Abstract
Holland's “hyperplane transform” of a “fitness landscape”, a random, real valued function of the verticies of a regular finite graph, is shown to be a special case of the Fourier transform of a function of a finite group. It follows that essentially all of the powerful Fourier theory, which assumes a simple form for commutative groups, can be used to characterize such landscapes. In particular, an analogue of the KarhunenLoève expansion can be used to prove that the Fourier coefficients of landscapes on commutative groups are uncorrelated and to infer their variance from the autocorrelation function of a random walk on the landscape. There is also a close relationship between the Fourier coefficients and Taylor coefficients, which provide information about the landscape's local properties. Special attention is paid to a particularly simple, but ubiquitous class of landscapes, so-called “AR(1) landscapes”.
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References
Dym H, McKean H (1972) Fourier series and integrals. Academic Press, San Diego, Calif
Feller W (1972) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New York
Fontana W, Griesmacher T, Schnabl W, Stadler P, Schuster P (1991) Statistics of landscapes based on free energies, replication and degradation rate constants of RNA secondary structures. Monatsh Chem (in press)
Fraser A, Swinney H (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A33:1134–1140
Goldberg D (1989a) Genetic algorithms and Walsh functions: Part I: a gentle introduction. Compl Syst 3:129–152
Goldberg D (1989b) Genetic algorithms and Walsh functions: Part II: deception and its analysis. Compl Syst 3:153–171
Holland J (1986) Escaping brittleness. In Michalski R, Carbonelli J, Mitchell T (eds) Machine learning, vol 2. Morgan Kauffmann, Los Altos, Calif
Holland J (1989) Lectures in the Science of Complexity. In Stein DL (ed) Proceedings of the 1988 Summer School on Complex Systems in Santa Fe, NM. Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley, Reading, Mass
Karlin S, Taylor H (1975) A first course in stochastic processes. Academic Press, New York
Papoulis A (1965) Probability, random variables, and stochastic processes. McGraw-Hill, New York
Priestley M (1981) Spectral analysis and time series. Academic Press, London
Sherrington D, Kirkpatrick S (1975) Phys Rev Lett 35:1792–1796
Sorkin G (1988) Combinatorial optimization, simulated annealing, and fractals. IBM Research Report RC13674 (No. 61253)
Stadler P, Schnabl W (1991) The landscape of the Travelling Salesman Problem. Phys Lett A (submitted for publication)
Voss R (1986) Characterization and measurement of random fractals. Phys Scr T13:257–260
Weinberger E (1990) Correlated and uncorrelated fitness landscapes and how o tell the difference. Biol Cybern 63:325–336
Weinberger E (1991a) The dynamics of schemas. (in preparation)
Weinberger E (1991b) Some properties of local optima in the N — k model, a tuneably rugged energy landscape. Phys Rev A (submitted for publication)
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Weinberger, E.D. Fourier and Taylor series on fitness landscapes. Biol. Cybern. 65, 321–330 (1991). https://doi.org/10.1007/BF00216965
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DOI: https://doi.org/10.1007/BF00216965