Hyperspace geography: Visualizing fitness landscapes beyond 4D

Human perception is finely tuned to extract structure about the 4D world of time and space as well as properties such as color and texture. Developing intuitions about spatial structure beyond 4D requires exploiting other perceptual and cognitive abilities. One of the most natural ways to explore complex spaces is for a user to actively navigate through them, using local explorations and global summaries to develop intuitions about structure, and then testing the developing ideas by further exploration. This article provides a brief overview of a technique for visualizing surfaces defined over moderate-dimensional binary spaces, by recursively unfolding them onto a 2D hypergraph. We briefly summarize the uses of a freely available Web-based visualization tool, Hyperspace Graph Paper (HSGP), for exploring fitness landscapes and search algorithms in evolutionary computation. HSGP provides a way for a user to actively explore a landscape, from simple tasks such as mapping the neighborhood structure of different points, to seeing global properties such as the size and distribution of basins of attraction or how different search algorithms interact with landscape structure. It has been most useful for exploring recursive and repetitive landscapes, and its strength is that it allows intuitions to be developed through active navigation by the user, and exploits the visual system's ability to detect pattern and texture. The technique is most effective when applied to continuous functions over Boolean variables using 4 to 16 dimensions

Rugged NK landscapes contain the highest peaks

NK models provide a family of tunably rugged fitness landscapes used in a wide range of evolutionary computation studies. It is well known that the average height of local optima regresses to the mean of the landscape with increasing ruggedness, K. This fact has been confirmed with both theoretical studies of landscape structure and empirical studies of evolutionary search. However, we show mathematically that the global optimum behaves quite differently: the expected value of the global optimum is highest in the maximally rugged case. Furthermore, we demonstrate that this expected value increases with K, despite the fact that the average fitness of the local optima decreases. We find the asymptotic value of the global optimum as N approaches infinity for both the smooth and maximally rugged cases. We interpret these results in the context of evolutionary search, and describe the relationship between the global optimum, local optima and found optima as search effort is geometrically increased

A comparison of neutral landscapes - NK, NKp and NKq

Recent research in molecular evolution has raised awareness of the importance of selective neutrality. Several different models of neutrality have been proposed based on Kauffman’s well-known NK landscape model. Two of these models, NKp and NKq, are investigated and found to display significantly different structural properties. The fitness distributions of these neutral landscapes reveal that their levels of correlation with non-neutral landscapes are significantly different, as are the distributions of neutral mutations. In this paper we describe a series of simulations of a hill climbing search algorithm on NK, NKp and NKq landscapes with varying levels of epistatic interaction. These simulations demonstrate differences in the way that epistatic interaction affects the 'searchability' of neutral landscapes. We conclude that the method used to implement neutrality has an impact on both the structure of the resulting landscapes and on the performance of evolutionary search algorithms on these landscapes. These model-dependent effects must be taken into consideration when modelling biological phenomena

An innovative learning model for computation in first year mathematics

MATLAB is a sophisticated software tool for numerical analysis and visualisation. The University of Queensland has adopted Matlab as its official teaching package across large first year mathematics courses. In the past, the package has met severe resistance from students who have not appreciated their computational experience. Several main factors contribute: Firstly, the software is numerical rather than symbolic, providing a departure from the thinking patterns presented in lectures and tutorials. Secondly, many students cannot see a direct connection between the laboratory exercises and core course material from lectures. Thirdly, the students find hurdles to entry as commands often return annoying error messages and don't execute, and programs are difficult to write and debug. Overall, the details of the mathematics are lost in trying to negotiate the software. After considerable effort in tuning, it appears that a sequence of innovations has captured student support and added considerable value to both the computational and traditional learning process