Spatial Dependency Modeling Using Spatial AutoRegression

1
Spatial Dependency Modeling Using Spatial
Auto-Regression

• Mete Celik 1,3, Baris M. Kazar 4, Shashi Shekhar
  1,3,
• Daniel Boley 1, David J. Lilja 1,2
• 1 CSE Department _at_ University of Minnesota, Twin
  Cities
• 2 ECE Department _at_ University of Minnesota, Twin
  Cities
• 3 Army High Performance Computing Research Center
  
• 4 Oracle USA

2
Outline of Todays Talk

• Motivation Background
• Problem Definition
• Related Work Contributions
• Proposed Approach
• Experimental Evaluation
• Conclusion Future Work

3
Motivation

• Widespread use of spatial databases
• Mining spatial patterns
• The 1855 Asiatic Cholera on London Griffith
• Fair Landing NYT, R. Nader
• Correlation of bank locations with loan
• activity in poor neighborhoods
• Retail Outlets NYT, Walmart, McDonald etc.
• Determining locations of stores by relating
• neighborhood maps with customer
• databases
• Crime Hot Spot Analysis NYT, NIJ CML
• Explaining clusters of sexual assaults by
• locating addresses of sex-offenders
• Ecology Uygar
• Explaining location of bird nests based on
  structural environmental variables

4
Spatial Auto-correlation (SA)

• Random Distributed Data (no SA) Spatial
  distribution satisfying assumptions of classical
  data

Pixel property with independent
identical distribution
Random Nest Locations

• Cluster Distributed Data Spatial distribution
  NOT satisfying assumptions of classical data

Pixel property with spatial auto-correlation
Cluster Nest Locations
5
Execution Trace
6th row
6th row

• Given
• Spatial framework
• Attributes

Space 4-neighborhood
Binary W
Row-normalized W

• W allows other neighborhood definitions
• distance based
• 8-neighbors

6
SDM Provides Better Model!

• Linear Regression ? SAR
• Spatial auto-regression (SAR) model has higher
  accuracy and removes IID assumption of linear
  regression

7
Data Structures in SAR Model

• Vectors y, ß, e
• Matrices W, x
• W is a large matrix

8
Computational Challenge

• Maximum-Likelihood Estimation MINimizing the
  log-likelihood Function
• Solving SAR Model
• 0 ? Least Squares Problem
• 0, 0 ? Eigen-value Problem
• General case ? Computationally expensive due to
  the log-det term in
  the ML Function

Log-det term Theorem 1
SSE term
9
Outline

• Motivation Background
• Problem Definition
• Related Work Contributions
• Proposed Approach
• Experimental Evaluation
• Conclusion Future Work

10
Problem Statement

• Given
• A spatial framework S consisting of sites s1, ,
  s?q for an underlying geographic space G
• A collection of explanatory functions fxk S ? ?k
  , k1,, K. ?k is the range of possible values
  for the explanatory functions
• A dependent function fy ? ? ?y
• A family of F (SAR equation) of learning model
  functions mapping ?1 x x ?k ? ?y
• A neighborhood relationship (4 and 8- neighbor)
  on the spatial framework
• Find
• The SAR parameter ? and the regression
  coefficient vector ? with a desired precision to
  save log-det computations.

11
Problem Statement Contd

• Objective
• Algebraic error ranking of approximate SAR model
  solutions.
• Constraints
• S is a multi-dimensional Euclidean Space,
• The values of the explanatory variables x and the
  dependent function (observed variable) y may not
  be independent with respect to those of nearby
  spatial sites, i.e., spatial autocorrelation
  exists.
• The domain of x and y are real numbers.
• The SAR parameter ? varies in the range 0,1),
• The error is normally distributed with unit
  standard deviation and zero mean, i.e., ?
  N(0,?2I) IID
• The neighborhood matrix W exhibits sparsity.

12
Related Work
13
Contributions

• A new approximate SAR model solution
  Gauss-Lanczos approximation method
• Key Idea Do not find all of the eigenvalues of W
• Error ranking of approximate SAR model solutions

14
Outline

• Motivation Background
• Problem Definition
• Related Work Contributions
• Proposed Approach
• Experimental Evaluation
• Conclusion Future Work

15
Gauss-Lanczos Approximation

• Log-det is approximated by transforming the
  eigenvalue problem to the quadratic form.
• Finally, Gauss-type quadrature rules are applied
  using Lanczos procedure

16
How does GL Method Work?

• GL (Algorithm 3.2) is repeated
• m (i.e., 400) times in our experiments
• Parameter r varies between 5 and 8 in our
  experiments.
• For large problem sizes, the effects of m and r
  for getting good solution are low.

17
Taylors Series Approximation

• Log-det term in terms of Taylors Series
• Trace is sum of eigen-values W is symmetrized
  neighborhood matrix

18
Chebyshev Polynomial Approximation

• Log-det term in terms of Chebyshev Polynomials
• Trace is sum of eigen-values, Ts are matrix
  polynomials, cs are Chebyshev polynomial
  coefficients

19
Outline

• Motivation Background
• Problem Definition
• Related Work Contributions
• Proposed Approach
• Experimental Evaluation
• Conclusion Future Work

20
Experiment Design
21
Exact and Approximate Values of Log-det

• GL gives better approximation while spatial
  autocorrelation
• increases

22
Absolute Relative Error of Approximations

• Absolute relative error of approximation goes
  down as
• spatial autocorrelation increases (GL Mean
  error 0.9, GL max error 1.78)

23
Conclusions

• GL is slightly more expensive than Taylor series
  and Chebyshev polynomials.
• GL gives better approximations when spatial
  autocorrelation is high and the problem size is
  large.
• GL quality depends on the number of iterations
  and the initial Lanczos vector and the random
  number generator.
• No need to compute all eigenvalues.

24
Acknowledgments

• AHPCRC
• Minnesota Supercomputing Institute (MSI)
• Spatial Database Group Members
• ARCTiC Labs Group Members
• Dr. Dan Boley
• Dr. Sanjay Chawla
• Dr. Vipin Kumar
• Dr. James LeSage
• Dr. Kelley Pace
• Dr. Pen-Chung Yew

THANK YOU VERY MUCH Q/A