k-NN Candlestick Pattern Search Extensions: Combining Forecasts

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:

choices

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

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

asdf

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:

asdf

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.

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:

Weighted average

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:

asdf

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).

asdf

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:

asdf

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:

stats

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.

k-NN Candlestick Pattern Search Extensions: More Data

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.

equity curves with without IBS

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), when IBS < 0.5

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  5 basis points. $0.005 per share in commissions.

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):

sensitivity analysis

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:

multi input short only

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:

multi input long short

Long and short strategies above combined. $0.005 per share in commissions.

stats

 

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:

AAPLMSFT

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.

Mining for Three Day Candlestick Patterns

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.

absolute difference 50 closest exp vs realized

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 squared

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:

closest count tests

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:

equity curves with without IBS

squared 50 closest 0.2pct limit ibs filter results

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:

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:

patterns 50th closest

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.

ETFs square 50 closest 0.2 ibs filter results

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.

Doing the Jaffray Woodriff Thing (Kinda), Part 1

Jaffray Woodriff, who runs QIM, a highly successful systematic fund, has provided enough details about his data mining approach in various interviews (particularly the one in the excellent book Hedge Fund Market Wizards) that I think I can approximate it. Even though QIM has been lagging a bit the last few years, they have an excellent track record, so their approach is certainly worthy of imitation if possible. They trade commodities, currencies, etc. so the approach seems to be highly portable. And while they suffer from significant price impact  issues (not to mention being forced into longer holding periods) due to their size, a small trader could probably do far better with the same strategies.

Introduction

The approach, as much as he has detailed it, goes as follows:

  • Generate random data.
  • Mine it for trading strategies.
  • The best strategies resulting from the random data are now the benchmark that you have to beat using real data.
  • Mine the real data, discard anything that isn’t better than the best models from the random data (this ensures that you have found an actual edge despite the excessive mining).
  • Use cross validation to more accurately estimate the performance of the models and avoid curve fitting.
  • Test the model out of sample, and retain it if it performs reasonably well compared to the in-sample results.

The point is essentially to generate an environment in which we know that we have no edge whatsoever, mine the data for the best possible results, and then use those as a benchmark that we have to clear in order to prove that an edge exists in the real data.

What they do after this is also quite interesting and important: checking the correlation between the newly discovered models and the models they already use. This ensures that any new “edge” they incorporate is a novel one and not simply a copy of something they already have. Supposedly this approach has yielded over 1500 different signals which they then use to trade, on medium-term horizons (if I remember correctly their average holding period is roughly one week). The issue of combining the predictions of 1500 signals into a decision to trade or not trade is beyond the scope of this post, but it’s a very interesting “ensemble model” problem.

It is clear that the approach requires not only rigorous statistical work, but also tons and tons of computing power (the procedure is highly parallelizable however, so you can just throw hardware at it to make it go faster). One potentially interesting way of tempering that requirement would be using genetic algorithms instead of brute force to search for new strategies. There are tricky issues with that approach, though: constructing the genome so that it can describe all possible trading models we want to look at, for example. How does one encode a wide array of chart patterns in a genome? There do not seem to be obvious/intuitive solutions.

Generating random data sets

There are several issues that have to be looked at here. Do we randomly sample the real data or do we use the parameters of that data and plug it into a known statistical distribution to generate completely new numbers? How many times do we repeat this procedure? In either case we are bound to lose some features of real financial time series, but this is probably a good thing since those features may result in true exploitable edges. It is important to generate a healthy number of data series. Some are simply going to be “better” than others for any one particular trading model, so testing over a single randomly generated series is not enough.

In general we want at least the semblance of a “real” data series. As such we can’t simply select random OHLC data; it would just result in a nonsensical time series with giant gaps all over the place. Instead I will use the following procedure:

  • Start by selecting a random day’s OHLC points. This forms our first data point.
  • Select any random day, and compute the day’s (close to close) percentage return from the previous day.
  • Use this value to generate the next fake closing price.
  • From that same (real) day, calculate the OHL prices in terms relative to the closing price.
  • Use those relative prices to generate the fake OHL prices.

I find this approach gives rather good results, producing series that look realistic and give the appearance of trends, different volatility regimes, etc. fake series

The models

Naturally I can’t test the billions upon billions of models that they test at QIM, and taking the model-agnostic approach is currently beyond my abilities. I can kind-of get around the issue by testing a very narrow range of models: moving average crossovers (another simple and interesting thing to test would be 1/2 day candlestick patterns). This still leaves a significant number of parameters to test:

  • The type of moving average to use (simple, exponential, or Hull)
  • The length of each moving average.
  • The values that the moving averages will be based on (open, high, low, or close).
  • The holding period. I’ll be using a technical entry, but a partially time-based exit. This may or may not be a good idea, but I’m running with it.
  • Trend-following vs contrarian (i.e. trade in the direction of the “fast” moving average or against it).

Evaluating the results

An important question remains: what metric do we use to evaluate the models? The use of cross validation presents unique problems in performance measurement, and we have to take these into account from this stage, because these results will be used for comparison to the real ones later on.

Drawdown is a problematic measure because drawdown extremes tend to be rare. When dividing a set into N folds for cross validation, a set of parameters may be rejected simply because a certain period generated a high drawdown, despite this drawdown being consistent with long-term expectations.

Another issue arises with the use of annualized returns: they may be rather meaningless if the signal fires very frequently. If what we care about is short-term predictability, it may be more prudent to look at average daily returns after a signal, instead of CAGR. This could also be ameliorated by taking trading costs into account, as weak but frequent signals would be filtered out.

In the end, many of these decisions depend on the trader’s choice of style. Every trader must decide for him or her self what risks they care about, and in what proportion to each other. As an attempt at a balanced performance metric, I will be using my Trading System Consistency, Drawdown, Return Asymmetry, Volatility, and Profit Factor Combination Metric (or TRASYCODRAVOPFACOM for short), which is calculated as follows:

 TRASYCODRAVOPFACOMSt. Dev. is the annualized standard deviation of daily returns, and the profit factor is calculated based on daily returns.

The TRASYCODRAVOPFACOM still has weaknesses: a set of parameters may pick only a tiny amount of trades over the years. If they’re successful enough, it can lead to a high score but a useless signal. To avoid this I’ll also be setting the minimum number of trades to 100, a reasonable hurdle given the 17 years long sample.

The random return results

Using a brute force approach, I collected approximately 704,000 results from 5 randomly generated series. It took several hours on my overclocked i5-2500K, so it’s definitely not a viable “real-world” approach (I am a terrible programmer, so some of the slowness is of my own making). The results look like you’d expect them to, with a few outliers at the top and to bottom:

random brute force CAGR random brute force PF random brute force TRASYCODRAVOPFACOM

Here are the best values achieved:

brute force random results maximums

Note that this isn’t a “universal” hurdle: it’s a hurdle for this specific subset of moving average signals, on the GBPUSD pair. I am certain that a wide array of signals and data would generate higher hurdles.

Genetic Algorithm?

Brute force takes ages, even for just 5 return series, which is far too low to draw any conclusions. Are there any faster ways than brute force to find the best possible results from our random data? If this were a “normal” dataset, I would say yes, of course! However I was not sure about this case due to the randomly generated data that we are dealing with.

If the data is random, does it follow that the optimal strategy parameters are also randomly distributed? Are they uniformly distributed or are there “clusters” that, due to somehow exploiting the structure of the time series, perform better or worse than the average? The question is: is the performance slope around local maxima smooth, or not? A simple method to make this thing go faster is to throw the problem into a genetic algorithm, but a GA will offer no performance improvement if the performance is uniformly randomly distributed.

Testing this is simple: I just ran a GA search on the same 5 series I brute forced above. If the GA results are similar to the brute force results, we can use the GA and save a lot of time. As long as there are enough populations, and they are large enough (I settled on 4 populations with 40 chromosomes each), the results are “close enough”: roughly 3-20% lower than the brute force (max CAGR was 7.043%, max avg. daily return was 0.176%). It might be a good idea to scale the GA results by, say, an additional 10-20% in order to make up for this deficit.

I then generated 100 series and put the GA to use. Here are the results:

random GA results maximums

And here are the distributions of maximum values achieved for each individual series:

random GA results

These results have set the bar rather high. One might imagine that throwing out everything below this hurdle will leave us with very little (nothing?) in the end. But if Woodriff is to be believed, he has found upwards of 1500 signals that perform better than the hurdle (and that’s 1500 signals that were uncorrelated enough with each other that they were added to their models). So there’s got to be a lot of interesting stuff to find!

In part 2 I will take a look at cross validation and what we can do with the real data.