Josep M. Oller

Global Genetic Change Tracks Global Climate Warming in Drosophila subobscura

Comparisons of recent with historical samples of chromosome inversion frequencies provide opportunities to determine whether genetic change is tracking climate change in natural populations. We determined the magnitude and direction of shifts over time (24 years between samples on average) in chromosome inversion frequencies and in ambient temperature for populations of the fly Drosophila subobscura on three continents. In 22 of 26 populations, climates warmed over the intervals, and genotypes characteristic of low latitudes (warm climates) increased in frequency in 21 of those 22 populations. Thus, genetic change in this fly is tracking climate warming and is doing so globally.

A distance between multivariate normal distributions based in an embedding into the Siegel group

This paper shows an embedding of the manifold of multivariate normal densities with informative geometry into the manifold of definite positive matrices with the Siegel metric. This embedding allows us to obtain a general lower bound for the Rao distance, which is itself a distance, and we suggest employing it for statistical purposes, taking into account the similitude of the above related metrics. Further-more, through this embedding, general statistical tests of hypothesis are derived, and some geometrical properties are studied too.

Computing the Rao distance for gamma distributions

This paper deals with the calculation of the Rao distance between Gamma distributions, which is the Riemannian distance induced by the Fisher information matrix on the Gamma statistical model. In this case no closed form expression of the Rao distance is available and a numerical approach is thus necessary to compute the distance. A computer program based on a simple shooting algorithm is presented and discussed.

A distance between elliptical distributions based in an embedding into the Siegel group

This paper describes two different embeddings of the manifolds corresponding to many elliptical probability distributions with the informative geometry into the manifold of positive-definite matrices with the Siegel metric, generalizing a result published previously elsewhere. These new general embeddings are applicable to a wide class of elliptical probability distributions, in which the normal, t-Student and Cauchy are specific examples. A lower bound for the Rao distance is obtained, which is itself a distance, and, through these embeddings, a number of statistical tests of hypothesis are derived.

SOME GEOMETRICAL ASPECTS OF DATA ANALYSIS AND STATISTICS

In this paper we discuss some desirable properties that a distance between probability spaces must satisfy, and from these considerations we introduce the information metric in several different approaches. Once this information metric is introduced, its applications to data analysis and statistics are explored. For instance, the classic analysis of variance methods may be obtained through this geometrical approach, and a geometrical interpretation of likelihood estimation may be given in terms of this metric, by defining in a natural way a distance between the individuals of a statistical population. Finally we consider several geometric properties of a general class of elliptic probability distributions, exhibiting the differential metric, the sectional curvatures, the Ricci tensor, the geodesic equations and, in some cases, the evaluation of the Riemannian distance, called the Rao distance, for this family of probability distributions. Some statistical hypothesis tests are also discussed.

Kernel-PCA data integration with enhanced interpretability

BackgroundNowadays, combining the different sources of information to improve the biological knowledge available is a challenge in bioinformatics. One of the most powerful methods for integrating heterogeneous data types are kernel-based methods. Kernel-based data integration approaches consist of two basic steps: firstly the right kernel is chosen for each data set; secondly the kernels from the different data sources are combined to give a complete representation of the available data for a given statistical task.ResultsWe analyze the integration of data from several sources of information using kernel PCA, from the point of view of reducing dimensionality. Moreover, we improve the interpretability of kernel PCA by adding to the plot the representation of the input variables that belong to any dataset. In particular, for each input variable or linear combination of input variables, we can represent the direction of maximum growth locally, which allows us to identify those samples with higher/lower values of the variables analyzed.ConclusionsThe integration of different datasets and the simultaneous representation of samples and variables together give us a better understanding of biological knowledge.

Kernel Methods for Dimensionality Reduction Applied to the «Omics» Data

Microarray technology has been advanced to the point at which the simultaneous monitoring of gene expression on a genome scale is now possible. Microarray experiments often aim to identify individual genes that are differentially expressed under distinct conditions, such as between two or more phenotypes, cell lines, under different treatment types or diseased and healthy subjects. Such experiments may be the first step towards inferring gene function and constructing gene networks in systems biology.

DISCRIMINANT ANALYSIS ALGORITHM BASED ON A DISTANCE FUNCTION AND ON A BAYESIAN DECISION

We propose a new algorithm for the allocation of an individual to one of several possible groups or populations. The algorithm enables us to define a finite partition over the sample space, based on distance function. This partition is used, jointly with the application of a standard Bayesian decision rule, to allocate individuals to the populations. The algorithm also provides a measure of the allocation confidence for each individual, in a similar manner to that of logistic regression. The error rates for classification are also computed using the leave-one-out method. Results are compared with those obtained with other discriminant analysis techniques previously reported: Fishers linear discriminant function, the quadratic discriminant function, logistic discrimination, and others.

Response to Comment on “Global Genetic Change Tracks Global Climate Warming in Drosophila subobscura”

Rodríguez-Trelles and Rodríguez advocate standardizing old and new collections by climate rather than by calendar and also propose that some of our samples were biased by inappropriate timing. Their first suggestion applies to few species, and its implementation alters photoperiodic cues. Their second point is valid, but our conclusions are robust: Observed genetic changes reflect global warming, not sampling artifacts.

A biplot method for multivariate normal populations with unequal covariance matrices

Some previous ideas about non-linear biplots to achieve a joint representation of multivariate normal populations and any parametric function without assumptions about the covariance matrices are extended. Usual restrictions on the covariance matrices (such as homogeneity) are avoided. Variables are represented as curves corresponding to the directions of maximum means variation. To demonstrate the versatility of the method, the representation of variances and covariances as an example of further possible interesting parametric functions have been developed. This method is illustrated with two different data sets, and these results are compared with those obtained using two other distances for the normal multivariate case: the Mahalanobis distance (assuming a common covariance matrix for all populations) and Rao’s distance, assuming a common eigenvector structure for all the covariance matrices.