An O(N) parallel method of computing the Log-Jacobian of the variable transformation for models with spatial interaction on a lattice

A parallel method for computing the log of the Jacobian of variable transformations in models of spatial interactions on a lattice is developed. The method is shown to be easy to implement in parallel and distributed computing environments. The advantages of parallel computations are significant even in computer systems with low numbers of processing units, making it computationally efficient in a variety of settings. The non-iterative method is feasible for any sparse spatial weights matrix since the computations involved impose modest memory requirements for storing intermediate results. The method has a linear computational complexity for datasets with a finite Hausdorff dimension. It is shown that most geo-spatial data satisfy this requirement. Asymptotic properties of the method are illustrated using simulated data, and the method is deployed for obtaining maximum likelihood estimates for the spatial autoregressive model using data for the US economy.

[1]  Oleg A. Smirnov.  Computation of the Information Matrix for Models With Spatial Interaction on a Lattice , 2005 .

[2]  R. J. Martin.  Approximations to the determinant term in gaussian maximum likelihood estimation of some spatial models , 1992 .

[3]  Robert E. Lucas,et al.  On the Size Distribution of Business Firms , 1978 .

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  Erricos John Kontoghiorghes,et al.  Efficient algorithms for estimating the general linear model , 2006, Parallel Comput..

[6]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[7]  Harry H. Kelejian,et al.  A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model , 1999 .

[8]  Gottfried Tappeiner,et al.  Performance contest between MLE and GMM for huge spatial autoregressive models , 2008 .

[9]  Lung-fei Lee,et al.  Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models , 2004 .

[10]  Luc Anselin,et al.  EFFICIENT ALGORITHMS FOR CONSTRUCTING PROPER HIGHER ORDER SPATIAL LAG OPERATORS , 1996 .

[11]  James P. LeSage,et al.  A matrix exponential spatial specification , 2007 .

[12]  D. Griffith.  Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses , 2000 .

[13]  Oleg A. Smirnov.  Modeling spatial discrete choice , 2010 .

[14]  James P. LeSage,et al.  Chebyshev approximation of log-determinants of spatial weight matrices , 2004, Comput. Stat. Data Anal..

[15]  Erricos John Kontoghiorghes,et al.  A graph approach to generate all possible regression submodels , 2007, Comput. Stat. Data Anal..

[16]  John Sutton,et al.  Technology and Market Structure: Theory and History , 1998 .

[17]  Lung-fei Lee,et al.  GMM and 2SLS estimation of mixed regressive, spatial autoregressive models , 2007 .

[18]  Erricos John Kontoghiorghes,et al.  Handbook of Parallel Computing and Statistics (Statistics, Textbooks and Monographs) , 2005 .

[19]  Leonhard Held,et al.  Gaussian Markov Random Fields: Theory and Applications , 2005 .

[20]  L. Anselin,et al.  Fast maximum likelihood estimation of very large spatial autoregression models: a characteristic polynomial approach , 2001 .

[21]  A. Pakes,et al.  Empirical Implications of Alternative Models of Firm Dynamics , 1989 .

[22]  John Harris,et al.  Handbook of mathematics and computational science , 1998 .

[23]  L. Anselin.  Spatial Econometrics: Methods and Models , 1988 .

[24]  Ingelin Steinsland,et al.  Parallel exact sampling and evaluation of Gaussian Markov random fields , 2007, Comput. Stat. Data Anal..

[25]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

[26]  K. Ord.  Estimation Methods for Models of Spatial Interaction , 1975 .

[27]  Man-Suk Oh,et al.  Bayesian analysis of regression models with spatially correlated errors and missing observations , 2002 .

[28]  Erricos John Kontoghiorghes,et al.  Handbook of Parallel Computing and Statistics , 2005 .

[29]  Xueyan Zhao,et al.  Testing the scale effect predicted by the Fujita-Krugman urbanization model , 2004 .