Using Quantum Mechanics to Cluster Time Series
This work addresses time series clustering for data analysis, but it is incremental as it builds on existing techniques like Keogh's work.
The paper tackles the problem of clustering time series by reducing them to 13-dimensional points to avoid excessive noise labeling, using quantum mechanics to estimate trends and a fast parameter fitting method. It achieves high accuracy and speed in clustering, though acknowledges other methods may be faster or more accurate.
In this article we present a method by which we can reduce a time series into a single point in $\mathbb{R}^{13}$. We have chosen 13 dimensions so as to prevent too many points from being labeled as "noise." When using a Euclidean (or Mahalanobis) metric, a simple clustering algorithm will with near certainty label the majority of points as "noise." On pure physical considerations, this is not possible. Included in our 13 dimensions are four parameters which describe the coefficients of a cubic polynomial attached to a Gaussian picking up a general trend, four parameters picking up periodicity in a time series, two each for amplitude of a wave and period of a wave, and the final five report the "leftover" noise of the detrended and aperiodic time series. Of the final five parameters, four are the centralized probabilistic moments, and the final for the relative size of the series. The first main contribution of this work is to apply a theorem of quantum mechanics about the completeness of the solutions to the quantum harmonic oscillator on $L^2(\mathbb{R})$ to estimating trends in time series. The second main contribution is the method of fitting parameters. After many numerical trials, we realized that methods such a Newton-Rhaphson and Levenberg-Marquardt converge extremely fast if the initial guess is good. Thus we guessed many initial points in our parameter space and computed only a few iterations, a technique common in Keogh's work on time series clustering. Finally, we have produced a model which gives incredibly accurate results quickly. We ackowledge that there are faster methods as well of more accurate methods, but this work shows that we can still increase computation speed with little, if any, cost to accuracy in the sense of data clustering.