Biplots of fuzzy coded data

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.

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.

A new light on Minkowski's? $(x)$ function

The well--known Minkowski's? $(x)$ function is presented as the asymptotic distribution function of an enumeration of the rationals in (0,1] based on their continued fraction representation. Besides, the singularity of ?$(x)$ is clearly proved in two ways: by exhibiting a set of measure one in which ?ï$(x)$ = 0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and viceversa. These sets are described by means of metrical properties of different systems for real number representation.

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.

Tying up the loose ends in simple correspondence analysis

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.

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.

Canvis i permanències a les closes de l'Alt Empordà

Una de les causes principals que van motivar la creació del Parc Natural dels Aiguamolls de l'Empordà va ser, indiscutiblement, la potencialitat de les zones marjalenques empordaneses d'acollir un gran nombre i varietat d'aus. El marc legislatiu que va fer efectiva la protecció, però, no limitava la seva acció a aquest grup, sinó que l'estenia al conjunt de flora i fauna present a la zona, i a tots els sistemes propers que n'havien de garantir la permanència. Així, a més de definir-se unes àrees de màxima protecció sota la denominació de Reserves Naturals Integrals, els límits del PNAE engloben àrees potser menys emblemàtiques, però que són clau per al manteniment del conjunt de sistemes d'aiguamolls a l'Alt Empordà. El present article se centra en un d'aquests sistemes, que tradicionalment ha realitzat un paper d'actor secundari, però que té unes qualitats que, almenys des d'una aproximació geogràfica, el converteixen en protagonista. Es tracta de les closes

Els Aiguamolls de la Puda de Banyoles: una proposta de gestió

La Puda, no fa pas tants anys, constituïa un espai emblemàtic de Banyoles i, en canvi, avui dia ha esdevingut un àmbit totalment al marge de la dinàmica de la ciutat. La realitat actual mostra que és un espai que no ha desenvolupat les seves potencialitats, fet que l'ha portat a no posseir plenament les característiques d'un parc urbà ni les pròpies d'una àrea natural. L'objectiu principal d'aquest article és proposar un model de gestió per als aiguamolls de la Puda, amb el propòsit de regenerar i protegir ambientalment la zona i recuperar per a la societat banyolina un espai oblidat. Es tracta, dones, de contemplar-ne tant la vessant natural com la social i de fer-ho prenent en consideració diferents escales d’anàlisi. Les raons que condueixen a tal objectiu es fonamenten en la convicció que tot i la precària salut ambiental que pateix la lona, encara està en condicions de desenvolupar plenament el seu alt valor ecològic, i que tot i la marginalitat que presenta respecte a la resta del municipi, manté la possibilitat d'esdevenir un espai singular i atractiu que participi activament de la vida banyolina

Morfogénesis del valle de Sant Miquel de Campmajor

A series of circumstances which converge in a little valley in the province of Girona, result in a peculiar type of morphogenetic evolution and a quite singular mechanical instability of its buttons. Very recent tectonic movements as well as dissolution phenomena of its subsoil due to pressurized underground water have played an important role in the morphogenesis. Such conditions have resulted in a fast morphological evolution which a progressive lowering of the valley floor as well as peculiar morphological features which favour the existence of numerous pits caused by sudden collapses distributed in altitude and time

L’Edició en lletra gòtica de l’Speculum al foder (1917). Història d’un misteriós exemplar d’infern

La filologia, tret de rares excepcions, acostuma a prestar poc interès a la bibliofília. Potser perquè es considera que un text acurat i fiable està renyit amb un llibre imprès en bon paper Japó o de fil, del amb una tipografia impecable, sovint il·lustrat i a un preu que sol ser alt, com correspon a un desplegament artesanal d’aquesta mena. Filologia i bibliofília, en una paraula, tenen vies diferents de difusió. I, amb tot, l’univers de la bibliofília custodia secrets molt ben guardats de la nostra història lingüística i literària, tan singular i plena de sotracs de diversa mena. Secrets que criden poderosament l’atenció del filòleg. El cens i descripció de testimonis del tractadet eròtic medieval conegut com a Speculum al foder, per exemple, amaga un curiós enigma bibliogràfic: un imprès en tipografia gòtica i sense peu editorial, un veritable incunable modern, fruit rar del mateix impuls que cent anys enrere es va ensenyorir de les arts plàstiques i constructives d’aquest país per omplir-lo de magnífiques fantasies medievalitzants. És d’aquest misteriós exemplar d’infern que s’ocupa el treballet que segueix

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.

Homotopy Batalin-Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.