This paper considers the problem of signal decomposition and data visualization. For this purpose, we introduce a new multiscale transform, termed `ensemble patch transformation' that enhances identification of local characteristics embedded in a signal and provides multiscale visualization according to different levels; hence, it is useful for data analysis and signal decomposition. In literature, there are data-adaptive decomposition methods such as empirical mode decomposition (EMD) by Huang et al. (1998). Along the same line of EMD, we propose a new decomposition algorithm that extracts meaningful components from a signal that belongs to a large class of signals, compared to the previous methods. Some theoretical properties of the proposed algorithm are investigated. To evaluate the proposed method, we analyze several synthetic examples and a real-world signal.

Ensemble Patch Transformation: A New Tool for Signal Decomposition

Using novel, nonparametric comovement measures based on the Thick Pen Transform, we study the OMXH25 stocks in the post-financial-crisis decade. The new measures allow us to work with stationary returns and with nonstationary volumes. The comovement can be monitored in time, it is possible to distinguish between comovement on different time scales, and even cross-term comovement can be quantified. The approach is visually-interpretable and multivariate in nature. The results indicate the presence of a cyclical pattern in the relatively strong comovement of returns on semi-annual and annual time scales, with more oscillations in the comovement on quarterly and monthly time scales, and the presence of a slight increasing pattern in the relatively weak comovement of volumes on semi-annual and annual time scales. Cross-term dependence between Nokia’s weekly and monthly features in returns and longer-term features in returns of other stocks is more variable than that based on volumes.

Short-, Long- and Cross-Term Comovement of OMXH25 Stocks

Traditional visualization of time series data often consists of plotting the time series values against time and ‘connecting the dots'. We propose an alternative, multiscale visualization technique, motivated by the scale-space approach in computer vision. In brief, our method also ‘connects the dots' but uses a range of pens of varying thicknesses for this. The resulting multiscale map, which is termed the thick pen transform, corresponds to viewing the time series from a range of distances. We formally prove that the thick pen transform is a discriminatory statistic for two Gaussian time series with distinct correlation structures. Further, we show interesting possible applications of the thick pen transform to measuring cross-dependence in multivariate time series, classifying time series and testing for stationarity. In particular, we derive the asymptotic distribution of our test statistic and argue that the test is applicable to both linear and non-linear processes under low moment assumptions. Various other aspects of the methodology, including other possible applications, are also discussed.

P. Fryzlewicz

Papers

Thick pen transformation for time series Series B Statistical methodology