19,595 research outputs found
Schemata as Building Blocks: Does Size Matter?
We analyze the schema theorem and the building block hypothesis using a
recently derived, exact schemata evolution equation. We derive a new schema
theorem based on the concept of effective fitness showing that schemata of
higher than average effective fitness receive an exponentially increasing
number of trials over time. The building block hypothesis is a natural
consequence in that the equation shows how fit schemata are constructed from
fit sub-schemata. However, we show that generically there is no preference for
short, low-order schemata. In the case where schema reconstruction is favoured
over schema destruction large schemata tend to be favoured. As a corollary of
the evolution equation we prove Geiringer's theorem. We give supporting
numerical evidence for our claims in both non-epsitatic and epistatic
landscapes.Comment: 17 pages, 10 postscript figure
An Overview of Schema Theory
The purpose of this paper is to give an introduction to the field of Schema
Theory written by a mathematician and for mathematicians. In particular, we
endeavor to to highlight areas of the field which might be of interest to a
mathematician, to point out some related open problems, and to suggest some
large-scale projects. Schema theory seeks to give a theoretical justification
for the efficacy of the field of genetic algorithms, so readers who have
studied genetic algorithms stand to gain the most from this paper. However,
nothing beyond basic probability theory is assumed of the reader, and for this
reason we write in a fairly informal style.
Because the mathematics behind the theorems in schema theory is relatively
elementary, we focus more on the motivation and philosophy. Many of these
results have been proven elsewhere, so this paper is designed to serve a
primarily expository role. We attempt to cast known results in a new light,
which makes the suggested future directions natural. This involves devoting a
substantial amount of time to the history of the field.
We hope that this exposition will entice some mathematicians to do research
in this area, that it will serve as a road map for researchers new to the
field, and that it will help explain how schema theory developed. Furthermore,
we hope that the results collected in this document will serve as a useful
reference. Finally, as far as the author knows, the questions raised in the
final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've
decided to put it on the arXiv as a more permanent home. The paper is
primarily expository, so I don't really know where to submit it, but perhaps
one day I will find an appropriate journa
Efficient regularized isotonic regression with application to gene--gene interaction search
Isotonic regression is a nonparametric approach for fitting monotonic models
to data that has been widely studied from both theoretical and practical
perspectives. However, this approach encounters computational and statistical
overfitting issues in higher dimensions. To address both concerns, we present
an algorithm, which we term Isotonic Recursive Partitioning (IRP), for isotonic
regression based on recursively partitioning the covariate space through
solution of progressively smaller "best cut" subproblems. This creates a
regularized sequence of isotonic models of increasing model complexity that
converges to the global isotonic regression solution. The models along the
sequence are often more accurate than the unregularized isotonic regression
model because of the complexity control they offer. We quantify this complexity
control through estimation of degrees of freedom along the path. Success of the
regularized models in prediction and IRPs favorable computational properties
are demonstrated through a series of simulated and real data experiments. We
discuss application of IRP to the problem of searching for gene--gene
interactions and epistasis, and demonstrate it on data from genome-wide
association studies of three common diseases.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS504 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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