Financial Time Series Clustering

Abstract

Financial time series clustering finds application in forecasting, noise reduction and enhanced index tracking. The central theme in all the available clustering algorithms is the dissimilarity measure employed by the algorithm. The dissimilarity measures, applicable in financial domain, as used or suggested in past researches, are correlation based dissimilarity measure, temporal correlation based dissimilarity measure and dynamic time wrapping (DTW) based dissimilarity measure. One shortcoming of these dissimilarity measures is that they do not take into account the lead or lag existing between the returns of different stocks which changes with time. Mostly, such stocks with high value of correlation at some lead or lag belong to the same cluster (or sector). The present paper, proposes two new dissimilarity measures which show superior clustering results as compared to past measures when compared over 3 data sets comprising of 526 companies. abstract environment.

A Appendix

1.1 A.1 Correlation Based Dissimilarity Measure (COR)

A simple dissimilarity measure for time series clustering is based on Pearson’s correlation factor between time series $X_{T}$ and $Y_{T}$ given by

$$\begin{aligned} COR(X_{T},Y_{T})=\frac{\sum _{t=1}^{T}(X_{t} - m_{X})(Y_{t} - m_{Y})}{\sqrt{\sum _{t=1}^{T}(X_{t} - m_{X})^{2}}\sqrt{\sum _{t=1}^{T}(Y_{t} - m_{Y})^{2}}} \end{aligned}$$

where $m_{X}$ and $m_{Y}$ are the average values of the time series $X_{T}$ and $Y_{T}$ respectively. The dissimilarity measure is then given by

$$\begin{aligned} Dissimilarity_{COR}(X_{T},Y_{T})=\sqrt{2(1-COR(X_{T},Y_{T}))} \end{aligned}$$

For more information regarding this distance measure, one may refer to [9].

1.2 A.2 Temporal Correlation Based Dissimilarity Measure (CORT)

As introduced by Douzal [12], the similarity between two time series is evaluated using first order temporal correlation coefficient [13] given by,

$$\begin{aligned} CORT(X_{T},Y_{T})=\frac{\sum _{t=1}^{T-1}(X_{t+1}-X_{t})(Y_{t+1}-Y_{t})}{\sqrt{\sum _{t=1}^{T-1}(X_{t+1}-X_{t})^{2}}\sqrt{\sum _{t=1}^{T-1}(Y_{t+1}-Y_{t})^{2}}} \end{aligned}$$

The dissimilarity proposed by Douzal [12] modulates the ‘dissimilarity value’ between $X_T$ and $Y_T$ using the coefficient $CORT(X_{T},Y_{T})$. Specifically, it is defined as follows.

$$\begin{aligned} Dissimilarity_{CORT}(X_{T},Y_{T})=\phi _{k}[CORT(X_{T},Y_{T})] \times Dissimilarity(X_{T},Y_{T}) \end{aligned}$$

where $\phi _{k}$ is an adaptive function given by,

$$\begin{aligned} \phi _{k}(u) = \frac{2}{1 + e^{ku}},\, k\ge 0 \end{aligned}$$

(6)

and Dissimilarity$(X_{T},Y_{T})$ refers to dissimilarity value computed using any of the available dissimilarity measures like Euclidean, DTW etc. In this paper, we choose DTW as the preferred dissimilarity measure. This is because DTW effectively takes into account slight shape distortions while calculating its dissimilarity measure value. For more details regarding DTW, readers are referred to [9, 11].