Shape-Sphere: A metric space for analysing time series by their shape

Abstract

Shape analogy is a key technique in analyzing time series. That is, time series are compared by how much they look alike. This concept has been applied for many years in geometry. Notably, none of the current techniques describe a time series as a geometric curve that is expressed by its relative location and form in space. To fill this gap, we introduce Shape-Sphere, a vector space where time series are presented as points on the surface of a sphere. We prove a pseudo-metric property for distances in Shape-Sphere. We show how to describe the average shape of a time series set using the pseudo-metric property of Shape-Sphere by deriving a centroid from the set. We demonstrate the effectiveness of the pseudo-metric property and its centroid in capturing the ‘shape’ of a time series set, using two important machine learning techniques, namely: Nearest Centroid Classifier and K-Means clustering, using 85 publicly available data sets. Shape-Sphere improves the nearest centroid classification results when the shape is the differentiating feature while keeping the quality of clustering equivalent to current state-of-the-art techniques.

Introduction

Comparing time series by their appearance or visual “shape” has diverse applications such as classifying heartbeat ECG records into normal and abnormal signals [21], clustering accelerometer records from wearables attached to a trainee into different sets of exercises [27], [28], [29], or categorising data received from a space shuttle [7]. In these problems, the relationships between points on the time series are more important than each individual absolute value. For example, Fig. 1 shows an abnormal and a normal heartbeat from the ECG200 database [37]. We can see that both heartbeats have peaks in the highlighted areas. However, the peaks in the normal heartbeat are more distinguishable than the peaks in the abnormal one. This detectability manifests itself in the relationship between the peak and its neighbourhood values.

Traditionally, shape analysis for time series data has focused on comparisons between pairs of time series. In such an approach, shape is indirectly characterised by a mapping between smaller pieces of the two time series to each other. For example, dynamic time warping (DTW) uses dynamic programming to find the “best” mapping between the two time series [24]. A DTW value is a numeric distance measure that represents the dissimilarity between two time series. An alternative approach identifies primitives from a time series where a shape is considered to be decomposable into peaks and troughs. In this approach, similar shapes are assumed to behave similarly through time, i.e., if one goes up the other goes up and if one goes down the other goes down as well. In this case, the similarity between two time series is defined through correlation [39]. Although these approaches can be used to define the similarity between two finite time series, they do not provide any insights into the internal relationships among the points of a single time series. This insight is crucial for the interpretation of time series data. As shown in Fig. 1, the notion of the shape of a time series would reveal the underlying relationship among the points on the time series, such as the smoothness of peaks shown in the figure. Our approach is to define the notion of shape in such a way that it enables us to reconstruct the original time series exactly. In this way, a time series is modelled as a unique geometric curve that can be presented to an observer like any other geometric object such as a line, circle or cycloid..

Mathematically, a time series can be described as a curve in space. The simplest curve is a line, an object with no curvature (peaks and troughs) (Fig. 2.1). Another simple object is one with constant curvature—a circle, whose curvature is the reciprocal of its radius (Fig. 2.2). Third, consider a cycloid; the curve generated by a point on the circumference of a circle that rolls along a straight line,² (Fig. 2.3). The cycloid’s curvature changes according to the fixed point on the circle. Differential geometry generalises this idea to uniquely determine the shape of the curve through its curvature [3]. The curvature shows the degree of deviation from a line while moving along the curve. By knowing the curvature at each point we can always generate the same curve regardless of the initial position [33]. This definition of shape brings three important characteristics for shape analysis, namely:

• Scale invariance,

• Translation invariance, and

• Rotation invariance.

Satisfying these properties through curvature enables us to define time series as geometric objects using their curvature. In this class, time series are compared by their appearance (shape) in space regardless of their scale, position, or rotation. Comparing time series is an important problem when trying to group similar time series. To determine whether a given time series is from a particular group, a common approach is to compare the new time series to every member of the group, which is an inefficient approach. This is the main reason for seeking a prototype to represent a group of time series. A centroid for a group of time series can be seen as the mean value for the given group, i.e., the centroid is a data point (time series) that has the minimum average distance to every other member of the group. Computing a centroid is coupled with designing a distance measure. The mean property of a centroid enables various important types of analysis on time series, such as: 1. Detecting anomalies in a semi-supervised fashion [27]. 2. Designing a fast indexing mechanism for data base retrieval [41]. 3. Generating synthetic time series data [20]. 4. Designing loss functions for optimisation problems with applications in deep neural networks [19] and clustering [30]. 5. Preserving privacy where a centroid represents a group of time series [36].

Our aim is to use differential geometric properties of curves to introduce a new representation of a time series that uniquely identifies the time series by its shape. The Fourier analysis of our representation leads us to design a vector space where each time series is uniquely defined by a vector. Our goal is to address two fundamental problems in time series analysis: 1. How to compute a prototype from a given set of time series?, and 2. How to compare a pair of time series?

We use two important tasks in time series analysis: 1. K-Means clustering and 2. Nearest Centroid Classifier (NCC), to demonstrate the effectiveness of our methods in addressing these two fundamental problems.