SPATIAL AUTOCORRELATIONBy: Ehsan Hamzei 810392121

INTRODUCTION

In its most general sense, spatial autocorrelation is concerned with the degree to which objects or activities at some place on the earth's surface are similar to other objects or activities located nearby.

Its existence is reflected in the proposition which Tobler (1970) has referred to as the "first law of geography: everything is related to everything else, but near things are more related than distant things."

MEASURE OF SPATIAL AUTOCORRELATIONGlobal Measures:

A single value which applies to the entire data set The same pattern or process occurs over the entire geographic area Example: An average for the entire area

Local Measures: A value calculated for each observation unit

Different patterns or processes may occur in different parts of the region Example: A unique number for each location

GLOBAL MEASURES (MORAN’S)

Formula for Moran’s I:

Where:N is the number of observations (points or polygons) is the mean of the variableXi is the variable value at a particular locationXj is the variable value at another locationWij is a weight indexing location of i relative to j

n

1i

2i

n

1i

n

1jij

n

1i

n

1jjiij

)x(x)w(

)x)(xx(xwNI

CORRELATION COEFFICIENT

Correlation Coefficient:

Moran’s I:

n

)x(x

n

)y(y

)/nx)(xy1(y

n

1i

2i

n

1i

2i

n

1iii

n

)x(x

n

)x(x

w/)x)(xx(xw

n

1i

2i

n

1i

2i

n

1i

n

1i

n

1jij

n

1jjiij

MORAN SCATTER PLOTS

Moran’s I can be interpreted as the correlation between variable, X, and the “spatial lag” of X formed by averaging all the values of X for the neighboring polygons.

We can then draw a scatter diagram between these two variables (in standardized form): X and lag-X (or W_X) (Example: Population density )

GLOBAL MEASURES (GEARY’S C)

Geary’s Contiguity Ratio”

For Geary, the cross-product uses the actual values themselves at each location…

Calculation is similar to Moran’s I.

n

1i

2i

n

1i

n

1jij

n

1i

n

1jjiij

)x(x)w(

)x)(xx(xwNI

n

1i

2i

n

1i

n

1jij

n

1i

n

1j

2jiij

)x(x)w(2

)x(xwNC

GLOBAL MEASURES

Hot Spots and Cold Spots: What is a hot spot?

A place where high values cluster together

What is a cold spot? A place where low values cluster together

LOCAL INDICATORS OF SPATIAL ASSOCIATION (LISA) local versions of Moran’s I, Geary’s C statistics.

Moran’s I is most commonly used, and the local version is often called Anselin’s LISA, or just LISA.

LISA: The statistic is calculated for each areal unit in the data. Example: For each polygon, the index is calculated based on neighboring

polygons with which it shares a border

CALCULATING ANSELIN’S LISA

The local Moran statistic for areal unit i is:zi is the original variable xi in “standardized form” or it can be in “deviation

form” j

jijii zwzI

x

ii SD

xxz

xxi

TNX FOR YOUR ATTENTION