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Summary Ordination and clustering methods are widely applied to ecological data that are non-negative, for example, species abundances or biomasses. These methods rely on a measure of multivariate proximity that quantifies differences between the sampling units (e.g. individuals, stations, time points), leading to results such as: (i) ordinations of the units, where interpoint distances optimally display the measured differences; (ii) clustering the units into homogeneous clusters or (iii) assessing differences between pre-specified groups of units (e.g. regions, periods, treatment–control groups). These methods all conceal a fundamental question: To what extent are the differences between the sampling units, computed according to the chosen proximity function, capturing the ‘size’ in the multivariate observations, or their ‘shape’? ‘Size’ means the overall level of the measurements: for example, some samples contain higher total abundances or more biomass, others less. ‘Shape’ means the relative levels of the measurements: for example, some samples have different relative abundances, i.e. different compositions. To answer this question, several well-known proximity measures are considered and applied to two datasets, one of which is used in a simulation exercise where ‘shape’ differences have been eliminated by randomization. For any dataset and any proximity measure, a quantification is achieved of the proportion of ‘size’ variance and ‘shape’ variance that the measure is capturing, as well as the proportion of variance that confounds ‘size’ and ‘shape’ together. The results consistently show that the Bray–Curtis coefficient incorporates both ‘size’ and ‘shape’ differences, to varying degrees. These two components are thus always confounded by this proximity measure in the determination of ordinations, clusters, group comparisons and relations to environmental variables. There are several implications of these results, the main one being that researchers should be aware of this issue when they choose a proximity measure. They should compute the ‘size’ and ‘shape’ components for their particular datasets, as this can radically affect the interpretation of their results. It is recommended to separate these components: analysing total abundances or other measures of ‘size’ by univariate methods, and using multivariate analysis on the relative abundances where size has been specifically excluded.

We construct a weighted Euclidean distance that approximates any distance or dissimilarity measure between individuals that is based on a rectangular cases-by-variables data matrix. In contrast to regular multidimensional scaling methods for dissimilarity data, our approach leads to biplots of individuals and variables while preserving all the good properties of dimension-reduction methods that are based on the singular-value decomposition. The main benefits are the decomposition of variance into components along principal axes, which provide the numerical diagnostics known as contributions, and the estimation of nonnegative weights for each variable. The idea is inspired by the distance functions used in correspondence analysis and in principal component analysis of standardized data, where the normalizations inherent in the distances can be considered as differential weighting of the variables. In weighted Euclidean biplots, we allow these weights to be unknown parameters, which are estimated from the data to maximize the fit to the chosen distances or dissimilarities. These weights are estimated using a majorization algorithm. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing the matrix and displaying its rows and columns in biplots.

WOS: 000295444200004, A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure-of-fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data, BBVA Foundation; Ministry of Science and Innovation [MTM2008-00642, MTM2009-09063], The second author thanks the BBVA Foundation for financial support in this research, as well as the Ministry of Science and Innovation Grants MTM2008-00642 and MTM2009-09063. The reports of two referees on the first version of this paper led to significant improvements and are hereby acknowledged with thanks.

The local polynomial estimator is particularly affected by the curse of dimensionality. So, the potentialities of such a tool become ineffective for large dimensional applications. Motivated by this, we propose a new estimation procedure based on the local linear estimator and a nonlinearity sparseness condition, which focuses on the number of covariates for which the gradient is not constant. Our procedure, called BID for Bias-Inflation-Deflation, is automatic and easily applicable to models with many covariates without any additive assumption to the model. It simultaneously gives a consistent estimation of a) the optimal bandwidth matrix, b) the multivariate regression function and c) the multivariate, bias-corrected, confidence bands. Moreover, it automatically identify the relevant covariates and it separates the nonlinear from the linear effects. We do not need pilot bandwidths. Some theoretical properties of the method are discussed in the paper. In particular, we show the nonparametric oracle property. For linear models, the BID automatically reaches the optimal rate $Op(n^{-1/2})$, equivalent to the parametric case. A simulation study shows a good performance of the BID procedure, compared with its direct competitor.

This paper introduces a new method of estimating indifference curves in the Marschak-Machina triangle. The method involves posing questions about indifference. Contrary to previous attempts, where subjects were required to identify those lotteries to which they were indifferent vis-à-vis a given lottery, the subjects are here required to determine its certainty equivalent. The procedure is repeated for a large number of lotteries inside the triangle. Simple, linear interpolation of certainty equivalent values between adjacent points representing the lotteries under consideration allows any indifference curve inside the triangle to be plotted. The experimental results presented in the paper shed new light on the shape of indifference curves inside the Marschak-Machina triangle, where curve parallelism, fanning-out, fanning-in and boundary effects, including (possibly discontinuous) jumps, are all common. As shown, those decision-making models, which can predict jumps on the triangle legs, offer the best econometric fit of the indifference curves obtained in the study.

Summary The singular value decomposition and its interpretation as a linear biplot have proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the specific case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to the special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is applied to a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.

Canonical correspondence analysis and redundancy analysis are two methods of constrained ordination regularly used in the analysis of ecological data when ordinations based on several response variables (for example, species abundances) are related linearly to several explanatory variables (for example, environmental variables, spatial positions of samples). In this report I demonstrate the advantages of the fuzzy coding of explanatory variables: first, nonlinear relationships can be diagnosed; second, more variance in the responses can be explained; and third, in the presence of categorical explanatory variables (for example, years, regions) the interpretation of the resulting triplot ordination is unified because all explanatory variables are measured at a categorical level.

In the present paper, we use a new generalization of the Hukuhara di¤erence and derivative for fuzzy-valued functions, and we study several properties of the new concepts in the setting of the LU-parametric representation of fuzzy numbers, as- sessed both from theoretical and computational points of view.

General equilibrium moels are usually represented as a system of levels equations (e.g., in North America) or a system of linearized equations (e.g., in Australia). Either representation can be used to obtain accurate solutions. General-purpose software is available in both cases-GAMS or MPS/GE is typically used by levels modellers and GEMPACK by linearizers. Some equations (notably accounting identities) are naturally expressed in the levels while others (especially behavioural equations) are naturally expressed in a linearized form. This paper describes the new GEMPACK facility for solving models represented as a mixture of levels and linearized equations and discusses the advantages to modellers of using such a representation.

Local (perturbation) methods compute solutions in one point and tend to deliver far lower accuracy levels than global solution methods. We develop a hybrid method that solves for some policy functions locally (using a perturbation method) and that solves for the other policy functions globally (using closed-form expressions and a numerical solver). We applied our hybrid method to solve large-scale RBC models used in the comparison analysis of Kollmann et al. (J Econ Dyn Control 35:186–202, 2011b). We obtain more accurate solutions than those produced by any other (either local or global) solution method participating in that comparison. Our running time is a few seconds. Lilia Maliar and Serguei Maliar acknowledge support from the Hoover Institution at Stanford University, the Ministerio de Ciencia e Innovación de España and FEDER funds under the project SEJ-2007-62656, the Ivie, and the Generalitat Valenciana under the grants BEST/2011/283 and BEST/2011/282, respectively.

Subcompositional coherence is a fundamental property of Aitchison's approach to compositional data analysis, and is the principal justification for using ratios of components. We maintain, however, that lack of subcompositional coherence, that is incoherence, can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods, which might be better suited to cope with problems such as data zeros and outliers, while being only slightly incoherent. The measure that we propose is based on the distance measure between components. We show that the two-part subcompositions, which are the most sensitive to subcompositional incoherence, can be used to establish a distance matrix which can be directly compared with the pairwise distances in the full composition. The closeness of these two matrices can be quantified using a stress measure that is common in multidimensional scaling, providing a measure of subcompositional incoherence. Furthermore, we strongly advocate introducing weights into this measure, where rarer components are weighted proportionally less than more abundant components. The approach is illustrated using power-transformed correspondence analysis, which has already been shown to converge to logratio analysis as the power transform tends to zero.

In this paper we have proposed a method to conduct the ordinal canonical correlation analysis (OCCA) that yields ordinal canonical variates and the coefficient of correlation between them, which is analogous to (and a generalization of) the rank correlation coefficient of Spearman. The ordinal canonical variates are themselves analogous to the canonical variates obtained by the conventional canonical correlation analysis (CCCA). Our proposed method is suitable to deal with the multivariable ordinal data arrays. Our examples have shown that in finding canonical rank scores and canonical correlation from an ordinal dataset, the CCCA is suboptimal. The OCCA suggested by us outperforms the conventional method. Moreover, our method can take care of any of the five different schemes of rank ordering. It uses the Particle Swarm Optimizer which is one of the recent and prized meta-heuristics for global optimization. The computer program developed by us is fast and accurate. It has worked very well to conduct the OCCA.

The use of simple and multiple correspondence analysis is well-established in social science research for understanding relationships between two or more categorical variables. By contrast, canonical correspondence analysis, which is a correspondence analysis with linear restrictions on the solution, has become one of the most popular multivariate techniques in ecological research. Multivariate ecological data typically consist of frequencies of observed species across a set of sampling locations, as well as a set of observed environmental variables at the same locations. In this context the principal dimensions of the biological variables are sought in a space that is constrained to be related to the environmental variables. This restricted form of correspondence analysis has many uses in social science research as well, as is demonstrated in this paper. We first illustrate the result that canonical correspondence analysis of an indicator matrix, restricted to be related an external categorical variable, reduces to a simple correspondence analysis of a set of concatenated (or “stacked”) tables. Then we show how canonical correspondence analysis can be used to focus on, or partial out, a particular set of response categories in sample survey data. For example, the method can be used to partial out the influence of missing responses, which usually dominate the results of a multiple correspondence analysis.

Many multivariate methods that are apparently distinct can be linked by introducing one or more parameters in their definition. Methods that can be linked in this way are correspondence analysis, unweighted or weighted logratio analysis (the latter also known as "spectral mapping"), nonsymmetric correspondence analysis, principal component analysis (with and without logarithmic transformation of the data) and multidimensional scaling. In this presentation I will show how several of these methods, which are frequently used in compositional data analysis, may be linked through parametrizations such as power transformations, linear transformations and convex linear combinations. Since the methods of interest here all lead to visual maps of data, a "movie" can be made where where the linking parameter is allowed to vary in small steps: the results are recalculated "frame by frame" and one can see the smooth change from one method to another. Several of these "movies" will be shown, giving a deeper insight into the similarities and differences between these methods

Keane’s bump function is considered as a standard benchmark for nonlinear constrained optimization. It is highly multi-modal and its optimum is located at the non-linear constrained boundary. The true minimum of this function is, perhaps, unknown. We intend in this paper to optimize Keane’s function of different dimensions (2 to 100) by the Repulsive Particle Swarm and Differential Evolution methods. The DE optimization program has gone a long way to obtain the optimum results. However, the Repulsive Particle Swarm optimization has faltered. We have also conjectured that the values of the decision variables diminish with the increasing index values and they form two distinct clusters with almost equal number of members. These regularities indicate whether the function could attain a minimum or (at least) has reached close to the minimum. We have used this conjecture to incorporate ordering of variable values before evalution of the function and its optimization at every trial. As a result, the performance of DE as well as the RPS has improved significantly. Our results are comparable with the best results available in the literature on optimization of Keane function. Our two findings are notable: (i) Keane’s envisaged min(f) = -0.835 for 50-dimensional problem is realizable; (ii) Liu-Lewis’ min(f) = -0.84421 for 200-dimensional problem is grossly sub-optimal.Computer programs (written by us in Fortran) are available on request.

Correlation matrices have many applications, particularly in marketing and financial economics - such as in risk management, option pricing and to forecast demand for a group of products in order to realize savings by properly managing inventories, etc. Various methods have been proposed by different authors to solve the nearest correlation matrix problem by majorization, hypersphere decomposition, semi-definite programming, or geometric programming, etc. In this paper we propose to obtain the nearest valid correlation matrix by the differential evaluation method of global optimization. We may draw some conclusions from the exercise in this paper. First, the ‘nearest correlation matrix problem may be solved satisfactorily by the evolutionary algorithm like the differential evolution method/Particle Swarm Optimizer. Other methods such as the Particle Swarm method also may be used. Secondly, these methods are easily amenable to choice of the norm to minimize. Absolute, Frobenius or Chebyshev norm may easily be used. Thirdly, the ‘complete the correlation matrix problem’ can be solved (in a limited sense) by these methods. Fourthly, one may easily opt for weighted norm or un-weighted norm minimization. Fifthly, minimization of absolute norm to obtain nearest correlation matrices appears to give better results. In solving the nearest correlation matrix problem the resulting valid correlation matrices are often near-singular and thus they are on the borderline of non-semi-positive-definiteness. One finds difficulty in rounding off their elements even at 6th or 7th places after decimal, without running the risk of making the rounded off matrix non-positive definite. Such matrices are, therefore, difficult to handle. It is possible to obtain more robust positive definite valid correlation matrices by constraining the determinant (the product of eigenvalues) of the resulting correlation matrix to take on a value significantly larger than zero. But this can be done only at the cost of a compromise on the criterion of ‘nearness.’ The method proposed by us does it very well.

Power transformations of positive data tables, prior to applying the correspondence analysis algorithm, are shown to open up a family of methods with direct connections to the analysis of log-ratios. Two variations of this idea are illustrated. The first approach is simply to power transform the original data and perform a correspondence analysis - this method is shown to converge to unweighted log-ratio analysis as the power parameter tends to zero. The second approach is to apply the power transformation to the contingency ratios, that is, the values in the table relative to expected values based on the marginals - this method converges to weighted log-ratio analysis, or the spectral map. Two applications are described: first, a matrix of population genetic data which is inherently two-dimensional, and second, a larger cross-tabulation with higher dimensionality, from a linguistic analysis of several books.

Correlation matrices have many applications, particularly in marketing and financial economics. The need to forecast demand for a group of products in order to realize savings by properly managing inventories requires the use of correlation matrices. In many cases, due to paucity of data/information or dynamic nature of the problem at hand, it is not possible to obtain a complete correlation matrix. Some elements of the matrix are unknown. Several methods exist that obtain valid complete correlation matrices from incomplete correlation matrices. In view of non-unique solutions admissible to the problem of completing the correlation matrix, some authors have suggested numerical methods that provide ranges to different unknown elements. However, they are limited to very small matrices up to order 4. Our objective in this paper is to suggest a method (and provide a Fortran program) that completes a given incomplete correlation matrix of an arbitrary order. The method proposed here has an advantage over other algorithms due to its ability to present a scenario of valid correlation matrices that might be obtained from a given incomplete matrix of an arbitrary order. The analyst may choose some particular matrices, most suitable to his purpose, from among those output matrices. Further, unlike other methods, it has no restriction on the distribution of holes over the entire matrix, nor the analyst has to interactively feed elements of the matrix sequentially, which might be quite inconvenient for larger matrices. It is flexible and by merely choosing larger population size one might obtain a more exhaustive scenario of valid matrices. Moreover, the Differential Evolution algorithm is amenable to parallelization.

The generalization of simple correspondence analysis, for two categorical variables, to multiple correspondence analysis where they may be three or more variables, is not straighforward, both from a mathematical and computational point of view. In this paper we detail the exact computational steps involved in performing a multiple correspondence analysis, including the special aspects of adjusting the principal inertias to correct the percentages of inertia, supplementary points and subset analysis. Furthermore, we give the algorithm for joint correspondence analysis where the cross-tabulations of all unique pairs of variables are analysed jointly. The code in the R language for every step of the computations is given, as well as the results of each computation.

The generalization of simple (two-variable) correspondence analysis to more than two categorical variables, commonly referred to as multiple correspondence analysis, is neither obvious nor well-defined. We present two alternative ways of generalizing correspondence analysis, one based on the quantification of the variables and intercorrelation relationships, and the other based on the geometric ideas of simple correspondence analysis. We propose a version of multiple correspondence analysis, with adjusted principal inertias, as the method of choice for the geometric definition, since it contains simple correspondence analysis as an exact special case, which is not the situation of the standard generalizations. We also clarify the issue of supplementary point representation and the properties of joint correspondence analysis, a method that visualizes all two-way relationships between the variables. The methodology is illustrated using data on attitudes to science from the International Social Survey Program on Environment in 1993.

A welfare analysis of policies for risk averse preferences is handicapped within linear or linearized models and their certainty equivalence property. I calculate the optimal policy in a nonlinear stochastic dynamic general equilibrium model. Then I estimate the posterior density of structural parameters and the marginal likelihood within a nonlinear state space model. The computational challenges are considerable and I concentrate on the numerics and statistics for a simple model of dynamic consumption and labor choice. It is estimated on simulated data in order to test the routines with known true parameters. The code is written for a general model class. The policy function is approximated by Smolyak Chebyshev polynomials and the rational expectation integral by Smolyak Gaussian quadrature. The Smolyak operator is used to extend univariate approximation and integration operators to many dimensions. It is also called boolean method, hyperbolic cross points, discrete or sparse grids, fully symmetrical integration rules or complete polynomials. It reduces the curse of dimensionality from exponential to polynomial growth. The likelihood integrals are evaluated by a Gaussian quadrature and Gaussian quadrature particle filter. The bootstrap or sequential importance resampling particle filter, based on a Monte Carlo integration, is used as an accuracy benchmark. The posterior is estimated by the Gaussian filter and a Metropolis-Hastings algorithm. I propose a genetic extension of the standard Metropolis-Hastings algorithm by parallel random walk sequences. The candidate parameter vectors are constructed by innovation from the parallel sequences in addition to the usual random walk shocks. This improves the robustness of start values and the global maximization properties. Moreover it simplifies a cluster implementation and the random walk variances decision is reduced to only two parameters so that almost no trial sequences are needed. Finally the marginal likelihood is calculated as a criterion for nonnested and quasi-true models in order to select between the nonlinear estimates and a first order perturbation solution combined with the Kalman filter.

The nearest correlation matrix problem is to find a valid (positive semidefinite) correlation matrix, R(m,m), that is nearest to a given invalid (negative semidefinite) or pseudo-correlation matrix, Q(m,m); m larger than 2. In the literature on this problem, 'nearest' is invariably defined in the sense of the least Frobenius norm. Research works of Rebonato and Jaeckel (1999), Higham (2002), Anjos et al. (2003), Grubisic and Pietersz (2004), Pietersz, and Groenen (2004), etc. use Frobenius norm explicitly or implicitly. However, it is not necessary to define 'nearest' in this conventional sense. The thrust of this paper is to define 'nearest' in the sense of the least maximum norm (LMN) of the deviation matrix (R-Q), and to obtain R nearest to Q. The LMN provides the overall minimum range of deviation of the elements of R from those of Q. We also append a computer program (source codes in FORTRAN) to find the LMN R from a given Q. Presently we use the random walk search method for optimization. However, we suggest that more efficient methods based on the Genetic algorithms may replace the random walk algorithm of optimization.

In simulation we often have to generate correlated random variables by giving a reference intercorrelation matrix, R or Q. The matrix R is positive definite and a valid correlation matrix. The matrix Q may appear to be a correlation matrix but it may be invalid (negative definite). With R(m,m) it is easy to generate X(n,m), but Q(m,m) cannot give real X(n,m). So, Q has to be converted into the near-most R matrix by some procedure. NJ Higham (2002) provides a method to generate R from Q that satisfies the minimum Frobenius norm condition for (Q-R). Ali Al-Subaihi (2004) gives another method, but his method does not produce an optimal R from Q. In this paper we propose an algorithm to produce an optimal R from Q by minimizing the maximum norm of (Q-R). A Computer program (in FORTRAN) also has been provided. Having obtained R from Q, the paper gives an algorithm to obtain X(n,m) from R(m,m). The proposed algorithm is based on factorization of R, yet it is different from the Kaiser Dichman (1962) procedure. A computer program also has been given.

Although correspondence analysis is now widely available in statistical software packages and applied in a variety of contexts, notably the social and environmental sciences, there are still some misconceptions about this method as well as unresolved issues which remain controversial to this day. In this paper we hope to settle these matters, namely (i) the way CA measures variance in a two-way table and how to compare variances between tables of different sizes, (ii) the influence, or rather lack of influence, of outliers in the usual CA maps, (iii) the scaling issue and the biplot interpretation of maps,(iv) whether or not to rotate a solution, and (v) statistical significance of results.

Dual scaling of a subjects-by-objects table of dominance data (preferences, paired comparisons and successive categories data) has been contrasted with correspondence analysis, as if the two techniques were somehow different. In this note we show that dual scaling of dominance data is equivalent to the correspondence analysis of a table which is doubled with respect to subjects. We also show that the results of both methods can be recovered from a principal components analysis of the undoubled dominance table which is centred with respect to subject means.

In this paper we propose a Pyramidal Classification Algorithm, which together with an appropriate aggregation index produces an indexed pseudo-hierarchy (in the strict sense) without inversions nor crossings. The computer implementation of the algorithm makes it possible to carry out some simulation tests by Monte Carlo methods in order to study the efficiency and sensitivity of the pyramidal methods of the Maximum, Minimum and UPGMA. The results shown in this paper may help to choose between the three classification methods proposed, in order to obtain the classification that best fits the original structure of the population, provided we have an a priori information concerning this structure.

The computer implementation of any large economic model is usually a very expensive and time consuming task. This paper describes a software package, called GEMPACK, which is being developed specifically to reduce dramatically the research time, effort and cost required to set up one solution method (the Johansen method) on an actual computer. Existing features of GEMPACK are described in detail, as are developments planned for the near future. The software is model-independent, in that it works for a wide class of economic models, irrespective of the form of the equations or underlying theory. GEMPACK is portable to most mini and mainframe computers, because it has been written in ANSI standard FORTRAN 77, with such portability as a primary requirement. Several models, including the Dervis, de Melo and Robinson model of Korea, have been implemented using GEMPACK.