# Randomly Sampling from one Random Signal

The Question

I was given a set of data from a local commodity market trader. The trader had employed two different trading algorithms on the same commodity. The price of the commodity, $$X_t$$, is clearly a random process. Sub-system-1 generates a trade signal based on a different algorithm (compared to sub-system-2), generating a series of signals $$s_{1t}$$ which lead to a cumulative profit of $$Y_{1t}$$. Similarly, the other algorithm has parameters given as $$s_{2t}$$ and $$Y_{2t}$$ respectively.

The trader wanted to ask if the trades on both sub-systems were correlated or not (essentially asking me to find the similarity between the trade signals). It is important to note here that the trade signals and the consecutive profit themselves are randomly sampled from the commodity price. For more context, one can assume that the first sub-system generates a trade signal after looking at price movements based on 15 min intervals while the other sub-system looks at 30 min intervals to generate a signal.

Approach I employed

Until now, to solve this problem, I have decided to look at the profit from both signals to evaluate the correlation between the sub-systems' working. In this, I tried to average the cumulative profit per day from each sub-system and found the cumulative profit on a particular day. On some days the system doesn't make any significant stride (essentially zero profit). I interpolate the values (using linear interpolation) for the zero profits to get a time series with no missing values. Repeating the same for the other sub-system's profit, I get two-time series that are uniformly sampled. Now I calculate the following:

1. Pearson's Correlation
2. Cross-Correlation
3. Dynamic Time Warping (the euclidean distance between the two signals)

I use these to comment if the time series are correlated or not.

Follow-Up Questions

1. Is my approach satisfactory?
2. Are there existing methods to judge the similarity of time series sampled at random intervals from a random signal?
3. I utilize an interpolation and averaging scheme to create similar sampling for comparing the two sub-systems. How safe is this?