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AGENDA

Today’s Class …

Geographic Variation
Spatial patterns
Spatial Autocorrelation
Moran’s I and LISA
Density
Nearest Neighbor

During Class …

Spatial Autocorrelation exercise

Next Class …

Bivariate Relationships

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Remember Tobler’s First Law ..

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

Values at locations near each other tend to be similar, with similarity decreasing over distance

Implies phenomena are NOT distributed randomly

Class 19

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Remember Geographic Variation

Variation in some phenomenon across space or from place to place

Observe in tables but view in maps

Events (disease cases)
Locations (hospitals)
Values (average income)

Class 19

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Geographic Variation

Visual observation is subjective

Patterns may be visible, but are eyes aren’t objective
More robust techniques to assist us in interpreting spatial patterns

Spatial Pattern – arrangement of objects in space in a regular or repeated way

Main concerns for geographers -> understand distributions

Class 19

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Remember Spatial Patterns - Events

Class 19

Clustered

Located or distributed near to another another

Random

Located or distributed such that there is no regular pattern

Ordered

Located or distributed in a regular or repeating fashion

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Remember Spatial Patterns - Attributes

Class 19

Clustered

Located or distributed near to another another

Random

Located or distributed such that there is no regular pattern

Ordered

Located or distributed in a regular or repeating fashion

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Spatial Cluster Analysis

Class 19

Global, does not tell us “where”

Identify whether events/values are clustered in space

local regions having ..
high/low values
higher density (unmarked points)

Identify clusters of events/values in space (deviations from expected)

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Types of Points

Class 19

Unmarked Points

location of an event

every point is a 1

Marked Points

location of event with measurable magnitude

number of events or magnitude of that event

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Correlation

The relationship between things that happen or change together

Relationship, association, interaction

Can measure using statistics

Class 19

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Spatial Autocorrelation

Degree of (self) similarity between objects that are located near each other
Arrangement or pattern of “values” within the landscape

Clustered, random, dispersed

Can be measured quantitatively

Region or globally

Class 19

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Spatial Autocorrelation

For areal (polygon), point, or raster data -> measure how values are arranged

Not just locations, but attributes associated with them

Not recommend for count data unless population is exactly same place to place

Class 19

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Pattern and Process

A spatial Pattern is generally the result of some spatial process

Clustering (autocorrelation) tells us about the patterns we see

Use statical models to understand and explain this process

Class 19

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Measuring Spatial Autocorrelation – Moran's I

Describes spatial autocorrelation within a region

Global (considers whole region)
Measures the magnitude

Returns single result (I)
Provided value -> probability associated with I

Class 19

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Measuring Spatial Autocorrelation – Moran's I

Global value

Ranges from -1 to 1 continuous
Perfectly dispersed = -1
Random = 0
Perfectly clustered = 1

Compares I of observed data to expected under complete spatial randomness

Class 19

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Measuring Spatial Autocorrelation – Moran's I

Class 19

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Measuring Spatial Autocorrelation – Moran's I

Interpreting output

Magnitude

Closer to 1 🡪 more clustered values
Closer to -1 🡪 more dispersed

Significance

Interpret –value (e.g., < 0.05)

Beware significant but unimportant deviation from random pattern

I = .04, p<.001
P is affected by number of observations

Class 19

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Measuring Spatial Autocorrelation – Moran's I

Robustness test

Multiple neighborhood definitions

Class 19

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Stationary vs Nonstationary

Class 19

Global

Assumes autocorrelation is stationary across space

Invariant from place to place

Local

Assumes autocorrelation is nonstationary across space

Varies from place to place

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Local Indicator of Spatial Association (LISA)

Local Version of Moran’s I

Iterates through each observation and provided a measure of autocorrelation and p value

Unlike global measurements, results can be mapped

Reveal throughout study area

Class 19

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Local Indicator of Spatial Association (LISA)

Observations can be

“hot” or “cold” spots
High or low outliers
Not significant

Useful for understanding “where” spatial autocorrelation is strong or weak

Class 19

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Local Indicator of Spatial Association (LISA)

Class 19

LISA	Observations	Neighbors
High-High	High	High
Low-Low	Low	Low
High Outlier	High	Low
Low Outlier	Low	High

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Local Indicator of Spatial Association (LISA) – Example ESRI

Class 19

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Spatial Point Pattern Tests - Density

Density is a measure of a point pattern

Divide the number of points/events by the area of the study region

200 points in 400-hectare region = .05 points per hectare

Global and local

Class 19

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Spatial Point Pattern Tests – Quadrat Analysis

Examines spatial distribution of points

Clustered, random, dispersed
Density of points located through region

Sections the study region into evenly sized quadrats
Evaluates variance among regions

Class 19

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Spatial Point Pattern Tests – Quadrat Analysis

Output

Variance to mean ratio (VMR)
p-Value (based on chi-square)
VMR < 1

More dispersed than random

VMR > 1

Clustered than random

Class 19

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Distance-based analysis

Consider how we can use distance between points to understand their spatial analysis

Class 19

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Nearest Neighbor Analysis

Class 19

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Nearest Neighbor Analysis - formula

Class 19

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Nearest Neighbor Analysis - output

Class 19

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Nearest Neighbor Analysis – interpretation

Class 19