Introduction to Spatial Statistics

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Introduction to Spatial Statistics

• Geostatistics Group
• Faye Belshe, Smitri Bhotika, Mike Gil,
• Mike Hyman, Kenny Lopiano, Jada White
• Slides contributed by Dr. Christman and Dr. Young
• October 9, 2009

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Types of Spatial Data

• Continuous Random Field
• Lattice Data
• Point Pattern Data
• Note Each type of data is analyzed differently

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Geostatistics

• Geostatistical analysis is distinct from other
  spatial models in the statistics literature in
  that it assumes the region of study is continuous

• Observations could be taken at any point within
  the study area
• Interpolation at points in between observed
  locations makes sense

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Spatial Autocorrelation

• Spatial modeling is based on the assumption that
  observations close in space tend to co-vary more
  strongly than those far from each other
• Positively co-vary values are similar in value
• E.g. elevation (or depth) tends to be similar for
  locations close together)
• Negatively co-vary values tend to be opposite in
  value
• E.g. density of an organism that is highly
  spatially clustered, where observations in
  between clusters are low and values within
  clusters are high

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Covariance

• Definition two variables are said to co-vary if
  their correlation coefficient is not zero
• where ? is the correlation coefficient between X
  and Y and ?X (?Y) is the standard deviation of
  X (Y)
• Consider this in the context of a single variable
• E.g. do nearest neighbors have non-zero
  covariance?

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Continuous Data Geostatistics

• Notation
•  
• Z(s) is the random process at location s(x, y)
• z(s) is the observed value of the process at
  location s(x, y)
• D is the study region
• The sample is the set z(s) s ? D . We say
  that it is a partial realization of the random
  spatial process Z(s) s ? D

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Conceptual Model

• where
• ?(s) is the mean structure called large-scale
  non-spatial trend
• W(s) is a zero-mean, stationary process whose
  autocorrelation range is larger than min si
  sj i,j 1, 2, , n called smooth
  small-scale variation
• ?(s) is a zero-mean, stationary process whose
  autocorrelation range is
• smaller than min si sj i,j 1, 2, ,
  n and which is independent of W(s) called
  micro-scale variation or measurement error
• ?(s) is the random noise term with zero-mean and
  constant variance and which is independent of
  W(s) and ?(s)

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Simpler Conceptual Model

• where
• ?(s) is the mean structure called large-scale
  non-spatial trend
• d(s) W(s) ?(s) is a zero-mean, stationary
  process with autocorrelation which combines the
  smooth small- scale and micro-scale variation
• ?(s) is the random noise term with zero-mean and
  constant variance which is independent of W(s)
  and ?(s)

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Graphical Concept with Trend
Red line indicates large-scale trend Green line
shows how the data are arranged around the
trend Note that there is a pattern to the points
around the red line. The pattern implies possible
positive autocorrelation in Z(x). Finally, there
is white noise.
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Graphical Concept without Trend
Red line indicates a constant mean, i.e. no
large-scale trend Green line shows how the data
are arranged around the trend Again, the pattern
of the green line implies possible positive
autocorrelation in RZ(x)
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Important Point

• The model indicates that Z can be decomposed
  into large-scale variation, small micro-scale
  variation, and noise
• The reality is that any estimated decomposition
  is not a unique
• E.g. in the graph just shown, we could have
  instead added a sinusoidal aspect to the
  large-scale trend and hence captured much of the
  apparent autocorrelation

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Example
Red line indicates large-scale trend captured by
a sinusoidal linear trend Green line shows how
the data are arranged around the trend Note that
now there is no obvious pattern and so the
remaining unexplained variation is likely white
noise in Z(x).
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Modeling

• Ultimately we want to do modeling of Z using the
  geostatistical model
• Requires estimates of the model components
• the mean
• the small-scale variation and the covariances
  among Z values at different locations
• Any leftovers, i.e. the unexplained or residual
  variability

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Important Point

• The choice of approach (detailed fit of a trend
  vs. large-scale trend autocorrelation) to
  estimating/predicting Z depends strongly on the
  reason for and uses of the model
• E.g. if you are interested in predicting Z at
  unsampled locations within the study area, then
  any model that uses covariates to estimate
  large-scale trend must also have the covariates
  known for the unsampled locations
• E.g. if you are interested in understanding the
  reasons for the spatial distribution of Z then
  you may or may not want to incorporate a spatial
  correlation component

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Correlation Structure (Semivariogram)

• Now, to assess spatial autocorrelation we look at
  the behavior of the following
• for every possible pair of locations in the
  dataset (N locations yields N(N-1)/2 pairs).
• Correlated we would expect Z(si) to be similar
  in value to Z(sj) and hence the squared
  difference to be small.
• Independent we would expect the squared
  difference to be relatively large since the two
  numbers would vary according to the population
  variability.

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Plot (Variogram Cloud)
Variogram cloud for a dataset of 400 observations
Looking for pattern, i.e. is there a trend in ?
with respect to distance between two locations
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Empirical Variogram

• The variogram cloud is usually very uninformative
• Difficult to discern trend or pattern
• More pertinent is to calculate the average values
  of ? for different distances
• Problem is we dont usually have discrete
  distances between locations (happens only when
  data are on a perfect grid).
• A common method for averaging ? at specific
  distances is to bin the distances into intervals
  (called lag distances), i.e. use all points
  within some bin width around a given distance
  value

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Continuous Data Geostatistics

• Because we do not usually have lots of values at
  discrete distances, a common method for averaging
  the values at discrete distances is to use all
  points within some bin width around a given
  distance value.
• So we choose several levels of h (distances) and
  calculate the empirical variogram
•  
• where N(h) is the set of all locations that are a
  distance of h apart within a tolerance region
  around h, i.e.
•   
• and N(h) is the number of pairs in N(h).
•  

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Empirical Semivariogram

• This plot is called an omnidirectional classical
  empirical semivariogram
• Omnidirectional because the direction between the
  pairs of locations was ignored,
• Classical because the equation used to estimate
  the mean (alternatives exist that are robust to
  outliers or to failure of assumptions of the
  model)
• Semi because of the division by 2 in the equation
  used

Graph based on a set of 20 distance lags
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Important Points

• The constantly increasing semi-variogram
  indicates that there is a problem with this
  dataset
• Ideally, it should at some distance level off at
  the variance of the process implying that at some
  distance the relationship between 2 locations is
  the same regardless of the distance between them
  (i.e. observations are independent at large
  distances)
• This graph indicates that
• The data imply correlation exists at all
  distances (and therefore the study region is
  small relative to the range of autocorrelation)
  or
• The data have a large-scale trend which may
  account for most of the seeming autocorrelation
  (small-scale trend)

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Semivariogram
Empirical semivariogram for different dataset in
which there was no large-scale trend but definite
autocorrelation
Note the rise and then leveling off of the ?(h)
values as distance increases
Well cover shapes for variograms in more detail
later
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Semivariogram
Empirical semivariogram for different dataset in
which there was no large-scale trend and no
autocorrelation
Note that the ?(h) values are more-or-less the
same regardless of distance
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Important Points

• If the empirical semivariogram increases in
  distance between locations, then the correlation
  between points is decreasing as distance
  increases
• The point at which it flattens to a constant
  value is the distance at which any two points
  that distance or larger apart are independent.
  The value of ? is the variance of the spatial
  process
• At this point in our analyses, the number of lag
  distances you use is not that critical but when
  we try to fit a curve to the empirical
  semivariogram later the number of lags becomes
  very important

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Important Point About Directionality

• Another point to consider is whether the pattern
  of autocorrelation, i.e. the shape of the curve
  describing the semivariogram, is the same in
  every direction.
• Cant tell from the omnidirectional plot.
• Need to check if there is a directional effect

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Directional Semivariograms

• To check directionality in the covariance, plot ?
  for each h for different directions
• Modify the sets of locations over which the
  averaging occurs
• Typically done using a set of binned directions
  (wedges of the compass)
• Requires that you modify the definition of
  neighborhood

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Directional Semivariograms
EXAMPLE calculate mean variability for the
angles 0, 22.5, 45, 67.5, 90, and 112.5? with a
tolerance of 11.25? on each side.
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Need for Assumptions in Order to Proceed Beyond
This Point

• The data that are collected are a partial
  observation of the spatial surface (e.g. map)
  that we are interested in
• In addition, it is usually assumed that there is
  some super process that created the particular
  surface for which we have this partial view
• To estimate the spatial autocorrelation we need
  to make some assumptions.
• Otherwise, we dont have sufficient information
  to make any inferences.

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Two Assumptions

• Stationarity, specifically second-order
  stationarity
• Isotropy

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Stationarity

• The mean of the process is constant, i.e. no
  trend
• ?(s) ? for all s ? D (1)
• The covariance between any pair of points depends
  only on the distance (and possibly direction) of
  the points NOT the location of the points in
  space
• where C(.) is the covariance function
• This implies that the variance of Z is constant
  everywhere
• If both points are met then the spatial process
  we are studying is said to be second-order
  stationary.

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Relationship between Semivariogram and Correlation

• Assuming intrinsic stationarity, we have
• Now, assuming that
  , we have
• where . Thus,

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Isotropy

• The covariance between any pair of points does
  not depend on direction but only distance

If this holds then the spatial process is said to
be isotropic
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Non-Constant Mean

• Two ways to handle a trend when it does exist
• Detrend the data using regression (or similar)
  with covariates and then use the residuals from
  the trend analysis for the spatial
  autocorrelation analysis
• E.g. disease rates as a function of population
  density
• Universal kriging (UK) which allows for
  estimating the trend as a global polynomial in s
  (x, y) and estimating the spatial
  autocorrelation simultaneously
• UK ignores other explanatory covariates which can
  be advantageous or not depending on the purpose
  of your study

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Non-Constant Variance

• To account for heterogeneity (non-constant
  variance),
• estimate variability in smaller subregions of
  the study area
• Need to make decisions about the size and extent
  of the subregions
• Need sufficient numbers of observations within
  each subregion
• Transform or standardize your data so that the
  variability of the transformed values is constant
  over the region

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Anisotropy

• Two types of anisotropy
• Geometric
• the range over which correlation is non-zero
  depends on direction
• The variance is constant over all directions
• This type can be adjusted for in geostatistical
  analyses
• Zonal
• Anything not geometric anisotropy
• Anisotropy implies that the spatial process
  evolves differentially throughout the study region

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Variography

• Fitting a valid semivariogram function to the
  empirical semivariogram
• Now we are interested in describing the variogram
  as an equation in which variance is a function
  of the distance.
• We shall assume that the spatial process is
  second-order stationary and isotropic in the
  following.

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Semivariogram

• We have already seen how to obtain the empirical
  variogram of
• is the semivariogram and is the primary
  quantity of interest because
•  
•  
• Now we are interested in describing the
  semivariogram as a function of the distance.
•  
• We shall assume that the spatial process is
  second-order stationary and isotropic in the
  following.

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Semivariogram

• Semivariogram Models have the following
  properties
•  
• 1) Many are not linear in their parameters
• 2) Must be conditionally negative-definite,
  i.e. the function must satisfy
  
• for any real numbers satisfying
•  
• 3) If as , there is
  microscale variation which is assumed to be due
  to measurement error (ME) or a process occurring
  at the microscale. ME is measurable only if we
  have replicate values at each location in the
  sample.

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Semivariogram

• Semivariogram Models have the following
  properties
• If ?(h) is constant for every h except h 0
  where ?(0) 0, then Z(s) and Z(t) are
  uncorrelated for any pair of locations s and t
• 
  , i.e. h2 is increasing faster than
  ?(h) as h increases

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A Typical Semivariogram
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Characteristics of the Semivariogram

• It is 0 when the separation distance is 0
  (Var(0)0).
• Nugget effect variation in two points very
  close together.
• May be measurement error
• May be indicative of erratic process (gold ore).
• The sill corresponds to the overall variance of
  the data.
• Data separated by distances less than the range
  are spatially autocorrelated (Less variation
  between close observations than between far
  observations.)

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Estimating the Semivariogram

• Take all pairwise differences in the data
• (Z(si)-Z(sj)), s (x, y), a point in the 2-D
  plane.
• Compute the Euclidean distance between the
  spatial locations
• Average pairs that have the same distance class
• Binning like a 2-D histogram.

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End Result Empirical Semivariogram
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Modeling the Semivariogram

• The semivariogram measures variation among units
  h units apart.
• Note We do not want negative standard errors.
• So, we model the semivariogram with selected
  parametric functions ensuring all standard errors
  are nonnegative.
• We estimate the nugget, sill, and range
  parameters of the model that best fit the
  empirical semivariogram (nonlinear least squares
  problem).

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Selected semivariogram models
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Covariogram Models
Spherical Model
Gaussian Model
Exponential Model
Power Model is simply a reparameterization of the
exponential model.
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Covariogram vs. Semivariogram
The covariogram and semivariogram are related
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The fitted semivariogram model
Estimates nugget0.084, sill0.269, range110.3
miles
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• Common methods for fitting these functions to a
  set of empirical semivariogram means
•  
• 1) choose the most likely candidate model
•  
• 2) Methods for estimating the parameters of the
  model
•  
• non-linear least squares estimation allows for
  the estimation of parameters that enter the
  equation non-linearly but ignores any dependences
  among the empirical variogram values
• non-linear weighted least-squares generalized
  least squares in which the variance-covariance
  of the variogram data points is accounted for in
  the estimation procedure
•  
• maximum likelihood assuming the data are Normally
  distributed but the estimators are likely to be
  highly biased, especially in small samples (the
  usual remedy is jackknifing)
•  
• restricted maximum likelihood maximize a
  slightly altered likelihood function which
  reduces the bias of the MLEs

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Properties of Variogram Models

• if as then there is
  microscale variation
• Usually assumed to be due to measurement error
  (ME)
• ME is measurable only if we have replicate values
  at each location in the sample
• When fitting a variogram function, may estimate a
  non-zero value for c0 even when you do not have
  replicate observations at sites. This is called
  the nugget.
• if ?(h) is constant for every h except h0 where
  ?(0) 0, then Z(si) and Z(sj) are uncorrelated
  for any pair of locations si and sj

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Properties of Variogram Models
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Choosing a Best Model

• Need to choose the variogram model that best fits
  the data
• Best minimum unexplained variation after
  fitting
• Look at a measure of deviance
• where is the empirical semivariogram
  for the ith lag and is the value
  predicted by the fitted semivariogram model

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Choosing a Best Model

• In the absence of comparing deviance (or similar)
  measures to determine if the model seems
  appropriate
• Compare fits visually
• Use prior knowledge from other studies to
  determine

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Next Steps

• Using the results of the variography to do
  statistical modeling of the spatial process
• kriging