### Visualizing the Similarity Between Multiple Time Series

Presenting the similarity between multiple time series in an intuitive manner is not an easy problem. The standard solution is a correlation matrix, but it’s a problematic approach. While it makes it easy to check the correlation between any two series, and (with the help of conditional formatting) the relation between one series and all the rest, it’s hard to extract an intuitive understanding of how all the series are related to each other. And if you want to add a time dimension to see how correlations have changed, things become even more troublesome.

The solution is multidimensional scaling (the “classical” version of which is known as Principal Coordinates Analysis). It is a way of taking a distance matrix and then placing each object in N dimensions such that the distances between each of them are preserved as well as possible. Obviously N = 2 is the obvious use case, as it makes for the simplest visualizations. MDS works similarly to PCA, but uses the dissimilarity matrix as input instead of the series. Here’s a good take on the math behind it.

It should be noted that MDS doesn’t care about how you choose to measure the distance between the time series. While I used correlations in this example, you could just as easily use a technique like dynamic time warping.

Below is an example with SPY, TLT, GLD, SLV, IWM, VNQ, VGK, EEM, EMB, using 252 day correlations as the distance measure, calculated every Monday. The motion chart lets us see not only the distances between each ETF at one point in time, but also how they have evolved.

Some interesting stuff to note: watch how REITs (VNQ) become more closely correlated with equities during the financial crisis, how distant emerging market debt (EMB) is from everything else, and the changing relationship between silver (SLV) and gold (GLD).

Here’s the same thing with a bunch of sector ETFs:

To do MDS at home: in R and MATLAB you can use cmdscale(). I have posted a C# implementation here.