PRESS statistic computation

Another criterion that is based on the predictive ability of PCA is the predicted sum of squares (PRESS) statistic. To compute the (cross validated) PRESS value at a certain k, we remove the ith observation from the original data set (for i = 1,. .., n), estimate the center and the k loadings of the reduced data set, and then compute the fitted value of the ith observation following Equation 6.16, now denoted as x, . Finally, we set... 

T. Schlick, SMM ]. Sci. Statist. Comput., in press. Modified Cholesky Factorizations for Sparse Preconditioners. 

Jelinski, Z. and Moranda, P. 1972. Software rehabihty research. In Statistical Computer Performance Evaluation, ed. W. Freiberger. Academic Press, New York. 

Chapter 3 presents the development of the PRESS statistic as a criterion for structure selection of dynamic process models which are linear-in-the-parameters. Computation of the PRESS statistic is based on the orthogonal decomposition algorithm proposed by Korenberg et al. (1988) and can be viewed as a by-product of their algorithm since very little additional computation is required. We also show how the PRESS statistic can be used as an efficient technique for noise model development directly fi-om time series data. 

This chapter consists of six sections. Section 3.2 introduces the orthogonal decomposition algorithm proposed by Korenberg et al. (1988). Section 3.3 describes the concept of the PRESS statistic. Section 3.4 shows how to compute the PRESS using the orthogonal decomposition algorithm. Section 3.5 applies the PRESS statistic to the problem of model structure selection for dynamic systems. Section 3.6 shows how the PRESS statistic... 

Computation of the true prediction errors e (A ), k = 1,2,is a tremendous task in dynamic system identification where we typically face a large amount of data (M) and possibly high dimensionality (n) of the parameter vector 6. It will be shown here that by using the orthogonal decomposition algorithm, the computation of the PRESS residuals is simplified to an extent that its calculation can be viewed as a byproduct of the algorithm. The following theorem presents the cornerstone for computation of the PRESS statistic. 

Jelinski and Moranda 1972] Z. Jelinski and P.B. Moranda. Software Reliability Research, in Statistical Computer Performance Evaluation, pp. 465-484, New York, Academic Press, 1972. 

D. Nicholson, N. D. Parsonage. Computer Simulation and the Statistical Thermodynamics of Adsorption. New York Academic Press, 1983. 

D. Nicholson, N. G. Parsonage. Computer Simulations and Statistical Mechanics of Adsorption. London Academic Press, 1982. 

Rodbard, D, Estimation of Molecular Weight by Gel Filtration and Gel Electrophoresis II. Statistical and Computational Considerations. In Methods of Protein Separation Catsimpoolas, ed. Plenum Press New York, 1976 Vol. 2, p 181. 

Models are often best understood relative to the situation they are designed to describe if their constitutive variables are allowed to fluctuate statistically in a realistic way. Once a variable has been assigned a suitable density of probability distribution and the parameters of this distribution have been chosen, the fluctuations can be conveniently produced by using random deviates from statistical tables. A random deviate is a particular value of a standard random variable. Many elementary books in statistics contain tables of deviates from uniform, normal, exponential,. .. distributions. Many high-level computation-oriented programming languages (e.g., MatLab) and spreadsheets, such as Microsoft Excel, also contain random number generators. The book by Press et al. (1986) contains software that produces random deviates for the most commonly used probability distributions. 

Methods in Computational Physics , Academic Press, NY, Vol 1(1963), "Statistical Physics Vol 2(1963), "Quantum Mechanics (1963)... 

Nicholson, D. Parsonage, N. G. Computer Simulation and. the Statistical Mechanics of Adsorption Academic Press New York, 1982. 

Refs. [i] Taur Y, McCulloch CE (1999) J Statistics Educ 7 [ii] Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C the art of scientific computing. Cambridge University Press, New York... 

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Comparison of numerical efficiencies of computing either directly or on the basis of thermodynamic integration leads us to the discussion of an important trick for these calculations stratification. We focus here on a specific example, but the general advantage of stratification can be considered in a much less specific context statistical uncertainties are mitigated by a nonstatistical subdivision of the problem, solution of the subdivided problems, and then recomposition of the whole (Hammersley and Handscomb, 1964, Section 5.3 Kalos and Whitlock, 1986, Section 4.5 Press etal, 1992, Section 7.8). 

For further reading on liquids in capillaries, see the reviews by R. Evans. J. Phys. Condens. Matter 2 (1990) 8989 and Microscopic Theories of Simple Fluids and Their Interfaces, in Liquids at Interfaces. J. Charvolin. J.F. Joanny and J. Zinn-Jusiin. Eds., Elsevier (1989). and D. Nicholson, N.G. Parsonage. Computer Simulation and the Statistical Mechanics of Adsorption. Academic Press (1982). 

E. dementi, G. Corongiu, J. H. Detrich, H. Khanmohammadbaigi, S. Chin, L. Domingo, A. Laaksonen, and H. L. Nguyen, in Structural Motion Membranes, Nucleic Acids and Proteins, Proc. Int. Symp. Struct. Dyn. Membr., Nucleic Acids, Proteins, E. dementi, Ed., Adenine Press, Guilderland, NY, 1985, pp. 59-86. Parallelism in Computational Chemistry. Applications in Quantum and Statistical Mechanics. 

Nicholson D. and Parsonage N. D., Computer Simulation and the Statistical Mechanics of Adsorption (Academic Press, London 1982). 

D. Nicholson and N. Parsonage, Computer simulation and the statistical mechanics of adsorption Acad. Press, London (1982). 

Box, G. E. P., and T. L. Henson, Some aspects of mathematical modeling in chemical engineering, Proc. Inaugural Conf. of the Scientific Computation Centre and the Institute of Statistical Studies and Research, Cairo Univ. Press, Cairo (1970), 548. 

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