A PARALLEL FORMULATION OF THE SPATIAL AUTO-REGRESSION MODEL FOR MINING LARGE GEO-SPATIAL DATASETS HPDM 2004 Workshop at SIAM Data Mining Conference Barış M. Kazar, Shashi Shekhar, David J. Lilja, Daniel Boley Army High Performance Computing and Research Center (AHPCRC) Minnesota Supercomputing Institute (MSI) Digital Technology Center (DTC) University of Minnesota

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets2 Overview Motivation Classical and New Data-Mining Techniques Problem Definition Our Approach Experimental Results Conclusions and Future Work

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets3 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

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets4 Key Concept: Neighborhood Matrix ( W ) W allows other neighborhood definitions distance based 8 and more neighbors Space + 4-neighborhood 6 th row Binary W 6 th row Row-normalized W Given: Spatial framework Attributes

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets5 Classical and New Data-Mining Techniques Solving Spatial Auto-regression Model  = 0, = 0 : Least Squares Problem  = 0, = 0 : Eigenvalue Problem  General case: Computationally expensive Maximum Likelihood Estimation Need parallel implementation to scale up

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets6 Related Work & Our Contributions Related work: Li, 1996  Limitations: Solved 1-D problem Our Contributions  Parallel solution for 2-D problems  Portable software  Fortran 77  An Application of Hybrid Parallelism »MPI messaging system »Compiler directives of OpenMP

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets7 A Serial Solution B Golden Section Search Calculate ML Function A Compute Eigenvalues C Least Squares Eigenvalues of W Compute Eigenvalues (Stage A )  Produces dense W neighborhood matrix,  Forms synthetic data y  Makes W symmetric  Householder transformation  Convert dense symmetric matrix to tri-diagonal matrix  QL Transformation  Compute all eigenvalues of tri-diagonal matrix

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets8 Serial Response Times (sec) Stage A is the bottleneck & Stage B and C contribute very small to response time

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets9 Problem Definition Given: A Sequential solution procedure: “Serial Dense Matrix Approach” for one-dimensional geo-spaces Find: Parallel Formulation of Serial Dense Matrix Approach for multi-dimensional geo-spaces Constraints:   N(0,  2 I) IID Reasonably efficient parallel implementation Parallel Platform Size of W (large vs. small and dense vs. sparse) Objective: Portable & scalable software

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets10 Our Approach – Parallel Spatial Auto-Regression Function vs. Data Partitioning  Function partitioning: Each processor works on the same data with different instructions  Data partitioning (applied): Each processor works on different data with the same instructions Implementation Platform:  Fortran with MPI & OpenMP API’s No machine-specific compiler directives  Portability  Help software development and technology transfer Other Performance Tuning  Static terms computed once

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets11 Contiguous P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4 Round-robin with chunk size Data Partitioning in a Smaller Scale 4 processors are used and chunk size can be determined by the user W is 16-by-16 and partitioned across processors P1- ( 40 vs. 58 ) P2- (36 vs. 42) P3- (32 vs. 26) P4- ( 28 vs. 10 )

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets12 A : Contiguous for rectangular loops & round-robin with chunk-size 4 B : Contiguous C : Contiguous The arrows are also synchronization points for parallel solution A B C There are synchronization points within the boxes as well Data Partitioning & Synchronization B Golden Section Search Calculate ML Function A Compute Eigenvalues C Least Squares Eigenvalues of W

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets13 Experimental Design

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets14 Experimental Results – Effect of Load Balancing

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets15 Experimental Results- Effect of Problem Size

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets16 Experimental Results- Effect of Chunk Size Critical value of the chunk size for which the speedup reaches the maximum. This value is higher for dynamic scheduling to compensate for the scheduling overhead. The workload is more evenly distributed across processors at the critical chunk size value.

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets17 Experimental Results- Effect of # of Processors

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets18 Summary Developed a parallel formulation of spatial auto-regression model Estimates maximum likelihood of regular square tessellation 1-D and 2-D planar surface partitionings for location prediction problems Used dense eigenvalue computation and hybrid parallel programming

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets19 Future Work 1.Understand reasons of inefficiencies –Algebraic cost model for speedup measurements on different architectures 2.Fine tune implemented parallel formulation –Consider alternate parallel formulations 3.Parallelize other serial solutions using sparse-matrix techniques −Chebyshev Polynomial approximation −Markov Chain Monte Carlo Estimator

A Parallel Formulation of The Spatial Auto-Regression Model for Mining Large Geo-spatial Datasets20 Acknowledgments & Final Word Army High Performance Computing Research Center-AHPCRC Minnesota Supercomputing Institute - MSI Digital Technology Center – DTC Spatial Database Group Members ARCTiC Labs Group Members Dr. Sanjay Chawla Dr. Kelley Pace Dr. James LeSage THANK YOU VERY MUCH Questions?