With the release of the R package `varycoef`

on CRAN (see) we enable you to analyze your spatial data and in a simple, yet versatile way. The underlying idea are *spatially varying coefficients*, short SVC, which have been studied over the last decades. These models are based on the linear regression model and are therefore very easy to interpret. Due to their design, SVC models offer a high flexibility. These two properties allow for better understanding of the data, gaining new insights and, ultimately, better predictive performance.

In this blog post, we will show what SVC models are and how to define them. Afterwards, we give a short and illustrative example on how to apply these tools in `varycoef`

using the well known data set `meuse`

from the package `sp`

.

**Disclaimer** This analysis is meant to be a show case of what is possible using the package `varycoef`

, not to write the most rigorous statistical analysis of a data set.

But first, let us start with the set up and let me show you where to find help:

```
# install from CRAN
# install.packages("varycoef")
# attach package
library(varycoef)
# general package help file
help("varycoef")
# vignette for detailed information on how to use varycoef
vignette("example", package = "varycoef")
```

`## Warning: vignette 'example' not found`

There is also a version of `varycoef`

on Github from which you can get the latest devel version.

Before we start, we want to give some prerequisites in order to understand the analysis below. Beside a classical linear regression model, we require the knowledge of geostatistics and tools thereof, like:

- what is the advantage of geostatistics?
- two-dimensional Gaussian processes, i.e.Â Gaussian random fields
- covariance functions and how the range and variance parameter influence them
- maximum likelihood estimation

SVC models are a generalization of the classical linear regression model. In matrix notation with a response vector \(\mathbf y\), a data matrix \(\mathbf X\), and a vector \(\boldsymbol \varepsilon\) containing the errors, a linear regression model is defined as

\[\mathbf y = \mathbf X \boldsymbol \beta + \boldsymbol \varepsilon.\]

In geostatistics, the error term usually is called the nugget. SVC models are introduced through a random effect \(\boldsymbol \eta(s)\) that is depending on the location \(s\). The underlying model takes the form

\[\mathbf y = \mathbf X \boldsymbol \beta + \mathbf W \boldsymbol \eta (s) + \boldsymbol \varepsilon\] where \(\mathbf W\) parses the covariates to the random effect \(\boldsymbol \eta(s)\). Specifically, each SVC is defined by a Gaussian process, more specifically a Gaussian random field (GRF), with mean zero and some stationary covariance function. This function usually depends on some marginal variance (sill) and a range. An example of 3 SVC, i.e.Â 3 GRFs, and the nugget \(\boldsymbol \varepsilon\) on a regular grid is given in the figure below.

```
# setting seed
set.seed(123)
# number of SVC
p <- 3
# sqrt of total number of observations
m <- 20
# covariance parameters
(pars <- data.frame(var = c(0.1, 0.2, 0.3),
scale = c(0.3, 0.1, 0.2)))
```

```
## var scale
## 1 0.1 0.3
## 2 0.2 0.1
## 3 0.3 0.2
```

```
nugget.var <- 0.05
# function to sample SVCs
sp.SVC <- fullSVC_reggrid(m = m, p = p,
cov_pars = pars,
nugget = nugget.var)
library(sp)
```

`## Warning: package 'sp' was built under R version 4.0.3`

Just by looking at the figure, one can see that

- SVC_1, SVC_2 and SVC_3 have some spatial clusteringâ€¦
- â€¦, whereas the nugget does not.
- The largest variance can be observed in SVC_2 and SVC_3.
- The largest spatial clusters are in SVC_1.
- They all vary around 0, which is expected as they are all zero mean GRF.

Given the response \(\mathbf y\), the data matrices \(\mathbf X\) and \(\mathbf W\), we want to estimate the coefficients \(\boldsymbol \beta\) as we would with a linear regression model, and further, try to understand the properties of \(\boldsymbol \eta(s)\). Another important aspect is that we are able to predict \(\boldsymbol \eta(s')\) for some new locations \(s'\). This is what we are about to do in the following sections.

As mentioned, the data set `meuse`

from the package `sp`

is quite well known.

```
# attach sp and load data
library(sp)
data("meuse")
# documentation
help("meuse")
# overview
summary(meuse)
```

```
## x y cadmium copper
## Min. :178605 Min. :329714 Min. : 0.200 Min. : 14.00
## 1st Qu.:179371 1st Qu.:330762 1st Qu.: 0.800 1st Qu.: 23.00
## Median :179991 Median :331633 Median : 2.100 Median : 31.00
## Mean :180005 Mean :331635 Mean : 3.246 Mean : 40.32
## 3rd Qu.:180630 3rd Qu.:332463 3rd Qu.: 3.850 3rd Qu.: 49.50
## Max. :181390 Max. :333611 Max. :18.100 Max. :128.00
##
## lead zinc elev dist
## Min. : 37.0 Min. : 113.0 Min. : 5.180 Min. :0.00000
## 1st Qu.: 72.5 1st Qu.: 198.0 1st Qu.: 7.546 1st Qu.:0.07569
## Median :123.0 Median : 326.0 Median : 8.180 Median :0.21184
## Mean :153.4 Mean : 469.7 Mean : 8.165 Mean :0.24002
## 3rd Qu.:207.0 3rd Qu.: 674.5 3rd Qu.: 8.955 3rd Qu.:0.36407
## Max. :654.0 Max. :1839.0 Max. :10.520 Max. :0.88039
##
## om ffreq soil lime landuse dist.m
## Min. : 1.000 1:84 1:97 0:111 W :50 Min. : 10.0
## 1st Qu.: 5.300 2:48 2:46 1: 44 Ah :39 1st Qu.: 80.0
## Median : 6.900 3:23 3:12 Am :22 Median : 270.0
## Mean : 7.478 Fw :10 Mean : 290.3
## 3rd Qu.: 9.000 Ab : 8 3rd Qu.: 450.0
## Max. :17.000 (Other):25 Max. :1000.0
## NA's :2 NA's : 1
```

`## [1] 155 14`

What we are interested in are the `cadmium`

measurements. On the plots below, the data is skewed, which is why we are going to work on the natural logarithm of `cadmium`

.

Adding the coordinate reference system (CRS, see `help(meuse)`

) and the sampling locations, one can obtain a spatial plot of the log cadmium measurements.

```
# construct spatial object
sp.meuse <- meuse
coordinates(sp.meuse) <- ~x+y
proj4string(sp.meuse) <- CRS("+init=epsg:28992")
# using package tmap
library(tmap)
# producing an interactive map
tmap_leaflet(tm_shape(sp.meuse) + tm_dots("l_cad", style = "cont"))
```

```
## Warning in CPL_crs_from_input(x): GDAL Message 1: +init=epsg:XXXX syntax is
## deprecated. It might return a CRS with a non-EPSG compliant axis order.
```

Generally, the values for `l_cad`

are highest close to the river Meuse. However, there is some spatial clustering comparing the center of all observations to the Northern part. We will now try model the `l_cad`

based on some of the covariates given in the data set `meuse`

.

As independent variables, we choose the following:

`dist`

, i.e.Â the normalized distance to the river Meuse.`lime`

, which is a 2-level factor indicating the presence of lime.`elev`

, i.e.Â the relative elevation above the local river bed.

I am not a geologist, but these covariates seem like a sensible choice, if I try to predict cadmium at a given location without taking further measurements of cadmium or other highly correlated heavy metals like copper, lead, or zinc.

Given the covariates, we continue to fit the data with different models and corresponding methods. Hence, we work our way up starting with a linear regression and geostatistical model.

We start with a linear regression model (LM) and ordinary least squares (OLS).

```
##
## Call:
## lm(formula = l_cad ~ 1 + dist + lime + elev, data = meuse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5731 -0.4291 0.1752 0.5395 1.4191
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.75085 0.56767 8.369 3.63e-14 ***
## dist -2.04497 0.39846 -5.132 8.69e-07 ***
## lime1 0.57632 0.16454 3.503 0.000606 ***
## elev -0.47304 0.07087 -6.675 4.42e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7685 on 151 degrees of freedom
## Multiple R-squared: 0.6141, Adjusted R-squared: 0.6064
## F-statistic: 80.09 on 3 and 151 DF, p-value: < 2.2e-16
```

The residual analysis shows:

The spatial distribution of the residuals is the following:

```
## cadmium copper lead zinc elev dist om ffreq soil lime landuse dist.m
## 1 11.7 85 299 1022 7.909 0.00135803 13.6 1 1 1 Ah 50
## 2 8.6 81 277 1141 6.983 0.01222430 14.0 1 1 1 Ah 30
## 3 6.5 68 199 640 7.800 0.10302900 13.0 1 1 1 Ah 150
## 4 2.6 81 116 257 7.655 0.19009400 8.0 1 2 0 Ga 270
## 5 2.8 48 117 269 7.480 0.27709000 8.7 1 2 0 Ah 380
## 6 3.0 61 137 281 7.791 0.36406700 7.8 1 2 0 Ga 470
## l_cad LM_res
## 1 2.4595888 0.8764648
## 2 2.1517622 0.1528248
## 3 1.8718022 0.4450312
## 4 0.9555114 0.2145201
## 5 1.0296194 0.3837506
## 6 1.0986123 0.7777244
```

```
# plot residuals at corresponding locations
tmap_leaflet(tm_shape(sp.meuse) + tm_dots("LM_res", style = "cont"))
```

One can observe that there is a spatial clustering of the residuals. This motivates us to use geostatistics.

The classical geostatistical model is a good reference and starting point for SVC models. Geostatistical models using a GRF are a special case of SVC models. More specifically, we want to model a single random effect and the data matrix \(\mathbf W\) only contains the intercept, such that the SVC model simplifies to the known geostatistical model

\[\mathbf y = \mathbf X \boldsymbol \beta + \boldsymbol \eta (s) + \boldsymbol \varepsilon.\]

Usually, we start by computing the empirical (semi-) variogramm and estimating a suitable model. Trying the exponential covariance function, we get the following:

`## Warning: package 'gstat' was built under R version 4.0.3`

```
# empirical variogram
eV <- variogram(LM_res ~ 1, sp.meuse)
# define variogram model with initial values
mV <- vgm(0.2, "Exp", 300, 0.4)
# fit model
(fV <- fit.variogram( eV, mV))
```

```
## model psill range
## 1 Nug 0.2343484 0.000
## 2 Exp 0.3845267 247.618
```

The fit looks reasonable and the estimated values are (literally) good starting points for our methodology coming up in the next section. So we can do ordinary kriging on the residuals.

```
# study area
data("meuse.grid")
coordinates(meuse.grid) <- ~x+y
proj4string(meuse.grid) <- proj4string(sp.meuse)
# kriging
GS.fit <- krige(LM_res ~ 1, sp.meuse,
newdata = meuse.grid, model = fV)
```

`## [using ordinary kriging]`