Download Spatial Statistics and Spatial Knowledge Discovery

Spatial Databases
First law of geography [Tobler]: Everything is related to everything, but
nearby things are more related than distant things.
Lecture 8 : Spatial Statistics 1
Autocorrelation & GWR
Pat Browne
Spatial autocorrelation
• Spatial autocorrelation is the degree of correlation
between neighbouring values.Spatial autocorrelation
is detected when the value of a variable in a
location is correlated with values of the same
variable in the neighbourhood (can be measured
with Moran I).
• Moran’s I measures the average correlation
between the value of a variable at one location and
the value at nearby locations. The essential idea is
to specify pairs of locations that influence each other
along with the relative intensity of interaction.
Moran’s I provides a global view of spatial
autocorrelation correlation.
Moran’s I
• The range of the Moran's I statistic
depends on the spatial weight matrix.
• When Moran's I is scaled by its bounds the
statistic is restricted to the range ±1
• Moran’s I can serve as a tool for modeling
spatial dependencies in many data mining
techniques.
Same Mean and SD but different
Moran’s I
Same Mean and SD but different
Moran’s I
Spatial Autocorrelation: Moran’s I example
Moran’s I - example
Figure 7.5, pp. 190
•Pixel value set in (b) and (c ) are same but their Moran Is are different.
•Q? Which dataset between (b) and (c ) has higher spatial autocorrelation?
Neigbourhood relationship
contiguity matrix
Spatial autocorrelation
Negative
Dispersed
Spatial
Independence
Spatial Clustering
Positive
Moran’s I
• Global Moran’s I
• What is the extent of clustering in the total area?
• Is this clustering significantly different from a
random spatial distribution?
• Local Moran’s I
• Do local clusters (high-high or low-low) or local
spatial outliers (high-low or low-high) exist?
• Are these local clusters and spatial outliers
statistically significant?
Moran Scatter Plot
Scatter Diagram between X and Lag-X, the “spatial lag” of X
formed by averaging all the values of X for the neighboring
polygons
Identifies which type of spatial autocorrelation exists.
Low/High
negative SA
Low/Low
positive SA
High/High
positive SA
High/Low
negative SA
Briggs Henan University 2010
11
Spatial autocorrelation
• Spatial autocorrelation is determined both by
similarities in position, and by similarities in
attributes
– Sampling interval
– Self-similarity
• Auto = self
• Correlation = degree of relatedness
correspondence
Moran’s I index
Statistical Spatial Data
• In this lecture we consider spatial data
contains an attribute e.g. house prices,
occurrences of disease, occurrences of
accidents, crop yield, poverty patterns,
crime rates, etc. Earlier parts of the course
covered the representation of physical
objects such as houses, counties, and
roads. These objects were arranged by
theme. Here we consider attributes of
those objects e.g. the population of an ED.
Spatial Statistics
• Spatial statistics is the statistical study of spatial
data that varies over discrete space e.g. crime
rates broken down by neighbourhood. Spatial
statistical models can be used for estimation,
description, and prediction based on probability
theory (not covered).
Standard statistical concepts: i.i.d
• A collection of two or more random
variables {X1, X2, … , } is independent
and identically distributed if the variables
have the same probability distribution, and
are independent.
Standard statistical concepts: Examples
• Example i.i.d: All other things being equal, a
sequence of dice rolls is i.i.d.
• Example of non i.i.d: bird nesting patterns in
wetlands, where the independent variables are
distance from water, length of grass, depth of
water and the dependent variable would be the
presence of a nest site. A uniform distribution of
these variables on a map would indicate an
even distribution, however a more complex
emerges where the variables are spatially
dependent.
Standard statistical concepts: Correlation
• Correlation: A correlation is a single number that
describes the degree of relationship between two
normally distributed variables. The variables are not
designated as dependent or independent. The value of a
correlation coefficient can vary from minus one to plus
one. A minus one indicates a perfect negative
correlation, while a plus one indicates a perfect positive
correlation. A correlation of zero means there is no
relationship between the two variables. When there is a
negative correlation between two variables, as the value
of one variable increases, the value of the other variable
decreases, and vice versa.
Standard statistical concepts: Variance and
covariance
• A measure of variation equal to the mean of the
squared deviations from the mean. The variance
is a measure of the amount of variation within
the values of that variable, taking account of all
possible values and their probabilities or
weightings.
• Covariance is measure of the variation between
variables, say X and Y. The range of covariance
values is unrestricted. However, if the X and Y
variables are first standardized, then covariance
is the same as correlation and the range of
covariance (correlation) values is from –1 to +1.
Standard statistical concepts: Correlation
• Correlation is a measure of the degree of linear
relationship between two variables, say X and Y. While
in regression the emphasis is on predicting one variable
from the other, in correlation the emphasis is on the
degree to which a linear model may describe the
relationship between two variables. In regression the
interest is directional, one variable is predicted and the
other is the predictor; in correlation the interest is nondirectional, the relationship is the critical aspect. The
correlation coefficient may take on any value between
plus and minus one (-1 < r < 1).
Standard statistical concepts: Regression
• Regression: takes a numerical dataset
and develops a mathematical formula that
fits the data. The results can be used to
predict future behaviour. Works well with
continuous quantitative data like weight,
speed or age. Not good for categorical
data where order is not significant, like
colour, name, gender, nest/no nest.
Example: plotting snowfall against height
above sea level.
Standard statistical concepts:
Regression
Y = A + BX; The response variable is y, and x is the
continuous explanatory variable. Parameter A is the
intercept. Parameter B is the slope. The difference
between each data point and the value predicted by
the line (the model) us called a residual
Standard statistical concepts: Null
hypothesis
• The null hypothesis, H0, represents a theory that has
been put forward, either because it is believed to be true,
but has not been proved. For example, in a clinical trial
of a new drug, the null hypothesis might be that the new
drug is no better, on average, than the current drug H0:
there is no difference between the two drugs on average.
• In general, the null hypothesis for spatial data is that
either the features themselves or of the values
associated with those features are randomly distributed
(e.g. no spatial pattern or bias).
Relation of i.i.d., regression, and correlation with
spatial phenomena.
• The first law of geography according to Waldo Tobler is
"Everything is related to everything else, but near things
are more related than distant things." In statistical terms
this is called autocorrelation where the traditional i.i.d.
assumption is not valid for spatially dependent variables
(e.g. temperature or crime rate) we need special
techniques to handle this type of data (e.g. Moran’s I).
These techniques usually involve including a weight
matrix which contains location information. The non-i.i.d.
nature of spatially dependent variables carries over into
regression and correlation which require spatial weights
Relation of i.i.d., regression, and
correlation with spatial database
• Spatial databases are used for spatial data mining,
which includes statistical techniques and more
specialised DM techniques such as association rules.. In
this case the data mining algorithms need to have a
spatial context. We must explicitly include location
information where previously with the i.i.d. assumption it
was not required Typical generic data mining activities
such as clustering, regression, classification, association
rules, all need a spatial context. Spatial DM is used in a
broad range scientific disciplines, such as analysis of
crime, modelling land prices, poverty mapping,
epidemiology, air pollution and health, natural and
environmental sciences, etc. The analyst must be aware
the special techniques required for SDM.
Relation of i.i.d., regression, and
correlation with spatial database
• Spatial databases are also used for pure
statistical research (e.g. environmental
studies). Those variables that are spatially
dependent (e.g. the PH of the soil) need to
be clearly identified and special
techniques applied to take into account
their spatial bias.
Unique features of spatial data Statistics
• General Statistics assumes the samples
are independently generated, which is
may not the case with spatial dependent
data, where:
– Like things tend to cluster together.
– Change tends to be gradual over space.
Spatial Autocorrelation1.
• Autocorrelation: degree of correlation between
neighbouring values.
• Spatial dependency: neighbouring values are
similar (i.e. positive spatial autocorrelation).
• Moran’s I enable assessment of the degree to
which values tend to be similar to neighbouring
values. We can observe how autocorrelation
varies with distance.
• The Moran scatter plot relates individual values
to weighted averages of neighbouring values.
The slope of a regression line fitted to the points
in the scatter plot gives the global Moran’s I.
Spatial Autocorrelation: Moran’s I
• Moran’s I measures the average correlation between
the value of a variable at one location and the value at
nearby locations. The essential idea is to specify pairs of
locations that influence each other along with the relative
intensity of interaction. Moran’s I provides a global view
of spatial autocorrelation correlation. We will look at
details later
• The range of the Moran's I statistic depends on the
spatial weight matrix.
• When Moran's I is scaled by its bounds the statistic is
restricted to the range ±1
• Moran’s I can serve as a tool for modeling spatial
dependencies in many data mining techniques.
Spatial Autocorrelation: Case Study
Nest locations
Distance to open water Vegetation durability
Water depth
Spatial Autocorrelation
Classical Statistical Assumptions
(i.i.d) do not hold for spatially
dependent data
Unique features of spatial data Statistics
First Law of Geography
• First law of geography [Tobler]:
– Everything is related to everything, but nearby
things are more related than distant things.
– People with similar backgrounds tend to live
in the same area
– Economies of nearby regions tend to be
similar
– Changes in temperature occur gradually over
space (and time) (equator V poles).
Spatial Autocorrelation: Moran’s I example
Moran’s I - example
Figure 7.5, pp. 190
•Pixel value set in (b) and (c ) are same but their Moran Is are different.
•Q? Which dataset between (b) and (c ) has higher spatial autocorrelation?
Moran’s I - example
• Moran I statistic for map 1 is 0.55316092
• Moran I statistic for map 2 is -0.76724138
Moran’s I - example
Spatial Autocorrelation : Moran
Scatterplot Map
São Paulo
WZ
Q4 = LH
Q1= HH
a
0
Q2= LL
Q3 = HL
0
z
Old-aged population
Spatial Heterogeneity.
• Spatial heterogeneity; Is there such a thing as an
average place with respect to some property (e.g.
vegetation). is difficult to imagine any subset of the
Earth’s surface being a representative sample of the
whole. GWR (later) addresses the localness of
spatial data.
Neigbourhood relationship
contiguity matrix
Spatial autocorrelation
• Spatial autocorrelation is determined both by
similarities in position, and by similarities in
attributes
– Sampling interval
– Self-similarity
• Auto = self
• Correlation = degree of relatedness
correspondence
Spatial autocorrelation
• In the following slide, each diagram contains 32
white cell and 32 blue cells = 64 cells.
• BB = Blue beside Blue
• BW = Blue beside White
• WW = White beside White.
Spatial autocorrelation
Negative
Dispersed
Spatial
Independence
Spatial Clustering
Positive
Spatial regression (SR)
• Spatial regression (SR) is a global spatial modeling
technique in which spatial autocorrelation among the
regression parameters are taken into account. SR is
usually performed for spatial data obtained from spatial
zones or areas. The basic aim in SR modeling is to
establish the relationship between a dependent variable
measured over a spatial zone and other attributes of the
spatial zone, for a given study area, where the spatial
zones are the subset of the study area. While SR is
known to be a modeling method in spatial data analysis
literature in spatial data-mining literature it is considered
to be a classification technique
Spatial autocorrelation
Negative
Dispersed
Spatial
The grids A and B represent two
different spatial resolutions over the same area.
Independence
Grid A contains 16 cells and Grid B contains 64 cells.
The strength of spatial autocorrelation is often a function of scale or spatial
resolution, as illustrated in above using black and white cells. High negative
spatial autocorrelation is exhibited in A since each cell has a different colour from
Positive
itsSpatial
neighbouring
Clustering
cells. In B each cell can be subdivided into four half-size cells,
assuming the cell’s homogeneity. Then, the strength of spatial autocorrelation
among the black and white cells increases, while maintaining
the same cell arrangement. his illustrates that spatial autocorrelation varies with
the study scale The strength of spatial autocorrelation is a function of scale,
increasing from 4-by-4 case to the 8-by-8 case.
Summary of spatial stats
• Moran’s I measures the average correlation between
the value of a variable at one location and the value at
nearby locations.
• Local Moran statistic measures spatial dependence on a
local basis, allowing the researcher to see its variation
over space, and by Geographically
• Geographically Weighted Regression allows the
parameters of a regression analysis to vary spatially.
GWR helps in detecting local variations in spatial
behavior and understanding local details, which may be
masked by global regression models. GWR, regression
coefficients are computed for every spatial zone.
Moran’s I
• A contiguity matrix may represent a
neighborhood relationship defined using
adjacency or Euclidean distance. There are
several definitions adjacency include a fourneighbourhood or an eight-neighborhood. Given
a gridded spatial framework, a fourneighborhood assumes that a pair of locations
influence each other if they share an edge
(rook). An eight-neighborhood assumes that a
pair of locations influence each other if they
share either an edge or a vertex (queen).
Moran’s I
• Using a normalised weight matrix the
values of I range from -1 to 1.
• Value = 1 : Perfect positive correlation
• Value = 0 : No autocorrelation
• Value = -1: Perfect negative correlation
• A Moran’s I may appear low (say 0.17) but
is statistically significant pattern is
clustered since index is above 0.
Moran’s I
• Global Moran’s I
• What is the extent of clustering in the total area?
• Is this clustering significantly different from a
random spatial distribution?
• Local Moran’s I
• Do local clusters (high-high or low-low) or local
spatial outliers (high-low or low-high) exist?
• Are these local clusters and spatial outliers
statistically significant?
Moran’s I: A measure of spatial
autocorrelation
• Given x  x1,...xn  sampled over n locations.
t
zWz
Moran I is defined as I 
zz t
Where




z   x1  x ,...,xn  x 


and W is a normalized contiguity matrix.
Fig. 7.5, pp. 190
Spatial autocorrelation
Negative
Dispersed
Spatial
The grids A and B represent two
different spatial resolutions over the same area.
Independence
Grid A contains 16 cells and Grid B contains 64 cells.
The strength of spatial autocorrelation is often a function of scale or spatial
resolution, as illustrated in above using black and white cells. High negative
spatial autocorrelation is exhibited in A since each cell has a different colour from
Positive
itsSpatial
neighbouring
Clustering
cells. In B each cell can be subdivided into four half-size cells,
assuming the cell’s homogeneity. Then, the strength of spatial autocorrelation
among the black and white cells increases, while maintaining
the same cell arrangement. his illustrates that spatial autocorrelation varies with
the study scale The strength of spatial autocorrelation is a function of scale,
increasing from 4-by-4 case to the 8-by-8 case.
How to decide the weight wij ?
The weight indicates the spatial interaction between entities.
1) Binary wij, also called absolute adjacency. Covers the general
case answering the question is a value in a region similar or
different to its neighbours.
wij = 1 if two geographic entities are adjacent; otherwise, wij = 0.
Choice of adjacency definition queens(8) or rooks(4).
How to decide the weight wij ?
The weight indicates the spatial interaction between entities.
2) The distance between geographic entities. Often the inverse
distance is used, further objects get less weight, near object get
more weight e.g. centre of epidemic.
wij = f(dist(i,j)), dist(i,j) is the distance between i and j.
3) The length of common boundary for area entities. Policing
borders, smaller borders less weight.
wij = f(leng(i,j)), leng(i,j) is the length of common boundary
between i and j.
How to decide the weight wij ?1
The choice of weights should ultimately be driven by a rationale for including
those areas as neighbors that have a spatial effect on a given location. This
rationale can be derived from theory or be the result of using ESDA to
experiment with different weights and connectivity orders. Since weights
matrices are used to create spatial lags that average neighboring values, the
choice of a weights matrix will determine which neighboring values will be
averaged. For instance, since rook weights will usually have fewer neighbors
than queen weights, on average, each neighboring observation has more
influence.
How to decide the weight wij ?
1
The question of which weights to choose is more pertinent in the context of
modeling than ESDA since modeling is based on substantive notions of
spatial effects while ESDA prioritizes the rejection of spatial randomness.
Therefore, if there are no substantive reasons to guide the choice of weights
in ESDA, using a weights file with as few neighbors as possible (such as
rook) makes sense. Especially with irregular areal units (as opposed to
grids), the difference between rook and queen weights is often minimal.
However, it is advisable to test how sensitive your results are to your weights
specifications by comparing multiple weights matrices.
Spatial Outlier Detection
• Global outliers are observations which
appear inconsistent with the remainder of
that data set.
• Global outliers deviate so much from other
observations that it may be possible that
they were generated by a different
mechanism.
• Spatial outliers are observations that
appear inconsistent with their neighbours.
Spatial Outlier Detection
• Detecting spatial outliers has important
applications in transportation, ecology,
public safety, public health, climatology
and location based services.
• Geographic objects have a spatial
(location, shape, metric & topological
properties) & non-spatial component
(house owner, sensor id., soil type).
Spatial Outlier Detection
• Spatial neighbourhoods may be defined using
spatial attributes & spatial relations.
• Comparisons between spatially referenced
objects can be based on non-spatial attributes.
• A spatial outlier is a spatially referenced object
whose non-spatial attribute values differ from
those of other spatially referenced objects in its
spatial neighbourhood.
Data for Outlier detection
In diagram on left G,P,S,Q show a big change in attribute for a small change in
location. The right hand diagram shows a normal distribution (corresponds to
attribute axis in left diagram)
Spatial Outlier Detection
• The upper left & lower
right quadrants of
figure 7.17 indicate a
spatial association of
dissimilar values; low
values surrounded by
high value neighbours
(P & Q) and high
values surrounded by
low values (S).
Spatial Outlier Detection
• Moranoutlier is a
point located in the
upper left or lower
right quadrant of a
Moran scatter plot.
Spatial Outlier Detection
• Moranoutlier is a
point located in the
upper left or lower
right quadrant of a
Moran scatter plot.
WZ
Q4 = LH
Db
0
Q2= LL
Q1= HH
Cb
a
Q3 = HL
z
0
values in a given location
Model Evaluation
• Consider the two-class classification problem
‘nest’ or ‘no-nest’. The four possible outcomes
(or predictions) are shown on the next slide. The
desired predictions are:
– 1) where the model says the should be a nest and
there is an actual nest (True Positive)
– 2) where the model says there is no nest and there is
no nest (True Negative)
• The other outcomes are not desirable and point
to a flaw in the model.
Model Evaluation
Spatial Statistical Models
• A Point Process is a model for the spatial
distribution of points in a point pattern.
Examples: the position of trees in a forest,
location of petrol stations in a city.
• Actual real world point patterns can be
compared (using distance) with a
randomly distributed point pattern random.
Case Study
Nest locations
Water depth
Distance to open water
Vegetation durability
Example showing different predictions: (a) the actual locations of nests; (b) pixels with actual nests;
(c) locations predicted by one model; and (d) locations predicted by another model. Prediction (d) is
spatially more accurate than (c).
Classical statistical assumptions do
not hold for spatially dependent
data
Case Study
• The previous maps illustrate two important
features of spatial data:
• Spatial Autocorrelation (not independent)
• Spatial data is not identically distributed.
• Two random variables are identically
distributed if and only if they have the
same probability distribution.
Geographically weighted
regression (GWR)
• GWR is an effective technique for exploring
spatial non-stationarity, which is characterized
by changes in relationships across the study
region leading to varying relations between
dependent and independent variables. Hence
there is a need for better understanding of the
spatial processes has emerged local modeling
techniques. GWR has been implemented in
various disciplines such as the natural,
environmental, social and earth sciences.
Spatial Regression1
• The assumption of i.i.d. underlying
ordinary least squares regression rarely
holds for spatial data. There are several
techniques that handle the spatial case;
– Moving window regression
– Geographic Weighted Regression (GWR)
• We will look at GWR
Geographic Weighted Regression (GWR) 1
•
The steps are;
1. Go to a location
2. Conduct regression using the raw data and
a geographic weighting scheme.
3. Move to next location go back to stage 2
until all locations have been visited.
•
The output is a set of regression
coefficients (e.g. slope and intercept) at
each location
Coords of observations, variables. distance from first
observation, and geographic weights
point
x
y
Var 1 Var 2 dist
Geo w
1
25
45
12
6
0
1
2
25
44
34
52
1
0.995
3
21
48
32
41
5
0.8825
4
27
52
12
25
8
0.7261
5
16
31
11
22
16
0.278
6
42
35
14
9
20
0.0889
7
9
65
56
43
26
0.034
8
29
76
75
67
32
0.006
9
61
66
43
32
42
0.0002
Location of points for previous table
Regression using previous table and locations, the geographic weighting pulls the
line towards the points with larger weights
Summary of spatial stats
• Moran’s I measures the average correlation between
the value of a variable at one location and the value at
nearby locations.
• Local Moran statistic measures spatial dependence on a
local basis, allowing the researcher to see its variation
over space, and by Geographically
• Geographically Weighted Regression allows the
parameters of a regression analysis to vary spatially.
GWR helps in detecting local variations in spatial
behavior and understanding local details, which may be
masked by global regression models. GWR, regression
coefficients are computed for every spatial zone.
© Oxford University Press, 2010. All rights reserved. Lloyd: Spatial Data Analysis
Two scatter plots and fitted lines for different aggregations of same value
© Oxford University Press, 2010. All rights reserved. Lloyd: Spatial Data Analysis
References
Lloyd: Spatial Data Analysis
Applied Spatial Data Analysis with R
Bivand, Pebesma, Gómez-Rubio
http://www.manning.com/obe/
http://www.spatial.cs.umn.edu/Book/