lecture 5 - weighted spatial autocorrelation

How to measure spatial autocorrelation?

Need to capture the spatial relationship between all pairs of locations.

With spatial weights or spatial structure matrix.

What is the spatial weights matrix?

Defines spatial relationship between one location and all other locations.

W_ij = relationship between location i and j

W represents a hypothesis, once spatial structure is determined → use measures (ex. morans i)

How to determine the weight?

1. adjacency

   1. rook’s case

   2. queen’s case

2. distance

   1. delauney triangulation

   2. spatial lag

3. other function

   1. non binary weights

Rooks case vs queens case

rooks case - one direction connection

• only when point to next or previous point = 1, otherwise = 0

queen’s case - one to many connections

• point to itself = 0, point to any point = 1

How does distance based spatial autocorrelation calculations work?

• can ignore adjacency → measure by distance 

• convert polygon to center points → measure distance between points → categorize points as adjacent based on threshold

• delauney triangulation: join pairs of events whose proximity polygons share a common boundary 

• spatial lag: time lag across space

  • neighbour’s neighbour = +1 

What are non binary weights?

To represent the strength of a relationship such that some relationships are stronger than others by using an inverse power relationship.

wij = 0 for weak, = 1 for strong

How to ensure symmetry for non binary weights?

Symmetry doesnt always happen, if different weights for same 2 points, can average 2 one way relationships to establish a pairwise two way relationship.

W_final = ½ (W + W^T)

What is Moran’s I?

It measures spatial autocorrelation based on feature locations and values

Comparing between every location + mean vs other locations + mean w/ a weight based on the diff between the two locations

[calculate…]

What are the variables for Moran’s I?

i and j = different areal units or zones

y = data value, ȳ = overall mean

when w = adjacency matrix, then covariance is only included for adjacent locations

(finding difference from the mean)

What do values of Moran’s I tell us about autocorrelation?

If locations are both above mean = pos, both below = pos correlation

opposite = neg correlation

0 = no relationship, > 0 = + auto

usually between -1 to +1

How is the autocorrelation relationship reflected in a Moran scatterplot?

Relationship between attribute values themselves (x axis) + local mean attribute value (avg value of adjacent locations, y axis)

• 4 quadrants; HH + LL (pos autocorrelation), HL + LH (outliers)

• dotted lines = global mean

Can we use p values to look at Moran’s I?

yes

Since there is spatial structure in moran’s i, it’s better to use monte carlo approach. 

• randomly assign attribute values 

• recalculate moran’s i for each scrambled map 

• test observed pattern from random 

moran’s i + spatial lag

• increases confidence in pos spa.auto. results 

• more lags (more steps) = getting more unrelated (gets further away) 

• applies to the whole study area (global)

How are local statistics applied?

• Getis-Ord Gi statistic

• Local Moran’s I

• GWR

Exploratory tools, sig can be difficult due to small # of cases for each local calc. Also repeatedly applying tests when tests assume independence can lead to problems.

What is the Getis-Ord Gi statistic? What are the key variables?

Detecting local concentration of low/high value in an attribute.

i = location

wij(d) = weights from W within distance d

xj = attribute values at location j, within distance d

!! denominator = sum of all values in study area (not local) !!

What does Gi say about the distribution?

Gi = proportion of clustering in the study area around location i

sum of neighbourhood as a proportion of the whole study area

• high Gi = high surrounding values

• low Gi = low surrounding values

What does the expected value of the Gi statistic represent?

• the expected value of Gi at distance d 

• the proportion of the study region by neighbourhood i where we assume 0 or 1 valued adjacency weights

How is the Z score applied to the Gi statistic?

Can use Z score to find the statistical significance by using standard deviation. Results need to be relative normal distribution.

the tails of normal distribution = unlikely that pattern is represented by null hyp  

Problems of the Gi statistic?

• be careful when making inferences from local statistics

• hard to assume normality (small area, few number of cases)

• edge effects

• can use simulation based approaches instead

What do the results of local moran’s i tell us?

• similar to global moran’s i

  • pos+ L_i = when low or high values are near each other

  • neg L_i = when low + high values are near each other

  • kinda like scatterplot

Difference between Ii and Gi

Ii = measures spatial autocorrelation w/o distinguishing between patterns dominated by concentration of high/low values

Gi = can determine if cluster = high values or low values (hot vs cold spots), more helpful to find pos+ spa.autocor.

inference with local staitsitics

cannot assume independence with Li and Gi statistics, local statistics assumes dependence on neighbouring values

How to fix the multiple testing problem?

Apply new threshold results in only areas that are stat. sig. → locations exhibit unusually high similarity with their neighbours → can strengthen findings but over conservative?

What is regression? What do the coefficients stand for?

Modeling the relationship between dependent variable (y) + independent variables (x_n)

What are residuals?

The difference between the observed value and predicted value of the model.

How is spatial dependence estimated in spatial regression?

considers spatial dependence uniform across study area, based on global estimates of spatial dependence.

What is GWR?

geographically weighted regression, assumes the relationship causes the variables to vary locally. Creates local models to understand spatial structure in the global regression model.

h = the bandwidth of the kernel, large h = includes more data points and lowers local weight

When is weighted linear regression used?

Used for local models instead of ordinary least squares. Each observation has a weight.

in a matrix: diagonals = w, non diagonals = 0

How are regression weights calculated? What is an issue?

Can use Gaussian Kernel, but bandwidth is chosen by user.

It is good for data that include significant variation in intensity of data points.

What are some criticims of GWR?

• not for stat inference

• bandwidths are adjusted

• can be difficult to determine how many observations are used to fit each model

• local models = based on subset of data = independent variables are not likely normally dis or non correlated

  • can overestimate coefficient variation across space