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.
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.
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.
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.
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.
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.
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.
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.
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.
I have gotten a couple of emails asking me about the topic of analyzing performance, so I decided to detail the tools I use. Measuring performance and attributing success or failure to the right factors is an extremely important part of the trading process. Actually trading a strategy will often reveal aspects that don’t come up in the research stage. Unexpected things happen, revealing previously hidden strengths or weaknesses. Strategies improve or deteriorate through time. Execution issues eat into returns. Patterns emerge that can be exploited to enhance returns or limit risk.
These situations, and performance evaluation in general, are a crucial part of the research/trading/performance loop:
A lack of attention to performance, and the underlying factors that drive it, will have a deleterious effect both on your long-term trading results and the things that you will discover in the research stage.
I’ll demonstrate the tools using two strategies, one of which has been going well, and the other not: 1) a rather generic GTAA momentum/trend-following strategy that has been running for a bit over a year, and 2) an AAPL swing trading strategy that’s been in “trial” mode for the last 6 months or so.
My performance analysis system, the QUSMA Portfolio and Trade Analytics Suite, is primarily based around the concept of a “trade”. A trade is a unit that can contain any number of orders and cash transactions (dividends, taxes, etc.), which are somehow related. A pair trade would include both legs in a single trade, for example. The underlying data is imported using IB’s flex queries which have a very simple and easy to handle XML structure.
Trades are assigned to a “strategy” and can also be assigned any number of tags. Some of the things that I use tags for are: trade direction (long/short/both), trade length, developed/developing country, asset class, etc. Notes with images can also be attached to trades, which is incredibly useful for reviews. Finally, the trades can be filtered on any number of criteria to produce reports, and compared against custom benchmarks.
A trade and its two associated orders.
There are some general principles that summarize my approach to performance measurement:
- Execution and commissions are extremely important.
- Separate timing from sizing.
- Statistics on trades in both dollar terms and % terms.
- Separate capital allocations to strategies from total capital.
- Statistics on returns both on capital allocated to a strategy (ROAC) and on total capital (ROTC).
- Always think probabilistically and in terms of expectations
- The more ways you can find to look at the data, the better.
Simple visual inspection is my starting point, and I think it’s very important. The simple act of staring at charts often leads to new research ideas.
GTAA strategy: a number of losing trades in TLT.
So let’s get started with the graphs and stats. At the top, the standard dollar PnL (daily and close-to-close) and equity curves (both in terms of ROAC and ROTC), which are also plotted against a benchmark:
GTAA strategy cumulative returns on allocated capital. Chart also comes in ROTC flavor.
AAPL strategy cumulative PnL.
Next up are the trade statistics. Commissions are right up there, it’s very important to keep in mind how much you are losing in those costs. A few basis points may not seem like much, but they can quickly eat up a significant portion of your profits. Note all the stats are given both in dollar and percentage terms, in order to separate timing effects from sizing effects.
Results by calendar month:
AAPL strategy. Also comes in ROTC flavor.
Probably the most important bit, statistics on daily returns, the standard ratios, and so forth. The MAR ratio is probably the most important number for me. The reason is simple: it determines my leverage constraints, and thus my returns. A high Sharpe ratio is meaningless if you can’t lever up. Note how the simple, static benchmark portfolio has destroyed the GTAA approach:
GTAA strategy. Benchmark is a 20/15/15/20/10/10/10 % mix of SPY/EFA/EEM/IEF/LQD/VNQ/DBC respectively. Stats are also available for ROTC.
Some simple benchmarking stuff:
GTAA strategy vs diversified benchmark.
Histograms of daily returns, and returns per trade. Again, it’s important to look at both dollar and percentage results:
Also, holding period histogram:
Position sizing vs trade returns. Naive risk parity seems to be doing alright:
Trade length vs returns chart, the relationship here is pretty clear.
The movement capture stats measure how good the strategy is at capturing returns. GU is gross upside, or the gross positive returns during the period. UC% is the percentage of that movement that was captured by being long, UM% is the percentage of the movement that was missed by being flat, while UL% is the percentage of the movement that was lost due to being short. The calculations are repeated for downside movement.
GTAA strategy. Being long-only, only upside movement has been captured.
Cumulative percent returns, by instrument. A similar chart with dollar PnL by instrument also exists.
Autocorrelation and partial autocorrelation stats based on daily returns:
GTAA strategy. High autocorrelation values can be exploited both to enhance returns and for risk management.
Standard value at risk calculations, based on resampled historical data. I’ll be adding the option to use parametric methods in the future.
GTAA strategy: 10-day value at risk.
Monte Carlo simulation. It simply uses historical data, either trades or daily returns (either ROAC or ROTC). Sampling can be done with replacement or without (the latter simply re-orders the existing equity curve). There is also an option to use N consecutive days/trades, which can capture volatility clustering and autocorrelation effects. The analysis returns confidence intervals for the equity curve, as well as the cumulative and point distributions of maximum drawdowns.
GTAA strategy: there is a 10% chance of a drawdown worse than 18% in the next 500 trading days.
Finally, some simple stats and charts on execution. All of my trades are either at the close or the open, so those are the prices I benchmark against. Below are stats from the AAPL strategy’s buy orders around the close.
Top: slippage vs time difference in seconds from benchmark. Middle: slippage by order type. Bottom: Slippage histogram.
I think that the biggest weakness in my toolset is the lack of interaction with backtesting results. These can be used in two main ways: 1) comparing theoretical results to real trading results, and 2) as an extended dataset for the risk management functions. Also, I don’t do any stock picking, but if I did that would entail several additions, mainly performance attribution by country, sector, etc. as well as analyzing value/size/momentum factor exposures.
Leave a comment and tell us what you like to use: is the standard stuff enough for you, or do you use any obscure ratios or unique charts?