advanced-algorithmic-trading

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This motivates the concept of a loss function, which quantitatively compares the difference
between the true values with the predicted values.
Assume that we have created an estimate ˆf of the underlying relationship f. ˆf might be a
linear regression or a Random Forest model, for instance. ˆf will have been trained on a particular
data set, τ, which contains predictor-response pairs. If there are n such pairs then τ is given by:
τ = {(x 1 , y 1 ), ..., (x n , y n )} (20.2)
The x i represent the prediction factors, which could be prior lagged prices for a series or
some other factors, as mentioned above. The y i could be the predictions for our stock prices
in the following period. In this instance, n represents the number of days of data that we have
available.
The loss function is denoted by L(y, ˆf(x)). Its job is to compare the predictions made by ˆf
at particular values of x to their true values given by y. A common choice for L is the absolute
error:
L(y, ˆf(x)) = |y − ˆf(x)| (20.3)
Another popular choice is the squared error:
L(y, ˆf(x)) = (y − ˆf(x)) 2 (20.4)
Note that both choices of loss function are non-negative. Hence the "best" loss for a model
is zero, that is, there is no difference between the prediction and the true value.
Training Error versus Test Error
Now that we have a loss function we need some way of aggregating the various differences between
the true values and the predicted values. One way to do this is to define the Mean Squared
Error (MSE), which is simply the average, or expectation value, of the squared loss:
MSE := 1 n∑
(y i −
n
ˆf(x i )) 2 (20.5)
i=1
The definition simply states that the Mean Squared Error is the average of all of the squared
differences between the true values y i and the predicted values ˆf(x i ). A smaller MSE means
that the estimate is more accurate.
It is important to realise that this MSE value is computed using only the training data. That
is, it is computed using only the data that the model was fitted on. Hence, it is actually known
as the training MSE.
In practice this value is of little interest to us. What we are really concerned about is how
well the model can predict values on new unseen data.
For instance, we are not really interested in how well the model can predict past stock prices of