K-Means on Time Series but each timestep is considered an individual point

Question

As stated in the question, I have a doubt about the possibility that K-Means would work if we apply it on one time series where each timestep is considered an individual data point. Please allow me to explain the problem as follows.

The dataset of interest comes are collected from sensor data. The dataset contains many multivariate time series, where one time series is denoted $T = \{ x_1, x_2, ..., x_N \}$ consisting of $N$ timesteps and of $P$ attributes. Domain experts tell us that there are 3 types of patterns that may be seen in any series. For example, in a series $T_i$ we can see the patterns $p_1$ and $p_2$, in another series $T_j$ we can see the patterns $p_1$ and $p_3$, etc. We want to extract these patterns, if any exists, from each time series.

Our team decided to run K-Means on each time series such that one timestep is seen as one data point. So the series $T$ becomes a matrix of $N$ rows and $P$ columns, each row is an individual data point. In other words, we've completely ignored the temporal dependence in the data.

Much to my surprise, K-Means has managed to find the patterns. The result is interpreted like this: a cluster label is assigned to each row in the matrix (i.e. that data point belongs to such cluster); consecutive data points that have the same cluster label form a pattern. Empirically, it works really well, the patterns are neatly found, but we're unable to provide any justification on why it works.

(Technically it's clear to us how K-Means finds/updates the centroids using Euclidean distance and so on but intuitively it isn't clear: why does it work on time series where one timestep = one data point??)

It really bugs me, being unable to understand why K-Means works but advanced algorithms that are dedicated to time series like motif discovery (using Matrix Profile for example) do not.

I'd appreciate it very much if anyone can provide some insights into this question.

Thank you very much in advance!

Answer 1

Well, one reason that I see is that probably your patterns are almost completely described by the information contained in the points, and are not necessarily dependent on the temporal factor.

Imagine a very simple time series with univariate points, with a sinusoidal shape. If we want to recognize patterns as in "points below 0" and "points above 0", K-means can easily do so, as the information is already contained in the points themselves, and the temporal component can be ignored.
However, if we want to separate patterns as "decreasing" and "increasing", we cannot do so with an algorithm that ignores the time dependency, unless make our points multivariate including some feature on the lagged inputs (change from last point, or mean of the past days).

In your case, if K-means manages to fully discern your patterns, it would probably mean that these are fully described by the information that your sensors supply, and that they do not depend on their lagged values, or on their momentum.