Part 6: Simulation Results and wrap-up

A fitness landscape is a 3-D (or sometimes 4-D) plot which shows how the fitness value varies with respect to other variables that it depends on. To obtain the fitness landscape for our problem I altered the population size from 2 to 20 for mutation probabilities within the range 0 to 1. I then ran the GA for 25 generations and recorded the average fitness of the entire population so as to ascertain optimum mutation rate and population size to use. Alternatively we could have recorded the fitness of the fittest organism within the population, but this would not have given us an indication of the overall quality of solutions evolved by the algorithm. Hence the reason for taking the overall population fitness into account. The plot below shows how the average fitness changes for different population size and mutation rate.

fLandSurf.png

We can also view the 3-D plot in the form of a contour plot:

fLandContour.png

We are interested in the dark regions because the value of the fitness function in these locations is closest to zero. It is important to point out that ideally the fittest organism in our problem should have a fitness of 0 as this represents that there in no difference between the predicted value and the actual value. We see that a mutation rate of alteast 0.5 and population size greater than 9 will yield fittest solutions within the first 25 generations.

mRate2.pngUsing the minimum mutation rate and population size (0.5 and 10 respectively) we run the GA for mating rates varying from 0.1 to 1 for 25 generations. This time we are interested in seeing how the fitness value depends on the mating rate The graph shows that for mating rates upto about 0.7, the algorithm produces fit organisms. For rates above 0.8 we see poor solutions as the overall fitness rapidly increases. The explanation for this is that organisms that high mating rates almost always results in child genes being dominated by that of the parents which gives little room for population diversity. Hence the population remains generally unfit for many generations into the future.

Hence for our experiment we use the following parameters:-

Population size: 10
Mutation rate: 0.5
Mating rate: 0.5

We let the GA run for 500 generations and get the fittest genome which we use in our prediction function. Recall that our prediction function is:

eq1.PNG

The paremeters of the fittest genome evolved by the GA after 500 generations are:

\alpha=0.1,\hspace{1}\beta=-0.2,\hspace{1}\gamma=0.3,\hspace{1}\delta=-0.2,\hspace{1}\theta=0.1,\hspace{1}\tau=-0.1,\hspace{1}\mu=6.6

a=25,\hspace{1}b=22,\hspace{1}c=7,\hspace{1}d=11,\hspace{1}e=55,\hspace{1}f=47,\hspace{1}

We can now apply our evolved prediction function to the 3 data sets of the FTSE 100 time series. However, to give more meaning to our results we can compare with the Random Walk model of an asset price, which takes the form:

P_{t+1}=P_{t}+\mu+Z_{t}

where \small Z_{t} is a normally distributed random variable and \small\mu is drift.

Comparative results are summarised below:

results.png

Data set 2 is the true out-of sample range with we intended to forecast via our prediction function. Clearly it under-performs relative to the random walk model suggesting that our prediction function provides a poor approximation of the FTSE 100 index. Note that it is not the GA which is at fault here, but the prediction function.

Key Points:

  • We have shown that the GA can evolve optimum parameters for a pre-defined prediction function
  • The performance of the GA is totally independent to the forecasting ability of the prediction function. The GA will almost always find the best solutions for the prediction function, which might be a poor approximator of the asset price as in our case.
  • GAs are very powerful tools for global search and optimisation. However, there is a trade-off between search coverage and simulation time. Analysing fitness landscapes based on a small number of evolved generations can help estimate suitable population size as well as mutation rate and mating rate hence reducing simulation time for larger generations.

Possible improvements:

  • Let the algorithm search for its own prediction function. This leads to Genetic Programming which helps generate very sophisticated prediction functions.
  • Use a range of other fundamental and technical indicators as inputs to the prediction function.
  • Apply filtering so as to reduce noise over-fitting.

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