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day4.qmd
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## Machine Learning methods applied to spatial data
### Learning goals
### Reading materials
From [Spatial Data Science: with applications in R](https://r-spatial.org/book/):
* Section 10.1: Mapping with non-spatial regression and ML models (you already read this)
From the [stars vignettes](https://r-spatial.github.io/stars/articles/):
* 7. Statistical modelling with stars objects
Area of Applicability:
* this [CAST vignette](https://hannameyer.github.io/CAST/articles/cast02-AOA-tutorial.html)
* [Machine learning-based global maps of ecological variables and the challenge of assessing them](https://www.nature.com/articles/s41467-022-29838-9)
::: {.callout-tip title="Summary"}
* Intro to prediction with R
* (functions of) Spatial coordinates as predictors
* Spatially correlated residuals
* Area of applicability
* RandomForestsGLS: Random forests [for dependent data](https://arxiv.org/abs/2007.15421)
:::
### Summary
* Data: coverages as predictors
* Pitfalls: independence, known predictors, clustered data, different support / spatial unalignment
* Model assessment and transferrability: cross validation strategies, area of applicability
* RandomForestsGLS
## Spatial coordinates as predictor
We'll rename coordinates to `x` and `y`:
```{r}
library(dplyr)
library(sf)
crs <- st_crs("EPSG:32632") # a csv doesn't carry a CRS!
no2 <- read.csv(system.file("external/no2.csv",
package = "gstat"))
no2 |> rename(x = station_longitude_deg, y = station_latitude_deg) |>
st_as_sf(crs = "OGC:CRS84", coords =
c("x", "y"), remove = FALSE) |>
st_transform(crs) -> no2.sf
# we need to reassign x and y:
cc = st_coordinates(no2.sf)
no2.sf$x = cc[,1]
no2.sf$y = cc[,2]
head(no2.sf)
"https://github.com/edzer/sdsr/raw/main/data/de_nuts1.gpkg" |>
read_sf() |>
st_transform(crs) -> de
```
```{r}
library(stars)
g2 = st_as_stars(st_bbox(de))
g3 = st_crop(g2, de)
g4 = st_rasterize(de, g3)
g4$ID_1[g4$ID_1 == 758] = NA
g4$ID1 = as.factor(g4$ID_1) # now a factor:
plot(g4["ID1"], reset = FALSE)
plot(st_geometry(no2.sf), add = TRUE, col = 'green')
no2.sf$ID1 = st_extract(g4, no2.sf)$ID1
no2.sf$ID1 |> summary()
```
### Simple ANOVA type predictor:
```{r}
lm1 = lm(NO2~ID1, no2.sf)
summary(lm1)
g4$NO2_aov = predict(lm1, as.data.frame(g4))
plot(g4["NO2_aov"], breaks = "equal", reset = FALSE)
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
### Simple linear models in coordinates: trend surfaces
```{r}
lm2 = lm(NO2~x+y, no2.sf)
summary(lm2)
cc = st_coordinates(g4)
g4$x = cc[,1]
g4$y = cc[,2]
g4$NO2_lm2 = predict(lm2, g4)
plot(g4["NO2_lm2"], breaks = "equal", reset = FALSE, main = "1st order polynomial")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
```{r}
lm3 = lm(NO2~x+y+I(x^2)+I(y^2)+I(x*y), no2.sf)
summary(lm3)
g4$NO2_lm3 = predict(lm3, g4)
plot(g4["NO2_lm3"], breaks = "equal", reset = FALSE, main = "2nd order polynomial")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
```{r}
lm4 = lm(NO2~x+y+I(x^2)+I(y^2)+I(x*y)+I(x^3)+I(x^2*y)+I(x*y^2)+I(y^3), no2.sf)
summary(lm4)
g4$NO2_lm4 = predict(lm4, g4)
plot(g4["NO2_lm4"], breaks = "equal", reset = FALSE, main = "3rd order polynomial")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
::: {.callout-note title="Breakout session 1"}
Discuss:
* how will the predicted surface look like when instead of (functions of) coordinates, the variable _elevation_ is used (e.g. to predict average temperatures)?
* what will be the value range, approximately, of the resulting predicted values?
:::
### regression tree
```{r}
library(rpart)
tree = rpart(NO2~., as.data.frame(no2.sf)[c("NO2", "x", "y")])
g4$tree = predict(tree, as.data.frame(g4))
plot(g4["tree"], breaks = "equal", reset = FALSE, main = "regression tree")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
### Random forest
```{r}
library(randomForest)
rf = randomForest(NO2~., as.data.frame(no2.sf)[c("NO2", "x", "y")])
g4$rf = predict(rf, as.data.frame(g4))
plot(g4["rf"], breaks = "equal", reset = FALSE, main = "random forest")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
Rotated coordinates:
```{r}
library(randomForest)
no2.sf$x1 = no2.sf$x + no2.sf$y
no2.sf$y1 = no2.sf$x - no2.sf$y
rf = randomForest(NO2~., as.data.frame(no2.sf)[c("NO2", "x1", "y1")])
g4$x1 = g4$x + g4$y
g4$y1 = g4$x - g4$y
g4$rf_rot = predict(rf, as.data.frame(g4))
plot(g4["rf_rot"], breaks = "equal", reset = FALSE, main = "random forest")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
### Using distance variables:
```{r}
st_bbox(de) |> st_as_sfc() |> st_cast("POINT") -> pts
pts = c(pts[1:4], st_centroid(st_geometry(de)))
d = st_distance(st_as_sfc(g4, as_points = TRUE), pts)
for (i in seq_len(ncol(d))) {
g4[[ paste0("d_", i) ]] = d[,i]
}
e = st_extract(g4, no2.sf)
for (i in seq_len(ncol(d))) {
no2.sf[[ paste0("d_", i) ]] = e[[16+i]]
}
(n = names(g4))
plot(merge(g4[grepl("d_", n)]))
```
```{r}
library(randomForest)
rf = randomForest(NO2~., as.data.frame(no2.sf)[c("NO2", n[grepl("d_", n)])])
g4$rf_d = predict(rf, as.data.frame(g4))
plot(g4["rf_d"], breaks = "equal", reset = FALSE, main = "random forest")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
### Adding more...
```{r}
pts = st_sample(de, 200, type = "regular")
d = st_distance(st_as_sfc(g4, as_points = TRUE), pts)
for (i in seq_len(ncol(d))) {
g4[[ paste0("d_", i) ]] = d[,i]
}
e = st_extract(g4, no2.sf)
for (i in seq_len(ncol(d))) {
no2.sf[[ paste0("d_", i) ]] = e[[16+i]]
}
(n = names(g4))
rf = randomForest(NO2~., as.data.frame(no2.sf)[c("NO2", n[grepl("d_", n)])])
g4$rf_dm = predict(rf, as.data.frame(g4))
plot(g4["rf_dm"], breaks = "equal", reset = FALSE, main = "random forest")
plot(st_cast(st_geometry(de), "MULTILINESTRING"), add = TRUE, col = 'green')
```
### Further approaches:
* use linear regression on Gaussian kernel basis functions, $\exp(-h^2)$
* use splines in $x$ and $y$, with a given degree of smoothing (or effective degrees of freedom)
* use additional, non-distance/coordinate functions as base function(s)
* provided they are available "everywhere" (as _coverage_)
* examples: elevation, bioclimatic variables, (values derived from) satellite imagery bands
::: {.callout-note title="Breakout session 2"}
Discuss:
* How would you assess whether residuals from your fitted model are spatially correlated?
* Does resampling using _random_ partitioning (as is done in random forest) implicitly assume that observations are independent?
:::
### Example from CAST / caret
```{r}
library(CAST)
library(caret)
data(splotdata)
class(splotdata)
r = read_stars(system.file("extdata/predictors_chile.tif",
package = "CAST"))
x = st_drop_geometry(splotdata)[,6:16]
y = splotdata$Species_richness
tr = train(x, y) # chooses a random forest by default
predict(split(r), tr) |> plot()
```
Clustered data?
```{r}
plot(r[,,,1], reset = FALSE)
plot(st_geometry(splotdata), add = TRUE, col = 'green')
```
## Cross validation: random or spatially blocked?
## Transferrability of models: "area of applicability"
Explained [here](https://doi.org/10.1111/2041-210X.13650);
```{r}
aoa <- aoa(r, tr)
plot(aoa)
plot(aoa$DI)
plot(aoa$AOA)
```
## Random Forests for Spatially Dependent Data
R package [RandomForestGLS](https://cran.r-project.org/web/packages/RandomForestsGLS/index.html)!
Combines the good parts of RF and Gaussian processes, in a very
[smart way](https://arxiv.org/abs/2007.15421)! (final paper,
paywalled, [here](https://doi.org/10.1080/01621459.2021.1950003)).
The discussion on variable selection / variable importance under
spatial correlated residuals is worth reading.
```{r}
library(RandomForestsGLS)
cc = st_coordinates(splotdata)
load("rfgls.rda")
if (!exists("rfgls")) {
rfgls = RFGLS_estimate_spatial(cc, as.double(y), x)
}
cc_pr = st_coordinates(split(r))
head(as.data.frame(split(r)))
pr = RFGLS_predict_spatial(rfgls, as.matrix(cc_pr), as.data.frame(split(r))[-(1:2)])
out = split(r)
out$rfgls = pr$prediction
out$rf = predict(split(r), tr)
plot(merge(out[c("rf", "rfgls")]), breaks = "equal")
```
## MaxEnt
* [A statistical explanation of MaxEnt for Ecologists](https://doi.org/10.1111/j.1472-4642.2010.00725.x)
* R package [maxnet](https://cran.r-project.org/web/packages/maxnet/index.html) does that using glmnet (lasso or elasticnet regularization on
* A paper detailing the equivalence and differences between point pattern models and MaxEnt, found [here](https://besjournals.onlinelibrary.wiley.com/doi/10.1111/2041-210X.12352).