Alternative measures of central tendency to the mean series of a set of time series based on Haar wavelet decomposition
When a representative is required as a measure of central tendency for a set of time series, the mean series is a usual choice. The mean series is formed merely by point-wise averaging. However, the mean series may over-smooth the important variation present in the set of time series. In this paper alternatives are proposed based on a measure called Ultimate Tamed Series (UTS). UTS is based on discrete Haar wavelet, which is subscribed by both point-wise concern and local trend given by an adjacent point.
UTS depends on the order of taming of the set of time series again adhering heterogeneities of data. Hence in situations where the order gives a meaningful interpretation, a predetermined order can be used to obtain one UTS as a representative to the set of time series. Otherwise uniqueness of the central tendency measure related to UTS is proposed by two ways: Point-wise Least Measure (PL) and Least Ultimate Tamed Series (L-UTS). These measures are derived using point-wise deviations between UTS and the original set of time series. Another two supportive measures: Point-wise Maximum Measure (PM) and Maximum Ultimate Tamed Series (M-UTS) are proposed as indicators of variability to identify the necessity of implementing PL and L-UTS instead of mean series. Further, computational complexity of UTS related measures is also analyzed.