spatial.Rmd

---
title: "Spatial statistics in R"
subtitle: "Global and local spatial autocorrelation"
author: "Andrew Rate"
date: "`r Sys.Date()`"
output: 
  html_document:
    highlight: kate
---

<style type="text/css">
  body{
  font-size: 12pt;
}
</style>

<div style="border: 2px solid #039; padding: 8px;">
<p style="text-align:center; font-size:12pt;">
<em>Maps / spatial pages</em>: [Maps](maps.html){style="color:#04b;"} | 
[Spatial statistics](spatial.html){style="color:#04b;"} | 
[Spatial interpolation](spat-interp.html){style="color:#04b;"}</p>
</div>

&nbsp;

## Introduction 
![](./images/spatial-art.png){align="right" alt="decorative" width="302" height="271"}

When we have data with a spatial component, which environmental data usually do,
an important spatial analysis is to investigate the effect of sample distance on
environmental variables. This could be considered a test of &ldquo;Tobler’s
first law of geography&rdquo; (Tobler, 2004), stating:

> Everything is related to everything else,  
> but near things are more related than distant things.

Whether the values of an environmental variable are spatially related (or not)
is expressed by the spatial autocorrelation or Moran’s I (Zhang
*et al*. 2008). The Moran’s I statistic is based on comparison of the values of
a variable at one point with neighbouring point &ndash; either a specified
number of neighbours, or the neighbours within a specified distance. A positive
Moran’s I autocorrelation suggests that locations with similar values of the
variable considered tend to cluster together. A Moran’s I close to zero (no
autocorrelation) suggests values of the variable considered are randomly
located, and a larger negative Moran’s I suggests that similar values of the
variable considered tend to be further apart than expected from a random spatial
distribution. The null hypothesis for the Moran’s I statistic is that there is
no spatial autocorrelation. Moran's I will vary with the number of neighbouring
points in the calculation, with more points giving Moran's I values closer to
zero (Kalogirou 2019).

Moran’s I is a global autocorrelation, across the whole spatial dataset being
analysed. A *local Moran’s I* shows the extent of significant spatial clustering
of similar values (of the variable considered) around each observation. We will
look at both Global and Local Moran's I.

This Chapter will make simple maps with added data (see
[mapData.html](mapData.html) if you haven't already), and help you to
understand and analyse **spatial autocorrelation** using **R**.

We will use a subset of the National Geochemical Survey of Australia (NGSA) data
documented by [Caritat and Cooper
(2011)](https://www.ga.gov.au/about/projects/resources/national-geochemical-survey){target="_blank"} 
to illustrate the various concepts and procedures involved. We supply data in 
`.csv` files for importing into **R**.

In this Chapter we use a range of R packages for different functions, shown in
Table 1.

```{r load-packages+environment-invisibly, message=FALSE, warning=FALSE, echo=FALSE, include=FALSE}
library(lctools)
library(spdep)
library(gstat)
library(sf)
library(maps)
library(maptiles)
library(prettymapr)
library(viridis)
library(flextable)

set_flextable_defaults(theme_fun = "theme_zebra", 
                       font.size = 10, fonts_ignore = TRUE)
BorderDk <- officer::fp_border(color = "#B5C3DF", style = "solid", width = 1)
BorderLt <- officer::fp_border(color = "#FFFFFF", style = "solid", width = 1)
```

```{r tabulate-packages, echo=FALSE}
pklist <- 
  data.frame(Package=c("sf","gstat","maps","spdep","lctools","maptiles",
                               "prettymapr","viridis"),
        Purpose=c("Spatial data handling, formatting, and storing spatial data",
                   "Geostatistics (variograms and variogram models, kriging)",
      "Simple map outlines","Spatial autocorrelation, LISA",
      "Spatial autocorrelation",
      "Retrieving and plotting tile-based map data",
      "Map annotations",
      "Colourblind-friendly colour palettes for maps"),
      Reference=c("Pebesma (2018)", "Pebesma (2004)", 
                  "Becker et al. (2021)",
                  "Pebesma and Bivand (2023)", "Kalogirou (2020)",
                  "Giraud (2021)", "Dunnington (2022)",
                  "Garnier et al. (2021)"))
flextable(pklist) |> width(width=c(1.2,4.5,2.5)) |>
  border_outer(border=BorderDk, part = "all") |>
  bold(j=1, bold = TRUE) |> 
  set_caption(caption="Table 1: R packages used for spatial data analysis and visualisation.")
```

<p>&nbsp;</p>

```{r load-packages-but-dont-run, echo=TRUE, eval=FALSE, message=FALSE, warning=FALSE}
library(sf)
library(maptiles)
library(prettymapr)
library(spdep)
library(gstat)
library(lctools)
library(maps)
library(viridis)
```

## Making a base map

We saw how to do this in [mapData.html](mapData.html). The comments in the code
below should help to remind us what the purpose of the code is.

```{r make-basemap, warning=FALSE, error=FALSE, results='hold'}

# define coordinate reference systems

LongLat <- st_crs(4326) # uses Earth ellipsis specs from WGS84 datum
UTM50S <- st_crs(32750) # just for Zone 50, S hemisphere!

# import the data
git <- "https://raw.githubusercontent.com/Ratey-AtUWA/Learn-R-web/main/"
ngsaWA <- read.csv(file=paste0(git,"ngsaWA.csv"), 
                   stringsAsFactors = TRUE)
NGSAWA_border <- read.csv(paste0(git,"NGSAWA_border.csv"))

# make spatial data objects

ngsaWAsf <- st_as_sf(x = ngsaWA, coords = c("LONGITUDE","LATITUDE"),
                     crs = LongLat)
ngsaWAbord_sf <- st_as_sf(NGSAWA_border, coords = c("Long.est","Lat.est"),
                          crs = LongLat)

# define map extent

WA_extent <- st_as_sf(data.frame(Longitude = c(108, 129),
                                 Latitude = c(-38, -13)),
                      coords=c("Longitude","Latitude"), crs=LongLat) 

```

## Getting and plotting the map tile data

We now need some functions from the `maptiles` package (Giraud 2021). The 
`get_tiles` function downloads map tiles from the provider selected into an 
object of class `SpatRaster`. Then we plot using the `plot_tiles` function, 
specifying `axes=TRUE` and setting margins using `mar=` 
(Figure 1).

```{r base-WA-map, fig.height=7.6, fig.width=6, fig.align='center', fig.cap="Figure 1: Map of Western Australia (rectangular Long-Lat projection) used subsequently as the base map for spatial analyses. Generated using the `maptiles` R package, with OpenTopoMap tiles.", out.width="55%", warning=FALSE, message=FALSE, results='hide'}
WAtiles <- get_tiles(WA_extent, provider = "OpenTopoMap", 
                     crop = TRUE, zoom = 5) # make map object

par(oma=c(0,0,0,0), mar=c(3,3,0.5,0.5), mgp=c(1.4,0.3,0), 
    lend=2, ljoin=1, tcl=0.3)
plot_tiles(WAtiles, axes=TRUE, mar=c(3,3,0.5,0.5))
mtext(side=1, line=1.6, text="Longitude (\u00B0 E)",
      font=2, cex=1.2)
mtext(side=2, line=1.6, text="Latitude (\u00B0 S)",
      font=2, cex=1.2)
```

<p>&nbsp;</p>

### Simple line-based base map

We can also make a very simple base map at country scales using the `maps`
package (Figure 2). The code makes use of the native
**R** pipe operator `|>` to sequence R functions without nesting. Map objects
made using the `sf` package also work well with `ggplot2`.

```{r simple-wa-map0, eval=FALSE, message=FALSE, warning=FALSE, results='hide'}
wa_sf <- map(regions = "Australia", plot=F) |> 
  st_as_sf(coords = c("x","y"), crs = LongLat) |> 
  st_crop(xmin=108,ymin=-38,xmax=129,ymax=-13)
library(ggplot2)
ggplot() +
  geom_sf(data=wa_sf) +
  labs(y="Latitude (\u00B0S)", x = "Longitude (\u00B0E)") + 
  theme_bw()
```

```{r simple-wa-map, fig.height=5.7, fig.width=4.5, fig.align='center', fig.cap="Figure 2: Map of Western Australia (rectangular Long-Lat projection) as a simple alternative base map for spatial analyses. Generated using the `maps`, `sf`, and `ggplot2` R packages.", out.width="40%", warning=FALSE, message=FALSE, results='hold', echo=FALSE}
wa_sf <- map(regions = "Australia", plot=F) |> 
  st_as_sf(coords = c("x","y"), crs = LongLat) |> 
  st_crop(xmin=108,ymin=-38,xmax=129,ymax=-13)
library(ggplot2)
xlabs = seq(115,125, 5)
ylabs = seq(-35, -15, 5)
ggplot() +
  geom_sf(data=wa_sf) +
  labs(y="Latitude (\u00B0S)", x = "Longitude (\u00B0E)") + 
  scale_x_continuous(breaks = xlabs, labels = paste0(xlabs,'°E')) +
  scale_y_continuous(breaks = ylabs, labels = paste0(ylabs,'°S')) +
  theme_bw()
```

<p>&nbsp;</p>

## A first look at the data

plot the locations of samples in 
the NGSA data (Figure 3):

```{r ngsa-points-map, fig.height=7.6, fig.width=6, fig.align='center', fig.cap="Figure 3: Sample locations for the Western Australia subset of the NGSA data plotted over a `maptiles` raster background.", out.width="55%"}
par(oma=c(0,0,0,0), mar=c(3,3,0.5,0.5), mgp=c(1.4,0.3,0), 
    lend=2, ljoin=1, tcl=0.3)
plot_tiles(WAtiles, axes=TRUE, mar=c(3,3,0.5,0.5))
mtext(side=1, line=1.6, text="Longitude (\u00B0 E)",
      font=2, cex=1.2)
mtext(side=2, line=1.6, text="Latitude (\u00B0 S)",
      font=2, cex=1.2)

points(st_coordinates(ngsaWAsf)[,1:2], pch=3)
legend("topleft", bty="n", legend="Sample locations for\nNGSA data in WA",
       pch = 3)
addnortharrow(pos="topright", border=1, lwd=1, text.col=1, padin=c(0.2,0.2))
legend("bottom", bty="n", 
       legend = c("Omitting scale bar, as scale on map varies by\nmore than 5%, so scale bar may be inaccurate\n \n "),
       y.intersp=0.9)
```

<p>&nbsp;</p>

We can also use `ggplot`/`ggmap` to plot the data using `geom_sf()`. In the
example in Figure 4, we also add information on soil pH by including
`aes(col=pH)` and `scale_colour_viridis_c()` in the code.

```{r ggmap-ngsa-points, fig.height=7.6, fig.width=6, fig.align='center', fig.cap="Figure 4: Sample locations and soil pH for the Western Australia subset of the NGSA data plotted usiing `ggmap` over a simple background.", out.width="40%"}
xlabs = seq(115,125, 5)
ylabs = seq(-35, -15, 5)
ggplot() +
  geom_sf(data=wa_sf, fill="grey80") +
  geom_sf(data=ngsaWAsf, aes(col=pH, size=2)) +
  labs(y="Latitude", x = "Longitude") + 
  scale_colour_viridis_c(option="H", direction=-1) +
  scale_x_continuous(breaks = xlabs, labels = paste0(xlabs,'°E')) +
  scale_y_continuous(breaks = ylabs, labels = paste0(-1*ylabs,'°S')) +
  theme_bw() +
  theme(axis.text=element_text(size=14, color="black"),
        axis.title=element_text(size=16,face="bold"),
        legend.title=element_text(size=16),
        legend.text=element_text(size=14),
        legend.key.height=unit(2.4, units="cm")) + 
  guides(col=guide_colourbar(title="Soil\npH"), size="none")
```

<p>&nbsp;</p>

### Plot a map with range-class symbols

First we make a simple features spatial data object:

```{r}
As.AR_sf <- st_as_sf(ngsaWA[,c("LONGITUDE","LATITUDE","As.AR")], 
                     coords = c("LONGITUDE","LATITUDE"),
                     crs = LongLat)
```

Identifying each point as belonging to a range, *if* suitable ranges are chosen,
is a convenient and relatively simple way to explore the spatial distribution of
values of a variable. In this example we use the Tukey boxplot thresholds as
these are specifically designed for exploratory data analysis and can easily
identify potentially unusual values (outliers). Percentile classes can also be
used, and there is a good discussion of the options in Chapter 5 of Reimann *et*
*al*. (2008). In this example we log~10~-transform As concentration, as its 
untransformed distribution has a substantial positive skew.

```{r map-range-classes, echo=1:19, fig.align='center', fig.cap="Figure 5: Map of arsenic (As) concentrations expressed as symbols for concentration ranges (UQR is 75th percentile, upWx is upper whisker, med is median, LQR is 25th percentile, loWx is lower whisker). Data are from from the WA subset of the NGSA data, -2mm tosoil fraction.", fig.height=7.5, out.width="55%", fig.width=6, message=FALSE, warning=FALSE}
palette(c("black",viridis::turbo(6, alp=0.7, end=0.9),"grey92", "white"))
par(oma=c(0,0,0,0), mar=c(3,3,0.5,0.5), mgp=c(1.4,0.3,0), 
    lend=2, ljoin=1, tcl=0.3)
plot_tiles(WAtiles, axes=TRUE, mar=c(3,3,0.5,0.5))
mtext(side=1, line=1.6, text="Longitude (\u00B0 E)",
      font=2, cex=1.2)
mtext(side=2, line=1.6, text="Latitude (\u00B0 S)",
      font=2, cex=1.2)
addnortharrow(pos="topleft", border=1, lwd=1, text.col=1,  
              padin=c(0.2,0.2), scale=1.2)
legend("bottomright", bty="n", 
       legend = c("Omitting scale bar, as scale","on map varies by more than 5%",
                  "so scalebar may be inaccurate"),
       y.intersp=0.9)
# construct and plot range classes in Tukey boxplot thresholds
As.AR_sf$As.log <- log10(As.AR_sf$As.AR)
As.AR_sf$tukey <- cut(As.AR_sf$As.log, 
    breaks = c(-999999,boxplot.stats(As.AR_sf$As.log)$stats,999999),
    labels = c("lo-outliers","loWx-LQR","LQR-med","med-UQR","UQR-upWx","hi+outliers"))
polygon(NGSAWA_border$Long.est, NGSAWA_border$Lat.est, lty=3, lwd=2,
        border="gray")
points(st_coordinates(As.AR_sf), pch = 21,
       cex=(c(.5,.65,.85,1,1.4,1.8)*2)[As.AR_sf$tukey], 
       bg=seq(2,7)[As.AR_sf$tukey])
     
legend("bottomleft", legend = rev(c("lo-outliers","loWx-LQR","LQR-med",
                                "med-UQR","UQR-upWx","hi+outliers")), 
       cex = 1.2, pch=21, pt.cex=rev(c(.5,.65,.85,1,1.4,1.8))*2, inset = 0.07, 
       bty="n", y.intersp=1.2, pt.bg=seq(7,2,-1), x.intersp = 1.)
text(par("usr")[1], par("usr")[3]+(0.38*(par("usr")[4]-par("usr")[3])), 
     labels=expression(bold("TOS -2mm\nAs.AR ranges")), pos = 4, cex = 1)
par(new=TRUE, fig=c(0.08,0.18,0.19,0.36), mar=c(0,0,0,0)+0.1,
    col.lab=1, col.axis=1, mgp=c(.9,0.2,0), xaxt="n", yaxt="n", bty="n", ann=F) 
boxplot(log10(As.AR_sf$As.log), col = "#A0A0A080")
```

<p>&nbsp;</p>

The map's symbol classes in Figure 5 are separated by
the standard Tukey box plot thresholds (minimum, lower-hinge, median,
upper-hinge, maximum) &ndash; the legend is roughly aligned with the relevant
box plot. We see a wide range of As concentration, with greater As probably
associated with iron-rich rocks in the Pilbara (northwest WA) and central
Yilgarn (central south WA).

<p>&nbsp;</p>

## Spatial Autocorrelation

Whether samples are spatially related or not, in terms of a particular measured
variable, is expressed by the *spatial autocorrelation* or **Moran's I**
(Zhang *et al*., 2008). The Moran's I statistic is based on comparison of the
values of a variable at one point with a specified number (or within a specified
distance) of neighbouring points. A positive Moran I autocorrelation suggests
that locations with similar values of the variable considered tend to cluster
together. A Moran's I close to zero suggests no autocorrelation (values of the
variable considered are randomly located), and a larger negative Moran's I
suggests that similar values of the variable considered tend to be further apart
than expected from a random spatial distribution. The Moran's I statistic is
tested against the null hypothesis of no spatial autocorrelation, and will vary
with the number of neighbouring points in the calculation, with more points
giving weaker autocorrelations (Kalogirou, 2019).

The basic Moran's I is a *global* autocorrelation, across the whole spatial
dataset being analysed. The *Local* Moran's I can also be calculated, and shows
the extent of significant spatial clustering of similar values (of the variable
considered) around each observation.

Spatial autocorrelation statistics, usually Moran's I, can be calculated using
various GIS and statistical software, including several packages which add
functionality to R (we use the `spdep` package).

### Calculate Global Moran's I

The subset of the National Geochemical Survey of Australia (NGSA) data we are
using is in the object `ngsaWAsf`.

We first use the `knearneigh()` function from the `spdep` R package to create a
matrix with the row numbers of points belonging to the set of the k nearest
neighbours of each other. We don't actually save the output but use the native
**R** pipe operator `|>` to run the output through (1) the `knn2nb()` function 
to convert to an object of class `nb` which includes neighbour region numbers, 
then (2) the `nb2listw` to create a list of weights (we can see what these look 
like by running `str(ngsaWA_nbwt)`, or `print(ngsaWA_nbwt)` for a summary).

Next we use the `moran.test()` function to generate our Moran's I statistic. The 
null hypothesis is that the observations are random with no spatial correlation.

```{r global-Moran, echo=TRUE, message=FALSE, warning=FALSE, paged.print=FALSE, results='hold'}
ngsaWA_nbwt <- knearneigh(x=ngsaWAsf, k=8) |> knn2nb() |> nb2listw()
ngsaWAsf$As.log <- log10(ngsaWAsf$As.AR)
(mI <- moran.test(ngsaWAsf$As.log, ngsaWA_nbwt)) # ; str(mI) # if needed
```

The `Expectation` in the output is the estimated value of Moran's I if the null
hypothesis is correct. We can try running the test with different numbers of
neighbour points, or use a distance-based criterion for defining neighbours.

### Calculate local Moran's I

In this example we use the `localmoran()` function from the `spdep` R package.
Once again we use the `knearneigh()` function to define which points are
neighbours (with `k=4` this time), and convert these to weights.

It's easy to modify the code to calculate Local Moran's I for another variable,
by editing the text string which is assigned to `var0`. For a list of
variables in the original data subset we could run `names(ngsaWA)`.

To make it easier to extract and map the output, we combine our data coordinates
`st_coordinates(ngsaWAsf)` with two extracts of our local Moran's I output 
object `lmi_As`: 

1.    `as.data.frame(lmi_As)`, which contains the values of local Moran's I 
(`Ii`), expected `Ii` if H~0~ is true (`E.Ii`), variance in Ii (`Var.Ii`), the 
standard normal score for `Ii` (`Z,Ii`), and the p-value (`Pr(z != E(Ii))`);  
2. `attributes(lmi_As)$quadr`, which contains the Local Indicators of Spatial 
Association (LISA) categories ('quadrants') for each observation.

```{r calc-local-mI, paged.print=FALSE}
var0 <- "As.log"
lmi_data <- na.omit(ngsaWAsf[,var0])
# lmi_data$As.log <- log10(lmi_data$As.AR)
lmi_wts <- knearneigh(ngsaWAsf,k=4) |> knn2nb() |> nb2listw()

lmi_As <- localmoran(lmi_data$As.log, lmi_wts)
lmiDF_As <- cbind(st_coordinates(ngsaWAsf),as.data.frame(lmi_As), attributes(lmi_As)$quadr)
colnames(lmiDF_As) <- c("Longitude","Latitude",
                        "Ii", "E.Ii", "Var.Ii", "Z.Ii", "Pvalue", 
                        "mean", "median", "pysal")
print(head(lmiDF_As[,3:10]), digits=3)
```

```{r rm-temp-lmi-objs, include=FALSE}
rm(list=c("var0", "lmi_data", "lmi_wts","lmi_As")) 
```

### Plot local Moran's I

We only want to display points having significant local Moran's I, so we subset
the results data frame from above using the `which()` function. We subset these
data further into positive and negative values of local Moran's I (using the
`subset()` function), since the `symbols()` function cannot input negative
values. Symbols are plotted with `sqrt()` values since human perception compares
areas better than diameters. 

```{r map-local-moransI, fig.align='center', out.width="50%", fig.cap="Figure 6: Map of Local Moran's I for arsenic (As) concentrations in the WA subset of the NGSA data, -2mm tosoil fraction. The Global Moran's I parameter is also shown above the legend.", message=FALSE, warning=FALSE, fig.height=6.4, fig.width=5, results='hold'}
lmi_signif <- lmiDF_As[which(lmiDF_As$Pvalue <= 0.05),c(1:3,7,9)]

par(oma=c(0,0,0,0), mar=c(4,4,1.5,1.5), mgp=c(1.4,0.3,0), 
    lend=2, ljoin=1, tcl=0.3, tcl = 0.5, lwd = 1, font.lab=2)
wa_outline <- subset(st_coordinates(wa_sf)[,c(1:2)], 
                     st_coordinates(wa_sf)[,4]==5) # subset omits islands
plot(wa_outline, asp=1.05, pch=".", col="grey", xlim=c(110,129), ylim=c(-38,-13), 
     xlab="Longitude (\u00b0E)", ylab="Latitude (\u00b0S)",xaxs="i")
polygon(rbind(wa_outline,wa_outline[1,]), col="grey88", border = "grey")
points(st_coordinates(ngsaWAsf),pch=3,col="grey60",cex=0.6)
neg_lmi <- subset(lmi_signif, lmi_signif$Ii<0)
pos_lmi <- subset(lmi_signif, lmi_signif$Ii>=0)
symbols(pos_lmi[,1:2], inches=F, circles=sqrt(pos_lmi$Ii*0.4), add=T, 
        fg="blue3", bg="#0000FF40")
symbols(neg_lmi[,1:2], inches=F, squares=sqrt(neg_lmi$Ii*-0.4), add=T, 
        fg="lightgoldenrod1", bg="#FFE00080")
addnortharrow(pos="topright", border=1, lwd=1, text.col=1,  
              padin=c(0.1,0.1), scale=0.5)
textMI=paste0("Global Moran's I = ",signif(mI$estimate[1],4))
legend("topleft", cex=0.9, bg="grey88", box.col = "grey", inset=0.02,
       legend=c("Local Moran's I \u2265 0","Local Moran's I < 0",
                "Sample locations"), 
       pch=c(21,22,3), col=c("blue3","lightgoldenrod1","grey60"), 
       pt.bg=c("#0000FF40","#FFE00080",NA),
       pt.cex=c(3,3,0.5), title = textMI)
legend("bottomright", bty="n", inset = 0.01, 
       legend = c("Not plotting scale bar as scale",
                  "on map varies by more than 5%,",
                  "so scalebar may be inaccurate"),
       y.intersp=0.9, cex = 0.8, text.col = "grey60")
box()
```

```{r tidy-up-mI, include=FALSE}
rm(list=c("neg_lmi","pos_lmi")) # tidy up
```

<p>&nbsp;</p>

The map in Figure 6 shows numerous sample locations
having positive Local Moran's I for the selected variable, and a significant
global spatial autocorrelation (Moran's I &cong; 0.46, with a very low p-value).

We can also use the information stored in our local Moran's I data frame to plot
a map of **l**ocal **i**ndicators of **s**patial **a**ssociation (LISA; see
Table 2).

<p>&nbsp;</p>

```{r tabulate-lisa, echo=FALSE}
lisalist<-data.frame(LISAQ=c("Low-Low","High-Low","Low-High","High-High"),
        Variable=c("< median","> median","< median","> median"),
      lmi=c("positive", "negative", "negative", "positive"))
flextable(lisalist,cwidth=c(2.4,2.4,2.4)) |> 
  border_outer(border=BorderDk, part = "all") |>
  set_header_labels(LISAQ = "LISA quadrant", 
    Variable = "Variable value", lmi = "Local Moran's I" ) |>
  footnote(i=1, j=2, value = as_paragraph("LISA quadrants can also be defined relative to the mean."), ref_symbols = "a", part="header") |>
  style(i=c(2,4), j=2, pr_t = fp_text_default(color="red")) |>
  style(i=c(1,3), j=2, pr_t = fp_text_default(color="blue")) |>
  style(i=c(1,4), j=3, pr_t = fp_text_default(color="red")) |>
  style(i=c(2,3), j=3, pr_t = fp_text_default(color="blue")) |>
  set_caption(caption="Table 2: Explanation of LISA (Local Indicators of Spatial Association) categories.")
```

<p>&nbsp;</p>

```{r map-lisa, fig.align='center', out.width="50%", fig.cap="Figure 7: Map of Local Indicators of Spatial Association (LISA) relative to median for arsenic (As) concentrations in the WA subset of the NGSA data, –2 mm topsoil fraction.", message=FALSE, warning=FALSE, fig.height=6.4, fig.width=5, results='hold'}
lmi_signif <- lmiDF_As[which(lmiDF_As$Pvalue <= 0.05),]
palette(c("black",RColorBrewer::brewer.pal(7,"RdBu")))
par(oma=c(0,0,0,0), mar=c(4,4,1.5,1.5), mgp=c(1.4,0.3,0), 
    lend=1, ljoin=0, tcl=0.3, tcl = 0.5, lwd = 1, font.lab=2)
wa_outline <- subset(st_coordinates(wa_sf)[,c(1:2)], 
                     st_coordinates(wa_sf)[,4]==5) # subset omits islands
plot(wa_outline, asp=1.05, pch=".", col="grey", xlim=c(110,129), ylim=c(-38,-13), 
     xlab="Longitude (\u00b0E)", ylab="Latitude (\u00b0S)",xaxs="i")
polygon(rbind(wa_outline,wa_outline[1,]), col="grey88", border = "grey")
points(st_coordinates(ngsaWAsf),pch=3,col="grey60",cex=0.6)
with(lmi_signif, 
     points(Longitude, Latitude, lwd=2,
            pch=c(21,25,24,22)[median], 
            col=c(8,8,2,2)[median], 
            bg=c(7,3,7,3)[median],
            cex = c(1.2,1,1,1.2)[median])
     )
addnortharrow(pos="topright", border=1, lwd=1, text.col=1,  
              padin=c(0.1,0.1), scale=0.5)
legend("topleft", cex=0.9, bg="grey88", box.col = "grey", inset=0.02,
       legend=levels(lmi_signif$median), title="LISA Quadrants",
       pch=c(21,25,24,22), col=c(8,8,2,2),
       pt.bg=c(7,3,7,3), pt.cex = c(1.2,1,1,1.2))
legend("bottomright", bty="n", inset = 0.01, 
       legend = c("Not plotting scale bar as scale",
                  "on map varies by more than 5%,",
                  "so scalebar may be inaccurate"),
       y.intersp=0.9, cex = 0.8, text.col = "grey60")
box()
```

```{r tidy-up-lmi, include=FALSE}
rm(list=c("mI", "ngsaWA_nbwt","lmi_signif")) # tidy up
```

<p>&nbsp;</p>

The LISA map enables us to identify localized map regions where values of the
chosen variable in or data are strongly positively or negatively associated with
one another, that is, different spatial clusters. Usually locations having
significant Low-High or High-Low association are isolated and less common, as we
see in Figure 7.

<div style="border: 2px solid #039; background-color:#fec; padding: 8px;"> 
## What's next?

In the [next Session](spat-interp.html), we build on the understanding of 
spatial autocorrelation to introduce spatial interpolation, in particular the
concepts of variograms and kriging.
</div>

<p>&nbsp;</p>


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Caritat, P., Cooper, M., 2011. *National Geochemical Survey of Australia: The Geochemical Atlas of Australia*. GA Record 2011/20 (2 Volumes). Geoscience Australia, Canberra, 557 pp. [https://www.ga.gov.au/about/projects/resources/national-geochemical-survey/atlas](https://www.ga.gov.au/about/projects/resources/national-geochemical-survey/atlas){target="_blank"}

Dunnington, D. 2017. *prettymapr: Scale Bar, North Arrow, and Pretty Margins in R*. R package version 0.2.2. [https://CRAN.R-project.org/package=prettymapr](https://CRAN.R-project.org/package=prettymapr){target="_blank"}

Garnier, Simon, Noam Ross, Robert Rudis, Antônio P. Camargo, Marco Sciaini, and Cédric Scherer (2021). *viridis - Colorblind-Friendly Color Maps for R*. R package version 0.6.2. [https://sjmgarnier.github.io/viridis/](https://sjmgarnier.github.io/viridis/){target="_blank"}

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Kalogirou, S., 2019. *Spatial Autocorrelation, Vignette for R package 'lctools'*, [https://cran.r-project.org/web/packages/lctools/vignettes/SpatialAutocorrelation.pdf](https://cran.r-project.org/web/packages/lctools/vignettes/SpatialAutocorrelation.pdf){target="_blank"}. 

Kalogirou, S. 2020. *lctools: Local Correlation, Spatial Inequalities, Geographically Weighted Regression and Other Tools*. R package version 0.2-8. [https://CRAN.R-project.org/package=lctools](https://CRAN.R-project.org/package=lctools){target="_blank"}.

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Pebesma, E., 2018. Simple Features for R: Standardized Support for Spatial Vector Data. *The R Journal* **10** (1), 439-446, [https://doi.org/10.32614/RJ-2018-009](https://doi.org/10.32614/RJ-2018-009){target="_blank"}. (package `sf`)

Pebesma E, Bivand R (2023). *Spatial Data Science With Applications in R*. Chapman & Hall.   [https://r-spatial.org/book/](https://r-spatial.org/book/){target="_blank"}. (package `spdep`)

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<p>&nbsp;</p>