Statistical Modelling I
HES 505 Fall 2023: Session 22
Revisiting neighbors and areal data
Getting Weights
Fit a model
Moran’s I coefficient is the slope of the regression of the lagged asthma percentage vs. the asthma percentage in the tract
More generally it is the slope of the lagged average to the measurement
cdc$casthma_cr
0.1670774 
Comparing observed to expected
We can generate the expected distribution of Moran’s I coefficients under a Null hypothesis of no spatial autocorrelation
Using permutation and a loop to generate simulations of Moran’s I
n <- 400L # Define the number of simulations
I.r <- vector(length=n) # Create an empty vector
for (i in 1:n){
# Randomly shuffle income values
x <- sample(cdc$casthma_cr, replace=FALSE)
# Compute new set of lagged values
x.lag <- lag.listw(lw.nearest, x)
# Compute the regression slope and store its value
M.r <- lm(x.lag ~ x)
I.r[i] <- coef(M.r)[2]
}
Boolean Overlays
Successive disqualification of areas
Series of “yes/no” questions
“Sieve” mapping
Boolean Overlays
Reclassifying
Which types of land are appropriate
Boolean Overlays
- Which types of land are appropriate?
Boolean Overlays
- Which types of land are appropriate?
Boolean Overlays
Why do we create distribution models?
To identify important correlations between predictors and the occurrence of an event
Generate maps of the ‘range’ or ‘niche’ of events
Understand spatial patterns of event co-occurrence
Forecast changes in event distributions
General analysis situation
From Long
Spatially referenced locations of events \((\mathbf{y})\) sampled from the study extent
A matrix of predictors \((\mathbf{X})\) that can be assigned to each event based on spatial location
Goal: Estimate the probability of occurrence of events across unsampled regions of the study area based on correlations with predictors
Modeling Presence-Absence Data
Random or systematic sample of the study region
The presence (or absence) of the event is recorded for each point
Hypothesized predictors of occurrence are measured (or extracted) at each point
Logistic regression
We can model favorability as the probability of occurrence using a logistic regression
A link function maps the linear predictor \((\mathbf{x_i}'\beta + \alpha)\) onto the support (0-1) for probabilities
Estimates of \(\beta\) can then be used to generate ‘wall-to-wall’ spatial predictions
\[ \begin{equation} y_{i} \sim \text{Bern}(p_i)\\ \text{link}(p_i) = \mathbf{x_i}'\beta + \alpha \end{equation} \]
An Example
Inputs from the dismo package
An Example
The sample data
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
Looking at correlations
An Example
Looking at correlations
An Example
Generating predictions
An Example
Generating predictions
An Example
Generating predictions
An Example
Generating predictions
The sampling situation
Opportunistic collection of presences only
Hypothesized predictors of occurrence are measured (or extracted) at each presence
Background points (or pseudoabsences) generated for comparison