Many users estimate spatial autoregressions to perform inference on regression parameters. However, as the sample size or the number of potential models rise, computational exigencies make exact computation of likelihood-based inferences tedious or even impossible. To address this problem, we introduce a lower bound on the likelihood ratio test that can allow users to conduct conservative maximum likelihood inference while avoiding the computationally demanding task of computing exact maximum likelihood point estimates. This form of inference, known as likelihood dominance, performs almost as well as exact likelihood inference for the empirical examples examined. We illustrate the utility of the technique by performing likelihood-based inference on parameters from a spatial autoregression involving 890,091 observations in less than a minute (given the spatial weight matrix).

Pace
and LeSage (2003), "Likelihood Dominance Spatial Inference," *Geographical Analysis* (pdf)

This chapter reviewed some of our techniques for handling spatial dependence in large data sets.

Spatial autocorrelation among regression residuals can arise from functional form misspecification. To address both functional form and spatial considerations, this paper simultaneously transforms the dependent and independent variables via B-splines as well as fitting a spatial autoregression (i.e., a spatial additive model). The log-likelihood contains two log-determinants, one for the dependent variable functional form transformation and one for the spatial dependent variable transformation. The paper applies these transformations to 11,006 observations on individual houses in Baton Rouge. The combined transformations greatly improve the pattern of the residuals and reduces their magnitude. On a Pentium Pro 200 MHz PC it took under a minute to calculate the spatial log-determinant and under 10 seconds to calculate the estimated joint transformations. I plan to include this type of function in the Spatial Statistics Toolbox.

Individual data arise over space as well as time, and usually show dependence in all of these dimensions. Modeling such dependence poses difficulties due to the different scales in the dimensions, and due to the unidirectional nature of time compared to the omnidirectional nature of space. This paper provided some computationally feasible means of modeling such spatial-temporal dependence for large data sets. In terms of performance, the spatial-temporal regression with 14 variables displayed 8% lower SSE than a regression using 211 variables attempting to control for the housing characteristics, time, and space via continuous and indicator variables. One-step ahead forecasts document the utility of the proposed spatial-temporal model. This estimator is implemented in the Spatial Statistics Toolbox 2.0.

(This is on Elsevier's site, and may require payment for institutions which do not subscribe to their services)

Using nearest neighbor spatial dependence
leads to a closed form for the eigenvalues and hence the log-determinant of the spatial
weight matrix. In turn, this simple result leads to a closed-form spatial maximum
likelihood estimator. Hence, one can find the neighbors and compute the maximum likelihood
estimates for 100,000 observations in under one minute (on a Pentium III 500 MHz machine)!
Using OLS for data known *a priori* to exhibit spatial dependence provides an
easily demolished "straw man" null hypothesis. The nearest neighbor maximum
likelihood estimator can serve as a more realistic null hypothesis for such spatial data.
This estimator is implemented in the Spatial Statistics Toolbox 1.1.

Ron Barry and I devised a means of
estimating the log-determinant of large, sparse matrices. Estimation of the
log-determinant of the variance-covariance matrix (or its inverse) allows maximum
likelihood estimation of large-scale spatial statistical problems. Most importantly, the
article shows a way of providing confidence intervals for the estimate and show these work
via a coverage study. To illustrate the potential of the estimator, we estimated the
log-determinant of a 1,000,000 by 1,000,000 matrix (which we did on a Pentium 133 MHz
machine!). The estimator has a simple form and its performance depends only upon the
degree of sparsity and not its pattern. Source code and executable code for it resides in
SpaceStatPack. The manuscript describing the estimator appeared in *Linear Algebra and
its Applications*. You can obtain a pdf version of the article by selecting the link
below or going to the Elsevier Science site.

Much of my computational approach to spatial
statistics appeared in *Geographical Analysis* (1997). The innovations in this
article include (1) permuting the rows and columns of the sparse spatial weight matrix to
vastly accelerate the computation of the log-determinant; (2) log-determinant reuse; and
(3) vectorized computation of the profile likelihoods. To give an idea about the speed
involved, on a Pentium II 400 MHz machine it takes under 10 seconds to compute a 3,107
observation simultaneous spatial autoregression (SAR) via maximum likelihood.

An additional article, which appeared in the
*Journal of Statistical Computation and Simulation*, discusses fast implementation
of Conditional Spatial Autoregressions (CAR).

Pace and Barry
(1997), "Fast CARs," *Journal of Statistical Computation and Simulation*
(pdf)

We discussed sparse kriging in *Communication in
Statistics, Simulation and *Computation. Using a published estimates on a spherical
variogram we solved the estimates 432 times as fast as using more conventional solution
techniques.

*Statistics and Probability Letters*
published an article where we analyzed a spatial data set of 20,640 observations using
normal maximum likelihood (SAR). The *Geographical Analysis* (1997) article
proposed a superior computational technology than used in the *Statistics and
Probability Letters* article, but the *Statistics and Probability Letters*
article used a larger data set. Moreover, it provided details on this data set which
resides in the Spatial Statistics Toolbox zipped files.

Pace
and Barry (1997), "Sparse Spatial Autoregressions," *Statistics and
Probability Letters* (pdf)

Otis Gilley and myself corrected a famous
data set and augmented it with spatial information. We described this in an article in the
*Journal of Environmental and Economic Management* (*JEEM*).

I have a variety of real estate manuscripts which apply spatial statistics.

Real Estate Spatial Statistics Articles

We have copyright permission from the publishers above to place these manuscripts on the web. In all cases the publishers own the copyright and have graciously granted us permission to place them on this website. More details appear in each manuscript.

Ohio
State University Press publishes
*Geographical
Analysis*
*.*

Elsevier Science* publishes Linear
Algebra and its Applications,* *
Journal of
Environmental and Economic Management (JEEM), * and *Statistics and Probability
Letters .*

Taylor &
Francis
publish the *Journal of Statistical Computation
and Simulation*.

Marcel Dekker publish
*Communications in Statistics*

You can download the articles above the spatial software packages via anonymous FTP (ftp.spatial-statistics.com/Spatial_Statistics_Toolbox or ftp.spatiotemporal.com). Some FTP clients perform downloading much better than browsers. For example, WS_FTP, CuteFTP, and FTP Explorer allow resumption of interrupted transfers and contain other features that make them ideal for downloading large files over the net.

For those without pdf viewers, I have supplied some of the manuscripts and documentation in html form. If possible, I strongly urge using the pdf versions. The conversion to html loses some of the formatting which makes the manuscripts and documentation harder to read. This is especially true for mathematical symbols and equations. In addition, it sometimes loses footnotes. You can obtain pdf file viewers for free from www.adobe.com. If you download the html files, for many of the manuscripts you need to erase the filename appearing at the end of the URL in the browser URL window and hit return. This will show all the html and gif files pertaining to that manuscript. You need to download all of them to a directory on your machine, if you wish to see the equations. This is far more work than simply downloading the pdf file version.

Geographical Analysis (1997) html

Geographical Analysis (2000) html

Geographical Analysis (2002) html

Advances in Spatial Econometrics (forthcoming) html

Statistics and Probability Letters (1997) html

Matlab Spatial Statistics Toolbox Documentation (1999) html

SpaceStatPack Documentation (1999) html

Matlab Spatial Statistics Toolbox 2.0 (2003) html

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