A general framework for never-ending learning from time series streams

Abstract

Time series classification has been an active area of research in the data mining community for over a decade, and significant progress has been made in the tractability and accuracy of learning. However, virtually all work assumes a one-time training session in which labeled examples of all the concepts to be learned are provided. This assumption may be valid in a handful of situations, but it does not hold in most medical and scientific applications where we initially may have only the vaguest understanding of what concepts can be learned. Based on this observation, we propose a never-ending learning framework for time series in which an agent examines an unbounded stream of data and occasionally asks a teacher (which may be a human or an algorithm) for a label. We demonstrate the utility of our ideas with experiments that consider real-world problems in domains as diverse as medicine, entomology, wildlife monitoring, and human behavior analyses.

Appendix 1: the relationship between buffer size and pattern discovery time

In the main text, we provided an equation to calculate the probability of discovering a repeated pattern within $n$ steps when the buffer size is $w$, but did not discuss how we derived the equation. We relegated the derivation to this appendix to enhance the flow of the paper.

1.1 Theoretical analysis

To calculate the probability of seeing at least two examples of the pattern in the buffer with no more than $n$ steps when the buffer size is $w$ under the assumptions given in the main text (c.f. Sect. 3.5), we model the problem with the classic urn and ball choice model (Dodge 2003) as follows.

An infinitely large urn contains colored balls in the ratio of one red ball to ninety-nine blue balls. At each time step, a ball is randomly sampled from the urn, and put into a box. The size of the box is $w$. When the box is full, we randomly discard a ball from the box, and replace it with the new sampled ball. The question at hand is: what are the chances of seeing at least two red balls in the box within $n$ samplings when the box size is $w$?

This model simulates our problem by having the red ball represent the pattern, and the blue balls represent random data. The box simulates the buffer, and the data stream is generated through sampling the balls at each time step. The pattern is discovered when exactly two red balls are seen in the box (recall that we stop when we see two red balls, so we will never see three or more), so the goal is to figure out the probability of discovering the pattern within $n$ steps when the box size is $w$.

According to the model, the probability of seeing at least two red balls in the next step is dependent on the current state of the box and independent of the previous states. For example, it is possible to see two red balls in the next step only when the box currently contains at least one red ball; and given the current state of balls in the box, the probability of seeing two red balls in the next step is independent of how many red balls there were previously. As such, this problem is a typical Markov process and is modeled using the Markov Chain (Norris 1998) as follows.

We take as states the number of red balls in the boxes $\hbox {S}_{0}$, $\hbox {S}_{1}$ and $\hbox {S}_{2}$, where $\hbox {S}_{0}/\hbox {S}_{1}$ denotes that there is zero/one red ball in the box, and $\hbox {S}_{2}$ denotes that there are at least two red balls. For simplicity, we explicitly assume that the box is full of blue balls at the beginning; that is, the initial state is $\hbox {S}_{0}$. As such, the initial probability vector $u$ is as follows:

$$\begin{aligned} u=\left[ {{ \begin{array}{ccc} 1&{} 0&{} 0 \\ \end{array}}}\right] \end{aligned}$$

We need to specify the transition matrix. The conditions that are required to transition from one state to another are listed in Table 13, where ‘impossible’ means with zero probability and ‘certainty’ means with hundred percent probability.

Table 13 Conditions for transitioning states

At each step, the probability of sampling a red ball from the urn is 1/100, and of sampling a blue ball is 99/100; the probability of discarding a red ball from the box is $\frac{n_{red}}{w}$, and of discarding a blue ball is $\frac{w-n_{red}}{w}$, where $n_{red}$ is the number of red balls currently in the box. Based on the conditions in Table 13, we can compute the transition probabilities and derive the transition matrix $T$ as follows:

$$\begin{aligned} T= \left[ {{ \begin{array}{ccc} {\frac{99}{100}}&{} {\frac{1}{100}}&{} 0 \\ {\frac{1}{w}*\frac{99}{100}}&{} {\frac{1}{w}*\frac{1}{100}+\frac{w-1}{w}*\frac{99}{100}}&{} {\frac{w-1}{w}*\frac{1}{100}} \\ 0&{} 0&{} 1 \\ \end{array}}}\right] \end{aligned}$$

According to the Markov Chain, the probability of seeing at least two red balls in the box (in state $S_{2}$) after $n$ steps is the $3^{\mathrm{rd}}$ entry in the vector $u^{(n)}=uT^{n}$, and thus, we have:

$$\begin{aligned} Pr(w,n)=\left( {\left[ {{ \begin{array}{ccc} 1&{} 0&{} 0 \\ \end{array}}}\right] \left[ {{\begin{array}{ccc} {\frac{99}{100}}&{} {\frac{1}{100}}&{} 0 \\ {\frac{1}{w}*\frac{99}{100}}&{} {\frac{1}{w}*\frac{1}{100}+\frac{w-1}{w}*\frac{99}{100}}&{} {\frac{w-1}{w}*\frac{1}{100}} \\ 0&{} 0&{} 1 \\ \end{array}}}\right] ^{n}}\right) (1,3) \end{aligned}$$

(1)

which is the equation we provided in the main text.

To verify our analysis, we also performed an empirical study with simulations. We did simulations for different values of $w$, and compare the simulation results with the theoretical results calculated using Eq. 1. Figure 32 shows the comparison results for $w = 10$. As can be seen, the two results are essentially identical.

Fig. 32

Comparison of simulation results (red/bold) with theoretical results (green/thin) for $w =10$