Online Segmentation of Time Series Based on Polynomial Least-Squares Approximations

Abstract

The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs.

Symbolic representation of smart meter data

“…For horizontal segmentation, there exist online algorithms which are able to select dynamically when a new segment has to be started [8,9,15]. But in our case, variation of the time dimension of a symbol would give it a different semantics and thus preventing us to run algorithms on top of symbolic data.…”

All for one or one for all? Combining heterogeneous features for activity spotting

“…Furthermore, an alternative feature type based on piecewise linear approximation [13] of inertial sensor data is introduced, compared, and analyzed.…”

“…This makes this technique well-suited for time-critical applications. More details can be found in [14], [13].…”

Determination of the Starting Point in Time Series for Trend Detection Based on Overlapping Trend

“…L i and Z h e n g [8] proposed a data representation method based on PPR to segment time series data. F u c h s et al [9] used the least squares method combined with orthogonal polynomials to fit the time series. X u e d o n g and K a n [10] redefine the concept of trend in sequence data and point out that sequences with inertia only can be called trend, and then extract the trend by means of piecewise polynomial representation and inertia test.…”