6. Local regression

Regression models are typically “global”. That is, all date are used simultaneously to fit a single model. In some cases it can make sense to fit more flexible “local” models. Such models exist in a general regression framework (e.g. generalized additive models), where “local” refers to the values of the predictor values. In a spatial context local refers to location. Rather than fitting a single regression model, it is possible to fit several models, one for each location (out of possibly very many) locations. This technique is sometimes called “geographically weighted regression” (GWR). GWR is a data exploration technique that allows to understand changes in importance of different variables over space (which may indicate that the model used is misspecified and can be improved).

There are two examples here. One short example with California precipitation data, and than a more elaborate example with house price data.

California precipitation

if (!require("rspatial")) devtools::install_github('rspatial/rspatial')

library(rspatial)
cts <- sp_data('counties')
p <- sp_data('precipitation')
head(p)
##      ID                 NAME   LAT    LONG ALT  JAN FEB MAR APR MAY JUN
## 1 ID741         DEATH VALLEY 36.47 -116.87 -59  7.4 9.5 7.5 3.4 1.7 1.0
## 2 ID743  THERMAL/FAA AIRPORT 33.63 -116.17 -34  9.2 6.9 7.9 1.8 1.6 0.4
## 3 ID744          BRAWLEY 2SW 32.96 -115.55 -31 11.3 8.3 7.6 2.0 0.8 0.1
## 4 ID753 IMPERIAL/FAA AIRPORT 32.83 -115.57 -18 10.6 7.0 6.1 2.5 0.2 0.0
## 5 ID754               NILAND 33.28 -115.51 -18  9.0 8.0 9.0 3.0 0.0 1.0
## 6 ID758        EL CENTRO/NAF 32.82 -115.67 -13  9.8 1.6 3.7 3.0 0.4 0.0
##   JUL  AUG SEP OCT NOV DEC
## 1 3.7  2.8 4.3 2.2 4.7 3.9
## 2 1.9  3.4 5.3 2.0 6.3 5.5
## 3 1.9  9.2 6.5 5.0 4.8 9.7
## 4 2.4  2.6 8.3 5.4 7.7 7.3
## 5 8.0  9.0 7.0 8.0 7.0 9.0
## 6 3.0 10.8 0.2 0.0 3.3 1.4

plot(cts)
points(p[,c('LONG', 'LAT')], col='red', pch=20)

image0

Compute annual average precipitation

p$pan <- rowSums(p[,6:17])

Global regression model

m <- lm(pan ~ ALT, data=p)
m
##
## Call:
## lm(formula = pan ~ ALT, data = p)
##
## Coefficients:
## (Intercept)          ALT
##      523.60         0.17

Create Spatial* objects with a planar crs.

alb <- CRS("+proj=aea +lat_1=34 +lat_2=40.5 +lat_0=0 +lon_0=-120 +x_0=0 +y_0=-4000000 +ellps=GRS80 +datum=NAD83 +units=m +no_defs")
sp <- p
coordinates(sp) = ~ LONG + LAT
crs(sp) <- "+proj=longlat +datum=NAD83"
spt <- spTransform(sp, alb)
ctst <- spTransform(cts, alb)

Get the optimal bandwidth

library( spgwr )
## Loading required package: spData
## To access larger datasets in this package, install the spDataLarge
## package with: `install.packages('spDataLarge',
## repos='https://nowosad.github.io/drat/', type='source'))`
## NOTE: This package does not constitute approval of GWR
## as a method of spatial analysis; see example(gwr)
bw <- gwr.sel(pan ~ ALT, data=spt)
## Bandwidth: 526221.1 CV score: 64886883
## Bandwidth: 850593.6 CV score: 74209073
## Bandwidth: 325747.9 CV score: 54001118
## Bandwidth: 201848.6 CV score: 44611213
## Bandwidth: 125274.7 CV score: 35746320
## Bandwidth: 77949.39 CV score: 29181737
## Bandwidth: 48700.74 CV score: 22737197
## Bandwidth: 30624.09 CV score: 17457161
## Bandwidth: 19452.1 CV score: 15163436
## Bandwidth: 12547.43 CV score: 19452191
## Bandwidth: 22792.75 CV score: 15512988
## Bandwidth: 17052.67 CV score: 15709960
## Bandwidth: 20218.99 CV score: 15167438
## Bandwidth: 19767.99 CV score: 15156913
## Bandwidth: 19790.05 CV score: 15156906
## Bandwidth: 19781.39 CV score: 15156902
## Bandwidth: 19781.48 CV score: 15156902
## Bandwidth: 19781.47 CV score: 15156902
## Bandwidth: 19781.47 CV score: 15156902
## Bandwidth: 19781.47 CV score: 15156902
## Bandwidth: 19781.47 CV score: 15156902
bw
## [1] 19781.47

Create a regular set of points to estimate parameters for.

r <- raster(ctst, res=10000)
r <- rasterize(ctst, r)
newpts <- rasterToPoints(r)

Run the gwr function

g <- gwr(pan ~ ALT, data=spt, bandwidth=bw, fit.points=newpts[, 1:2])
g
## Call:
## gwr(formula = pan ~ ALT, data = spt, bandwidth = bw, fit.points = newpts[,
##     1:2])
## Kernel function: gwr.Gauss
## Fixed bandwidth: 19781.47
## Fit points: 4087
## Summary of GWR coefficient estimates at fit points:
##                    Min.    1st Qu.     Median    3rd Qu.      Max.
## X.Intercept. -702.40121   79.54254  330.48807  735.42718 3468.8702
## ALT            -3.91270    0.03058    0.20461    0.41542    4.6133

Link the results back to the raster

slope <- r
intercept <- r
slope[!is.na(slope)] <- g$SDF$ALT
intercept[!is.na(intercept)] <- g$SDF$'(Intercept)'
s <- stack(intercept, slope)
names(s) <- c('intercept', 'slope')
plot(s)

image1

California House Price Data

We will use house prices data from the 1990 census, taken from “Pace, R.K. and R. Barry, 1997. Sparse Spatial Autoregressions. Statistics and Probability Letters 33: 291-297.” You can download the data here

houses <- sp_data("houses1990.csv")
dim(houses)
## [1] 20640     9
head(houses)
##   houseValue income houseAge rooms bedrooms population households latitude
## 1     452600 8.3252       41   880      129        322        126    37.88
## 2     358500 8.3014       21  7099     1106       2401       1138    37.86
## 3     352100 7.2574       52  1467      190        496        177    37.85
## 4     341300 5.6431       52  1274      235        558        219    37.85
## 5     342200 3.8462       52  1627      280        565        259    37.85
## 6     269700 4.0368       52   919      213        413        193    37.85
##   longitude
## 1   -122.23
## 2   -122.22
## 3   -122.24
## 4   -122.25
## 5   -122.25
## 6   -122.25

Each record represents a census “blockgroup”. The longitude and latitude of the centroids of each block group are available. We can use that to make a map and we can also use these to link the data to other spatial data. For example to get county-membership of each block group. To do that, let’s first turn this into a SpatialPointsDataFrame to find out to which county each point belongs.

library(sp)
coordinates(houses) <- ~longitude+latitude
plot(houses, cex=0.5, pch=1, axes=TRUE)

image2

Now get the county boundaries and assign CRS of the houses data matches that of the counties (because they are both in longitude/latitude!).

library(raster)
counties <- readRDS("data/counties.rds")
## Warning in readRDS("data/counties.rds"): invalid or incomplete compressed
## data
## Error in readRDS("data/counties.rds"): error reading from connection
crs(houses) <- crs(counties)
## Error in crs(counties): object 'counties' not found

Do a spatial query (points in polygon)

cnty <- over(houses, counties)
## Error in over(houses, counties): object 'counties' not found
head(cnty)
## Error in head(cnty): object 'cnty' not found

Summarize

We can summarize the data by county. First combine the extracted county data with the original data.

hd <- cbind(data.frame(houses), cnty)
## Error in cbind(data.frame(houses), cnty): object 'cnty' not found

Compute the population by county

totpop <- tapply(hd$population, hd$NAME, sum)
## Error in tapply(hd$population, hd$NAME, sum): object 'hd' not found
totpop
## Error in eval(expr, envir, enclos): object 'totpop' not found

Income is harder because we have the median household income by blockgroup. But it can be approximated by first computing total income by blockgroup, summing that, and dividing that by the total number of households.

# total income
hd$suminc <- hd$income * hd$households
## Error in eval(expr, envir, enclos): object 'hd' not found
# now use aggregate (similar to tapply)
csum <- aggregate(hd[, c('suminc', 'households')], list(hd$NAME), sum)
## Error in aggregate(hd[, c("suminc", "households")], list(hd$NAME), sum): object 'hd' not found
# divide total income by number of housefholds
csum$income <- 10000 * csum$suminc / csum$households
## Error in eval(expr, envir, enclos): object 'csum' not found
# sort
csum <- csum[order(csum$income), ]
## Error in eval(expr, envir, enclos): object 'csum' not found
head(csum)
## Error in head(csum): object 'csum' not found
tail(csum)
## Error in tail(csum): object 'csum' not found

Regression

Before we make a regression model, let’s first add some new variables that we might use, and then see if we can build a regression model with house price as dependent variable. The authors of the paper used a lot of log tranforms, so you can also try that.

hd$roomhead <- hd$rooms / hd$population
## Error in eval(expr, envir, enclos): object 'hd' not found
hd$bedroomhead <- hd$bedrooms / hd$population
## Error in eval(expr, envir, enclos): object 'hd' not found
hd$hhsize <- hd$population / hd$households
## Error in eval(expr, envir, enclos): object 'hd' not found

Ordinary least squares regression:

# OLS
m <- glm( houseValue ~ income + houseAge + roomhead + bedroomhead + population, data=hd)
## Error in is.data.frame(data): object 'hd' not found
summary(m)
##
## Call:
## lm(formula = pan ~ ALT, data = p)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -638.4 -281.2 -115.7  187.4 1793.5
##
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 523.60251   26.50338  19.756  < 2e-16 ***
## ALT           0.16997    0.03505   4.849  1.7e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 425.4 on 454 degrees of freedom
## Multiple R-squared:  0.04925,    Adjusted R-squared:  0.04715
## F-statistic: 23.52 on 1 and 454 DF,  p-value: 1.704e-06
coefficients(m)
## (Intercept)         ALT
## 523.6025073   0.1699685

Geographicaly Weighted Regression

By county

Of course we could make the model more complex, with e.g. squared income, and interactions. But let’s see if we can do Geographically Weighted regression. One approach could be to use counties.

First I remove records that were outside the county boundaries

 hd2 <- hd[!is.na(hd$NAME), ]
## Error in eval(expr, envir, enclos): object 'hd' not found

Then I write a function to get what I want from the regression (the coefficients in this case)

regfun <- function(x)  {
  dat <- hd2[hd2$NAME == x, ]
  m <- glm(houseValue~income+houseAge+roomhead+bedroomhead+population, data=dat)
  coefficients(m)
}

And now run this for all counties using sapply:

countynames <- unique(hd2$NAME)
## Error in unique(hd2$NAME): object 'hd2' not found
res <- sapply(countynames, regfun)
## Error in lapply(X = X, FUN = FUN, ...): object 'countynames' not found

Plot of a single coefficient

dotchart(sort(res['income', ]), cex=0.65)
## Error in res["income", ]: object of type 'closure' is not subsettable

There clearly is variation in the coefficient (\(beta\)) for income. How does this look on a map?

First make a data.frame of the results

resdf <- data.frame(NAME=colnames(res), t(res))
## Error in t.default(res): argument is not a matrix
head(resdf)
## Error in head(resdf): object 'resdf' not found

Fix the counties object. There are too many counties because of the presence of islands. I first aggregate (‘dissolve’ in GIS-speak’) the counties such that a single county becomes a single (multi-)polygon.

dim(counties)
## Error in eval(expr, envir, enclos): object 'counties' not found
dcounties <- aggregate(counties, vars='NAME')
## Error in aggregate(counties, vars = "NAME"): object 'counties' not found
dim(dcounties)
## Error in eval(expr, envir, enclos): object 'dcounties' not found

Now we can merge this SpatialPolygonsDataFrame with data.frame with the regression results.

cnres <- merge(dcounties, resdf, by='NAME')
## Error in merge(dcounties, resdf, by = "NAME"): object 'dcounties' not found
spplot(cnres, 'income')
## Error in spplot(cnres, "income"): object 'cnres' not found

To show all parameters in a ‘conditioning plot’, we need to first scale the values to get similar ranges.

# a copy of the data
cnres2 <- cnres
## Error in eval(expr, envir, enclos): object 'cnres' not found

# scale all variables, except the first one (county name)
# assigning values to a "@data" slot is risky, but (I think) OK here
cnres2@data = data.frame(scale(data.frame(cnres)[, -1]))
## Error in data.frame(cnres): object 'cnres' not found
spplot(cnres2)
## Error in spplot(cnres2): object 'cnres2' not found

Is this just random noise, or is there spatial autocorrelation?

library(spdep)
## Loading required package: Matrix
##
## Attaching package: 'Matrix'
## The following object is masked from 'package:spam':
##
##     det
nb <- poly2nb(cnres)
## Error in extends(class(pl), "SpatialPolygons"): object 'cnres' not found
plot(cnres)
## Error in plot(cnres): object 'cnres' not found
plot(nb, coordinates(cnres), add=T, col='red')
## Error in plot(nb, coordinates(cnres), add = T, col = "red"): object 'nb' not found

lw <- nb2listw(nb)
## Error in nb2listw(nb): object 'nb' not found
moran.test(cnres$income, lw)
## Error in moran.test(cnres$income, lw): object 'lw' not found
moran.test(cnres$roomhead, lw, na.action=na.omit)
## Error in moran.test(cnres$roomhead, lw, na.action = na.omit): object 'lw' not found

By grid cell

An alternative approach would be to compute a model for grid cells. Let’s use the ‘Teale Albers’ projection (often used when mapping the entire state of California).

TA <- CRS("+proj=aea +lat_1=34 +lat_2=40.5 +lat_0=0 +lon_0=-120 +x_0=0 +y_0=-4000000
              +datum=NAD83 +units=m +no_defs +ellps=GRS80 +towgs84=0,0,0")
countiesTA <- spTransform(counties, TA)
## Error in spTransform(counties, TA): object 'counties' not found

Create a RasteLayer using the extent of the counties, and setting an arbitrary resolution of 50 by 50 km cells

library(raster)
r <- raster(countiesTA)
## Error in raster(countiesTA): object 'countiesTA' not found
res(r) <- 50000

Get the xy coordinates for each raster cell:

xy <- xyFromCell(r, 1:ncell(r))

For each cell, we need to select a number of observations, let’s say within 50 km of the center of each cell (thus the data that are used in different cells overlap). And let’s require at least 50 observations to do a regression.

First transform the houses data to Teale-Albers

housesTA <- spTransform(houses, TA)
## Error in spTransform(xSP, CRSobj, ...): No transformation possible from NA reference system
crds <- coordinates(housesTA)
## Error in coordinates(housesTA): object 'housesTA' not found

Set up a new regression function.

regfun2 <- function(d)  {
 m <- glm(houseValue~income+houseAge+roomhead+bedroomhead+population, data=d)
 coefficients(m)
}

Run the model for al cells if there are at least 50 observations within a radius of 50 km.

res <- list()
for (i in 1:nrow(xy)) {
    d <- sqrt((xy[i,1]-crds[,1])^2 + (xy[i,2]-crds[,2])^2)
    j <- which(d < 50000)
    if (length(j) > 49) {
        d <- hd[j,]
        res[[i]] <- regfun2(d)
    } else {
        res[[i]] <- NA
    }
}
## Error: object 'crds' not found

For each cell get the income coefficient:

inc <- sapply(res, function(x) x['income'])

Use these values in a RasterLayer

rinc <- setValues(r, inc)
## Error in setValues(r, inc): values must be numeric, integer, logical or factor
plot(rinc)
## Error in plot(rinc): object 'rinc' not found
plot(countiesTA, add=T)
## Error in plot(countiesTA, add = T): object 'countiesTA' not found
Moran(rinc)
## Error in Moran(rinc): object 'rinc' not found

So that was a lot of ‘home-brew-GWR’.

Question 1: Can you comment on weaknesses (and perhaps strengths) of the approaches I have shown?

Question 2: Can you do it the easier and more professional way for these data, using the spgwr package?

spgwr package

Now use the spgwr package (and the the gwr function) to fit the model. You can do this with all data, as long as you supply and argument fit.points (to avoid estimating a model for each observation point. You can use a raster similar to the one I used above (perhaps disaggregate with a factor 2 first).

This is how you can get the points to use:

Create a RasterLayer with the correct extent

r <- raster(countiesTA)
## Error in raster(countiesTA): object 'countiesTA' not found

Set to a desired resolution. I choose 25 km

res(r) <- 25000

I only want cells inside of CA, so I add some more steps.

ca <- rasterize(countiesTA, r)
## Error in rasterize(countiesTA, r): object 'countiesTA' not found

Extract the coordinates that are not NA.

fitpoints <- rasterToPoints(ca)
## Error in nlayers(x): object 'ca' not found

I don’t want the third column

fitpoints <- fitpoints[,-3]
## Error in eval(expr, envir, enclos): object 'fitpoints' not found

Now specificy the model

gwr.model <- ______

gwr returns a list-like object that includes (as first element) a SpatialPointsDataFrame that has the model coeffients. Plot these using spplot, and after that, transfer them to a RasterBrick object.

To extract the SpatialPointsDataFrame:

sp <- gwr.model$SDF
spplot(sp)

To reconnect these values to the raster structure (etc.)

cells <- cellFromXY(r, fitpoints)
dd <- as.matrix(data.frame(sp))
b <- brick(r, values=FALSE, nl=nrow(dd))
b[cells] <- dd
names(b) <- colnames(dd)
plot(b)

Question 3: Briefly comment on the results and the differences (if any) with the two home-brew examples.