Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Ecological data are often lognormally distributed. Nutrient concentrations, population densities and biomasses, rates of production and other flows are always positive, and generally have standard deviations that increase as the mean increases. Lognormally distributed variables have these characteristics, whereas normally distributed variables can be negative and have a standard deviation that does not change as the mean changes. Lognormal errors arise when sources of variation accumulate multiplicatively, whereas normal errors arise when sources of variation are additive. Given a normally distributed random variable Y, one can calculate a lognormally distributed random variable Z = exp(Y) where exp stands for the exponential function (exp(x) = e x). The mean Z and the standard deviation s Z of the lognormal variable are related to the mean Y and standard deviation s Y of the normal variable by) 2 / exp() exp(2 Y s Y Z = [1] 5. 0 2 ] 1) [exp(− = Y Z s Z s [2] Equation 1 can be used to correct for transformation bias in logarithmic regression. Suppose that lognormally-distributed observations Z have been log transformed as Y = log(Z) to fit a regression model such as ε + =) , (ˆ b X f Y [3] where Y is the log-transformed response variable which is predicted to be Y ˆ computed from the function f, X is a matrix of predictors, b is a vector of parameters, and the errors ε are normally distributed with mean zero and standard deviation s ε. Predictions Z ˆ in the original units are calculated using equation 1 as ] 2) ˆ exp[(ˆ 2 ε s Y Z + = [4] Note that estimates the median prediction of Z, which will be smaller than the mean for a lognormally distributed variate. Thus it makes sense to adjust the median upward, as in equation 4.) ˆ exp(Y Equation 1 is also used in drawing random numbers from a lognormal distribution. Generators for normally-distributed random variables Y are common. Suppose we draw many values of Y with mean zero and standard deviation s Y. Then from equation 1, the mean of exp(Y) will not be 1 = e 0 ; instead the mean of exp(Y) will be. Generally, however, one would prefer to have the mean of a set of lognormally distributed random numbers be 1. This can be accomplished by shifting the random numbers to Y) 2 / exp(2 Y …

対数正規分布(Lognormal Distribution)のあてはめについて

Multidimensional scaling (MDS) is a technique used to extract a set of independent variables from a proximity matrix or matrices. Applications of MDS are found in a wide range of areas, including visualization, pattern analysis, data preprocessing, scale development, cybernetics, and localization. The overall rationale behind the paper is to help share innovations across disciplines. We survey and synthesize MDS methods from the academic areas of psychometrics, statistics, and computing. We define classical MDS and distance-based MDS. We then introduce basic MDS formulations and functions. We survey MDS techniques designed for nonlinear data and describe distance-based MDS in terms of minimizing the energy function in a spring system. We describe completely nonmetric MDS techniques for ordinal input data and describe how MDS solutions can be compared using ordinal neighborhood information. We describe optimization methods for fitting MDS models, covering both continuous optimization techniques and combinatorial techniques. We give several illustrative applications of MDS from the areas of cybernetics, air traffic control, molecular chemistry, robotics, and network localization. We link this work to the techniques described in the previous sections of the paper. We list a wide range of currently available MDS software and discuss possible future work in the area.

Two-Way Multidimensional Scaling: A Review

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Nuevos métodos de análisis multivariante

Judging goodness of fit in multidimensional scaling requires a comprehensive set of diagnostic tools instead of relying on stress rules of thumb. This article elaborates on corresponding strategies and gives practical guidelines for researchers to obtain a clear picture of the goodness of fit of a solution. Special emphasis will be placed on the use of permutation tests. The second part of the article focuses on goodness-of-fit assessment of an important variant of multidimensional scaling called unfolding, which can be applied to a broad range of psychological data settings. Two real-life data sets are presented in order to walk the reader through the entire set of diagnostic measures, tests, and plots. R code is provided as supplementary information that makes the whole goodness-of-fit assessment workflow, as presented in this article, fully reproducible.

Goodness-of-Fit Assessment in Multidimensional Scaling and Unfolding

It is well known that considering a non-Euclidean Minkowski metric in Multidimensional Scaling, either for the distance model or for the loss function, increases the computational problem of local minima considerably. In this paper, we propose an algorithm in which both the loss function and the composition rule can be considered in any Minkowski metric, using a multivariate randomly alternating Simulated Annealing procedure with permutation and translation phases. The algorithm has been implemented in Fortran and tested over classical and simulated data matrices with sizes up to 200 objects. A study has been carried out with some of the common loss functions to determine the most suitable values for the main parameters. The experimental results confirm the theoretical expectation that Simulated Annealing is a suitable strategy to deal by itself with the optimization problems in Multidimensional Scaling, in particular for City-Block, Euclidean and Infinity metrics.

Global Optimization in Any Minkowski Metric: A Permutation-Translation Simulated Annealing Algorithm for Multidimensional Scaling

Given a set of objects and a symmetric matrix of dissimilarities between them, Unidimensional Scaling is the problem of finding a representation by locating points on a continuum. Approximating dissimilarities by the absolute value of the difference between coordinates on a line constitutes a serious computational problem. This paper presents an algorithm that implements Simulated Annealing in a new way, via a strategy based on a weighted alternating process that uses permutations and point-wise translations to locate the optimal configuration. Explicit implementation details are given for least squares loss functions and for least absolute deviations. The weighted, alternating process is shown to outperform earlier implementations of Simulated Annealing and other optimization strategies for Unidimensional Scaling in run time efficiency, in solution quality, or in both.

A Permutation-Translation Simulated Annealing Algorithm for L1 and L2 Unidimensional Scaling

A global simulated annealing heuristic for the three-parameter lognormal maximum likelihood estimation

In this paper, we propose a cluster-MDS model for two-way one-mode continuous rating dissimilarity data. The model aims at partitioning the objects into classes and simultaneously representing the cluster centers in a low-dimensional space. Under the normal distribution assumption, a latent class model is developed in terms of the set of dissimilarities in a maximum likelihood framework. In each iteration, the probability that a dissimilarity belongs to each of the blocks conforming to a partition of the original dissimilarity matrix, and the rest of parameters, are estimated in a simulated annealing based algorithm. A model selection strategy is used to test the number of latent classes and the dimensionality of the problem. Both simulated and classical dissimilarity data are analyzed to illustrate the model.

J. Fernando Vera

Papers

Global Optimization in Any Minkowski Metric: A Permutation-Translation Simulated Annealing Algorithm for Multidimensional Scaling

A Permutation-Translation Simulated Annealing Algorithm for L1 and L2 Unidimensional Scaling

A global simulated annealing heuristic for the three-parameter lognormal maximum likelihood estimation

A Latent Class Multidimensional Scaling Model for Two-Way One-Mode Continuous Rating Dissimilarity Data

A dual latent class unfolding model for two-way two-mode preference rating data