Depth-based inference for functional data

We propose robust inference tools for functional data based on the notion of depth for curves. We extend the ideas of trimmed regions, contours and central regions to functions and study their structural properties and asymptotic behavior. Next, we introduce a scale curve to describe dispersion in a sample of functions. The computational burden of these techniques is not heavy and so they are also adequate to analyze high-dimensional data. All these inferential methods are applied to different real data sets

On the concept of depth for functional data

The statistical analysis of functional data is a growing need in many research areas. We propose a new depth notion for functional observations based on the graphic representation of the curves. Given a collection of functions, it allows to establish the centrality of a function and provides a natural center-outward ordering of the sample curves. Robust statistics such as the median function or a trimmed mean function can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally very fast and turns out to be convenient to study high-dimensional observations. The natural properties are established for the new depth and the uniform consistency of the sample depth is proved. Simulation results show that the trimmed mean presents a better behavior than the mean for contaminated models. Several real data sets are considered to illustrate this new concept of depth. Finally, we use this new depth to generalize to functions the Wilcoxon rank sum test. It allows to decide whether two groups of curves come from the same population. This functional rank test is applied to girls and boys growth curves concluding that they present different growth patterns

Regular solutions to a supercritical elliptic problem in exterior domains

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal than 3. We prove that, for \lambda small, this problem admits infinitely many regular solutions

Depth-based classification for functional data

Classification is an important task when data are curves. Recently, the notion of statistical depth has been extended to deal with functional observations. In this paper, we propose robust procedures based on the concept of depth to classify curves. These techniques are applied to a real data example. An extensive simulation study with contaminated models illustrates the good robustness properties of these depth-based classification methods

Simplicial similarity and its application to hierarchical clustering

In the present document, an extension of the statistical depth notion is introduced with the aim to allow for measuring proximities between pairs of points. In particular, we will extend the simplicial depth function, which measures how central is a point by using random simplices (triangles in the two-dimensional space). The paper is structured as follows: In first place, there is a brief introduction to statistical depth functions. Next, the simplicial similarity function will be defined and its properties studied. Finally, we will present a few graphical examples in order to show its behavior with symmetric and asymmetric distributions, and apply the function to hierarchical clustering.Statistical depth, Similarity measures, Hierarchical clustering

Unemployment and Inflation Persistence in Spain: Are There Phillips Trade-Offs?

This paper studies the dynamic behavior of inflation and unemployment in Spain during the period 1964?1997. In particular, we analyze the implications of high persistence in both unemployment and inflation dynamics for inference regarding the size of Phillips trade-offs and sacrifice ratios in the Spanish economy, in response to a demand shock. To do so we use a Stuctural VAR approach with several identification outlines which give rise to alternative interpretations of the joint unemployment-inflation dynamics. When using a bivariate VAR we cannot reject the existence of a permanent output loss of one-half of one percentage point for each percentage point of permanent disinflation. However, when the VAR is augmented with a third variable, in order to disentangle monetary from non-monetary shocks within the demand class, the evidence favours a lower and marginally permanent trade-off with an output loss of about one-fourth of one percentage point.Publicad

A half-graph depth for functional data

A recent and highly attractive area of research in statistics is the analysis of functional data. In this paper a new definition of depth for functional observations is introduced based on the notion of "half-graph" of a curve. It has computational advantages with respect to other concepts of depth previously proposed. The half-graph depth provides a natural criterion to measure the centrality of a function within a sample of curves. Based on this depth a sample of curves can be ordered from the center outward and L-statistics are defined. The properties of the half-graph depth, such as the consistency and uniform convergence, are established. A simulation study shows the robustness of this new definition of depth when the curves are contaminated. Finally real data examples are analyzed

Detecting synchronization in spatially extended discrete systems by complexity measurements

The synchronization of two stochastically coupled one-dimensional cellular automata (CA) is analyzed. It is shown that the transition to synchronization is characterized by a dramatic increase of the statistical complexity of the patterns generated by the difference automaton. This singular behavior is verified to be present in several CA rules displaying complex behavior.Comment: 4 pages, 2 figures; you can also visit http://add.unizar.es/public/100_16613/index.htm