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.

[1]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[2]  D. F. Andrews,et al.  PLOTS OF HIGH-DIMENSIONAL DATA , 1972 .

[3]  George E. P. Box,et al.  Time Series Analysis: Forecasting and Control , 1977 .

[4]  A. Buse.  Cubic Splines as a Special Case of Restricted Least Squares , 1977 .

[5]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[6]  R. Sibson.  Studies in the Robustness of Multidimensional Scaling: Procrustes Statistics , 1978 .

[7]  Adi Raveh,et al.  Finding common seasonal patterns among time series: An MDS approach , 1980 .

[8]  Adi Raveh,et al.  A Graphic Display for Characterization of Seasonal Pattern Similarities of Time Series , 1981 .

[9]  K. Kosmelj,et al.  Aspect temporel des relations entre les variables hydriques du Haut-Rhône français , 1983 .

[10]  R. A. Harshman,et al.  Data preprocessing and the extended PARAFAC model , 1984 .

[11]  A. M. Marquez.  L'analyse des données évolutives , 1985 .

[12]  A. Carlier,et al.  Factor Analysis of Evolution and Cluster Methods on Trajectories , 1986 .

[13]  Katarina Košmelj.  A two‐step procedure for clustering time varying data , 1986 .

[14]  A. Agresti,et al.  Multiway Data Analysis , 1989 .

[15]  K. Kosmelj,et al.  Cross-sectional approach for clustering time varying data , 1990 .

[16]  D. Piccolo.  A DISTANCE MEASURE FOR CLASSIFYING ARIMA MODELS , 1990 .

[17]  R. Summers,et al.  The Penn World Table (Mark 5): An Expanded Set of International Comparisons, 1950-1987 , 1991 .

[18]  John F. MacGregor,et al.  Process monitoring and diagnosis by multiblock PLS methods , 1994 .

[19]  S. Joly,et al.  Three-way distances , 1995 .

[20]  John F. MacGregor,et al.  Multivariate SPC charts for monitoring batch processes , 1995 .

[21]  Elizabeth Ann Maharaj,et al.  A SIGNIFICANCE TEST FOR CLASSIFYING ARMA MODELS , 1996 .

[22]  Robert H. Shumway,et al.  Discrimination and Clustering for Multivariate Time Series , 1998 .

[23]  Pierpaolo D'Urso,et al.  Dissimilarities Between Trajectories of a Three-Way Longitudinal Data Set , 1998 .

[24]  Hans-Hermann Bock,et al.  Advances in data science and classification , 1998 .

[25]  Elizabeth Ann Maharaj,et al.  Comparison and classification of stationary multivariate time series , 1999, Pattern Recognit..

[26]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[27]  Pierpaolo D'Urso,et al.  Fuzzy Time Arrays and Dissimilarity Measures For Fuzzy Time Trajectories , 2000 .

[28]  Elizabeth Ann Maharaj,et al.  Cluster of Time Series , 2000, J. Classif..

[29]  Richard J. Martin.  A metric for ARMA processes , 2000, IEEE Trans. Signal Process..

[30]  Pierpaolo D'Urso,et al.  The Geometric Approach to the Comparison of Multivariate Time Trajectories , 2001 .