## Spatial Autocorrelation

The
instantiation of Tobler’s
first law of geography

Everything is related to everything
else, but near things are more related than distant things.

Correlation
of a variable with itself through space.

The
correlation between an observation’s value on a variable and the value of
close-by observations on the same variable

The
degree to which characteristics at one location are similar (or dissimilar) to
those nearby.

Measure
of the extent to which the occurrence of an event in an areal unit constrains,
or makes more probable, the occurrence of a similar event in a neighboring
areal unit.

Several
measures available:

Join Count Statistic

Moran’s I

Geary’s C ratio

General (Getis-Ord) G

Anselin’s
Local Index of Spatial Autocorrelation (LISA)

__Positive spatial autocorrelation__

- high values

surrounded by nearby high values

- intermediate values surrounded

by nearby intermediate values

- low values surrounded by

nearby low values

__Negative spatial autocorrelation__

- high values

surrounded by nearby low values

- intermediate values surrounded

by nearby intermediate values

- low values surrounded by

nearby high values

### Why Spatial Autocorrelation Matters

•Spatial autocorrelation is of interest in
its own right because it suggests the operation of a spatial process

•Additionally,
most statistical analyses are based on the assumption that the values of
observations in each sample are independent of one another

–Positive
spatial autocorrelation violates this, because samples taken from nearby areas
are related to each other and are

__not__independent## Moran’s I

X is the mean of the variable

Xi is the variable value at a
particular location

Xj is the variable value at another
location

Wij is a weight indexing location of i relative to j

•Applied to a continuous variable for
polygons or points

•Similar to correlation coefficient:
varies between –1.0 and + 1.0

–Value 0 or close to 0: indicates no spatial autocorrelation or random data

–High values close to 1 or -1: high auto-correlation

•Positive value: clustered data

•Negative value: dispersed / uniform data

–Negative/positive values indicate negative/positive
autocorrelation

–

•Differences from correlation coefficient are:

–Involves
one variable only, not two variables

–Incorporates
weights (wij)
which
index relative location

–Think
of it as “the correlation between neighboring values on a variable”

–More
precisely, the correlation between variable, X,
and the “spatial lag” of X formed by averaging all
the values of X for the neighboring polygons

### Interpolation

Interpolation is the process of using points
with known values or sample points to

estimate values at other unknown
points. It can be used to predict unknown values

for any geographic point
data, such as elevation, rainfall, chemical concentrations, noise levels, and
so on.

It predicts
values for cells in a raster from a limited number of sample data points.

Interpolation
is based on the assumption that spatially distributed objects are spatially
correlated; in other words, things that are close together tend to have similar
characteristics.

Why interpolate?

Visiting
every location in a study area to measure any data is usually difficult, time
consuming and costly. Instead, measurement can be done for some sample input
data points, that can be used to predict the values of all other locations.
Input points can be either randomly, strategically, or regularly spaced
points.

•Point based

Given
a number of points whose locations and values are known, determine the values
of other points; e.g. weather station readings, spot heights, oil well
readings, porosity measurements

•
Lines
to points

Line
data for interpolation; e.g. contours to elevation grids

•
Areal
interpolation

Given
a set of data mapped on one set of source zones determine the values of the
data for a different set of target zones; e.g. given population counts for
census tracts, estimate populations for electoral districts

Types

Spatial
Interpolation method can be categorized in several ways.

First
they can be grouped into global and local methods.

1. Global
Interpolation: It maps across a whole region; uses
every known point available to estimate an unknown value. It produces smother surface with less
abrupt variations. – e.g. Trend surface, regression models

2. Local
Interpolation: It repeatedly applies to small portion of the whole region; uses
a sample of known points to estimate an unknown value. This method is designed
to capture the local or short range variation. – e.g. IDW, Thiessen polygon, Spline

Second,
spatial interpolation methods can be grouped into exact and inexact
interpolation.

1.
An Exact interpolation predicts
a value at the point location that is the same as is known value; honors the
data input data points, passes through all the points . -e.g. Kriging

2.
An Inexact interpolation (or
approximate) predicts a value at the point location that differ from its known
value; used
when there is some uncertainty about the surface, believes that in many data
sets there are global trends that varies slowly and overlain local
fluctuations.

Third:
spatial interpolation methods may be deterministic or stochastic

1.Deterministic Models use a
mathematical function to predict unknown values and result in hard
classification of the value of features.

●

2. Statistical
Techniques produce
confidence limits to the accuracy of a prediction but are more difficult to
execute since more parameters need to be set.

Deterministic Models :

1.Trend
surface analysis / Polynomial

2.Minimum
Curvature Spline

3.Inverse
Distance Weighted

4.Natural neighbourhood

Rectangular

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