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AnIntroductiontoMappingandSpatialModellinginR

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An Introduction to Mapping and Spatial Modelling in R. © Richard Harris, 2013 1

An Introduction to Mapping and Spatial Modelling R

By and © Richard Harris, School of Geographical Sciences, University of Bristol

An Introduction to Mapping and Modelling R by Richard Harris is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.Based on a work at www.social-statistics.org.

You are free:

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Attribution — You must attribute the work in the following manner: Based on An Introduction to Mapping and Spatial Modelling R by Richard Harris (www.social-statistics.org).

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(Document version 0.1, November, 2013. Draft version.)

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Introduction and contents

This document presents a short introduction to R highlighting some geographical functionality. Specifically, it provides:

• A basic introduction to R (Session 1)

• A short 'showcase' of using R for data analysis and mapping (Session 2)

• Further information about how R works (Session 3)

• Guidance on how to use R as a simple GIS (Session 4)

• Details on how to create a spatial weights matrix (Session 5)

• An introduction to spatial regression modelling including Geographically Weighted Regression (Session 6)

Further sessions will be added in the months (more likely, years) ahead.

The document is provided in good faith and the contents have been tested by the author. However, use is entirely as the user's risk. Absolutely no responsibility or liability is accepted by the author for consequences arising from this document howsoever it is used. It is is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (see above).

Before starting the following should be considered.

First, you will notice that in this document the pages and, more unusually, the lines are numbered. The reason is educational: it makes directing a class to a specific part of a page easier and faster. For other readers, the line numbers can be ignored.

Second, the sessions presume that, as well as R, a number of additional R packages (libraries) have been installed and are available to use. You can install them by following the 'Before you begin' instructions below.

Third, each session is written to be completed in a single sitting. If that is not possible, then it would normally be possible to stop at a convenient point, save the workspace before quitting R, then reload the saved workspace when you wish to continue. Note, however, that whereas the additional packages (libraries) need be installed only once, they must be loaded each time you open R and require them. Any objects that were attached before quitting R also need to be attached again to take you back to the point at which you left off. See the sections entitled 'Saving and loading workspaces', 'Attaching a data frame' and 'Installing and loading one or more of the packages (libraries)' on pages 10, 31 and 37 for further information.


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Before you begin

Install R. It can be downloaded from http://cran.r-project.org/.I currently am using version 3.0.2.

Start R. Use the drop-down menus to change your working directory to somewhereyou are happy to download all the files you need for this tutorial.

At the > prompt type,download.file("http://dl.dropboxusercontent.com/u/214159700/RIntro.zip", "Rintro.zip")

and press return.

Next, type unzip("Rintro.zip")

All the data you need for the sessions are now available in the working directory.

If you would like to install all the libraries (packages) you need for these practicals, typeload(“begin.RData”)

and theninstall.libs()

You are advised to read Installing and loading one or more of the packages (libraries) on p. 37before doing so.

Please note:this is a draft version of the document and has not as yet

been thoroughly checked for typos and other errors.


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http://cran.r-project.org/

Session 1: Getting Started with R

This session provides a brief introduction to how R works and introduces some of the more common commands and procedures. Don't worry if not everything is clear at this stage. The purpose is to get you started not to make you an expert user. If you would prefer to jump straight to seeing R in action, then move on the Session 2 (p.13) and come back to this introduction later.

1.1 About R

R is an open source software package, licensed under the GNU General Public Licence. You can obtain and install it for free, with versions available for PCs, Macs and Linux. To find out what is available, go to the Comprehensive R Archive Network (CRAN) at http://cran.r-project.org/

Being free is not necessarily a good reason to use R. However, R is also well developed, well documented, widely used and well supported by an extensive user community. It is not just software for 'hobbyists'. It is widely used in research, both academic and commercial. It has well developed capabilities for mapping and spatial analysis.

In his book R in a Nutshell (O'Reilly, 2010), Joseph Adler writes, “R is very good at plotting graphics, analyzing data, and fitting statistical models using data that fits in the computer's memory.” Nevertheless, no software provides the perfect tool for every job and Adler adds that “it's not good at storing data in complicated structures, efficiently querying data, or working with data that doesn't fit in the computer's memory.”

To these caveats it should be added that R does not offer spreadsheet editing of data of the type found, for example, in Microsoft Excel. Consequently, it is often easier to prepare and 'clean' data prior to loading them into R. There is an add-in to R that provides some integration with Excel. Go to http://rcom.univie.ac.at/ and look for RExcel.

A possible barrier to learning R is that it is generally command-line driven. That is, the user types a command that the software interprets and responds to. This can be daunting for those who are used to extensive graphical user interfaces (GUIs) with drop-down menus, tabs, pop-up menus, left or right-clicking and other navigational tools to steer you through a process. It may mean that R takes a while longer to learn; however, that time is well spent. Once you know the commands it is usually much faster to type them than to work through a series of menu options. They can be easily edited to change things such as the size or colour of symbols on a graph, and a log or script of the commands can be saved for use on another occasion or for sharing with others.

Saying that, a fairly simple and platform independent GUI called R Commander can be installed (see http://cran.r-project.org/web/packages/Rcmdr/index.html). Field et al.'s book Discovering Statistics Using R provides a comprehensive introduction to statistical analysis in R using both command-lines and R Commander.

1.2 Getting Started

Assuming R has been installed in the normal way on your computer, clicking on the link/shortcut to R on the desktop will open the RGui, offering some drop-down menu options, and also the R Console, within which R commands are typed and executed. The appearance of the RGui differs a little depending upon the operating system being used (Windows, Mac or Linux) but having used one it should be fairly straightforward to navigate around another.


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http://cran.r-project.org/web/packages/Rcmdr/index.html

http://rcom.univie.ac.at/


Figure 1.1. Screen shot of the R Gui for Windows

1.2.1 Using R as a calculator

At its simplest, R can be used as a calculator. Typing 1 + 1 after the prompt > will (after pressing the return/enter key, ↵) produce the result 2, as in the following example:

> 1 + 1[1] 2

Comments can be indicated with a hash tag and will be ignored

> # This is a comment, no need to type it

Some other simple mathematical expressions are given below.

> 10 - 5[1] 5> 10 * 2[1] 20> 10 - 5 * 2 # The order of operations gives priority to

# multiplication[1] 0> (10 - 5) * 2 # The use of brackets changes the order[1] 10> sqrt(100) # Uses the function that calculates the square root[1] 10> 10^2 # 102

[1] 100> 100^0.5 # 100.5, i.e. the square root again[1] 10> 10^3[1] 1000> log10(100) # Uses the function that calculates the common log[1] 2> log10(1000)[1] 3> 100 / 5[1] 20> 100^0.5 / 5[1] 2


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1.2.2 Incomplete commands

If you see the + symbol instead of the usual (>) prompt it is because what has been typed is incomplete. Often there is a missing bracket. For example,

> sqrt(+ 100 # The + symbol indicates that the command is incomplete+ )[1] 10> (1 + 2) * (5 - 1+ )[1] 12

Commands broken over multiple lines can be easier to read.

> for (i in 1:10) { # This is a simple loop+ print(i) # printing the numbers 1 to 10 on-screen+ }[1] 1[1] 2[1] 3[1] 4[1] 5[1] 6[1] 7[1] 8[1] 9[1] 10

1.2.3 Repeating or modifying a previous command

If there is a mistake in a line of a code that needs to be corrected or if some previously typed commands will be repeated then the ↑ and ↓ keys on the keyboard can be used to scroll between previous entries in the R Console. Try it!

1.3 Scripting and Logging in R

1.3.1 Scripting

You can create a new script file from the drop down menu File → New script (in Windows) or File → New Document (Mac OS). It is basically a text file in which you could write, for example,

a <- 1:10print(a)

In Windows, if you move the cursor up to the required line of the script and press Ctrl + R, then it will be run in the R Console. So, for example, move the cursor to where you have typed a <- 1:10 and press Ctrl + R. Then move down a line and do the same. The contents of a, the numbers 1 to 10, should be printed in the R Console. If you continue to keep the focus on the Scripting window and go to Edit in the RGui you will find an option to run everything. Similar commands are available for other Operating Systems (e.g. Mac key + Return). You can save files and load previously saved files.

Scripting is both good practice and good sense. It is good practice because it allows for reproducibility of your work. It is good sense because if you need to go back and change things you can do so easily without having to start from scratch.

Tip: It can be sensible to create the script in a simple text editor that is independent of R, such as


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Notepad. Although you will not be able to use Ctrl + R in the same way, if R crashes for any reason you will not lose your script file.

1.3.2 Logging

You can save the contents of the R Console window to a text file which will then give you a log file of the commands you have been using (including any mistakes). The easiest way to do this is to click on the R Console (to take the focus from the Scripting window) and then use File → Save History (in Windows) or File → Save As (Mac). Note that graphics are not usually plotted in the R Console and therefore need to be saved separately.

1.4 Some R Basics

1.4.1 Functions, assignments and getting help

It is helpful to understand R as an object-oriented system that assigns information to objects within the current workspace. The workspace is simply all the objects that have been created or loaded since beginning the session in R. Look at it this way: the objects are like box files, containing useful information, and the workspace is a larger storage container, keeping the box files together. A useful feature of this is that R can operate on multiple tables of data at once: they are just stored as separate objects within the workspace.

To view the objects currently in the workspace, type

> ls()character(0)

Doing this runs the function ls(), which lists the contents of the workspace. The result, character(0), indicates that the workspace is empty. (Assuming it currently is).

To find out more about a function, type ? or help with the function name,

> ?ls()> help(ls)

This will provide details about the function, including examples of its use. It will also list the arguments required to run the function, some of which may be optional and some of which may have default values which can be changed as required. Consider, for example,

> ?log()

A required argument is x, which is the data value or values. Typing log() omits any data and generates an error. However, log(100) works just fine. The argument base takes a default value of e1

which is approximately 2.72 and means the natural logarithm is calculated. Because the default is assumed unless otherwise stated so log(100) gives the same answer as log(100, base=exp(1)). Using log(100, base=10) gives the common logarithm, which can also be calculated using the convenience function log10(100).

The results of mathematical expressions can be assigned to objects, as can the outcome of many commands executed in the R Console. When the object is given a name different to other objects within the current workspace, a new object will be created. Where the name and object already exist, the previous contents of the object will be over-written, without warning – so be careful!

> a <- 10 – 5> print(a)[1] 5> b <- 10 * 2


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> print(b)[1] 20> print(a * b)[1] 100> a <- a * b> print(a)[1] 100

In these examples the assignment is achieved using the combination of < and -, as in a <- 100. Alternatively, 100 -> a could be used or, more simply, a = 100. The print(..)command can often be omitted, though it is useful, and sometimes necessary (for example, when what you hope should appear on-screen doesn't).

> f = a * b> print(f)[1] 2000> f[1] 2000> sqrt(b)[1] 4.472136> print(sqrt(b), digits=3) # The additional parameter now specifies[1] 4.47 # the number of significant figures> c(a,b) # The c(...) function combines its arguments[1] 100 20> c(a,sqrt(b))[1] 100.000000 4.472136> print(c(a,sqrt(b)), digits=3)[1] 100.00 4.47

1.4.2 Naming objects in the workspace

Although the naming of objects is flexible, there are some exceptions,

> _a <- 10Error: unexpected input in "_"> 2a <- 10Error: unexpected symbol in "2a"

Note also that R is case sensitive, so a and A are different objects

> a <- 10> A <- 20> a == A[1] FALSE

The following is rarely sensible because it won't appear in the workspace, although it is there,

> .a <- 10> ls()[1] "a" "b" "f"> .a[1] 10> rm(.a, A) # Removes the objects .a and A (see below)

1.4.3 Removing objects from the workspace

From typing ls() we know when the workspace is not empty. To remove an object from the workspace it can be referenced to explicitly – as in rm(A) – or indirectly by its position in the workspace. To see how the second of these options will work, type

> ls()


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[1] "a" "b" "f"

The output returned from the ls() function is here a vector of length three where the first element is the first object (alphabetically) in the workspace, the second is the second object, and so forth. We can access specific elements by using notation of the form ls[index.number]. So, the first element – the first object in the workspace – can be obtained using,

> ls()[1] # Get the brackets right! some rounded some square[1] "a"> ls()[2][1] "b"

Note how the square brackets […] are used to reference specific elements within the vector. Similarly,

> ls()[3][1] "f"> ls()[c(1,3)][1] "a" "f"> ls()[c(1,2,3)][1] "a" "b" "f"> ls()[c(1:3)] # 1:3 means the numbers 1 to 3[1] "a" "b" "f"

Using the remove function, rm(...), the second and third objects in the workspace can be removed using

> rm(list=ls()[c(1,3)])> ls()[1] "b"

Alternatively, objects can be removed by name

> rm(b)

To delete all the objects in the workspace and therefore empty it, type the following code but – be warned! – there is no undo function. Whenever rm(...) is used the objects are deleted permanently.

> rm(list=ls())> ls()character(0) # In other words, the workspace is empty

1.4.4 Saving and loading workspaces

Because objects are deleted permanently, a sensible precaution prior to using rm(...) is to save the workspace. To do so permits the workspace to be reloaded if necessary and the objects recovered. One way to save the workspace is to use

> save.image(file.choose(new=T))

Alternatively, the drop-down menus can be used (File → Save Workspace in the Windows version of the RGui). In either case, type the extension .RData manually else it risks being omitted, making it harder to locate and reload what has been saved. Try creating a couple of objects in your workspace and then save it with the names workspace1.RData

To load a previously saved workspace, use

> load(file.choose())

or the drop-down menus.

When quitting R, it will prompt to save the workspace image. If the option to save is chosen it will


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be saved to the file .RData within the working directory. Assuming that directory is the default one, the workspace and all the objects it contains will be reloaded automatically each and every time R is opened, which could be useful but also potentially irritating. To stop it, locate and delete the file. The current working directory is identified using the get working directory function, getwd() and changed most easily using the drop-down menus.

> getwd()[1] "/Users/rich_harris"

(Your working directory will differ from the above)

Tip: A good strategy for file management is to create a new folder for each project in R, saving the workspace regularly using a naming convention such as Dec_8_1.RData, Dec_8_2.RData etc. That way you can easily find and recover work.

1.5 Quitting R

Before quitting R, you may wish to save the workspace. To quit R use either the drop-down menus or

> q()

As promised, you will be prompted whether to save the workspace. Answering yes will save the workspace to the file .RData in the current working directory (see section 1.4.4, 'Saving and loading workspaces', on page 10, above). To exit without the prompt, use

> q(save = "no")

Or, more simply,

> q("no")

1.6 Getting Help

In addition to the use of the ? or help(…) documentation and the material available at CRAN, http://cran.r-project.org/, R has an active user community. Helpful mailing lists can be accessed from www.r-project.org/mail.html.

Perhaps the best all round introduction to R is the An Introduction to R which is freely available at CRAN (http://cran.r-project.org/manuals.html) or by using the drop-down Help menus in the RGui. It is clear and succinct.

I also have a free introduction to statistical analysis in R which accompanies the book Statistics for Geography and Environmental Science. It can be obtained from http://www.social-statistics.org/?p=354.

There are many books available. My favourite, with a moderate level statistical leaning and written with clarity is,

Maindonald, J. & Braun, J., 2007. Data Analysis and Graphics using R (2nd edition). Cambridge: CUP.

I also find useful,

Adler, J., 2010. R in a Nutshell. O'Reilly: Sebastopol, CA.

Crawley, MJ, 2005. Statistics: An Introduction using R. Chichester: Wiley (which is a shortened version of The R Book by the same author).


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http://www.social-statistics.org/?p=354


http://cran.r-project.org/manuals.html

http://www.r-project.org/mail.html


Field, A., Miles, J. & Field, Z., 2012. Discovering Statistics Using R. London: Sage

However, none of these books is about mapping or spatial analysis (of particular interest to me as a geographer). For that, the authoritative guide making the links between geographical information science, geographical data analysis and R (but not really written for R newcomers) is,

Bivand, R.S., Pebesma, E.J. & Gómez-Rubio, V., 2008. Applied Spatial Data Analysis with R. Berlin: Springer.

Also helpful is,

Ward, M.D. & Skrede Gleditsch, K., 2008. Spatial Regression Models. London: Sage. (Which uses R code examples).

And

Chun, Y. & Griffith, D.A., 2013. Spatial Statistics and Geostatistics. London: Sage. (I found this book a little eccentric but it contains some very good tips on its subject and gives worked examples in R).

The following book has a short section of maps as well as other graphics in R (and is also, as the title suggests, good for practical guidance on how to analyse surveys using cluster and stratified sampling, for example):

Lumley, T., 2010. Complex Surveys. A Guide to Analysis Using R. Hoboken, NJ: Wiley.

Springer publish an ever-growing series of books under the banner Use R! If you are interested in visualization, time-series analysis, Bayesian approaches, econometrics, data mining, …, then you'll find something of relevance at http://www.springer.com/series/6991. But you may well also find what you are looking for for free on the Internet.


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http://www.springer.com/series/6991

Session 2: A Demonstration of R

This session provides a quick tour of some of R's functionality, with a focus on some geographical applications. The idea here is to showcase a little of what R can do rather than providing a comprehensive explanation to all that is going on. Aim for an intuitive understanding of the commands and procedures but do not worry about the detail. More information about the workings of R is given in the next session. More details about how to use R as a GIS and for spatial analysis are given in Sessions 4, 5 and 6.

Note: this session assumes the libraries RgoogleMaps, png, sp and spdep are installed and available for use. You can find out which packages you currently have installed by using

> row.names(installed.packages())

If the packages cannot be found then they can be installed using install.packages(c("RgoogleMaps","png","sp","spdep")). Note that you may need administrative rights on your computer to install the package (see Section 3.5.1, Installing and loading one or more of the packages (libraries), p.37).

2.1 Getting Started

As the focus of this session is on showing what R can do rather than teaching you how to do it. instead of requiring you to type a series of commands, they can instead be executed automatically from a previously written source file (a script: see Section 1.3.1, page 7). As the commands are executed we will ask R to echo (print) them to the screen so you can following what is going on. At regular intervals you will be prompted to press return before the script continues.

To begin, type,

> source(file.choose(), echo=T)

and load the source file session2.R. After some comments that you should ignore, you will be prompted to load the .csv file schools.csv:

> ## Read in the file schools.csv file> wait()Please presss return

schools.data <- read.csv(file.choose())

Assuming there is no error, we will now proceed to a simple inspection of the data. Remember: the commands you see written below are the ones that appear in the source file. You do not need to type them yourself for this session.

2.2 Checking the data

It is always sensible to check a data table for any obvious errors.

> head(schools.data) # Shows the first few rows of the data> tail(schools.data) # Shows the bottom few rows of the data

We can produce a summary of each column in the data table using

> summary(schools.data)

In this instance, each column is a continuous variable so we obtain a six-number summary of the centre and spread of each variable.

The names of the variables are


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> names(schools.data)

Next the number of columns and rows; and a check – row-by-row – to see if the data are complete (have no missing data).

> ncol(schools.data)> nrow(schools.data)> complete.cases(schools.data)

It is not the most comprehensive check but everything appears to be in order.

2.3 Some simple graphics

The file schools.csv contains information about the location and some attributes of schools in Greater London (in 2008). The locations are given as a grid reference (Easting, Northing). The information is not real but is realistic. It should not, however, be used to make inferences about real schools in London.

Of particular interest is the average attainment on leaving primary school (elementary school) of pupils entering their first year of secondary school. Do some schools in London attract higher attaining pupils more than others? The variable attainment contains this information.

A stripchart and then a histogram will show that (not surprisingly) there is variation in the average prior attainment by school.

> attach(schools.data)> stripchart(attainment, method="stack", xlab="Mean Prior Attainment by School")> hist(attainment, col="light blue", border="dark blue", freq=F, ylim=c(0,0.30),+ xlab=”Mean attainment)

Here the histogram is scaled so the total area sums to one. To this we can add a rug plot,

> rug(attainment)

also a density curve, a Normal curve for comparison and a legend.

> lines(density(sort(attainment)))> xx <- seq(from=23, to=35, by=0.1)> yy <- dnorm(xx, mean(attainment), sd(attainment))> lines(xx, yy, lty="dotted")> rm(xx, yy)> legend("topright", legend=c("density curve","Normal curve"),+ lty=c("solid","dotted"))

If would be interesting to know if attainment varies by school type. A simple way to consider this is to produce a box plot. The data contain a series of dummy variables for each of a series of school types (Voluntary Aided Church of England school: coe = 1; Voluntary Aided Roman Catholic: rc = 1; Voluntary controlled faith school: vol.con = 1; another type of faith school: other.faith = 1; a selective school (sets an entrance exam): selective = 1). We will combine these into a single, categorical variable then produce the box plot showing the distribution of average attainment by school type.

First the categorical variable:

> school.type <- rep("Not Faith/Selective", times=nrow(schools.data))# This gives each school an initial value which will# then be replaced with its actual type

> school.type[coe==1] <- "VA CoE"# Voluntary Aided Church of England schools are given# the category VA CoE


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> school.type[rc==1] <- "VA RC"# Voluntary Aided Roman Catholic schools are given# the category VA RC [etc.]

> school.type[vol.con==1] <- "VC"> school.type[other.faith==1] <- "Other Faith"> school.type[selective==1] <- "Selective"> school.type <- factor(school.type)> levels(school.type) # A list of the categories[1] "Not Faith/Selective" "Other Faith" "Selective" [etc.]

Now the box plots:

> par(mai=c(1,1.4,0.5,0.5)) # Changes the graphic margins> boxplot(attainment ~ school.type, horizontal=T, xlab="Mean attainment", las=1,+ cex.axis=0.8) # Includes options to draw the boxes and labels horizontally> abline(v=mean(attainment), lty="dashed") # Adds the mean value to the plot> legend("topright", legend="Grand Mean", lty="dashed")

Not surprisingly, the selective schools (those with an entrance exam) recruit the pupils with highest average prior attainment.

Figure 2.1. A histogram with annotation in R


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Figure 2.2. Mean prior attainment by school type

2.4 Some simple statistics

It appears (in Figure 2.2) that there are differences in the levels of prior attainment of pupils in different school types. We can test whether the variation is significant using an analysis of variance.

> summary(aov(attainment ~ school.type)) Df Sum Sq Mean Sq F value Pr(>F) school.type 5 479.8 95.95 71.42 <2e-16 ***Residuals 361 485.0 1.34

It is, at a greater than 99.9% confidence (F = 71.42, p < 0.001).

We might also be interested in comparing those schools with the highest and lowest proportions of Free School Meal eligible pupils to see if they are recruiting pupils with equal or differing mean prior attainment. We expect a difference because free school meal eligibility is used as an indicator of a low income household and there is a link between economic disadvantage and educational progress in the UK.

> attainment.high.fsm.schools <- attainment[fsm > quantile(fsm, probs=0.75)]# Finds the attainment scores for schools with the highest proportions of FSM pupils> attainment.low.fsm.schools <- attainment[fsm < quantile(fsm, probs=0.25)]# Finds the attainment scores for schools with the lowest proportions of FSM pupils> t.test(attainment.high.fsm.schools, attainment.low.fsm.schools)

Welch Two Sample t-test

data: attainment.high.fsm.schools and attainment.low.fsm.schools t = -15.0431, df = 154.164, p-value < 2.2e-16alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -3.437206 -2.639240 sample estimates:mean of x mean of y 26.58352 29.62174


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It comes as little surprise to learn that those schools with the greatest proportions of FSM eligible pupils are also those recruiting lower attaining pupils on average (mean attainment 26.6 Vs 29.6, t = -15.0, p < 0.001, the 95% confidence interval is from -3.44 to 2.64).

Exploring this further, the Pearson correlation between the mean prior attainment of pupils entering each school and the proportion of them that are FSM eligible is -0.689, and significant (p < 0.001):

> round(cor(fsm, attainment),3)> cor.test(fsm, attainment)

Pearson's product-moment correlation

data: fsm and attainment t = -18.1731, df = 365, p-value < 2.2e-16alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: -0.7394165 -0.6313939 sample estimates: cor -0.6892159

Of course, the use of the Pearson correlation assumes that the relationship is linear, so let's check:

> plot(attainment ~ fsm)> abline(lm(attainment ~ fsm)) # Adds a line of best fit (a regression line)

There is some suggestion the relationship might be curvilinear. However, we will ignore that here.

Finally, some regression models. The first seeks to explain the mean prior attainment scores for the schools in London by the proportion of their intake who are free school meal eligible. (The result is the line of best fit added to the scatterplot above).

The second model adds a variable giving the proportion of the intake of a white ethnic group.

The third adds a dummy variable indicating whether the school is selective or not.

> model1 <- lm(attainment ~ fsm, data=schools.data)> summary(model1)

Call:lm(formula = attainment ~ fsm, data = schools.data)

Residuals: Min 1Q Median 3Q Max -2.8871 -0.7413 -0.1186 0.5487 3.6681

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 29.6190 0.1148 258.12 <2e-16 ***fsm -6.5469 0.3603 -18.17 <2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.178 on 365 degrees of freedomMultiple R-squared: 0.475,Adjusted R-squared: 0.4736 F-statistic: 330.3 on 1 and 365 DF, p-value: < 2.2e-16

> model2 <- lm(attainment ~ fsm + white, data=schools.data)> summary(model2)


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Call:lm(formula = attainment ~ fsm + white, data = schools.data)


Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 30.1250 0.1979 152.21 < 2e-16 ***fsm -7.2502 0.4214 -17.20 < 2e-16 ***white -0.8722 0.2796 -3.12 0.00196 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.164 on 364 degrees of freedomMultiple R-squared: 0.4887, Adjusted R-squared: 0.4859 F-statistic: 173.9 on 2 and 364 DF, p-value: < 2.2e-16

> model3 <- update(model2, . ~ . + selective)# Means: take the previous model and add the variable 'selective'

> summary(model3)

Call:lm(formula = attainment ~ fsm + white + selective, data = schools.data)

Residuals: Min 1Q Median 3Q Max -2.6262 -0.5620 0.0537 0.5607 3.6215

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 29.1706 0.1689 172.712 <2e-16 ***fsm -5.2381 0.3591 -14.586 <2e-16 ***white -0.2299 0.2249 -1.022 0.307 selective 3.4768 0.2338 14.872 <2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Looking at the adjusted R-squared value, each model appears to be an improvement on the one that precedes it (marginally so for model 2). However, looking at the last (model 3), we may suspect that we could drop the white ethnicity variable with no significant loss in the amount of variance explained. An analysis of variance confirms that to be the case.

> model4 <- update(model3, . ~ . - white)# Means: take the previous model but remove the variable 'white'

> anova(model4, model3)Analysis of Variance Table

Model 1: attainment ~ fsm + selectiveModel 2: attainment ~ fsm + white + selective Res.Df RSS Df Sum of Sq F Pr(>F)


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1 364 307.42 2 363 306.54 1 0.88222 1.0447 0.3074

The residual error, measured by the residual sum of squares (RSS), is not very different for the two models, and that difference, 0.882, is not significant (F = 1.045, p = 0.307).

2.5 Some simple maps

For a geographer like myself, R becomes more interesting when we begin to look at its geographical data handling capabilities.

The schools data contain geographical coordinates and are therefore geographical data. Consequently they can be mapped. The simplest way for point data is to use a 2-dimensional plot, making sure the aspect ratio is fixed correctly.

> plot(Easting, Northing, asp=1, main="Map of London schools")# The argument asp=1 fixes the aspect ratio correctly

Amongst the attribute data for the schools, the variable esl gives the proportion of pupils who speak English as an additional language. It would be interesting for the size of the symbol on the map to be proportional to it.

> plot(Easting, Northing, asp=1, main="Map of London schools",+ cex=sqrt(esl*5))

It would also be nice to add a little colour to the map. We might, for example, change the default plotting 'character' to a filled circle with a yellow background.

> plot(Easting, Northing, asp=1, main="Map of London schools",+ cex=sqrt(esl*5), pch=21, bg="yellow")

A more interesting option would be to have the circles filled with a colour gradient that is related to a second variable in the data – the proportion of pupils eligible for free school meals for example.

To achieve this, we can begin by creating a simple colour palette:

> palette <- c("yellow","orange","red","purple")

We now cut the free school meals eligibility variable into quartiles (four classes, each containing approximately the same number of observations).

> map.class <- cut(fsm, quantile(fsm), labels=FALSE, include.lowest=TRUE)

The result is to split the fsm variable into four groups with the value 1 given to the first quarter of the data (schools with the lowest proportions of eligible pupils), the value 2 given to the next quarter, then 3, and finally the value 4 for schools with the highest proportions of FSM eligible pupils.

There are, then, now four map classes and the same number of colours in the palette. Schools in map class 1 (and with the lowest proportion of fsm-eligible pupils) will be coloured yellow, the next class will be orange, and so forth.

Bringing it all together,

> plot(Easting, Northing, asp=1, main="Map of London schools",+ cex=sqrt(esl*5), pch=21, bg=palette[map.class])

It would be good to add a legend, and perhaps a scale bar and North arrow. Nevertheless, as a first map in R this isn't too bad!


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Figure 2.3. A simple point map in R

Why don't we be a bit more ambitious and overlay the map on a Google Maps tile, adding a legend as we do so? This requires us to load an additional library for R and to have an active Internet connection.

> library(RgoogleMaps)

If you get an error such as the following

Error in library(RgoogleMaps) : there is no package called ‘RgoogleMaps’

it is because the library has not been installed.

Assuming that the data frame, schools.data, remains in the workspace and attached (it will be if you have followed the instructions above), and that the colour palette created above has not been deleted, then the map shown in Figure 2.4 is created with the following code:

> MyMap <- MapBackground(lat=Lat, lon=Long)> PlotOnStaticMap(MyMap, Lat, Long, cex=sqrt(esl*5), pch=21, bg=palette[map.class])> legend("topleft", legend=paste("<",tapply(fsm, map.class, max)), pch=21, pt.bg=palette, pt.cex=1.5, bg="white", title="P(FSM-eligible)")> legVals <- seq(from=0.2,to=1,by=0.2)> legend("topright", legend=round(legVals,3), pch=21, pt.bg="white", pt.cex=sqrt(legVals*5), bg="white", title="P(ESL)")

(If you are running the script for this session then the code you see on-screen will differ slightly. That is because it has some error trapping included in it incase there is no Internet connection available)


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Remember that the data are simulated. The points shown on the map are not the true locations of schools in London. Do not worry about understanding the code in detail – the purpose is to see the sort of things R can do with geographical data. We will look more closely at the detail in later sessions.

Figure 2.4. A slightly less simple map produced in R

2.6 Some simple geographical analysis

Remember the regression models from earlier? It would be interesting to test the assumption that the residuals exhibit independence by looking for spatial dependencies. To do this we will consider to what degree the residual value for any one school correlates with the mean residual value for its six nearest other schools (the choice of six is completely arbitrary).

First, we will take a copy of the schools data and convert it into an explicitly spatial object in R:

> detach(schools.data)> schools.xy <- schools.data> library(sp)> attach(schools.xy)> coordinates(schools.xy) <- c("Easting", "Northing")> # Converts into a spatial object> class(schools.xy)> detach(schools.xy)> proj4string(schools.xy) <- CRS("+proj=tmerc datum=OSGB36")


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> # Sets the Coordinate Referencing System

Second, we find the six nearest neighbours for each school.

> library(spdep)> nearest.six <- knearneigh(schools.xy, k=6, RANN=F)> # RANN = F to override the use of the RANN package that may not be installed

We can learn from this that the six nearest schools to the first school in the data (row 1) are schools 5, 38, 2, 40, 223 and 6:

> nearest.six$nn[1,][1] 5 38 2 40 223 6

The neighbours object, nearest.six, is an object of class knn:

> class(nearest.six)

It is next converted into the more generic class of neighbours.

> neighbours <- knn2nb(nearest.six)> class(neighbours)[1] "nb"> summary(neighbours)Neighbour list object:Number of regions: 367 Number of nonzero links: 2202 Percentage nonzero weights: 1.634877 Average number of links: 6 [etc.]

The connections between each point and its neighbours can then be plotted. It may take a few minutes.

> plot(neighbours, coordinates(schools.xy))

Having identified the six nearest neighbours to each school we could give each equal weight in a spatial weights matrix or, alternatively, decrease the weight with distance away (so the first nearest neighbour gets most weight and the sixth nearest the least). Creating a matrix with equal weight given to all neighbours is sufficient for the time being.

> spatial.weights <- nb2listw(neighbours)

(The other possibility is achieved by creating then supplying a list of general weights to the function, see ?nb2listw)

We now have all the information required to test whether there are spatial dependencies in the residuals. The answer is yes (Moran's I = 0.218, p < 0.001, indicating positive spatial autocorrelation).

> lm.morantest(model4, spatial.weights)

Global Moran's I for regression residuals

data: model: lm(formula = attainment ~ fsm + selective, data = schools.data)weights: spatial.weights Moran I statistic standard deviate = 7.9152, p-value = 1.235e-15alternative hypothesis: greater sample estimates:Observed Moran's I Expectation Variance 0.2181914682 -0.0038585704 0.0007870118


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2.7 Tidying up

It is better to save your workspace regularly whilst you are working (see Section 1.4.4, 'Saving and loading workspaces', page 10) and certainly before you finish. Don't forget to include the extension .RData when saving. Having done so, you can tidy-up the workspace.

> save.image(file.choose(new=T))> rm(list=ls()) # Be careful, it deletes everything!

2.8 Further Information

A simple introduction to graphics and statistical analysis in R is given in Statistics for Geography and Environmental Science: An Introduction in R, available at http://www.social-statistics.org/?p=354.


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Session 3: A Little More about the workings of R

This session provides a little more guidances on the 'inner workings' of R. All the commands are contained in file session3.R and can be run using it (see the section 'Scripting' on p.7). You can, if you wish, skip this session and move straight on to the sessions on mapping and spatial modelling in R, returning to this later to better understand some of the commands and procedures you will have used.

3.1 Classes, types and coercion

Let us create two objects, each a vector containing ten elements. The first will be the numbers from one to ten, recorded as integers. The second will be the same sequence but now recorded as real numbers (that is, 'floating point' numbers, those with a decimal place).

> b <- 1:10> b [1] 1 2 3 4 5 6 7 8 9 10> c <- seq(from=1.0, to=10.0, by=1)> c [1] 1 2 3 4 5 6 7 8 9 10

Note that in the second case, we could just type,

> c <- seq(1, 10, 1)> c [1] 1 2 3 4 5 6 7 8 9 10

This works because if we don't explicitly define the argument (by omitting from=1 etc.) then R will assume we are giving values to the arguments in their default order, which in this case is in the order from, to and by.Type ?seq and look under Usage for this to make a little more sense.

In any case, the two objects, b and c, appear the same on screen but one is an object of class integer whereas the other is an object of class numeric and of type double (double precision in the memory space).

> class(b)[1] "integer"> class(c)[1] "numeric"> typeof(c)[1] "double"

Often it possible to coerce an object from one class and type to another.

> b <- 1:10> class(b)[1] "integer"> b <- as.double(b)> class(b)[1] "numeric"> typeof(b)[1] "double"> class(c)> c <- as.integer(c)> class(c)[1] "integer"> c


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[1] 1 2 3 4 5 6 7 8 9 10> c <- as.character(c)> class(c)[1] "character"> c [1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10"

The examples above are trivial. However, it is important to understand that seemingly generic functions like summary(...) can produce outputs that are dependent upon the class type. Try, for example,

> class(b)[1] "numeric"> summary(b) Min. 1st Qu. Median Mean 3rd Qu. Max. 1.00 3.25 5.50 5.50 7.75 10.00 > class(c)[1] "character"> summary(c) Length Class Mode 10 character character

In the first instance, a six number summary of the centre and spread of the numeric data is given. That makes no sense for character data. The second summary gives the length of the vector, its class type and its storage mode.

A more interesting example is provided if we consider the plot(...) command, used first with a single data variable, secondly with two variables in a data table, and finally on a model of the relationship between those two variables.

The first variable is created by generating 100 observations drawn randomly from a Normal distribution with mean of 100 and a standard deviation of 20.

> var1 <- rnorm(n=100, mean=100, sd=20)

Being random, the data assigned to the variable will differ from user to user. Usually we would want this. However, in this case it would be easier to ensure we get the same by ensuring we each get the same 'random' draw:

> set.seed(1)> var1 <- rnorm(n=100, mean=100, sd=20)

Always check the data,

> class(var1)[1] "numeric"> length(var1) # The number of elements in the vector [1] 100> summary(var1) Min. 1st Qu. Median Mean 3rd Qu. Max. 55.71 90.12 102.30 102.20 113.80 148.00 > head(var1) # The first few elements[1] 87.47092 103.67287 83.28743 131.90562 106.59016 83.59063> tail(var1) # The last few elements[1] 131.73667 111.16973 74.46816 88.53469 75.50775 90.53199

They seem fine! Returning to the use of the plot(...) command, in this instance it simply plots the data in order of their position in the vector.

> plot(var1)


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Figure 3.1. A simple plot of a numeric vector

To demonstrate a different interpretation of the plot command, a second variable is created that is a function of the first but with some random error.

> set.seed(101)> var2 <- 3 * var1 + 10 + rnorm(100, 0, 25)# which, because n, mean and sd are the first three arguments into rnorm# is the same as writing var2 <- 3 * var1 + 10 + rnorm(n=100, mean=100, sd=20)> head(var2)[1] 264.2619 334.8301 242.9887 411.0758 337.5397 290.1211

Next, the two variables are gathered together in a data table, of class data frame, where each row is an observation and each column is a variable. There is more about data frames on page 29, in Section 3.2 ('Data frames')

> mydata <- data.frame(x = var1, y = var2)> class(mydata)[1] "data.frame"> head(mydata) x y1 87.47092 264.26192 103.67287 334.83013 83.28743 242.98874 131.90562 411.07585 106.59016 337.53976 83.59063 290.1211> nrow(mydata) # The number of rows in the data[1] 100> ncol(mydata) # The number of columns[1] 2

In this case, plotting the data frame will produce a scatter plot (to which the line of best fit shown in Figure 3.2 will be added shortly).

> plot(mydata)

If there had been more than two columns in the data table, or if they had not been arranged in x, y order, then the plot could be produced by referencing the columns directly. All the following are equivalent:


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> with(mydata, plot(x, y)) # Here the order is x, y> with(mydata, plot(y ~ x)) # Here it is y ~ x> plot(mydata$x, mydata$y)> plot(mydata[,1], mydata[,2]) # Plot using the first and second columns> plot(mydata[,2] ~ mydata[,1])

The attach(...) command could also be used. This is introduced in Section 3.2.2, 'Attaching a data frame' on page 31.

Figure 3.2. A scatter plot. A line of best fit has been added.

The line of best fit in Figure 3.2 is a regression line. To fit the regression model, summarising the relationship between y and x, use

> model1 <- lm(y ~ x, data=mydata) # lm is short for linear model> class(model1)[1] "lm"

model1 is an object of class lm, short for linear model. Using the summary(...) function summarises the relationship between y and x.

> summary(model1)Call:lm(formula = y ~ x, data = mydata)Residuals: Min 1Q Median 3Q Max -57.102 -16.274 0.484 15.188 47.290 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 8.6462 13.6208 0.635 0.527 x 3.0042 0.1313 22.878 <2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


We created variable y as a function of x and it shows: x is a significant predictor of y at a greater than 99.9 confidence, 84% of the variance in y is explained by x, and the equation of the regression


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line is y = 8.65 + 3.00x.

Now using the plot(...) function on the object of class lm has an effect that is different from the previous two cases. It produces a series a diagnostic plots to help check the assumptions of regression have been met.

> plot(model1)

The first plot is a check for non-constant variance and outliers, the second for normality of the model residuals, the third is similar to the first, and the fourth identifies both extreme residuals and leverage points.

These four plots can be viewed together, changing the default graphical parameters to show the plots in a 2-by-2 array (as in Figure 3.3).

> par(mfrow = c(2,2)) # Sets the graphical output to be 2 x 2> plot(model1)

Finally, we might like to go back to our previous scatter plot and add the regression line of best fit to it,

> par(mfrow = c(1,1)) # Resets the window to a single graph> plot(mydata)> abline(model1)

Figure 3.3.Default plots for an object of class linear model

3.2 Data frames

The preceding section introduced the data frame as a class of object containing a table of data where the variables are the columns of the data and the rows are the observations.


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> class(mydata)> summary(mydata)

Looking at the data summary, the object mydata contains two columns, labelled x and y. These column headers can also be revealed by using

> names(mydata)[1] "x" "y"

or with

> colnames(mydata)[1] "x" "y"

The row names appear to be the numbers from 1 to 100 (the number of rows in the data), though actually they are character data:

> rownames(mydata) [1] "1" "2" "3" "4" "5" "6" "7" "8" [etc.]> class(rownames(mydata))[1] "character"

The column names can be changed either individually or together. Individually:

> names(mydata)[1] <- "v1"> names(mydata)[2] <- "v2"> names(mydata)[1] "v1" "v2"

All at once:

> names(mydata) <- c("x","y")> names(mydata)[1] "x" "y"

… as can the row names,

> rownames(mydata)[1] <- "0"> rownames(mydata) [1] "0" "2" "3" "4" "5" "6" "7" "8" [etc.]> rownames(mydata) = seq(from=0, by=1, length.out=nrow(mydata))> rownames(mydata) [1] "0" "1" "2" "3" "4" "5" "6" "7" "8" [etc.]

The above can be especially useful when merging data tables with GIS shapefiles in R (because the first entry in an attribute table for a shapefile usually is given an ID of 0). Otherwise, it is usually easiest for the first row in a data table to be labelled 1, so let's put them back to how they were.

> rownames(mydata) = 1:nrow(mydata)> rownames(mydata) [1] "1" "2" "3" "4" "5" "6" "7" "8" [etc.]

3.2.1 Referencing rows and columns in a data frame

The square bracket notation can be used to index specific row, columns or cells in the data frame. For example:

> mydata[1,] # The first row of data x y1 87.47092 264.2619> mydata[2,] # The second row of data x y2 103.6729 334.8301> round(mydata[2,],2) # The second row, rounded to 2 decimal places x y


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2 103.67 334.83> mydata[nrow(mydata),] # The final row of the data x y100 90.53199 261.236> mydata[,1] # The first column of data [1] 87.47092 103.67287 83.28743 131.90562 [etc.]> mydata[,2] # The second column, which here is also … [1] 264.2619 334.8301 242.9887 411.0758 337.5397 [etc.]> mydata[,ncol(mydata)] # … the final column of data [1] 264.2619 334.8301 242.9887 411.0758 337.5397 [etc.]> mydata[1,1] # The data in the first row of the first column[1] 87.47092> mydata[5,2] # The data in the fifth row of the second column[1] 337.5397> round(mydata[5,2],0)[1] 338

Specific columns of data can also be referenced using the $ notation

> mydata$x # Equivalent to mydata[,1] because the column name is x [1] 87.47092 103.67287 83.28743 131.90562 106.59016 [etc.]> mydata$y [1] 264.2619 334.8301 242.9887 411.0758 337.5397 290.1211 [etc.] > summary(mydata$x) Min. 1st Qu. Median Mean 3rd Qu. Max. 55.71 90.12 102.30 102.20 113.80 148.00 > summary(mydata$y) Min. 1st Qu. Median Mean 3rd Qu. Max. 140.4 284.1 314.1 315.6 355.7 447.6 > mean(mydata$x)[1] 102.1777> median(mydata$y)[1] 314.1226> sd(mydata$x) # Gives the standard deviation of x[1] 17.96399> boxplot(mydata$y)> boxplot(mydata$y, horizontal=T, main="Boxplot of variable y")

(Boxplots are sometimes said to be easier to read when drawn horizontally)

One way to avoid the use of the $ notation is to use the function with(...) instead:

> with(mydata, var(x)) # Gives the variance of x[1] 322.7048> with(mydata, plot(y, xlab="Observation number"))

However, even this is cumbersome so the attach function may be preferred...

3.2.2 Attaching a data frame

Sometimes any of the ways to access a specific part of a data table becomes tiresome and it is useful to reference the column or variable name directly. For example, instead of having to type mean(mydata[,1]), mean(mydata$x) or with(mydata, mean(x)) it would be easier just to refer to the variable of interest, x, as in mean(x).

To achieve this the attach(...) command is used. Compare, for example,

> mean(x)


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Error in mean(x) : object 'x' not found

(which generates an error because there is not an object called x in the workspace; it is only a column name within the data frame mydata) with

> attach(mydata)> mean(x)[1] 102.1777

(which works fine). If, to use the earlier analogy, objects in R's workspace are like box files, then now you have opened one up and its contents (which include the variable x) are visible.

To detach the contents of the data frame use detach(...)

> detach(mydata)> mean(x)Error in mean(x) : object 'x' not found

It is sensible to use detach when the data frame is no longer being used or else confusion can arise when multiple data frames contain the same column names, as in the following example:

> attach(mydata)> mean(x) # This will give the mean of mydata$x[1] 102.1777> mydata2 = data.frame(x = 1:10, y=11:20)> head(mydata2) x y1 1 112 2 123 3 134 4 145 5 156 6 16> attach(mydata2)The following object(s) are masked from 'mydata': x, y> mean(x) # This will now give the mean of mydata2$x[1] 5.5> detach(mydata2)> mean(x)[1] 102.1777> detach(mydata)> rm(mydata2)

3.2.3 Sub-setting the data table and logical queries

Subsets of a data frame can be created by referencing specific rows within it. For example, imagine we want a table only of those observations that have a a value above the mean of some variable.

> attach(mydata)> subset <- which(x > mean(x))> class(subset)[1] "integer"> subset [1] 2 4 5 7 8 9 11 12 15 18 19 20 21 22 25 30 31 33 [etc.] > mydata.sub <- mydata[subset,]> head(mydata.sub) x y2 103.6729 334.83014 131.9056 411.07585 106.5902 337.5397


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7 109.7486 354.71558 114.7665 351.48119 111.5156 367.4726

Note how the row names of this subset have been inherited from the parent data frame.

A more direct approach is to define the subset as a logical vector that is either true or false dependent upon whether a condition is met.

> subset <- x > mean(x)> class(subset)[1] "logical"> subset [1] FALSE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE [etc.]> mydata.sub <- mydata[subset,]> head(mydata.sub) x y2 103.6729 334.83014 131.9056 411.07585 106.5902 337.53977 109.7486 354.71558 114.7665 351.48119 111.5156 367.4726

A yet more succinct way of achieving the same is:

> mydata.sub <- mydata[x > mean(x),]# Selects those rows that meet the logical condition, and all columns

> head(mydata.sub) x y2 103.6729 334.83014 131.9056 411.07585 106.5902 337.53977 109.7486 354.71558 114.7665 351.48119 111.5156 367.4726

In the same way, to select those rows where x is greater than or equal to the mean of x and y is greater than or equal to the mean of y

> mydata.sub <- mydata[x >= mean(x) & y >= mean(y),]# The symbol & is used for and

Or, those rows where x is less than the mean of x or y is less than the mean of y

> mydata.sub <- mydata[x < mean(x) | y < mean(y),]# The symbol | is used for or

3.2.4 Missing data

Missing data is given the value NA. For example,

> mydata[1,1] = NA> mydata[2,2] = NA> head(mydata) x y1 NA 264.26192 103.67287 NA3 83.28743 242.98874 131.90562 411.07585 106.59016 337.53976 83.59063 290.1211


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R will, by default, report NA or an error when some calculations are tried with missing data:

> mean(mydata$x)[1] NA> quantile(mydata$y)Error in quantile.default(mydata$y) : missing values and NaN's not allowed if 'na.rm' is FALSE

To overcome this, the default can be changed or the missing data removed.

To ignore the missing data in the calculation,

> mean(mydata$x, na.rm=T)[1] 102.3263> quantile(mydata$y, na.rm=T) # Divides the data into quartiles 0% 25% 50% 75% 100% 140.4475 282.1064 313.7536 356.5862 447.6020

Alternatively, there are various ways to remove the missing data. For example …

> subset <- !is.na(mydata$x)

… creates a logical vector which is true where the data values of x are not missing (the ! in the expresion means not):

> head(subset)[1] FALSE TRUE TRUE TRUE TRUE TRUE

Using the subset,

> x2 <- mydata$x[subset]> mean(x2)[1] 102.3263

More succinctly,

> with(mydata, mean(x[!is.na(x)]))[1] 102.3263

Alternatively, a new data frame can be created without any missing data whereby any row with any missing value is omitted.

> subset <- complete.cases(mydata)> head(subset)[1] FALSE FALSE TRUE TRUE TRUE TRUE> mydata.complete = mydata[subset,]> head(mydata.complete) x y3 83.28743 242.98874 131.90562 411.07585 106.59016 337.53976 83.59063 290.12117 109.74858 354.71558 114.76649 351.4811

3.2.5 Reading data from a file into a data frame

The accompanying file schools.csv (used in Session 2) contains information about the location and some attributes of schools in Greater London (in 2008). The locations are given as a grid reference (Easting, Northing). The information is not real but is realistic.

A standard way to read a file into a data frame, with cases corresponding to lines and variables to fields in the file, is to use the read.table(...) command.


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> ?read.table

In the case of schools.csv, it is comma delimited and has column headers. Looking through the arguments for read.table the data might be read into R using

> schools.data <- read.table("schools.csv", header=T, sep=",")

This will only work if the file is located in the working directory, else the location (path) of the file will need to be specified (or the working directory changed). More conveniently, use file.choose()

> schools.data <- read.table(file.choose(), header=T, sep=",")

Looking through the usage of read.table in the R help page, a variant of the command is found where the defaults are for comma delimited data. So, most simply, we could use,

schools.data <- read.csv(file.choose())

Having read-in the data, some basic checks of it are helpful,

> head(schools.data, n=3) FSM EAL SEN white blk.car blk.afr indian pakistani [etc.]1 0.659 0.583 0.031 0.217 0.032 0.222 0.002 0.0202 0.391 0.424 0.001 0.350 0.087 0.126 0.003 0.0123 0.708 0.943 0.038 0.048 0.000 0.239 0.000 0.004# Views the first three lines of the data> ncol(schools.data)[1] 17> nrow(schools.data)[1] 366> summary(schools.data) FSM EAL SEN [etc.] Min. :0.0000 Min. :0.0000 Min. :0.00000 1st Qu.:0.1323 1st Qu.:0.1472 1st Qu.:0.00800 Median :0.2500 Median :0.3165 Median :0.02000 Mean :0.2702 Mean :0.3491 Mean :0.02308 3rd Qu.:0.3897 3rd Qu.:0.5122 3rd Qu.:0.03400 Max. :0.7730 Max. :1.0000 Max. :0.11300

It seems to be fine.

For more about importing and exporting data in R, consult the R help document, R Data Import/Export (see under the Help menu in R or http://cran.r-project.org/manuals.html).

3.3 Lists

A list is a little like a data frame but offers a more flexible way to gather objects of different classes together. For example,

> mylist <- list(schools.data, model1, "a")> class(mylist)[1] "list"

To find the number of components in a list, use length(...),

> length(mylist)[1] 3

Here the first component is the data frame containing the schools data. The second component is the linear model created earlier. The third is the character “a”. To reference a specific component, double square brackets are used:

> head(mylist[[1]], n=3)


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FSM EAL SEN white blk.car blk.afr indian pakistani [etc.]1 0.659 0.583 0.031 0.217 0.032 0.222 0.002 0.0202 0.391 0.424 0.001 0.350 0.087 0.126 0.003 0.0123 0.708 0.943 0.038 0.048 0.000 0.239 0.000 0.004

> summary(mylist[[2]])Call:lm(formula = y ~ x, data = mydata)Residuals: Min 1Q Median 3Q Max -57.102 -16.274 0.484 15.188 47.290 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 8.6462 13.6208 0.635 0.527 x 3.0042 0.1313 22.878 <2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


> class(mylist[[3]])[1] "character"

The double square brackets can be combined with single ones. For example,

> mylist[[1]][1,] FSM EAL SEN white blk.car blk.afr indian pakistani [etc.]1 0.659 0.583 0.031 0.217 0.032 0.222 0.002 0.020

is the first row of the schools data. The first cell of the same data is

> mylist[[1]][1,1][1] 27

3.4 Writing a function

In brief, a function is written in R in the follow way,

> function.name <- function(list of arguments) {+ function code+ return(result)+ }

So, a simple function to divide the product of two numbers by their sum could be,

> my.function <- function(x1, x2) {+ result <- (x1 * x2) / (x1 + x2)+ return(result)+ }

Now running the function

> my.function(3, 7)[1] 2.1


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3.5 R packages for mapping and spatial data analysis

By default, R comes with a base set of packages and methods for data analysis and visualization. However, there are many other packages available, too, that greatly extend R's value and functionality. These packages are listed alphabetically at http://cran.r-project.org/web/packages/available_packages_by_name.html.

Because there are so many, it can be useful to browse the packages by topic (at http://cran.r-project.org/web/views/). The topic, or 'task view' of particular interest here is the analysis of spatial data: http://cran.r-project.org/web/views/Spatial.html

3.5.1 Installing and loading one or more of the packages (libraries)

Note: If reading this in class it is likely that the packages have been installed already or you will not have the administrative rights to install them. If so, this section is for information only.There also is no need to install the packages if you have done so already (when following the instructions under 'Before you begin' on p.3).

To install a specific package the install.packages(...) command is used, as in:

> install.packages("ctv")Installing package(s) into ‘/Users/ggrjh/Library/R/2.13/library’(as ‘lib’ is unspecified)trying URL 'http://cran.uk.r-project.org/bin/macosx/leopard/contrib/2.13/ctv_0.7-4.tgz'Content type 'application/x-gzip' length 289693 bytes (282 Kb)opened URL==================================================downloaded 282 Kb

The package needs to be installed once but loaded each time R is started, using the library(...) command

> library("ctv")

In this case what has been installed is a package that will now allow all the packages associated with the spatial task view to be installed together, using:

> install.views("Spatial")

Note that installing packages may, by default, require access to a directory/folder for which administrative rights are required. If necessary, it is entirely possible to install R (and therefore the additional packages) in, for example, 'My Documents' or on a USB stick.

3.5.2 Checking which packages are installed

You can see which packages are installed by using

> row.names(installed.packages())

To see whether a specific package is installed use

> is.element("sp",installed.packages()[,1])# replace sp with another package name to check if that is installed

3.6 Tidying up and quitting

You may want to save and/or tidy up your workspace before quitting R. See sections 1.5 and 2.7 on pages 11 and 23.


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http://cran.r-project.org/web/views/Spatial.html

http://cran.r-project.org/web/views/

http://cran.r-project.org/web/views/

http://cran.r-project.org/web/packages/available_packages_by_name.html

http://cran.r-project.org/web/packages/available_packages_by_name.html

3.7 Further Information

See An Introduction to R, available at CRAN (http://cran.r-project.org/manuals.html) or by using the drop-down Help menus in the RGui.



Session 4: Using R as a simple GIS

This session introduces some of the libraries available for R that allow it to be used as a simple GIS. Examples of plotting (x, y) point data, producing choropleth maps, creating a raster, and undertaking a spatial join are given. The code required may seem quite difficult on first impression but it becomes easier with practice, plus often it is a case of 'recycling' existing code, making small changes where required. The advantages of using R instead of a normal standalone GIS are addressed with an example of mapping the residuals from a simple econometric (regression) model and looking for a correlation between neighbours.

For this session you will need to have the following libraries installed: sp, maptools, GISTools, classInt, RColorBrewer, raster and spdep (see 'Installing and loading one or more of the packages (libraries)', p. 37).

4.1 Simple Maps of XY Data

4.1.1 Loading and Mapping XY Data

We begin by reading some XY Data into R. The file landprices.csv is in a comma separated format and contains information about land parcels in Beijing, including a point georeference – a centroid, marking the centre of the land parcel. The data are simulated and not real but they are realistic.

> landdata <- read.csv(file.choose())> head(landdata, n=3) x y LNPRICE LNAREA DCBD DELE DRIVER DPARK Y0405 Y0607 Y08091 454393.1 4417809 9.29 11.84 8.28 7.25 7.38 8.09 0 1 02 442744.9 4417781 8.64 10.97 9.04 5.61 8.41 7.51 0 1 03 444191.7 4416996 6.69 7.92 8.89 5.62 8.22 7.27 0 1 0

Given the geographical coordinates, it is possible to map the data as they are in much the same way as we did in Section 2.5 ('Some simple maps'). At its simplest,

> with(landdata, plot(x, y, asp=1))

However, given an interest in undertaking spatial analysis in R, it would be better to convert the data into what R will explicitly recognise as a spatial object. For this we will require the sp library, Assuming it is installed,

> library(sp)

We can now coerce landdata into an object of class spatial (sp) by telling R that the geographical coordinates for the data are found in columns 1 and 2 of the current data frame.

> coordinates(landdata) = c(1,2)> class(landdata)[1] "SpatialPointsDataFrame"attr(,"package")[1] "sp"

The locations of the data points are now simply plotted using

> plot(landdata)

or, if we prefer a different symbol,

> plot(landdata, pch=10)> plot(landdata, pch=4)

(type ?points and scroll down to below 'pch values' to see the options for different types of point


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character)

4.1.2 The Coordinate Reference System (CRS)

If you type summary(landdata) you will find it is an object of class SpatialPointsDataFrame meaning it is comprised of geographic data (the Spatial Points) and also attribute data (information about the land parcels at those points). The bounding box – the rectangle enclosing all the points – has the coordinates (428554.2, 4406739.2) at its bottom-left corner and the coordinates (463693.8, 4440346.6) at its top-right. A six number summary is given for each of the attribute variables to show their average and range. Above these summaries you will find some NAs, indicating missing information. What are missing are the details about the coordinate reference system (CRS) for the data. If the geographical coordinates were in (longitude, latitude) format then we could use the codeproj4string(landdata) <- CRS(“+longlat”) to set the CRS. In our case, the data use the Xian 1980 / Gauss-Kruger CM 117E projection. If you go to http://www.epsg-registry.org/ you will find various search options to retrieve the EPSG (European Petroleum Survey Group) code for this projection. It is 2345.

> crs <- CRS("+init=epsg:2345")> proj4string(landdata) <- crs> summary(landdata)> plot(landdata, pch=21, bg="yellow", cex=0.7)

# cex means character expansion: it controls the symbol size

4.1.3 Adding an additional map layer

The map still lacks a sense of geographical context so we will add a polygon shapefile giving the boundaries of districts in Beijing. The file is called 'beijing_districts.shp'. This first needs to be loaded into R which in turn requires the maptools library:

> library(maptools)> districts <- readShapePoly(file.choose())> summary(districts)

As before, the coordinate reference system is missing but is the same as for the land price data.

> proj4string(districts) <- crs> summary(districts)

We can now plot the boundaries of the districts and then overlay the point data on top:

> plot(districts)> plot(landdata, pch=21, bg="yellow", cex=0.7, add=T)

4.2 Creating a choropleth map using GISTools

A choropleth map is one where the area units (the polygons) are shaded according to some measurement of them. Looking at the attribute data for the districts shapefile (which is accessed using the code @data) we find it includes a measure of the population density,

> head(districts@data, n=3) SP_ID POPDEN JOBDEN0 0 0.891959 1.4951501 1 1.246510 0.4662642 2 5.489660 2.787890

That is a variable we can map. The easiest way to do this is using the GIS Tools library,

> library(GISTools)

As with most libraries, if we want to know more about what it can do, type ? followed by its name


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http://www.epsg-registry.org/

(so here ?GISTools) and follow the link to the main index. If you do, you will find there is a function called choropleth with the description: “Draws a choropleth map given a spatialPolygons object, a variable and a shading scheme.”

Currently we have the spatialPolygons object. It is the object districts.

> class(districts)[1] "SpatialPolygonsDataFrame"attr(,"package")[1] "sp"

The difference between a spatialPolygons object and a SpatialPolygonsDataFrame is the first contains only the geographical information required to draw the boundaries of a map whereas the second also contains attribute data that could be used to shade it (but doesn't have to be; another variable could be used). We will use the variable POPDEN. All that is left is a shading scheme. In the first instance, let's allow another function, auto.shading, to do the work for us.

> shades <- auto.shading(districts@data$POPDEN)> choropleth(districts, districts@data$POPDEN)> plot(landdata, pch=21, bg="yellow", cex=0.7, add=T)

It would be good to add a legend to the map. To do so, use the following command and then click towards the bottom-right of your map in the place where you would like the legend will go:

> locator(1)$x[1] 462440$y[1] 4407000

We can now place the legend at those map units,

> choro.legend(462440,4407000,shades,fmt="%4.1f",title='Population density')

In practice, it may take some trial-and-error to get the legend in the right place.

Similarly, we can add a north arrow and a map scale,

> north.arrow(461000, 4445000, "N", len=1000, col="light gray")> map.scale(425000,4400000,10000,"km",5,subdiv=2,tcol='black',scol='black',+ sfcol='black')

The result is shown in Figure 4.1, below.


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Figure 4.1. An example of a choropleth map produced in R

4.2.1 Some alternative choropleth maps

What we see on a choropleth map and how we interpret it is a function of the classification used to shade the areas. Typing ?auto.shading we discover that the default for autoshading is to use a quantile classification with five (n + 1) categories and a red shading scheme. We might like to compare this with using a standard deviation based classification and one based on the range.

> x <- districts@data$POPDEN> shades2 <- auto.shading(x, cutter=sdCuts, cols=brewer.pal(5,"Greens"))> shades3 <- auto.shading(x, cutter=rangeCuts, cols=brewer.pal(5,"Blues"))

We could now plot these maps in the same way as before. It will work; however, it may also become tiresome typing the same code each time to overlay the point data and to add the annotations. We could save ourselves the trouble by writing a simple function to do it:

> map.details <- function(shading) {+ plot(landdata, pch=21, bg="yellow", cex=0.7, add=T)+ choro.legend(461000,4407000,shading,fmt="%4.1f",title='Population density')+ north.arrow(473000, 4445000, "N", len=1000, col="light gray")+ map.scale(425000,4400000,10000,"km",5,subdiv=2,tcol='black',+ scol='black',sfcol='black')+ }

Now we can produce the maps,

> # The quantile classification, as before> choropleth(districts, x, shades)> map.details(shades)

> # The standard deviation classification> choropleth(districts, x, shades2)> map.details(shades2)


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> # The range-based classification> choropleth(districts, x, shades3)> map.details(shades3)

4.3 XY Maps with the point symbols shaded

In Section 4.1.1, the maps we produced of the land price data were not very elegant. It would be nice to alter the size or shading of the points according to some attribute of the data; for example the land value. This information is contained in the dataset,

> head(landdata@data, n=3) LNPRICE LNAREA DCBD DELE DRIVER DPARK Y0405 Y0607 Y08091 9.29 11.84 8.28 7.25 7.38 8.09 0 1 02 8.64 10.97 9.04 5.61 8.41 7.51 0 1 03 6.69 7.92 8.89 5.62 8.22 7.27 0 1 0

Altering the size is achieved simply enough by passing some function of the variable (LNPRICE) to the character expansion argument (cex). For example,

> x <- landdata@data$LNPRICE> plot(landdata, pch=21, bg="yellow", cex=0.2*x)

Shading them according to their value is a little harder. The process is to cut the values into groups (using a quintile classification, for example). Then create a colour palette for each group. Finally, map the points and shade them by group.

4.3.1 Creating the groups

There are various ways this can be done. The first stage is to find the lower and upper values (the break points) for each group and an easy way to do that is to use the classInt library that is designed for the purpose.

> library(classInt)

For example, for a quantile classification with five groups the break points are:

> classIntervals(x, 5, "quantile")

(we can see the number of land parcels in each group is approximately but not exactly equal)

For a 'natural breaks' classification they are

> classIntervals(x, 5, "fisher")

or

> classIntervals(x, 5, "jenks")

For an equal interval classification

> classIntervals(x, 5, "equal")

For a standard deviation classification

> classIntervals(x, 5, "sd")

and so forth (see ?classIntervals for more options)

Let's use a natural breaks classification, extracting the break points from the classIntervals(...) function and storing them in an object called break.points,

> break.points <- classIntervals(x, 5, "fisher")$brks# Specifying $brks gives just the values without the number in each group

> break.points[1] 4.850 6.300 7.105 7.865 8.820 11.060


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The second stage is to assign each of the land price values to one of the five groups. This is done using the cut(...) function.

> groups <- cut(x, break.points, include.lowest=T, labels=F)

We can check the number of land parcels in each group by counting them

> table(groups)groups 1 2 3 4 5 159 267 352 227 112

Which should be the same as for

> classIntervals(x, 5, "fisher")style: fisher [4.85,6.3) [6.3,7.105) [7.105,7.865) [7.865,8.82) [8.82,11.06] 159 267 352 227 112

4.3.2 Creating the colour palette and mapping the data

This is most easily done using the RColorBrewer library which is based on the ColorBrewer website, http://www.colorbrewer.org and is designed to create “nice looking colour palettes especially for thematic maps.”

> library(RColorBrewer)

In fact, you have used this library already when creating the choropleth maps. It is implicit in the command auto.shading(x, cutter=sdCuts, cols=brewer.pal(5,'Greens')), for example, where the function brewer.pal(...) is a call to RColorBrewer asking it to create a sequential palette of five colours going from light to dark green.

We can create a colour palette in the same way,

> palette <- brewer.pal(5, "Greens")# Use ?brewer.pal to find out about other colour schemes

map the data

> plot(districts)> plot(landdata, pch=21, bg=palette[groups], cex=0.2*x, add=T)

and add the north arrow and scale bar

> north.arrow(473000, 4445000, "N", len=1000, col="light gray")> map.scale(425000,4400000,10000,"km",5,subdiv=2,tcol='black',scol='black',+ sfcol='black')

Adding the legend is a little harder and uses the legend(...) function,

> legend("bottomright", legend=c("4.85 to <6.3", "6.3 to <7.105",+ "7.105 to <7.865","7.865 to <8.82","8.82 to 11.06"), pch=21, pt.bg=palette,+ pt.cex = c(0.2*5.6, 0.2*6.7, 0.2*7.5, 0.2*8.3, 0.2*9.9), title="Land value (log)")

See ?legend for further details.

Tip When creating maps in colour, be wary of choosing colours that cannot be distinguished from one another by those with colour-blindness. Red-green combinations are particularly problematic. Be aware, also, that many publications still require graphics to be in grayscale. You could use, for example, palette <- brewer.pal(5, "Greys")


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http://www.colorbrewer.org/

Figure 4.2 A shaded point map produced in R

4.4 Creating a raster grid

A problem with the map shown in Figure 5.2 is that of over-plotting: of the multiple points occupying the same space on the map and obscuring each other. One solution is to convert the points into a raster grid, giving the average land value per grid cell.

We begin by loading the raster library

> library(raster)

then define the length of each raster cell (here giving a 1km by 1km cell size as the units are metres)

> cell.length <- 1000

We will allow the grid to complete cover the districts of Beijing so will base its dimensions on the bounding box (the minimum enclosing rectangle) for the districts. The bounding box is found using

> bbox(districts) min maxx 418358.5 473517.4y 4391094.2 4447245.9

and this information will be used to calculate the number of columns (ncol) and the number of rows (nrow) for the grid:

> xmin <- bbox(districts)[1,1]> xmax <- bbox(districts)[1,2]> ymin <- bbox(districts)[2,1]> ymax <- bbox(districts)[2,2]> ncol <- round((xmax - xmin) / cell.length, 0)> nrow <- round((ymax - ymin) / cell.length, 0)> ncol


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[1] 55> nrow[1] 56

We then create a blank 55 by 56 raster grid,

> blank.grid <- raster(ncols=ncol, nrows=nrow, xmn=xmin, xmx=xmax, ymn=ymin, ymx=ymax)

The next stage is to define the (x, y) and attribute values of the points that we are going to aggregate by averaging into the bank grid.

> xs <- coordinates(landdata)[,1]> ys <- coordinates(landdata)[,2]> xy <- cbind(xs, ys)> x <- landdata@data$LNPRICE> land.grid = rasterize(xy, blank.grid, x, mean)

We can now plot the grid using the default values,

> plot(land.grid)> plot(districts, add=T)

or customise it a little

> break.points <- classIntervals(x, 8, "pretty")$brks> palette <- brewer.pal(8, "YlOrRd")> plot(land.grid, col=palette, breaks=break.points, main="Mean log land value")> plot(districts, add=T)> map.scale(426622,4394549,10000,"km",5,subdiv=2,tcol='black',scol='black',+ sfcol='black')> north.arrow(468230, 4394708, "N", len=1000, col="white")

Figure 4.3. The land price data rasterised


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4.5 Spatially joining data

A further example of the sort of GIS functionality available in R is given by assigning each of the land parcel points the attribute data for the districts within which they are located – a spatial join based on a point-in-polygon operation.

This is simple to achieve with the basis being the over(...) function, here overlaying the points on the polygons. For example,

> head(over(landdata, districts), n=3) SP_ID POPDEN JOBDEN1 57 17.9478 19.33222 8 22.9676 20.44683 15 32.4849 65.3001

shows that the first point in the land price data is located in the 58 th of the districts. The reason that it is the 58th and not the 57th is that the IDs (SP_ID) are numbered beginning from zero not one, which is common for GIS. We can easily check this is correct:

> plot(districts[58,])> plot(landdata[1,], pch=21, add=T)

- the point is indeed with the polygon.

To join the data we will create a new data frame containing the attribute data from landdata and the results of the overlay function above.

> joined.data <- data.frame(slot(landdata, "data"), over(landdata, districts))# could also use data.frame(landdata@data, over(landdata, districts)

> head(joined.data, n=3) LNPRICE LNAREA DCBD DELE DRIVER DPARK Y0405 Y0607 Y0809 SP_ID POPDEN JOBDEN1 9.29 11.84 8.28 7.25 7.38 8.09 0 1 0 57 17.9478 19.33222 8.64 10.97 9.04 5.61 8.41 7.51 0 1 0 8 22.9676 20.44683 6.69 7.92 8.89 5.62 8.22 7.27 0 1 0 15 32.4849 65.3001

These data cannot be plotted as they are. What we have is just a data table. It is not linked to any map.

> class(joined.data)[1] "data.frame"

However, they can be linked to the geography of the existing map to create a new Spatial Points-with-attribute data object that can then be mapped in the way described in Section 4.3, p.43.

> combined.map <- SpatialPointsDataFrame(coordinates(landdata), joined.data)> class(combined.map)[1] "SpatialPointsDataFrame"attr(,"package")[1] "sp"> proj4string(combined.map) <- crs> head(combined.map@data)

> x <- combined.map@data$POPDEN# Select the variable to map

> break.points <- classIntervals(x, 5, "quantile")$brks# Find the breaks between the map classes

> groups <- cut(x, break.points, include.lowest=T, labels=F)# Place the observations into the groups (map classes)

> palette <- brewer.pal(5, "Blues")# Create a colour palette

> plot(districts)> plot(combined.map, pch=21, bg=palette[groups], cex=0.8, add=T)


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# Plot the data> map.scale(426622,4396549,10000,"km",5,subdiv=2,tcol='black',scol='black',+ sfcol='black')> north.arrow(473000, 4445000, "N", len=1000, col="white")

# Add the map scale and north arrow> classIntervals(x, 5, "quantile")

# Look at the break points to include in the legend> legend("bottomright", legend=c("0.87 to <3.32", "3.32 to <10.2",+ "10.2 to <17.7","17.7 to <25.4","25.4 to 283"), pch=21, pt.bg=palette,+ pt.cex=0.8, title="Population density")

# Add the legend to the map

Figure 4.4. The population density of the districts at each of the land parcel points(the map is a result of a spatial join operation)

4.6 Why is any of this useful?

A question you may ask at this stage is why use R when it appears to involve a reasonable amount of code to do things that are easily completed using the point-and-click interface of a standard GIS. It's a good question.

There are several answers.

The first is that R produces graphics of publishable quality and you have considerable control over their design. That might be an advantage.

Secondly, the code used to complete the tasks and produces the maps allows for reproducibility: it can be shared with (and checked by) other people. It can also be easily changed if, for example, you wanted to slightly alter the point sizes or change the classes from coloured to grayscale. Making a


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few tweaks to a script can be much faster than having to go through a number of drop-down menus, tabs, right-clicks, etc. to achieve what you want.

Third, precisely because R's spatial capabilities do not standalone from the rest of its functionality, they allow for the integration of statistical and spatial ways of working. Three examples follow.

4.6.1 Example 1: mapping regression residuals

We can use the combined dataset to fit a hedonic land price model, estimating some of the predictors of land price at each of the locations. The variables are:

LNPRICE: Log of the land price: RMB per square metre LNAREA: Log of the land parcel size DCBD: distance to the CBD on a log scale DELE: distance to the nearest elementary school on a log scale DRIVER: distance to the nearest river on a log scale DPARK: distance to the nearest park on a log scaleY0405: A dummy variable (1 or 0) indicating whether the land was sold in the period 2004-5Y0607: A dummy variable indicating whether the land was sold in the period 2006-7Y0809: A dummy variable indicating whether the land was sold in the period 2007-8(All of the remaining land parcels were sold before 2004)SP_ID: An ID for the polygon (district) in which the land parcel is locatedPOPDEN: the population density in each district. data source: the fifth census data in 2000. JOBDEN the jobden density in each district. data source: the fifth census data in 2000.

Remember: the data are simulated and not genuine.

To fit the regression model we use the function lm(...), short for linear model.

> model1 <- lm(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map@data)> summary(model1)

Call:lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map@data)


Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 11.916563 0.575031 20.723 < 2e-16 ***DCBD -0.259700 0.055202 -4.705 2.87e-06 ***DELE -0.079764 0.032784 -2.433 0.01513 * DRIVER 0.063370 0.029665 2.136 0.03288 * DPARK -0.294466 0.046747 -6.299 4.31e-10 ***POPDEN 0.004552 0.001047 4.346 1.51e-05 ***JOBDEN 0.006990 0.003009 2.323 0.02037 * Y0405 -0.183913 0.057289 -3.210 0.00136 ** Y0607 0.152825 0.087152 1.754 0.07979 . Y0809 0.803620 0.118338 6.791 1.81e-11 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


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To obtain the residuals (errors) from this model we can use any of the functions residuals(...), rstandard(...) or rstudent(...) to obtain the 'raw' residuals, the standarised residuals and the Studentised residuals, respectively.

> residuals <- rstudent(model1)> summary(residuals)

An advantage of using R is that we can now map the residuals to look for geographical patterns that, if they exist, would violate the assumption of independent errors (and potentially affect both the estimate of the model coefficients and their standard errors).

> break.points <- c(min(residuals), -1.96, 0, 1.96, max(residuals))> groups <- cut(residuals, break.points, include.lowest=T, labels=F)> palette <- brewer.pal(4, "Set2")> plot(districts)> plot(combined.map, pch=21, bg=palette[groups], cex=1, add=T)> map.scale(426622,4396549,10000,"km",5,subdiv=2,tcol='black',scol='black',+ sfcol='black')> north.arrow(473000, 4445000, "N", len=1000, col="white")> legend("bottomright", legend=c("<1.96", "-1.96 to 0", "0 to 1.96",">1.96"),+ pch=21, pt.bg=palette, pt.cex=0.8, title="Studentised residuals")

Figure 4.5. Map of the regression residuals from a model predicting the land parcel prices


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4.6.2 Example 2: Comparing values with those of a nearest neighbour

Is there a geographical pattern to the residuals in Figure 4.5? Perhaps, although this raises the question of what actually a random pattern would like. What there definitely is, is a significant correlation between the residual value at any one point and that of its nearest neighbouring point:

> library(spdep) # Loads the spatial dependence library> knn1 <- knearneigh(combined.map, k=1, RANN=F)$nn # Finds the first nearest neighbour to each point> head(knn1, 3) [,1][1,] 10 # The nearest neighbour to point 1 is point 10[2,] 172 # The nearest neighbour to point 2 is point 172[3,] 152 [etc.]> cor.test(residuals, residuals[knn1])


data: residuals and residuals[knn1]t = 13.0338, df = 1115, p-value < 2.2e-16alternative hypothesis: true correlation is not equal to 095 percent confidence interval: 0.3116038 0.4134505sample estimates: cor 0.3636132 # Calculates the correlation between a point and its nearest neighbour

4.6.3 Example 3: Exporting the data as a new shapefile

If we wished, we could now save the residual values as a new shapefile to be used in other GIS. This is straightforward and uses the same procedure to create a Spatial Points-with-attribute data object that we used in Section 4.5.

residuals.map <- SpatialPointsDataFrame(coordinates(landdata), data.frame(residuals))getwd() # This is the working directory where the map will be saved.

# You may want to change it using the drop down menus.writePolyShape(residuals.map, "residuals.shp")

4.7 R Geospatial Data Abstraction Library

Before ending this session we note the package rgdal providing an interface to many of the open source projects gathered in the Open Source Geospatial Foundation (www.osgeo.org); in particular, to Frank Warmerdam's Geospatial Data Abstraction Library (http://www.gdal.org). This package is not installed with the other spatial packages under the Spatial Task View (see 'Installing and loading one or more of the packages (libraries),' p.37) but can be installed in the normal way from CRAN1:

> install.packages("rgdal")

Once installed, rgdal can be used for spatial data import and export, and projection and transformation, as documented in Chapter 4 of Bivand et al.

1 Note: it used to be the case that Mac Intel OS X binaries were not provided on CRAN, but could be installed from the CRAN Extras repository with> setRepositories(ind=1:2)> install.packages("rgdal")

However, at the time of writing the Mac binaries are provided on CRAN and can be downloaded in the normal way without having to change the source repository.


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http://www.gdal.org/

http://www.osgeo.org/

Reference:

Bivand, R.S., Pebesma, E.J. & Gómez-Rubio, V., 2008. Applied Spatial Data Analysis with R. Berlin: Springer.

4.8 Getting Help

There is a mailing list for discussing the development and use of R functions and packages for handling and analysis of spatial, and particularly geographical, data. It can be subscribed to at www.stat.math.ethz.ch/mailman/listinfo/r-sig-geo.

The spatial cheat sheet by Barry Rowlingson at Lancaster University is really helpful: http://www.maths.lancs.ac.uk/~rowlings/Teaching/UseR2012/cheatsheet.html

There are some excellent R spatial tips and tutorials on Chris Brunsdon's Rpubs site, http://rpubs.com/chrisbrunsdon, and on James Cheshire's website, http://spatial.ly/r/.

Perhaps the most hardest thing is to remember which library to use when. At the risk of over-simplification:

Use library(maptools) to import and export shapefileUse library(sp) to create and manipulate spatial objects in RUse library(GISTools) to help create mapsUse library (RColorBrewer) to create colour schemes for mapsUse library(raster) to create rasters and rasterisations of dataUse library(spdep) to look at the spatial dependencies in data and (as we shall see in the next session) to create spatial weights matrices

Tip: if you load begin by loading library(GISTools) you will find maptools, sp and RColorBrewer (amongst others) are automatically loaded too.


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http://spatial.ly/r/

http://rpubs.com/chrisbrunsdon

http://www.maths.lancs.ac.uk/~rowlings/Teaching/UseR2012/cheatsheet.html

http://www.stat.math.ethz.ch/mailman/listinfo/r-sig-geo

Session 5: Defining Neighbours

This session demonstrates more of R's capability for spatial analysis by showing how to create spatial weightings based on contiguity, nearest neighbours and by distance. Methods of inverse distance weighting are also introduced and used to produce Moran plots and tests of spatial dependency in data. We find that the assumption of independence amongst the residuals of the regression model fitted in the previous session appears to be unwarranted.

5.1 Identifying neighbours

5.1.1 Getting Started

If you have closed and restarted R since the last session, load the workspace session5.RData which contains the districts polygons and combined map created previously in Session 4 (see Section4.1.3, p.40 and Section 4.5, p.47). All the code for this session is contained in the file Session5.R

> load(file.choose()) # Load the workspace session5.RData

Also load the spdep library,

> library(spdep)

5.1.2 Creating a contiguity matrix

A contiguity matrix is one that identifies polygons that share boundaries and (in what is called the Queen's case) corners too. In other words, it identifies neighbouring areas. To do this we use the poly2nb(...) function – polygons to an object of class neighbours.

> contig <- poly2nb(districts)

A summary of the neighbours object shows that there are 134 regions (districts, which can be confirmed using nrow(districts)) with each being linked to 5.46 others, on average. There are two regions with no links (use plot(districts) and you can see them to the east of the map), and two regions with 10 links. The Queen's case is assumed by default, see ?poly2nb.

> summary(contig)Neighbour list object:Number of regions: 134 Number of nonzero links: 732 Percentage nonzero weights: 4.076632 Average number of links: 5.462687 2 regions with no links:20 80Link number distribution:

0 1 2 3 4 5 6 7 8 9 10 2 1 6 8 23 29 26 21 9 7 2 1 least connected region:129 with 1 link2 most connected regions:90 100 with 10 links

It is helpful to learn a little more about the structure of the contiguity object. It is an object of class nb which is itself a type of list.

> class(contig)[1] "nb"> typeof(contig)


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[1] "list"

Looking at the first part of this list we find that the first district has two neighbours: polygons 52 and 54.

> contig[[1]][1] 52 54

The second district has five neighbours: polygons 3, 4, 6, 99, 100

> contig[[2]][1] 3 4 6 99 100

We can confirm this is correct by plotting the district and its neighbours on a map

> plot(districts)> plot(districts[2,], col="red", add=T)> plot(districts[c(3,4,6,99,100),], col="yellow", add=T)

5.1.3 K nearest neighbours (knn)

It would not make sense to evaluate contiguity of point data. Instead, we could find, for example, the six nearest neighbours to each point:

> knear6 <- knearneigh(combined.map, k=6, RANN=F)

This produces an object of class knn. Looking at its parts we find that the six nearest neighbours (nn) to point 1 are points 10, 11, 1077, 1076, 93, 453, where 10 is the closest; that there are 1117 points in total (np); that we searched for the six nearest neighbours (k); that the points exist in a two-dimensional space (dimension); and that we can see the coordinates of the points (labelled x).

> names(knear6)[1] "nn" "np" "k" "dimension" "x" > head(knear6$nn) [,1] [,2] [,3] [,4] [,5] [,6][1,] 10 11 1077 1076 93 453[2,] 172 110 155 162 153 156[3,] 152 149 150 151 169 168[4,] 135 143 679 148 678 674[5,] 32 164 166 165 167 168[6,] 432 920 969 1024 919 968> head(knear6$np)[1] 1117> head(knear6$k)[1] 6> head(knear6$dimension)[1] 2> head(knear6$x, n=3) x y[1,] 454393.1 4417809[2,] 442744.9 4417781[3,] 444191.7 4416996

Imagine we are interested in calculating the correlation between some variable (call it x) at each of the points and at each of the points' closest neighbour. From the above [head(knear6$nn)] we can see this is the correlation between x1, x2, x3, x4, x5, x6, (etc.) and x10, x172, x152, x135, x32, x432, (etc.). For the correlation with the second closest neighbours it would be with x11, x110, x149, x143, x164, x920, (etc.), for the third closest, x1077, x155, x150, x679, x166, x969, (etc.), and so forth. Using the simulated data about the price of land parcels in Beijing, we can calculate these correlations as follows:

> x <- combined.map$LNPRICE # Or, combined.map@data$LNPRICE


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> cor(x, x[knear6$nn[,1]]) # Correlation with the 1st nearest neighbour[1] 0.5275507> cor(x, x[knear6$nn[,2]]) # Correlation with the 2nd nearest neighbour[1] 0.4053884> cor(x, x[knear6$nn[,3]]) # Correlation with the 3rd nearest neighbour[1] 0.4073915> cor(x, x[knear6$nn[,6]]) # Correlation with the 6th nearest neighbour[1] 0.3163395

What these values suggest is that even at the sixth nearest neighbour, the value of a land parcel at any given point tends to be similar to the value of the land parcels around it – an example of positive spatial autocorrelation.

An issue is that the threshold of six nearest neighbours is purely arbitrary. An interesting question is how far – how many neighbours away – we can typically go from a point and still find a similarity in the land price values. One way to determine this would be to carry on with the calculations above, repeating the procedure until we get to, say, the 250 th nearest neighbour. This is, in fact, what we will do but automating the procedure. One way to achieve this is to use a for loop:

> knear100 <- knearneigh(combined.map, k=250, RANN=F)> # Find the 250 nearest neighbours> correlations <- vector(mode="numeric", length=250)> # Creates an object to store the correlations> for (i in 1: 250) {+ correlations[i] <- cor(x, x[knear250$nn[,i]])+ }> correlations [1] 0.52755068 0.40538835 0.40739150 0.35804349 0.33960454 0.31633952 [etc.]

Another way – which amounts to the same thing – is to make use of R's ability to apply a function sequentially to columns (or rows) in an array of data:

> correlations <- apply(knear250$nn, 2, function(i) cor(x, x[i]))# The 2 means to apply the correlation function to the columns of knear100$nn

> correlations [1] 0.52755068 0.40538835 0.40739150 0.35804349 0.33960454 0.31633952 [etc.]

In either case, once we have the correlations they can be plotted,

> plot(correlations, xlab="nth nearest neighbour", ylab="Correlation")> lines(lowess(correlations)) # Adds a smoothed trend line

Looking at the plot (Figure 5.1) we find that the land prices become more dissimilar (less correlated) the further away we go from each point, dropping to zero correlation from after about the 200th nearest neighbour. The rate of decrease in the correlation is greatest to about the 35 th neighbour, after which it begins to flatten.

We can also determine the p-values associated with each of these correlations and identify which are not significant at a 99% confidence,

> pvals <- apply(knear250$nn, 2, function(i) cor.test(x, x[i])$p.value)> which(pvals > 0.01) [1] 63 88 110 115 121 125 134 136 137 138 140 142 145 146 [etc.]

It is from about the 100th neighbour that the correlations begin to become insignificant. Whether this is usefully information or not is a moot point: a measure of statistical significance is really only an indirect measure of the sample size. It may be better to make a decision about the threshold at which the neighbours are not substantively correlated based on the actual correlations (the effect sizes) rather than their p-values. Whilst it remains a subjective choice, here we will here use the 35 th

neighbour as the limit, before which the correlations are typically equal to r = 0.20 or greater.


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> knear35 <- knearneigh(combined.map, k=35, RANN=F)

To now convert this object of class knn to the same class of object that we had in Section 5.1.2 ('Creating a contiguity matrix') we use

> knear35nb <- knn2nb(knear35)> class(knear35nb)[1] "nb"> head(knear35nb, n=1)[[1]] [1] 8 10 11 14 19 54 56 57 82 91 92 93 [etc.]

Note that these are now in numeric order.

> knear35nbNeighbour list object:Number of regions: 1117 Number of nonzero links: 39095 Percentage nonzero weights: 3.133393 Average number of links: 35 Non-symmetric neighbours list

Figure 5.1. The Pearson correlation between the land parcel values ateach point and their nth nearest neighbour

5.1.4 Identifying neighbours by (physical) distance apart

It is also possible to identify the neighbours of points by their Euclidean or Great Circle distance apart using the function dnerneigh(...). For example, if we wanted to identify all points between 100 and 1000 metres of each other:

> d100to1000 <- dnearneigh(combined.map, 100, 1000)> class(d100to1000)[1] "nb"> d100to1000Neighbour list object:Number of regions: 1117 Number of nonzero links: 9822 Percentage nonzero weights: 0.7872154


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Average number of links: 8.793196 48 regions with no links:31 45 51 52 139 160 193 276 277 289 291 313 336 403 429 508 511 512 516 518 520 535 559 565 567 706 710 716 717 792 796 805 811 817 818 819 860 862 877 878 899 900 947 965 988 992 1037 1039

See ?dnerneigh(...) for further details.

5.2 Creating a Spatial Weights matrix

What we created in Section 5.1 was a list of neighbours where we had flexibility to decide about what counts as a neighbour. The next stage will be to convert it into a spatial weights matrix so we can use it for various methods of spatial analysis. This extra stage of conversion may seem like an unnecessary additional chore. However, the creation of the spatial weights matrix allows us to define the strength of relationship between neighbours. For example, we may want to give more weight to neighbours that are located closer together and less weight to those that are further apart (decreasing to zero beyond a certain threshold).

5.2.1 Creating a binary list of weights

We could create a simple binary 'matrix' from any of our existing lists of neighbours. In principle:

> spcontig <- nb2listw(contig, style="B")Error in nb2listw(contig, style = "B") : Empty neighbour sets found

Note, however, the error message, which arises because two of the Chinese districts do not share a boundary with others,

> contig

In this case, we shall have to instruct the function to permit an empty set

> spcontig <- nb2listw(contig, style="B", zero.policy=T)

The same problem does not arise for the one hundred nearest neighbours (by definition, it cannot – each point has neighbours) but it does for the distance based list:

> spknear35 <- nb2listw(knear35nb, style="B")> spd100to1000 <- nb2listw(d100to1000, style="B")Error in nb2listw(d100to1000, style = "B") : Empty neighbour sets found> spd100to1000 <- nb2listw(d100to1000, style="B", zero.policy=T)

What we create are objects of class listw (a list of spatial weights):

> class(spcontig)[1] "listw" "nb" > class(spknear35)[1] "listw" "nb" > class(spd100to1000)[1] "listw" "nb"

Looking at the first of these objects we can see how it has been constructed. It contains binary weights (style B); district 1 has two neighbours, districts 52 and 54; and both of those have been given a weight of one (all other districts therefore have a weight of zero with district 1). Similarly, district 2 has neighbours 3, 4, 6, 999 and 100, each with a weight of one.

> names(spcontig)[1] "style" "neighbours" "weights" > spcontig$style[1] "B"> head(spcontig$neighbours, n=2)[[1]]


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[1] 52 54

[[2]][1] 3 4 6 99 100

> head(spcontig$weights, n=2)[[1]][1] 1 1

[[2]][1] 1 1 1 1 1

The other spatial weights 'matrices' have the same form.

5.2.2 Creating a row-standardised list of weights

Using binary weights where (1) indicates two places are neighbours, and (0) indicates they are not, may create a problem when different places have different numbers of neighbours (as is the case for both the contiguity and distance-based approaches). Imagine a calculation where the result is in someway dependent upon the sum of the weights involved. For example,

yi=∑n

wij x j

All things being equal, we expect places with more neighbours to generate larger values of y simply because they have more non-zero values contributing to the sum. A way around this problem is to scale the weights so that for any one place they sum to one – a process known as row-standardisation and actually the default option:

> spcontig <- nb2listw(contig, zero.policy=T)> spknear35 <- nb2listw(knear35nb)> spd100to1000 <- nb2listw(d100to1000, zero.policy=T)> names(spcontig)[1] "style" "neighbours" "weights" > spcontig$style[1] "W"> head(spcontig$neighbours, n=2)[[1]][1] 52 54

[[2]][1] 3 4 6 99 100

> head(spcontig$weights, n=2)[[1]][1] 0.5 0.5

[[2]][1] 0.2 0.2 0.2 0.2 0.2

In the case of the contiguity matrix, district 1 still has neighbours 52 and 54, and district 2 still has neighbours 3, 4, 6, 99 and 100, but the weights are now row-standardised (style W) and in each case they sum to one.

5.2.3 Creating an inverse distance weighting (IDW)

A more ambitious undertaking is to decrease the weighting given to two points according to their distance apart, reducing to zero, for example, beyond the 35th nearest neighbour. To achieve this, we


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begin by calculating the distances between each of the points, using the spDists(...) function. This calculate the distances between two sets of points where the points' locations are defined by an (x, y) coordinates (or by longitude and latitude: see ?spDists). To obtain the (x, y) coordinates of all theland parcels contained in our combined map we could use the function coordinates(...), therefore obtaining the distance-between-points matrix using

> d.matrix <- spDists(coordinates(combined.map), coordinates(combined.map))

However, since combined.map is a SpatialPoints(DataFrame) object we can just reference it directly without explicitly supplying the point coordinates,

> d.matrix <- spDists(combined.map, combined.map)

Either way, the same result is achieved: an np by np matrix where np is the number of points and the matrix contains the distances between them:

> d.matrix # The full matrix. It's too large to show on screen.> head(d.matrix[1,1:10]) # The distances from point 1 to the first 10 others[1] 0.000 11648.171 10233.788 10360.952 9345.806 3618.377> nrow(d.matrix)[1] 1117

This is showing that the distance from point 1 to point 2 is 11.6km. The distance from point 1 to itself is, of course, zero and the matrix is symmetric,

> d.matrix[1,2] # The distance from point 1 to point 2 ...[1] 11648.17> d.matrix[2,1] # ... is the same as from point 2 to point 1[1] 11648.17

If we are going to reduce the weighting to zero beyond the 35th nearest neighbour then we don't actually need the full distance matrix, only the distances from each point to those 35 neighbours. We have already identified the nearest 35 neighbours for each point but for the sake of completeness let's do it again:

> knear35 <- knearneigh(combined.map, k=35, RANN=F) # Find the 35 nearest neighbours for each point> np <- knear35$np # The total number of points

Looking at the results we know, for example, that the nearest neighbours to point 1 are points 10, 11, 1077, etc. so the distances we need to extract from the distance matrix are row 1, columns 10, 11, 1077, and so forth. For point 2 the nearest neighbours are 172, 110, 155, etc. so from the distance matrix we need row 2, columns 172, 110, 155, …

> head(knear35$nn, n=2) [,1] [,2] [,3] [,4] [,5] [etc.][1,] 10 11 1077 1076 93 [etc.][2,] 172 110 155 162 153 [etc.]

For point 1 we may obtain the distances to its 35 nearest neighbours using,

> i <- knear35$nn[1,]> head(i)[1] 10 11 1077 1076 93 453> distances <- d.matrix[1,i]> head(distances)[1] 411.0010 566.2565 570.7371 760.7815 784.0760 788.7454

For point 2:

> i <- knear35$nn[2,]


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> head(i)[1] 172 110 155 162 153 156> distances <- d.matrix[2,i]> head(distances)[1] 220.4186 293.3992 827.5340 844.6190 845.0816 868.7554

The same logic underpins the following code except instead of manually obtaining the distances for each point in turn, it loops through them all sequentially. It is also calculates weights that are

inversely related to the distance from a point to its neighbours (here wij=1

d ij0.5 is used).

> d.weights <- vector(mode="list", length=np) # Creates an empty list to store the inverse distance weights> for (i in 1:np) { # A loop taking each point in turn+ neighbours <- knear35$nn[i,] # Finds the neighbours for the ith point+ distances <- d.matrix[i,neighbours] # Calculates the distance between the

# ith point and its neighbours+ d.weights[[i]] <- 1/distances^0.5 # Calculates the IDW+ }>

We can now create the list of neighbours (an object of class nb) and have a corresponding list of general weights (based on inverse distance weighting) that together allow for the final spatial weights matrix to be created:

> knear35nb <- knn2nb(knear35)> head(knear35nb, n=2) # The list of neighbours[[1]] [1] 10 11 91 92 [etc.]

[[2]] [1] 3 61 66 67 77 [etc.]

> head(d.weights, n=2) # The corresponding list of general weights[[1]] [1] 0.04932630 0.04202362 0.04185833 0.03625518 0.03571255 [etc.]

[[2]] [1] 0.06735594 0.05838087 0.03476219 0.03440880 0.03439938 [etc.]

> spknear35IDW <- nb2listw(knear35nb, glist=d.weights) # Creates the spatial weights matrix, now with IDW

Looking at the result we find that point 1 still has points 10, 11, 91, 92 and so forth as its neighbours (as it should, it would be worrying if that had changed!) but looking at their weighting, it decreases with distance away.

> head(spknear35IDW$neighbours, n=1)[[1]] [1] 10 11 91 92 [etc.]

> head(spknear35IDW$weights, n=1)[[1]] [1] 0.04698452 0.04002853 0.03987110 0.03453395 [etc.]

Note, however, that the weights are not actually wij=1

d ij0.5 but are rescaled to wij

STD=wij

∑ wi

because they are row-standardised. This means that the inverse distance weighting is a function of the local distribution of points around each point not just how far away away they are. For example,


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consider a point where all its neighbours are quite far from it. Using a strictly distance-based weighting each of those neighbours should receive a low weighting. However, once row standardisation is applied those low weights will be scaled upwards to sum to one. Reciprocally, imagine a point where all its neighbours are very close. Using a distance-based weighting those neighbours should receive a high weighting; in effect, though, they will be scaled downwards by the row standardisation. This may sound undesirable and counter to the objectives of inverse distance weighting, and can be prevented by changing the weights style

> spknear35IDWC <- nb2listw(knear35nb, glist=d.weights, style="C")

However, imagine the points sample across both urban and rural areas. The distances between points will most likely be smaller in the urban regions (where the density of points is greater, reflecting the greater population density), with greater distances between points in the rural regions. If row standardisation is not applied then the net result will be to give more weight to the urban parts of the region such that any subsequent calculation dependent upon the sum of the weights will be more strongly influenced by the urban areas than by the rural ones. Therefore careful though needs to be given to the style of weights to use.

5.2.4 Variants of the above

Common forms of inverse distance weighting include the bisquare and Gassian functions. These are, respectively,

wij=(1−d ij

2

d MAX2 )

2

where dMAX is the threshold beyond which the weights are set to zero; and

wij=exp(−0.5×d ij

2

d MAX2 )

The R code to calculate these weightings is below.

# Calculates the bisquare weighting to the 35th neighbourknear35 <- knearneigh(combined.map, k=35, RANN=F)np <- knear35$npd.weights <- vector(mode="list", length=np)for (i in 1:np) {

neighbours <- knear35$nn[i,]distances <- d.matrix[i,neighbours]dmax <- distances[35]d.weights[[i]] <- (1 - distances^2/dmax^2)^2

}spknear35bisq <- nb2listw(knn2nb(knear35), glist=d.weights, style="C")

# Calculates the Gaussian weighting to the 35th neighbourknear35 <- knearneigh(combined.map, k=35, RANN=F)np <- knear35$npd.weights <- vector(mode="list", length=np)for (i in 1:np) {

neighbours <- knear35$nn[i,]distances <- d.matrix[i,neighbours]dmax <- distances[35]d.weights[[i]] <- exp(-0.5*distances^2/dmax^2)

}spknear35gaus <- nb2listw(knn2nb(knear35), glist=d.weights, style="C")


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5.3 Using the spatial weights

5.3.1 Creating a spatially lagged variable

Once we have the spatial weights we can use them to create a spatially lagged variable. For example, if xi is the value of the land parcel at point i, then its spatial lag is the mean value of the land parcels that are the neighbours of i, where those neighbours are defined by the spatial weights. More precisely, it is the weighted mean value if, for example, inverse distance weighting has been employed. It is straightforward to calculate the spatially lagged variable. For example,

> x <- combined.map$LNPRICE> lagx <- lag.listw(spknear35gaus, x)

Having done so, the correlation between points and their neighbours can be calculated,

> cor.test(x, lagx)


data: x and lagxt = 14.1361, df = 1115, p-value < 2.2e-16alternative hypothesis: true correlation is not equal to 095 percent confidence interval: 0.3389422 0.4384758sample estimates: cor 0.389847

Here there is evidence of significant positive spatial autocorrelation – that the land price at one point tends to be similar to the land prices of its neighbours. This can be seen if we plot the two variables on a scatter plot, although the relationship is also somewhat noisy and may not be linear.

> plot(lagx ~ x)> best.fit <- lm(lagx ~ x)> abline(best.fit)> summary(best.fit)

Call:lm(formula = lagx ~ x)


Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.70723 0.12272 46.51 <2e-16 ***x 0.23176 0.01639 14.14 <2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1



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Figure 5.1. The relationship between the land price values and the spatial lag of those values

5.3.2 A Moran plot and test

What we created in Figure 5.1 is known as a Moran plot. A more direct way of producing it is to use the moran.plot(...) function,

> moran.plot(x, spknear35gaus)

which flags potential outliers / influential observations. To suppress their labelling, include the argument labels=F.

The Moran coefficient and related test provide a measure of the spatial autocorrelation in the data, given the spatial weightings.

> moran.plot(x, spknear35gaus)> moran.test(x, spknear35gaus)

Moran's I test under randomisation

data: x weights: spknear35gaus

Moran I statistic standard deviate = 39.663, p-value < 2.2e-16alternative hypothesis: greatersample estimates:Moran I statistic Expectation Variance 0.2657983392 -0.0008960573 0.0000452122

Essentially the Moran statistic is a correlation value, although it need not be exactly zero in the presence of no correlation (here the expected value is not zero but slightly negative) and can go beyond the range -1 to +1. The interpretation though is that the price of the land parcels and their neighbours are positively correlated: there is a tendency for like-near-like values.

More strictly, we should acknowledge the note found under ?moran.test(...) that the derivation of the tests assumes the matrix is symmetric, which it is not (because where A is one of the nearest


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neighbours to B it does not follow that B is necessarily one of the nearest neighbours to A):

> spknear35gausCharacteristics of weights list object:Neighbour list object:Number of regions: 1117 Number of nonzero links: 39095 Percentage nonzero weights: 3.133393 Average number of links: 35 Non-symmetric neighbours list # Note that the 'matrix' is not symmetric

Weights style: C Weights constants summary: n nn S0 S1 S2C 1117 1247689 1117 58.55285 4623.092

To correct for this we can use a helper function, listw2U(...),

> moran.test(x, listw2U(spknear35gaus))

5.4 Checking a regression model for spatially correlated errors

We end this session by fitting re-fitting the regression model of Section 4.6.1, 'Example 1: mapping regression residuals', p. 49 but this time using a Moran test to check whether the assumption of spatial independence in the residuals is warranted.

First the regression model,

> model1 <- lm(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map)

Now a Moran plot and test.

> moran.plot(residuals(model1), spknear35gaus)> lm.morantest(model1, listw2U(spknear35gaus))

# listw2U is used because the weights are not symmetrical


data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map)weights: listw2U(spknear35gaus)

Moran I statistic standard deviate = 16.0962, p-value < 2.2e-16alternative hypothesis: greatersample estimates:Observed Moran's I Expectation Variance 9.683406e-02 -3.893216e-03 3.916053e-05

The estimated correlation is about 0.097. Not huge, perhaps, but significant enough to question the assumption of independence.

Note that this result is dependent on the spatial weightings. If we change them, then the results of the Moran test will change also. For example, using the bi-square weighings (from Section 5.2.4, p.61):

> lm.morantest(model1, listw2U(spknear35bisq))



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data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map)weights: listw2U(spknear35bisq)


Here the change is slight, largely because the rate of decay of the inverse distance weighting matters rather less than the number of neighbours it decays to. It both cases above the threshold is 35, a number we obtained by judgement from Figure 5.1. Imagine we had chosen 150 instead. The results we then get (using a Gaussian decay function) are,


which, although still statistically significant, reduces the Moran's I value to about a third of its previous value.

5.5 Summary

The general process for creating spatial weights in R is as follows:

(a) read-in the (X, Y) data or shapefile into the R workspace and (in doing-so) convert it into a spatial object (see Session 4).

(b) Decide how neighbouring observations will be defined: by nearest neighbour, by distance, by contiguity for example.

(c) Convert the object of class nb into an object of class listw (spatial weights). For k-nearest neighbours there is a prior stage of converting the knn object into class nb.

(d) At the time of creating the spatial weights object you need to decide what type of weights to use, for example binary or row-standardised. You may also supply a list of general weights to produce inverse distance weighting.


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Session 6: Spatial regression analysis

In this session we draw together some of what we have learned about mapping and creating spatial weights in R to undertake spatial forms of regression, specifically a spatially lagged error model, a lagged y model, a mixed model, Geographically Weighted Regression and a multilevel model. The session is more an introduction to how to fit these sorts of models in R rather than an in-depth tutorial into their specification and use. Inevitably some of the models fit the (simulated) data better than others. However, that is not to say they would necessarily be preferred when using other data and in other analytical contexts. Which model to choose will depend on the purpose of the analysis and the theoretical basis for the modelling.

This session requires the spdep, Gwmodel and lme4 libraries to have been installed.

6.1 Introduction

6.1.1 Getting Started

If you have closed and restarted R since the last session, load the workspace session6.RData which contains the districts map and the combined (synthetic) land parcel and district data from Session 4 as well as the spatial weights with a Gaussian decay to the 35th neighbour created in Session 5.

> load(file.choose())> library(spdep)> ls()[1] "combined.map" "districts" "spknear35gaus"

Recall that the (log) of the land price values show significant spatial variation,

> moran.test(combined.map$LNPRICE, spknear35gaus)

Moran's I test under randomisation

data: combined.map$LNPRICE weights: spknear35gaus

Moran I statistic standard deviate = 39.663, p-value < 2.2e-16alternative hypothesis: greatersample estimates:Moran I statistic Expectation Variance 0.2657983392 -0.0008960573 0.0000452122

Our challenge is to try and explain some of that variation within a regression framework.

6.1.2 OLS regression

We begin by re-fitting the regression model from the end of the previous session, noting once again the apparent spatial dependencies in the residuals that violates the assumption of independence:

> model1 <- lm(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map)> lm.morantest(model1, listw2U(spknear35gaus))


data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 +


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Y0607 + Y0809, data = combined.map)weights: listw2U(spknear35gaus)


In addition to their apparent lack of independence we may also note that the residuals appear to show evidence of heteroskedasticity – of non-constant variance (they are therefore not independent nor identically distributed):

> plot(residuals(model1) ~ fitted(model1))# Plot the residuals against the fitted values

> abline(h=0, lty="dotted") # Add a horizontal line at residual value = 0> lines(lowess(fitted(model1), residuals(model1)), col="red") # Add a trend line. We are hoping not to find a trend but # to see that the residuals are random noise around 0.

# They are not.

More simply, a similar plot can be produced using

> plot(model1, which=3)

however that same code won't work for the spatial models we produce below. Instead, we can write a function to produce what we need,

> hetero.plot <- function(model) {+ plot(residuals(model) ~ fitted(model))+ abline(h=0, lty="dotted")+ lines(lowess(fitted(model), residuals(model)), col="red")+ }> hetero.plot(model1)

There are no quick fixes for the violated assumption of independent and identically distributed errors. The violation suggests we cannot take on trust the standard errors, t- and p-values shown under the model summary. It is likely that the standard errors for at least some of the predictor variables have been under-estimated (because if we have spatial dependencies in the residuals then they likely arise from spatial dependencies in the data which in turn mean we have less degrees of freedom than we think we have).

> summary(model1)

Call:lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map)


Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 11.916563 0.575031 20.723 < 2e-16 ***DCBD -0.259700 0.055202 -4.705 2.87e-06 ***DELE -0.079764 0.032784 -2.433 0.01513 * DRIVER 0.063370 0.029665 2.136 0.03288 * DPARK -0.294466 0.046747 -6.299 4.31e-10 ***POPDEN 0.004552 0.001047 4.346 1.51e-05 ***JOBDEN 0.006990 0.003009 2.323 0.02037 *


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Y0405 -0.183913 0.057289 -3.210 0.00136 ** Y0607 0.152825 0.087152 1.754 0.07979 . Y0809 0.803620 0.118338 6.791 1.81e-11 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


If we have doubts about this model because of the spatial patterning of the errors (which are likely but not necessarily caused by the patterning of the Y variable) then we need to consider other approaches.

6.2 Spatial Econometric Methods

6.2.1 Spatial Error Model

One option is to fit a spatial simultaneous autoregressive error model which decomposes the error into two parts: a spatially lagged component and a remaining error: y=X β+λW ξ+ε

Fitting the model and comparing it with the standard regression model we find that two of the predictor variables (DELE and POPDEN) no longer are significant at a conventional level and that the standard errors for many have risen. The lambda (a measure of spatial autocorrelation) is significant. The model fits the data better than the previous model (the AIC score is lower and the log likelihood value greater, as is the pseudo-R2 value):

> model2 <- errorsarlm(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map, spknear35gaus)> summary(model2)

Call:errorsarlm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map, listw = spknear35gaus)


Type: error Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(>|z|)(Intercept) 12.7929865 1.0803367 11.8417 < 2.2e-16DCBD -0.4528868 0.1179775 -3.8388 0.0001237 ***DELE -0.0430191 0.0414930 -1.0368 0.2998391DRIVER 0.0805519 0.0389786 2.0666 0.0387749 *DPARK -0.2225801 0.0640377 -3.4758 0.0005094 ***POPDEN 0.0040076 0.0012339 3.2479 0.0011624 **JOBDEN 0.0032269 0.0035468 0.9098 0.3629286Y0405 -0.2185441 0.0548818 -3.9821 6.831e-05 ***Y0607 0.2488569 0.0832995 2.9875 0.0028127 **Y0809 0.9301437 0.1131248 8.2223 2.220e-16 ***

Lambda: 0.68882, LR test value: 84.635, p-value: < 2.22e-16Asymptotic standard error: 0.053718 z-value: 12.823, p-value: < 2.22e-16Wald statistic: 164.43, p-value: < 2.22e-16


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Log likelihood: -1379.548 for error modelML residual variance (sigma squared): 0.67987, (sigma: 0.82455)Number of observations: 1117 Number of parameters estimated: 12 AIC: 2783.1, (AIC for lm: 2865.7)

(I have added the asterisks manually).

The AIC is given above but we can also obtain it using,

> AIC(model1)[1] 2865.731> AIC(model2)[1] 2783.096

The log likelihood scores using,

> logLik(model1)'log Lik.' -1421.865 (df=11)> logLik(model2)'log Lik.' -1379.548 (df=12)

And the pseudo-R2 for the spatial error model as,

> cor(combined.map$LNPRICE, fitted(model2))^2[1] 0.3590577

The problem of heteroskedasticity remains, however:

> hetero.plot(model2)

6.2.2 A spatially lagged y model

Although the spatial error model (above) fits the data better than the standard OLS model, it tells us only that there is an unexplained spatial structure to the residuals, not what caused them. It may offer better estimates of the model parameters and their statistical significance but it does not presuppose any particular spatial process generating the patterns in the land price values. A different model that explicitly tests for whether the land value at a point is functionally dependent on the values of neighbouring points is the spatially lagged y model: Y=ρW y+X β+ε

The model is fitted in R using,

> model3 <- lagsarlm(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map, spknear35gaus)> summary(model3)

Call:lagsarlm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map, listw = spknear35gaus)


Type: lag Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(>|z|)(Intercept) 9.3098717 0.8336866 11.1671 < 2.2e-16 ***DCBD -0.1753427 0.0585675 -2.9939 0.0027547 **DELE -0.0706011 0.0323879 -2.1799 0.0292677 *DRIVER 0.0334482 0.0304653 1.0979 0.2722435DPARK -0.2591509 0.0467432 -5.5441 2.954e-08 ***


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POPDEN 0.0042982 0.0010362 4.1482 3.351e-05 ***JOBDEN 0.0057660 0.0029799 1.9350 0.0529953Y0405 -0.1869054 0.0565474 -3.3053 0.0009488 ***Y0607 0.1928584 0.0860329 2.2417 0.0249819 **Y0809 0.8452182 0.1167707 7.2383 4.545e-13 ***

Rho: 0.23634, LR test value: 18.273, p-value: 1.9136e-05Asymptotic standard error: 0.055848 z-value: 4.2318, p-value: 2.3185e-05Wald statistic: 17.908, p-value: 2.3185e-05

Log likelihood: -1412.729 for lag modelML residual variance (sigma squared): 0.73357, (sigma: 0.85648)Number of observations: 1117 Number of parameters estimated: 12 AIC: 2849.5, (AIC for lm: 2865.7)LM test for residual autocorrelationtest value: 109.4, p-value: < 2.22e-16

The model is an improvement on the OLS model but does not appear to fit the data as well as the error model (the lagged y model has a greater AIC, lower log likelihood and lower pseudo-R2):

> AIC(model3)[1] 2849.458> logLik(model3)'log Lik.' -1412.729 (df=12)> cor(combined.map$LNPRICE, fitted(model3))^2[1] 0.3074552

The heteroskedasticity remains,

> hetero.plot(model3)

Moreover (and possibly related to the heteroskedasticity) significant autocorrelation remains in the residuals from this model. We can, of course, plot these residuals (see Session 4, 'Using R as a simple GIS' for further details). Here we will write a simple function to do so.

> # A function to draw a quick map> quickmap <- function(x, subset=NULL) {+ # The subset is of use for the later GWR model+ if(!is.null(subset)) {+ x <- x[subset]+ combined.map <- combined.map[subset,]+ }+ library(classInt)+ break.points <- classIntervals(x, 5, "fisher")$brks+ groups <- cut(x, break.points, include.lowest=T, labels=F)+ library(RColorBrewer)+ palette <- brewer.pal(5, "Spectral")+ plot(districts)+ plot(combined.map, pch=21, bg=palette[groups], add=T)+ # Some code to create the legend automatically...+ break.points <- round(break.points, 2)+ n <- length(break.points)+ break.points[n] <- break.points[n] + 0.01+ txt <- vector("character",length=n-1)+ for (i in 1:(length(break.points) - 1)) {+ txt[i] <- paste(break.points[i],"to <",break.points[i+1])+ }+ legend("bottomright",legend=txt, pch=21, pt.bg=palette)+ }


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> quickmap(x)

Figure 6.1. The residuals from the spatial lagged y model stilldisplay evidence of positive spatial autocorrelation

Note that the beta estimates of the lagged y-model cannot be interpreted in the same way as for a standard OLS model. For example, the beta estimate of 0.004 for the POPDEN variable does not mean that if (hypothetically) we increased that variable by one unit at each location we should then expect the (log) of the land parcel to everywhere increase by 0.004 even holding the other X variables constant. The reason is because if we did raise the value it would start something akin to a 'chain reaction' through the feedback of Y via the lagged Y values which will have a different overall effect at different locations. That (equilibrium) effect is obtained from premultiplying by(I−ρW )−1β a given change in x at a location, holding x constant for other locations. The code

below, based on Ward & Gleditsch (2008, pp.47) will do that, taking each location in turn. However I would advise against running it here as it takes a long time. For further details see Ward & Gleditsch pp. 44-50.

> ## You are advised not to run this. Based on Ward & Gleditsch pp.47> n <- nrow(combined.map)> I <- matrix(0, nrow=n, ncol=n)> diag(I) <- 1> rho <- model3$rho> weights.matrix <- listw2mat(spknear35gaus)> results <- rep(NA, times=10)> for (i in 1:10) {+ cat("\nCalculating for point",i," of ",n)+ xvector <- rep(0, times=n)+ xvector[i] <- 1+ impact <- solve(I - rho * weights.matrix) %*% xvector * 0.004+ results[i] <- impact[i]+ }


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6.2.3 Choosing between the models using Lagrange Multiplier (LM) Tests

Before fitting the spatial error and lagged y models (above), we could have looked for evidence in support of them using the function lm.LMtests(...). This tests the basic OLS specification against the more general spatial error and lagged y models. Robust tests also are given. There is evidence for both of the spatial models in favour of the simpler OLS model but it is stronger (in purely statistical terms) for the error model.

> lm.LMtests(model1, spknear35gaus, test="all")

Lagrange multiplier diagnostics for spatial dependence

data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map)weights: spknear35gaus

LMerr = 199.8088, df = 1, p-value < 2.2e-16



LMlag = 22.6309, df = 1, p-value = 1.963e-06



RLMerr = 182.3297, df = 1, p-value < 2.2e-16



RLMlag = 5.1518, df = 1, p-value = 0.02322



SARMA = 204.9606, df = 2, p-value < 2.2e-16


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Warning message:In lm.LMtests(model1, spknear35gaus, test = "all") : Spatial weights matrix not row standardized

6.2.4 Including lagged X variables

So far we have accommodated the spatial dependencies in the data by consideration to the error and with consideration to the dependent (y) variable. Attention now turns to the predictor variables. An extension to the lagged y model is to lag all the included x variables too.

> model4 <- lagsarlm(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map, spknear35gaus, type="mixed")> summary(model4)

Call:lagsarlm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map, listw = spknear35gaus, type = "mixed")


Type: mixed Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(>|z|)(Intercept) 14.42342251 2.30649891 6.2534 4.017e-10 ***DCBD -0.70355874 0.22856175 -3.0782 0.0020826 **DELE -0.00261626 0.04397004 -0.0595 0.9525531DRIVER 0.09144563 0.04468328 2.0465 0.0407043 *DPARK -0.06440409 0.08154334 -0.7898 0.4296362POPDEN 0.00313902 0.00130863 2.3987 0.0164527 *JOBDEN -0.00063843 0.00377879 -0.1690 0.8658360Y0405 -0.21727049 0.05452762 -3.9846 6.760e-05 ***Y0607 0.22959603 0.08351013 2.7493 0.0059719 **Y0809 0.90803383 0.11300773 8.0351 8.882e-16 ***lag.(Intercept) -9.63048411 2.68285336 -3.5896 0.0003311 ***lag.DCBD 0.65596211 0.23856166 2.7497 0.0059658 **lag.DELE -0.03552211 0.06711110 -0.5293 0.5965952lag.DRIVER -0.04970303 0.07485862 -0.6640 0.5067167lag.DPARK -0.12152627 0.13681302 -0.8883 0.3743980lag.POPDEN 0.00164266 0.00261945 0.6271 0.5305917lag.JOBDEN 0.01011556 0.00665176 1.5207 0.1283262lag.Y0405 0.23932841 0.25856976 0.9256 0.3546615lag.Y0607 -0.62537354 0.35590512 -1.7571 0.0788947lag.Y0809 -1.36572890 0.53207778 -2.5668 0.0102646 *

Rho: 0.57914, LR test value: 49.317, p-value: 2.1777e-12Asymptotic standard error: 0.066316 z-value: 8.7331, p-value: < 2.22e-16Wald statistic: 76.267, p-value: < 2.22e-16

Log likelihood: -1364.385 for mixed modelML residual variance (sigma squared): 0.66612, (sigma: 0.81616)Number of observations: 1117 Number of parameters estimated: 22 AIC: 2772.8, (AIC for lm: 2820.1)LM test for residual autocorrelation


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test value: 0.43053, p-value: 0.51173

What we find is that the lag of the distance to the CBD (measured on a lag scale) is significant but note that the direction of the relationship is different from that for the original DCBD variable – the sign has reversed. The same has happened to the dummy variable indicating the sale of the land parcel in the years 2008 to 2009. Taking the first case, what it suggests is that the relationship between land price value and the (log of) distance to the CBD is not linear. Adding the square of this variable to the original OLS model improves the model fit:

> model1b <- update(model1, . ~ . + I(DCBD^2)) # Adds the square of the variable> summary(model1b)

Call:lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + I(DCBD^2), data = combined.map)


Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.633565 2.691502 1.722 0.08543 . DCBD 1.376557 0.593375 2.320 0.02053 * DELE -0.066188 0.033051 -2.003 0.04547 * DRIVER 0.065121 0.029583 2.201 0.02792 * DPARK -0.271174 0.047360 -5.726 1.33e-08 ***POPDEN 0.004193 0.001052 3.985 7.20e-05 ***JOBDEN 0.010106 0.003204 3.154 0.00165 ** Y0405 -0.187146 0.057129 -3.276 0.00109 ** Y0607 0.139812 0.087018 1.607 0.10840 Y0809 0.809708 0.118003 6.862 1.13e-11 ***I(DCBD^2) -0.095239 0.034389 -2.769 0.00571 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


> anova(model1, model1b)Analysis of Variance Table

Model 1: LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809Model 2: LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + I(DCBD^2) Res.Df RSS Df Sum of Sq F Pr(>F) 1 1107 834.13 2 1106 828.39 1 5.7448 7.67 0.005708 **---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

However, doing the same in the spatial model results in a situation where the distance to CBD variable no longer is significant under the spatial error model nor under the lagged y model though in the latter case it is more borderline and the square of the variable remains significant:

> model2b <- update(model2, . ~ . + I(DCBD^2))


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> summary(model2b)

Call:errorsarlm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + I(DCBD^2), data = combined.map, listw = spknear35gaus)


Type: error Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(>|z|)(Intercept) 9.2141717 3.7866485 2.4333 0.014961 **DCBD 0.3841081 0.8585578 0.4474 0.654595DELE -0.0366181 0.0420046 -0.8718 0.383337DRIVER 0.0813644 0.0389130 2.0909 0.036534 **DPARK -0.2145776 0.0645093 -3.3263 0.000880 ***POPDEN 0.0039035 0.0012375 3.1543 0.001609 **JOBDEN 0.0040846 0.0036396 1.1223 0.261754Y0405 -0.2191549 0.0548720 -3.9939 6.499e-05 ***Y0607 0.2466920 0.0833052 2.9613 0.003063 **Y0809 0.9292095 0.1130977 8.2160 2.220e-16 ***I(DCBD^2) -0.0500019 0.0509827 -0.9808 0.326710

Lambda: 0.68506, LR test value: 77.873, p-value: < 2.22e-16Asymptotic standard error: 0.05426 z-value: 12.625, p-value: < 2.22e-16Wald statistic: 159.4, p-value: < 2.22e-16

Log likelihood: -1379.069 for error modelML residual variance (sigma squared): 0.67948, (sigma: 0.82431)Number of observations: 1117 Number of parameters estimated: 13 AIC: 2784.1, (AIC for lm: 2860)

> model3b <- update(model3, . ~ . + I(DCBD^2))> summary(model3b)

Call:lagsarlm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + I(DCBD^2), data = combined.map, listw = spknear35gaus)


Type: lag Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(>|z|)(Intercept) 4.0409714 2.6745551 1.5109 0.1308153DCBD 1.0508575 0.5886085 1.7853 0.0742086 DELE -0.0611436 0.0326664 -1.8718 0.0612404DRIVER 0.0373019 0.0304562 1.2248 0.2206611DPARK -0.2445848 0.0472913 -5.1719 2.318e-07 ***POPDEN 0.0040491 0.0010413 3.8884 0.0001009 ***JOBDEN 0.0082186 0.0031842 2.5811 0.0098486 ***Y0405 -0.1890889 0.0564567 -3.3493 0.0008102 ***


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Y0607 0.1796598 0.0860161 2.0887 0.0367368 *Y0809 0.8462846 0.1165821 7.2591 3.897e-13 ***I(DCBD^2) -0.0717869 0.0342566 -2.0956 0.0361209 **

Rho: 0.21633, LR test value: 14.876, p-value: 0.00011479Asymptotic standard error: 0.056612 z-value: 3.8212, p-value: 0.00013282Wald statistic: 14.601, p-value: 0.00013282

Log likelihood: -1410.567 for lag modelML residual variance (sigma squared): 0.73092, (sigma: 0.85494)Number of observations: 1117 Number of parameters estimated: 13 AIC: 2847.1, (AIC for lm: 2860)LM test for residual autocorrelationtest value: 100.58, p-value: < 2.22e-16

6.2.5 Including lagged X variables in the OLS model

We can, if we wish, include specific lagged X variables in the OLS model. The process is to create them then include them in the model. The lag of DCBD and the lag of Y0809 are the most obvious candidates to include (from model 4, above). To create the lagged variables,

> lag.DCBD <- lag.listw(spknear35gaus, combined.map$DCBD)> lag.Y0809 <- lag.listw(spknear35gaus, combined.map$Y0809)

and then add them to the model,

> model1c <- update(model1, .~. + lag.DCBD + lag.Y0809)

The result seems to fit the data better than the original OLS model (AIC score of 2848.3 Vs 2865.7 – remember, the lower the better) but actually the lag of DCBD appears not to be significant in this model whilst significant spatial autocorrelation appears to remain:

> summary(model1c)

Call:lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + lag.DCBD + lag.Y0809, data = combined.map)


Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 11.742616 0.572254 20.520 < 2e-16 ***DCBD -0.226122 0.075021 -3.014 0.002636 ** DELE -0.068113 0.032621 -2.088 0.037026 * DRIVER 0.105230 0.030779 3.419 0.000652 ***DPARK -0.282638 0.046536 -6.074 1.72e-09 ***POPDEN 0.004162 0.001042 3.995 6.89e-05 ***JOBDEN 0.008902 0.003012 2.955 0.003188 ** Y0405 -0.198322 0.056897 -3.486 0.000510 ***Y0607 0.161288 0.087123 1.851 0.064399 . Y0809 0.852457 0.118536 7.192 1.18e-12 ***lag.DCBD -0.056987 0.054031 -1.055 0.291789 lag.Y0809 -2.079764 0.476501 -4.365 1.39e-05 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


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> AIC(model1c)[1] 2848.332> lm.morantest(model1c, spknear35gaus)


data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + lag.DCBD + lag.Y0809,data = combined.map)weights: spknear35gaus


6.3 Discussion

Although the revised spatial models are presenting different results in regard to the distance to CBD variable and its effect on land price, and these differ again from the revised OLS models, there may be an argument that, in a general sense, the models are indicating the same thing. Distance to CBD is a geographic measure. It tries to explain something about land values by their distance from the CBD. The spatial error and lagged y models introduce geographical considerations in other ways (and in addition to the distance to CBD variable). Looking at the results of the Lagrange Multiplier tests (but we could also use the AIC or Log Likelihood scores) the spatial error model remains the 'preferred' model. And that, again, is another clue: the complexity of the spatial patterning of the land parcel prices has yet to be fully captured by our model; its causes remain largely unexplained.

> lm.LMtests(model1b, spknear35gaus, test=c("LMerr", "LMlag"))


data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + I(DCBD^2), data =combined.map)weights: spknear35gaus

LMerr = 169.6169, df = 1, p-value < 2.2e-16


data: model: lm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + I(DCBD^2), data =combined.map)weights: spknear35gaus

LMlag = 18.0596, df = 1, p-value = 2.141e-05


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Warning message:In lm.LMtests(model1b, spknear35gaus, test = c("LMerr", "LMlag")) : Spatial weights matrix not row standardized

6.4 Geographically Weighted Regression (GWR)

An alternative way of attempting to explain the spatial variation in the land value prices is to allow the effect sizes of the predictor variables to themselves vary over space. Geographically Weighted Regression (GWR) offers this where the estimate of βx at point location i is not simply the global estimate for all points in the study region but a local estimate based on surrounding points weighted by the inverse of their distance away.

6.4.1 Fitting the GWR model

The stages of fitting a Geographically Weighted Regression model are first to load the GWR library, calculate a distance matrix containing the distances between points, calibrate the bandwidth for the local model fitting, fit the model, then look for evidence of spatial variations in the estimates.

First the library. There are two we could use. The first is library(spgwr). However, we will use the more recently developed library(GWmodel) which contains a suite of tools for geographically weighted types of analysis.

> library(GWmodel)

Next the distance matrix:

> distances <- gw.dist(dp.locat=coordinates(combined.map))

Now the bandwidth, here using a nearest neighbours metric and a Gaussian decay for the inverse distance weighting (for other options, see ?bw.gwr). The bandwidth is found using a cross-validation optimisation procedure.

> bw <- bw.gwr(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map, adaptive=T, dMat=distances)> bw[1] 30

Here the bandwidth decreases to zero at the 30th neighbour, encouragingly similar to the 35 th neighbour value we have used for our spatial weights throughout this session. Next the model is fitted:

>gwr.model <- gwr.basic(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map, adaptive=T, dMat=distances, bw=bw)

Looking at the results we can see that the model has a better fit to the data than any other fitted thus far. The summary of the GWR estimates is of how the (local) beta estimates vary across the study region. For example, from the inter-quartile range, the effect of distance to CBD on land value prices is typically found to be from -0.682 to -0.030.

> gwr.model [..] *********************************************************************** * Results of Geographically Weighted Regression * ***********************************************************************

*********************Model calibration information********************* Kernel function: gaussian Adaptive bandwidth: 30 (number of nearest neighbours)


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Regression points: the same locations as observations are used. Distance metric: A distance matrix is specified for this model calibration.

****************Summary of GWR coefficient estimates:****************** Min. 1st Qu. Median 3rd Qu. Max. X.Intercept. -2.992e+01 8.776e+00 1.163e+01 1.432e+01 36.7400 DCBD -2.452e+00 -6.818e-01 -3.166e-01 -3.005e-02 3.6800 DELE -9.924e-01 -1.839e-01 -2.105e-02 1.184e-01 0.5259 DRIVER -9.369e-01 -5.216e-02 1.076e-01 2.344e-01 0.7053 DPARK -1.161e+00 -2.895e-01 -1.756e-01 -3.797e-02 0.7852 POPDEN -2.440e-02 -7.219e-04 3.197e-03 1.001e-02 0.0293 JOBDEN -1.965e-02 -2.778e-03 5.276e-03 1.744e-02 0.1056 Y0405 -8.825e-01 -3.227e-01 -2.218e-01 -1.028e-01 0.4346 Y0607 -9.749e-01 -2.447e-01 2.268e-01 7.012e-01 1.3980 Y0809 -1.016e+00 -3.069e-01 8.426e-01 1.869e+00 2.7530 ************************Diagnostic information************************* Number of data points: 1117 Effective number of parameters (2trace(S) - trace(S'S)): 178.6996 Effective degrees of freedom (n-2trace(S) + trace(S'S)): 938.3004 AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 2555.443 AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 2381.741 Residual sum of squares: 489.1524 R-square value: 0.58657 Adjusted R-square value: 0.5077481

*********************************************************************** Program stops at: 2013-10-23 15:03:25

The estimates themselves for each of the land parcel points are found within the gwr model's spatial data frame:

> names(gwr.model$SDF)

The parts of this data frame with the original variable names are the local beta estimates, those with the suffice _SE are the corresponding standard errors, together giving the t-values, marked _TV.

6.4.2 Mapping the estimates

If we map the local beta estimates for the distance to CBD variable, the spatially variable effect becomes clear,

> x <- gwr.model$SDF$DCBD> par(mfrow=c(1,2)) # This will allow for two maps to be drawn side-by-side> quickmap(x)

However, we might wish to ignore those that are locally insignificant at a 95% confidence.

> quickmap(x, subset=abs(gwr.model$SDF$DCBD_TV) > 1.96)

Doing so, what we appear to find is that there are clusters of land parcels where distance from the CBD has a greater effect on their value than is true for surrounding locations.


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Figure 6.2. The local beta estimates for the distance to CBD variable estimatedusing Geographically Weighted Regression. In the right-side plot the 'insignificant'

estimates are omitted.

6.4.3 Testing for significance of the GWR parameter variability

(Note: at the time of the writing there is an error in the function montecarlo.gwr(...) included in the GWmodel package version 1.2-1 that will be changed in future updates. Load the file montecarlo.R using the source(file.choose()) function to import a corrected version.)

The function montecarlo.gwr(...) uses a randomisation approach to undertake significance testing for the variability of the estimated local beta estimates (the regression parameters). The default number of simulations is 99 (which, with the actual estimates, gives 100 sets of values in total). That is quite a low number but can be used for illustrative purposes. In practice it would be better to raise it to 999, 9999 or even more.

> montecarlo.gwr(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data=combined.map, adaptive=T, dMat=distances, bw=bw)

Tests based on the Monte Carlo significance test

p-value(Intercept) 0.00DCBD 0.00DELE 0.00DRIVER 0.00DPARK 0.00POPDEN 0.00JOBDEN 0.01Y0405 0.03Y0607 0.00Y0809 0.00

All of the variables appear to show significant spatial variation.


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6.5 A Simple Multilevel Model

A final modelling approach we might consider here is a multilevel one. At its simplest, a multilevel approach allows the unexplained residual variance from a linear regression model to be partitioned at different levels – here the individual land price values (at the lower level) and the census districts (at the higher – more aggregate – level). We can then assess what amount of the unexplained variance in the land prices is due to some sort of higher-level 'contextual' effect.

6.5.1 Null models

The simplest multilevel model is one that does nothing more than estimate the mean of the land price values, uses that as the sole predictor of the land price values, and then partitions the errors in the way described above. In other words, it is a regression model for which there is only an intercept term (no slope) and the residual variances are estimated at the lower and higher geographical levels.

There are a number of packages to fit multilevel models in R. We shall use...

> library(lme4)

To fit the model with an intercept-only using standard OLS estimation and no partitioning of the residual variance we would type,

> nullLMmodel <- lm(LNPRICE ~ 1, data=combined.map)

Obtaining the log likelihood value as,

> logLik(nullLMmodel)'log Lik.' -1617.09 (df=2)

To fit the corresponding model using a multilevel approach where the residual variance is assumed to be random at both the land parcel and district scales the notation is similar but includes the parentheses identifying the scales. 1 indicates the lower level whilst the variable SP_ID arises from the overlay of the point and polygonal data undertaken in Section 4.5, p. 47, 'Spatially joining data' and gives a unique ID for each district.

> nullMLmodel <- lmer(LNPRICE ~ (1 | SP_ID), data=combined.map@data)> summary(nullMLmodel)Linear mixed model fit by REML ['lmerMod']Formula: LNPRICE ~ (1 | SP_ID) Data: combined.map@data

REML criterion at convergence: 2983.876

Random effects: Groups Name Variance Std.Dev. SP_ID (Intercept) 0.3537 0.5947 Residual 0.7222 0.8498 Number of obs: 1117, groups: SP_ID, 111

Fixed effects: Estimate Std. Error t value(Intercept) 7.53186 0.06523 115.5

The key thing to note here is the proportion of the residual variance that is at the district level,

> 0.3537 / (0.3537 + 0.7222)

[1] 0.328748

- almost one third. This is a sizeable amount and is suggestive of the spatial patterning of the land


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parcel values. The log likelihood of this model is greater than for the OLS model,

> logLik(nullMLmodel)'log Lik.' -1491.938 (df=3)

6.5.2 The likelihood ratio test

The likelihood ratio test statistic is two times the difference in the log likelihood values for two models, here

> 2 * as.numeric((logLik(nullMLmodel) - logLik(nullLMmodel)))[1] 250.3039

Assessing against a chi-squared distribution with 1 degree of freedom (the difference in the degrees of freedom for the two models, arising from the estimation of the additional, higher-level error variance) we find 'the probability the result (i.e. the improved likelihood value) has arisen by chance' is essentially zero:

> 1 - pchisq(250, 1)[1] 0

6.5.3 A random intercepts model

Having established that there is an appreciable amount of variation in the land parcel prices at the district scale, our next stage will be to refit our predictive regression mode but again with a multilevel framework. Recall, for example, (OLS) model1,

> model1$calllm(formula = LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809, data = combined.map)

Its multilevel equivalent is,

> MLmodel <- lmer(LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + (1 | SP_ID), data=combined.map@data)> summary(MLmodel)Linear mixed model fit by REML ['lmerMod']Formula: LNPRICE ~ DCBD + DELE + DRIVER + DPARK + POPDEN + JOBDEN + Y0405 + Y0607 + Y0809 + (1 | SP_ID) Data: combined.map@data

REML criterion at convergence: 2817.831

Random effects: Groups Name Variance Std.Dev. SP_ID (Intercept) 0.1481 0.3848 Residual 0.6322 0.7951 Number of obs: 1117, groups: SP_ID, 111[…]

Even with the predictor variables now included there remains an appreciable amount of variation between districts,

> 0.1481 / (0.1481 + 0.6322)[1] 0.1897988

Consequently there is strong support in favour of the multilevel model over the OLS one,

> 2 * as.numeric((logLik(MLmodel) - logLik(model1)))[1] 25.90044> 1 - pchisq(25.9, 1)[1] 3.595691e-07


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6.5.4 Mapping the district-level residuals

There is much more we could undertake in regard to the multilevel model, including allowing the effect of each predictor variable to vary from one district to another (a random intercepts and slopes model). Here, however, we shall confide ourselves to mapping the district level residuals – to identify those districts where the land parcel values are higher or lower than average.

The process of doing so begins by obtaining the residuals at the higher level of the hierarchy, using the function ranef(...) (short for random effects)

> district.resids <- ranef(MLmodel)

The output from this function is a list, in this case of length 1 (this being the number of levels above the land parcel level for which that variance has been estimated – i.e. the district level).

> typeof(district.resids)[1] "list"> length(district.resids)[1] 1

Inspecting its contents we find that it is a data frame containing the IDs of the districts and information telling us whether the district-level effect is one of raising or decreasing the land parcel prices:

> head(district.resids[[1]], n=3) (Intercept)10 0.22619895100 0.09476389101 -0.22773190> summary(district.resids[[1]][,1]) Min. 1st Qu. Median Mean 3rd Qu. Max. -0.78800 -0.21480 0.03126 0.00000 0.21810 0.61750

Note that not every one of the census districts in Beijing will be included in this output. That is because not every district contains a land parcel that was sold in the period of the data. We therefore have to match those districts for which we do have data back to the original map of all districts, which we can then use to map the residuals. First the matching,

> ids1 <- as.numeric(rownames(district.resids[[1]])) > ids2 <- as.numeric(as.numeric(districts$SP_ID))> matched <- match(ids1, ids2)> districts2 <- districts[matched,]

Second, the mapping,

districts2 <- districts[matched,]x <- district.resids[[1]][,1]group <- cut(x, quantile(x, probs=seq(0,1,0.2)), include.lowest=T, labels=F)palette <- brewer.pal(5, "RdYlBu")plot(districts)plot(districts2, col=palette[group], add=T)round(quantile(x, probs=seq(0,1,0.2)),3)legend("bottomright", legend=c("-0.788 to < -0.261","-0.261 to < -0.069",+ "-0.069 to < 0.105","0.105 to < 0.250",+ "0.250 to < 0.618"), pch=21, pt.bg=palette, cex=0.8)

Looking at the map, there appear to be clusters of districts with higher than expected land parcel prices, some contiguous or close to districts with lower than expected prices. Not all these residual values are necessarily significantly different from zero. Nevertheless, what we seem to have evidence for – again – is the complexity of the geographical patterning.


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Figure 6.3. The district-level residual estimates (white = no data)

6.6 Consolidation and Conclusion

What this session has demonstrated is a number of spatial models, all within a regression framework, that can be applied to model land parcel prices in Beijing (but remember that the data are simulated). Purely in terms of Akaike's Information Criterion (the AIC scores) the GWR and spatial error models are the best fits to the data although in many respects they (and the multilevel model) tell us only that geographical complexities exist, not actually about the processes that cause them. Even so, precisely because they do exist, it should warn us against using standard regression techniques that assume the residual errors are independently and identically distributed.

6.7 Getting Help

There is an excellent workbook on spatial regression analysis in R by Luc Anselin. It is available at http://openloc.eu/cms/storage/openloc/workshops/UNITN/20110324-26/Basile/Anselin2007.pdf

The definitive text for Geographically Weighted Regression is Fotheringham et al. (2002): Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. For more information about the GWmodel package, see http://arxiv.org/pdf/1306.0413.pdf

An introduction to multilevel modelling in R is offered by Camille Szmaragd and George Leckie at the Centre for Multilevel Modelling, University of Bristol: http://www.bristol.ac.uk/cmm/learning/module-samples/5-r-sample.pdf


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http://www.bristol.ac.uk/cmm/learning/module-samples/5-r-sample.pdf

http://arxiv.org/pdf/1306.0413.pdf

http://openloc.eu/cms/storage/openloc/workshops/UNITN/20110324-26/Basile/Anselin2007.pdf

More to follow. Check at www.social-statistics.org for updates.

