Seasonal Disorder in Urban Traffic Patterns: A Low Rank Analysis

Abstract

This article proposes several advances to sparse nonnegative matrix factorization (SNMF) as a way to identify large-scale patterns in urban traffic data. The input to our model is traffic counts organized by time and location. Nonnegative matrix factorization additively decomposes this information, organized as a matrix, into a linear sum of temporal signatures. Penalty terms encourage this factorization to concentrate on only a few temporal signatures, with weights which are not too large. Our interest here is to quantify and compare the regularity of traffic behavior, particularly across different broad temporal windows. In addition to the rank and error, we adapt a measure introduced by Hoyer to quantify sparsity in the representation. Combining these, we construct several curves which quantify error as a function of rank (the number of possible signatures) and sparsity; as rank goes up and sparsity goes down, the approximation can be better and the error should decreases. Plots of several such curves corresponding to different time windows leads to a way to compare disorder/order at different time scalewindows. In this paper, we apply our algorithms and procedures to study a taxi traffic dataset from New York City. In this dataset, we find weekly periodicity in the signatures, which allows us an extra framework for identifying outliers as significant deviations from weekly medians. We then apply our seasonal disorder analysis to the New York City traffic data and seasonal (spring, summer, winter, fall) time windows. We do find seasonal differences in traffic order.

Appendix

For completeness, let’s write down the calculations leading to the algorithm of Sect. 2.5.

Writing out ${\mathcal{E}}_{\beta ,\eta }$ of (5), we get

$$\begin{aligned}&{\mathcal{E}}_{\beta ,\eta }(W,H)= \sum _{(t,\ell )\in {\mathcal{I}}}\left| D_{t,\ell }-(WH)_{t,\ell }\right| ^2 \\&\quad + \beta \sum _{n=1}^N\left( \sum _{\ell =1}^L H_{n,\ell }\right) ^2 + \eta \sum _{n=1}^N\sum _{t=1}^T W_{\ell ,t}^2. \end{aligned}$$

We seek to minimize this by alternating between minimization problems in W and H. Namely, if we start with a fixed $(W,H)\in \mathbb {R}^{T\times N}_+\times \mathbb {R}^{N\times L}_+$, we can construct a descent step for the function ${\mathcal{E}}_{\beta ,\eta }(W,\cdot )$ and then, letting $H'$ be the result, we can construct a descent step for ${\mathcal{E}}_{\beta ,\eta }(\cdot ,H')$. This should decrease the value of ${\mathcal{E}}_{\beta ,\eta }$, and we can then proceed iteratively.

The gradients of ${\mathcal{E}}_{\beta ,\eta }$ in the directions of W and H are given by

$$\begin{aligned} \frac{\partial {\mathcal{E}}_{\beta ,\eta }}{\partial W_{\hat{t},\hat{n}}}(W,H)&=-2 \sum _{\ell : (\hat{t},\ell )\in {\mathcal{I}}}\left( D_{\hat{t},\ell }-\sum _{n=1}^N W_{\hat{t},n}H_{n,\ell }\right) H_{\hat{n},\ell }\\&\quad +\eta W_{\hat{t},\hat{n}}\\&= -2 \left( \left[ D-WH\right] _{\mathcal{I}}H^T+\eta W\right) _{\hat{t},\hat{n}} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial {\mathcal{E}}_{\beta ,\eta }}{\partial H_{\hat{n},\hat{\ell }}}(W,H)&=-2 \sum _{t: (t,\hat{\ell })\in {\mathcal{I}}}\left( D_{t,\hat{\ell }}-\sum _{n=1}^N W_{t,n}H_{n,\hat{\ell }}\right) W_{t,\hat{n}} \\&\quad + 2\beta \left( \sum _{n=1}^NH_{n,\hat{\ell }}\right) \\&= -2 \left( W^T\left[ D-WH\right] _{\mathcal{I}}\right) _{\hat{n},\hat{\ell }}\\&\quad + 2\beta (\mathbf {1}_{N\times N} H)_{\hat{n},\hat{\ell }}. \end{aligned}$$

As in Kim and Park (2008), we want to iteratively find the critical points of ${\mathcal{E}}_{\beta ,\eta }$, i.e. the solutions of

$$\begin{aligned} \left[ WH\right] _{\mathcal{I}}H^T - [D]_{\mathcal{I}}H^T + \eta W&= 0\\ W^T\left[ WH\right] _{\mathcal{I}}- W^T[D]_{\mathcal{I}}+ \beta \mathbf {1}_{N\times N} H&= 0 \end{aligned}$$

The above formulae suggest a multiplicative descent rule (which need not be gradient descent; see Lee and Seung (2001)). Fix $(W,H)\in \mathbb {R}_+^{T\times N}\times \mathbb {R}_+^{N\times L}$. Assume that

$$\begin{aligned} \frac{\partial {\mathcal{E}}_{\beta ,\eta }}{\partial H_{n,\ell }}>0; \end{aligned}$$

(14)

we can then decrease the value of ${\mathcal{E}}_{\beta ,\eta }$ by decreasing $H_{n,\ell }$. Rewriting (14) as

$$\begin{aligned} -2 \left( W^T\left[ D-WH\right] _{\mathcal{I}}\right) _{n,\ell }+ 2\beta (\mathbf {1}_{N\times N} H)_{n,\ell }>0 \end{aligned}$$

or rather

$$\begin{aligned} \left( W^T\left[ WH\right] _{\mathcal{I}}\right) _{n,\ell }+ \beta (\mathbf {1}_{N\times N} H)_{n,\ell }>\left( W^T\left[ D\right] _{\mathcal{I}}\right) _{n,\ell }, \end{aligned}$$

since W, H, and D all have nonnegative entries, both sides of this equation are nonnegative. This in turn can be written as ${\chi _{n,\ell }^h(W,H)<1}$ where

$$\begin{aligned} \chi _{n,\ell }^h(W,H) \overset{\text {def}}{=}\frac{\left( W^T\left[ D\right] _{\mathcal{I}}\right) _{n,\ell }}{\left( W^T\left[ WH\right] _{\mathcal{I}}\right) _{n,\ell }+ \beta (\mathbf {1}_{N\times N} H)_{n,\ell }}. \end{aligned}$$

Thus, another way to decrease $H_{n,\ell }$ while still retaining nonnegativity is to multiply it by $\chi _{n,\ell }^h(W,H)$. Reviewing these steps, we also see that if $\frac{\partial {\mathcal{E}}_{\beta ,\eta }}{\partial H_{n,\ell }}<0$, we want to increase $H_{n,\ell }$, and can again multiply by $\chi _{n,\ell }^h(W,H)$. Finally, if $\frac{\partial {\mathcal{E}}^\beta }{\partial H_{n,\ell }}=0$ (i.e., we have found a critical point) $\chi _{n,\ell }^h(W,H)=1$, so multiplying $H_{n,\ell }$ by $\chi _{n,\ell }^h(W,H)$ leaves $H_{n,\ell }$ unchanged.

The update rule for $W_{t,n}$ is similar. To start, assume that

$$\begin{aligned} \frac{\partial {\mathcal{E}}_{\beta ,\eta }}{\partial W_{t,n}}>0; \end{aligned}$$

(15)

then we can decrease ${\mathcal{E}}_{\beta ,\eta }$ by decreasing $W_{t,n}$. We can rewrite (15) as

$$\begin{aligned} -2 \left( \left[ D-WH\right] _{\mathcal{I}}{\mathcal{I}}H^T+\eta W\right) _{t,n}>0. \end{aligned}$$

We can again rewrite this as the comparison of two nonnegative quantities;

$$\begin{aligned} \left( \left[ WH\right] _{\mathcal{I}}H^T+\eta W\right) _{t,n} >\left( \left[ D\right] _{\mathcal{I}}H^T\right) _{t,n}; \end{aligned}$$

This in turn is equivalent to $\chi _{t,n}^w(W,H)<1$ where

$$\begin{aligned} \chi ^w_{t,n}(W,H)\overset{\text {def}}{=}\frac{\left( \left[ D\right] _{\mathcal{I}}H^T\right) _{t,n}}{\left( \left[ WH\right] _{\mathcal{I}}H^T+\eta W\right) _{t,n}} \end{aligned}$$

In other words, we can decrease $W_{t,n}$ by multiplying by $\chi _{t,n}^w(W,H)$. One can similarly see that if $\frac{\partial {\mathcal{E}}^\beta }{\partial W_{t,n}}<0$, gradient descent again increases or decreases W with the same sign as multiplying by $\chi _{t,n}^w(W,H)$.

Our proposed update rule for W and H is now

$$\begin{aligned} W'_{t,n}&=W_{t,n}\chi _{t,n}^w(W,H)\\ H'_{n,\ell }&= H_{n,\ell }\chi _{n,\ell }^h(W,H). \end{aligned}$$

which is equivalent to (6).