Clustering Activity–Travel Behavior Time Series using Topological Data Analysis

Abstract

Over the last few years, traffic data have been exploding and the transportation discipline has entered the era of big data. It brings out new opportunities for doing data-driven analysis, but it also challenges traditional analytic methods. This paper proposes a new divide and combine-based approach to do K-means clustering on activity–travel behavior time series using features that are derived using tools in time series analysis and topological data analysis. Our approach facilitates a case study, where each individual’s daily activity–travel behavior is characterized as a categorical time series consisting of three different levels. Clustering data from five waves of the National Household Travel Survey ranging from 1990 to 2017 suggests that activity–travel patterns of individuals over the last 3 decades can be grouped into three clusters. Results also provide evidence in support of recent claims about differences in activity–travel patterns of different survey cohorts. The proposed method is generally applicable and is not limited only to activity–travel behavior analysis in transportation studies. Driving behavior, travel mode choice, household vehicle ownership, when being characterized as categorical time series, can all be analyzed using the proposed method.

Appendix: TDA and the First-order Persistence Landscape

We start with a brief review of topological data analysis (TDA), which is now an emerging area for analyzing big data with complex structures. Using computational homology, TDA is aimed at analyzing the topological features of data and representing these features using low-dimensional representations (Carlsson 2009). The input to TDA is often a set of data points (point cloud) or a function, and persistence homology distills essential topological features in the data, which can then be used together with suitable dissimilarity measures to identify patterns in the data sets. We discuss TDA on functions, which is the approach developed in Sections 2 and 3.

Computational Procedure for TDA on Functions

We look at the method to construct persistence diagrams on functions using the sublevel set filtration. Figure 10 shows the simple procedure of extracting a persistence diagram from a function. Suppose $y_j = f(j), j=1,\dots , 10$ and let the sublevel set be $L_r = \{y_j|y_j \le r\}$. TDA is used to construct the persistence diagram based on $L_r$.

• (i) When $r = 0$, a connected component is identified (marked as a blue dot, which is the oldest connected component). The vertical slash line of the second plot records the “birth time $= 0$” and the horizontal slash line indicates r. There is no point on the birth/death plot, since no connected components died at $r=0$.

• (ii) When $r = 0.5$, there are two more connected components coming out (indicated in blue); the blue dot in the middle with a blue line connecting it to the dark green dot indicates that the oldest connected component “enlarges” and is “still alive”. The other black vertical slash line in the second plot gives the “birth time” for the other two new connected components. There is no connected component dead yet, and hence no points are shown on the birth/death plot.

• (iii) When $r = 1$, all old components “enlarge” and there is one newer component “killed” by the older one. Therefore, there is a “black dot with birth $= 0.5$ and death $= 1$” shown on the second plot.

• (iv) When $r = 2$, the last component is “killed, birth $= 0$, death $= 2$”, which is the black dot on the location (0, 2). The other black dot corresponding to (0.5, 1.5) of the second plot tells the “birth and death” of another connected component.

Fig. 10

Four pairs of plots, in order from top left to bottom right, to illustrate the procedure of getting the persistence diagram on a function

First-Order Persistence Landscape

First, in the persistence diagram obtained using the sublevel set filtration, the furthest point away from the diagonal line is always born at the minimum value of the function and dies at the maximum value of the function.

Second, referring to the definition of persistence landscape in Section 2.3 from Bubenik (2015), given a persistence diagram $\{(b_i, d_i), \forall i\}$, the first-order persistence landscape is

$$\begin{aligned} \text{ PL }(\ell ) = \max _i\{ \min (\ell -b_i, d_i-\ell )_+ \}, \end{aligned}$$

where $\ell$ is a real number. Because the persistence diagram uses a sublevel set filtration, it has the point $(d_{\min }, d_{\max })$. For all $(b_i, d_i)$ that belong to the persistence diagram, $d_{\min }\le b_i \le d_i \le d_{\max }$. Therefore, for any real number $\ell$, $\ell -d_{\min } \ge \ell -b_i$ and $d_{\max } - \ell \ge d_i - \ell$, which implies that

$$\begin{aligned} \min (\ell -d_{\min }, d_{\max }-\ell )_+ \ge \min (\ell -b_i, d_i-\ell )_+ , \end{aligned}$$

which in turn implies that

$$\begin{aligned} \text{ PL }(\ell )= & {} \max _i\{ \min (\ell -b_i, d_i-\ell )_+ \} \\= & {} \min (\ell -d_{\min }, d_{\max }-\ell )_+. \end{aligned}$$

Finally, let $(d_{\min }, d_{\max }) \subset (D_{\min }, D_{\max })$ and taking grids $\{D_{\min }+\frac{(\ell -1)*(D_{\max }-D_{\min })}{L-1},\ell =1, 2, \ldots , L\}$, we have

$$\begin{aligned} \text{ PL }(\ell ) = \min (V_1 (\ell ), V_2 (\ell ))_{+}, \end{aligned}$$

where,

$$\begin{aligned} V_1(\ell )= & {} D_{\min } + \frac{(\ell -1) (D_{\max }-D_{\min })}{L-1} - d_{\min } \\ V_2(\ell )= & {} d_{\max } - D_{\min } - \frac{(\ell -1) (D_{\max } - D_{\min })}{L-1}. \end{aligned}$$

These expressions will be used on the WFT function obtained from each time series $n=1,\ldots ,N$ in Section 2.