DERIVATION OF BIOPHYSICAL VARIABLES FROM FINE RE SOLUTION IMAGERY

FOR CO-PROCESSING WITH SOCIO-ECONOMIC DATA

IN AN URBAN AREA

Gabor Zsigoïics

A thesis submitted in conformity with the requirements for the degree of Master of Science,

Graduate Department of Geography, University of Toronto

Q Copyright by Gabor Zsigovics 2000

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Abstract

Derivation of biophysical variables from fine resolution imagery for CO-processing

with socio-econornic data in an urban area.

Master of Science, 2000,

Gabor Zsigovics,

Graduate Department of Geography, University of Toronto.

The integration of the social and physical sciences is a prevalent issue,

especially in urban studies. Recent advances in remote sensing and G.I.S.

provide the technological background to derive spatially and temporally detailed

information about the biophysical environment (such as land use ana land

cover). There is great potential in using such detailed and up-to-date information

in modelliny phenomena related to 'quality of life'.

This study demonstrates the feasibility of deriving land cover

characteristics from remotely sensed data and explores their relationships with

some socio-economic characteristics. Vegetation amount and pattern measures

are derived from both satellite images and aerial photography (at various

resolutions between lm and 100m). Relationships between these biophysical

variables and socio-economic variables are then investigated using various

qualitative and quantitative tools. Finally, prediction of some of the socio-

economic characteristics are explored applying statistical model selection

criteria.

Acknowledgements

I would like to express my sincere gratitude to Dr. Ferenc Csillag who

"coached" me so well during the last two years. Although I entered the program

with very little computational experience and even less statistical background,

with perseverance and Dr. Csillag's unwavering belief that I would pull through

the technical roadblocks, I have mastered a small chunk of a very exciting part of

the current geographical expertise.

The GUESS Research groups constant support was much appreciated.

Many thanks to Scott, Hannah, Zuzu, Marcy, Rebecca, Andy, Richard, Peter,

and Ken who al1 helped me a great deal.

It is impossible not to thank those who have helped me corne this far:

llona and Frank, my parents. They gave me that last little push to finally

complete the writing of the thesis. A big thank-you to my sister, Agnes, who

helped me refresh some of those trigonometric equations, and even looked over

some of my bibliography.

And to Monica, rny girlfriend, I owe the world. Let it be visits to PGB or

coffee breaks. her radiance made my "bad cornputer days" seem quite

acceptable.

This research was in part supported by the generous contribution of the

National Sciences and Engineering Research Council of Canada ("NSERC)

through an Industrial Scholarship.

iii

Table of Contents

Page #

Abstract ........................................................................................ ii

Acknowledgements.. ...................................................................... iii

Table of Contents.. ........................................................................ .iv

List of Figures.. .............................................................................. vii

List of Tables.. .............................................................................. .x

Abbreviations ................................................................................ .xi

I Introduction

1.1 Importance of the integration of the social and physical

sciences.. ...................................................................... -1

1 -2 Rationale.. ..................................................................... -3 1 -3 Objectives.. ................................................................... ..4

................................................................. 1.4 Site description 5

1 -4.1 Physical environment.. .......................................... -5 1 -4.2 Social environment.. ............................................. .6 1 -4.3 Biological environment.. ........................................ ..6

II Derivation of vegetation cover and pattern

2.1 Remote sensing of urban environments. .............................. .8

2.2 Sources of remotely sensed data.. ..................................... .ll 2.2.1 Aerial photographs.. .............................................. 1 1

2.2.2 Satellite imagery .................................................. .12

Page #

2.3 Resampling .................................................................... -13

2.4 Texture layers ................................................................. -13

2.4.1 Optimal kernel size for calculating texture ................... 15

2.5 Vegetation indices ............................................................ -17

2.6 Image classification .......................................................... -18

2.6.1 Selection of training sites ....................................... -19

2.6.2 Training signature statistics and analysis .................... 20

2.6.3 Supervised classification ......................................... 21

2.6.4 ISODATA clustering .............................................. 23

2.6.5 Hybrid classification ............................................... 23

2.6.6 Classified images ................................................. -24

2.7 Accuracy assessment ....................................................... -29

2.7.1 Producer's and user's accuracy ................................ 31

2.8 Satellite image (LANDSAT Thematic Mapper) ......................... 33

......................... 2.9 Contagion: A measure of spatial arrangement 35

III Exploriny the relationship between biophysical and socio-economic variables

3.1 Sources of socio-economic data ........................................... 38

3.1 . 1 Census Data ......................................................... 38

3.1.2 Social Deprivation Index ......................................... -39 3.2 Exploratory Data Analysis (EDA) .......................................... -40

3.2.1 QualitativeNisual assessrnent of variables ................... 40

3.2.2 Univariate statistics ................................................ -47

3.2.2.1 Descriptives ............................................... 47

...................... 3.2.2.2 Spatial dependence: Moran's I -49

3.2.2.3 Spatial dependence: Semi-variograms ............ 50

3.2.3 Multivariate statistics ............................................... 54

3.2.3.1 Scatterplots ............................................... 54

Page #

3.2.3.2 Decile plots .............................................. 58

3.2.3.3 Pearson's correlation tables ........................ 61

IV Co-processing of biophysical and socio-economic data using simple statistical

rnodels

Spatial linear regression model (SLM) .................................. 63

Regression trees ............................................................ -64

Residual diagnostics ....................................................... -65

Which model? - Model selection strategies ........................... 74

4.4.1 Akaike's Information Criterion (AIC) ......................... 74

Finding the appropriate resolution ....................................... 76

............................................. Census Tract Level -77

Non-spatial scaling of socio-economic variables to

a regular grid ...................................................... 83

References ................................................................................. -91

Appendix 1 ................................................................................... 98

List of Figures

Page #

Figur .e 1 : The study area as seen from the air. This area is the subject of a large research initiative nained the Sout h-east Toronto (SETO) Health Mapping Project. .............................................................. ..7

Figure 2: Flowchart of steps to derive vegetation cover from aerial photographs and satellite imagery of different resolutions ....................................... .........9

Figure 3: False-colour composite of the study area taken by the LANDSAT TM satellite in July of 1992.. ........................................................... 12

Figure 4: Accuracy of supervised classification at 1 m resolution as a function of kernel size used to derive the texture layer. ................................... 16

Figure 5: Accuracy of supervised classification at 5m resolution as a function of kernel size used to derive the texture layer.. .................................. -1 6

Figure 6: Assessrnent of the effect of changing the threshold for the vegetation classes in supervised classification. .................................. ..22

Figure 7(a-f.): Vegetation cover maps at 2m, Sm, 10m, 25m, Som, and 1 OOm resolutions ............................................................................... 25-7

Figure 8: Percentage vegetation cover as a function of resolution calculated for the entire SETO area and three smaller subsets (Subset A - Rosedale; Subset B -East York; Subset C - Downtown).. ...................... 28

Figure 9: Location of subsets used for accuracy assessment.. ........................ -29

Figure 10: Vegetation cover derived from the satellite image ............................ 34

Figure 1 1 : Contagion as a function of vegetation cover computed for the East York subset on a square grid using classified images at different resolutions.. ................................................................................ -36

Figure 12: Contagion as a function of vegetation cover computed for the three subsets (Subset A- Rosedale, Subset B-East York, Subset C- Downtown). ............ .. ................................................................. -37

Figure 13 (a-e): Spatial distribution of selected census variables.. ................... 42-4

Figure 14: Histograms of selected census and biophysical variables.. .............. 45-6

Figure 1 5 (a-b): Variograms for census and biophysical variables. ................ -52-3

Figure 16: Scatterplot matrix of selected socio-economic and biophysical variables.. .................................................................................... 56

vii

Paçe #

Figure

Figure

Figure

17: 3-0 bar graph displaying the relationship between SINGLE. ..................................................................... POPDENS. and VEG 56

18: Scatterplot of vegetation cover derived from 2m resolution aerial photograph and median family income ............................................... 57

19 (a-f): Decile plots of census variables as a function of vegetation cover ......................................................................................... -58

Figure 20: Spatial interaction of neighbouring units taken into consideration in a SLM ........................................................................................... 63

Figure 21 (a-d): Reduction in deviance plot. in addition to three pruned tree models ................................................................................. 66

Figure 22: Residual plots for the SLM predicting LNSINGLE from LNTM25M at the EA level ................... .... ......... - 6 8

Figure 23: Residual plots for the RT model predicting LNSINGLE from LNTM25M at the EA level ............................................................... 69

............ Figure 24 (a-b): Predicted maps of LNSINGLE for both the RT and SLM 70

Figure 25: Residual plots for the SLM predicting LNPOPDENS from LNTM25M at the EA level ............. ...... ..................................................... 71

Figure 26: Residual plots for the RT model predicting LNPOPDENS from LNTM25M at the EA level ............................................................... 72

Figure 27 (a-b): Predicted maps of LNPOPDENS for both the RT and SLM ....... 73

Figure 28: AIC as a function of the number of terminal nodes .......................... 75

Figure 29: Scatterplot matrix for selected variables at the CT level ................... 78

Figure 30: Residual plots for the SLM predicting LNSINGLE from LNVEG2M at the CT level ............................................................................. 79

Figure 31 : Residual plots for the RT model predicting LNSINGLE from LNVEG2M at the CT level ....................... ........ ........................... 80

Figure 32: Predicted maps of LNSINGLE using the RT and SLM models ............................................... at the CT level .......................... .. 81

Figure 33: Observed map of LNSINGLE at the CT level ................................. 82

Figure 34: SINGLE scaled to a lOOm grid using a tree regression determined at the EA level ............................................................................. 84

viii

Page #

Figure 35: Observed SINGLE for the area corresponding to that of Fig . 34 ...... -85

Figure 36: The spatial distribution of high vegetation amount area in sclcio-economic ................................................... feature space located in figure 17 89

List of fables Page #

'able 1 : Digital number (DN) differences of means. between the two rows of images calculated over a 50 pixal wide and 500 pixel long area located

.................................. on the adjacent sides of the two rows of images 11

Table 2: Classification scheme used in supervised classification of the fine resolution imagery ................................................................. -19

Table 3: Transformed Divergence separability values for classes of the ................................................................ supervised classification -21

................ Table 4: Comparison of Khat (Kappa) accuracy with overall accuracy 31

Table 5: Accuracy assessment according to user's and producer's accuracies ..... 32

Table 6: Jarman's (1 983) weights for a social deprivation index ........................ -40

Table 7: Descriptive summaries for all of the variables selected for the study ....... 48

Table 8: Moran's I coefficients for selected variables ...................................... -50

Table 9: Pearson's correlation coefficients for selected biophysical and socio-economic variables .......................... .. .................... -61

Table 10: Pearson's correlation coefficients for a smaller subset ........................ 60

Table 1 1 : .4 1C values for various model runs ................................................. 75

Table 1 2: Pearson's correlation coefficients for selected socio-economic and biophysical variables at the CT level .......................................... -78

Abbreviations

Abbreviation iûescri~tion of variable - - - - - 1

POPDENS l~ooulation densitv - - - - 1

l MM1 POP ,

UNIDEG INCOME SINGLE INDEX

I Percent vegetation cover derived from the satellite image using clusterina

Proportion of population who are immigrants Proportion of population with a university diploma Median family income Proportion of population living in single-detached homes Social de~rivation index . . . . .

C . -

NDVl IPercent vegetation cover derived from the satellite image using NOVI LN(NAME) l ~ h e LN preceding variable names denotes that the natural logarithrn

VEG (2M- 1 OOM)

1 Itransformation is applied.

Percent vegetation cover at different resolutions from aerial ohotoara~hs

I

Introduction

1.1 Importance of the integration of the social and physical sciences

Recently, interest has grown substantially in the study of urban

ecosystems. This is not surprising given the percentage of people living in large

urban centres. Scientists from both the physical and social sciences are

recognising that in order to gain a more complete understanding of these highly

heterogeneous systems, it is necessary to integrate their respective fields.

Their prominent research question is: "How do the spatial structure of economic,

ecological and physical patterns in an urban area relate to one another, and

how do they change over time?" (Pickett et al., 1997). Physical scientists need

"to learn from social scientists what the institutional, organisational, and

interactive features of humans are that should be added to ecosystem models to

make them more complete and useful" (Pickett et al., 1997) while social

scientists require "that the principles of bidogy to be taken into consideration in

an interpretation of social phenomena; that human society is not entirely an

artificial creation.." (Machalis et al., 1996, citing Sorokin 1928, 207). The major

current research agendas reflect this need for integration. For example, the

U.S. National Science Foundation has set up two of its new long term ecological

research (LTER) sites in urtian areas (Baltimore and Phoenix), while the

Canadian National Centres of Excellence (NCE) program is strictly funding

interdisciplinary research.

A promising field, aiding the integration of the physical with the social, is

remote sensing that now provides unprecedented detail of the physical

environment. Some of the more specific research agendas include "pixelisation

of the social" and vice versa (Geoghegan et al., 1998). For example, Wood and

Skole (1 998) linked satellite, census, and survey data to study deforestation in

the Amazon. The use of census data has also found use in image classification,

whereby socio-economic data are used in the selection of the training sites

(Mesev, 1998).

Biophysical variables derived from remote sensing images have been

found to be useful in estimating census data (Lo, 1995). Grove (1996), studying

the relationship between patterns of social stratification and vegetation structure

of an urbanhural watershed, found that selected socio-economic features at the

census block level explained a large proportion of the variance in vegetation

cover and percent imperviousness. A cluster analysis by Fernandes et al. (In

Press) has reinforced the findings of Grove ( 1 996). Although Grove's research

indicated a positive association between socio-economic status and vegetation

cover, Ryznar (1998) found a negative relationship between socio-economic

status and the percentage change in vegetation over a longer time period in the

city of Detroit. This negative relationship was attributed to the fact that

abandoned urban areas experienced a rapid proliferation of vegetation, such as

weeds.

The integration will lead not only to more varied and perhaps more

appropriate methodology, but it will allow the full description of the highly

heterogeneous urban environmen: with more appropriate concepts. Concepts

such as the ecosystem concept have been redefined to "human ecosystemsn fcr

applications to human populations (Machilis et al., 1997). The ecological

gradient paradigm has been a very useful analytical tool to provide cross-

sectional analysis of urban areas (McDonell and Pickett, 1990; Foresman et ai.

1997). Increasingly. investigators are using biophysical data to measure the

quaiity of life in neighbourhoods (Lo and Faber, 1997). From these examples it

is evident that interdisciplinary studies provide insight into the patterns and

processes of urban ecosystems.

Although previous research has shown significant relationships between

biophysical and socio-economic data, the relationships are not at al1 trivial.

Fu rther, exploratory analyses are needed to assess the relationships between

biophysical and socio-economic data at various scales. This study

demonstrates the feasibility of deriving land cover characteristics from remotely

sensed data and explores their relationships with some socio-economic

charactaristics derived from the census. Vegetation amount and pattern

measures are derived from both satellite images and aerial photography (at

various resolutions between 1 m and 1 OOm). Relationships between these

biophysical variables and census variables are then investigated using various

qualitative and quantitative tools. Finally, modelling of some of the socio-

economic characteristics are explored applying statistical mode1 selection

criteria.

1.2 Rationale

One of the major limitations of studies employing socio-economic data

(collected through censuses) is that, due to confidentiality issues, the finest unit

at which data are available is at the enurneration area (EA) level. This means

that a single value of a certain variable, such as median family income, is

associated to an area that could include a very diverse population. The

variance is effactively reduced to zero within a unit, potentially leading to issues

of ecological fallacy when relationships determined at one scale are associated

with associations at another scale (Openshaw, 1984). Furthermore, the census

units have fixed boundaries that do not necessarily reflect the patterns of human

or physical processes.

The second major limitation of census data is the collection frequency. In

Canada, a nation-wide census is conducted every five years. However, many

urban processes (e.g. urban sprawl) operate under shorter time scales. Thus,

the need for of up-to-date socio-economic data at various scales is becoming

more apparent (Lo and Faber, 1997).

On the other hand, remotely sensed imagery is now becoming not only

more frequently obtained (3 days in the case of the new IKONOS satellite

(Mulroney, 2000)) but also at a spatial resolution where one can almost discern

individual humans. The imagery is relatively inexpensive (in comparison to

ground suweys), therefore, it has the potential to become the ideal source of

data for a wide variety of applications.

The largest limitation of the imagery is that the variables studied by social

scientists are not readily derived from such images. The relationships are

cornplex, however, variables such as land use, urban temperature, vegetation

indices. and building structure derived from remotely sensed images have

already been demonstrated to have some association with socio-economic

variables such as median income (Lo and Faber, 1997; Grove, 1996). This

study will further explore the relationship between biophysical and socio-

economic variables in a smalfer part of a large metropolitan area.

1.3 Objectives

The purpose of this study is to:

1. Derive vegetation cover and pattern from satellite imagery and aerial

photographs of various resolution.

II. Explore the relationships between biophysical and socio-economic

variables.

I 1 1 . lmplement simple statistical models using both biophysical and socio-

econornic data.

1 -4 Site description

The study area is situated just east of the downtown core of the City of

Toronto, Canada (Fig. 1). Covering an area of about 16 km2, it has a

heterogeneous mix of both social and physical environments. This area is in

part chosen because of this great heterogeneity and in part due to the fact that

a large health project (Southeast Toronto (SETO) Health Mapping Project) is

focusing on the area, which might benefit from the results of this study.

1.4.1 Physical environment

The physical environment is characterised by a gently rising lake bed (of

ancient Lake Iroquois), divided by the Don River Valley which empties into Lake

Ontario. The relatively flat lakebed rises from the present Lake Ontario

shoreline at 75m a.s.1. to the old Iroquois shoreline at 150m a.s.l..at the northern

edge of the study area (Westgate et a/., :999). Wide remnant valleys incise this

northern area (Rosedale). The built physical environment adds another layer to

the relief. The area adjacent to Yonge Street is mostly covered with high rise

offices and apartmentslcondominiums. Moving east, Cabbagetown and

Riverdale, on the east side of the Don River, are characteristic of tree covered

residential areas. Regent Park to the south and Rosedale to the north create a

polarised axis of wooded estates and apartment blocks. The main

thoroughfares for traffic are the GardinerIDon Valley Park expressways, Yonge,

BloodDanforth, Mt. Pieasant /Jarvis, Queen, College streets.

The macroclimate of Toronto can be described as continental, with cold

winters (-3OC) and warm, humid surnmers (26OC) with winds dominated by the

westerlies (Munn et al., 1999). However, the macroclimate is significantly

altered by the urban structure. The urban heat island is well established,

especially over the western part of the study area (Munn et al., 1999). Heat

islands are not only associated with higher temperatures but also give rise to air

circulation patterns that can trap pollutants (Oke, 1987).

1.4.2 Social environment

The study area comprises a great mixture of communities, some of the

wealthiest (Rosedale) and poorest (Regent Park) neighbourhoods of Toronto

and of Canada as well. According to the 1991 Census of Canada. there were

122,830 people living in the area. The arbitrarily determined enurneration areas

(EA) divide this total population into 335 EAs, each consisting of on average

400 persons. In turn these 335 EAs are grouped into 28 nested census tracts

(CT), with an average population of 4200 persons.

The socio-economic environment conforms in many aspects to the

physical environment. The wealthiest neighbourhoods are those of Rosedale,

Riverdale, and Cabbagetown with tree covered single story houses (average

income $68,000). On the other hand, the poorest neighbourhoods are Regent

Park and Don Mount Court, which are public housing initiatives (average income

<$20,000). Many of the tenants in these public housing units are recent

immigrants to Canada.

1.4.3 Biological environment

Most of the original tree cover has been removed by human activities,

except for large tracts of woodlands in the northern parts of the study area

(Rosedale). Deciduous trees such as sugar maple, beech, and basswood and

conifers such as white pine and white cedar dominate these forests

(Eckenwalder, 1999). The other well-established neighbourhoods such as

Riverdale also have considerable tree cover with many gardens of introduced

species from various parts of the world. However, these vegetation

cornmunities created by people are not as structurally complex and do not fomi

a closed canopy. The marshlands still occupying the shoreline in the 19"

century around Ashbridge's Bay have been converted to industrial land-use,

wiping out a whole ecosystem (Zimmerman, 1999).

Figure 1: The study area (marked with the thick red line) as seen from the air. This area is the subject of a large research initiative named the South-east Toronto (SETO) Health Mapping Project.

II

Derivation of vegetation covei and pattern

The aim of this chapter is to present the methods of deriving vegetation

cover and vegetation pattern from satellite imagery and aerial photography. A

quick overview of the methods is included as a flowchart in Figure 2. First, the

usefulness of remote sensing will be discussed in the urban context. Sections

will follow on pre-processing, including resampling and texture layer calculations.

The classification of imagery, using supervised and unsupervised methods, will

be presented next, including accuracy assessment of the binary vegetation

maps. Finally, the importance and computation of a measure of vegetation

pattern will be discussed.

2.1 Remote sensing of urban environments

With the introduction of commercially available remotely sensed data,

manÿ areas of science have embraced this new data source with great

enthusiasm. One of the great advantages of rernotely sensed data is that it

aliows the researcher to study a rnuch larger region than traditional field studies.

Fields such as landscape ecology, whose focus is the larger ecological unit

covering many hectares, have just began to mature with the spread of this new

data source. Remotely sensed images not only cover a larger area, but they

also offer greater detail (1 m pixel resolution in the case of the new IKONOS

satellite). For most regions of the Earth, one is able to choose from imagery

collected by sensors of variable resolutions (1 m to 1 km). This scalability allows

for issues of scale to be addressed explicitly. The fine spatial resolution is often

matched by high temporal resolution (3 days in the case of the IKONOS

satellite). Although remotely sensed images offer a much wider geographical

scale, fine spatial and temporal resolutions, in most cases they do not directly

Figure 2: Flowchart of steps to derive vegetation cover frorn aerial photographs and satellite imagery of different resolutions. Of importance is the loop from accuracy assessrnent back to the classification process. The final vegetation cover maps are derived after many iterations of the classification process in which training signatures are continually redef ined,

provide the researcher with the variables that they are studying. These variables

of interest need to be derived from the images that are often represented by

digital numbers (DN), corresponding to reflectance. For example, land use/land

cover studies classify images using a classification system of interest (Gong and

Howarth, 1992); landscape ecologists derive leaf area index (LAI) or biomass

information through vegetation indices (Peterson et al., 1987); social scientist

estimate population counts (Lo, 1995), etc.

This study also harnesses the advantages of remotely sensed images and

evaluates their utility for urban studies. The main purpose of this derivation is to

obtain biophysical variables (vegetation cover, vegetation fragmentation and a

vegetation index) which would have been very difficult to obtain with ground level

collection and then evaluate their usefulness in CO-processing with socio-

economic data.

Althoug h vegetation cover has been mapped f rom remotely sensed

images (Gaydos, 1992), the mapping of vegetation in urban areas has not

received as much attention by the remote sensing community as it warrants,

pointed out Nowak (1994). Yet a more accurate knowledge of the spatial

distribution of vegetation cover can be vital for understanding the hydrology,

climatology of urban systems and as this study is hypothesisiny, it can help in the

identification of socio-economic status. Unfortunately, one of the most popular

classification systems, the USGS defined systern (Anderson, 1976) is very

deficient in labelling vegetation in urban areas. It is very much biased towards

classes of residential and commercial even if there were a significant canopy

layer. This bias is understandable given the fact that this system was designed

to help in the classification of the entire United States of which only 1% is

urbanised. The approach adopted in this study is to classify rernotely sensed

images obtained both from aeroplane and satellite platforms, using a land cover

classification based on tone and its spatial variation (texture).

2.2 Sources of remotely sensed data

2.2.1 Aerial photographs

Aerial photography at a scale of 1 :20,000 (pixel resolution of lm) is

obtained from the Colour Orthophoto Collection of the University of Toronto Map

LiSrary7s online database. The pictures were taken by the Triathlon Mapping

Corporation (1 995) who scanned, oriented, and rectified the diapositives. Seven

images are identified in the photo-collection that cover the south-east part of the

City of Toronto. A quick visual inspection reveals that the digital numbers are not

norrnalised between the images in the two rows (the southern row, covering part

of the lake, being somewhat brighter). A quantitative cornparison of a 50 cell

wide band on the edge of the two rows reveals that in the green and blue bands,

the digital numbers (DN) are higher

Table 1. Digital number (DN) differences of means, between the two rows of images calculated over a 50 pixel wide and 500 pixel long area located on the adjacent sides of the two rows of images. The numbers in brackets indicate the standard deviation.

Row 2 Row 1

in the southern row (Table 1). However, subsequent work with the images also

shows variation in intensity within the same photograph. These are important

considerations in the classification process since tonal differences in the same

image will require well-defined training areas for the various classes.

Due to the reflectance differences and to avoid working with very large

databases (>50MB) the photos from the two rows are kept separate. However,

photographs of the same row are merged (mosaicked) into one larger database.

RED 150.1 (46.1) 150.1 (37.8)

GREEN 141.57 (60.1 8) 147.9 (37.9)

BLUE 1 38.2 (53.94) 141 -5 (34.2)

N 25896 261 29

2.2.2 Satellite imagery

The satellite image was taken by the LANDSAT THEMATIC MAPPER

(TM) satellite, with a resolution of 30m, in July of 1992 (Fig. 3). This image,

comprising three bands (TM2-green:0.52-0.6pm; TM3-red:0.63-0.69pm; TM40

NI R:O.76-O.gpm), has been resampled to 25m resolution.

Satellite imagery provides a useful way of studying the resolution

dependence of the classification of vegetation. For CO-processing with socio-

economic data at the €A level, it may turn out that the use of very fine resolution

(1 m - 1 0m) is not warranted. Although the images were taken a few years before

the aerial photographs, it is assumed that the vegetation cover between the two

years remained relatively constant. This assumption is reasonable given the fact

t hat the study area includes long established neighbourhoods.

Figure 3: False-colour composite of the study area taken by the LANDSAT TM satellite in July of 1992. With this cotour coding , vegetation appears as red.

2.3 Resampling

To assess the effect of resolution on accuracy and percent cover, the

aerial photographs are resampled from l m resolution to Zm, Sm, 10m, 25m,

50m, and lOOm resolutions. The resampling algorithm uses an arithmetic

average over the specified window size that moves across the image without

overlapping. Although this is a very crude approximation in what truly happens in

sensors of coarser resolutions, it has been used widely in remote sensing studies

of scale (G raniero, 1 999; Burchfield, 2000).

2.4 Texture layers

A major limitation of traditional classification algorithms is that the spatial

structure of the data is not taken into consideration. However, as Tobler's First

Law of Geography states, it is to be expected that pixels near each other will

belong to more similar classes than pixels farther away frorn each other.

Therefore, classes that overlap in spectral space could be separated based on

spatial information on who are the neighboun. A great deal of effort has been

lately devoted to improve per-pixel classification by the inclusion of

neig h bou rhood information (Pesaresi, 2000; Chica-Olmo and Abaraca-

Hernandez, 2000; Graniero, 1999). It should be noted that most classification

implementations still use per-pixel methods, however, they niake use of

contextual layers that explicitly define the spatial structure.

There are several ways to obtain contextual information. Semivariograms,

correllelograms. etc. are used extensively (Arai, 1 993; Miranda et al., 1998;

Woodcock et al. 1988), however, probably the most often used technique is the

derivation of texture measures using some predefined algorithm (Hudak and

Wessman, 1998; Graniero. 1999; Pesaresi, 2000). Texture is closely linked to

tone and its spatial distribution. Image areas with tonal homogeneity are more

easily described by their characteristic tone (colour), whereas highly

heterogeneous areas in terris of tone are more easily described by their texture

(Haralick, 1 979).

Texture can be computed in several ways. Usually, it is computed with

the use of a grey-level CO-occurrence matrix that gives a count of tonal

adjacency. This matrix is then computed under a moving average filter with a

certain kernel size, and the central value of the filter is assigned a value

calculated with an equation such as entropy, contrast or angular moment. A

recent study has shown that most of these measures are highly correlated, with

contrast the most successful at improving classification (Pesaresi, 2000).

Interestingly, the early suggestions of using larger kernel sizes to rnimic the way

humans perceive texture, have not been followed because users have found

these iarger kernels to be inappropriate for heterogeneous areas such as cities

(Hodgson, 1998). This study makes use of a contrast measure given by Haralick

et al. (1 973) (Eq. 1).

CONTRAST = 1 (i-j12 ' Pi, / Z Po ;

where Pii is the estimated probability of going frorn grey level i to grey level j.

Contrast calculates the probability of occurrence of two brightness values

separated by a given distance in a given direction and within a specified kernel or

window size. The algorithm implementing the measure was written by Graniero

(1996). It takes a raw 8-bit image and under a specified moving kernel

calculates contrast using a horizontal displacement vector. Edge areas are

important in texture calculation because the algorithm can not compute the

measure where more than half of the kernel is displaced off the image.

Consequently, a band of zeros is created at the edge, which is as thick as one-

half the size of the kernel. This can seriously effect the analysis if the images are

small or if the images are later mosaicked.

As it will become evident later, vegetation, especially, tree cover is

spectrally very similar to shadows cast by trees and buildings. It is hypothesised

that the texture layers will improve the separability of these two classes.

2.4.1 Optimal kernel size for calculating texture

Contrast is computed over a range of kernel sizes from 3 to 25 pixels.

The most appropriate kernel size is detemined during the classification process

using accuracy assessment by the Kappa statistic. However, the ideal kernel

size can also be deduced theoretically by considering the needs of the

classification process. Although vegetation cover exhibits great heterogeneity in

the urban landscape, it is safe to assume that tree canopies are in the range of

3m to 20m in diameter. Thus, the optimal kernel size will most likely be in this

range.

The ideal kernel size is assessed on a smaller subset by performing

supervised classification using texture layers calculated at different kernel sizes.

The resülts indicate that the accuracy is maximised at kernel sizes of seven and

nine, at 1 rn resolution (Fig. 4). The effect of texture is also assessed for coarser

resolutions. At 5m resolution, where kernel sizes of three and five correspond to

15m and 25m wide areas, reveal that at a kernel of three, accuracy is improved,

however, at a kernel of five, it decreases (Fig. 5). These findings support results

from the literature which emphasise that texture is most useful for fine resolution

imagery (Graniero, 1999). Consequently, texture layers are computed only for

1 ml 2m, and 5rn resolutions.

Kemal Jze

Figure 4: Accuracy of supervised classification at l m resolution as a function of kernel size used to derive the texture layer. The accuracy is 0.52 without the use of texture layers. The ideal kernel size seems to be seven or nine pixels wide.

Figure 5: Accuracy of supervised classification at 5m resolution as a function of kernel size used to derive the texture layer. At a kernel of five (25m) accuracy is actually reduced from the case where no texture layer is used.

2.5 Vegetation indices

Indices of vegetation amount have been widely employed by ecologists,

however, they have also been utilised in mapping of vegetation as well. For

example, Gaydos (1992) classified aerial photographs by computing the

Normalised Difference Vegetation Index (NDVI) and then selecting a threshold

for the vegetation class. NDVl is based on the fact that chlorophyll absorbs red

light (RED) and mesophyll tissue strongly reflects infrared radiation (IR).

Therefore, the 1R.R ratio can provide a measure of the abundance of "green"

vegetation within a particular pixel. NDVl is a useful parameter for estimating

biomass or vegetation vigour. It is easily computed using the following equation

(Eq-2).

NDVl is computed for the satellite image using the TM3 and TM4 bands. The

values are scaled to O - 100. To convert the NDVl layer into a map of vegetation

a cut-off of 30 is arbitrarily specified as the boundary between vegetated and

non-vegetated. Deriving vegetation maps in this fashion eliminates the need to

perform classification on the map. The big limitation in this study is the lack of a

near infrared band for the aerial photographs. However, even with the help of a

IR band, Gaydos (1992) found a high percentage of confusion between the

vegetation and shadow classes.

2.6 Image classification

The literature abounds with studies whose main objective is the derivation

of land use from remotely sensed images (Gong and Howarth, 1992; Gong and

Howarth, 1990a; Haack, 1 987; Green et al. 1994). In contrast, the needs of this

study dictate classification according to land cover. While the derivation of land

use can be a much more complicated issue due to the difficulty of assigning land

covers s hared between land uses to culturally defined classes, the derivation of

land cover can be equally challenging in some instances. Although the objective

is sirnply to derive a binary map of vegetated and non-vegetated surfaces, the

problern of inadequate spectral resolution and the effect of shadows, create a

similar situation faced by researchers primarily interested in land use.

All classification schemes should incorporate two phases in their

implementation (Gong and Howarth, 1990a). In the first phase, an appropriate

classification scheme should be chosen. The scheme should either implement

an already existing scheme such as the very popular Anderson classification

system (Campbell, 1996) cr introduce a new or modified scheme that will be

ideal for the project at hand. Choosing an existing system will allow future users

to compare classifications. The nature of this study dictates that instead of land

use categories, land cover categories should be chosen. Moreover, the nature of

the data (colour photograph with red, green, and blue bands) limits the

classification to be based on colour (tonal variation) and texture. Consequentlÿ,

classes are chosen based on surface types and colour (Table 2). For some of

the coarser resolution imagery, the number of classes is reduced because of the

inability to find homogeneous sites for some of the classes like roads and grey

rooftops. The second phase of a classification process comprises the actual

implementation of the classification using the categories decided upon in phase

one.

Tree canopy Dark-g reen

Grass Light-green

Shadow (large) Dark-grey Shadows of large buildings

Shadow (thin) Light-grey Shadows of small buildings

Urban(highly Orange Mainly concrete, white paint ref lect ive) Urban (grey rooftop) Red A great mix of surfaces that looked grey

Urban (Roads) Purple

Table 2: Classification scheme used in supervised classification of the fine resolution imagery.

2.6.1 Selection of training sites

A good classification is predicated on the quality of the training data. One

of the first impressions during the classification is the large geographic sîale of

the study area, even when it is partitioned into two halves. To capture the slight

differences in reflectivity within the same class over the whole image, training

pixels are collected from across the whole image. Large images can have quite

dramatic changes of intensity moving from one end of the image to the other.

This in turn is due to camera limitations and bidirectional reflectance effects.

One of the assumptions of the Maximum Likelihood Classifier is that the

training classes are normal. In order to achieve such ideal distributions, training

sites are located in homogenous areas, in most cases as groups of pixels.

Selecting groups of pixels may be a violation of independently selected pixels,

because blocks of training data contain cells which have neighbouring cslls of

similar values and thus they are autocorrelated to some distance and not totally

independent. Many producers of supervised classification collect training data in

contiguous blocks that may violate the independent assumption of the Gaussian

classifier. In order to obtain training data where each pixel contributes

information independently to the class signature, training data should be

collected randomly by single pixels. Campbell (1 981 ) found a significant

difference between block collected training data and single pixel training data.

Moreover, Gong and Howarth (1990a) also found that single pixel, randomly

collected training data produced better separability between classes. However,

the effect of the difference of using randomly picked field data as opposed to

collecting contiguous training data on the classification results varied throughout

the growing season (Campbell, 1981). in fact, the difference between

classification from June and September resulted in only a slight difference. The

effect of spatial autocorrelation on the training areas will most likely be

dependent on the resolution of the imagery. In this study, the assumption is

made that the error due to the difference between single pixel versus block

training is orders of magnitude smaller than that of the error caused by the

confusion of vegetation and shadows.

2.5.2 Training signature statistics and analysis

A good classification is rooted in the quality of the signatures. Hixson et

al. (1 980) provide evidence that differences in the selection of training data are

more important influences on accuracy than are differences among different

classification procedu res. Signature separability is a powerful anal ysis tool that

can assess the quality of signatures. If the separability between two classes is

high, they wiil most likely produce an accurate classification with minimal

confusion. A common measure of separability is the Transformed Divergence

index, which is calculated from the means and covariance matrices of each

spectral class or training site. Statistical distances can be calculated not only

between classes but also between training sites of the same class. This

separability is an indirect estimate of the Iikelihood of correct classification

between groups of different band combinations (Haack et al., 1987). A

separability of 1.5 or higher is acceptable, but in order to have very good

classification results 1.9 or higher is recommended (max. = 2.0) (Gong and

Howarth, 1990b).

The key to an accurate vegetation map is to obtain high signature

separabilities between vegetation and shadow classes (Table 3). Unfortunately,

the separability is quite low (1.45) between the tree class and large shadow class

due to the fact the two classes had very similar spectral properties and in many

cases similar textural properties. However, the separability is even lower

between the trees and grass classes. For this reason, the grass and tree

classes are later merged into one, vegetation class. It appears that the texture

layer is successful at discriminating between vegetation and thin shadows (1.99).

These thin shadows are cast by smaller, one or two storey buildings.

Table 3: Transformed Divergence separability values for classes of the supervised classification. The lowest separability (boldface) is observed between trees and grass, and trees and large shadows.

2 6.3 Supervised classification

Separability

Vegetation jG rass) Urban (Highly ref lec tive) Urban (Grey rooftop) S hadow(large) Urban (Roads) Shadow(thin)

The classifier chosen for this study is the popular Maximum Likelihood

Classification (MLC). The MLC is a per-pixel classifier that assigns each pixel to

a class with the highest likelihood of belonging to its probability density function

that is Gaussian. The probability density function is calculated from the means

Urban (Grey rooftop)

1.99 1.86 1.99

Urban (Highly ref Iect ive)

1 -85

2.00 1.98 2.00

Shadow (larqe)

2.00 _ 1.99

Vegetation (Trees)

1.22

2.00

1.95

1.45 1 -99 1.99

Urban (Roads)

- 2.00

Vegetation (Grass)

2.00

1 -98

1.94 1.99 2.00

and variances of the training data. The classifier does not take into consideration

the location of pixels, or the spectral characteristics of surrounding pixels. Error

may enter into this classification method if the classes do not display multivariate

normal frequency distributions (Campbell, 1996). Studies of fine resolution

images have highlighted the fact that per-pixel classifien such as MLC, which

were developed for coarse resolution MSS data, are not appropriate at finer

scales and that new classifiers should be developed (Gong and Howarth,

1990a). However, until new algorithms are developed and tested, most

classifications of fine resolution imagery are dependent on the use of contextual

information derived from the images to improve the traditional classifiers.

Maximum Iikelihood classification allows the specification of the threshold

for the probability of accepting a pixel into a class. The effect of the change of

this variable is assessed similar to the selection of the optimal kernel size. Of

course, as the threshold is increased the amount of correctly classified pixels

increases, however, the errors of commission rise along with it and the errors of

omission decrease. The threshold for which there is a comparatively similar

decrease of omission errors and an increase of commission errors is 4 (Fig. 6).

This threshold is implemented in al1 classifications.

U>

60000 50000 - . / - -Commission % 40000 . - +-Omission O g 30000 . - Correct

Thnshold for vegetation class

Figure 6: Assessrnent of the effect of changing the threshold (in spectral space) for the vegetation class in supewised classification. The errors due to commission steeply rise beyond THRS.

2.6.4 ISODATA clustering

Unsupervised classification entails an unguided approach to image

segmentation. It is automated and it uses an iterative procedure whereby pixels

are assigned to the closest class (minimum Euclidean distance), after which the

means of the classes are recalculated and the pixels reassigned to the closest

classes until a pre-specified confidence limit is reached for al1 pixels (Jensen,

1996). One of the big limitations of clustering is the lack of control over class

selection and the subsequent assignment of classes to various land use or land

cover classes. This assignment is especially difficult for images of 10m

resolution or coarser.

2.6.5 Hybrid classification

Preliminary classifications of a subset in the downtown portion of the study

area reveal that the optimal classification is that using a supervised approach.

The user-guided approach allows for a better separation of the shadow and

vegetation classes. On the other hand, during the classification of the entire

study area, clustering proves more effective for the areas outside the downtown

core, where the confusion between shadow and vegetated pixels is much lower

than in the downtown core. Thus, a hybrid approach is implemented. A bitmap

is created for the downtown core, under which a supervised classifica!ion is

carried out, whereas the other parts of the image are classified by ISODATA

clustering. Depending on the resolution, 15-25 clusters are created by the

ISODATA algorithm, which are then assigned to either the vegetated or non-

vegetated classes. This assignment of classes gets increasingly difficult as the

resolution is coarsened. Due to the difficulty of finding homogeneous classes, al1

images with 25m or coarser resolution are classified using the unsupervised

approach. For classification of the finer resolution imagery (1 m. 2m, 5m), four

layers are used: red, green, blue and texture layers. For the coarser imagery,

only the three visible bands are ernployed. The use of al1 three spectral bands is

most likely superfluous as they contain redundant (multicollinearity, ~~>0 .98 )

information.

Although there are several classes created during the classification

process, they are al1 either assigned to the vegetation class or the urban class.

These final classified images for the two rows of imagery are mosaicked to

produce the final classified map for the whole study area (F ig. 7a-f).

2.6.6 Classified images

The proportion of vegetation to urban exhibits great variability over the

study area. Vegetation cover increases from zero cover in some of the downtown

blocks to over 100% in areas of Rosedale. This proportion, however, is scale

sensitive. There is evidence of the vegetation class to shrink in percentage

cover as the resolution coarsens (mainfy due to the mixed pixel effect), in areas

where vegetation at a fine resolution is fragmented and of low total cover (e-g.

East York) (Fig. 8). On the other hand, in areas of high vegetation cover. the

proportion actually increases (Fig. 8). It is interesting to observe that the

proportion of vegetation to urban increases and then levels off for the downtown

subset (Fig. 8). This could be caused by some of the parks in the subset being

aggregated with other nearby treed areas. The percentage cover of vegetation

over the whole study area is constant (-23%) until about 10m resolution, then

dips to 17% (at 50m res.) before rising again to 20% (at 100m) (Fig. 8).

Generally, classes that are clumped at a fine resolution will grow in proportion, as

they are aggregated (Cao and Lam, 1997). It should be stressed that these

proportions can be very much influenced by the training class selection in a

supervised approach and by cluster assignment in unsupervised classification.

For example, the assignment of clusters to the

Figure 7 (a.): Vegetation cover at 2m resolution.

7 (b.) Vegetation cover at 5m resolution.

25

7 (c.) Vegetation cover at 10m resolution.

7 (d.) Vegetation cover at 25m resolution.

7 (e.) Vegetation cover at 50m resolution.

7 (f.) Vegetation cover at 100m resolution.

27

correct land cover class is especially difficult at coarse resolutions due to the

mixed pixel effect. Also, at coarse resolutions, the availability of homogeneous

areas for training class selection is rninimised. Since separate classifications are

perforrned on the two rows of images, there is some discrepancy between the

two rows in the final classified images.

A 100 f à? V 8

8 o i m jWm . SET0

O r SUBSETA c 60 O - - i A SUBSEl-8

40 hA Q) x SUESETC Ul a3 X

X I*' p , 20 t 0 0 0 lxX A

t

O ' A * Y

Resolution (m)

Figure 8: Percentage vegetation cover as a function of resolution calcukted for the entire SET0 area and three smaller subsets (Subset A - Rosedale; Subset B - East York; Subset C - Downtown (see Fig. 9 for the location of the subsets)). The percent cover rises in areas (Subset A) where at a fine resolution the vegetation is clumped and decreases in areas where it is fragmented (Subset 6).

2.7 Accuracy assessment

Accuracy is assessed by the comparison of the classified images with

those of three visually classified areas (Fig. 9). Two of these areas are in the

downtown core and the other is in a suburban area. Visual classification is

su perior to automated classification, because trained interpreters can directly

Figure 9: Location of subsets used for accuracy assessment (Subset D, E,F). Subsets A,B, and C are used to calculate contagion (see section 2.9).

identify the classes of interest (Martin and Howarth, 1989). However, error

enters the visual interpretation in several ways. First, vegetation is delineated

with the use of a cursor, which sometimes goes one or two pixels over or under

the perceived boundary of vegetation. Second, even at l m resolution, some of

the vegetation is hard to distinguish from other objets of similar colour such as

green roofs. In addition, if vegetation is concealed by shadows, it is omitted.

However, some of the shadows cast by trees that are true shadow pixels could

be included in the ?ruew image. These error sources are minimised, yet they still

have an effect in the accuracy assessment of the classified images.

The coincidence matrix (also called contingency or confusion matrix) is a

universal tool in remote sensing accuracy assessment. The elements on the

main diagonal of the matrix represent the number of pixels of the same class that

overlap and the elements off the main diagonal are the pixels of the same class

that do not overlap (Congalton et a/., 1983). In most cases, column headings are

those of the reference data and row headings of the classified data. Reference

images are often collected in situ or derived from high resolution photography or

imagery. Due to the immensity of data volume for high resolution databases,

most analyses rely on stratified random sampling to obtain the reference data

(Congalton, 1991). However, as pointed out before, the present study utilised

small subsets as reference maps obtained by visual classification. This could

bias the estimates in areas further away from the subsets but overall it provides a

very large number of pixels to be used in the assessment.

Accuracy can be computed in several ways. For the combined accuracy

of al1 the classes, overall accuracy ana the Kappa coefficient have been widely

implemented. Overall accuracy has been the traditional coefficient of accuracy,

however, it overestimates accuracy by only taking into consideration the

elements on the main diagonal (Table 4). Congalton et a/. (1 983) was one of the

first to recommend the use of Cohen's Kappa coefficient (Eq. 3), as it also takes

into consideration the elements off the main diagonal, the errors of omission and

commission. The Khat statistic gives an indication of accuracy after random

agreement is removed from consideration.

Kappa coefficient: r

N ' x x , - 1 (x*' x,,) 1=1 1=1

,where N is the total number of classified pixels, Xii is the element on the main

diagonal, Xi+and X+; are the row and column totals respectively.

Table 4: Cornparison of Khat (Kappa) accuracy with overall accuracy. By not considering the off-diagonal celfs, overall accuracy overestimates the more reasonabie accuracy given by the Khat statistic.

SET0 KHAT OVERALL

2.7.1 Producer's and user's accuracy

While the Kappa statistic gives an overall estimation of accuracy, it can be

very biased when there are only two classes with disproportionate coverages. In

such cases the larger class will naturally have very high agreement, which then

biases the Kappa statistic. Therefore, individual class accuracies as catculated

by the producer's and user's accuracies might give a better indication of how

close the classification is to the "truth". Producer's accuracy indicates how many

of the classified pixels of a reference class actually belonged to that reference

class. These are the errors of omission and usually occur when class

boundaries (in feature space) are underestimated. On the other hand, user's

accuracy is a measure of how usable the classified map is, that is are the pixels

on the classified maps accurate (these are the erron of commission).

Resolution(m) 1

2 1 S 0.671 1 0.620 0.897 1 0.916

25 0.385

, 0.798

1 O 0.657 . 0.912

50 1 100 0.389 1 0.295 0.846 1 0.830

Table 5 indicates that from a user's perspective for fine resolutions, the

probability that a pixel on the classified map is vegetation in reality as well is

around 80 O h in âll three subsets. This probability decreases to 40 % for the

coarser resolutions. It is interesting to observe that the user's accuracy for the

25m resolution satellite image the accuracy is comparable to that of the 25m

resolution aerial photography, and in two of the subsets even superior. This is

quite encouraging for the satellite imagery since it means that for many

applications, gains may not be realised by the use of fine resolution aerial

photography. In terms of producer's accuracy, the values are much lower than

for the user's accuracy because of the confusion created by the shadow class

and the errors that entered into the delineation of the "true" map. Both user's

and producer's accuracies are influenced by the percentage of coverage. For

example, the producer's accuracy for Subset E at 2m, with 6.1% vegetation

cover, is less than in either Subset D or F, where the percentage cover is much

higher. The higher the percentage cover of a class, the higher the likelihood for

agreement.

Table 5: Accuracy assessrnent according to user's and producer's accuracies computed for three subsets (please see Fig. 9 for the location of the subsets). Of interest are the relatively high accuracies of the satellite image (TM-25).

PRODUCER'S (Omission) SUBSET D SUBSET E

% Vegetation Cover at 2m

19.5 6.1

Resoiution of Vegetation 2m

0.52 0.38 0.69

0.76

0.83 0.77

SUBSETF USER'S (Commission) SUBSET D

SUBSET E SUBSET F

. 34.8

19.5

6.1 34.8

5m

0.61 0.58 0.66

0.70

0.79 0.80

50rn

0.81 0.50 0.34

10m

0.83 0.72 0.59

0.64

0.70 0.77

i

lOOm

0.87 0.39 0.35

25m

0.49 0.70 0.37

0.51

0.39 0.m

TM-25m

0.36 0.34 0.60

0.82

0.57 0.78

0.49

0.46 0.71

0.33

0.40 0.53

2.8 Satellite image

The costlbenefit ratio, in terms of time and finances, is in favour of the

derivation of vegetation cover from the satellite irnagery. In addition, the

accuracy assessrnent indicates that vegetation cover derived from satellite

imagery is just as accurate as finer resolution aerial photography.

Although the process of classification is the same for satellite and aerial

photographs, the coarser resolution and the availability of an IR band merit sorne

further discussion. Due to the relatively coarse resolution, the number of land

cover classes used is five as opposed to eight for the fine resolution aerial

photography.

The false colour composite image provides great aid in the delineation of

vegetated versus non-vegetated or non-photosynthesising surfaces (Fig. 3).

Light blue signifies cernent, asphalt, or bare soi1 surfaces that are the

characteristic of primary materials in urban areas. Red areas are vegetated

areas, with higher intensities of red signifying higher percentage of vegetation.

Brownish red areas are mixed pixel areas where urban materials are mixing with

vegetation. Sometimes, an analyst can be at odds with what to do with such

mixed pixels. Of course, if one trains for these areas as vegetation al1 of the

mixed pixels will be classified as vegetation, if one does not train for them,

vegetation will be omitted in the resulting classification which are contained by

the mixed pixels.

Supervised and unsupervised classifications result in very similar results

(Fig. 10). Therefore, to Save time, clustering is chosen for the TM images. Two

of the five classes identified by the clustering are easily assigned to the tree and

grass classes, however, a third class contains the mixed pixels of residential

areas. The inclusion of these pixels significantly increases the percentage of

tree cover. This can be a very serious limitation of coarse imagery for tasks such

as vegetation mapping because individual trees blend into the urban fabric. The

near-infrared band does eliminate most of the shadow-vegetation confusion,

however, at 25m resolution shadows are also considerably blended into the

urban fabric.

A comparison of the 25m satellite and the 25m aerial photo derived

classifications show very low consistency (-40%). This could be caused by

producer's errors during the classification or by inconsistencies in the geometric

registration of either of the images to the UTM grid.

Figure 10: Vegetation cover derived from the satellite image at 25m resolution.

2.9 Contagion: A measure of spatial arrangement

One way of characterising urban vegetation structure is to assess its

cover. However, spatial arrangement of this vegetation cover may also be an

important variable to consider when studying relationships between biophysical

and socio-economic characteristics. "Configuration of land use is one of the

major contributors to the quality of life (Geoghegan et ai., 1997)".

Many different landscape indices have been proposed by ecologists to

measure habitat fragmentation, species diversity, etc (Riitters et al., 1995). They

have also found some application in urban studies (Burchfield, 2000; Geoghegan

et al., 1997). One commonly used index is contagion, which measures the

degree to which classes are clumped within a landscape. At low contagion, the

landscape exhibits high fragmentation whereas high contagion indicates low

fragmentation. Contagion is a modification of the entropy equation and is similar

to the first-order join count statistics.

m m

Contagion = 2 ' ln (m) - Z Z Pij ' ln (Pij) ; i=1: j=l

,where m is the total number of land classes and Pi, is the total number of times

attribute i is adjacent to attribute j. The values of Pu form the entries of the so

called grey-level CO-occurrence matrix (GLCM). The second term of the equation

is the entropy equation, which is a maximum when al1 pixels of attribute i are as

far apart from one another as possible. (Baker and Gai, 1992)

Contagion is calculated in the GRASS GIS program (Baker and Cai. 1992)

for three different subsets of the classified image (Fig. 9). All three areas

measure 600m by 600m. These areas are divided up into 36, 100m wide,

squares. Contagion, as well as vegetation cover, is calculated within each

square. In one of the subsets, the effect of resolution is assessed by calculating

contagion and vegetation cover using different resolutions.

As is evidenced by the graphs in figures 1 1 and 12, the relationship

between contagion and cover is very predictable which is an artefact of the

contagion algorithm. Contagion is a minimum at approximately 50% cover

because this is when the likelihood of maximally mixing two classes is the

highest. These findings have also been documented in the literature (Gustafson

and Parker, 1992; Benson and MacKenzie, 1995) and point to the fact that one

should be careful when using indices such as contagion, when characterising the

spatial arrangement of landscapes. Therefore, contagion is not used later in the

modelling.

Figure 11: Contagion as a function of vegetation cover computed for the East York subset (Subset 6) on a square grid using classified images at different resolutions. It appears that contagion is highest at 2m resolution.

O 20 40 60 80 1 0 0 120

Vegetation cover (%)

Figure 12: Contagion as a function of vegetation cover cornputed for the three subsets (Subset A- Rosedale, Subset B- East York, Subset C- Downtown). The shape of the plot is characteristic of contagion/cover relationships.

As presented in this chapter the derivation of biophysical variables such

as vegetation cover and vegetation pattern are feasible from both satellite

imagery and aerial photography. The time and effort required to process the

aerial photographs are in excess as that for the satellite image. Despite the

coarser resolution of the satellite image the accuracy of the classified maps is

comparable to that of the accuracy of maps derived from aerial photographs.

However, it remains to be seen whether there is significant difference in the

results of statistical analyses between different resolutions when these

vegetation rnaps are aggregated to the €A level.

111

Exploring the relationship between

biophysical and socio~onomic variables

The following chapter first describes the sources of socio-economic data

and the reason why the variables are chosen. Other sections follow on

exploratory data analysis, including qualitative and quantitative assessment.

Spatial dependence is measured using Moran's I statistic and by the semi-

variogram. Multivariate analysis includes both scatterplot analysis and

correlation table calculation.

3.1 Sources of socio-economic data

3.1 .1 Census data

A number of variables are selected from the 1991 Census of Canada

database at both the enumeration area (€A) and census tract (CT) level of

census geography. The choice of variables is dictated by suggestions from the

literature. For example, Grove (1 996) in his study of the reciprocal relationships

between vegetation structure and social stratification in an urban watershed

selected variables suggested by Shevky and Bell (1 955) that were classified as

belonging to one of three different groups of indices, namely a socio-economic,

household and ethnicity index. The socio-economic index included variables

reflecting income and education levels. The household index captured variables

such as marital status, home-ownership, and type of household. The ethnicity

index reflected the race and immigrant status. Finally, based on these three

indices a fourth composite social area index was built. Ryznar (1998), who

studied the relationship between vegetation change and the social environment,

in addition to socio-econornic variables included demographic indicators such as

fertility and age structure. Lo and Faber (1997) assessed the feasibility of

integrating Landsat Thematic Mapper and census data for quality of life

assessment using the following socio-economic variables: population density,

per capita income, median home value, and percentage of college graduates.

The variables selected for this analysis are as follows: population density,

number of single-detached homes, number of immigrants, number of university

graduates, and median household income.

The count variables are divided by the total population of the EA and then

multiplied by 100. One of the limitations of EA level data is the problern of

missing or suppressed values, especially for income related variables. Out of

302 EAs analysed for this study 94 have their values suppressed due mainly to

conf identiality issues.

3.1 -2 Social deprivation index

Socio-economic indices may be much better measures of the socio-

economic status of a geographical area than a single census variable. Social

deprivation indices have been snown to correlate with health in residential areas

(Mays and Chinn, 1989; Carstairs and Morris, 1989). Frohlich and Mustard

(1 996) computed a socio-economic index from six census variables. They found

a high correlation between the socio-econornic index and a health index.

However, Grove (1996) found that the utility of composite indices in the study of

the physical environment and the social environment needs a lot more research.

Grove (1996) found much higher correlations between single census variables

and biophysical variables than a composite index and biophysical variables.

The choice of variables for a social deprivation index should reflect the

main domains of socio-economic characteristics: dwelling, education,

employment, income, and mobility (Frohlich and Mustard, 1996). Jarman's

(1983) index, which reflects four of these characteristics, is calculated using

weights that were originally derived from a survey of General Practitioners (GPs)

in the UK.

Census category description Household of 4 or more people Lone parent families Rental housina units

Table 6: Jarman's (1 983) weights for the social deprivation index. Weights derived from a survey of General Practitioners in the UK.

--

' Weights 2.88 3.01 3.6 -

People 15+ without secondary education People 15+ unemployed People 15+ not in labour force Famifies with low incorne

All of the variables are first standardised and then assessed for

skewness. The removal of skewness has been identified as a prerequisite for

the generation of a composite index (Gilthorpe, 1995). Families with low income

are not calculated into the equation because of the many (94) suppressed

values.

2.9 3.34 3.34 3

3.2 Exploratory data analysis (EDA)

3.2.1 QualitativeNisual assessrnent of variables

A quick way to gain soma understanding about the spatial distribution of

the variables is to map them. The maps presented on the next few pages are

created using quartiles (Fig. 13 a-e). lt should be stressed that the pattern

shown is heavily influenced by the boundaries of the EAs with larger units

tending to dominate. The chloropleth maps also make it seem as if the variable

being mapped applies to every part of the study region, when it is evident from

the aerial photograph (Fig. 1). that there is great heterogeneity in land uses, let

alone socio-econom ic characteristics. In addition, chloroplet h maps can be

misleading because alternative choices of shading and class intervals can lead

to widely different visual interpretation (Bailey and Gatrell, 1995, p.256; Langford

and Unwin, 1994).

Keeping these reservations in mind, there are very evident trends on the

maps presented in figures 13(a-e). Population density displays a pattern where

the highest densities are observed in €As, which occupy apartment buildings

(Fig. 13a). Outside these very small EAs, the density is somewhat higher on the

east side of the study area (Riverdale) as opposed to areas of Rosedale to the

north and Cabbagetown, just south of it. SINGLE shows a highly clumped

distribution (Fig.13b). The downtown sections of the study area have almost no

single-detached homes, whereas, Rosedale and Riverdale have much higher

percentages of single-detached homes. A somewhat different spatial

distribution, yet still highly autocorrelated, can be observed in the case of the

proportion of university graduates (Fig. 13 c). EAs in the highest quartile are

found mainly in Rosedale, Cabbagetown and a few in Riverdale. Those in the

lowest quartile are found near the public housing projects and the apartment

EAs. The distribution of the percentage of immigrants exhibits more of a random

pattern, however, recent immigrants also tend to congregate in the public

housing projects, apartment blocks and the south-east portion of the study area

(Fig. 13d). Last, the social deprivation index manifests a distribution of deprived

areas in the south of the study area and much better off areas in the north,

Rosedale, Cabbagetown (Fig. 13e). It is interesting to observe one of the

Rosedale EAs to be in the third quartile, which might indicate that the index with

these weights is not really suitable for this area.

The histograms of the variables reveal their one-dimensional distribution.

Three of the socio-economic variables (POPDENS, INCOME, and SINGLE) and

al1 of the biophysical variables (VEG) reveal skewness to the right (Fig. 14).

SINGLE 0.01 0.01 - 6.26

-5-5 6.26 - 18.88 W 18.88 - 06.58 Suppressed

Figure 13 (a-e): Spatial distribution of selected census variables. a. Population density is highest in small EAs containing apartments. b. The lowest quartile of SINGLE is al1 zeros since these represent the values of the apartment and downtown EAs.

Figure 13 (a-e): Spatial distribution of selected census variables. c. UNlDEG is highest in Rosedale and Cabbagetown. d. The distribution of IMMIPOP displays a pattern of higher proportions near public housing projects and in the southeast part of the study area.

Figure 13 (a-e): Spatial distribution of selected census variables. e. Socially deprived areas are those in the highest quartile. These are found near the public housing projects and near the lake.

. . . . - . OS2 OSL OS O

r . . . . . i

OZ1 08 OV O

O 20 40 60 80

VEG lOM

Figure 14 (Cont.): Histograms of vegetation cover variables at various resolutions. They are al1 skewed to the right and subsequently transformed with the natural log.

3.2.2 Univariate statistics

3.2.2.1 Descriptives

The distribution with the largest range is that of POPDENS. Its maximum

is at 1891299 pers./km2 and minimum at 408.4 pers./km2 (Table 7). Such high

population densities are observed because €As covering only apartment

buildings have relatively high populations and very small areas. One should be

careful not to attribute these maximums of population densities to larger areas of

the city. The standard deviation of POPDENS is 249145.7 pers./km2. SINGLE.

IMMIPOP, and UNIDEG have roughly similar ranges. The mean value of

SINGLE is 5.0 %, not surprising given the fact that a large proportion of the

study area covers the downtown core. The mean total immigrant population is

quite high at 36.9 %, reflecting a high influx of recent immigrants. The highest

value UNIDEG is 62.5 % with a mean of 21.0 %. Median family income ranges

from about $ 5000 to $ 194074. These extremes represent the poorest and

richest people living not only in Toronto but also in Canada. The mean of

JNCOME is $ 27014 with a standard deviation of $ 25719.7. The social

deprivation index ranges from -364.9 (higher standard of living) to 631 -4 (lower

standard of living). The mean, median, and standard deviation are 33.5, 18.4,

and 206.1 respectively.

The descriptive summaries for the biophysical variables are also included

in Table 8. For VEG, derived from the aerial photographs, the range widens

from 75.7 to 100 with coarsening resolution as a result of the derived cover to

aggregate at coarse resolutions. The mean of these same variables increases

from 17.3 (at 2m resolution) to 23.1 (at 25m res.) and then decreases to 16.2 (at

1 OOm res.). The standard deviation increases with coarsening resolution from

13.9 to 29.8.

The descriptive summaries like their corresponding histograms reveal that

a majority of the variables are heavily skewed to the right. Correcting for

1 1 1 I 1 1 1 1

SKEWNESS 1 2.85 1 3.38 I 0.70 1 0.44 1 1.61 1 0.53 1 1

CENSUS

MIN

MAX

TOTAL

STD

INCOME

5216.45

INDEX

-364.94

POPDENS !SINGLE

408.35 1 0.01

1891 299.00

302

2491 45.70

KURTOSIS

VEGCOVER

MIN

IMMIPOP

0.01

86.58

302

1 1.81

12.23

VEG2M

0 .O0

1

TOTAL

STD

Table 7: Descriptive summaries for al1 of the variables selected for this study. For INCOME the 94 suppressed EAs are rernoved.

UNIDEG

0.01

14.73

73.43

MEDl A N

MAX

SKEWNESS

KURTOSIS

skewness is important, since most of the traditional statistical techniques,

including Pearson correlations require variables to have normal distributions.

The variables exhibiting high skewness (A) are transformed using the natural

logarithm. To avoid taking the logarithm of zero a small increment is added to

variables with zero entries. In some cases (SINGLE, VEG), the transformed

variables still exhibit a non-normal distribution, due to a large number of EAs

having values close to zero. Other transformations are attempted, however, the

distributioris are still not close to normal, so the variables are left using the log

transformation.

122.23

302

17.66

14.83

75.71

302

13.90

1

19.98

88.89

1.29

2.28

62.51

302

14.66

4.48

VEG2SM

0.00

14.79

VEGSM

0.00

302

15.09

2.00

VEGlûM

0.00

13.68

100.00

194074.01

208

2571 9.72

I 7.03 1 4.25

VEG5ûM 1VEG100M

0.00 1 0.00

1 1 1

1.75

100.00 I

1.81

1 -97

631 -41

302

206.1 1

TMVEG25M

0.00

302

16.69

1.61

1 -44

1.13 i 0.86

1.20 1 0.67

1

AVGNOVl

0.01

0.00

100.00

302

10.48

1 1

302

27.59

1 -85

3.32

1 -38

1.24

302

30.04

0.78

-0.01

2.53

100.00

10.65

47.91

302

29.80

302

20.89

3.2.2.2 Spatial dependence: Moran's I

The notion that many of the socio-economic and biophysical variables are

spatially dependent should not be surprisiny, especially to geographers. For

exarnple, wealthier individuals will most likely live closer to other wealthy

individuais than to poorer. Such dependence on neighbouring values violates

the assumption of independence required for rnodels such as linear regression.

Therefore. it is a good to have a firm idea of the magnitude and the significance

of spatial autocorrelation. Unfortunately, different measures of spatial

autocorrelation yield different values for its magnitude as well as significance.

One of the most established methods to calculate spatial dependence is to

compute Moran's I (Eq. 5) which is similar to the Pearson correlation statistic

with the addition of a neighbourhood matrix, W. Thus, Moran's I and its close

relative Geary's C are very much influenced by the selection of the

neighbourhood matrix. It should be defined as tightly as possible and with the

correct weighting to accurately reflect the relationship between neighbours. The

neighbourhood matrix in this study is defined based on first order neighbours.

Moran's I has mean of -f/(n-1) when there is no association. A value near one

attests to very strong positive autocorrelation whereas a value near negative one

shows very strong negative autocorrelation. For socio-economic variables, it is

normal to observe correlation coefficients of 0.1 5-0.20 (Griffith, 1996). This

coefficient is useful only for summarising autocorrelation and is not suitable for

prediction.

where Wij is the neighbourhood matrix.

The autocorrelation coefficients for al1 of the variables indicate significant

association (Table 8). Proportion of single detached houses (SINGLE) and

TM25M exhibited the highest autocorrelations, 0.701 and 0.435 respectively.

b

Table 8: Moran's I autocorrelation values for selected variables. The neighbourhood matrix is defined based on first order adjacency. All coefficients are highly significant, SINGLE exhibiting the highest coefficient. For the VEG variable, the coefficients decrease but then rise as the resolution is coarsened from 2rn to 1 OOm.

Moran's 1 LNPOPDENS LNSINGLE

INDEX LNVEG2M LNVEGSM LNVEG10M LNVEG25M LNVEG50M LNVEG1 OOM LNTM25M LNAVGNDVI

3.2.2.3 Spatial dependence: Semi-variance

Another way of visualising and determining the covariance of values in

Coefficient 0.295 0.701

0.41 5 0.341 0.299 0.202 0.1 98 0.293 -

0.327 0.435 0.1 75

space is by the plotting of variograms. Empirical variograms display the change

Signif icance I

0.0000 0.0000

0.6600 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 O .O000 0.0000

of semivariance with distance. The theory behind semivariance is based on the

assumption that the variance calculated between two points at a fixed distance

will increase and reach a global limit at a certain range (see Atkinson and Lewis

(2000) for an excellent review). Beyond the range the values are said not to be

dependent on each other. Also, it is assumed that this covariance structure is

constant throughout the study area and does not change with direction.

In order to reduce the effect of outliers, the variogram is estimated using a

robust method (Cressie and Hawkins, 1980) (Eq. 6).

where N(h) is the set of al1 pairwise Euclidean distances i-j = hl (N(h)l is the

number of distinct pairs in N(h), and 4 and zj are data values at spatial locations

i and j, respectively.

The variograrns are well defined for al1 of the variables except for SINGLE

and INDEX, which have linearly increasing semi-variance plots (Fig. 15a-b).

The POPDENS, INCOME, UNIDEG, and IMMlPOP census variables exhibit

ranges (distance at which the plots level) in the 600m to 800m range. These

values are reasonable given the fact that at these distances neighbourhoods

share similar characteristics. The semi-variance plot for VEG2M seems to be

cyclic with 400m cycles. This type of variogram is sometimes observed for

urban areas where the pattern of land use repeats every city block (Gaydos,

1 992).

, 1 r 1 --7

OOOOS OOOOE OOOOL O

OOE 002 001 O

ewurcb

3.2.3 Multivariate statistics

3.2.3.1 Scatterplots

Although scatterplots are visuaf depictions of the relationship between

two variables rather than a quantitative assessment, they are presented in this

section because they are a good first approximation of bivariate reiationships.

The scatterplot matrix exhibits somewhat dense data clouds, however, clear

linear relationships can be discerned between a number of variables (Fig. 16). It

is not surprising that there are linear relationships amongst a number of socio-

economic variables. For example, there is a positive linear relationship between

INCOME, SINGLE, and UNIDEG. One would expect high-income families to

live in single-detached homes and possessing post-secondary education than to

live in apartments and have lower levels of education. There is also a positive

linear relationship between IMMIPOP and INDEX, explained by the fact that

recent immigrants live in areas of lower socio-economic conditions. On the

other hand, negative linear relationships can be observed between UNIDEW

INCOME and IMMIPOP/ INDEX.

In assessing the relationship between the biophysical and socio-

economic variables, clear associations can only be discerned between VEG and

SINGLE, and VEG and INCOME, both of them positive. The positive

relationship between VEG and SINGLE is evident even from the aerial

photograph (Fig. 1 ) where single-detached homes are found in vegetated areas.

A less apparent, negative relationship can be obseived between population

density and vegetation cover. Such a relationship can be expfained by the fact

that areas with high population densities are those which contain apartment

buildings which are found in areas with less vegetation cover than single-

detached houses.

The three dimensional bar graph in figure 17 gives a sense of how

vegetation cover is related to POPDENS and SINGLE in the study area. As

expected the highest percentage of vegetation cover is characteristic of €As with

a high proportion of SINGLE. However, there is a srnaller increase of TM2SM

values further down along the SINGLE axis indicating that there is not

necessarily a ctear linear relationship between the variables.

Figure 17: 3-0 bar graph displaying the relationship between SINGLE, POPDENS, and VEG. The increase in TM25M moderately low proportion of single-detached homes is interesting to observe. It could be a result of EAs with a high percentage of parks and public green-spaces.

1- LNPOPDENS

Figure 16: Scatterplot matrix of selected socio-economic and biophysical variables. VEG seems to be related to INCOME, SINGLE, and LNPOPDENS,

When the 94 EAs with zero entries for INCOME and two more outliers

located in Rosedale are removed from the database, the relationship between

vegetation cover and income is quite clearly Iinear.

Figure 18: Scatterplot of vegetation cover derived from 2m resolution aerial photograph and median family income. All 94 suppressed €As and two outliers are removed. The relationship is clearly linear.

3.2.3.2 Decile plots

An alternative way to plat the relationship between the biophysical and

socio-economic variables is to plot the average of a dependent variable for every

decile of the independent variable. Some extra insight is gained by such

analysis. For example, the negative association between VEG and POPDENS

is only evident from the median (5" decile) (Fig. 19 a-1). It should be noted that

the median for the VEG variable is 14.8; first and third quartiles are 7.0 and 25.4

respectively; the maximum is 75.7. The positive association between VEG and

SINGLE is also most apparent from the median up, and it is very similar to a

quadratic function. Likewise, the positive relationship between VEG and

INCOME is only evident from the 7" decile of the vegetation variable. There is

no clear linear association between VEG and UNIDEG and VEG and IMMIPOP.

Figure 19 (a-9: Decile plots showing averages of selected socio- economic variables for every decile of VEGZM. The linear associations observed on the scatterplots (Fig. 16) are apparent only in the upper half of the distribution for UNIDEG, POPDENS, INCOME, SINGLE, and IMMIPOP.

It is interesting to observe the cyclic relationship between VEG and INDEX.

Lows in the deprivation index are observed at both ends of the VEG distribution.

There is also a high at the 7m decile.

3.2.3.3 Pearson's correlation tables

As indicated by the scatterplots there are significant linear correlations

between a number of the socio-economic variables. The highest significant

negative correlations are obsewed between UNIDEG and INDEX (-0.68) and

SINGLE and POPDENS (-0.62) (Table 9) . These numbers are self-evident as

university graduates are more likely to be found in socially privileged areas than

in socially deprived areas. Also, higher proportions of single-detached houses

c m be found in areas with lower population density. The correlations between

the biophysical and socio-economic variables reflect the following trends.

Vegetation cover (derived from imagery at al1 resolutions) is highly correlated

with proportion of single houses (SINGLE) (-0.40 to -0.63) and population

density (0.32 to 0.68). The highest correlations are observed between

vegetation cover layer derived from the satellite imagery and SINGLE (0.68) and

POPDENS (-0.63). It appears that the strength of the correlation between VEG

(derived from the aerial photographs) and SINGLE/ POPDENS weakens as the

resolution in coarsened from 2m to 25d50m and then increases as the

resolution reaches 1 00m.

The correlations between INCOME and the other variables is presented

for the smaller database where those EAs with zero income and the two outlying

EAs of Rosedale have been removed (Table 10). Again the highest correlation

is between UNIDEG and INDEX (-0.73). Moderately high and highly significant

correlations are obsewed between INCOME and UNIDEG (0.66), SINGLE

(0.51 ), INDEX(-0.5)' VEG2M(0.46), and POPDENS (-0.43). Vegetation cover is

correlated highest with SINGLE (0.61 ), INCOME (0.46). and POPDENS (-0.41 ).

It is fair to conclude that most of the variables can be classified into two

categories. I NCOME, SINGLE, UNIDEG, and VEG show positive correlations

between each other whereas they are negatively associated with fMMlPOP and

I N DEX. Thus, based on these correlations and scatterplots high vegetation

cover appears to be a characteristic of well-to-do areas. Based on similar

results, Lo and Faber (1997) suggest the use of greeness as a desirable quality

and as a important determinant of quality of life. Of course, this does not at al1

imply that low-income people and recent immigrants prefer areas with less

vegetation, however, it is an indication that vegetation is a resource that is not

distributed equally along the urban landscape (Grove 1 996).

1 1 INCOME 1 SINGLE 1 IMMlPOP 1 UNIDEG 1 INDEX2 / VEGPMT 1 LNDENS 1

SINGLE 0.51 1 .O0 -0.21 0.21 -0.03 0.61 -0.50

IMMlf'OP -0.28 4.21 1 .O0 -0.43 0.32 -0.1 2 0.43

UNIDEG 0.66 0.21 4.43 1 .O0 -0.73 0.23 -0.19

INDEX2 4.50 1 -0.03 0.32 4.73 1 .O0 0.00 -0.07 VEG2MT 0.46 0.6 t -0.12 0.23 0.00 1 .O0 -0.41 LNDENS 4.43 4.50 0.43 4.19 -0.07 -0.41 1 .O0

Table 10: Pearson's correlation coefficients for a subset where the suppressed INCOME EAs are removed. Boldfaced values are significant at p=0.001. INCOME is significantly correlated to al1 other variables. VEG is positively correlated to INCOME, SINGLE, and UNIDEG and negatively to POPDENS.

IV

Co-processing of biophysical and socio-economic data using simple

statistical models

There are a large number of public, private, and academic interests,

including city planning offices, health organisations, and universities, who rely on

up-to-date socio-economic data (Jensen, 1996; Treitz et al., 1 992). However, the

high cost of door to door surveys implies that the national censuses, taken every

five years in Canada, are usually the rnost up-to-date sources of socio-economic

data. One way to alleviate this data shortage is to take advantage of up-to-date,

cost-effective, and scalable remotely sensed data and model socio-economic

characteristics based on documented relationships between biophysical and

socio-economic data. Many studies, including this one have, shown significant

relationships between biophysical data and socio-economic characteristics

(Grove, 1996; Ryznar, 1998, Lo and Faber, 1997).

Based on the exploratory work presented in Chapter 3, significant

correlations are present between vegetation cover and proportion of single-

detached homes and between vegetation cover and population densiiy.

Therefore, as a potential application for CO-processing biophysical with socio-

economic data, a conceptual model will be implemented whereby the two socio-

economic variables will be predicted from the vegetation cover variable.

This prediction will be tested with two statistical models, which hold great

potential in the rnodelling of urban phenornena. The first model presented is a

spatial linear model (SLM), followed by a regression tree (RT) model. The SLM

is ideal for modelling variables with a significant amount of spatial dependence

because it explicitly controls for the effect of spatial autocorrelation. As identified

in the exploratory data analysis, there is a significant leve! of spatial

autocorrelation (according to Moran's I statistic) in the variables selected for this

study, justifying the use of this model. However, not al1 of the relationships

plotted in the €DA section show a clear linear dependence and therefore the

SLM might not be appropriate always. Although the RT model is a piecewise

linear model, it is applicable to non-linear but monotonic relationships. These

two models will be compared with a model selecting strategy, which accounts for

model cornplexity in addition to goodness of fit. Finally, sections will follow on

finding the appropriate resolution for such modelling.

4.1 Spatial linear regression model (SLM)

Many mainstream scientific studies fail to account for the spatial

relationship between units of the same variable. However, it is hard to find

phenornena, which are not spatially dependent at some scale. This neglect for

spatial autocorrelation may lead to biased, more optimistic results because of the

extra strength, values near each other spatially impart to the regression (Griffith

and Can, 1996). Griffith (1996) estimates that when a variable exhibits a spatial

autocorrelation value of 0.25, the efficiency of the Ordinary Least Squared

estimator is zero.

Spatial modelling diffen from aspatial modelling by the inclusion of

neighbouring values into the regression equation. Therefore, the response

variable is not only a fundion of the predictors but of the neighbouring predictand

values as well (Fig. 2O).The way these neighbours are defined is a crucial step in

Figure 20: Spatial interaction of neighbouring units taken into consideration in a S M .

63

the modelling exercise (Haining, 1990). Griffith (1 996) suggests a conservative

matrix with as few neighbours as possible with appropriate weights. It is best to

experiment with different matrices. The neighbourhood chosen for this study is a

first order matrix with weights equal to one.

The other important specification in a spatial autoregressive model is the

type of covariance model used. The Conditional autoregressive (CAR)

covariance model is used as opposed to the Sirnultaneous autoregressive (SAR)

mode1 because the residuals from the SAR model are correfated with the

neighbouring data values, resulting in inconsistent least-squares parameter

estimateç, while the CAR model does not have this problem. The CAR model

requires symmetry, which is specified in the construction of the neighbourhood

matrix.

CAR: E = (I-~N)''D$ ;

where p and o are scalar parameters to be estimated, N is a weighted neighbour

matrix, and D is a diagonal matrix used to account for non-homogeneous

variance of the marginal distributions. For the CAR model, N is required to be

symmetrical, meaning that the effect of one neighbour on another should be

equal in magnitude both ways.

4.2 Regression Trees (RT)

Classification and Regression Tree (CART) is a relatively novel statistical

model (Breiman, et al. 1984). It can be very useful because it can take both

categorical and interval data. In the case of categorical data, it is called a

classification tree and in the case of ratio data, it is called a regression tree (RT).

RT rnodels fit a piecewise Iinear model to the data. They have not been widely

employed, even though, in many applications they are superior to linear models,

which can be very biased to non-tinear relationships. Trees take advantage of a

splitting procedure that partitions the data according to some predictor variables

so that the deviance of the response variable is minimised in the different

branches of the tree. The partitioning algorithm chooses only those predictor

variables, which can actually reduce the deviance of the model. Thus, a model

with a large number of variables is input into a tree, the CART acts as a forward

selecting regression. The leaves of a tree correspond to bins for the predictand

variable. That is there are as many unique predicted values as there are leaves.

For example, in a bivariate regression, 6.59 is assigned to the predictand when

the range for the predictor is 4.0 to 14.5, 3.45 when the range is 14.6 to 50.7, and

so on.

To reduce the complexity of the tree, the tree can be pruned back to

nodes beyond which the deviance is only minirnally reduced. The final tree size

in the case of the SINGLE model is 13 leaves (Fig. 21) and for the POPDENS

model seven leaves.

4.3 Residual diagnostics

The maps of the residuals of the models are presented along with the

common diagnostic plots such as histograms and QQ-plots of residual

distributions, and plots of observed values against the predicted values to reveal

the fit of the regression. In the case of the autoregressive mode1 (SLM) the

variation in the response variable is modelled with a linear component, a

neighbour covariance component (termed "signaln(Haining, 1 990, p.258). and a

residual variation component. Therefore, the predicted values are computed as

the fitted values from the linear component plus the signal.

Figure 21: a. This plot shows how the deviance decreases as the size of the tree is increased in a mode1 where SINGLE is predicted from TM25M. b. Pruned tree with 13 leaves. c. Pruned tree with five leaves. d. Pruned tree with two leaves.

CONCEPTUAL MODEL: LNSlNGLE - LNTM25M

The map of residuals from the SLM show a pattern where the €As around

the downtown core have positive residuals whereas the downtown EAs have

negative residuals which are larger in magnitude (Fig.22). Also some of the

highest residuals are observed in EAs with small areas. The qq-plot reveals that

towards the ends of the distribution, as well as in the centre, the fit is poor (Fig.

22). This lack of fit can be explained by the fact that for EAs where SINGLE is

very low, there is no linear relationship between it and VEG. The residuals of the

RT model show a somewhat better fit (Fig. 23). However, there are still a

number of EAs where the model fit is very poor. Comparing the observed and

predicted maps, it is evident that the RT model prediction looks much more like

the original distribution (Fig. 24).

CONCEPTUAL MODEL: LNPOPDENS - LNTM2SM

The SLM fit for the POPDENS model seems to be better than for the

SINGLE model. Again, the largest residuals are found in the downtown core,

however, those residuals outside of the core are mainly negative, as opposed to

positive residuals in the case of the SINGLE model. There is a high

concentration of positive residuals around the Cabbagetown area. These are

most likely apartment EAs with very high population densities. The residuals of

the RT model resernble the residual distribution of the SLM, also exhibiting a high

concentration of positive residuals in the Cabbagetown area. As with the

SINGLE prediction, the RT residuals are smaller in magnitude than those of the

SLM. However, a visual comparison of the observed versus predicted spatial

distributions reveals a closer resemblance by the SLM than by the RT model.

SLM - SINGLE

Quantiles of standard nomal

Figure 22: Residual diagnostic plots for the SLM predicting LNSINGLE from LNTM25M at the EA level. The residuals (both negative and positive) are higher in magnitude in the downtown core.

RT - SINGLE

Quantiies of standard normal

Figure 23: Residual diagnostic plots for the RT mode1 predicting LNSINGLE frorn LNTM25M at the €A levet. The fit is still poor towards the ends of the distribution, however, the residuals are smaller in magnitude in most EAs as compared to the SLM in Fig. 22.

(a.) - RT-Predided

SINGLE

O 1000 a1W) Men

(b.) - SLM - Predicted

Figure 24 (a-b): Predicted maps of LNSINGLE using the RT and SLM models. in cornparison to the observed distribution (Fig. 1 3b), the RT map more closely resembles the original pattern.

Observeci vs. predicted

Figure 25: Residual diagnostic plots for the SLM predicting LNPOPDENS from LNTM25M at the €A level. In comparison to the prediction of LNSINGLE, the fit is much better. Clumping of positive residuals in the Cabbagetown is evident. These are probably EAs with apartments.

- POPDENS

Observed a. predicted

---rm."

Figure 26: Residual diagnostic plots for the RT model predicting LNPOPDENS from LNTM25M at the EA level. The residuals are smaller in magnitude, however, the high concentration of positive residuals in Cabbagetown is still present.

(b.)- SLM - Predcted 900 O 900 1800 Meters -

Figure 27: Predicted maps of LNPOPDENS using the RT and SLM models.

4.4 Which model? - Model selection strategies

4.4.1 Akai ke's l nformation Criterion (AC)

The residual sum of squares has been the traditional way to compare

different statistical rnodels. However, the main limitation of this approach is that

it does not consider the size of the model. So, a model using 15 variables

showing a remarkably good fit may be selected as the better model than a more

restricted model. Thus, as Linhart and Zuchini (1 986) suggest. models should be

parsimonious and only include as many variables as can be reliabfy estimated.

Therefore, the discrepancy between a selected segmentation and the observed

phenomena, should not only be measured by the estimation component

(measuring the goodness of fit between observed and predicted) but the

approximation cornponent (measuring model complexity) (Csillag and Kabos,

1997). A simple way to express this duality of model discrepancy is through the

Akaike Information Criterion (AIC)(Eq.8) (Akaike, 1974). Model selection using

the AIC assures that complex models with too many variables will pay a penalty

for a decrease of the neg-loglikelihood. The equation looks simpls, however, it

has deep mathematical roots which are beyond the scope of this thesis.

AIC = 2 * neg-loglikelihood + 2 ' parameters ; (8)

To compare the results of the SLM and RT in a quantitative way the AIC is

computed using the neg-loglikelihood estimates of the models. The model with

the smallest AIC is deemed the best model. For predicting SINGLE, the SLM is

the better model at al1 resolutions whereas the RT is the better model for

estimating POPDENS at the coaner resolutions of the aerial photographs and

the satellite image derived VEG layers (Table 11). The lowest AIC values are

observed when the vegetation layer is derived from the satellite imagery. There

seems to be a change in AIC values with coarsening resolution, however, this

change is not predictable. In the case of the StM-LNSINGLE model, the AIC

increases and decreases as the resolution is coarsened from 2m to 100m

resolution. This pattern is reversed in the case of the CART-LNPOPDENS

model.

AIC SLM CART Resolution (m) LNSINGLE 1 LNPOPDENS LNSINGLE 1 LNPOPDENS

Table 11: AIC values for various model runs. According to these values, the RT outperforms the SLM in predicting LNPOPDENS, however, the SLM is better at predicting LNSINGLE. THE lowest AIC values are observed when predicting from the TM25M vegetation variable. In this case the RT is superior for both LNSINGLE and LNPOPDENS.

The change in AIC is also assessed as the complexity of the tree is

increased from a basic two branch tree îo the full sized tree. The AIC decreases

much like the deviance plot (Fig. 29), however, not as abruptiy.

Figure 28: AIC as a function of the number of terminal nodes. For best prediction the tree should be pruned at around 10 nodes because beyond this point the AIC is only marginally improved.

4.5 Finding the appropriate resolution

A pressing issue in geography, as well as most other disciplines studying

spatially distributed phenomena, is the question of scale. What happens to a

relationship when it is studied at various levels of aggregation? Is there an

optimal scale at which certain processes operate? Can relationships from one

level be used to explain relationships at another level? These questions fit under

the umbrella of the Modifiable Unit Area Problem (MAUP) which has two

important components.

The first component deals with the scale issue that is the change in

strength and significance of relationships when going from one level of

aggregation (scale) to another. Some of the first researchers to study this

aggregation problem were Gehlke and Biehl (1934) who realised that the

magnitude of the correlation they were studying increased dramatically when

they aggregated their data. Later, Robinson (1 954) showed mathematically what

can happen when correlations are computed at different levels of aggregation.

Since then, many papers in geography as well as fields such as epidemiology

have addressed this issue (Fotheringham and Wong, 1991 ;Openshaw, 1984;

Amrhein, 1995). The aggregation problem is also analogous to the term labelled

ecological bias. If one transplants a determined relationship from one scale to

another scale an ecological fallacy is most likely to occur since in most cases the

joint distributions of variables changes when going from one scale to the other.

The severity of this ecological fallacy depends on the methods of analysis

(Openshaw, 1984) and it could be entirely an artefact of violated assumptions

and misspecified models (Amrhein, 1995). Nonetheless. it is very tempting to

draw individual level associations from group level research. Under some

circumstances, it might be permissible. For instance, when the groups are small

enough, with homogeneous populations, the ecological bias might not be that

high. Ecological bias is severe if the variance of the underlying population is

high, then it is obvious that there is little chance that an association determined at

the group level can be attributed to an individual. Ecological bias might be

somewhat balanced by 'regression-dilution' bias (random errors in measuring the

modelled variables) which may be greater in individuals than for populations

(English, 1998). Due to the heterogeneous nature of the study area in terms of

its human population, the ecological bias is probably severe. Therefore,

individual level associations should not be inferred from this research.

The second aspect of MAUP research is the aggregation effect. This

effect is manifest when partitioned data at one scale are repartitioned using a

different set of regions. Openshaw and Taylor (1 979) has shown that it is

possible to obtain a wide range of correlations from the same dataset, by

partitioning the data differently. Unfortunately, the partitions of socio-economic

investigations are firmly set by the data collecting agencies who set boundaries

that are in most cases arbitrary and do not follow any ecological or socio-

economic boundaries (Openshaw, 1 984; Bracken, 1 994).

4.5.1 Census Tract Level

The exploration of the relationship between socio-economic and

biophysical characteristics has been so far carried out at the €A level of census

geography. This is the lowest level of census collection and with the smallest

fevel of ecological bias of individual relationships. However, for some lines of

research group level analysis is more appropriate (Logan and Molotch, 1 987).

For this reason, it might be useful to assess the relationship between socio-

economic and biophysical characteristics at another scale. Logically, the next

scale above the EA level would be at the census tract (CT) level. These two

levels occupy the bottom of a nested hierarchy of collection areas.

Many researchers have observed that as data are aggregated, the

strength of the association between variables increases (Clark and Avery, 1976).

This is also the case for the variables under study. The scatterplots reveal linear

associations between al1 variables except for IMMl and VEG (Fig. 29). The

Pearson correlation coefficients increase with aggregation (Table 11). This

stronger association is perhaps due to the fewer degrees of freedom.

Table 1 1 : Pearson's correlation coefficients for selected socio-economic and biophysical variables at the CT level. Boldfaced numbers indicate significance at p=0.001. The coefficients are higher for most variables than at the EA level.

-0.38 0.78

LNINCOME LNVEG2M

The results of the modelling indicate that despite stronger linear

-0-63 0.54

L

relationships, there are still a few very high residuals in some of the CTs. A

better fit is obsewed with the RT model than the SLM rnodel (Fig. 30-33).

LNPOPDENS LNSINGLE

-0.63 -0.38

Figure 29: Scatterplot matrix for selected variables at the CT level. Linear relationships are apparent between VEG and INCOME, VEG and SCHOOL (UNIDEG), VEG and SINGLE, and VEG and POPDENS. There are also other linear relationships between the socio-economic variables.

1 -00 -0.43

0.54 0.78

-0.43 1 -00

-0.65 -0.08

0.69 -0.35

-0.57 0.29

0.48 1 .O0

1

0.88 0.1 5

1.00 0.48

Quantiles of standard noma[

Figure 30: Residual diagnostic plots for the SLM predicting LNSINGLE from LNVEGZM at the CT level. The residuals are skewed to the left indicating a few very high negative residuals, which can be seen on the map.

CART - SINGLE

fi Observed vs. predicted

Quantiles of standard normal

Figure 31 : Residual diagnostic plots for the RT model predicting LNSINGLE from LNVEGZM at the CT level. Residuals are skewed to the left and they are higher near the lake.

(a .) - SLM - Pl

SINGLE

900 O 900 1800 Wtew - (b.) - CART - PREOlCTED

Figure 32: Predicted maps of LNSINGLE using the RT and SLM models at the CT level. (see fig. 33 for cornparison).

Figure 33: Observed map of LNSINGLE at the CT ievel.

4.5.2 Non-spatial scaling of socio-economic variables to a regular grid

As pointed out at the beginning of section 4.5 one of the big limitations of

census related research is the lack of data at scales finer than the €A level. A

possible solution to this lack of scalability of socio-economic data is the use of

biophysical data for scaling. It is evident from Chapter Two that biophysical data

obtained from remotely sensed images are readily scalable (from 0.5m to 1 km

resolutions). One possible way to interpolate socio-economic data to a grid is to

use the relationship determined at the €A level and predict the socio-economic

variable at a scale where there is biophysical data available. The results of such

a prediction are shown in Figure 34 (a-d). A tree model is specified at the EA

level for SINGLE and VEG. This tree is then extrapolated to a 100 m grid for

which VEG is available. The effect of the size of the tree is also assessed. The

results of the gridded layers can be compared to the EA level map layer (Fig. 35).

Of course this kind of interpolation is subject to ecologic bias. The

relationship determined at the EA level rnight not be appropriate at the 100m

level.

b. 2 nodes a. 5 nodes

d. 21 nodes

Figure 34: SINGLE scaled to a 100m grid using a tree regression determined at the EA level and then extrapolated to a 1 OOm grid for which VEG is catculated. Each of the images displayed here are created with different sizes of trees. The lighter pink and blue indicate high values of SINGLE.

Allvar3-region .sh p 0.01 0.01 - 4.66 4.66 - 10.88 10.88 - 20.46 20.46 - 86.58 Suppressed

Figure 35: Observed SINGLE for the scaled area at the EA level. As opposed to the grids in fig. 34, dark colours indicate high SINGLE values.

Conclusion

This final chapter will offer a review of the findings and discuss the

potential for the CO-processing of biophysical and socio-economic data in an

urban context. The integration of the social and physical sciences is at the

forefront of many research initiatives in North America. 60th sides have a lot to

offer to the other and many studies have already taken advantage of the wealth

of data and techniques offered by interdisciplinary research . The nature of this study is also interdisciplinary, utilising the tools of

geographic information systems. remote sensing, and statistics. It assesses the

feasibility of deriving land-cover characteristics from scalable remotely sensed

data and explores their relationships with some socio-economic characteristics.

There is indication in the literature that biophysical variables such as vegetation

cover have significant associations with quality of life indicators such as income,

household characteristics, and education status (Grove, 1996; Lo and Faber,

1997). While the nature of these relationships Is in question (causality versus

spu rious correlations), the advantage of frequently obtained, cost-effective, fine

resolution data derived from remote sensing to characterise the biophysical

environment is worthwhile to be investigated.

The area selected in this study is highly heterogeneous in terms of the

social environment. Some of the richest and poorest people of Canada inhabit

th is area, creating very sharp boundaries between the different socio-economic

g roups. This study explores the relationships between socio-economic and

biophysical variables for this very complex area.

Vegetation cover is derived from both aerial photography and satellite

imagery at various resolutions using supervised and unsupervised classification.

For the finer resolution aerial photography, the main obstacle in accurate

classification is the presence of shadows, which are spectrally very similar to

vegetation. At coarser resolutions, the mixed pixel effect is the limiting factor in

accurate classification. Despite its coaner resolution, the accuracy of the

vegetation cover layer derived from the satellite imagery (-75%) is comparable to

that of the accuracy of the vegetation layer derived from the finer resolution aerial

photography (-80%). The implication of this result is that much time and effort

can be saved by utilising satellite images due to ease of processing as opposed

to aerial photographs. The vegetation cover layen are al1 aggregated to the EA

level where each EA is assigned a value of percent cover.

The exploration of the relationship between vegetation amount and

selected socio-economic variables such as population density, proportion of

u niversity graduates, etc., reveals that vegetation amount is significantly

associated with population density, median farnily income, and proportion of

single-detached homes (see Fig. 19 (b,c,d) for the clear monotone relationships

exhibited on the decile plots). For these monotone relationships, quite high

Pearson's correlation coefficients are observed (-0.41 between vegetation cover

and population density; 0.46 between vegetation cover and median family

income; 0.61 between vegetation cover and proportion of single homes),

indicatirig tha'r the relationships are perhaps linear.

These findings are theri the basis of prediction of two of the socio-

eco nom ic characteristics. Two statistical models, a spatial linear model and a

tree regression model, are implemented. The spatial linear model is chosen

because it explicitly deals with the effect of spatial autocorrelation and the tree

regression model is selected because it is well suited for non-linear monotone

relationships. The residual diagnostics plots indicate a better fit of the regression

tree model results, however, quantitative assessment, using the Akaike

Information Criterion, indicates that the spatial linear model is in sorne instances

superior. The prediction is repeated at the census tract level with slightly better

results. The non-spatial interpolation of one of the socio-economic variables to a

lOOm resolution grid using tree regression serves as a potential solution to the

scaling of census data.

Some of the conclusions of this study are the following. First, it

demonstrates the feasibility of deriving vegetation amount and pattern from fine

resolution imagery. This result alone can aid in specific studies concerning the

distribution of urban vegetation. Rowntree and Nowak (1991) point out that

inventories of vegetation (trees) located on both private and public land are

lacking in most North American cities. Yet, the accurate knowledge of the

distribution of vegetation can be important for studies of carbon sequestration, for

models of hydrology, microclimates (trees are excellent at reducing the heat

islands created by buildings (Oke and Roth, 1989)) and air quality. Furthermore,

accurate knowledge of vegetation cover, as well as other urban land uses and

land covers, is much in need for city planners and those modelling the quality of

life. In fact, Lo and Faber (1997) advocate the use of biophysical variables as

measures of quality of Iife since these data provide the rneans of better relating

environmental quality to social quality. Ryznar (1998) goes as far as to Say that

"it may be possible to eliminate the use of census data altogether and rely on the

satellite images to detenine neighbourhood statusn. Of course, relying on only

environmental data may be too premature, as the exact nature of the

relationships is still unclear.

Second, there are clear monotone relationships between vegetation cover

and a number of socio-economic variables including population density and

income. This finding reinforces the works of Grove (1 996), Lo and Faber (1 997).

and Weber and Hirsch (1992) who have also found such relationships. The

exact nature of the relationships is not clear, however they are al1 monotonic.

The resultant relationships are most likely vety sensitive to scale as the census

tract level analysis indicates.

The third major conclusion is that vegetation cover derived from rernotely

sensed images and then aggregated to the EA level can be used to model

scarce socio-economic data. Although the implementation of the statistical

models is rather simple, it serves as a good template for further prediction. With

more accurate characterisation of the relationships, such predictions could be

used to update databases of socio-economic variables and allow for the scaling

of socio-economic data.

Further research must ascertain whether more precise relationships can

be found between socio-economic and biophysical variables. One potential way

of better characterising the relationship between biophysical and socio-economic

variables is to partition the study area into homogenous units in which there are

more definite linear or exponential associations present. lnstead of finding al1

encompassing relationships, many, locally specific relationships should be

sought. For example, the distinct regions identified in socio-economic feature

space on the 3-D plot in figure 17 could be used as a basis for partitioning the

urban landscape (Fig. 36).

egionl egion2 est

Figure 36: The spatial distribution of high vegetation amount area in socio-economic feature space located in figure 17. Region 1 corresponds to the smaller distinct region in fig. 17 and Region 2 corresponds to the larger one. The partitioning of feature space based on such three dimensional plotting technique could be a possible way of finding areas where there are clear relationships between biophysical and socio- economic variables.

The goal is to identify regions in feature space, which correspond to

geographically contiguous areas (Region 2 could be identified as such a region).

As urban ecosystems become the primary focus of both social and

physical scientists, the repeated assessment of biophysical environment and

understanding its relationship with complex socio-economic variables will be

essential for the successful management cf urban areas.

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Appendix 1 : Statistical analysis using S-PLUS and S-PLUS-SpatialStats

Expioratory Data Analysis (EDA)

Classification and Regression Trees (CART)

#pick the lowest deviance ea.p.tree<-prune.tree(ea.tree1k=708) ea,tree.predict<-predict(ea.p.tree) #count number of leaves (=LEAF)

eatrsec-tree(S ING LE-VEG2M) summary(ea.p.tree) ea.p.tree AIC1 summary(ea.tree.predict)

par(mfrow=c(3,2)) plot(ea.p.tree) text(ea.p.tree) title("S1NGLE-VEG2M. k=708") plot(prune.tree(ea.tree)) title("Reduction of Deviance") image(interp(X,Y,predict(ea.p.tree))) title("CART prediction for SINGLE") ea.tree.residc-(SINGLE-ea.tree.predict) image(interp(X,Y,ea.tree.resid)) title("CART residuals for SINGLE") hist(ea.tree.resid) title("Histogram of CART residualsu) qqnorm(ea.tree.resid) qqline(ea.tree.resid) title("QC1-plot of CART residuals")

Spatial Linear Regression (SLM)

Xbuild neighborhood matrix ea.place<-cbind(X,Y) ea-quadc-quad.tree(eâ.place) ea.nhbr~-find.neighbor(x=ea.place,quadtree=ea.quad,m~.dist=400) ea.n hbrc-ea,nhbr[ea,nhbr[,3]!=0,] ea-snh brc-spatial.neighbor(row.id=ea.nhbr[l ],col.id=ea.nhbr[,2],symmetric=TRUE)

#get rid of extra pairs from neigborhood matrix created by Arcview ea2.adjust.snhbr~-spatial.condense(ea.adjust.snhbrlsymmet~=T)

#Check for islands check.islands(ea.snhbr,remap=F)

Spatial linear model ea.slmc-slm(log(SINGLE)- log(VEG2M),cov.family=CAR,spatial.arglist=list(neighbor=ea.snhbr))

summary(ea.slm) Irt(ea.slm ,pararneters=O) AIC

XVisuals for slm par(mfrow=c(3,2)) image(interp(X,Y,ea.pred)) titie("SLM prediction for log(SlNGLE)") image(interp(X,Y,ea.slm$residuals)) title("SLM residuals for log(S1NGLE)") hist(ea.slmSresiduals) title("Histogram of SLM residuals") qqnorm(ea.slmSresiduals) qqline(ea.slmSresiduals) title("QQ-plot of SLM residuals") plot(ea.slmSfitted,ea.slmSresiduals) abline(h=O) title("Residuals vs. fitted values") ea.signal.slrn c-log(S1NGLE)-ea.slm$fitted-ea.slm$residuals plot(log(SINGLE),fitted(ea.slm)+ea.signal.slm) abline(0.1)

#Histogram with various options hist(SINGLE,nclass=l O,xlab=aSINGLE",main=wEAw,xlirn~(O,30)lylim=c(O,1 15))

#Makes 30 perspective piot persp(interp(X,Y,LAKE),zlab="SlNG LE",eye=c(632000,4830000,5000))

#Partitions data geographically par(mfrow=c(2,2)) eda.treec-tree(S1NGLE-X+Ylmindev=.OO1 ) plot(prune.tree(eda.tree)) eda2.treec-prune. tree(eda.tree, best=l O) plot(eda2.tree,type="uU) text(eda2.tree.srt=90) par(pty="s0 j partition.tree(eda2.tree) points(X,Y)

#To select outliers plot(fitted(ea.lm),residuaIs(ea.lm)) outIiers~-identify(fitted(ea.lm),residuals~ea.lm),labels=EA.ID) outliers ea.fit2.lmc-lm(Y-X,subset=-outliers)