWetQuarter 
 Min.   :  1.00   Min.   :-3.3729   Min.   :-1.3377   Min.   :-1.6926  
 1st Qu.: 25.75   1st Qu.:-0.4594   1st Qu.:-0.7980   1st Qu.:-0.6895  
 Median : 50.50   Median : 0.2282   Median :-0.2373   Median :-0.2224  
 Mean   : 50.50   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
 3rd Qu.: 75.25   3rd Qu.: 0.7118   3rd Qu.: 0.7140   3rd Qu.: 0.6508  
 Max.   :100.00   Max.   : 1.4285   Max.   : 2.4843   Max.   : 2.2713  
 PrecipDryQuarter   MinTempCold        TempRange      
 Min.   :-1.0828   Min.   :-3.9919   Min.   :-2.7924  
 1st Qu.:-0.7013   1st Qu.:-0.0598   1st Qu.:-0.5216  
 Median :-0.3770   Median : 0.3582   Median : 0.2075  
 Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
 3rd Qu.: 0.4290   3rd Qu.: 0.5495   3rd Qu.: 0.6450  
 Max.   : 3.1713   Max.   : 1.1092   Max.   : 2.0407  

An example

• Fitting the classification tree

library(tree)
pts.df <- cbind(pts.df, pres.abs$y)
colnames(pts.df)[8] <- "y"
pts.df$y <- as.factor(ifelse(pts.df$y == 1, "Yes", "No"))
tree.model <- tree(y ~ . , pts.df)
plot(tree.model)
text(tree.model, pretty=0)

An example

• Fitting the classification tree

summary(tree.model)

Classification tree:
tree(formula = y ~ ., data = pts.df)
Variables actually used in tree construction:
[1] "TempRange"        "PrecipWetQuarter" "ID"               "MeanAnnTemp"     
Number of terminal nodes:  8 
Residual mean deviance:  0.3164 = 29.11 / 92 
Misclassification error rate: 0.07 = 7 / 100 

Benefits and drawbacks

Random Forests

An example

• Fitting the Random Forest

library(randomForest)
class.model <- y ~ .
rf2 <- randomForest(class.model, data=pts.df)
varImpPlot(rf2)

The sampling situation

The Challenge with Background Points

• What constitutes background?

• Not measuring probability, but relative likelihood of occurrence

• Sampling bias affects estimation

• The intercept

\[ \begin{equation} y_{i} \sim \text{Bern}(p_i)\\ \text{link}(p_i) = \mathbf{x_i}'\beta + \alpha \end{equation} \]

MaxEnt

Maximum Entropy models

Challenges with MaxEnt

• Not measuring probability, but relative likelihood of occurrence

• Sampling bias affects estimation (but can be mitigated using tuning parameters)

• Theoretical issues with background points and the intercept

• Recent developments relate MaxEnt (with cloglog links) to Inhomogenous Point Process models

Extensions

• Polynomial, splines, piece-wise regression

• Neural nets, Support Vector Machines, many many more