Overlays

• Methods for identifying optimal site selection or suitability

• Apply a common scale to diverse or dissimilar outputs

Getting Started

Boolean Overlays

Boolean Overlays

• Reclassifying

• Which types of land are appropriate

nlcd <-  rast(system.file("raster/nlcd.tif", package = "spDataLarge"))
plot(nlcd)

Boolean Overlays

• Which types of land are appropriate?

nlcd.segments <- segregate(nlcd)
names(nlcd.segments) <- levels(nlcd)[[1]][-1,2]
plot(nlcd.segments)

Boolean Overlays

• Which types of land are appropriate?

srtm <- rast(system.file("raster/srtm.tif", package = "spDataLarge"))
slope <- terrain(srtm, v = "slope")

Boolean Overlays

• Make sure data is aligned!

suit.slope <- slope < 10
suit.landcov <- nlcd.segments["Shrubland"]
suit.slope.match <- project(suit.slope, suit.landcov)
suit <- suit.slope.match + suit.landcov

Boolean Overlays

Challenges with Boolean Overlays

1. Assume relationships are really Boolean

2. No measurement error

3. Categorical measurements are known exactly

4. Boundaries are well-represented

A more general approach

• Define a favorability metric

\[ \begin{equation} F(\mathbf{s}) = \prod_{M=1}^{m}X_m(\mathbf{s}) \end{equation} \]

• Treat \(F(\mathbf{s})\) as binary
• Then \(F(\mathbf{s}) = 1\) if all inputs (\(X_m(\mathbf{s})\)) are suitable
• Then \(F(\mathbf{s}) = 0\) if not

Estimating favorability

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

• \(F(\mathbf{s})\) does not have to be binary (could be ordinal or continuous)

• \(X_m(\mathbf{s})\) could also be extended beyond simply ‘suitable/not suitable’

• Adding weights allows incorporation of relative importance

• Other functions for combining inputs (\(X_m(\mathbf{s})\))

Weighted Linear Combinations

\[ \begin{equation} F(\mathbf{s}) = \frac{\sum_{i=1}^{m}w_iX_i(\mathbf{s})}{\sum_{i=1}^{m}w_i} \end{equation} \]

• \(F(s)\) is now an index based on the values of \(X_m(\mathbf{s})\)

• \(w_i\) can incorporate weights of evidence, uncertainty, or different participant preferences

• Dividing by \(\sum_{i=1}^{m}w_i\) normalizes by the sum of weights

Model-driven overlay

\[ \begin{equation} F(\mathbf{s}) = w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon \end{equation} \]

• If we estimate \(w_i\) using data, we specify \(F(s)\) as the outcome of regression

• When \(F(s)\) is binary → logistic regression

• When \(F(s)\) is continuous → linear (gamma) regression

• When \(F(s)\) is discrete → Poisson regression

• Assumptions about \(\epsilon\) matter!!

Why do we create distribution models?