The Signed Cumulative Distribution Transform for 1-D signal analysis and classification

Abstract

<p style='text-indent:20px;'>This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [<xref ref-type="bibr" rid="b29">29</xref>] to arbitrary (signed) signals on <inline-formula><tex-math id="M1">\begin{document}$ \overline {\mathbb{R}} $\end{document}</tex-math></inline-formula>. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.</p>

Geodesic Properties of a Generalized Wasserstein Embedding for Time Series Analysis

“…Since (1−α)id+αg −1 is strictly increasing and (s (Aldroubi et al, 2022)). By the inverse formula in Proposition A.3, it is not hard to see that the expression in ( 8) is the Jordan decomposition of p α , i.e.,…”

“…Now for a (non-zero) signed signal, the Jordan decomposition (Royden & Fitzpatrick, 1988) is applied to s(t) = s + (t) − s − (t), 4 where s + (t) and s − (t) are the absolute values of the positive and negative parts of the signal s(t). Given a fixed L 1 -normalized positive reference signal s 0 defined on Ω s0 , the signed cumulative distribution transform (SCDT) (Aldroubi et al, 2022) of s(t) is then defined as the 2 The fact that s0 and s have finite second moments guarantees the existence of a unique optimal transport map between them.…”

“…In recent years, many transport transform-based techniques have been developed to leverage Wasserstein distances and linearized (L 2 ) embeddings to facilitate the application of many standard data analysis (Kolouri et al, 2017). In particular, the cumulative distribution transform (CDT), based on the 1D Wasserstein embedding, was introduced in (Park et al, 2018) as a means of classifying normalized non-negative signals, and has been extended to general signed signals via the the signed cumulative distribution transform (SCDT) in (Aldroubi et al, 2022).…”

“…Wasserstein embeddings based on the cumulative distribution transform (CDT) (Park et al, 2018;Rubaiyat et al, 2020;Aldroubi et al, 2022) have recently emerged as a robust, computationally efficient, and accurate end-to-end classification method for time series (1D signal) classification. They are particularly effective for classifying data emanating from physical processes where signal classes can be modeled as observations of a particular set of template signals under some unknown, possibly random, temporal deformation or transportation (Park et al, 2018;Shifat-E-Rabbi et al, 2021;Rubaiyat et al, 2022b).…”

“…As the CDT (Park et al, 2018) is defined for probability measures (or their associated density functions) which are non-negative and normalized, in this manuscript we elucidate the geometry of general time series (signed, nonnormalized) with a generalized Wasserstein metric in the SCDT embedding space with a L 2 -type metric. In addition to the convexity and isometric embedding properties of the SCDT (Aldroubi et al, 2022), which are shared by the CDT, we look at the geodesic properties of the SCDT and illustrate the differences between geometry of data in CDT and SCDT space. In particular, the SCDT embedding S ⊆ L 2 (s 0 ) × R 2 is not a geodesic space while geodesics exist between signals within a generative model (see Figure 1).…”