An algorithm for continuous piecewise linear bounding of discrete time series data

Abstract

An algorithm for enclosing a given set of time series data inside a continuous piecewise linear band of varying height subject to certain constraints is presented. The band is defined by two piecewise linear curves that lie above and below the data respectively. Segments of these curves are constrained to start and end at one of the data points, and those whose slope does not lies between its neighbours' slopes are required to be at least as wide as a user-specified value. The algorithm yields a band which accurately preserves the general trends of the data, while enclosing the inherent measurement noise. This band is typically obtained in O(n log K) time, where n is the number of data points and K is the number of linear segments. The algorithm is described and its capabilities are tested on four data sets. Comparisons are made with alternative algorithms.

Frequent State Transition Patterns of Multivariate Time Series

“…likelihood estimators (MLE) [32]) and model evaluation (see, e.g., F-, R-, S-, and T-testings [33]). To obtain symbolic sequences with better description, higher query efficiency, less noise, and less redundancy, methods such as symbolic aggregate approximation (SAX) [34], discrete fourier transform(DFT) [35], discrete wavelet transform (DWT) [36], piecewise linear representation (PLR) [37] and minimum description length (MDL) [38] are proposed. Pattern discovery techniques such as temporal association rules discovery (TARD) [39] and discovery of temporal associations (DTA) [40] consist of 2 stages, namely feature clustering [41] and block sequence discovery.…”

Parameter Range Reduction in Ordinary Differential Equation Models

“…Converting the discrete time series datas to a continuous representation may be accomplished using a variety of methods. In this paper, we use an algorithm presented in [10] that replaces time series data with a continuous piecewise linear band that encloses all data points. The results of the range reduction algorithm depend on the representation and thus on what the user thinks is a valid continuous representation that encloses the true solution.…”

“…We added to the sample data normally distributed noise with a standard deviation equal to one-percent of the maximum amplitude of each variable. Continuous bands for the data were generated by the algorithm described in [10] and are plotted in Figure 1. Fig.…”

“…Fig. 1: Simulated data for the nonlinear pendulum and continuous bands generated by the algorithm described in [10] using t min = 0.18 and min height = 0.15 for x and using using t min = 0.18 and min height = 0.40 for y. The natural interval extension of the discretization of equation ( 4) is…”

“…In addition to the improvements made in this paper to the parameter range reduction algorithm, we also switched from using the banding algorithm in [11] to the banding algorithm in [10]. To differentiate between the improved results obtained by the modifications to the parameter range reduction algorithm and those obtained by changing the banding algorithm, we also tested the algorithm in this paper with the prior banding algorithm.…”