The Standard Biplot

Biplots are graphical displays of data matrices based on the decomposition of a matrix as the product of two matrices. Elements of these two matrices are used as coordinates for the rows and columns of the data matrix, with an interpretation of the joint presentation that relies on the properties of the scalar product. Because the decomposition is not unique, there are several alternative ways to scale the row and column points of the biplot, which can cause confusion amongst users, especially when software packages are not united in their approach to this issue. We propose a new scaling of the solution, called the standard biplot, which applies equally well to a wide variety of analyses such as correspondence analysis, principal component analysis, log-ratio analysis and the graphical results of a discriminant analysis/MANOVA, in fact to any method based on the singular-value decomposition. The standard biplot also handles data matrices with widely different levels of inherent variance. Two concepts taken from correspondence analysis are important to this idea: the weighting of row and column points, and the contributions made by the points to the solution. In the standard biplot one set of points, usually the rows of the data matrix, optimally represent the positions of the cases or sample units, which are weighted and usually standardized in some way unless the matrix contains values that are comparable in their raw form. The other set of points, usually the columns, is represented in accordance with their contributions to the low-dimensional solution. As for any biplot, the projections of the row points onto vectors defined by the column points approximate the centred and (optionally) standardized data. The method is illustrated with several examples to demonstrate how the standard biplot copes in different situations to give a joint map which needs only one common scale on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot readable. The proposal also solves the problem in correspondence analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important.

Lyfe-cycle effects on household expenditures: A latent-variable approach

Using data from the Spanish household budget survey, we investigate life-cycle effects on several product expenditures. A latent-variable model approach is adopted to evaluate the impact of income on expenditures, controlling for the number of members in the family. Two latent factors underlying repeated measures of monetary and non-monetary income are used as explanatory variables in the expenditure regression equations, thus avoiding possible bias associated to the measurement error in income. The proposed methodology also takes care of the case in which product expenditures exhibit a pattern of infrequent purchases. Multiple-group analysis is used to assess the variation of key parameters of the model across various household life-cycle typologies. The analysis discloses significant life-cycle effects on the mean levels of expenditures; it also detects significant life-cycle effects on the way expenditures are affected by income and family size. Asymptotic robust methods are used to account for possible non-normality of the data.

Contribution biplots

In order to interpret the biplot it is necessary to know which points usually variables are the ones that are important contributors to the solution, and this information is available separately as part of the biplot s numerical results. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic directly into the graphical display, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses such as correspondence analysis, principal component analysis, log-ratio analysis and the graphical results of a discriminant analysis/MANOVA, in fact to any method based on the singular-value decomposition. In the contribution biplot one set of points, usually the rows of the data matrix, optimally represent the spatial positions of the cases or sample units, according to some distance measure that usually incorporates some form of standardization unless all data are comparable in scale. The other set of points, usually the columns, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that usually only one common scale for row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, this version of the biplot also solves the problem in correspondence analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution.

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.

Forest fires: Evaluation of government intervention measures

The purpose of this paper is to examine the relation between government measures, volunteer participation, climate variables and forest fires. A number of studies have related forest fires to causes of ignition, to fire history in one area, to the type of vegetation and weathercharacteristics or to community institutions, but there is little research on the relation between fire production and government prevention and extinction measures from a policy evaluation perspective.An observational approach is first applied to select forest fires in the north east of Spain. Taking a selection of fires with a certain size, a multiple regression analysis is conducted to find significant relations between policy instruments under the control of the government and the number of hectares burn in each case, controlling at the same time the effect of weather conditions and other context variables. The paper brings evidence on the effects of simultaneity and the relevance of recurring to army soldiers in specific days with extraordinary high simultaneity. The analysis also brings light on the effectiveness of twopreventive policies and of helicopters for extinction tasks.

Country effects in ISSP-1993 environmental data: Comparison of SEM approaches

Structural equation models (SEM) are commonly used to analyze the relationship between variables some of which may be latent, such as individual ``attitude'' to and ``behavior'' concerning specific issues. A number of difficulties arise when we want to compare a large number of groups, each with large sample size, and the manifest variables are distinctly non-normally distributed. Using an specific data set, we evaluate the appropriateness of the following alternative SEM approaches: multiple group versus MIMIC models, continuous versus ordinal variables estimation methods, and normal theory versus non-normal estimation methods. The approaches are applied to the ISSP-1993 Environmental data set, with the purpose of exploring variation in the mean level of variables of ``attitude'' to and ``behavior''concerning environmental issues and their mutual relationship across countries. Issues of both theoretical and practical relevance arise in the course of this application.

Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise

We study the dynamics of reaction-diffusion fronts under the influence of multiplicative noise. An approximate theoretical scheme is introduced to compute the velocity of the front and its diffusive wandering due to the presence of noise. The theoretical approach is based on a multiple scale analysis rather than on a small noise expansion and is confirmed with numerical simulations for a wide range of the noise intensity. We report on the possibility of noise sustained solutions with a continuum of possible velocities, in situations where only a single velocity is allowed without noise.

A Neanderthal Lower Incisor from Cova del Gegant (Sitges, Barcelona, Spain)

Cova del Gegant is located near the city of Sitges (Barcelona, Spain). The cave is a small karst system which contains Upper Pleistocene archaeological and paleontological material (DauRa et al., 2005). The site was first excavated in 1954 and then in 1972 and 1974- (Viñas, 1972; Viñas & Villalta, 1975) and in 1985 and 1989 (maRtínez et al., 1985; moRa, 1988; maRtínez et al., 1990). Finally, in 2007, Grup de Recerca del Quaternari has restarted the archaeological research at Cova del Gegant (DauRa, 2008; DauRa et al., 2010). A human mandible was recovered during the first field season in 1954 and was recently published by DauRa et al. (2005). In the present study, we describe a new human tooth (left I2) that appeared, like the mandible, in a revision of the faunal material recovered from the site in 1974-1975. The specimen preserves the entire crown and the cervical two thirds of the root (Figure 1). The lack of the root apex makes it difficult to determine if the tooth was fully developed at the time of death. However, CT analysis reveals a pulp cavity that could be still open, suggesting root formation was incomplete. The specimen shows only slight dental wear corresponding to stage 2 of Molnar (1971 en Hillson, 1996). Morphologically, the crown shows slight shovelling and a lingual tubercle and appears similar to Neandertal incisors. Standard crown measurements (buccolingual diameter=7.7 mm; mesiodistal diameter= 7.3 mm) (Figure 2) suggest a fairly large tooth, particularly in the BL dimension, again resembling Neandertals in this regard. Discriminant analysis classified the Gegant incisor as Neandertal with a 99.8% posterior probability (Table 2). Association of this tooth with the previously described mandible is considered unlikely given the different ages at death estimated for each. Thus, there appear to be two individuals preserved in the sediments of the Gegant cave, one adult and one subadult (around 8-10 years old).

Predictions of the MCMI-III personality disorders from NEO-PI-R domains and facets: Comparison between American and Spanish samples

Este estudio ex post facto analiza las relaciones entre las dimensiones y facetas del NEO-PI-R y los 14 trastornos de personalidad del MCMI-III en una muestra no clínica española (N = 674). Se exploran las diferencias y similitudes con los resul- tados de Dyce y O’Connor en una muestra americana con los mismos instrumentos. Como se esperaba, los análisis factoriales de facetas reteniendo cinco factores mostraron un modelo de relaciones muy similar entre ambas muestras, con un coeficiente de la congruencia total de 0,92, y coeficientes de congruencia de factor aceptables, salvo para el factor Apertura (0,68). En consonancia con las predicciones de Widiger y Widiger et al. los porcentajes de correlaciones significativas estaban alrededor de 60% en ambas muestras, y la mayoría coincidían. El análisis de regresión múltiple con dimensiones también reveló un gran parecido entre los resultados americanos y españoles, Neuroticismo fue el predictor más relacionado con los trastornos de personalidad. Se encontraron diferencias en las regresiones por facetas, aunque la varianza explicada fue prácticamente la misma que en las dimensiones. Se discute la validez transcultural y el valor predictivo del NEO-PI-R sobre los trastornos de personalidad del MCMI-III, junto con las ventajas relativas de las facetas sobre las dimensiones.

Factores de explotación asociados a la duración del anestro postparto en vacas nodrizas de razas Parda de Montaña y Pirenaica

Para determinar los factores de explotación relacionados con la reactivación ovárica postparto en vacas nodrizas se realizó un análisis global de una serie de indicadores productivos y la duración del anestro postparto (APP) de 549 vacas explotadas en condiciones extensivas. Debido a la naturaleza multifactorial del proceso en estudio se eligió la metodología estadística multivariante (Análisis Factorial de Correspondencias Múltiples y Análisis Cluster). La duración del APP estuvo asociada a cuatro factores que explicaron el 59% de la heterogeneidad inicial de la muestra y que se definieron como: «Alimentación preparto» (19% de la inercia), «Alimentación postparto-Edad» (16.4%), «Manejo del ternero» (13%) y «Dificultad al parto» (10.5%). Estos factores se introdujeron en un Análisis Cluster que identificó cinco grupos de vacas con características productivas y reproductivas diferentes, y que denominamos como: «Primíparas», «Acceso restringido», «Acceso Libre-Parda de Montaña», «Parto de otoño» y «Parto de primavera». La raza no estuvo relacionada con la duración del APP, aunque el análisis Cluster asoció los largos APP inducidos por la crianza libre con la raza Parda de Montaña. En la raza Parda de Montaña, la duración del APP fue mayor en primavera que en otoño debido a diferencias nutricionales más que a un efecto estacional en sí. El parto de otoño se adaptó mejor a las condiciones de montaña seca.

Preferred Spatial Frequencies for Human Face Processing Are Associated with Optimal Class Discrimination in the Machine

Psychophysical studies suggest that humans preferentially use a narrow band of low spatial frequencies for face recognition. Here we asked whether artificial face recognition systems have an improved recognition performance at the same spatial frequencies as humans. To this end, we estimated recognition performance over a large database of face images by computing three discriminability measures: Fisher Linear Discriminant Analysis, Non-Parametric Discriminant Analysis, and Mutual Information. In order to address frequency dependence, discriminabilities were measured as a function of (filtered) image size. All three measures revealed a maximum at the same image sizes, where the spatial frequency content corresponds to the psychophysical found frequencies. Our results therefore support the notion that the critical band of spatial frequencies for face recognition in humans and machines follows from inherent properties of face images, and that the use of these frequencies is associated with optimal face recognition performance.