Piecewise Trend Approximation: A Ratio-Based Time Series Representation

and Applied Analysis 3 can identify segments of variable length. Also, the APCA algorithm is able to produce high quality approximations of a time series by resorting to solutions adopted in the wavelet domain. In SAX method, dimensionality of original time series is first reduced by applying PAA, then the PAA coefficients are quantized, and finally each quantization level is represented by a symbol so that SAX is a symbolic representation of time series. TheDSA representation is based on the derivative version of the original time series. DSA entails derivative estimation, segmentation, and segment modeling to map a time series into a different value domain which allows for maintaining information on the significant features of the original series in a dense and concise way. For representing a time series of n points, it can be performed inO(n) by using DWT, SD, (the fastest version of) PLA, PAA, SAX, and DSA, whereas the complexity of APCA is O(n log(n)). There are some other kinds of time series representations applying continuous polynomial functions to approximate time series, include Singular Value Decomposition (SVD) [15, 16], Discrete Fourier Transforms (DFT) [17, 18], splines, nonlinear regression, and Chebyshev polynomials [19, 20], of which the details are kindly referred to the references. In contrast to conventional representations based on raw data, a time series representation based on ratios between any two consecutive data points in a given time series is proposed by applying piecewise segment approximation to reduce dimensionality in Section 3. 3. PTA: Piecewise Trend Approximation Given a time series Y = {(y 1 , t 1 ), . . . , (y n , t n )}, where y i is a real numeric value and t i is the timestamp, it can be represented as a PTA representation