Time-series similarity measures

Question

Suppose I have two time series $X$ and $Y$ of stock prices. How do I measure the "similarity" of $X$ and $Y$?

(I'm being deliberately vague as I don't have a particular application, and I'm curious about different approaches in general. But I guess you can imagine that there's some stock x that I don't want to trade directly, for whatever reason, so I want to find a similar stock y to trade in its place.)

One method is to take a Pearson or Spearman correlation. To avoid problems of spurious correlation (since the price series likely contain trends), I should take these correlations on the differenced or returns series (which should be more stationary).

What are other similarity methods and their pros/cons?

Answer 1

I assume you're using returns (or log returns) instead of actual stock prices. In practice, you may also want to smooth the data by using a moving average.

There are several correlation coefficients:

• Pearson's $r$ - most commonly used definition of correlation:

\begin{equation} r = \frac{\sigma_{x,y}}{\sigma_x \sigma_y} \end{equation}

• Spearman's $\rho$ - uses the rank of each data set (array index if data had been sorted); less sensitive to outliers in the sample as it's non-parametric:

\begin{equation} d_i = x_i - y_i \end{equation}

\begin{equation} \rho = 1 - \frac{6\Sigma d_{i}^{2}}{n(n^{2}-1)} \end{equation}

• Kendall's $\tau$ - also based on ranking, but represents the probability that the two data sets are in the same order vs. the probability that they are in different orders:

\begin{equation} \tau = \frac{C - D}{\frac{1}{2} n(n - 1)} \end{equation}

• Kruskal's $\Gamma$ - similar to Kendall's, but acknowledges ties in the ordering:

\begin{equation} \Gamma = \frac{C - D}{C + D} \end{equation}

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$C$ is the number of concordant pairs and $D$ is the number of discordant pairs.

Answer 2

You can look at cointegration.

Answer 3

If you use a Box-Jenkins model, look at this research which uses an ARIMA framework to define clusters, and then measures the similarity of the time series via a cepstral coefficient based upon the autoregressive parameters.

http://www.csee.umbc.edu/~kalpakis/homepage/papers/ICDM01.pdf