Two-way Matrices Research Articles

MacroPARAFAC for handling rowwise and cellwise outliers in incomplete multiway data

Multiway data extend two-way matrices into higher-dimensional tensors, often explored through dimensional reduction techniques. In this paper, we study the Parallel Factor Analysis (PARAFAC) model for handling multiway data, representing it more compactly through a concise set of loading matrices and scores. We assume that the data may be incomplete and could contain both rowwise and cellwise outliers, signifying cases that deviate from the majority and outlying cells dispersed throughout the data array. To address these challenges, we present a novel algorithm designed to robustly estimate both loadings and scores. Additionally, we introduce an enhanced outlier map to distinguish various patterns of outlying behavior. Through simulations and the analysis of fluorescence Excitation-Emission Matrix (EEM) data, we demonstrate the robustness of our approach. Our results underscore the effectiveness of diagnostic tools in identifying and interpreting unusual patterns within the data.

Least-squares bilinear clustering of three-way data

A least-squares bilinear clustering framework for modelling three-way data, where each observation consists of an ordinary two-way matrix, is introduced. The method combines bilinear decompositions of the two-way matrices with clustering over observations. Different clusterings are defined for each part of the bilinear decomposition, which decomposes the matrix-valued observations into overall means, row margins, column margins and row–column interactions. Therefore up to four different classifications are defined jointly, one for each type of effect. The computational burden is greatly reduced by the orthogonality of the bilinear model, such that the joint clustering problem reduces to separate problems which can be handled independently. Three of these sub-problems are specific cases of k-means clustering; a special algorithm is formulated for the row–column interactions, which are displayed in clusterwise biplots. The method is illustrated via an empirical example and interpreting the interaction biplots are discussed. Supplemental materials for this paper are available online, which includes the dedicated R package, lsbclust.

Quadratically Optimal Bi-Matrix Games

In this paper, we introduce the class of quadratically optimal (bi-matrix) games, which are bi-matrix games whose set of equilibrium points contain all pairs of probability vectors which maximize the expected pay-off of some pay-off matrix. We call the equilibrium points obtained in this way, quadratically optimal equilibrium points. We prove the existence of quadratically optimal equilibrium points of identical bi-matrix games, i.e. bi-matrix games for which the two pay-off matrices are equal, from which it easily follows that weakly potential bi-matrix games (a generalization of potential bimatrix games) are quadratically optimal. We also show that those weakly potential square bi-matrix games which have potential matrices that are two-way matrices are quadratically and symmetrically solvable games (i.e. there exists a square pay-off matrix whose expected pay-off maximizing probability vectors subject to the requirement that the two probability vectors (row probability vector and column probability vector) being equal) are equilibrium points of the bi-matrix game. We show by means of an example of a 2×2 identical symmetric potential bi-matrix game that for every potential matrix of the game, the set of pairs of probability distributions that maximizes the expected pay-off of the potential matrix is a strict subset of the set of equilibrium points of the potential game.

On the potential for using selected PCA-based methods to analyze the crime rate in Poland

The aim of the paper is to assess the potential for using some selected PCA-based methods to analyze the spatial diversity of crime in Poland during 2000-2017. Classical principal components analysis (PCA) deals with two-way matrices, usually taking into account objects and variables. In the case of data analyzed in the study, apart from two dimensions (objects – voivodships, variables – criminal offences), there is also the dimension of time, so the dataset can be seen as data cube: objects × variables × time. Therefore, this type of data requires the use of methods handling three-way data structures. In the paper the variability of some selected categories of criminal offences in time (2000–2017) and space (according to voivodships) is analyzed using the between-class and the within-class principal component analysis. The advantage of these methods is, among others, the possibility of the graphical presentation of the results in two-dimensional space with the use of factorial maps.

Dissecting the space-time structure of tree-ring datasets using the partial triadic analysis.

Tree-ring datasets are used in a variety of circumstances, including archeology, climatology, forest ecology, and wood technology. These data are based on microdensity profiles and consist of a set of tree-ring descriptors, such as ring width or early/latewood density, measured for a set of individual trees. Because successive rings correspond to successive years, the resulting dataset is a ring variables × trees × time datacube. Multivariate statistical analyses, such as principal component analysis, have been widely used for extracting worthwhile information from ring datasets, but they typically address two-way matrices, such as ring variables × trees or ring variables × time. Here, we explore the potential of the partial triadic analysis (PTA), a multivariate method dedicated to the analysis of three-way datasets, to apprehend the space-time structure of tree-ring datasets. We analyzed a set of 11 tree-ring descriptors measured in 149 georeferenced individuals of European larch (Larix decidua Miller) during the period of 1967–2007. The processing of densitometry profiles led to a set of ring descriptors for each tree and for each year from 1967–2007. The resulting three-way data table was subjected to two distinct analyses in order to explore i) the temporal evolution of spatial structures and ii) the spatial structure of temporal dynamics. We report the presence of a spatial structure common to the different years, highlighting the inter-individual variability of the ring descriptors at the stand scale. We found a temporal trajectory common to the trees that could be separated into a high and low frequency signal, corresponding to inter-annual variations possibly related to defoliation events and a long-term trend possibly related to climate change. We conclude that PTA is a powerful tool to unravel and hierarchize the different sources of variation within tree-ring datasets.

On the potential for the Partial Triadic Analysis to grasp the spatio-temporal variability of groundwater hydrochemistry

Standard multivariate statistical techniques, such as principal components analysis and hierarchical cluster analysis, have been widely used as unbiased methods for extracting meaningful information from groundwater quality data. However, these classical multivariate methods deal with two-way matrices, usually parameters×sites or parameters×time, while often the dataset resulting from qualitative water monitoring programs should be seen as a datacube parameters×sites×time. Three-way matrices, such as the one proposed here, are difficult to handle and to analyse by classical multivariate statistical tools and thus should be treated with approaches dealing with three-way data structures. One possible analysis approach is the use of Partial Triadic Analysis (PTA). Applied to the dataset of the Luxembourg Sandstone aquifer, the PTA appears to be a new promising statistical instrument for hydrogeologists, for characterization of temporal or spatial hydrochemical variations induced by natural and anthropogenic factors. This new approach for groundwater management offers potential for (1) identifying a common multivariate spatial structure, (2) untapping the different hydrochemical patterns and explaining their controlling factors and (3) analyzing the temporal variability of this structure and grasping hydrochemical changes.

Spatial and temporal scales of environmental forcing of Acartia populations (Copepoda: Calanoida) in the Canal de Mira (Ria de Aveiro, Portugal)†

Abstract Zooplankton and hydrological data were collected from August 2000 and June 2002 at six stations distributed throughout Canal de Mira (Ria de Aveiro, Portugal). The abundance of Acartia clausi and Acartia tonsa adults and juveniles of Acartia spp. for each station and month were combined in a three-way data matrix, which was decomposed into three two-way matrices corresponding to different modes: biological, time, and space. Cluster analysis applied on the space mode revealed the existence of three different zones as a consequence of zooplankton composition. At each mode, principal component analysis showed strong seasonal variations in zones 1 and 2. A different spatial pattern was found between the periods November 2000–April 2001 and November 2001–April 2002, with the displacement of the highest abundance levels from the middle estuary to near the mouth. The congeneric populations were segregated in space: the A. clausi population was restricted to downstream stations (zone 1), whereas the A. tonsa population dominated the middle estuary. Significant correlations between hydrological parameters and copepod abundance were found to differ from zone to zone. The statistical methodology was a valuable tool to (i) discriminate spatial and seasonal distribution patterns, (ii) define estuarine sections based on the faunistic composition, and (iii) evaluate delayed effects of phytoplankton.

The SIMCLAS Model: Simultaneous Analysis of Coupled Binary Data Matrices with Noise Heterogeneity Between and Within Data Blocks

In many research domains different pieces of information are collected regarding the same set of objects. Each piece of information constitutes a data block, and all these (coupled) blocks have the object mode in common. When analyzing such data, an important aim is to obtain an overall picture of the structure underlying the whole set of coupled data blocks. A further challenge consists of accounting for the differences in information value that exist between and within (i.e., between the objects of a single block) data blocks. To tackle these issues, analysis techniques may be useful in which all available pieces of information are integrated and in which at the same time noise heterogeneity is taken into account. For the case of binary coupled data, however, only methods exist that go for a simultaneous analysis of all data blocks but that do not account for noise heterogeneity. Therefore, in this paper, the SIMCLAS model, being a Hierarchical Classes model for the simultaneous analysis of coupled binary two-way matrices, is presented. In this model, noise heterogeneity between and within the data blocks is accounted for by downweighting entries from noisy blocks/objects within a block. In a simulation study it is shown that (1) the SIMCLAS technique recovers the underlying structure of coupled data to a very large extent, and (2) the SIMCLAS technique outperforms a Hierarchical Classes technique in which all entries contribute equally to the analysis (i.e., noise homogeneity within and between blocks). The latter is also demonstrated in an application of both techniques to empirical data on categorization of semantic concepts.

A Brief Survey of Asymmetric MDS and Some Open Problems

A brief review is made of a body of extant asymmetric MDS models and methods, given a one-mode, two-way asymmetric square relational data matrix whose elements are similarity or dissimilarity measures between objects, or a special two-mode, three-way asymmetric relational data matrix which is composed of one-mode, two-way asymmetric square relational data matrices, and several open problems are discussed.

Standard error of prediction for multilinear PLS: 2. Practical implementation in fluorescence spectroscopy

In Part 1 of this series, a new simplified expression was derived for estimating sample-specific standard error of prediction in inverse multivariate regression. The focus was on the application of this expression in multilinear partial least squares (N-PLS) regression, but its scope is more general. In this paper, the expression is applied to a fluorescence spectroscopic calibration problem where N-PLS regression is appropriate. Guidelines are given for how to cope in practice with the main assumptions underlying the proposed methodology. The sample-specific uncertainty estimates yield coverage probabilities close to the stated nominal value. Similar results were obtained for standard (i.e., linear) PLS regression and principal component regression on data rearranged to ordinary two-way matrices. The two-way results highlight the generality of the proposed expression.

Standard error of prediction for multilinear PLS2. Practical implementation in fluorescence spectroscopy

In Part 1 of this series, a new simplified expression was derived for estimating sample-specific standard error of prediction in inverse multivariate regression. The focus was on the application of this expression in multilinear partial least squares (N-PLS) regression, but its scope is more general. In this paper, the expression is applied to a fluorescence spectroscopic calibration problem where N-PLS regression is appropriate. Guidelines are given for how to cope in practice with the main assumptions underlying the proposed methodology. The sample-specific uncertainty estimates yield coverage probabilities close to the stated nominal value. Similar results were obtained for standard (i.e., linear) PLS regression and principal component regression on data rearranged to ordinary two-way matrices. The two-way results highlight the generality of the proposed expression.

Evaluating South Africa's various alternative futures via the AHP

This paper describes a pilot study in which Saaty's Analytic Hierarchy Process (AHP) is used to try and gain some insight into what form of society will prevail in South Africa in the foreseeable future. The problem is structured in a hierarchy in which the actors, or power-groups, are at the top and the alternative future scenarios are at the bottom. Panellists are used to establish the relative influences of the different actors as well as the preferences which each of the actors have for the different scenarios, by means of two-way matrices of paired comparisons. The results of the exercise are discussed, as is the use of this approach in a more serious attempt to forecast South Africa's future political, social and economic system.

Multivariate image regression (MIR): implementation of image PLSR?first forays

In the effort of analysing multivariate images, image PLS has been considered interesting. In this paper, image PLS (MIR) is compared with image PCA (MIA) by studying a comparison data set. While MIA has been commercially available for some time, image PLS has not. The kernel PLS algorithm of Lindgren has been implemented in a development environment which is a combination of G (LabVIEW) and MATLAB. In this presentation the power of this environment, as well as an early example in image regression, will be demonstrated. With kernel PLS, all PLS vectors (eigenvectors and eigenvalues) can be calculated from the joint variance–covariance (X′Y and Y′X) and association (Y′Y and X′X) matrices. The dimensions of the kernel matrices X′YY′X and Y′XX′Y are K × K (K is the number of X-variables) and M × M (M is the number of Y-variables) respectively. Hence their size is dependent only on the number of X and Y-variables and not on the number of observations (pixels), which is crucial in image analysis. The choice of LabVIEW as development platform has been based on our experience of a very short implementation time combined with user-friendly interface possibilities. Integrating LabVIEW with MATLAB has speeded up the decomposition calculations, which otherwise are slow. Also, algorithms for matrix calculations are easier to formulate in MATLAB than in LabVIEW. Applying this algorithm on a representative test image which shows many of the typical features found in technical imagery, we have shown that image PLS (MIR) decomposes the data differently than image PCA (MIA), in accordance with chemometric experience from ordinary two-way matrices. In the present example the Y-reference texture-related image used turned out to be able to force a rather significant ‘tilting’ compared with an ‘ordinary MIA’ of the primary structures in the original, spectral R/G image. Copyright © 2000 John Wiley & Sons, Ltd.

Probability matrix decomposition models

In this paper, we consider a class of models for two-way matrices with binary entries of 0 and 1. First, we considerBoolean matrix decomposition, conceptualize it as alatent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models. Second,probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models. Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is presented. This algorithm is an EM-algorithm, and is a special case of a more general algorithm that can be used for the whole class of LRMs. And fourth, as an example, a PMD model is applied to data on decision making in psychiatric diagnosis.

A new receptor model: A direct trilinear decomposition followed by a matrix reconstruction

AbstractIn many cases, monitoring data for ambient airborne particles can be organized in the form of a three‐way data table with one way for chemical species, one for sampling periods and one for sites. A direct trilinear decomposition followed by a matrix reconstruction (DTDMR) is developed to analyze such a data table as a whole. The three‐way data set is composed into three two‐way matrices by a direct trilinear decomposition (DTD). The column vectors of each of the matrices are called ‘source profiles’, ‘emission patterns’ and ‘site coefficients’ respectively. Particulate sources are identified by examining both their source profiles and emission patterns. After the sources have been identified, emission patterns and site coefficients are used to produce a three‐way matrix that gives estimates of mass contributions of sources to the samples collected at every site in every period. By simulation study, not only has the method been verified, but a good indicator has been found that shows the number of factors (i.e. sources) in the system. Unlike other receptor models, DTDMR does not require source profile data and does not involve trial‐and‐error procedures. Since DTDMR identifies sources based on variations in two dimensions, it has a higher potential to distinguish two sources that have similar chemical compositions. The DTDMR model has provided excellent results with simulated data and has been applied in a real world three‐way data set.

Fluorescence lifetime-enhanced spectral characterization of human serum

Frequency-domain fluorescence lifetime techniques were used for the characterization of pooled human serum, including normal serum, hyperlipid serum, and sera that had been stripped of various components. Fluorescence lifetime measurements of normal human serum revealed lifetime components primarily in the regions of 10(2) ps, 1-2 ns, 4-7 ns, and 9-10 ns. Phase-resolved fluorescence spectroscopy (PRFS), a frequency-domain technique that combines spectral and lifetime information, in measurements of phase-resolved fluorescence intensity (PRFI), provided the basis for comparison of the various sera. Measurements of PRFI vs excitation wavelength and emission wavelength yield a phase-resolved excitation-emission matrix (PREEM) at a given modulation frequency. Multifrequency measurements yield a three-way excitation-emission-frequency array. The multifrequency PREEMs of the various sera were compared with each other and with the corresponding two-way excitation-emission matrices (EEMs) that are obtained using conventional, steady-state fluorescence spectroscopy. Application of matrix-based analysis techniques to the steady-state and PRFS data arrays allowed direct comparison between the two approaches. Results demonstrate the enhanced discrimination among samples that is achieved through the additional dimension of fluorescence lifetime in PRFS.