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.
Read more k-NN Candlestick Pattern Search Extensions: Combining Forecasts
This is a followup to the Mining for Three Day Candlestick Patterns post. If you haven’t read the original post, do so now because I’m not going to repeat the basic mechanics of the strategy. While the approach was somewhat fruitful, it also had some obvious problems: it only seems to work in bearish or high volatility market regimes, and it couldn’t produce good short signals. The main idea I had to resolve these issues was simply to get more data.
Original strategy using only SPY data. Note long stretches of flat results.
That is easier said than done. Could we use mutual funds or index values to extend the dataset backwards? No, because the daily high/low values are inaccurate. The only alternative we are left with is using data from other instruments. So I picked a broad selection of equity ETFs to include: EWY, EWD, EWC, EWQ, EWU, EWA, EWP, EWH, EWL, EFA, EPP, EWM, EWI, EWG, EWO, IWM, QQQ, EWS, EWT, and EWJ.
The selection was comprehensive and unoptimized. I think you could do some sort of walk-forward optimization that picks the best combination of securities to include in the data set. I’m not sure how much that would help.
The additional data worked fantastically well, resolving both problems. The number of opportunities to trade increased significantly, long signals work very nicely under all market conditions, and predicting negative returns works far better. There was also an unexpected benefit: far less time is needed before the forecasts become usable. In the original implementation I waited 2000 days before starting to use the forecasts. With the extended data set this can be cut to 500, thus letting the backtest cover a longer period.
Performance-wise there were no problems, as the Accord .NET k-d tree implementation that I use is very quick. Finding the nearest 75 points in a data set of approximately 100,000, in 11 dimensions, takes less than 2 milliseconds on my overclocked 2500K.
The settings used in the search are simple: the length of the patterns is 3 days, the 75 closest ones are used to construct a forecast by averaging their next-day returns, and distance is calculated as the sum of squared distances in every dimension. Trades are taken when the forecast is above/below a certain threshold. They are then passed through a filter which only allows long positions when IBS < 0.5 and short positions only when IBS > 0.5.
It should be noted that using traditional measures of “fit” does not work very well with pattern matching. Adding the above instruments actually increases the RMSE, despite significantly increasing the trading performance of the forecasts.
A look at forecasts vs realized next-day returns:
PatternFinderMultiInput (x-axes) vs next day returns (y-axies), for IBS < 0.5 and forecast > 0
An important aspect to note is that even marginally positive forecasts work very well. For example, with the extended dataset, forecasts between 5 and 10 basis points resulted in an average 21 bp return the next day. On the other hand, using SPY data only, the return for those forecasts was just 5 basis points. What this means is that there are many more trades to take, which is what allows the strategy to do well in all market environments. Here’s the long-only equity curve:
Long position taken when IBS < 0.5 and forecast > 5 basis points. $0.005 per share in commissions.
A couple of charts to analyze the sensitivity of the long-only strategy’s results to changes in inputs (IBS limit and minimum forecast limit):
The additional data also has the benefit of making shorting possible. The equity curve doesn’t look as good, but it’s still a giant improvement over zero predictive ability on the short side:
Short position taken when IBS > 0.5 and forecast < -20 basis points. $0.005 per share in commissions.
Finally, the long and short strategies combined, along with the stats:
Long and short strategies above combined. $0.005 per share in commissions.
The concept also seems to work for stocks. For example, I tested a long-only strategy on AAPL, using the same settings as above, both with and without the addition of MSFT data. The Microsoft data improved every aspect of the results, with surprisingly consistent performance over nearly 20 years:
It would be interesting to try to apply this on a more massive scale, by increasing the data set to something like all S&P 500 stocks. Some technical restrictions prevent me from doing that right now, but I’ll come back to the idea in the future.
Read more k-NN Candlestick Pattern Search Extensions: More Data
I’ve been thinking a lot about candlestick patterns lately but grew tired of trying to generate ideas and instead decided to mine for them. I must confess I didn’t expect much from such a simplistic approach, so I was pleasantly surprised to see it working well. Unfortunately I wasn’t able to discover any short set-ups. The general bias of equity markets toward the upside makes it difficult to find enough instances of patterns that are followed by negative returns.
The idea is to mine past data for similar 3 day patterns, and then use that information to make trading decisions. There are several choices we must make:
- The size of the lookback window. I use an expanding window that starts at 2000 days.
- Once we find similar patterns, how do we choose which ones to use?
- How do we measure the similarity between the patterns?
To fully describe a three day candlestick pattern we need 11 numbers. The close-to-close percentage change from day 1 to day 2, and from day 2 to day 3, as well as the positions of the open, high, and low relative to the close for each day.
To measure the degree of similarity between any two 3-day patterns, I tried both the sum of absolute differences and the sum of the squared differences between those 11 numbers; the results were quite similar. It would be interesting to try to optimize individual weights for each number, as I imagine some are more important than others.
The final step is to select a number of the closest patterns we find, and simply average their next-day returns to arrive at an expected return.
Expected vs realized returns for SPY, 50 closest patterns by absolute difference. Numbers above the bars indicate the number of instances in each bucket.
How do we choose which patterns are “close enough” to use? Choose too few and the sample will be too small. Choose too many and you risk using irrelevant data. That’s a number that we’ll have to optimize.
Histogram of expected return estimates for different sample sizes.
When comparing the results we also run into another problem: the smaller the sample, the more spread out the expected return estimates will be, which means more trades will be chosen given a certain minimum limit for entry. My solution was to choose a different limit for trade entry, such that all sample sizes would generate the same number of days in the market (300 in this case). Here are the walk-forward results:
The trade-off between sample size and relevance is clear, and the “sweet spot” appears to be somewhere in the 50-150 range or so, for both the absolute difference and squared difference approaches. Depending on how selective you want to be, you can decrease the limit and trade off more trades for lower expected returns. For me, 30 bp is a reasonable area to aim for.
A nice little addition is to use IBS by filtering out any trades with IBS > 50%. Using squared differences, I select the 50 closest patterns. When their average next-day return is greater than 0.2%, a long position is taken. The results are predictably great:
The IBS filter removes close to 40% of days in the market yet maintains essentially the same CAGR, while also more than halving the maximum drawdown.
Let’s take a look at some of the actual patterns. Using squared differences, the 50 closest patterns, and a 0.2% limit, the last successful trade was on February 26, 2013. The expected return on that day was 0.307%. Here’s what those 3 days looked like, as well as the 5 closest historical patterns:
As you can see below, even the 50th closest pattern seems to be, based on visual inspection, rather close. The “main idea” of the pattern seems to be there:
Here are the stats from a bunch of different equity index ETFs, using square differences, the 50 closest patterns, 0.2% expected return limit and the IBS < 0.5 filter.
The 0.2% limit seems to be too low for some of them, producing too many trades. Perhaps setting an appropriate limit per-instrument would be a good idea.
The obvious path forward is to also produce 2-day, 4-day, 5-day, etc. versions, perhaps with optimized distance weighting and some outlier filtering, and combine them all in a nice little ensemble to get your predictions out of. The implementation is left as an exercise for the reader.
Read more Mining for Three Day Candlestick Patterns
