Machine learning principles

A learning algorithm is fed with data samples, normally derived in some way from historical prices. Each sample consists of n variables x₁ .. x_n, commonly named predictors, features, signals, or simply input. These predictors can be the price returns of the last n bars, or a collection of classical indicators, or any other imaginable functions of the price curve (I’ve even seen the pixels of a price chart image used as predictors for a neural network!). Each sample also normally includes a target variable y, like the return of the next trade after taking the sample, or the next price movement. In the literature you can find y also named label or objective. In a training process, the algorithm learns to predict the target y from the predictors x₁ .. x_n. The learned ‘memory’ is stored in a data structure named model that is specific to the algorithm (not to be confused with a financial model for model based strategies!). A machine learning model can be a function with prediction rules in C code, generated by the training process. Or it can be a set of connection weights of a neural network.

Training:     x₁ .. x_n, y  =>  model

Prediction:   x₁ .. x_n, model  =>  y

The predictors, features, or whatever you call them, must carry information sufficient to predict the target y with some accuracy. They must also often fulfill two formal requirements. First, all predictor values should be in the same range, like -1 .. +1 (for most R algorithms) or -100 .. +100 (for Zorro or TSSB algorithms). So you need to normalize them in some way before sending them to the machine. Second, the samples should be balanced, i.e. equally distributed over all values of the target variable. So there should be about as many winning as losing samples. If you do not observe these two requirements, you’ll wonder why you’re getting bad results from the machine learning algorithm.

Regression algorithms predict a numeric value, like the magnitude and sign of the next price move. Classification algorithms predict a qualitative sample class, for instance whether it’s preceding a win or a loss. Some algorithms, such as neural networks, decision trees, or support vector machines, can be run in both modes.

A few algorithms learn to divide samples into classes without needing any target y. That’s unsupervised learning, as opposed to supervised learning using a target. Somewhere inbetween is reinforcement learning, where the system trains itself by running simulations with the given features, and using the outcome as training target. AlphaZero, the successor of AlphaGo, used reinforcement learning by playing millions of Go games against itself. In finance there are few applications for unsupervised or reinforcement learning. 99% of machine learning strategies use supervised learning.

Whatever signals we’re using for predictors in finance, they will most likely contain much noise and little information, and will be nonstationary on top of it. Therefore financial prediction is one of the hardest tasks in machine learning. More complex algorithms do not necessarily achieve better results. The selection of the predictors is critical to the success. It is no good idea to use lots of predictors, since this simply causes overfitting and failure in out of sample operation. Therefore data mining strategies often apply a preselection algorithm that determines a small number of predictors out of a pool of many. The preselection can be based on correlation between predictors, on significance, on information content, or simply on prediction success with a test set. Practical experiments with feature selection can be found in a recent article on the Robot Wealth blog.

Here’s a list of the most popular data mining methods used in finance.

1. Indicator soup

Most trading systems we’re programming for clients are not based on a financial model. The client just wanted trade signals from certain technical indicators, filtered with other technical indicators in combination with more technical indicators. When asked how this hodgepodge of indicators could be a profitable strategy, he normally answered: “Trust me. I’m trading it manually, and it works.”

It did indeed. At least sometimes. Although most of those systems did not pass a WFA test (and some not even a simple backtest), a surprisingly large number did. And those were also often profitable in real trading. The client had systematically experimented with technical indicators until he found a combination that worked in live trading with certain assets. This way of trial-and-error technical analysis is a classical data mining approach, just executed by a human and not by a machine. I can not really recommend this method – and a lot of luck, not to speak of money, is probably involved – but I can testify that it sometimes leads to profitable systems.

2. Candle patterns

Not to be confused with those Japanese Candle Patterns that had their best-before date long, long ago. The modern equivalent is price action trading. You’re still looking at the open, high, low, and close of candles. You’re still hoping to find a pattern that predicts a price direction. But you’re now data mining contemporary price curves for collecting those patterns. There are software packages for that purpose. They search for patterns that are profitable by some user-defined criterion, and use them to build a specific pattern detection function. It could look like this one (from Zorro’s pattern analyzer):

int detect(double* sig)
{
  if(sig[1]<sig[2] && sig[4]<sig[0] && sig[0]<sig[5] && sig[5]<sig[3] && sig[10]<sig[11] && sig[11]<sig[7] && sig[7]<sig[8] && sig[8]<sig[9] && sig[9]<sig[6])
      return 1; 
  if(sig[4]<sig[1] && sig[1]<sig[2] && sig[2]<sig[5] && sig[5]<sig[3] && sig[3]<sig[0] && sig[7]<sig[8] && sig[10]<sig[6] && sig[6]<sig[11] && sig[11]<sig[9])
      return 1;
  if(sig[1]<sig[4] && eqF(sig[4]-sig[5]) && sig[5]<sig[2] && sig[2]<sig[3] && sig[3]<sig[0] && sig[10]<sig[7] && sig[8]<sig[6] && sig[6]<sig[11] && sig[11]<sig[9])
      return 1;
  if(sig[1]<sig[4] && sig[4]<sig[5] && sig[5]<sig[2] && sig[2]<sig[0] && sig[0]<sig[3] && sig[7]<sig[8] && sig[10]<sig[11] && sig[11]<sig[9] && sig[9]<sig[6])
      return 1;
  if(sig[1]<sig[2] && sig[4]<sig[5] && sig[5]<sig[3] && sig[3]<sig[0] && sig[10]<sig[7] && sig[7]<sig[8] && sig[8]<sig[6] && sig[6]<sig[11] && sig[11]<sig[9])
      return 1;
  ....
  return 0;
}

This C function returns 1 when the signals match one of the patterns, otherwise 0. You can see from the lengthy code that this is not the fastest way to detect patterns. A better method, used by Zorro when the detection function needs not be exported, is sorting the signals by their magnitude and checking the sort order. An example of such a system can be found here.

Can price action trading really work? Just like the indicator soup, it’s not based on any rational financial model. One can at best imagine that sequences of price movements cause market participants to react in a certain way, this way establishing a temporary predictive pattern. However the number of patterns is quite limited when you only look at sequences of a few adjacent candles. The next step is comparing candles that are not adjacent, but arbitrarily selected within a longer time period. This way you’re getting an almost unlimited number of patterns – but at the cost of finally leaving the realm of the rational. It is hard to imagine how a price move can be predicted by some candle patterns from weeks ago.

Still, a lot effort is going into that. A fellow blogger, Daniel Fernandez, runs a subscription website (Asirikuy) specialized on data mining candle patterns. He refined pattern trading down to the smallest details, and if anyone would ever achieve any profit this way, it would be him. But to his subscribers’ disappointment, trading his patterns live (QuriQuant) produced very different results than his wonderful backtests. If profitable price action systems really exist, apparently no one has found them yet.

3. Linear regression

The simple basis of many complex machine learning algorithms: Predict the target variable y by a linear combination of the predictors  x₁ .. x_n.

[latex]y = a_0 + a_1 x_1 + … + a_n x_n[/latex]

The coefficients a_n are the model. They are calculated for minimizing the sum of squared differences between the true y values from the training samples and their predicted y from the above formula:

[latex]Minimize(\sum (y_i-\hat{y_i})^2)[/latex]

For normal distributed samples, the minimizing is possible with some matrix arithmetic, so no iterations are required. In the case n = 1 – with only one predictor variable x – the regression formula is reduced to

[latex]y = a + b x[/latex]

which is simple linear regression, as opposed to multivariate linear regression where n > 1. Simple linear regression is available in most trading platforms, f.i. with the LinReg indicator in the TA-Lib. With y = price and x = time it’s often used as an alternative to a moving average. Multivariate linear regression is available in the R platform through the lm(..) function that comes with the standard installation. A variant is polynomial regression. Like simple regression it uses only one predictor variable x, but also its square and higher degrees, so that x_n == xⁿ:

[latex]y = a_0 + a_1 x + a_2 x^2 + … + a_n x^n[/latex]

With n = 2 or n = 3, polynomial regression is often used to predict the next average price from the smoothed prices of the last bars. The polyfit function of MatLab, R, Zorro, and many other platforms can be used for polynomial regression.

4. Perceptron

Often referred to as a neural network with only one neuron. In fact a perceptron is a regression function like above, but with a binary result, thus called logistic regression. It’s not regression though, it’s a classification algorithm. Zorro’s advise(PERCEPTRON, …) function generates C code that returns either 100 or -100, dependent on whether the predicted result is above a threshold or not:

int predict(double* sig)
{
  if(-27.99*sig[0] + 1.24*sig[1] - 3.54*sig[2] > -21.50)
    return 100;
  else
    return -100;
}

You can see that the sig array is equivalent to the features x_n in the regression formula, and the numeric factors are the coefficients a_n.

5. Neural networks

Linear or logistic regression can only solve linear problems. Many do not fall into this category – a famous example is predicting the output of a simple XOR function. And most likely also predicting prices or trade returns. An artificial neural network (ANN) can tackle nonlinear problems. It’s a bunch of perceptrons that are connected together in an array of layers. Any perceptron is a neuron of the net. Its output goes to the inputs of all neurons of the next layer, like this:

Like the perceptron, a neural network also learns by determining the coefficients that minimize the error between sample prediction and sample target. But this requires now an approximation process, normally with backpropagating the error from the output to the inputs, optimizing the weights on its way. This process imposes two restrictions. First, the neuron outputs must now be continuously differentiable functions instead of the simple perceptron threshold. Second, the network must not be too deep – it must not have too many ‘hidden layers’ of neurons between inputs and output. This second restriction limits the complexity of problems that a standard neural network can solve.

When using a neural network for predicting trades, you have a lot of parameters with which you can play around and, if you’re not careful, produce a lot of selection bias:

• Number of hidden layers
• Number of neurons per hidden layer
• Number of backpropagation cycles, named epochs
• Learning rate, the step width of an epoch
• Momentum, an inertia factor for the weights adaption
• Activation function

The activation function emulates the perceptron threshold. For the backpropagation you need a continuously differentiable function that generates a ‘soft’ step at a certain x value. Normally a sigmoid, tanh, or softmax function is used. Sometimes it’s also a linear function that just returns the weighted sum of all inputs. In this case the network can be used for regression, for predicting a numeric value instead of a binary outcome.

Neural networks are available in the standard R installation (nnet, a single hidden layer network) and in many packages, for instance RSNNS and FCNN4R.

6. Deep learning

Deep learning methods use neural networks with many hidden layers and thousands of neurons, which could not be effectively trained anymore by conventional backpropagation. Several methods became popular in the last years for training such huge networks. They usually pre-train the hidden neuron layers for achieving a more effective learning process. A Restricted Boltzmann Machine (RBM) is an unsupervised classification algorithm with a special network structure that has no connections between the hidden neurons. A Sparse Autoencoder (SAE) uses a conventional network structure, but pre-trains the hidden layers in a clever way by reproducing the input signals on the layer outputs with as few active connections as possible. Those methods allow very complex networks for tackling very complex learning tasks. Such as beating the world’s best human Go player.

Deep learning networks are available in the deepnet and darch R packages. Deepnet provides an autoencoder, Darch a restricted Boltzmann machine. I have not yet experimented with Darch, but here’s an example R script using the Deepnet autoencoder with 3 hidden layers for trade signals through Zorro’s neural() function:

library('deepnet', quietly = T) 
library('caret', quietly = T)

# called by Zorro for training
neural.train = function(model,XY) 
{
  XY <- as.matrix(XY)
  X <- XY[,-ncol(XY)] # predictors
  Y <- XY[,ncol(XY)]  # target
  Y <- ifelse(Y > 0,1,0) # convert -1..1 to 0..1
  Models[[model]] <<- sae.dnn.train(X,Y,
      hidden = c(50,100,50), 
      activationfun = "tanh", 
      learningrate = 0.5, 
      momentum = 0.5, 
      learningrate_scale = 1.0, 
      output = "sigm", 
      sae_output = "linear", 
      numepochs = 100, 
      batchsize = 100,
      hidden_dropout = 0, 
      visible_dropout = 0)
}

# called by Zorro for prediction
neural.predict = function(model,X) 
{
  if(is.vector(X)) X <- t(X) # transpose horizontal vector
  return(nn.predict(Models[[model]],X))
}

# called by Zorro for saving the models
neural.save = function(name)
{
  save(Models,file=name) # save trained models
}

# called by Zorro for initialization
neural.init = function()
{
  set.seed(365)
  Models <<- vector("list")
}

# quick OOS test for experimenting with the settings
Test = function() 
{
  neural.init()
  XY <<- read.csv('C:/Project/Zorro/Data/signals0.csv',header = F)
  splits <- nrow(XY)*0.8
  XY.tr <<- head(XY,splits) # training set
  XY.ts <<- tail(XY,-splits) # test set
  neural.train(1,XY.tr)
  X <<- XY.ts[,-ncol(XY.ts)]
  Y <<- XY.ts[,ncol(XY.ts)]
  Y.ob <<- ifelse(Y > 0,1,0)
  Y <<- neural.predict(1,X)
  Y.pr <<- ifelse(Y > 0.5,1,0)
  confusionMatrix(Y.pr,Y.ob) # display prediction accuracy
}

7. Support vector machines

Like a neural network, a support vector machine (SVM) is another extension of linear regression. When we look at the regression formula again,

[latex]y = a_0 + a_1 x_1 + … + a_n x_n[/latex]

we can interpret the features x_n as coordinates of a n-dimensional feature space. Setting the target variable y to a fixed value determines a plane in that space, called a hyperplane since it has more than two (in fact, n-1) dimensions. The hyperplane separates the samples with y > o from the samples with y < 0. The a_n coefficients can be calculated in a way that the distances of the plane to the nearest samples – which are called the ‘support vectors’ of the plane, hence the algorithm name – is maximum. This way we have a binary classifier with optimal separation of winning and losing samples.

The problem: normally those samples are not linearly separable – they are scattered around irregularly in the feature space. No flat plane can be squeezed between winners and losers. If it could, we had simpler methods to calculate that plane, f.i. linear discriminant analysis. But for the common case we need the SVM trick: Adding more dimensions to the feature space. For this the SVM algorithm produces more features with a kernel function that combines any two existing predictors to a new feature. This is analogous to the step above from the simple regression to polynomial regression, where also more features are added by taking the sole predictor to the n-th power. The more dimensions you add, the easier it is to separate the samples with a flat hyperplane. This plane is then transformed back to the original n-dimensional space, getting wrinkled and crumpled on the way. By clever selecting the kernel function, the process can be performed without actually computing the transformation.

Like neural networks, SVMs can be used not only for classification, but also for regression. They also offer some parameters for optimizing and possibly overfitting the prediction process:

• Kernel function. You normally use a RBF kernel (radial basis function, a symmetric kernel), but you also have the choice of other kernels, such as sigmoid, polynomial, and linear.
• Gamma, the width of the RBF kernel
• Cost parameter C, the ‘penalty’ for wrong classifications in the training samples

An often used SVM is the libsvm library. It’s also available in R in the e1071 package. In the next and final part of this series I plan to describe a trading strategy using this SVM.

8. K-Nearest neighbor

Compared with the heavy ANN and SVM stuff, that’s a nice simple algorithm with a unique property: It needs no training. So the samples are the model. You could use this algorithm for a trading system that learns permanently by simply adding more and more samples. The nearest neighbor algorithm computes the distances in feature space from the current feature values to the k nearest samples. A distance in n-dimensional space between two feature sets (x₁ .. x_n) and (y₁ .. y_n) is calculated just as in 2 dimensions:

[latex display=”true”]D = \sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + … + (x_n-y_n)^2}[/latex]

The algorithm simply predicts the target from the average of the k target variables of the nearest samples, weighted by their inverse distances. It can be used for classification as well as for regression. Software tricks borrowed from computer graphics, such as an adaptive binary tree (ABT), can make the nearest neighbor search pretty fast. In my past life as computer game programmer, we used such methods in games for tasks like self-learning enemy intelligence. You can call the knn function in R for nearest neighbor prediction – or write a simple function in C for that purpose.

9. K-Means

This is an approximation algorithm for unsupervised classification. It has some similarity, not only its name, to k-nearest neighbor. For classifying the samples, the algorithm first places k random points in the feature space. Then it assigns to any of those points all the samples with the smallest distances to it. The point is then moved to the mean of these nearest samples. This will generate a new samples assignment, since some samples are now closer to another point. The process is repeated until the assignment does not change anymore by moving the points, i.e. each point lies exactly at the mean of its nearest samples. We now have k classes of samples, each in the neighborhood of one of the k points.

This simple algorithm can produce surprisingly good results. In R, the kmeans function does the trick. An example of the k-means algorithm for classifying candle patterns can be found here: Unsupervised candlestick classification for fun and profit.

10. Naive Bayes

This algorithm uses Bayes’ Theorem for classifying samples of non-numeric features (i.e. events), such as the above mentioned candle patterns. Suppose that an event X (for instance, that the Open of the previous bar is below the Open of the current bar) appears in 80% of all winning samples. What is then the probability that a sample is winning when it contains event X? It’s not 0.8 as you might think. The probability can be calculated with Bayes’ Theorem:

[latex display=”true”]P(Y \vert X) = \frac{P(X \vert Y) P(Y)}{P(X)}[/latex]

P(Y|X) is the probability that event Y (f.i. winning) occurs in all samples containing event X (in our example, Open(1) < Open(0)). According to the formula, it is equal to the probability of X occurring in all winning samples (here, 0.8), multiplied by the probability of Y in all samples (around 0.5 when you were following my above advice of balanced samples) and divided by the probability of X in all samples.

If we are naive and assume that all events X are independent of each other, we can calculate the overall probability that a sample is winning by simply multiplying the probabilities P(X|winning) for every event X. This way we end up with this formula:

[latex display=”true”]P(Y | X_1 .. X_n) ~=~ s~P(Y) \prod_{i}{P(X_i | Y)}[/latex]

with a scaling factor s. For the formula to work, the features should be selected in a way that they are as independent as possible, which imposes an obstacle for using Naive Bayes in trading. For instance, the two events Close(1) < Close(0) and Open(1) < Open(0) are most likely not independent of each other. Numerical predictors can be converted to events by dividing the number into separate ranges.

The Naive Bayes algorithm is available in the ubiquitous e1071 R package.

11. Decision and regression trees

Those trees predict an outcome or a numeric value based on a series of yes/no decisions, in a structure like the branches of a tree. Any decision is either the presence of an event or not (in case of non-numerical features) or a comparison of a feature value with a fixed threshold. A typical tree function, generated by Zorro’s tree builder, looks like this:

int tree(double* sig)
{
  if(sig[1] <= 12.938) {
    if(sig[0] <= 0.953) return -70;
    else {
      if(sig[2] <= 43) return 25;
      else {
        if(sig[3] <= 0.962) return -67;
        else return 15;
      }
    }
  }
  else {
    if(sig[3] <= 0.732) return -71;
    else {
      if(sig[1] > 30.61) return 27;
      else {
          if(sig[2] > 46) return 80;
          else return -62;
      }
    }
  }
}

How is such a tree produced from a set of samples? There are several methods; Zorro uses the Shannon information entropy, which already had an appearance on this blog in the Scalping article. At first it checks one of the features, let’s say x₁. It places a hyperplane with the plane formula x₁ = t into the feature space. This hyperplane separates the samples with x₁ > t from the samples with x₁ < t. The dividing threshold t is selected so that the information gain – the difference of information entropy of the whole space, to the sum of information entropies of the two divided sub-spaces – is maximum. This is the case when the samples in the subspaces are more similar to each other than the samples in the whole space.

This process is then repeated with the next feature x₂ and two hyperplanes splitting the two subspaces. Each split is equivalent to a comparison of a feature with a threshold. By repeated splitting, we soon get a huge tree with thousands of threshold comparisons. Then the process is run backwards by pruning the tree and removing all decisions that do not lead to substantial information gain. Finally we end up with a relatively small tree as in the code above.

Decision trees have a wide range of applications. They can produce excellent predictions superior to those of neural networks or support vector machines. But they are not a one-fits-all solution, since their splitting planes are always parallel to the axes of the feature space. This somewhat limits their predictions. They can be used not only for classification, but also for regression, for instance by returning the percentage of samples contributing to a certain branch of the tree. Zorro’s tree is a regression tree. The best known classification tree algorithm is C5.0, available in the C50 package for R.

For improving the prediction even further or overcoming the parallel-axis-limitation, an ensemble of trees can be used, called a random forest. The prediction is then generated by averaging or voting the predictions from the single trees. Random forests are available in R packages randomForest, ranger and Rborist.

Conclusion

There are many different data mining and machine learning methods at your disposal. The critical question: what is better, a model-based or a machine learning strategy? There is no doubt that machine learning has a lot of advantages. You don’t need to care about market microstructure, economy, trader psychology, or similar soft stuff. You can concentrate on pure mathematics. Machine learning is a much more elegant, more attractive way to generate trade systems. It has all advantages on its side but one. Despite all the enthusiastic threads on trader forums, it tends to mysteriously fail in live trading.

Every second week a new paper about trading with machine learning methods is published (a few can be found below). Please take all those publications with a grain of salt. According to some papers, phantastic win rates in the range of 70%, 80%, or even 85% have been achieved. Although win rate is not the only relevant criterion – you can lose even with a high win rate – 85% accuracy in predicting trades is normally equivalent to a profit factor above 5. With such a system the involved scientists should be billionaires meanwhile. Unfortunately I never managed to reproduce those win rates with the described method, and didn’t even come close. So maybe a lot of selection bias went into the results. Or maybe I’m just too stupid.

Compared with model based strategies, I’ve seen not many successful machine learning systems so far. And from what one hears about the algorithmic methods by successful hedge funds, machine learning seems still rarely to be used. But maybe this will change in the future with the availability of more processing power and the upcoming of new algorithms for deep learning.

Papers

1. Classification using deep neural networks: Dixon.et.al.2016
2. Predicting price direction using ANN & SVM:  Kara.et.al.2011
3. Empirical comparison of learning algorithms: Caruana.et.al.2006
4. Mining stock market tendency using GA & SVM: Yu.Wang.Lai.2005

The next part of this series will deal with the practical development of a machine learning strategy.

⇒ Build Better Strategies – Part 5