Andrés M. Alonso

Overview of object oriented data analysis.

Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where a variety of Euclidean approaches, such as principal components analysis, have been very successful. Challenges in modern medical image analysis motivate the statistical analysis of populations of more complex data objects that are elements of mildly non-Euclidean spaces, such as lie groups and symmetric spaces, or of strongly non-Euclidean spaces, such as spaces of tree-structured data objects. These new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics. The notion of object oriented data analysis also impacts data analysis, through providing a framework for discussion of the many choices needed in many modern complex data analyses, especially in interdisciplinary contexts.

Time series clustering based on forecast densities

A new clustering method for time series is proposed, based on the full probability density of the forecasts. First, a resampling method combined with a nonparametric kernel estimator provides estimates of the forecast densities. A measure of discrepancy is then defined between these estimates and the resulting dissimilarity matrix is used to carry out the required cluster analysis. Applications of this method to both simulated and real life data sets are discussed.

Seasonal Dynamic Factor Analysis and Bootstrap Inference: Application to Electricity Market Forecasting

In this work, we propose the Seasonal Dynamic Factor Analysis (SeaDFA), an extension of Nonstationary Dynamic Factor Analysis, through which one can deal with dimensionality reduction in vectors of time series in such a way that both common and specific components are extracted. Furthermore, common factors are able to capture not only regular dynamics (stationary or not) but also seasonal ones, by means of the common factors following a multiplicative seasonal VARIMA(p, d, q) × (P, D, Q)s model. Additionally, a bootstrap procedure that does not need a backward representation of the model is proposed to be able to make inference for all the parameters in the model. A bootstrap scheme developed for forecasting includes uncertainty due to parameter estimation, allowing enhanced coverage of forecasting intervals. A challenging application is provided. The new proposed model and a bootstrap scheme are applied to an innovative subject in electricity markets: the computation of long-term point forecasts and prediction intervals of electricity prices. Several appendices with technical details, an illustrative example, and an additional table are available online as Supplementary Materials.

Non-linear time series clustering based on non-parametric forecast densities

The problem of clustering time series is studied for a general class of non-parametric autoregressive models. The dissimilarity between two time series is based on comparing their full forecast densities at a given horizon. In particular, two functional distances are considered: L^1 and L^2. As the forecast densities are unknown, they are approximated using a bootstrap procedure that mimics the underlying generating processes without assuming any parametric model for the true autoregressive structure of the series. The estimated forecast densities are then used to construct the dissimilarity matrix and hence to perform clustering. Asymptotic properties of the proposed method are provided and an extensive simulation study is carried out. The results show the good behavior of the procedure for a wide variety of nonlinear autoregressive models and its robustness to non-Gaussian innovations. Finally, the proposed methodology is applied to a real dataset involving economic time series.

Discrimination of locally stationary time series using wavelets

Time series are sometimes generated by processes that change suddenly from one stationary regime to another, with no intervening periods of transition of any significant duration. A good example of this is provided by seismic data, namely, waveforms of earthquakes and explosions. In order to classify an unknown event as either an earthquake or an explosion, statistical analysts might be helped by having at their disposal an automatic means of identifying, at any time, which pattern prevails. Several authors have proposed methods to tackle this problem by combining the techniques of spectral analysis with those of discriminant analysis. The goal is to develop a discriminant scheme for locally stationary time series such as earthquake and explosion waveforms, by combining the techniques of wavelet analysis with those of discriminant analysis.

On sieve bootstrap prediction intervals

In this paper we consider a sieve bootstrap method for constructing nonparametric prediction intervals for a general class of linear processes. We show that the sieve bootstrap provides consistent estimators of the conditional distribution of future values given the observed data.

Discriminant analysis of multivariate time series: Application to diagnosis based on ECG signals

In analysing ECG data, the main aim is to differentiate between the signal patterns of healthy subjects and those of individuals with specific heart conditions. We propose an approach for classifying multivariate ECG signals based on discriminant and wavelet analyses. For this purpose we use multiple-scale wavelet variances and wavelet correlations to distinguish between the patterns of multivariate ECG signals based on the variability of the individual components of each ECG signal and on the relationships between every pair of these components. Using the results of other ECG classification studies in the literature as references, we demonstrate that our approach applied to 12-lead ECG signals from a particular database compares favourably. We also demonstrate with real and synthetic ECG data that our approach to classifying multivariate time series out-performs other well-known approaches for classifying multivariate time series.

Supervised classification for functional data: A weighted distance approach

A natural methodology for discriminating functional data is based on the distances from the observation or its derivatives to group representative functions (usually the mean) or their derivatives. It is proposed to use a combination of these distances for supervised classification. Simulation studies show that this procedure performs very well, resulting in smaller testing classification errors. Applications to real data show that this technique behaves as well as-and in some cases better than-existing supervised classification methods for functions.

Clustering Time Series of Sea Levels: Extreme Value Approach

In this paper, long (>40 years) hourly tide gauge records from the North Atlantic are analyzed. A new time series clustering approach which combines Bayesian methodology, extreme value theory, and classification techniques is adopted for the analysis of the regional variability of sea-level extremes. The tide gauge records are clustered on the basis of their corresponding predictive distributions for 25-, 50-, and 100-year return values. The results of the cluster analysis show a clear distinction between the higher latitude stations for which the return values are largest and the remaining locations. This distinction reflects in the U.S. east coast the transition between the Scottian shelf and Gulf of Maine area and the mid-Atlantic Bight area. For the stations at lower latitudes the results show a grouping based on return levels that is not a function of geographical proximity but reflects local effects in extreme sea levels associated with the specific location of each tide gauge.

Comparison of time series using subsampling

In this paper we propose a procedure based on the subsampling techniques for the comparison of stationary time series that are not necessarily independent. We study a test based on the Euclidean distance between the autocorrelation functions of two series. Consistency of the proposed method is established. We present a Monte Carlo study with the size and the power of the proposed test.