The second, and probably final, followup to the Mining for Three Day Candlestick Patterns post. Previously, we improved performance by adding more data to the search. In this post we’ll try to improve the system further by combining multiple predictors. The central question is how to combine the forecasts. I test averaging, weighted averaging, regression, and a voting scheme and compare them against a baseline one-predictor strategy.

Set-Up

Combining predictors is a standard tactic in machine learning, but the case of k-NN predictors is a bit of an outlier. Typical ensemble methods depend on generating variations in the data set in order to generate different and complementary predictors (as in the cases of boosting and bagging). This doesn’t work very well with nearest neighbor predictors, however, because they tend to be insensitive to variations in the data set. So what can we vary? The choice of k, the choice of inputs, the choice of distance measure for the nearest neighbors, and some pre-processing options such as whether to adjust for volatility or not.

I am not going to make any variation in outputs as that’s reserved for a post of its own. The idea is pretty simple: it’s essentially a random forest with k-NN predictors instead of decision trees (here’s an interesting paper on it).

So we’re left with k, sum of absolute or sum of square distances, and volatility adjustment. I picked 10 combinations of these options:

The k values were picked at random and I’m sure it’s possible to do better by optimizing them using cross validation.

The signals obviously overlap significantly, and have similar stats when used one-by-one:

Long signal stats. Long position threshold: forecast > 5 basis points & IBS < 0.5.

Short signal stats. Short position threshold: forecast < -10 basis points & IBS > 0.5.

The instrument traded is SPY. Additional data is taken from the following instruments for the pattern search: EWY, EWD, EWC, EWQ, EWU, EWA, EWP, EWH, EWL, EFA, EPP, EWM, EWI, EWG, EWO, IWM, QQQ, EWS, EWT, and EWJ. The thresholds in each case are adjusted to result in a similar length of time spent in the market. Position sizing is done based on the 10-day realized volatility of SPY, as described in this post: leverage is equal to 20% divided by 10-day realized annualized standard deviation, with a maximum leverage of 200%. Finally, an IBS filter is applied that allows long positions only when IBS < 0.5 and short positions only when IBS > 0.5.

The baseline is the PF3 predictor: k = 75, square distance measure, no volatility adjustment. Here’s the equity curve:

PF3 predictor equity curve. $0.005 per share in commissions.

Averaging

The simplest approach is obviously to just average the 10 forecasts and then use the average value to generate trades. A long position is taken when the average forecast is greater than 15 basis points, and a short position when the average is smaller than -12.5 basis points. Here’s what the equity curve looks like:

Equity curve using average forecast. $0.005 per share in commissions.

It’s interesting to note that the dispersion of forecasts is inversely related to the accuracy of the average: the smaller the standard deviation of the forecasts, the more accurate they are. Unfortunately effect is marginal and thus not particularly useful for improving the strategy.

Weighted Averaging

A simple extension, that generates slightly better stats, is to weigh each forecast before averaging. There’s a wide array of stats one can use here (Sharpe/Sortino/MAR ratios are obvious candidates); I picked the mean square error. The inverse of the MSE becomes the forecast’s weight, so that smaller errors result in greater weights. The same thresholds as above are used to generate signals. The weights provide a slight improvement both in terms of Sharpe and MAR ratios. The equity curve:

Equity curve using weighted average forecast, with weights equal to the inverse of the mean square error. $0.005 per share in commissions.

Voting

Using a threshold for each forecast, (>5 basis points for a “long” vote, and <-10 basis points for a “short” vote), each predictor is assigned a long or short vote. The overlap between the votes is significant, between 88% and 97% for different estimators. How many votes should we require for a trade? It quickly becomes obvious that simple majority voting isn’t enough, as only near-unanimous decisions provide worthwhile predictions. The average next-day return when there are between 1 and 8 long votes is 0.4 basis points. The average return after 9 or 10 long votes is 23 basis points.

The resulting equity curve looks like this:

Equity curve using voting system. 9 or more votes required to take a position. $0.005 per share in commissions.

Ordinary Least Squares

It’s also possible to combine the forecasts using regression, with next-day returns as the dependent variable and the k-NN predictor forecasts as the independent ones.

The distribution of forecasts with OLS is very tightly clustered around 0, and for some reason higher forecasts are not associated with higher next-day returns (as they are for the 3 methods above). I don’t really understand why this is the case. The thresholds for trades are 0.5 basis points for a long trade, and -0.5 basis points for a short trade.

An issue here is, of course, multicollinearity due to the similarity of the independent variables. This can lead to, among other problems, overfitting (which is usually characterized by very large absolute values of the coefficients). Using ridge regression solves that issue by limiting the absolute value of coefficients.

A potentially interesting idea would be to constrain the coefficients to positive values, which might lessen the overfitting effects and also make much more sense on an intuitive level (after all, we know all the forecasts are similarly accurate, so negative coefficients don’t make much sense).

Equity curve using OLS regression. $0.005 per share in commissions.

Ridge Regression

If multicollinearity is a significant problem, we can use ridge regression to solve it. It offer significant improvement over the OLS approach, but it still fares badly compared to the one-predictor case. The same thresholds as in the OLS approach are used. Here’s the equity curve:

Equity curve using ridge regression. $0.005 per share in commissions.

Stats

Here are the stats for the single-predictor base case and all the combination methods:

All of them other than the voting failed horribly. I’m not sure why, but it’s good to know. The improvement provided by the voting system is sizable, however. Not only does the voting-based strategy achieve significantly higher risk-adjusted returns, it does it while spending 15% less time in the market. Those results are also easy to improve on by simply adding more predictors. The marginal gain from each new predictor will be diminishing, but there is definitely more value to wring out of it. And this is just with 3-day patterns: we can easily add 2 and 4 day patterns into the mix as well.

Other Possibilities

A wide array of machine learning methods can be used to combine predictions. Especially if the number of forecasts grew larger, techniques such as random forests or ANNs would be interesting to investigate. As long as simpler methods work very well I think there is little reason to increase the complexity (not to mention the opaqueness) of the strategy.