Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

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

Analysis of matched matrices

We consider the joint visualization of two matrices which have common rowsand columns, for example multivariate data observed at two time pointsor split accord-ing to a dichotomous variable. Methods of interest includeprincipal components analysis for interval-scaled data, or correspondenceanalysis for frequency data or ratio-scaled variables on commensuratescales. A simple result in matrix algebra shows that by setting up thematrices in a particular block format, matrix sum and difference componentscan be visualized. The case when we have more than two matrices is alsodiscussed and the methodology is applied to data from the InternationalSocial Survey Program.

From correspondence analysis to multiple and joint correspondence analysis

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.

Biplots of compositional data

The singular value decomposition and its interpretation as alinear biplot has proved to be a powerful tool for analysing many formsof multivariate data. Here we adapt biplot methodology to the specifficcase of compositional data consisting of positive vectors each of whichis constrained to have unit sum. These relative variation biplots haveproperties relating to special features of compositional data: the studyof ratios, subcompositions and models of compositional relationships. Themethodology is demonstrated on a data set consisting of six-part colourcompositions in 22 abstract paintings, showing how the singular valuedecomposition can achieve an accurate biplot of the colour ratios and howpossible models interrelating the colours can be diagnosed.

Weighted Euclidean biplots

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, the method 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.

A Small Baseline DIFSAR Approach for Investigating Deformations on Full Resolution SAR Interferograms

This paper presents a differential synthetic apertureradar (SAR) interferometry (DIFSAR) approach for investigatingdeformation phenomena on full-resolution DIFSAR interferograms.In particular, our algorithm extends the capabilityof the small-baseline subset (SBAS) technique that relies onsmall-baseline DIFSAR interferograms only and is mainly focusedon investigating large-scale deformations with spatial resolutionsof about 100 100 m. The proposed technique is implemented byusing two different sets of data generated at low (multilook data)and full (single-look data) spatial resolution, respectively. Theformer is used to identify and estimate, via the conventional SBAStechnique, large spatial scale deformation patterns, topographicerrors in the available digital elevation model, and possibleatmospheric phase artifacts; the latter allows us to detect, onthe full-resolution residual phase components, structures highlycoherent over time (buildings, rocks, lava, structures, etc.), as wellas their height and displacements. In particular, the estimation ofthe temporal evolution of these local deformations is easily implementedby applying the singular value decomposition technique.The proposed algorithm has been tested with data acquired by theEuropean Remote Sensing satellites relative to the Campania area(Italy) and validated by using geodetic measurements.

Robust power allocation algorithms for MIMO OFDM systems with imperfect CSI

This paper presents a Bayesian approach to the design of transmit prefiltering matrices in closed-loop schemes robust to channel estimation errors. The algorithms are derived for a multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) system. Two different optimizationcriteria are analyzed: the minimization of the mean square error and the minimization of the bit error rate. In both cases, the transmitter design is based on the singular value decomposition (SVD) of the conditional mean of the channel response, given the channel estimate. The performance of the proposed algorithms is analyzed,and their relationship with existing algorithms is indicated. As withother previously proposed solutions, the minimum bit error rate algorithmconverges to the open-loop transmission scheme for very poor CSI estimates.

Simultaneous measurement of regional cerebral blood flow by perfusion CT and stable xenon CT: a validation study.

BACKGROUND AND PURPOSE: Knowledge of cerebral blood flow (CBF) alterations in cases of acute stroke could be valuable in the early management of these cases. Among imaging techniques affording evaluation of cerebral perfusion, perfusion CT studies involve sequential acquisition of cerebral CT sections obtained in an axial mode during the IV administration of iodinated contrast material. They are thus very easy to perform in emergency settings. Perfusion CT values of CBF have proved to be accurate in animals, and perfusion CT affords plausible values in humans. The purpose of this study was to validate perfusion CT studies of CBF by comparison with the results provided by stable xenon CT, which have been reported to be accurate, and to evaluate acquisition and processing modalities of CT data, notably the possible deconvolution methods and the selection of the reference artery. METHODS: Twelve stable xenon CT and perfusion CT cerebral examinations were performed within an interval of a few minutes in patients with various cerebrovascular diseases. CBF maps were obtained from perfusion CT data by deconvolution using singular value decomposition and least mean square methods. The CBF were compared with the stable xenon CT results in multiple regions of interest through linear regression analysis and bilateral t tests for matched variables. RESULTS: Linear regression analysis showed good correlation between perfusion CT and stable xenon CT CBF values (singular value decomposition method: R(2) = 0.79, slope = 0.87; least mean square method: R(2) = 0.67, slope = 0.83). Bilateral t tests for matched variables did not identify a significant difference between the two imaging methods (P >.1). Both deconvolution methods were equivalent (P >.1). The choice of the reference artery is a major concern and has a strong influence on the final perfusion CT CBF map. CONCLUSION: Perfusion CT studies of CBF achieved with adequate acquisition parameters and processing lead to accurate and reliable results.

Novel Regression Methods For Spectral Data

Singular Value Decomposition (SVD), Principal Component Analysis (PCA) and Multiple Linear Regression (MLR) are some of the mathematical pre- liminaries that are discussed prior to explaining PLS and PCR models. Both PLS and PCR are applied to real spectral data and their di erences and similarities are discussed in this thesis. The challenge lies in establishing the optimum number of components to be included in either of the models but this has been overcome by using various diagnostic tools suggested in this thesis. Correspondence analysis (CA) and PLS were applied to ecological data. The idea of CA was to correlate the macrophytes species and lakes. The di erences between PLS model for ecological data and PLS for spectral data are noted and explained in this thesis. i

Forecasting financial weather - can we foresee market sentiment? Spectrum of stock price behavior - NYSE case study

The desire to create a statistical or mathematical model, which would allow predicting the future changes in stock prices, was born many years ago. Economists and mathematicians are trying to solve this task by applying statistical analysis and physical laws, but there are still no satisfactory results. The main reason for this is that a stock exchange is a non-stationary, unstable and complex system, which is influenced by many factors. In this thesis the New York Stock Exchange was considered as the system to be explored. A topological analysis, basic statistical tools and singular value decomposition were conducted for understanding the behavior of the market. Two methods for normalization of initial daily closure prices by Dow Jones and S&P500 were introduced and applied for further analysis. As a result, some unexpected features were identified, such as a shape of distribution of correlation matrix, a bulk of which is shifted to the right hand side with respect to zero. Also non-ergodicity of NYSE was confirmed graphically. It was shown, that singular vectors differ from each other by a constant factor. There are for certain results no clear conclusions from this work, but it creates a good basis for the further analysis of market topology.

Static waves in corporate space : characterizing oscillating trading patterns in New York stock exchange

Various researches in the field of econophysics has shown that fluid flow have analogous phenomena in financial market behavior, the typical parallelism being delivered between energy in fluids and information on markets. However, the geometry of the manifold on which market dynamics act out their dynamics (corporate space) is not yet known. In this thesis, utilizing a Seven year time series of prices of stocks used to compute S&P500 index on the New York Stock Exchange, we have created local chart to the corporate space with the goal of finding standing waves and other soliton like patterns in the behavior of stock price deviations from the S&P500 index. By first calculating the correlation matrix of normalized stock price deviations from the S&P500 index, we have performed a local singular value decomposition over a set of four different time windows as guides to the nature of patterns that may emerge. I turns out that in almost all cases, each singular vector is essentially determined by relatively small set of companies with big positive or negative weights on that singular vector. Over particular time windows, sometimes these weights are strongly correlated with at least one industrial sector and certain sectors are more prone to fast dynamics whereas others have longer standing waves.

Projection de la mortalité aux âges avancées au Canada : comparaison de trois modèles

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

On some Density Theorems in Number Theory and Group Theory

Gowers, dans son article sur les matrices quasi-aléatoires, étudie la question, posée par Babai et Sos, de l'existence d'une constante $c>0$ telle que tout groupe fini possède un sous-ensemble sans produit de taille supérieure ou égale a $c|G|$. En prouvant que, pour tout nombre premier $p$ assez grand, le groupe $PSL_2(\mathbb{F}_p)$ (d'ordre noté $n$) ne posséde aucun sous-ensemble sans produit de taille $c n^{8/9}$, il y répond par la négative. Nous allons considérer le probléme dans le cas des groupes compacts finis, et plus particuliérement des groupes profinis $SL_k(\mathbb{Z}_p)$ et $Sp_{2k}(\mathbb{Z}_p)$. La premiére partie de cette thése est dédiée à l'obtention de bornes inférieures et supérieures exponentielles pour la mesure suprémale des ensembles sans produit. La preuve nécessite d'établir préalablement une borne inférieure sur la dimension des représentations non-triviales des groupes finis $SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ et $Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Notre théoréme prolonge le travail de Landazuri et Seitz, qui considérent le degré minimal des représentations pour les groupes de Chevalley sur les corps finis, tout en offrant une preuve plus simple que la leur. La seconde partie de la thése à trait à la théorie algébrique des nombres. Un polynome monogéne $f$ est un polynome unitaire irréductible à coefficients entiers qui endengre un corps de nombres monogéne. Pour un nombre premier $q$ donné, nous allons montrer, en utilisant le théoréme de densité de Tchebotariov, que la densité des nombres premiers $p$ tels que $t^q -p$ soit monogéne est supérieure ou égale à $(q-1)/q$. Nous allons également démontrer que, quand $q=3$, la densité des nombres premiers $p$ tels que $\mathbb{Q}(\sqrt[3]{p})$ soit non monogéne est supérieure ou égale à $1/9$.

Modeling and Analysis of Some Heavy Tailed Time Series

The thesis has covered various aspects of modeling and analysis of finite mean time series with symmetric stable distributed innovations. Time series analysis based on Box and Jenkins methods are the most popular approaches where the models are linear and errors are Gaussian. We highlighted the limitations of classical time series analysis tools and explored some generalized tools and organized the approach parallel to the classical set up. In the present thesis we mainly studied the estimation and prediction of signal plus noise model. Here we assumed the signal and noise follow some models with symmetric stable innovations.We start the thesis with some motivating examples and application areas of alpha stable time series models. Classical time series analysis and corresponding theories based on finite variance models are extensively discussed in second chapter. We also surveyed the existing theories and methods correspond to infinite variance models in the same chapter. We present a linear filtering method for computing the filter weights assigned to the observation for estimating unobserved signal under general noisy environment in third chapter. Here we consider both the signal and the noise as stationary processes with infinite variance innovations. We derived semi infinite, double infinite and asymmetric signal extraction filters based on minimum dispersion criteria. Finite length filters based on Kalman-Levy filters are developed and identified the pattern of the filter weights. Simulation studies show that the proposed methods are competent enough in signal extraction for processes with infinite variance.Parameter estimation of autoregressive signals observed in a symmetric stable noise environment is discussed in fourth chapter. Here we used higher order Yule-Walker type estimation using auto-covariation function and exemplify the methods by simulation and application to Sea surface temperature data. We increased the number of Yule-Walker equations and proposed a ordinary least square estimate to the autoregressive parameters. Singularity problem of the auto-covariation matrix is addressed and derived a modified version of the Generalized Yule-Walker method using singular value decomposition.In fifth chapter of the thesis we introduced partial covariation function as a tool for stable time series analysis where covariance or partial covariance is ill defined. Asymptotic results of the partial auto-covariation is studied and its application in model identification of stable auto-regressive models are discussed. We generalize the Durbin-Levinson algorithm to include infinite variance models in terms of partial auto-covariation function and introduce a new information criteria for consistent order estimation of stable autoregressive model.In chapter six we explore the application of the techniques discussed in the previous chapter in signal processing. Frequency estimation of sinusoidal signal observed in symmetric stable noisy environment is discussed in this context. Here we introduced a parametric spectrum analysis and frequency estimate using power transfer function. Estimate of the power transfer function is obtained using the modified generalized Yule-Walker approach. Another important problem in statistical signal processing is to identify the number of sinusoidal components in an observed signal. We used a modified version of the proposed information criteria for this purpose.

Enhancement of Calcifications in Mammograms using Volterra Series based Quadratic Filter

The paper summarizes the design and implementation of a quadratic edge detection filter, based on Volterra series, for enhancing calcifications in mammograms. The proposed filter can account for much of the polynomial nonlinearities inherent in the input mammogram image and can replace the conventional edge detectors like Laplacian, gaussian etc. The filter gives rise to improved visualization and early detection of microcalcifications, which if left undetected, can lead to breast cancer. The performance of the filter is analyzed and found superior to conventional spatial edge detectors