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Pikaia Gracilens, a little worm-like beast that crawled in the mud of
a long gone seafloor of the Cambrian era, 530 million years
ago. While not particularly impressive in the tooth and claw department,
Pikaia is believed to be the founder of the phylum Chordata, whose
subsequent evolution had consequences still very much felt today by the
rest of the ecosystem. Image digitized from the excellent book
The Rise of Fishes, by John A. Long (1995, The Johns Hopkins University
Press).
Welcome to the PIKAIA Homepage.
PIKAIA (pronounced ``pee-kah-yah'') is a general purpose
function optimization FORTRAN-77 subroutine based on a genetic algorithm.
PIKAIA is a public domain software available electronically
from the anonymous ftp archive of the High Altitude Observatory.
The subroutine is particularly useful (and robust) in treating
multimodal optimization problems.
The development of genetic algorithm-based inversion methods
is but one aspect of the research in helioseismology
carried out in the
Solar
Interior Section
of the
High Altitude Observatory,
a scientific division of the
National Center for Atmospheric Research
in Boulder, Colorado.
PIKAIA 1.0 was written in 1995 by Paul Charbonneau and Barry Knapp,
both then at HAO/NCAR. Version 1.2 was released in April 2002.
- Genetic Algorithms
- The PIKAIA subroutine
- An example of PIKAIA usage
- Obtaining a copy of PIKAIA
- Registering as a PIKAIA User
- List of PIKAIA Users
- Bug Reports
- PIKAIA in IDL and f90, and parallel PIKAIA
- References and further readings
- Oslo Tutorial (a.k.a. NCAR TN-450+IA)
1. Genetic Algorithms
Genetic algorithms (hereafter GAs) are a class of search techniques
inspired from the biological process of evolution by means of natural
selection. They can be used to construct numerical optimization
techniques that perform robustly on problem characterized by
ill-behaved search spaces.
Consider the following generic modeling task: a model that depends
on a set of adjustable parameters is used to fit a
given dataset; the task consists in finding the single parameter set
that minimizes the difference between the model's predictions and the
data. A top-level view of a canonical genetic algorithm for this task
could be as follows: start by generating a set (``population'') of trial
solutions, usually by choosing random values for all model parameters;
then:
- Evaluate the goodness of fit (``fitness'') of each member
of the current population (through a chi square measure with the data,
for example).
- Select pairs of solutions (``parents'') from the current
population, with the probability of a given solution being selected
made proportional to that solution's fitness.
- Breed the two solutions selected in (2) and produce two
new solutions (``offspring'').
- Repeat steps (2)-(3) until the number of offspring
produced equals the number of individuals in the current population.
- Use the new population of offspring to replace the old population.
- Repeat steps (1) through (5) until some termination criterion
is satisfied (e.g., the best solution of the current population reaches
a goodness of fit exceeding some preset value).
Superficially, this may look like some peculiar variation on the Monte
Carlo theme. There are two crucial differences: First, the probability
of a given solution being selected to participate in a breeding event
is made proportional to that solution's fitness (step 2); better trial solutions
breed more often, the computational equivalent of
natural selection.
Second, the production of new trial solutions from existing
ones occurs through
breeding.
This involves
encoding
the parameters defining each solution as a string-like
structure (``chromosome''), and performing genetically inspired operations of
crossover
and
mutation
to the pair of chromosomes encoding the two parents,
the end result of these operations being
two new chromosomes defining the two offspring. Applying the
reverse process of decoding those strings into solution parameters
completes the breeding process and yields two new offspring solutions
that incorporate information from both parents.
Technical details:
A genetic-algorithm based approach to a given optimization task,
as defined above, amounts to a form of forward modeling.
Generally speaking, adopting
a forward modeling approach has both advantages and drawbacks;
Advantages:
- No derivatives of the goodness of fit
function with respect to model parameters need be computed;
it matters little whether the relationship between the model
and its parameters is linear or nonlinear.
- Nothing in the procedure outlined above depends critically on
using a least-squares statistical estimator; any other robust
estimator can be substituted, with little or no changes to the overall
procedure.
Drawbacks:
- In most real applications, the model will need to be evaluated
(i.e., given a parameter set, compute a synthetic
dataset and its associated goodness of fit) a great many times;
if this evaluation is computationally expensive, the forward modeling
approach can become impractical.
GA-based optimization retains the advantageous features of forward
modeling, while reducing the number of required function
evaluations to a level that is often much more computationally manageable.
2. The PIKAIA subroutine
PIKAIA is a fully self-contained, general
purpose optimization subroutine. The routine maximizes
a user-supplied FORTRAN function, the name of which is
passed in as an argument. Generally, we have tried to
emulate the general feel and ease of use of the
Numerical Recipes optimization subroutine (for example
amobea). A call to PIKAIA looks like
call pikaia(ff,n,ctrl,xb,fb,status)
where ff is the name of the FORTRAN function
to be maximized, n the dimension of the search
space, and ctrl is a control vector whose elements
determine the behavior of the algorithm (robust default values
are built in). The output vector xb contains the
optimal solution parameters, namely the best solution of the
final population at the end of the evolution.
The scalar fb is the fitness of the
solution (i.e., the value returned by ff with
xb as input). The output variable status
codes successful termination, or error conditions associated
with illegal input parameter values.
The (real) function ff must be declared as
function ff(n,x)
dimension x(n)
where the floating-point array x defines one
point in the search space, and the function returns a
(positive-definite) measure
of fitness (or goodness of fit, or whatever) associated with
that point. And that is all there is to it.
Technical details:
PIKAIA incorporates only the two basic genetic operators: uniform
one-point crossover, and uniform one-point mutation.
Unlike many GA packages available commercially or in the public domain,
the encoding within PIKAIA is based on a decimal
alphabet made of the 10 simple integers (0 through 9);
this is because binary
operations are usually carried out through platform-dependent
functions in FORTRAN. Three reproduction
plans are available: Full generational replacement, Steady-State-Delete-Random,
and Steady-State-Delete-Worst. Elitism is available and is a default option.
The mutation rate can be dynamically controlled by monitoring the difference
in fitness between the current best and median in the population
(also a default option).
Selection is rank-based and stochastic, making use of the Roulette Wheel
Algorithm.
PIKAIA is supplied with a ranking subroutine based on the Quicksort
algorithm, and a random number generator based on the minimal standard
Lehmer multiplicative linear congruential generator of Park and Miller
(1988, Communications of the ACM, 31, 1192).
3. An example of PIKAIA usage
Consider an optimization problem that consists of maximizing
a function of two variables f(x,y), with x
and y bounded between 0 and 1. The function defines a
2-D landscape, in which one is seeking the highest elevation
point. If the landscape is smooth and simple this problem is
readily treated with conventional hill-climbing methods.
However a landscape such as the following
would be a much harder task:
This is a surface plot of the function f(x,y),
and the inset in the upper right is a color-coded version of the
same function. The global maximum (indicated by the arrow and
located at (x,y)=(0.5,0.5), where f=1) is surrounded
by concentric rings
of secondary maxima, where a simple hill-climbing method
would most likely get stuck.
This problem is easily solved with PIKAIA. An individual is
an (x,y) pair, and fitness can be directly
defined as the altitude in the 2-D landscape, i.e., the value
returned by f(x,y). Examination of the corresponding
fitness function
and
driver code
shows how simple the use of PIKAIA is for such a problem.
The following animation illustrates the evolution of the
population's distribution in parameter space. Each individual
is shown as a solid green dot, and the best of each generation
as a larger, yellow dot.
Observe how the ``best solution remains
stuck, for a little while, on the innermost ring of secondary
extrema, but eventually ``finds'' the thin, central global maximum.
The small plot in the upper right shows
the variation with generation count
of the error (defined as log(1-f) for the best (yellow)
and median (purple) individuals in the population. The sudden
drops in the error reflect the appearance of a favorable
mutation, and its subsequent
spread in the population.
This solution was obtained by running PIKAIA for 50 generations
under its default settings.
Click here to view animation [Size: 574 KB]
4. Obtaining a copy of PIKAIA
PIKAIA is public domain software and is
available electronically on the High Altitude Observatory
anonymous ftp archive.
To get there click on
the preceding anchor; save all files as Plain text, except the User's Guide,
which should be saved as Postscript.
Alternately, type on your own system:
ftp ftp.hao.ucar.edu 122
Use anonymous as a username and your
e-mail address as a password. Once logged in type
cd archive/pikaia
Depending on how your computer system is firewalled, you might experience
problems with ftp; if so point your browser to
http://download.hao.ucar.edu/archive/pikaia/
One way or the other, you are now in the main PIKAIA directory.
The files in this directory are:
README.txt
This file contains a description of the directory's content and
last minute information. Please do read it.
pikaia.f
This file contains the PIKAIA subroutine itself, together
with an appropriate random number generator, ranking
routine, and driver and fitness function for an
installation check problem. This file is completely
self-contained, and can be compiled/linked immediately.
userguide.ps
This is a Postscript file containing the User's Guide to
PIKAIA 1.0, otherwise known as NCAR Technical Note
418+IA [Size: about 2MB].
userguide_errata.txt
Some minor errors and typos discovered in the User's
Guide. These have been corrected in the current
userguide.ps, but may be present in earlier
versions of the Guide, depending on the date at which
the typos were reported. Note that the typos and errors
are also present in the hard copy of the Guide published
by NCAR in December 1995.
examples
Subdirectory containing drivers and fitness functions
for the example problems discussed in chapter 5 of the
User's Guide.
examples.tar
UNIX tar file of the examples subdirectory. A convenient
way to grab all the examples files in one shot if you
are working on, or have access to, a UNIX system.
In case of difficulties installing or running the code, send e-mail to
pikaia@hao.ucar.edu.
5. Registering as a PIKAIA user
We invite prospective users of PIKAIA to send a brief
e-mail message to
pikaia@hao.ucar.edu
(if possible, please include
a few words on what problem is to be tackled with
the subroutine; we're curious). This will allow us to maintain an e-mail list,
to be used to send bug reports and/or news/upgrade announcements.
Note that by registering as a PIKAIA user, you will
not be added to some open e-mailing list, with the
usual consequence of being submerged under mostly unwanted e-mail
messages. Again, the list is used exclusively to send
bug reports and other news/upgrade announcements.
6. List of registered PIKAIA Users
Current and Past Users:
Frank Crary CU/LASP
Eugene Lavely Blackhawk Geosc. Geoacoustic waveform inversion
Guillermo Torres Harvard/CfA
Alex Razoumov UBC/Vancouver Nonlinear function minimization
Mark Fardal CU/CASA Game theory
Mihaly Horanyi CU/LASP Ozone inverse problem
Isodoros Doxas CU/APAS Magnetotail modeling
Kelsey Johnson CU/APAS Solar magnetogram analysis
Dieter Hartmann Clemson U.
Hal Levison SWRI/Boulder
Deborah Haber CU/Boulder Sunspot helioseismology
Adrian Webster Roy.Obs. Edinburgh Data Modeling
Martin Sperl U. Vienna Period fitting
Alberto Cappi U. Bologna
Ian Howarth Univ. Coll. London
Richard Boivin CERCA/Montreal Engineering design
Jacques Richer CERCA/Montreal Nonlinear semi-empirical fits
Gilles Fontaine U. de Montreal Statistical Thermodynamics
Claude Carignan U. de Montreal Nonlinear least squares
Robert Lamontagne U. de Montreal Nonlinear function minimization
Paul Seagraves HAO/NCAR Stokes profiles fitting
Tim Brown HAO/NCAR Orbital Parameter fits
Karel Schrijver Lockheed/Palo Alto Emission measure analysis
Ignaz Wanders Ohio State U. Reverberation Mapping
Burkhart Fuchs U. Heidelberg Dynamic Modelling
Russeil Delphine CNRS/Paris Data Modeling
Luciano Mantegazza INFN/Pavia
Tuck Stebbins CU/Boulder
Andreas Bobinger U. Muenchen Inverse Modeling
Paul Cally Monash U.
Th. Appourchaux ESA Inverse Modeling
Sarah Gibson NASA/Goddard Coronal Structure Inversion
Stuart Jeffries NOAO
Ronnie Hoogerwerf Leiden Observatory
Reyco Henning U. of Denver Helioseismic inversion
Sami Solanki ETH/Zuerich Stokes profiles fitting
Barry Lapthorn QMW College, U.K. Helioseismic inversion
Daniel Carpintero U. La Plata, Arg.
Misha Haywood Obs. Paris
Steve Arendt CORA/Boulder
Davis Valls-Gabaud Obs. Strasbourg fitting Color-Mag diagrams
Andreas Lagg Johns Hopkins Spectral Data analysis
Bill Chaplin ESTEC/Noordwwijk Helioseismic inversions
Travis Metcalfe U. Texas/Austin White dwarf seismology
Christian Theis U. Kiel, Ger. Modeling galaxy encounters
Hardi Peter Kiepenheuer Inst. Spectral line analysis
Ladislav Hejna Czech Acad. Sc. Orbital fitting
Jose Adolfo Brazil U.? Thermo-resistive sensorsa
R. Lantosoa Madagascar Obs. Gravity/magnetism inversions
Olof Friberg Chalmers U., Sweden vehicule design (acoustics)
Thierry Nieus Belgium biophysical modeling
Antonio Emolo U. Naples seismic analysisa
Luca Teriaca Armagh Obs. Spectral line analysis
Katherine Manson Australian Nat. U. magnetic spectra analysis
S.L. Hidalgo Rguez IAC/Spain modeling interactive galaxies
Steven Hale U. Birmingham, UK helioseismic data processing
Jackie Schoendorf Data modeling
Lidong Xia MPI Aeronomie SUMER spectral analysis
Simon Casassus U. Chile
Gordon Chin NASA Goddard
Andrea Fontana CERN spectroscopy of antihydrogen
Bastian Liebrecht RWTH-Aachen Aircraft/spacecraft design
Waleed Hamdy NRIAG/Egypt
Stephane Courteau UBC/Canada galactic light curve decomp.
Charlie Meyer STMicroelectronics engineering design
Kevin Flood SLAC/Stanford particle data analysis
Ian Robinson U. Birmingham/UK solar wind energetic events
Eric Depagne DASGAL/Obs. Meudon
Scott McIntosh ESA Emission measure modeling
Alan Miller CSIRO Math.& Inf.
Frank Timmes U. Chicago Stellar Opacities
Cesar Aaron Moya Kyoto U. Geoseismology
Mohamed Abdelwahed NRIAG/Egypt Earthquakes
Mark Alston U. Strathclyde Spectral Modelling
I.R. De Souza Unicamp/Brazil Chemical engineering design
Bev Smith East Tennessee S.U.
Lee Rottler UC/Santa Cruz Period fitting & spectral analysis
Roman Petryk UBC/Physics & Astr.
Daniel Kubas Univ. Potsdam Gravitational microlensing
Lapo Boschi U. Federico II Seismic Imaging
Antonio Piersanti INGV/Rome Mantle Rheology
Volker Rath U. Aachen/RWTH Geothermal Inverse modeling
Ivan Zevallos U. Sao Paulo Geophysical Inversion
Luciano Lamberti Politecnico di Bari Structural optimization
Kiran Solanki Mississippi State U Constrained optimization
Paola Rogata GMV S.A./Madrid Interplanetary trajectories
Omur Cakirli Ege U. Observastory Asteroseismology
Jos de Bruijne ESA/ESTEC Engineering design
K. Gozdziewski Torun CfA Orbital Element Fitting
Liz Humphreys Harvard CfA
S.A. Klein U. Wisconsin Equation-solving
Michiel Min Astro. Inst. IR spectral analysis
PIKAIA Publications in refereed Journals:
- Kennelly, E.J., and 12 co-authors, 1996, The oscillation modes
of theta^2 Tauri,
Astronomy & Astrophysics 313, 571-580.
- Kaastra, J.S., et al. 1996,
Emission measure analysis methods: the corona of AR Lacertae revisited,
Astronomy & Astrophysics, 314, 547-557.
- Noyes, R.W., et al. 1997,
A planet orbiting the star rho Coronae Borealis,
The Astrophysical Journal, 483, L111-L114
- Charbonneau, P., Tomczyk, S., Schou, J., and Thompson, M.J., 1998
The Rotation of the Solar Core inferred by genetic forward modeling,
The Astrophysical Journal, 496, 1015-1030.
- Gibson, S., and Charbonneau, P. 1998, Empirical Modeling of the
Solar Corona using Genetic Algorithms, Journal of Geophysical
Research, 103(A7), 14511-14521.
- McIntosh, S.W., Diver, D.A., Judge, P.G., Charbonneau, P.,
Ireland, J., and Brown, J.C. 1998, Spectral Decomposition
by Genetic Forward Modeling
Astronomy & Astrophysics (Supplements), 132, 145-153.
- Theis, C. 1998, Modeling Encounters of Galaxies: the case of NGC4449,
Review of modern astronomy, 12, in press.
- Peter, H. 1999,
Analysis of transition-region emission-line profiles from full-disk
scans of the sun using the SUMER instrument on SOHO,
The Astrophysical Journal, 516, 490-504.
- Charbonneau, P., Christensen-Dalsgaard, J., Henning, R.,
Larsen, R.M., Schou, J., Thompson, M.J., and Tomczyk, S. 1999
Helioseismic constraints on the structure of the solar tachocline,
The Astrophysical Journal, 527, 445-460.
- Peter, H. 1999, The chromosphere in coronal holes and the quiet-sun
network: an HeI (584 A) full-disk scan by SUMER/SOHO,
The Astrophysical Journal, 522, L77-L80.
- Sevenster, M., Saha, P., Valls-Gabaud, D., and Fux, R. 1999,
New constraints on a triaxial model of the Galaxy,
Monthly Notices of the Royal Astronomical Society, 307, 584-594.
- Peter, H., and Judge, P.G. 1999,
On the Doppler shifts of solar ultraviolet emission lines,
The Astrophysical Journal, 522, 1148-1166
- Teriaca, L., Banerjee, D., and Doyle, J.G. 1999,
SUMER observations of Doppler shift in the quite sun and in active region,
Astronomy and Astrophysics, 349, 636-648
- Doyle, J.G., Teriaca, L., and Banerjee, D. 1999,
Coronal hole diagnostics out to 8R,
Astronomy and Astrophysics, 349, 956-960.
- McIntosh, S.W., Charbonneau, P., and Brown, J.C. 2000,
The Astrophysical Journal, 529, 1115-1130.
- Billieres, M., Fontaine, G., Brassard, P., Charpinet, S., Liebert, J.,
and Saffer, R.A. 2000,
Detection of p-mode pulsations and possible ellipsoidal luminosity variations
in the hot subdwarf B star KPD 1930+2752,
The Astrophysical Journal, 530, 441-453.
- Doyle, J.G., Teriaca, L., and Banerjee, D. 2000,
Solar transition region line broadening: limb to limb measurements,
Astronomy and Astrophysics, 356, 335-338.
- Bobinger, A. 2000,
Genetic eclipse mapping and the advantage of black sheep,
Astronomy and Astrophysics, 357, 1170-1180.
- McIntosh, S.W. 2000,
On the inference of differential emission measures using diagnostic line ratios,
The Astrophysical Journal, 533, 1043-1052.
- Skartlien, R., and Rast, M.P. 2000,
p-mode intensity-velocity phase differences and convective sources,
The Astrophysical Journal, 535, 464-472.
- Peter, H. 2000,
Multi-component structure of solar and stellar transition regions,
Astronomy and Astrophysics, 360, 761-776.
- Harries, T.J., and Howarth, I.D. 2000,
Spectropolarimetric orbits of symbiotic stars: five S-type systems,
Astronomy and Astrophysics, 361, 139-152.
- Banerjee, D., Teriaca, L., Doyle, J.G., and Lemaire, P. 2000,
Polar plumes and inter-plume regions as observed by SUMER on SOHO,
Solar Physics, 194, 43-58.
- Metcalfe, T.S., Nather, R.E., and Winget, D.E. 2000,
Genetic-algorithm-based asteroseismological analysis of the
DBV white dwarf GD358,
The Astrophysical Journal, 545, 974-981
- Metcalfe, T.S., Winget, D.E., and Charbonneau, P. 2001,
Preliminary constraints on 12C(alpha,gamma)16O from white dwarf seismology,
The Astrophysical Journal, 557, 1021-1027
- Theis, Ch., and Kohle, S. 2001,
Multi-method-modeling for interacting galaxies.
I. A unique scenario for NGC 4449?
Astronomy and Astrophysics, 370, 365-383.
- Torres, G. 2001,
The change in the inclination angle of the noneclipsing binary SS Lacertae:
future eclipses,
The Astronomical Journal, 121, 2227-2238.
- Peter, H. 2001,
On the nature of the transition region from the chromosphere to the corona
of the Sun,
Astronomy and Astrophysics, 374, 1108-1120.
- Bartzakos, P., Moffat, A.F.J., and Miemela, V.S. 2001,
Magellanic Cloud WC.WO Wolf-Rayet stars. II. Colliding winds in binaries,
Monthly Notices of the Royal Astronomical Society, 324, 33-50.
- Severino, G., Magri, M., Oliviero, M., Straus, Th.,
and Jefferies, S.M. 2001,
The solar intensity-velocity cross spectrum: a powerful diagnostic for
helioseismology,
The Astrophysical Journal, 561, 444-449.
- Gelino, D.M., Harrison, T.E., and Orosz, J.A. 2001,
A multiwavelength, multiepoch study of the soft X-ray transient prototype,
V616 Monocerotis,
The Astronomical Journal, 122, 2668-2678.
- Metcalfe, T.S. 2003,
White dwarf asteroseismology and the
12C(alpha,gamma)16O rate,
The Astrophysical Journal (Letters), in press.
Please send preprints/reprints of any work you have carried out
where PIKAIA was used in a non-trivial way. Like we said,
we're curious.
7. Bug Reports
Since the official release of PIKAIA 1.0 in January 1996
several minor bugs and a few not-so-minor ones
have been reported by users. All users who were
registered at the time were notified of the corrective actions to
be taken. The following give access to the original texts of the
bug reports issued up to the present time. All bugs have been corrected
as per the issue date of the corresponding bug report, as listed below:
REPORT ISSUED ON CONCERNING
No. 1 01/20/96 Incorrect declaration ordering in
subroutine report
No. 2 02/14/96 Minor bug in random number initialization
No. 3 02/14/96 Assorted minor and one not-so-minor bugs
in some of the example drivers and
fitness functions
No. 4 05/20/96 Additional safeguard needed in
random number generator routine
8. PIKAIA in other computing languages
Thanks to the kind efforts and generosity of various PIKAIA users, versions
in IDL and f90 are now available publicly for general use.
An IDL transliteration of the original FORTRAN-77
has been produced by Sarah Gibson, and is available directly by
clicking here. A more streamlined and efficient IDL version has also been produced
by Scott McIntosh, and can be accessed by
clicking here.
Scott is currently (April 2003) fixing some last remaining odds and ends,
but the code seems to be running OK.
For a FORTRAN-90 version, go to
Alan Miller's software page
and scroll down to Miscellaneous Software, looking for pikaia.f90
A parallel version of PIKAIA is now available, thanks to the efforts
of Travis Metcalfe. Versions exist in both MPI and PVM, and can be
found at Travis'
Parallel PIKAIA Web Page. [Technical detail:
only the Full-Generational-Replacement
reproduction plan can be used in these parallel versions, as the steady-state
plans are much harder to parallelize in a robust and efficient manner].
9. References and further readings
On the subroutine PIKAIA itself see
- Charbonneau, P., 1995, Genetic Algorithms in Astronomy
and Astrophysics,
The Astrophysical Journal (Supplements), 101, 309
[abstract]
- Charbonneau, P., and Knapp, B. 1995, A User's guide
to PIKAIA 1.0,
NCAR Technical Note 418+IA (Boulder: National Center for
Atmospheric Research).
[Table of Content]
- Charbonneau, P., 2002, Release Notes for PIKAIA 1.2,
NCAR Technical Note 451+STR (Boulder: National Center for
Atmospheric Research).
[full text, postscript]
- Charbonneau, P., 2002, An Introduction to Genetic Algorithms
for Numerical Optimization,
NCAR Technical Note 450+IA (Boulder: National Center for
Atmospheric Research).
[full text, postscript]; previously
known as the Oslo Tutorial
On genetic algorithms in general, see
- Davis, L. 1991, Handbook of Genetic Algorithms
(New York: Van Nostrand Reinhold).
- Goldberg, D.E. 1989, Genetic Algorithms in Search,
Optimization and Machine Learning (Reading: Addison-Wesley)
- Holland, J.H. 1975, Adaptation in Natural and Artificial Systems
(Ann Arbor: The University of Michigan Press; Second Edition 1992, MIT Press).
- Mitchell, M. 1996, An Introduction to Genetic Algorithms
(Cambridge: The MIT Press).
- Baeck, T. 1996, Genetic Algorithms in Theory and Practice
(Oxford: The Oxford University Press).
- Michalewicz, Z. 1994, Genetic Algorithms + Data Structures
= Evolution Programs (Berlin: Springer).
And finally, our not particularly well-informed choice for the top
five books dealing with relevant aspects of evolutionary biology:
- Dawkins, R. 1986, The Blind Watchmaker
(New York: W.W. Norton).
- Eldredge, N. 1985, Time Frames
(New York: Simon and Schuster).
- Gould, S. J. 1989, Wonderful Life
(New York: W.W. Norton).
- Maynard Smith, J. 1989, Evolutionary Genetics
(Oxford: The Oxford University Press).
- Stebbins, G. L. 1966, Processes of organic Evolution
(Englewood Cliffs: Prentice-Hall).
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