# 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')
##      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)


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)  ## 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)  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
# 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), ]
tail(csum)


## 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
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), ]


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.