Pierpaolo D’Urso

Dissimilarity measures for time trajectories

In this paper, by considering one of the possible geometric representations of time arrays in the ‘object space’ (space of the units), we analyze different dissimilarity measures between multivariate time trajectories of the units, which are classified, systematically, in various approaches, by taking into account their features. In particular, we define three classes of dissimilarities: the ‘geometric’ class, in which dissimilarities are built according to the geometrical features of the trajectories (instantaneous position, slope of the inter-temporal segments (velocity), concavity and convexity of each pair of inter-temporal segments (acceleration), polygonal line (shape), portion of area between each pair of trajectories); the ‘correlative’ class, in which dissimilarity measures that analyze the autocorrelation and cross-correlation functions of the univariate components of each multivariate trajectory are classified; the ‘structural’ class, containing dissimilarities which consider the structural aspects of the trajectories, such as the linear or polynomial trends and the seasonality of each univariate component. An empirical comparison is also included.

A Fuzzy Clustering Model for Multivariate Spatial Time Series

Clustering of multivariate spatial-time series should consider: 1) the spatial nature of the objects to be clustered; 2) the characteristics of the feature space, namely the space of multivariate time trajectories; 3) the uncertainty associated to the assignment of a spatial unit to a given cluster on the basis of the above complex features. The last aspect is dealt with by using the Fuzzy C-Means objective function, based on appropriate measures of dissimilarity between time trajectories, by distinguishing the cross-sectional and longitudinal aspects of the trajectories. In order to take into account the spatial nature of the statistical units, a spatial penalization term is added to the above function, depending on a suitable spatial proximity/ contiguity matrix. A tuning coefficient takes care of the balance between, on one side, discriminating according to the pattern of the time trajectories and, on the other side, ensuring an approximate spatial homogeneity of the clusters. A technique for determining an optimal value of this coefficient is proposed, based on an appropriate spatial autocorrelation measure. Finally, the proposed models are applied to the classification of the Italian provinces, on the basis of the observed dynamics of some socio-economical indicators.

Wavelet-based Fuzzy Clustering of Time Series

Traditional procedures for clustering time series are based mostly on crisp hierarchical or partitioning methods. Given that the dynamics of a time series may change over time, a time series might display patterns that may enable it to belong to one cluster over one period while over another period, its pattern may be more consistent with those in another cluster. The traditional clustering procedures are unable to identify the changing patterns over time. However, clustering based on fuzzy logic will be able to detect the switching patterns from one time period to another thus enabling some time series to simultaneously belong to more than one cluster. In particular, this paper proposes a fuzzy approach to the clustering of time series based on their variances through wavelet decomposition. We will show that this approach will distinguish between time series with different patterns in variability as well identifying time series with switching patterns in variability.

The Geometric Approach to the Comparison of Multivariate Time Trajectories

To compare time trajectories different approaches might be envisaged. In this paper, considering the geometric approach, several dissimilarity measures between time trajectories are taken into account. An empirical comparison of the dissimilarity measures is also shown.

Dissimilarities Between Trajectories of a Three-Way Longitudinal Data Set

This paper deals with the problem of evaluating dissimilarities between trajectories in a three-way longitudinal data set (a set of multiple time series). The dissimilarity between trajectories is defined as a conic combination of the dissimilarities between trends, velocities and accelerations of the pair of trajectories. The coefficients of the linear combination are estimated maximizing its variance. The proposed methodology is applied on a real data set to classify trajectories of Italian regions by their employment dynamic.

Fuzzy Time Arrays and Dissimilarity Measures For Fuzzy Time Trajectories

In this paper we define a fuzzy extension of a time array. The algebraic and geometric characteristics of the fuzzy time array are analyzed. Furthermore, considering the objects space ℜ J+1, where J is the number of variables and the remaining dimension is related to time, we suggest different dissimilarity measures for fuzzy time trajectories.

Change point analysis of imprecise time series

Abstract In this paper we describe how to conduct a change-point analysis when dealing with time series imprecisely or vaguely observed, i.e. time ordered observations whose values are not known exactly, such as interval or ordinal time series (imprecise time series). In order to treat such time series, we propose to employ a fuzzy approach i.e. data are parameterized in the form of fuzzy variables. Then, to detect the number and location of change points we employ a deviation measure for fuzzy variables in the framework of Atheoretical Regression Trees (ART). We present simulation results pertaining to the behavior of the proposed approach as well as two empirical applications to real imprecise time series.

Fuzzy K-medoids clustering models for fuzzy multivariate time trajectories

Following the fuzzy approach, the clustering problem concerning a set of fuzzy multivariate time trajectories is addressed. The obtained clusters are characterized by observed typical LR fuzzy time trajectories, medoids, belonging to the data set at hand. Two different clustering models are proposed according to the cross-sectional or longitudinal aspects of the time trajectories. An application to air pollution data is carried out.

Clustering seasonal time series using extreme value analysis: An application to Spanish temperature time series

ABSTRACT A challenging aspect of grouping together regional temperature time series is that some regions have similar summer temperatures but different winter temperatures and vice versa. We explore this by applying cluster analysis to regional temperature time series in Spain using as features the parameter estimates of location, scale, and shape, obtained from fitting the generalized extreme value (GEV) distribution to the block maxima and block minima of the series. Using this approach, our findings reveal that the identified clusters can be meaningfully interpreted and are well validated. The motivation for using this approach is that each time series is represented by just three easily extracted features. If features were to be extracted as a result of conventional time series modeling, they are likely to be impacted upon by the uncertainty of model selection. This is not the case with GEV modeling. Furthermore, GEV modeling enables long – term projections of the maxima and minima that cannot otherwise be achieved from conventional time series modeling. For comparison purposes, we also explore clustering the block maxima and block minima of the times series. In addition, we explore the performance of this approach using simulated data.

Mixtures of Autoregressions with an Improper Component for Panel Data

An EM algorithm for fitting mixtures of autoregressions of low order is constructed and the properties of the estimators are explored on simulated and real datasets. The mixture model incorporates a component with an improper density, which is intended for outliers. The model is proposed as an alternative to the search for the order of a single-component autoregression. The methods can be adapted to other patterns of dependence in panel data. An application to the monthly records of income of the outlets of a retail company is presented.