Spatial Statistics : Relationships

1
Lecture 3

• Spatial Statistics Relationships

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Relationship statements

• Male moderate drinkers are less likely to suffer
  from insulin dependent diabetes than nondrinkers.
• A better economy has more potential for people to
  be employed.
• The number of watches someone wears is directly
  proportional to the number of arms they have.
• Mountains cause rainfall. Smoking causes cancer.
• Girls are better than boys, so there.
• The number of teapots in China has no effect on
  the frequency of volcanic eruptions in Italy.

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This lecture

• Correlation how much do two variables vary
  together?
• Regression what is their relationship?
• Spatial autocorrelation and crosscorrelation
• Semi-variograms
• Geographically Weighted Regression

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Correlation

• As one variable changes, how closely do others
  follow it?
• Usually represented on graphs.
• Can plot unrelated pairs of data from datasets of
  different sizes (q-q plots)
• rank both datasets and plot 10 value against 10
  value, 90 value against 90 value.
• Or plot linked pairs of data in scatterplots.
• Correlation can be positive or negative.

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Positive correlations

• Attractiveness of chosen gender vs. alcohol
  intake.
• Bus trip time from Headingley vs. importance of
  travel reason.
• Ashs cumulative Pokémon losses vs. matches
  played.

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Negative correlations

• Ability to perform with chosen gender vs. alcohol
  intake.
• Money vs. clubs visited.
• Will to live vs. time in statistics lectures.

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Correlation is one of the most useful and used
statistical techniques.

• Correlation is an essential part of science.
• Correlation is an essential part of politics.

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Examples
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Correlation is one of the most abused statistical
techniques.

• Correlation is an essential part of dodgy
  science.
• Correlation is an essential part of political
  misinformation.
• Data can be selectively correlated.
• There is no cause and effect link just because
  two variables correlate.

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Examples
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What can we do about this?

• Tricky, but one start is to build convincing
  cause-and-effect models that demonstrate the same
  behavior.
• This gives us something concrete to investigate.
• But we then have to test our predictions.

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Correlation can be strong or weak.

• Strong Weak

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How do we measure correlation?

• We use Correlation coefficients.
• These are usually denoted as r and vary between
  1 and 1. 
• -1 very strong negative correlation.
• 0 no correlation.
• 1 very strong positive correlation.

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Which one depends on the type of data

•  Parametric tests used for data that is
• Interval or ratio.
• Normally distributed.
• Sample populations have the same standard
  deviations.
• Non-parametric tests used for all other data,
  including
• Ranked data.
• Categorized data.

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One Parametric test Pearsons Correlation
Coefficient.

• Idea is to calculate the average covariance how
  much one variable varies as the other varies.
• Deviation value mean
• The product of the variables deviations gives a
  measure of covariance.
• (valueOne meanOne) x (valueTwo meanTwo)
• If both variables values are far from the mean
  the product is large. If one deviation is large
  and the other small, the number will be smaller.

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Pearsons Correlation Coefficient

• Pearsons correlation coefficient r is the sum
  of these products normalised by the standard
  deviations.
• The simplest way of calculating this is
• r ((Sxy) / n) xmym
• sxsy
• where x and y are samples, xm and ym are
  sample means, sx and sy are sample standard
  deviations, and n is the sample sizes.

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One non-parametric test Spearman Rank
Correlation Coefficient.

• Given x and y sample pairs, we convert the xs
  into their rank in all the xs, and the ys into
  their rank in all the ys.
• Spearmans coefficient is then calculated using
• rs 1 - 6Sd2
• n3 n
• where d is the difference between the ranks for
  any given pair (a measure of the covariance).

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Testing the significance of the coefficients

• When the data can be assumed normal, we can test
  the null hypothesis that there is no correlation
  (r 0) using the following statistic
• t r (n 2)0.5
• (1 r2)0.5
• which has a t distribution and n-2 degrees of
  freedom.

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Problem correlations
Bizarre
Strong but non-linear
With many non-linear relationships we can
transform the data to a linear form. For example
exponential data can be made linear by taking the
natural log of the data.
Very bizarre
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Regression

• Quantifying the relationship between two or more
  variables.
• Linear regression with two variables.

We aim to produce a single line that quantifies
the relationship.
?y
Dependent variable (y)
?x
The equation for such a line is y a bx where
b is the slope (?y/?x on the figure). We can
use this line to predict new values given an
independent value.
a
Independent variable (x)
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Finding the regression line

• We take the line that minimizes some measure of
  how well the line fits the data.
• In the case of two variable linear regression, we
  try to minimize the deviations between the data
  and the line, or residuals.

The equation for such a line is given by b
S(x-x)(y-y) S(x-x)2 a y - bx
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How much the line explains the data

• The sum of the squared residuals gives us a
  measure of how much of the data is not explained
  by the line.
• This value, divided by the total variation in the
  data (the sum of the values squared) gives a
  fraction of how badly the line matches.
• One minus this gives how well it matches - the
  coefficient of determination.
• Conveniently, this value is the square of the
  correlation coefficient r, and is also therefore
  known as r2.
• Thus, the significance test for the r value also
  gives us the significance of our line.

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Multiple regression

• We can still do regression when there is more
  that one independent variable.
• For example, in the case of three variables (two
  independent) we are looking for a solution sheet,
  not a line

We can do the same thing with a computer for as
many dimensions as we like, but more than three
become hard to visualize as graphs. Were
essentially trying to fit a line with the
equation y a b1x1 b2x2 b3x3
y
x2
x1
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Polynomial regression

• In some cases we may want to fit a curve through
  non-linear data in multi-variable space.
• For one independent variable, the equation (which
  a type known as a polynomial) for the line is
• y a bx b2x2 b3x3
• Excel, for example, will fit this for you.

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Polynomial curve fitting

• The degree of the polynomial is the power of the
  last term. Higher degree curves fit the sample
  data better.
• However, weve seen that a sample doesnt
  necessarily have the same distribution as the
  population.
• Our curve should reflect a general population
  model not the sample data, with all its
  measurement and random errors.
• When we look at AI techniques well see that
  predictive models based on data can become less
  accurate about the world as they increasingly
  match our samples and not the population.
• Therefore we have to make a judgement as to the
  polynomial degree, and not necessarily pick the
  highest.

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Summary

• Correlation measures covariance, but doesnt say
  anything about causal relationships.
• We can measure correlation and test its
  significance.
• We can quantify relationships using regression
  equations and use these to predict.

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Spatial autocorrelation

• One of the major issues in dealing with
  geographical data.
• The idea that values at one point may be
  correlated with values of the same variable
  nearby (or cross-correlated with another variable
  nearby).
• Geodemographics people living near each other
  may have the same interests because they have the
  same opportunities and self-cluster.
• Rainfall in one geographical area stops rainfall
  in another.
• Crime spots cause social decay which in turn
  causes more crime in a limited geographical area.
  
• All graded or clustered information suffers from
  this.

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Frog averaging

• Say we want to know what the average number of
  frogs in the country is.
• We take a sample of six points.
• Three of them are normal and fall across the
  whole country but three fall in an small area
  where theres a hidden pond.
• Its like weve only really taken four samples.

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How does this effect significance testing /
correlation / regression?

• Essentially if our data is spatially correlated,
  we arent sampling as randomly as we would like
  in our attempts to get an overview of the
  population.
• i.e. some of our samples are the same (not
  independent) / dont count. This is the
  equivalent of taking a smaller sample.
• In correlation, it is possible that all our
  correlation is due to geography, and none to our
  variables.

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How do we test for it?

• First, plot the data and look for geographical
  trends.
• Particularly plot the residuals of any
  regressions.

For example, the plot to the left might represent
murder rate residuals in an area after
deprivation and policing levels are taken into
account. / regressed. Anyone want to guess where
Dr Lecter lives?

• Cluster analysis (two weeks) looks for these
  kinds of trends.

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But what about if its more pervasive?

• How do we test if, for example, its a constant
  relationship between neighbours? Statistics that
  junk individuals together are useless.
• Example ring speciation.
• If you start eastwards from Alaska theres no
  real difference between Herring gulls in one area
  of the Arctic Ocean and the next. But the minor
  differences build up around the globe so that
  Alaskan and Siberian gulls cant interbreed.
• Theres negative spatial autocorrelation in the
  fertility you wouldnt understand if you mixed
  the whole population up.
• Example factors in the spread of Ebola.
• We need a measure of the covariance between
  neighbours.

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Plotting Autocorrelations

• Imagine we had the following map of mineral
  deposits, just showing one variable.

NNE

• Obviously there is an spatial autocorrelation in
  the NNE direction and not in the others.

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h-scattergrams

• One way of displaying of autocorrelation is to
  plot the values of points against the value of a
  neighbour distance h away in some direction.

• Usually the correlation will decrease with
  distance.
• Correlation may vary with direction as well.

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Correlogram

• We can get a number of h-scatterplots for
  different h, and work out their correlation
  coefficients.
• This gives the strength of the correlation as
  distance from a set of points drops off.

We can plot these against each other for one
direction.
Or as a contour plot for all directions.
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Moment of Inertia

• If a point x1 and its neighbour x2 were identical
  and plotted against each other, theyd fall on
  the 45 line x1 x2.
• A measure of how much data does this is the
  moment of inertia.
• m 1/2n S(x1-x2)2
• Unlike the correlation coefficient, m increases
  as the data gets further spread.

36
Variograms

• A plot of the moment of inertia vs. h is called a
  Semi-variogram, or, more usually, just a
  Variogram.

m 100
m
h-distance
m 200
h
h-angle
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Problems

• Variograms cant use all the data values without
  additional assumptions e.g. what is North or the
  Northernmost data point? Usually we ignore the
  boundaries.
• All the correlation plots can suffer badly from a
  few unusual values, which can badly reduce the
  correlations.
• h-scatterplots allow us to see which unusual
  points are causing the problems and let us decide
  whether to remove them.

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Multi-variant Plots

• We may be interested in the relationship between
  two variables and whether they are spatially
  cross-correlated.
• We can plot h-scatterplots for a variable x and a
  variable y, but shift the y location by h.

y, h11/2
x
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Multi-variant Correlation

• We can also calculate the cross-correlation for
  this h-scatterplot.
• r (for some h) ((Sxy) / n) xmym
• sxsy
• Where the means and standard deviations are just
  for the variable points used, i.e. x at one
  position, y at another.

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Multi-variant Variograms

• Equally the equation for the moment of inertia
  can be extended to
• m 1/2n S((x1-x2) (y1-y2))
• Note that this is no longer the moment of
  inertia as the line can be off 45.
• Also note that it uses both x and y at positions
  1 and 2.

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The Use of Variograms

• As well see in later lectures, variograms can be
  very useful.
• They represent the variability at different
  distances from a point.
• You can therefore use them to construct
  probability models of a landscape and predict the
  value of missing areas.
• This is known as kriging, and well look at it in
  later sessions.
• However, it might be nice to have a single
  statistic we can use to assess autocorrelation.
  One way is using Joint Count Statistics.

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Joint Count Statistics

• Moran and Geary in the 1950s.
• Defines a binary variable something is either
  present (white) or not (black).
• Calculate the number of B-B W-W and B-W
  connections.
• These totals can then be compared with the normal
  distribution, which is what wed get if the
  process was random.

D
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Developments of this

• Morans I for contiguous areas.
• Gearys c for contiguous areas.
• However, this is strongly dependent on
• which directions you take as contiguous,
• variation in the size of areas and boundaries.

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Cliff and Ords Morans I test

• Core values are the deviations from the mean at
  two locations.
• These are then multiplied by an a priori weight
  which represents how much two areas might effect
  each other.
• This is then normalized by the variation and
  sample-number-to-weights ratio.

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How do we define the weights?

• Various options
• One or zero depending on whether the areas are
  adjacent.
• Each area has a total of one, and this is
  divided up between its adjacent neighbours
  dependant on the number of them.
• Exponentially related to the distance between the
  areas (its possible to assess the relationship
  between each area and all the others).
• We have to pick the most reasonable.

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Geographically Weighted Regression

• Pioneered by the Newcastle United team of
  Fotheringham, Brunsdon, and Charlton.
• A bit like Morans for regression, only even more
  arduous.

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The Core Idea

• A standard regression has the same parameters
  wherever you are geographically.
• E.g. relationship between socioeconomics and
  secondary school performance.
• Usually the residuals tell you where youve gone
  wrong.
• GWR allows the parameters to vary spatially, so
  you can look at these.
• Assumes a link between where you are, and the
  strength of a relationship.

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Locally weighted regression

• Run a standard regression for each point, but
  weight near points as more important.
• Often the weights are an exponential function of
  distance and/or limited to a fixed number of
  nearest neighbours.
• Weakly dependent on the form of the weights.
  Strongly dependent on how far weights stretch
  around an area.
• Can try to find the best distance. This is the
  one that gives the best prediction for each
  point, if that point is excluded from GWR
  calculations.
• Run, run and run again.

49
GWR Software

• Derive local t statistics.
• Perform tests to assess the significance of the
  spatial variation in the local parameter
  estimates.
• Perform tests to determine if the local model
  performs better than the global one, accounting
  for differences in degrees of freedom.
• http//www.ncl.ac.uk/geography/GWR

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GWR ExampleSpatial Variations in School
Performance

• Did a global regression on Primary School Maths
  results vs. demographics.
• Then did a GWR regression. Derived the weights
  for each factor for each point and plotted them.
• Divided by an error variation term to give a
  rough idea of the significance of the weights,
  and plotted these.

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GWR Example Results

• In Leeds / Bradford, school size was much more
  important than elsewhere (inverse relationship).
• In Manchester, middle-class children do
  proportionally better than their social group
  elsewhere.
• While the combination of variables and unknowns
  is complex, GWR does suggest interesting avenues
  of investigation.

Weights for school size
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Summary

• Correlation measures covariance, but doesnt say
  anything about causal relationships.
• We can measure correlation and test its
  significance.
• We can quantify relationships using regression
  equations and use these to predict.

53
Summary

• Spatial autocorrelation means our sampling
  strategies arent as random / large as wed like.
• Correlations can be due to geographical
  correlations, not the ones weve tested for.
• Plotting residuals geographically may let us see
  autocorrelation.
• Variograms and h-scatterplots are another good
  way.
• Morans I test allows us to quantify spatial
  autocorrelation for a given weight scheme.
• Geographically Weighted Regression helps us to
  take autocorrelation into account and investigate
  the weight it has.

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Next lecture

• Interpolation
• Homework
• Read handout on Autocorrelation Stats.
• View GWR talk.
• http//www.geocomputation.org/2001/
• Keynote.