IS415 Geospatial Analytics for Business Intelligence Lesson 10: Geospatial Data Analysis Part 1: Lattice Analysis

What will you learn from this lesson The differences between GIS analysis and geospatial data analysis Challenges face in analysing geospatial data The basic concepts of point patterns and point patterns analysis techniques

Core Competencies Capable to setup a Geospatial Web Server Capable to design and implement web map services that are conformed to OCG WMS standard Capable to design and implement web feature services that are conformed to OCG WFS standard

Tobler’s First law of Geography Everything is related to everything else, but near things are more related than distant things Structure of Spatial Dependence Distance decay Closeness = Similarity Source: http://en.wikipedia.org/wiki/Tobler%27s_first_law_of_geography and http://en.wikipedia.org/wiki/Waldo_R._Tobler

Question?

Defining Spatial Weight Matrices Adjacency criterion: 1 if location j is adjacent to i, wij =  0 if location j is not adjacent to i.   Distance criterion: 1 if location j is within distance d from i, wij (d) =  0 otherwise. A general spatial distance weight matrices: wij (d) = dij-ab

Spatial Weight Matrix: Contiguity Weight

Identifying Global Pattern of Spatial Distribution Moran I: Geary C: Moran I (Z value) is positive: observations tend to be similar; negative: observations tend to be dissimilar; approximately zero: observations are arranged randomly over space. Geary C: large C value (>>1): observations tend to be dissimilar; small C value (<<1) indicates that they tend to be similar.

Identifying Local Patterns of Spatial Distribution Local Moran: significant and negative if location i is associated with relatively low values in surrounding locations; significant and positive if location i is associated with relatively high values of the surrounding locations. Local Geary: significant and small Local Geary (t<0) suggests a positive spatial association (similarity); significant and large Local Geary (t>0) suggests a negative spatial association (dissimilarity).

Modifiable Areal Unit Problem (MAUP) Aggregation Problem Special case of ecological fallacy Spatial heterogeneity A million spatial allocarrelation coefficients Zonation Problem Both in size and spatial arrangement

Why do relationships vary spatially? Sampling variation Nuisance variation, not real spatial non-stationarity Relationships intrinsically different across space Real spatial non-stationarity Model misspecification Can significant local variations be removed?

Some definitions Spatial non-stationarity: the same stimulus provokes a different response in different parts of the study region Global models: statements about processes which are assumed to be stationary and as such are location independent Local models: spatial decompositions of global models, the results of local models are location dependent – a characteristic we usually anticipate from geographic (spatial) data

Regression Regression establishes relationship among a dependent variable and a set of independent variable(s) A typical linear regression model looks like: yi=0 + 1x1i+ 2x2i+……+ nxni+i With yi the dependent variable, xji (j from 1 to n) the set of independent variables, and i the residual, all at location i When applied to spatial data, as can be seen, it assumes a stationary spatial process The same stimulus provokes the same response in all parts of the study region Highly untenable for spatial process

Geographically weighted regression Local statistical technique to analyze spatial variations in relationships Spatial non-stationarity is assumed and will be tested Based on the “First Law of Geography”: everything is related with everything else, but closer things are more related

GWR Addresses the non-stationarity directly Allows the relationships to vary over space, i.e., s do not need to be everywhere the same This is the essence of GWR, in the linear form: yi=i0 + i1x1i+ i2x2i+……+ inxni+i Instead of remaining the same everywhere, s now vary in terms of locations (i)

A Hedonic House Pricing Model Housing hedonic model in Milwaukee Data: MPROP 2004 – 3430+ samples used Dependent variable: the assessed value (price) Independent variables: air conditioner, floor size, fire place, house age, number of bathrooms, soil and Impervious surface (remote sensing acquired)

The global model

The global model 62% of the dependent variable’s variation is explained All determinants are statistically significant Floor size is the largest positive determinant; house age is the largest negative determinant Deteriorated environment condition (large portion of soil&impervious surface) has significant negative impact

GWR run: summary Number of nearest neighbors for calibration: 176 (adaptive scheme) AIC: 76317.39 (global: 81731.63) GWR performs better than global model

GWR run: non-stationarity check F statistic Numerator DF Denominator DF* Pr (> F) Floor Size 2.51 325.76 1001.69 0.00 House Age 1.40 192.81 1 001.69 Fireplace 1.46 80.62 0.01 Air Conditioner 1.23 429.17 Number of Bathrooms 2.49 262.39 Soil&Imp. Surface 1.42 375.71 Tests are based on variance of coefficients, all independent variables vary significantly over space

General conclusions Except for floor size, the established relationship between house values and the predictors are not necessarily significant everywhere in the City Same amount of change in these attributes (ceteris paribus) will bring larger amount of change in house values for houses locate near the Lake than those farther away

General conclusions In the northwest and central eastern part of the City, house ages and house values hold opposite relationship as the global model suggests This is where the original immigrants built their house, and historical values weight more than house age’s negative impact on house values

Calibration of GWR Local weighted least squares Weights are attached with locations Based on the “First Law of Geography”: everything is related with everything else, but closer things are more related than remote ones

Weighting schemes Determines weights Most schemes tend to be Gaussian or Gaussian-like reflecting the type of dependency found in most spatial processes It can be either Fixed or Adaptive Both schemes based on Gaussian or Gaussian-like functions are implemented in GWR3.0 and R

Fixed weighting scheme Bandwidth Weighting function

Problems of fixed schemes Might produce large estimate variances where data are sparse, while mask subtle local variations where data are dense In extreme condition, fixed schemes might not be able to calibrate in local areas where data are too sparse to satisfy the calibration requirements (observations must be more than parameters)

Adaptive weighting schemes Weighting function Bandwidth

Adaptive weighting schemes Adaptive schemes adjust itself according to the density of data Shorter bandwidths where data are dense and longer where sparse Finding nearest neighbors are one of the often used approaches

Calibration Surprisingly, the results of GWR appear to be relatively insensitive to the choice of weighting functions as long as it is a continuous distance-based function (Gaussian or Gaussian-like functions) Whichever weighting function is used, however the result will be sensitive to the bandwidth(s)

Calibration An optimal bandwidth (or nearest neighbors) satisfies either Least cross-validation (CV) score CV score: the difference between observed value and the GWR calibrated value using the bandwidth or nearest neighbors Least Akaike Information Criterion (AIC) An information criterion, considers the added complexity of GWR models