# Fitness Landscape Analysis of Product Unit Neural Networks
## 3 Fitness Landscape Analysis
Larger values of \(G_{dev}\) indicate that the gradients of certain walks deviate a lot from \(G_{avg}\). This is an indication of \"cliffs\" or sudden \"peaks\" or \"valleys\" present in the landscape [23]. The variability of the fitness values or ruggedness of the landscape is estimated with the first entropic measure (\(FEM\)) [22]. [PERSON] and [PERSON] [22] proposed two measures based on the \(FEM\): namely, micro ruggedness (\(FEM_{0.01}\)), where the step sizes of the PRWs are 1% of the search space, and macro ruggedness (\(FEM_{0.1}\)), where the step sizes of the PRWs are 10% of the search space. The \(FEM\) measures provide a value in \([0,1]\), where 0 indicates a flat landscape, and larger values indicate a more rugged landscape. For a detailed description of the \(FEM\) measures and pseudocode, see [23]. The fitness-distance correlation (\(FDC\)) was introduced by [PERSON] [24] as a measure of global problem hardness. The \(FDC\) measure is based on the premise that for a landscape to be easily searched, error should decrease as distance to the optimum decreases in the case of minimization problems. The \(FDC\) measures the covariance between the fitness of a solution and its distance to the nearest optimum. Fitness should therefore correlate well with the distance to the optimum if the optimum is easy to locate. However, the \(FDC\) requires knowledge of the global optima, which is often unknown for optimization problems. Therefore, this measure was extended by [PERSON] [20] by making use of the fittest points in the sample instead of the global optima (\(FDC_{s}\)). Instead of estimating how well the landscape guides the search towards the optimum, the \(FDC_{s}\) quantifies how well the problem guides the search towards areas of better fitness. Therefore, \(FDC_{s}\) changes the focus from a measure of problem hardness to searchability. The \(FDC_{s}\) measure gives a value in \([-1,1]\), where 1 indicates a highly searchable landscape, \(-1\) indicates a deceptive landscape, and 0 indicates a lack of information in the landscape to guide the search. The dispersion metric (\(DM\)) [25] is calculated by comparing the overall dispersion of uniformly sampled points to a subset of the fittest points. The \(DM\) describes the underlying structure of the landscape by estimating the presence of funnels. A funnel in a landscape is a global basin shape that consists of clustered local minima [20]. A single-funnel landscape has an underlying unimodal \"basin\"-like structure, whereas a multi-funnel landscape has an underlying multimodal-modal structure. Multi-funnel landscapes can present problems for optimization algorithms because they may become trapped in sub-optimal funnels [20]. A positive value for \(DM\) indicates the presence of multiple funnels. Neutrality of the landscape can be characterized by the \(M_{1}\) and \(M_{2}\) measures [26]. \(M_{1}\) calculates the proportion of neutral structures in a PRW in order to estimate the overall neutrality of the landscape. \(M_{2}\) estimates the relative size of the largest neutral region. The \(M_{1}\) and \(M_{2}\) measures both produce values in \([0,1]\), where 1 indicates a completely neutral landscape, and 0 indicates that the landscape has no neutral regions.