advanced-algorithmic-trading

227
ŷ = ˆf(x) = argmax z∈R p(y = z | x) (16.3)
Common regression techniques include Linear Regression, Support Vector Regression and
Random Forests.
Regression will be utilised in this book to estimate future asset prices in an intraday trading
strategy, give previously known information.
16.3.1 Financial Example
To make this concrete consider an example of modelling the next days price of the London-based
equity Royal Dutch Shell (LSE:RDSB) via the historical prices of crude oil and natural gas.
Each day, which is indexed by i, has an associated pair (x i , y i ) representing the feature vector
and response for that day. The feature vector x i represents the historical prices of crude oil (c ij )
at current day i and historical lag j; similarly for natural gas (g ij ), over a period of N days, itself
indexed by j ∈ 1, . . . , N. y i represents the price of RDSB tomorrow, that is at i + 1.
In this example the goal is to estimate the function f that relates tomorrows price for RDSB
with the historical prices of crude oil and natural gas.
16.4 Training
Now that the probabilistic formulation has been defined it remains to discuss how it is possible
to "supervise" or "train" the model with a specific set of data.
In order to train the model it is necessary to define a loss function between the true value
of the response y and its estimate from the model ŷ, given by L(y, ŷ).
In the classification setting common loss models include the 0-1 loss and the cross-entropy.
In the regression setting a common loss model is given by the Mean Squared Error (MSE):
MSE = 1 N
N∑
|y i − ŷ i | 2 (16.4)
i=1
This states the the total error of a model given a particular set of data is the average of the
sum of the squared differences between all of the training values y i and their associated estimates
ŷ i .
This loss function essentially penalises estimate values far from their true values quite heavily,
as the differences are squared. Notice also that the important aspect is the squared distance
between values, not whether they are positive or negative deviations. MSE will be discussed in
more depth in the next chapter on Linear Regression.
Equipped with this loss function it is now possible to make estimates of ˆf, and thus ŷ,
by carrying out "fitting" algorithms on particular machine learning techniques that attempt to
minimise the value of these loss functions, by adjusting the parameters θ of the model.
A minimal value of a loss function states that the errors between the true values and the
estimated values are not too severe. This leads to the hope that the model will perform similarly
when exposed to data that it has not been "trained" on.