library(sf)
library(compositions)
data(juraset)
d <- st_as_sf(juraset, coords = c("X", "Y"))
plot(d[3])
1 Introduction to spatial mapping
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1.1 What is spatial mapping?
Spatial mapping: predicting or visualizing how a variable of interest varies across space.
Main goal: predict values at unsampled locations.
- Common in:
- Environmental science, especially soil
- Ecology & conservation
- Public health & epidemiology
- Urban & socio-economic studies
- Environmental science, especially soil
1.2 Ingredients for spatial mapping
Point data (outcome, response, target variable)
- Measurements or observations at known sampled locations
- Examples: soil samples, bird sightings, hospital admissions
Optional: Covariates (predictors, explanatory variables)
- Continuous spatial layers available for the whole area of interest
- Often available as gridded / spatial raster, or can be transformed into raster
- Examples: elevation, climate, land use, population density, remote sensing indices
Prediction function
- A model with a pre-defined structure or an algorithm that establishes a structure from the relationships of the response and covariates (machine learning)
- Examples: linear regression, ordinary kriging, random forest
NoteSpatial mapping
Spatial mapping links point data to raster covariates using a prediction function. Mapping can be done without raster covariates, then only the neighborhood relationships of the point data are used.
1.3 Examples of point data
Example of geostatistical data: Soil pollution
Example of geostatistical data: Air pollution
library(gstat)
library(ggplot2)
no2 <- read.csv(system.file("external/no2.csv",package = "gstat"))
v.no2 <- st_as_sf(no2, crs = "OGC:CRS84", coords = c("station_longitude_deg", "station_latitude_deg"))
v.de <- read_sf("https://github.com/edzer/sdsr/raw/main/data/de_nuts1.gpkg")
ggplot() + geom_sf(data = v.de) + geom_sf(data = v.no2, mapping = aes(col = NO2)) 
1.4 Examples of predictor data
Predictor engineering from elevation model
Predictor engineering using different scales
These are now just derivatives from one data set (elevation model). We could also have different datasets, and not be sure which is best to include. An example is meteorological data, where often different versions exist (e.g. average yearly rainfall over different periods, computed by different models or based on a different set of rainfall stations).
Predictors from vector data
Also areal or lattice data can be possible predictors. Spatial distribution within areas is unknown and we have crisp boundaries, reflecting large uncertainty in the spatial allocation of the data.
A practical approach: transform to raster and apply smoothing.
library(sf)
library(mapview)
mapviewOptions(basemaps = "OpenStreetMap")
d <- st_read(system.file("shape/nc.shp", package = "sf"), quiet = TRUE)
mapview(d, zcol = "SID79")Number of sudden infant deaths per counties of North Carolina (USA) in 1979.
1.5 Prediction functions
Two main branches:
Geostatistical model (kriging approaches)
\(Y = \mathbf{X}\beta + E + Z\)
where
\(Y\) : observed point data at location \(i\)
\(\mathbf{X}\beta\) : general trend, e.g. a linear regression with covariates \(\mathbf{X}\) (raster data)
\(E\) : small scale fluctuations, spatial autocorrelation specified by a variogram
\(Z\) : random error (e.g. measurement error, part we cannot explain with model)
→ Explicit model for:
Points closer to each other are more similar, than points further away.
→ Yields prediction uncertainty directly, many special cases solved.
→ Classical statistics, relies on e.g. normally distributed residuals.
→ Selecting a trend model is often challenging.
(Spatial) machine learning
\(Y = f(\mathbf{X}) + Z~~\) or \(~~Y = f(\mathbf{X}, I) + Z~~\) or \(~~Y = f_i(\mathbf{X}) + Z\)
where
\(Y\) : observed point data at locations \(i\)
\(f(X)\) : complex function, established by an algorithm using covariates \(\mathbf{X}\)
\(I\) : location \(i\) of the data points
\(Z\) : random error (e.g. measurement error, part we cannot explain with model)
→ Uses non-spatial prediction approach, sometimes integrates spatial position somehow.
→ Easy to find a model, relaxed statistical assumptions.
→ Spatial autocorrelation is less explicitly handled, mostly ad-hoc.
→ Drawback for special cases (e.g. uncertainty for a mean prediction of a region)
1.6 Predictive mapping example
Workflow digital soil mapping
Mapping of soil organic carbon stocks
Swiss government needed data for greenhouse gas inventory.
→ Mapping of carbon stocks in forest soils using kriging with a linear trend model
Map update
Some years later, 850 new soil point data became available.
→ Mapping with random forest
1.7 Goal of spatial predictive analysis
→ Point sampling is sparse & expensive.
→ End-users need actionable spatial information, not just point samples.
→ Supports decision-making, environmental assessment, resource or risk management.
Our objective is to
Compute spatial predictions of response variable for
- a location without measurement, i.e. creating spatially continuous map,
- spatial means or totals for pre-defined area (e.g. province),
Quantify uncertainty of these predictions,
Estimate relationships of spatial response variable and covariates.
1.8 Outlook
In short, you will learn:
Handling spatial data in R
Find a prediction function / model
Investigate the spatial structure / auto-correlation of the data
Interpretation of the prediction model
Assess the performance of the predictions