Extension Of Principal Component Analysis Research Articles

Characterization and discrimination of different aged ‘Aceto Balsamico Tradizionale di Modena’ products by head space mass spectrometry and chemometrics

This work can be seen as an attempt to develop an analytical procedure in the context of quality control and authenticity assessment of typical food. To this aim, head-space mass spectrometry (HS-MS) coupled with multivariate data analysis, is proposed as a fast technique for furnishing a clear visualization and a suitable interpretation of the ageing process of ‘Aceto Balsamico Tradizionale di Modena’ (ABTM) and, for classifying products of different age. Considering the complexity of this food matrix, due to its traditional making procedure, the obtained instrumental data have first been analysed by parallel factor analysis (PARAFAC), an extension of principal component analysis to higher order arrays, in order to visualise the ‘natural’ grouping of vinegar samples and to inspect producers similarity/dissimilarity. On the basis of the PARAFAC results a reasonable class partition with respect to ageing was accomplished and both linear discriminant analysis (LDA) and soft independent modelling of class analogy (SIMCA) were applied as classification tools. Furthermore, it has been shown that discrimination on age basis can be improved by using feature selection in the wavelet domain through WPTER algorithm.

Principal component analysis on interval data

Real world data analysis is often affected by different types of errors as: measurement errors, computation errors, imprecision related to the method adopted for estimating the data. The uncertainty in the data, which is strictly connected to the above errors, may be treated by considering, rather than a single value for each data, the interval of values in which it may fall: the interval data. Statistical units described by interval data can be assumed as a special case of Symbolic Object (SO). In Symbolic Data Analysis (SDA), these data are represented as boxes. Accordingly, purpose of the present work is the extension of Principal Component analysis (PCA) to obtain a visualisation of such boxes, on a lower dimensional space pointing out of the relationships among the variables, the units, and between both of them. The aim is to use, when possible, the interval algebra instruments to adapt the mathematical models, on the basis of the classical PCA, to the case in which an interval data matrix is given. The proposed method has been tested on a real data set and the numerical results, which are in agreement with the theory, are reported.

“Spaghetti” PCA analysis: An extension of principal components analysis to time dependent interval data

Interval data are generally defined by the upper and the lower value assumed by a unit for a continuous variable. In our approach we introduce a special type of interval description depending on time. Each observation is characterized by an oriented interval of values with a starting and an ending value for each period of observation: for example, the opening and the closing price of a stock in a financial market in a day or a week, the initial and the final expression of a gene at the beginning and at the ending of an experiment. Several factorial techniques have been developed in order to treat interval data, but not yet for oriented intervals. In this paper we present an extension of principal component analysis to time dependent interval data, or, in general, to oriented intervals. From a geometrical point of view, the proposed approach can be considered as an analysis of oriented segments (nicely called “ spaghetti”) defined in a multidimensional space identified by periods. We introduce the formulas for the standardization of data, the calculation of matrices and the interpretation of the results.

Construction of data-adaptive orthogonal wavelet bases with an extension of principal component analysis

We show that orthonormal bases of functions with multiscale compact supports can be obtained from a generalization of principal component analysis. These functions, called multiscale principal components (MPCs), are eigenvectors of the correlation operator expressed in different vector subspaces. MPCs are data-adaptive functions that minimize their correlation with the reference signal. Using MPCs, we construct orthogonal bases which are similar to dyadic wavelet bases. We observe that MPCs are natural wavelets, i.e. their average is zero or nearly zero if the signal has a dominantly low-pass spectrum. We show that MPCs perform well in simple data compression experiments, in the presence or absence of singularities. We also introduce concentric MPCs, which are orthogonal basis functions having multiscale concentric supports. Use as kernels in convolution products with a signal, these functions allow to define a wavelet transform that has a striking capacity to emphasize atypical patterns.

SOM-based algorithms for qualitative variables

It is well known that the SOM algorithm achieves a clustering of data which can be interpreted as an extension of Principal Component Analysis, because of its topology-preserving property. But the SOM algorithm can only process real-valued data. In previous papers, we have proposed several methods based on the SOM algorithm to analyze categorical data, which is the case in survey data. In this paper, we present these methods in a unified manner. The first one (Kohonen Multiple Correspondence Analysis, KMCA) deals only with the modalities, while the two others (Kohonen Multiple Correspondence Analysis with individuals, KMCA_ind, Kohonen algorithm on DISJonctive table, KDISJ) can take into account the individuals, and the modalities simultaneously.

Principal Component Analysis of Fuzzy Data Using Autoassociative Neural Networks

This paper describes an extension of principal component analysis (PCA) allowing the extraction of a limited number of relevant features from high-dimensional fuzzy data. Our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as PCA, without explicit matrix diagonalization. Fuzzy input values are propagated through the network using fuzzy arithmetics, and the weights are adjusted to minimize a suitable error criterion, the inputs being taken as target outputs. The concept of correlation coefficient is extended to fuzzy numbers, allowing the interpretation of the new features in terms of the original variables. Experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data.

Twinned principal curves

Principal Curves are extensions of Principal Component Analysis and are smooth curves, which pass through the middle of a data set. We extend the method so that, on pairs of data sets which have underlying non-linear correlations, we have pairs of curves which go through the ‘centre’ of data sets in such a way that the non-linear correlations between the data sets are captured. The core of the method is to iteratively average the current local projections of the data points which produces an increasingly sparsified set of nodes. The Twinned Principal Curves are generated in three ways: by joining up the nodes in order, by performing Local Canonical Correlation Analysis and by performing Local Exploratory Correlation Analysis (Koetsier et al., 2002). The latter two are shown to improve the forecasting capability of the method but at an increased computational load. We show that it is crucial to terminate the algorithm after a small number of iterations for the first method and investigate several criteria for doing so.

Robust Linear Clustering with Least Squares Criterion and Its Application to Collaborative Filtering

Non-linear extensions of Principal Component Analysis (PCA) have been developed for detecting the lower-dimensional representations of real world data sets. Fuzzy c-Varieties (FCV) is the linear fuzzy clustering algorithm that can be regarded as a Local PCA technique. However least squares techniques often fail to account for “outliers”. This paper proposes a technique for making the FCV algorithm robust to intra-sample outliers. The objective function based on the lower rank approximation of the data matrix is minimized by a robust M-estimation algorithm that is similar to FCM-type iterative procedures. The new method is also useful for estimating missing values and a numerical experiment of Collaborative Filtering reveals an improvement in recommendation performance.

ON THE FOUNDATIONS OF GRANULAR COMPUTING PARADIGM

This paper investigates the foundations of granular computing, which is a paradigm oriented towards capturing and processing meaningful pieces of information, the so-called information granules. Each level of granulation induces a higher complexity in the semantic structure of the universe of discourse we have to deal with. Equipping the space of such granules with suitable metric concepts is the key to disclose the cognitive structures underlying granular information systems and to abstract or highlight their hidden meaning. We first interrogate on the epistemological foundations of granular computing and address its hard core: the representation formalism. As far as we are concerning with representing fuzzy granules, the accent is shifted from norms and metrics that equip numeric spaces to those equip-ping function spaces. The Hilbert space of square-integrable functions is the most suitable choice when specific goals are assumed, such as deriving second order statistics (variance or covariance) for random fuzzy-valued variables, or using the projection theorem for solving minimum norm problems. We interrogate on the foundations of univariate and multivariate granular statistics, fuzzy extension of principal component analysis, granular clustering, granular graphical models, granular least squares estimates and discuss some related approaches.

Unsupervised learning in neural computation

In this article, we consider unsupervised learning from the point of view of applying neural computation on signal and data analysis problems. The article is an introductory survey, concentrating on the main principles and categories of unsupervised learning. In neural computation, there are two classical categories for unsupervised learning methods and models: first, extensions of principal component analysis and factor analysis, and second, learning vector coding or clustering methods that are based on competitive learning. These are covered in this article. The more recent trend in unsupervised learning is to consider this problem in the framework of probabilistic generative models. If it is possible to build and estimate a model that explains the data in terms of some latent variables, key insights may be obtained into the true nature and structure of the data. This approach is also briefly reviewed.

Statistical assessment of bilateral symmetry of shapes

SUMMARY This paper formulates the problem of measuring bilateral symmetry of objects analytically for landmark data, and develops various new testing procedures and exploratory data analyses. The development is linked with the fundamental biological problem of measuring directional asymmetry and fluctuating asymmetry. We distinguish two types of symmetry, object symmetry and matching symmetry, and provide tests under assumptions of isotropic landmark variability as well as non-isotropy. The tests require novel statistical and geometrical analyses of the Procrustes shape manifold. For describing components of symmetry and asymmetry within samples, an extension of principal component analysis is introduced and illustrated by appropriate deformation techniques. Various real examples are used to illustrate the practical relevance of this work.

A simultaneous non-symmetrical principal component analysis with a group structure

The aim of this paper is to propose an extension of principal component analysis onto a reference subspace (PCAR) to the case where the same dependent variables have been measured on the same statistical units under two, or more, different observational conditions. As the units belong to the same multidimensional space, we profitably apply the orthogonal Procrustean rotations, jointly with PCAR, so as to enrich the interpretability of patterns on factorial planes. The proposed technique is applied to a problem of agreement in the area of sensory data analysis for representing evaluation gaps between the perception of quality by wine experts and ordinary consumers. The proposed approach allows to explain the eventually detected gaps in terms of the physical–chemical characteristics of wines. Copyright © 1999 John Wiley & Sons, Ltd.

RBF principal manifolds for process monitoring

This paper describes a novel means for creating a nonlinear extension of principal component analysis (PCA) using radial basis function (RBF) networks. This algorithm comprises two distinct stages: projection and self-consistency. The projection stage contains a single network, trained to project data from a high- to a low-dimensional space. Training requires solution of a generalized eigenvector equation. The second stage, trained using a novel hybrid nonlinear optimization algorithm, then performs the inverse transformation. Issues relating to the practical implementation of the procedure are discussed, and the algorithm is demonstrated on a nonlinear test problem. An example of the application of the algorithm to data from a benchmark simulation of an industrial overheads condenser and reflux drum rig is also included. This shows the usefulness of the procedure in detecting and isolating both sensor and process faults. Pointers for future research in this area are also given.

From neural learning to independent components

Independent component analysis (ICA) is a signal processing technique in which a set of random variables are represented in terms of a set of underlying independent component variables. The most central application is the blind source separation for time domain signals. Most approaches to the ICA problem start from information theoretic criteria like maximum likelihood or maximum entropy, and result in numerical on-line learning algorithms. We emphasize here the connection of ICA to neural learning, especially constrained Hebbian learning rules that are nonlinear extensions of principal component analysis learning rules introduced by the author. We review results showing that the nonlinearities in the learning rules are not critical but can be chosen so that the learning rules not only produce independent components, but also have other desirable properties like robustness or fast convergence, contrary to the often used polynomial functions ensuing from cumulant expansions. Also fast batch versions of the learning rules have been developed. Some results are given on the stationary points and their asymptotic stability. It is pointed out that sigmoid-shaped nonlinear functions are a good choice from several points of view.

A Nonlinear Extension of Principal Component Analysis for Clustering and Spatial Differentiation

The limitations in the use of linear principal component analysis (PCA) for identifying morphological features of data sets are discussed. A nonlinear extension of PCA is introduced to provide this capability. It differs from ordinary PCA methods only in that the objective function involves a nonlinear transformation of the principal components. The procedure is shown to be closely related to the exploratory projection pursuit (EPP) algorithm proposed by Friedman (1987), and thus its usefulness in clustering and spatial differentiation applications is apparent. An efficient gradient ascent algorithm is proposed for implementation, based upon the use of a stochastic approximation. The inherent computational advantages in the suggested implementation over other EPP methods are evident, given that EPP methods require the estimation of a density function at every iteration. The nonlinear extension is evaluated by using several well-known datasets, including Fisher’s (1936) Iris data. An industrial application...

Resolution of the fluorescence spectra of plant pigment-complexes using trilinear models

The intensity of fluorescence from a pigment is separately linear in functions of excitation wavelength, emission wavelength, and any chemical treatment which alters overall fluorescence yield. This multiple linearity permits the use of an extension of principal components analysis to resolve overlapping spectra without the use of any additional information. The method is used to resolve the spectra of pigment complexes in pea thylakoids, using the concentration of Mg 2+ as the chemical treatment variable. Two components could be resolved accurately. One has little or no dependence on Mg 2+ ; the other, with an ecitation spectrum resembling LHC II, has a dependence on Mg 2+ which follows the Hill equation with a binding constant of 0.4–0.6 mM and a Hill coefficient of 2.4–3.1.

EDS Image Classification by Projection after Partitioning

In the analysis of multispectral x-ray dispersive images of geological materials, it is found that the large number of mineral phases typically present, leads to a large number of significant principal component images. This large dimensionality produces difficulties in graphically presenting the data, and interactively partitioning die image. A further problem is in the nature of the mineral phase distributions. Some phases will be well separated, in at least one projection direction, and other phases are continuous mixtures, with two or more end points. Both types of mineral phase distributions can be found intimately mixed in a geological sample. The very different types of these distributions implies that routines for searching n-space for clustering will have difficulty producing a good projection of the continuous mixture phase. Also for this reason it is difficult to see how alternative projection techniques can give a meaningful weight to a mixed phase in the presence of well separated components.Here we describe a multi-step image analysis which circumvents the above problem, and promises to be a convenient, and relatively simple extension of principal component analysis (PCA). The analysis method relies on the fact that there is very often, some clear separation of an image after PCA, and that this clear separation can be readily observed in the first few principal component images. Thus the information in the first few principal component scatter diagrams can be partitioned interactively into regions that are well separated from each other, but may be individually complex. PCA can then be applied to the data subsets, and the process can be repeated until a meaningful projection of the data can be viewed in a low dimensional scatter diagram. The real image can then be formed using either density or false color imaging, depending on which is appropriate for the subset alone. The method is thus finding projections after partitioning.

Some mathematical notes on three-mode factor analysis.

The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.