6 Analysis of Results

Figure 5.9 shows the structure and behavioural sampling for the Ant problem. There is a distinct trend of the highest number of unique structures sampled toward sizes of 45.

Figure 5.10: Parity results, cumulative sampling of unique structures and behaviours.

The bottom graph in Figure 5.9 shows the sampled behaviours of each size for the Ant problem. Note that the number of unique behaviours sampled of all sizes in each generation is greatly reduced after approximately 10 generations. Also, after two unusual peaks at behaviours of size 20 and 24, the number of unique sampled behaviours greatly decreases for behaviours of large size (which represent more ``fit'' solutions).

The sampling of structures for the Parity problem is shown in the top graph of Figure 5.10. This problem samples fewer unique structures but at larger sizes.

Figure 5.11: Regression results, cumulative sampling of unique structures and behaviours.

The number of unique behaviours sampled in the Parity problem are shown in the bottom graph of Figure 5.10. A behaviour has a maximum length of 32, which represents all $2^5$ correct classifications. However, there are ${32 \choose k}$ possible unique behaviours for a length of $k$. For the expected random strategy classification of size 16, nearly 2500 unique behaviours are sampled over the course of a run. Genetic programming spends a large amount of effort searching neutral behaviours equivalent to a random strategy, with slight peaks at fitness 18 and 20. Symmetry in the Parity bit-string instances probably rewards the solving of an additional instance with another symmetrical instance also solved correctly, explaining why fitness is concentrated on the random (16) strategy initially, followed by one instance additionally solved (17+1=18) and then another (19+1=20).

The Regression problem's sampling of structures is shown in the top graph of Figure 5.11. A distinct trend is seen toward sampling unique structures of sizes near 40. Fewer unique structures of larger sizes are sampled. The number of unique behaviours sampled in the Regression problem, depicted in the bottom graph of Figure 5.11, shows a strong attraction toward behaviours of size 12. All generations during the run sample unique behaviours of this size. As behaviour does not directly reflect the fitness, these behaviours may or may not have neutral fitness. However, the Binomial-3 fitness function contains 50 equidistant $x$ values that generate the target $y$ values for testing an individual. The angle gradient between successive $y$ values is nearly always greater than 1. Thus, for our behaviour measure, if a behaviour is to represent the function ideally, it will need close to 50 angle changes between points.

The 95% confidence bars for each of the average distributions from Figures 5.9 to 5.11 are shown in Figure 5.12. From left to right, the Ant, Parity and Regression structures are shown on the top row, and behaviours on the bottom row. The sampling between the 30 runs is fairly uniform with the greatest variations occurring near the peaks in the Regression problem.