Determination of underlying components of a cyclical time series by means of two-way and three-way factor analytic techniques

Abstract

A technique is presented for determining the underlying components in a cyclical time series which is influenced by one prominent cycle (the diurnal or the yearly cycle). The separation of the components is based on their different shapes within this period, assuming that the shape of each component stays approximately constant with time and that the amplitude of each component is a slowly changing function. The series is folded into matrix shape so that each cycle forms one column. The matrix is factorized by principal component analysis or by positive matrix factorization (non-negatively constrained factor analysis with individual weighting of data values), resulting in the shape and amplitude functions for the underlying components. Synthetic two-way demonstration examples are analyzed. As a real-life example, traffic-induced carbon monoxide concentrations in urban air are analyzed. The CO has a diurnal concentration cycle which changes shape on weekends. This behavior is explained by two factors, identified with work-related and other traffic. The CO data in fact contain another multiplicative cycle, the weekly workdays/weekend pattern. Arranging the data according to time of day, day of week, and week of the year creates a three-way array. The method is extended to the analysis of such arrays. Existing software for the well-known PARAFAC model is used for solving the three-way model. Two factors are again obtained. Their diurnal and weekly cycles correspond to the work-related and weekend-related traffic patterns. Analysis of cyclical multivariate data is discussed: such data are also governed by the three-way PARAFAC model. The advantage of the PARAFAC model relative to customary two-way methods is emphasized: there is usually no rotational ambiguity in PARAFAC results. Copyright © 2000 John Wiley & Sons, Ltd.