## 6 Experimental Investigation: Hyperplane Estimation Techniques
### Search Environment Properties  
As previously discussed, a hyperplane can be thought of as a line in two dimensions, a sheet, or plane, in three dimensions, and so on for higher dimensions. The crucial idea is that a hyperplane can be described by a linear equation, determined by, for example, in three dimensions:  
\[ax+by+cz+d=0 \tag{7}\]  
where the solution (_a,b,c,d_) determines the orientation of the plane in the three-dimensional space. This linear description of a plane holds as the dimensions increase. For example, in five dimensions, the hyperplane can be described:  
\[av+bw+cx+dy+ez+f=0 \tag{8}\]  
To uniquely describe a hyperplane, one requires a minimal set of points of size equal to the number of dimensions of the space containing the hyperplane. For three dimensions, three points make up the minimal set; for five dimensions, five points.  
It was necessary to generate synthetic data with which to test the algorithms. Hence, some data set was needed, residing within which a hyperplane whose points were perturbed orthogonally by Gaussian noise (the inliers) could be found. All points not associated with the hyperplane are thus termed outliers.  
Generation of this data is relatively straight forward once the singular value decomposition technique previously described is known. Random minimal sets can be determined, the system solved for (_a,b,c,d,_...) and, consequently, a set of inlier points generated with each axis co-ordinate perturbed by Gaussian noise at an appropriate level.  
The ratio of inliers to outliers determines the overall noisiness of the environment (distinct from the noise perturbations on the inliers themselves).  
Thus, two main variables can be evaluated: the number of dimensions and the overall noisiness of the environment. The minimal set size is equivalent to the number of dimensions. Increasing the dimensions increases the search space size, without the need for additional data points. There is a combinatorial explosion in the search space size as the dimensions increase.  
For all the trials, dimensions and noise detailed below, 1000 data points were used as the search space. A randomly positioned hyperplane was placed within this space for each trial. The level of environmental noise determines the proportion of inliers to outliers; a level of 50% implies 500 data points making up the inliers (the generated hyperplane) and the remaining 500 evenly distributed outliers. The inliers were disturbed orthogonally from the true hyperplane by Gaussian noise, as described previously. For each measure, 10 trials were run and the average measurement taken (of both convergence time and accuracy). SDS convergence occurred at the 100% agent activity level.  
The convergence times of the trials have been used as a guide to computational expense. Another method is to compare the number of singular value decompositions and residual calculations needed for convergence; however, upon analysis, the correlation between the time to convergence in milliseconds and the number of SVDs (the most computationally-intensive process) performed is very close, and hence, the actual time taken is used as a more illustrative measure of overall performance.  
To measure the accuracy of the algorithms, the hypothesis converged as the optimal solution was tested against all points in the data space; the percentage classed as inliers are seen to be the score of the hypothesis. The accuracy of the hypothesis is a measure of the percentage of inliers in the data that are described by the solution.  
The size of the population of agents in standard SDS has a bearing on the number of iterations required for SDS to converge; similarly for data-driven variants. Too large a population, for a given search, will require SDS to perform far too many SVDs, thus reducing its speed of convergence; too small a population may lead to a lack of accuracy, with the benefit of speeding up the algorithm.