Spatial Autocorrelation

Geographer and statistician Waldo Tobler (1970) summarized a key component affecting any analysis of spatially referenced data through his first law of geography:

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

This succinctly defines the statistical notion of (positive) spatial autocorrelation, in which pairs of observations taken nearby are more alike than those taken farther apart. i.e. they are correlated.

Such data violate the independence assumption that many standard statistical methods hinge on and therefore spatial autocorrelation must be accounted for.

Consider a map of the 171 regions (i.e. census tracts) in Multnomah county. For each region we mark its geographic centroid

• i.e. its 2-dimensional mean
• i.e. the point at which if you held it by a needle underneath, the region would balance

There are:

• 882 total links i.e. pairs of regions that share a border
• Distribution of the number of links. Ex: There are 11 regions that have 3 neighbors.

 3  4  5  6  7  8  9 10 11 
11 45 62 31 12  6  1  1  2 

A global index of spatial autocorrelation provides a summary over the entire study area of the level of spatial similarity observed among neighboring observations.

Say we have \( n \) regions each with a measurement of some variable of interest \( Y_i \) for \( i=1, \ldots, n \)

Today we'll investigate \( Y_i \) = proportion of the \( n=171 \) census tract's population that are hispanic.

Let

• \( w_{ij} \) be a weight describing the proximity of regions \( i \) and \( j \)
• \( \mbox{sim}_{ij} \) be a measure of similarity between \( Y_i \) and \( Y_j \)

Most global indices are of the form:

\[ \frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij} \mbox{sim}_{ij}}{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}} \]

i.e. it's a weighted average of the similarity of the variable over all pairs of regions.

Moran's \( I \) statistic (1950) is among the most used measures

\[ \frac{1}{s^2}\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij} (Y_i - \overline{Y})(Y_j - \overline{Y})}{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}} \] where \[ s^2 = \frac{1}{N} \sum_{i=1}^{N} (Y_i - \overline{Y})^2 \]

This is used as hypothesis test statistic.

• If neighboring regions tend to
  • have similar values (pattern is clustered): \( I \) will be positive
  • have different values (pattern is regular): \( I \) will be negative
• Under the null hypothesis of no spatial correlation, the expected value of \( I \) is \[ -\frac{1}{N-1} \]
• Unlike a traditional correlation coefficient, \( I \) is not restricted to be between \( [-1,1] \)
• A \( p \)-value is computed.

You're going to evaluate Moran's \( I \) assuming proximity weights

\[ w_{ij} = \left\{ \begin{array}{cl} 1 & \mbox{if regions $i$ and $j$ share a border}\\ & \mbox{(not including corners)}\\ 0 & \mbox{otherwise} \end{array} \right. \]

to see if Hispanic populations in Mulnomah County tend to cluster i.e. exhibit positive spatial autocorreleation.

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