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27
Strong Markov Random Field Model
"... The strong Markov random field (MRF) model is a submodel of the more general MRFGibbs model. The strongMRF model defines a system whereby not only is the field Markovian with respect to a defined neighbourhood, but all subneighbourhoods also define a Markovian system. A checkerboard pattern is a ..."
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The strong Markov random field (MRF) model is a submodel of the more general MRFGibbs model. The strongMRF model defines a system whereby not only is the field Markovian with respect to a defined neighbourhood, but all subneighbourhoods also define a Markovian system. A checkerboard pattern is a perfect example of a strong Markovian system. Although the strong Markovian system requires a more stringent assumption about the field, it does have some very nice mathematical properties. One mathematical property, is the ability to define the strong Markov random field model with respect to its marginal distributions over the cliques. This property allows a direct equivalence to the ANOVA loglinear construction to be proved. From this proof, the general ANOVA loglinear construction formula is derived.
Simulate and Reject Monte Carlo Exact Conditional Tests for Quasiindependence
 In Proceedings of COMPSTAT
, 1994
"... this paper, we propose improvements to a naive simulate and reject procedure for generating r \Theta c tables under quasiindependence for an arbitrary pattern of fixed cells. Although some of the algorithmic improvements are described for generating under QI for the offdiagonal cells of a square t ..."
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this paper, we propose improvements to a naive simulate and reject procedure for generating r \Theta c tables under quasiindependence for an arbitrary pattern of fixed cells. Although some of the algorithmic improvements are described for generating under QI for the offdiagonal cells of a square table, the ideas are applicable to other patterns of fixed cells. Apart from complete enumeration, which is only viable for small tables, the simulate and reject procedure is currently the only method for generating independent tables from the exact null distribution under QI. Our improvements to the naive procedure greatly increase its efficiency. Smith, McDonald and Forster (1994) discuss another method for generating tables under QI using a Gibbs sampling approach, based on theoretical results in Forster, McDonald and Smith (1994). However, the generated tables are not necessarily independent and are only realizations from an approximation to the exact null distribution. When using a single Markov chain, the observed table is the obvious starting value. For multiple chains, obtaining other starting values with the same sufficient statistics for the nuisance parameters as the observed data is problematic. A possible solution is to generate a small number of independent starting values using the simulate and reject algorithms proposed. Acknowledgements
Exact Tests for TwoWay Symmetric Contingency Tables
"... this paper, we review exact tests and the computing problems involved. We propose new recursive algorithms for exact goodnessoffit tests of quasiindependence, quasisymmetry, linearbylinear association and some related models. We propose that all computations be carried out using symbolic comput ..."
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this paper, we review exact tests and the computing problems involved. We propose new recursive algorithms for exact goodnessoffit tests of quasiindependence, quasisymmetry, linearbylinear association and some related models. We propose that all computations be carried out using symbolic computation and rational arithmetic in order to calculate the exact pvalues accurately and describe how we implemented our proposals. Two examples are presented.
Exact Tests via Complete Enumeration: A Distributed Computing Approach. Ph. D. thesis under submission
, 1997
"... The analysis of categorical data often leads to the analysis of a contingency table. For large samples, asymptotic approximations are sufficient when calculating pvalues, but for small samples the tests can be unreliable. In these situations an exact test should be considered. This bases the test o ..."
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The analysis of categorical data often leads to the analysis of a contingency table. For large samples, asymptotic approximations are sufficient when calculating pvalues, but for small samples the tests can be unreliable. In these situations an exact test should be considered. This bases the test on the exact distribution of the test statistic. Sampling techniques can be used to estimate the distribution. Alternatively, the distribution can be found by complete enumeration. This thesis develops a number of new algorithms for complete enumeration of various models. Recursive algorithms are developed to test for independence in r × c tables. The algorithm is extended for multidimensional tables. One algorithm is extended to enumerate tables under the model of quasiindependence, and a rejection stage enables testing of models such as quasisymmetry and uniform association. A new algorithm is developed that enables a model to be defined by a model matrix, and all tables that satisfy the model are found. This provides a more efficient enumeration mechanism for complex models and extends the range of models that can be tested. The technique can lead to large calculations and a distributed version of the algorithm is developed that enables a number of machines to work efficiently on the same problem.
Monte Carlo Exact Conditional Tests for Quasiindependence using Gibbs Sampling
, 1994
"... this paper, is the hypothesis of QI for the offdiagonal cells of a r \Theta r square table, where the sufficient statistics for the nuisance parameters are x i+ ; x +j and x ii , for i; j = 1; : : : ; r. ..."
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this paper, is the hypothesis of QI for the offdiagonal cells of a r \Theta r square table, where the sufficient statistics for the nuisance parameters are x i+ ; x +j and x ii , for i; j = 1; : : : ; r.
Stat/Library
"... this document is subject to change without notice. VISUAL NUMERICS, INC., MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MATERIAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. Visual Numerics, Inc., shall not be liable for errors c ..."
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this document is subject to change without notice. VISUAL NUMERICS, INC., MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MATERIAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. Visual Numerics, Inc., shall not be liable for errors contained herein or for incidental, consequential, or other indirect damages in connection with the furnishing, performance, or use of this material. All rights are reserved.No part of this document may be photocopied or reproduced without the prior written consent of Visual Numerics, Inc.
S. W. Zucker and K. Kant, "Multiplelevel representation for texture discrim
"... p. 329, Mar. 1993. [398] P. Zinzindohoue, "Use of a new original concept for improving texture discrim ination and image segmentation in presence of noise," Optik, vol. 90, p. 97, May 1992. [377] D. Wang, V. HaeseCoat, and J. Ronsin, "A nonlinear decomposition algo rithm for noisy texture segme ..."
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p. 329, Mar. 1993. [398] P. Zinzindohoue, "Use of a new original concept for improving texture discrim ination and image segmentation in presence of noise," Optik, vol. 90, p. 97, May 1992. [377] D. Wang, V. HaeseCoat, and J. Ronsin, "A nonlinear decomposition algo rithm for noisy texture segmentation," Conference publication, vol. 354, p. 319, 1992. [378] H. Wechsler, "Texture analysis a survey," Signal Processing, vol. 2, pp. 271 282, 1980. [379] L. Wei, "Deterministic texture analysis and synthesis using tree structure vec tor quantization," in Xll Brazilian Symposium on Computer Graphics and Image Processing, pp. 20213, Oct 1999. [380] J. S. Weszka, C. R. Dyer, and A. Rosenfeld, "A comparative study of texture measures for terrain classification," IEEE Transactions on Systems, Man and Cybernetics, vol. 6, pp. 269 285, Apr. 1976. [381] R. G. White and C. J. Oliver, "Data driven texture segmentation of SAR imagery," Conference publication, vol. 365, p. 415, 1992. [3
C functions for statistical analysis C/Stat/Library
"... this document is governed by a Visual Numerics Software License Agreement. This document contains confidential and proprietary information constituting valuable trade secrets. No part of this document may be reproduced or transmitted in any form without the prior written consent of Visual Numerics. ..."
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this document is governed by a Visual Numerics Software License Agreement. This document contains confidential and proprietary information constituting valuable trade secrets. No part of this document may be reproduced or transmitted in any form without the prior written consent of Visual Numerics. RESTRICTED RIGHTS LEGEND: This documentation is provided with RESTRICTED RIGHTS. Use, duplication, or disclosure by the U.S. Government is subject to the restrictions set forth in subparagraph (c)(1)(ll) of the Rights in Technical Data and Computer Software clause at DFAR 252.2277013, and in subparagraphs (a) through (d) of the Commercial Computer Software  Restricted Rights clause at FAR 52.22719, and in similar clauses in the NASA FAR Supplement, when applicable. Contractor/Manufacturer is Visual Numerics, Inc., 2500 Wilcrest Drive, Ste 200, Houston, Texas 77042
Estimation of Exact Pvalues*
, 1989
"... The traditional practice in the analysis of contingency table has been one of the goodnessoffit tests (e.g. Pearson's X 2 or the likelihood ratio 0 2) which lean heavily on asymptotic machinery. It is well known that these methods can give spurious results when the table is sparse or when the samp ..."
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The traditional practice in the analysis of contingency table has been one of the goodnessoffit tests (e.g. Pearson's X 2 or the likelihood ratio 0 2) which lean heavily on asymptotic machinery. It is well known that these methods can give spurious results when the table is sparse or when the sample size of the table is small. Although various algorithms have been proposed for exact tests in twodimensional tables, exact tests in three and higherdimensional tables is almost an untouched area. In fact, even the minimal extension from twodimension to threedimension poses conceptually new problems. Viewing the Pvvalue as the expectation of a indicator function on a Markov chain whose equilibrium distribution is the same as the distribution of contingency tables with same margins, we propose a method to estimate the Pvalue by simulating the Markov chain which is constructed by the Metropolis algorithm. Probability functions for multidimensional contigency tables are develeped in the process. The method considerably extends the bounds of computational feasibility of the exact test for contingency table of virtually any dimension. It is fast and can be modified to calculate other statistics of the table. It also has the advantage that it can be to parallel processing. Numerical examples are given to illustrate the method.