Spatial Autocorrelation Flashcards
Spatial Autocorrelation Defn
- Stat measure of spatial dependence
- Test for double similarity (similarity in location and in attribute)
- One of the few indices to consider attribute and location jointly
Spatial Autocorr strength
- Quantitative assessment of sign and value
Spatial Autocorr weakness
- No casual explanation of spatial process
- Why is there an observed relationship?
- Does location affect attribute or vice versa, or both?
- Is observed relationship the process or indication of another process?
- Is observed process interactive or reactive?
Positive spatial Autocorr
- Features which are similar in location also tend to have attributes of similar value
- Nearby things are similar
- Clustered
Negative spatial Autocorr
- Features which are close together in space tend to have attributes that are dissimilar
- Dispersed
Zero spatial Autocorr
- Occurs when attributes are independent of location
- Implies spatial randomness
What is the default Spatial Autocorr statistic?
Moran’s I Statistic
Moran’s I
- Classic/common way of measuring degree of spatial autocross
- Can be simplified to z-score (z = x -xmean/s)
- Spatial equivalent to Pearson Product Moment Correlation Coefficient
Moran’s I eqn
- Complex
- Transformed to z-score = n x Sum of i x Sum of j for WijiZj/[(n-1) x sum of i x sum of j for Wij]
Moran’s I close to 1
- Similar attributes tend to cluster in space
- Nearby is similar
- Contiguous zones
Moran’s I close to -1
- Dissimilar attributes tend to cluster in space
- Nearby is dissimilar
- Checkerboard pattern
- High ‘competition’
Moran’s I close to 0
- Attributes are randomly located in space
Geary’s c Stat
- Paired comparison of spatial autocorr that relates closely to semivariogram (Variance vs. distance)
Geary’s c stat less than 1
- Similar attributes tend to cluster in space
- Regionalized, smooth
Geary’s c stat greater than 1
- Dissimilar attributes tend to cluster in geographic space
- Checkerboard pattern
- Contrasting
Geary’s c stat close to 1
- Attributes are randomly located in geographic space
- Random
What is a desirable combination of spatial autocross stats?
- Corroboration of Moran’s I and Geary’s c is desirable
Geary < 1, Moran >0
Similar, Clustered, Smooth, Regionalized
Geary close to 1 and Moran close to 0
- Random, Independent, Uncorrelated
Geary > 1, Moran < 0
- Dissimilar, Dispersed, Contrasting, Checkerboard
Spatial proximity (W or C)
- Weights are a measure of spatial proximity between regions i and j
- Basically a weighting matrix for data, W = [Wij]
How can weights be defined?
- Binary connectivity (Wij = 1 for contiguous regions if polygon i and polygon j are adjacent and wii=0)
- Distance between i and j (Wij = 1 if point j is w/in distance of point i and Wii=0)
LISA stands for?
Local Indicators of Spatial Autocorrelation
LISA definition
- Set of tools for visualizing spatial association
- Helps ID features seen in data
- Utilizes local indicators to indicate significant spatial clustering
- Sum of LISA’s for all observations is proportional to global indicators of spatial association
- Calculate a global statistic but reality can be different
Main LISA tools
- Local Moran’s I
- Boxplots
- Histograms
- LISA Maps
- Can use linked outputs to select and see where outliers are on all plots
LISA Cluster map
- High-High = Strong positive spatial autocorr
- Low-Low = Strong negative spatial autocorr
= High-Low = Some positive but not significant
= Low-High = Some negative but not significant
What are possible reasons that a low income neighbourhood would have significant pattern/relationships?
- Age, Occupation, Education Level, etc.
What are 4 things to consider for the effect of spatial autocorrelation?
- Relationship btwn independent x and dependent variables is linear
- Homoscedasticity, residuals w/ mean = 0 and constant variance (no trend in residuals)
- Residuals not autocorrelated (value of one error affects the value of another area), Durbin-Watson test
- Errors follow normal distribution
What would you want to do when testing spatial autocorrelation?
- Test that residuals are not autocorrelated (Ex. Durbin-Watson test)
- Residuals have mean = 0 and constant variance, i.e. no trend (Homoscedasticity)
BLUE
Best Linear Unbiased Estimator
Effects of spatial inefficiency Assumes?
- Constant variance and normal distribution
- Errors are independently, identically distributed (BLUE)
BLUE, B
- Best
- Most efficient result
- Assess improvement by doing variables around
BLUE, U
- Unbiased
- Constant variance and normal distribution
What happens when the Independence assumption is violated?
- BLUE best is not acceptable
- Variance is greater than minimum
Effects of spatial autocorrelation, Independently distributed observations vs. dependent
- Independent, n observations = n units of information
- Spatially dependent, autocorrelated, n obs = less than n units of information
- Independent has tall, narrow peaked graph
- Dependent has lower, more gradual peaked graph
Spatial dependency = ?
- Spatial autocorrelation
- Reduces sample size and gets further from actual population
- Increases chance for type I and II errors
- Variance not consistent over area, errors not equal (i.e. error good for high values but more error in low values for example)
BLUE variance matrix
- Has lots of 0’s
- Sigma^2 I
- Independently distributed observations
Autocorrelated Variance CoVariance Matrix
- Has lots of greek symbols with number subscripts
- Sigma^2 = Omega
- Spatial dependence, autocorrelated error
