FAQ, Part 7 of 7: Hardware

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Warren Sarle

Jul 29, 2002, 2:30:02 PM7/29/02
Archive-name: ai-faq/neural-nets/part7
Last-modified: 2002-05-31
Maintainer: (Warren S. Sarle)

Copyright 1997, 1998, 1999, 2000, 2001, 2002 by Warren S. Sarle, Cary, NC,
USA. Answers provided by other authors as cited below are copyrighted by
those authors, who by submitting the answers for the FAQ give permission for
the answer to be reproduced as part of the FAQ in any of the ways specified
in part 1 of the FAQ.

This is part 7 (of 7) of a monthly posting to the Usenet newsgroup See the part 1 of this posting for full information
what it is all about.

========== Questions ==========

Part 1: Introduction
Part 2: Learning
Part 3: Generalization
Part 4: Books, data, etc.
Part 5: Free software
Part 6: Commercial software
Part 7: Hardware and miscellaneous

Neural Network hardware?
What are some applications of NNs?
Face recognition
Finance and economics
Games, sports, gambling
Materials science
Weather forecasting
What to do with missing/incomplete data?
How to forecast time series (temporal sequences)?
How to learn an inverse of a function?
How to get invariant recognition of images under translation, rotation,
How to recognize handwritten characters?
What about pulsed or spiking NNs?
What about Genetic Algorithms and Evolutionary Computation?
What about Fuzzy Logic?
Unanswered FAQs
Other NN links?


Subject: Neural Network hardware?

Overview articles:

o Clark S. Lindsey and Thomas Lindblad (1998), "Review of hardware neural
networks: A user's perspective",

o P. D. Moerland and E. Fiesler (1997), "Neural Network Adaptations to
Hardware Implementations", in Handbook of Neural Computation,

The journal, IEEE Transactions on Neural Networks, plans to have a
special issue on neural networks hardware implementations in September,

Various NN hardware information can be found at the following web sites:

o Pacific Northwest National Laboratory:
o Dr. Denise Gorse, University College London:
o Neural Chips and Evolvable Hardware:


Subject: What are some applications of NNs?

There are vast numbers of published neural network applications. If you
don't find something from your field of interest below, try a web search.
Here are some useful search engines:


o The Pacific Northwest National Laboratory: including a list of
commercial applications at
o The Stimulation Initiative for European Neural Applications:
o The DTI NeuroComputing Web's Applications Portfolio:
o The Applications Corner, NeuroDimension, Inc.:
o The BioComp Systems, Inc. Solutions page:
o Chen, C.H., ed. (1996) Fuzzy Logic and Neural Network Handbook, NY:
McGraw-Hill, ISBN 0-07-011189-8.
o The series Advances in Neural Information Processing Systems containing
proceedings of the conference of the same name, published yearly by
Morgan Kauffman starting in 1989 and by The MIT Press in 1995.


o P.H. Heinemann, Automated Grading of Produce:
o Deck, S., C.T. Morrow, P.H. Heinemann, and H.J. Sommer, III. 1995.
Comparison of a neural network and traditional classifier for machine
vision inspection. Applied Engineering in Agriculture. 11(2):319-326.
o Tao, Y., P.H. Heinemann, Z. Varghese, C.T. Morrow, and H.J. Sommer III.
1995. Machine vision for color inspection of potatoes and apples.
Transactions of the American Society of Agricultural Engineers.


o "No Hands Across America Journal" - steering a car:


o PNNL, General Applications of Neural Networks in Chemistry and Chemical
o Prof. Dr. Johann Gasteiger, Neural Networks and Genetic Algorithms in
o Roy Goodacre, pyrolysis mass spectrometry: and Fourier transform
infrared (FT-IR) spectroscopy: contain applications
of a variety of NNs as well as PLS (partial least squares) and other
statistical methods.
o Situs, a program package for the docking of protein crystal structures to
single-molecule, low-resolution maps from electron microscopy or small
angle X-ray scattering:
o An on-line application of a Kohonen network with a 2-dimensional output
layer for prediction of protein secondary structure percentages from UV
circular dichroism spectra:


o Computer Aided Tracking and Characterization of Homicides and Sexual
Assaults (CATCH):

Face recognition

o Face Recognition Home Page:
o Konen, W., "Neural information processing in real-world face-recognition
o Jiang, Q., "Principal Component Analysis and Neural Network Based Face
o Lawrence, S., Giles, C.L., Tsoi, A.C., Back, A.D. (1997), "Face
Recognition: A Convolutional Neural Network Approach," IEEE Transactions
on Neural Networks, 8, 98-113,

Finance and economics

o Athanasios Episcopos, References on Neural Net Applications to Finance
and Economics:
o Franco Busetti, Heuristics and artificial intelligence in finance and
o Trippi, R.R. & Turban, E. (1993), Neural Networks in Finance and
Investing, Chicago: Probus.
o Zirilli, J.S. (1996), Financial Prediction Using Neural Networks,
International Thomson Publishing, ISBN 1850322341,
o Andreas S. Weigend, Yaser Abu-Mostafa, A. Paul N. Refenes (eds.) (1997)
Decision Technologies for Financial Engineering: Proceedings of the Fourth
International Conference on Neural Networks in the Capital Markets (Nncm
'96) Publisher: World Scientific Publishing Company, ISBN: 9810231245

Games, sports, gambling

o General:

Jay Scott, Machine Learning in Games:

METAGAME Game-Playing Workbench:

R.S. Sutton, "Learning to predict by the methods of temporal
differences", Machine Learning 3, p. 9-44 (1988).

David E. Moriarty and Risto Miikkulainen (1994). "Evolving Neural
Networks to Focus Minimax Search," In Proceedings of Twelfth National
Conference on Artificial Intelligence (AAAI-94, Seattle, WA), 1371-1377.
Cambridge, MA: MIT Press,

Games World '99 at

o Backgammon:

G. Tesauro and T.J. Sejnowski (1989), "A Parallel Network that learns to
play Backgammon," Artificial Intelligence, vol 39, pp. 357-390.

G. Tesauro and T.J. Sejnowski (1990), "Neurogammon: A Neural Network
Backgammon Program," IJCNN Proceedings, vol 3, pp. 33-39, 1990.

G. Tesauro (1995), "Temporal Difference Learning and TD-Gammon,"
Communications of the ACM, 38, 58-68,

Pollack, J.P. and Blair, A.D. (1997), "Co-Evolution in the Successful
Learning of Backgammon Strategy," Brandeis University Computer Science
Technical Report CS-97-193,

o Bridge:


He Yo, Zhen Xianjun, Ye Yizheng, Li Zhongrong (19??), "Knowledge
acquisition and reasoning based on neural networks - the research of a
bridge bidding system," INNC '90, Paris, vol 1, pp. 416-423.

M. Kohle and F. Schonbauer (19??), "Experience gained with a neural
network that learns to play bridge," Proc. of the 5th Austrian Artificial
Intelligence meeting, pp. 224-229.

o Checkers/Draughts:

Mark Lynch (1997), "NeuroDraughts: an application of temporal difference
learning to draughts," Software
available at

K. Chellapilla and D. B. Fogel, "Co-Evolving Checkers Playing Programs
using Only Win, Lose, or Draw," SPIE's AeroSense'99: Applications and
Science of Computational Intelligence II, Apr. 5-9, 1999, Orlando,
Florida, USA,

David Fogel (1999), Evolutionary Computation: Toward a New Philosophy
of Machine Intelligence (2nd edition), IEEE, ISBN: 078035379X

David Fogel (2001), Blondie24: Playing at the Edge of AI, Morgan Kaufmann
Publishers, ISBN: 1558607838
According to the publisher, this is:

... the first book to bring together the most advanced work in the
general use of evolutionary computation for creative results. It is
well suited for the general computer science audience.

Here's the story of a computer that taught itself to play checkers
far better than its creators ever could. Blondie24 uses a program
that emulates the basic principles of Darwin evolution to discover on
its own how to excel at the game. Through this entertaining story,
the book provides the reader some of the history of AI and explores
its future.

Unlike Deep Blue, the celebrated chess machine that beat Garry
Kasparov, the former world champion chess player, this evolutionary
program didn't have access to other games played by human grand
masters, or databases of moves for the endgame. It created its own
means for evaluating the patterns of pieces that it experienced by
evolving artificial neural networks--mathematical models that loosely
describe how a brain works.

See for a variety
of online papers by Fogel.

Not NNs, but classic papers:

A.L. Samuel (1959), "Some studies in machine learning using the game of
checkers," IBM journal of Research and Development, vol 3, nr. 3, pp.

A.L. Samuel (1967), "Some studies in machine learning using the game of
checkers 2 - recent progress," IBM journal of Research and Development, vol
11, nr. 6, pp. 601-616.

o Chess:

Sebastian Thrun, NeuroChess:

Luke Pellen, Octavius:

Louis Savain (AKA Nemesis), Animal, a spiking neural network that the author
hopes will learn to play a passable game of chess after he implements the
motivation mechanism:

o Dog racing:

H. Chen1, P. Buntin, L. She, S. Sutjahjo, C. Sommer, D. Neely (1994),
"Expert Prediction, Symbolic Learning, and Neural Networks: An Experiment on
Greyhound Racing," IEEE Expert, December 1994, 21-27,

o Football (Soccer):

Kuonen Diego, "Statistical Models for Knock-out Soccer Tournaments", (not neural nets, but relevant)

o Go:

David Stoutamire (19??), "Machine Learning, Game Play, and Go," Center for
Automation and Intelligent Systems Research TR 91-128, Case Western Reserve

David Stoutamire (1991), Machine Learning Applied to Go, M.S. thesis, Case
Western Reserve University,

Schraudolph, N., Dayan, P., Sejnowski, T. (1994), "Temporal Difference
Learning of Position Evaluation in the Game of Go," In: Neural Information
Processing Systems 6, Morgan Kaufmann 1994,

P. Donnelly, P. Corr & D. Crookes (1994), "Evolving Go Playing Strategy in
Neural Networks", AISB Workshop on Evolutionary Computing, Leeds, England, or

Markus Enzenberger (1996), "The Integration of A Priori Knowledge into a Go
Playing Neural Network,"

Norman Richards, David Moriarty, and Risto Miikkulainen (1998), "Evolving
Neural Networks to Play Go," Applied Intelligence, 8, 85-96,

Dahl, F. A. (1999), "Honte, a Go-playing program using neural nets",

o Go-Moku:

Freisleben, B., "Teaching a Neural Network to Play GO-MOKU," in I.
Aleksander and J. Taylor, eds, Artificial Neural Networks 2, Proc. of
ICANN-92, Brighton UK, vol. 2, pp. 1659-1662, Elsevier Science Publishers,

Katz, W.T. and Pham, S.P. "Experience-Based Learning Experiments using
Go-moku", Proc. of the 1991 IEEE International Conference on Systems, Man,
and Cybernetics, 2: 1405-1410, October 1991.

o Olympics:

E.M.Condon, B.L.Golden, E.A.Wasil (1999), "Predicting the success of nations
at the Summer Olympics using neural networks", Computers & Operations
Research, 26, 1243-1265.

o Pong:


o Reversi/Othello:

David E. Moriarty and Risto Miikkulainen (1995). Discovering Complex Othello
Strategies through Evolutionary Neural Networks. Connection Science, 7,

Yoshioka, T., Ishii, S., and Ito, M., Strategy acquisition for the game
``Othello'' based on reinforcement learning, IEICE Transactions on
Information and Systems E82-D 12, 1618-1626, 1999,

o Tic-Tac-Toe/Noughts and Crosses:

Fogel, David Bb (1993), "Using evolutionary programming to construct neural
networks that are capable of playing tic-tac-toe," Intern. Conf. on Neural
Networks 1993, IEEE, San Francisco, CA, pp. 875-880.

Richard S. Sutton and Andrew G. Barto (1998), Reinforcement Learning: An
Introduction The MIT Press, ISBN: 0262193981,

Yongzheng Zhang, Chen Teng, Sitan Wei (2000), "Game playing with
Evolutionary Strategies and Modular Neural Networks: Tic-Tac-Toe,"

Rob Ellison, "Neural Os and Xs," (An online Javascript demo,
but you may not live long enough to teach the network to play a mediocre
game. I'm not sure what kind of network it uses, but maybe you can figure
that out if you read the source.)


o PNNL, Neural Network Applications in Manufacturing:
o PNNL, Applications in the Electric Power Industry:
o PNNL, Process Control:
o Raoul Tawel, Ken Marko, and Lee Feldkamp (1998), "Custom VLSI ASIC for
Automotive Applications with Recurrent Networks",
o Otsuka, Y. et al. "Neural Networks and Pattern Recognition of Blast
Furnace Operation Data" Kobelco Technology Review, Oct. 1992, 12
o Otsuka, Y. et al. "Applications of Neural Network to Iron and Steel
Making Processes" 2. International Conference on Fuzzy Logic and Neural
Networks, Iizuka, 1992
o Staib, W.E. "Neural Network Control System for Electric Arc Furnaces"
M.P.T. International, 2/1995, 58-61
o Portmann, N. et al. "Application of Neural Networks in Rolling
Automation" Iron and Steel Engineer, Feb. 1995, 33-36
o Gorni, A.A. (2000), "The modelling of hot rolling processes using neural
networks: A bibliographical review",
o Murat, M. E., and Rudman, A. J., 1992, Automated first arrival picking: A
neural network approach: Geophysical Prospecting, 40, 587-604.

Materials science

o Phase Transformations Research Group (search for "neural"):


o PNNL, Applications in Medicine and Health:


o Mozer, M. C. (1994), "Neural network music composition by prediction:
Exploring the benefits of psychophysical constraints and multiscale
processing," Connection Science, 6, 247-280,
o Griffith, N., and Todd, P.M., eds. (1999), Musical Networks: Parallel
Distributed Perception and Performance, Cambridge, MA: The MIT Press,
ISBN 0-262-07181-9.


o Institute of Robotics and System Dynamics:
o UC Berkeley Robotics and Intelligent Machines Lab:
o Perth Robotics and Automation Laboratory:
o University of New Hampshire Robot Lab:

Weather forecasting and atmospheric science

o UBC Climate Prediction Group:
o Artificial Intelligence Research In Environmental Science:
o MET-AI, an mailing list for meteorologists and AI researchers:
o Caren Marzban, Ph.D., Research Scientist, National Severe Storms
o David Myers's references on NNs in atmospheric science:


Zaknich, Anthony and Baker, Sue K. (1998), "A real-time system for the
characterisation of sheep feeding phases from acoustic signals of jaw
sounds," Australian Journal of Intelligent Information Processing Systems
(AJIIPS), Vol. 5, No. 2, Winter 1998.

This paper describes a four-channel real-time system for the detection and
measurement of sheep rumination and mastication time periods by the analysis
of jaw sounds transmitted through the skull. The system is implemented using
an 80486 personal computer, a proprietary data acquisition card (PC-126) and
a custom made variable gain preamplifier and bandpass filter module. Chewing
sounds are transduced and transmitted to the system using radio microphones
attached to the top of the sheep heads. The system's main functions are to
detect and estimate rumination and mastication time periods, to estimate the
number of chews during the rumination and mastication periods, and to
provide estimates of the number of boli in the rumination sequences and the
number of chews per bolus. The individual chews are identified using a
special energy threshold detector. The rumination and mastication time
periods are determined by neural network classifier using a combination of
time and frequency domain features extracted from successive 10 second
acoustic signal blocks.


Subject: What to do with missing/incomplete data?

The problem of missing data is very complex.

For unsupervised learning, conventional statistical methods for missing data
are often appropriate (Little and Rubin, 1987; Schafer, 1997; Schafer and
Olsen, 1998). There is a concise introduction to these methods in the
University of Texas statistics FAQ at

For supervised learning, the considerations are somewhat different, as
discussed by Sarle (1998). The statistical literature on missing data deals
almost exclusively with training rather than prediction (e.g., Little,
1992). For example, if you have only a small proportion of cases with
missing data, you can simply throw those cases out for purposes of training;
if you want to make predictions for cases with missing inputs, you don't
have the option of throwing those cases out! In theory, Bayesian methods
take care of everything, but a full Bayesian analysis is practical only with
special models (such as multivariate normal distributions) or small sample
sizes. The neural net literature contains a few good papers that cover
prediction with missing inputs (e.g., Ghahramani and Jordan, 1997; Tresp,
Neuneier, and Ahmad 1995), but much research remains to be done.


Donner, A. (1982), "The relative effectiveness of procedures commonly
used in multiple regression analysis for dealing with missing values,"
American Statistician, 36, 378-381.

Ghahramani, Z. and Jordan, M.I. (1994), "Supervised learning from
incomplete data via an EM approach," in Cowan, J.D., Tesauro, G., and
Alspector, J. (eds.) Advances in Neural Information Processing Systems
6, San Mateo, CA: Morgan Kaufman, pp. 120-127.

Ghahramani, Z. and Jordan, M.I. (1997), "Mixture models for Learning from
incomplete data," in Greiner, R., Petsche, T., and Hanson, S.J. (eds.)
Computational Learning Theory and Natural Learning Systems, Volume IV:
Making Learning Systems Practical, Cambridge, MA: The MIT Press, pp.

Jones, M.P. (1996), "Indicator and stratification methods for missing
explanatory variables in multiple linear regression," J. of the American
Statistical Association, 91, 222-230.

Little, R.J.A. (1992), "Regression with missing X's: A review," J. of the
American Statistical Association, 87, 1227-1237.

Little, R.J.A. and Rubin, D.B. (1987), Statistical Analysis with Missing
Data, NY: Wiley.

McLachlan, G.J. (1992) Discriminant Analysis and Statistical Pattern
Recognition, Wiley.

Sarle, W.S. (1998), "Prediction with Missing Inputs," in Wang, P.P.
(ed.), JCIS '98 Proceedings, Vol II, Research Triangle Park, NC, 399-402,

Schafer, J.L. (1997), Analysis of Incomplete Multivariate Data, London:
Chapman & Hall, ISBN 0 412 04061 1.

Schafer, J.L., and Olsen, M.K. (1998), "Multiple imputation for
multivariate missing-data problems: A data analyst's perspective," or

Tresp, V., Ahmad, S. and Neuneier, R., (1994), "Training neural networks
with deficient data", in Cowan, J.D., Tesauro, G., and Alspector, J.
(eds.) Advances in Neural Information Processing Systems 6, San Mateo,
CA: Morgan Kaufman, pp. 128-135.

Tresp, V., Neuneier, R., and Ahmad, S. (1995), "Efficient methods for
dealing with missing data in supervised learning", in Tesauro, G.,
Touretzky, D.S., and Leen, T.K. (eds.) Advances in Neural Information
Processing Systems 7, Cambridge, MA: The MIT Press, pp. 689-696.


Subject: How to forecast time series (temporal sequences)?

In most of this FAQ, it is assumed that the training cases are statistically
independent. That is, the training cases consist of pairs of input and
target vectors, (X_i,Y_i), i=1,...,N, such that the conditional
distribution of Y_i given all the other training data, (X_j,
j=1,...,N, and Y_j, j=1,...i-1,i+1,...N) is equal to the
conditional distribution of Y_i given X_i regardless of the values in the
other training cases. Independence of cases is often achieved by random

The most common violation of the independence assumption occurs when cases
are observed in a certain order relating to time or space. That is, case
(X_i,Y_i) corresponds to time T_i, with T_1 < T_2 < ... <
T_N. It is assumed that the current target Y_i may depend not only on
X_i but also on (X_i,Y_i) in the recent past. If the T_i are equally
spaced, the simplest way to deal with this dependence is to include
additional inputs (called lagged variables, shift registers, or a tapped
delay line) in the network. Thus, for target Y_i, the inputs may include
X_i, Y_{i-1}, X_{i-1}, Y_{i-1}, X_{i-2}, etc. (In some
situations, X_i would not be known at the time you are trying to forecast
Y_i and would therefore be excluded from the inputs.) Then you can train
an ordinary feedforward network with these targets and lagged variables. The
use of lagged variables has been extensively studied in the statistical and
econometric literature (Judge, Griffiths, Hill, Lütkepohl and Lee, 1985). A
network in which the only inputs are lagged target values is called an
"autoregressive model." The input space that includes all of the lagged
variables is called the "embedding space."

If the T_i are not equally spaced, everything gets much more complicated.
One approach is to use a smoothing technique to interpolate points at
equally spaced intervals, and then use the interpolated values for training
instead of the original data.

Use of lagged variables increases the number of decisions that must be made
during training, since you must consider which lags to include in the
network, as well as which input variables, how many hidden units, etc.
Neural network researchers have therefore attempted to use partially
recurrent networks instead of feedforward networks with lags (Weigend and
Gershenfeld, 1994). Recurrent networks store information about past values
in the network itself. There are many different kinds of recurrent
architectures (Hertz, Krogh, and Palmer 1991; Mozer, 1994; Horne and Giles,
1995; Kremer, 199?). For example, in time-delay neural networks (Lang,
Waibel, and Hinton 1990), the outputs for predicting target Y_{i-1} are
used as inputs when processing target Y_i. Jordan networks (Jordan, 1986)
are similar to time-delay neural networks except that the feedback is an
exponential smooth of the sequence of output values. In Elman networks
(Elman, 1990), the hidden unit activations that occur when processing target
Y_{i-1} are used as inputs when processing target Y_i.

However, there are some problems that cannot be dealt with via recurrent
networks alone. For example, many time series exhibit trend, meaning that
the target values tend to go up over time, or that the target values tend to
go down over time. For example, stock prices and many other financial
variables usually go up. If today's price is higher than all previous
prices, and you try to forecast tomorrow's price using today's price as a
lagged input, you are extrapolating, and extrapolating is unreliable. The
simplest methods for handling trend are:

o First fit a linear regression predicting the target values from the time,
Y_i = a + b T_i + noise, where a and b are regression
weights. Compute residuals R_i = Y_i - (a + b T_i). Then
train the network using R_i for the target and lagged values. This
method is rather crude but may work for deterministic linear trends. Of
course, for nonlinear trends, you would need to fit a nonlinear

o Instead of using Y_i as a target, use D_i = Y_i - Y_{i-1} for
the target and lagged values. This is called differencing and is the
standard statistical method for handling nondeterministic (stochastic)
trends. Sometimes it is necessary to compute differences of differences.

For an elementary discussion of trend and various other practical problems
in forecasting time series with NNs, such as seasonality, see Masters
(1993). For a more advanced discussion of NN forecasting of economic series,
see Moody (1998).

There are several different ways to compute forecasts. For simplicity, let's
assume you have a simple time series, Y_1, ..., Y_99, you want to
forecast future values Y_f for f > 99, and you decide to use three
lagged values as inputs. The possibilities include:

Single-step, one-step-ahead, or open-loop forecasting:
Train a network with target Y_i and inputs Y_{i-1}, Y_{i-2},
and Y_{i-3}. Let the scalar function computed by the network be
designated as Net(.,.,.) taking the three input values as arguments
and returning the output (predicted) value. Then:
forecast Y_100 as Net(Y_99,Y_98,Y_97)
forecast Y_101 as Net(Y_100,Y_99,Y_98)
forecast Y_102 as Net(Y_101,Y_100,Y_99)
forecast Y_103 as Net(Y_102,Y_101,Y_100)
forecast Y_104 as Net(Y_103,Y_102,Y_101)
and so on.

Multi-step or closed-loop forecasting:
Train the network as above, but:
forecast Y_100 as P_100 = Net(Y_99,Y_98,Y_97)
forecast Y_101 as P_101 = Net(P_100,Y_99,Y_98)
forecast Y_102 as P_102 = Net(P_101,P_100,Y_99)
forecast Y_103 as P_103 = Net(P_102,P_101,P_100)
forecast Y_104 as P_104 = Net(P_103,P_102,P_101)
and so on.

N-step-ahead forecasting:
For, say, N=3, train the network as above, but:
compute P_100 = Net(Y_99,Y_98,Y_97)
compute P_101 = Net(P_100,Y_99,Y_98)
forecast Y_102 as P_102 = Net(P_101,P_100,Y_99)
forecast Y_103 as P_103 = Net(P_102,P_101,Y_100)
forecast Y_104 as P_104 = Net(P_103,P_102,Y_101)
and so on.

Direct simultaneous long-term forecasting:
Train a network with multiple targets Y_i, Y_{i+1}, and Y_{i+2}
and inputs Y_{i-1}, Y_{i-2}, and Y_{i-3}. Let the vector
function computed by the network be designated as Net3(.,.,.),
taking the three input values as arguments and returning the output
(predicted) vector. Then:
forecast (Y_100,Y_101,Y_102) as Net3(Y_99,Y_98,Y_97)

Which method you choose for computing forecasts will obviously depend in
part on the requirements of your application. If you have yearly sales
figures through 1999 and you need to forecast sales in 2003, you clearly
can't use single-step forecasting. If you need to compute forecasts at a
thousand different future times, using direct simultaneous long-term
forecasting would require an extremely large network.

If a time series is a random walk, a well-trained network will predict Y_i
by simply outputting Y_{i-1}. If you make a plot showing both the target
values and the outputs, the two curves will almost coincide, except for
being offset by one time step. People often mistakenly intrepret such a plot
to indicate good forecasting accuracy, whereas in fact the network is
virtually useless. In such situations, it is more enlightening to plot
multi-step forecasts or N-step-ahead forecasts.

For general information on time-series forecasting, see the following URLs:

o Forecasting FAQs:
o Forecasting Principles:
o Investment forecasts for stocks and mutual funds:


Elman, J.L. (1990), "Finding structure in time," Cognitive Science, 14,

Hertz, J., Krogh, A., and Palmer, R. (1991). Introduction to the Theory of
Neural Computation. Addison-Wesley: Redwood City, California.

Horne, B. G. and Giles, C. L. (1995), "An experimental comparison of
recurrent neural networks," In Tesauro, G., Touretzky, D., and Leen, T.,
editors, Advances in Neural Information Processing Systems 7, pp.
697-704. The MIT Press.

Jordan, M. I. (1986), "Attractor dynamics and parallelism in a
connectionist sequential machine," In Proceedings of the Eighth Annual
conference of the Cognitive Science Society, pages 531-546. Lawrence

Judge, G.G., Griffiths, W.E., Hill, R.C., Lütkepohl, H., and Lee, T.-C.
(1985), The Theory and Practice of Econometrics, NY: John Wiley & Sons.

Kremer, S.C. (199?), "Spatio-temporal Connectionist Networks: A Taxonomy
and Review,"

Lang, K. J., Waibel, A. H., and Hinton, G. (1990), "A time-delay neural
network architecture for isolated word recognition," Neural Networks, 3,

Masters, T. (1993). Practical Neural Network Recipes in C++, San Diego:
Academic Press.

Moody, J. (1998), "Forecasting the economy with neural nets: A survey of
challenges and solutions," in Orr, G,B., and Mueller, K-R, eds., Neural
Networks: Tricks of the Trade, Berlin: Springer.

Mozer, M.C. (1994), "Neural net architectures for temporal sequence
processing," in Weigend, A.S. and Gershenfeld, N.A., eds. (1994) Time
Series Prediction: Forecasting the Future and Understanding the Past,
Reading, MA: Addison-Wesley, 243-264,

Weigend, A.S. and Gershenfeld, N.A., eds. (1994) Time Series Prediction:
Forecasting the Future and Understanding the Past, Reading, MA:


Subject: How to learn an inverse of a function?

Ordinarily, NNs learn a function Y = f(X), where Y is a vector of
outputs, X is a vector of inputs, and f() is the function to be learned.
Sometimes, however, you may want to learn an inverse of a function f(),
that is, given Y, you want to be able to find an X such that Y = f(X).
In general, there may be many different Xs that satisfy the equation Y =

For example, in robotics (DeMers and Kreutz-Delgado, 1996, 1997), X might
describe the positions of the joints in a robot's arm, while Y would
describe the location of the robot's hand. There are simple formulas to
compute the location of the hand given the positions of the joints, called
the "forward kinematics" problem. But there is no simple formula for the
"inverse kinematics" problem to compute positions of the joints that yield a
given location for the hand. Furthermore, if the arm has several joints,
there will usually be many different positions of the joints that yield the
same location of the hand, so the forward kinematics function is many-to-one
and has no unique inverse. Picking any X such that Y = f(X) is OK if
the only aim is to position the hand at Y. However if the aim is to
generate a series of points to move the hand through an arc this may be
insufficient. In this case the series of Xs need to be in the same "branch"
of the function space. Care must be taken to avoid solutions that yield
inefficient or impossible movements of the arm.

As another example, consider an industrial process in which X represents
settings of control variables imposed by an operator, and Y represents
measurements of the product of the industrial process. The function Y =
f(X) can be learned by a NN using conventional training methods. But the
goal of the analysis may be to find control settings X that yield a product
with specified measurements Y, in which case an inverse of f(X) is
required. In industrial applications, financial considerations are
important, so not just any setting X that yields the desired result Y may
be acceptable. Perhaps a function can be specified that gives the cost of X
resulting from energy consumption, raw materials, etc., in which case you
would want to find the X that minimizes the cost function while satisfying
the equation Y = f(X).

The obvious way to try to learn an inverse function is to generate a set of
training data from a given forward function, but designate Y as the input
and X as the output when training the network. Using a least-squares error
function, this approach will fail if f() is many-to-one. The problem is
that for an input Y, the net will not learn any single X such that Y =
f(X), but will instead learn the arithmetic mean of all the Xs in the
training set that satisfy the equation (Bishop, 1995, pp. 207-208). One
solution to this difficulty is to construct a network that learns a mixture
approximation to the conditional distribution of X given Y (Bishop, 1995,
pp. 212-221). However, the mixture method will not work well in general for
an X vector that is more than one-dimensional, such as Y = X_1^2 +
X_2^2, since the number of mixture components required may increase
exponentially with the dimensionality of X. And you are still left with the
problem of extracting a single output vector from the mixture distribution,
which is nontrivial if the mixture components overlap considerably. Another
solution is to use a highly robust error function, such as a redescending
M-estimator, that learns a single mode of the conditional distribution
instead of learning the mean (Huber, 1981; Rohwer and van der Rest 1996).
Additional regularization terms or constraints may be required to persuade
the network to choose appropriately among several modes, and there may be
severe problems with local optima.

Another approach is to train a network to learn the forward mapping f()
and then numerically invert the function. Finding X such that Y = f(X)
is simply a matter of solving a nonlinear system of equations, for which
many algorithms can be found in the numerical analysis literature (Dennis
and Schnabel 1983). One way to solve nonlinear equations is turn the problem
into an optimization problem by minimizing sum(Y_i-f(X_i))^2. This
method fits in nicely with the usual gradient-descent methods for training
NNs (Kindermann and Linden 1990). Since the nonlinear equations will
generally have multiple solutions, there may be severe problems with local
optima, especially if some solutions are considered more desirable than
others. You can deal with multiple solutions by inventing some objective
function that measures the goodness of different solutions, and optimizing
this objective function under the nonlinear constraint Y = f(X) using
any of numerous algorithms for nonlinear programming (NLP; see Bertsekas,
1995, and other references under "What are conjugate gradients,
Levenberg-Marquardt, etc.?") The power and flexibility of the nonlinear
programming approach are offset by possibly high computational demands.

If the forward mapping f() is obtained by training a network, there will
generally be some error in the network's outputs. The magnitude of this
error can be difficult to estimate. The process of inverting a network can
propagate this error, so the results should be checked carefully for
validity and numerical stability. Some training methods can produce not just
a point output but also a prediction interval (Bishop, 1995; White, 1992).
You can take advantage of prediction intervals when inverting a network by
using NLP methods. For example, you could try to find an X that minimizes
the width of the prediction interval under the constraint that the equation
Y = f(X) is satisfied. Or instead of requiring Y = f(X) be satisfied
exactly, you could try to find an X such that the prediction interval is
contained within some specified interval while minimizing some cost

For more mathematics concerning the inverse-function problem, as well as
some interesting methods involving self-organizing maps, see DeMers and
Kreutz-Delgado (1996, 1997). For NNs that are relatively easy to invert, see
the Adaptive Logic Networks described in the software sections of the FAQ.


Bertsekas, D. P. (1995), Nonlinear Programming, Belmont, MA: Athena

Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.

DeMers, D., and Kreutz-Delgado, K. (1996), "Canonical Parameterization of
Excess motor degrees of freedom with self organizing maps", IEEE Trans
Neural Networks, 7, 43-55.

DeMers, D., and Kreutz-Delgado, K. (1997), "Inverse kinematics of
dextrous manipulators," in Omidvar, O., and van der Smagt, P., (eds.)
Neural Systems for Robotics, San Diego: Academic Press, pp. 75-116.

Dennis, J.E. and Schnabel, R.B. (1983) Numerical Methods for
Unconstrained Optimization and Nonlinear Equations, Prentice-Hall

Huber, P.J. (1981), Robust Statistics, NY: Wiley.

Kindermann, J., and Linden, A. (1990), "Inversion of Neural Networks by
Gradient Descent," Parallel Computing, 14, 277-286,

Rohwer, R., and van der Rest, J.C. (1996), "Minimum description length,
regularization, and multimodal data," Neural Computation, 8, 595-609.

White, H. (1992), "Nonparametric Estimation of Conditional Quantiles
Using Neural Networks," in Page, C. and Le Page, R. (eds.), Proceedings
of the 23rd Sympsium on the Interface: Computing Science and Statistics,
Alexandria, VA: American Statistical Association, pp. 190-199.


Subject: How to get invariant recognition of images under
translation, rotation, etc.?


Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press, section 8.7.

Masters, T. (1994), Signal and Image Processing with Neural Networks: A
C++ Sourcebook, NY: Wiley.

Soucek, B., and The IRIS Group (1992), Fast Learning and Invariant Object
Recognition, NY: Wiley.

Squire, D. (1997), Model-Based Neural Networks for Invariant Pattern

Laurenz Wiskott, bibliography on "Unsupervised Learning of Invariances in
Neural Systems"


Subject: How to recognize handwritten characters?


o Don Tveter's The Pattern Recognition Basis of AI at
o Andras Kornai's homepage at
o Yann LeCun's homepage at
Data sets of handwritten digits can be found at

Other references:

Hastie, T., and Simard, P.Y. (1998), "Metrics and models for handwritten
character recognition," Statistical Science, 13, 54-65.

Jackel, L.D. et al., (1994) "Comparison of Classifier Methods: A Case
Study in Handwritten Digit Recognition", 1994 International Conference on
Pattern Recognition, Jerusalem

LeCun, Y., Jackel, L.D., Bottou, L., Brunot, A., Cortes, C., Denker,
J.S., Drucker, H., Guyon, I., Muller, U.A., Sackinger, E., Simard, P.,
and Vapnik, V. (1995), "Comparison of learning algorithms for handwritten
digit recognition," in F. Fogelman and P. Gallinari, eds., International
Conference on Artificial Neural Networks, pages 53-60, Paris.

Orr, G.B., and Mueller, K.-R., eds. (1998), Neural Networks: Tricks of
the Trade, Berlin: Springer, ISBN 3-540-65311-2.


Subject: What about pulsed or spiking NNs?

The standard reference is:

Maass, W., and Bishop, C.M., eds. (1999) Pulsed Neural Networks,
Cambridge, MA: The MIT Press, ISBN: 0262133504.

For more information on this book, see the section on "Pulsed/Spiking
networks" under "Other notable books" in part 4 of the FAQ. Also see
Professor Maass's web page at

Some other interesting URLs include:

o Laboratory of Computational Neuroscience (LCN) at the Swiss Federal
Institute of Technology Lausanne,

o The notoriously hyped Berger-Liaw Neural Network Speaker-Independent
Speech Recognition System,


Subject: What about Genetic Algorithms?

There are a number of definitions of GA (Genetic Algorithm). A possible one

A GA is an optimization program
that starts with
a population of encoded procedures, (Creation of Life :-> )
mutates them stochastically, (Get cancer or so :-> )
and uses a selection process (Darwinism)
to prefer the mutants with high fitness
and perhaps a recombination process (Make babies :-> )
to combine properties of (preferably) the succesful mutants.

Genetic algorithms are just a special case of the more general idea of
"evolutionary computation". There is a newsgroup that is dedicated to the
field of evolutionary computation called It has a detailed
FAQ posting which, for instance, explains the terms "Genetic Algorithm",
"Evolutionary Programming", "Evolution Strategy", "Classifier System", and
"Genetic Programming". That FAQ also contains lots of pointers to relevant
literature, software, other sources of information, et cetera et cetera.
Please see the FAQ for further information.

For an entertaining introduction to evolutionary training of neural nets,

David Fogel (2001), Blondie24: Playing at the Edge of AI, Morgan Kaufmann
Publishers, ISBN: 1558607838

There are other books and papers by Fogel and his colleagues listed under
"Checkers/Draughts" in the "Games, sports, gambling" section above.

For an extensive review, see:

Yao, X. (1999), "Evolving Artificial Neural Networks," Proceedings of the
IEEE, 87, 1423-1447,

Here are some other on-line papers about evolutionary training of NNs:

o Backprop+GA:


o Very long chromosomes:

More URLs on genetic algorithms and NNs:

o Omri Weisman and Ziv Pollack's web page on "Neural Network Using Genetic
Algorithms" at

o Christoph M. Friedrich's web page on Evolutionary algorithms and
Artificial Neural Networks has a bibloigraphy and links to researchers at

o Andrew Gray's Hybrid Systems FAQ at the University of Otago at

o Differential Evolution:

For general information on GAs, try the links at and


Subject: What about Fuzzy Logic?

Fuzzy logic is an area of research based on the work of L.A. Zadeh. It is a
departure from classical two-valued sets and logic, that uses "soft"
linguistic (e.g. large, hot, tall) system variables and a continuous range
of truth values in the interval [0,1], rather than strict binary (True or
False) decisions and assignments.

Fuzzy logic is used where a system is difficult to model exactly (but an
inexact model is available), is controlled by a human operator or expert, or
where ambiguity or vagueness is common. A typical fuzzy system consists of a
rule base, membership functions, and an inference procedure.

Most fuzzy logic discussion takes place in the newsgroup
(where there is a fuzzy logic FAQ) but there is also some work (and
discussion) about combining fuzzy logic with neural network approaches in

Early work combining neural nets and fuzzy methods used competitive networks
to generate rules for fuzzy systems (Kosko 1992). This approach is sort of a
crude version of bidirectional counterpropagation (Hecht-Nielsen 1990) and
suffers from the same deficiencies. More recent work (Brown and Harris 1994;
Kosko 1997) has been based on the realization that a fuzzy system is a
nonlinear mapping from an input space to an output space that can be
parameterized in various ways and therefore can be adapted to data using the
usual neural training methods (see "What is backprop?") or conventional
numerical optimization algorithms (see "What are conjugate gradients,
Levenberg-Marquardt, etc.?").

A neural net can incorporate fuzziness in various ways:

o The inputs can be fuzzy. Any garden-variety backprop net is fuzzy in this
sense, and it seems rather silly to call a net "fuzzy" solely on this
basis, although Fuzzy ART (Carpenter and Grossberg 1996) has no other
fuzzy characteristics.
o The outputs can be fuzzy. Again, any garden-variety backprop net is fuzzy
in this sense. But competitive learning nets ordinarily produce crisp
outputs, so for competitive learning methods, having fuzzy output is a
meaningful distinction. For example, fuzzy c-means clustering (Bezdek
1981) is meaningfully different from (crisp) k-means. Fuzzy ART does not
have fuzzy outputs.
o The net can be interpretable as an adaptive fuzzy system. For example,
Gaussian RBF nets and B-spline regression models (Dierckx 1995, van
Rijckevorsal 1988) are fuzzy systems with adaptive weights (Brown and
Harris 1994) and can legitimately be called neurofuzzy systems.
o The net can be a conventional NN architecture that operates on fuzzy
numbers instead of real numbers (Lippe, Feuring and Mischke 1995).
o Fuzzy constraints can provide external knowledge (Lampinen and Selonen

More information on neurofuzzy systems is available online:

o The Fuzzy Logic and Neurofuzzy Resources page of the Image, Speech and
Intelligent Systems (ISIS) research group at the University of
Southampton, Southampton, Hampshire, UK:
o The Neuro-Fuzzy Systems Research Group's web page at Tampere University
of Technology, Tampere, Finland: and
o Marcello Chiaberge's Neuro-Fuzzy page at
o The homepage of the research group on Neural Networks and Fuzzy Systems
at the Institute of Knowledge Processing and Language Engineering,
Faculty of Computer Science, University of Magdeburg, Germany, at
o Jyh-Shing Roger Jang's home page at with
information on ANFIS (Adaptive Neuro-Fuzzy Inference Systems), articles
on neuro-fuzzy systems, and more links.
o Andrew Gray's Hybrid Systems FAQ at the University of Otago at


Bezdek, J.C. (1981), Pattern Recognition with Fuzzy Objective Function
Algorithms, New York: Plenum Press.

Bezdek, J.C. & Pal, S.K., eds. (1992), Fuzzy Models for Pattern
Recognition, New York: IEEE Press.

Brown, M., and Harris, C. (1994), Neurofuzzy Adaptive Modelling and
Control, NY: Prentice Hall.

Carpenter, G.A. and Grossberg, S. (1996), "Learning, Categorization, Rule
Formation, and Prediction by Fuzzy Neural Networks," in Chen, C.H.
(1996), pp. 1.3-1.45.

Chen, C.H., ed. (1996) Fuzzy Logic and Neural Network Handbook, NY:
McGraw-Hill, ISBN 0-07-011189-8.

Dierckx, P. (1995), Curve and Surface Fitting with Splines, Oxford:
Clarendon Press.

Hecht-Nielsen, R. (1990), Neurocomputing, Reading, MA: Addison-Wesley.

Klir, G.J. and Folger, T.A.(1988), Fuzzy Sets, Uncertainty, and
Information, Englewood Cliffs, N.J.: Prentice-Hall.

Kosko, B.(1992), Neural Networks and Fuzzy Systems, Englewood Cliffs,
N.J.: Prentice-Hall.

Kosko, B. (1997), Fuzzy Engineering, NY: Prentice Hall.

Lampinen, J and Selonen, A. (1996), "Using Background Knowledge for
Regularization of Multilayer Perceptron Learning", Submitted to
International Conference on Artificial Neural Networks, ICANN'96, Bochum,

Lippe, W.-M., Feuring, Th. and Mischke, L. (1995), "Supervised learning
in fuzzy neural networks," Institutsbericht Angewandte Mathematik und
Informatik, WWU Muenster, I-12,

Nauck, D., Klawonn, F., and Kruse, R. (1997), Foundations of
Neuro-Fuzzy Systems, Chichester: Wiley, ISBN 0-471-97151-0.

van Rijckevorsal, J.L.A. (1988), "Fuzzy coding and B-splines," in van
Rijckevorsal, J.L.A., and de Leeuw, J., eds., Component and
Correspondence Analysis, Chichester: John Wiley & Sons, pp. 33-54.


Subject: Unanswered FAQs

o How many training cases do I need?
o How should I split the data into training and validation sets?
o What error functions can be used?
o How can I select important input variables?
o Should NNs be used in safety-critical applications?


Subject: Other NN links?

o Search engines
o ++++++++++++++

o Yahoo:
o Neuroscience Web Search:

o Archives of NN articles and software
o ++++++++++++++++++++++++++++++++++++

o Neuroprose ftp archive site
o --------------------------- This directory
contains technical reports as a public service to the connectionist
and neural network scientific community.

o Finnish University Network archive site
o ---------------------------------------

A large collection of neural network papers and software at Contains all the public domain
software and papers that they have been able to find. All of these
files have been transferred from FTP sites in U.S. and are mirrored
about every 3 months at fastest. Contact:

o SEL-HPC Article Archive
o -----------------------

o Machine Learning Papers
o -----------------------

o Plain-text Tables of Contents of NN journals
o ++++++++++++++++++++++++++++++++++++++++++++

Pattern Recognition Group, Department of Applied Physics,
Faculty of Applied Sciences, Delft University of Technology,

o The Collection of Computer Science Bibliographies:
o ++++++++++++++++++++++++++++++++++++++++++++++++++
Bibliographies on Neural Networks

o BibTeX data bases of NN journals
o ++++++++++++++++++++++++++++++++

The Center for Computational Intelligence maintains BibTeX data bases of
various NN journals, including IEEE Transactions on Neural Networks,
Machine Learning, Neural Computation, and NIPS, at or

o NN events server
o ++++++++++++++++

There is a WWW page for Announcements of Conferences, Workshops and Other
Events on Neural Networks at IDIAP in Switzerland. WWW-Server:

o Academic programs list
o ++++++++++++++++++++++

Rutvik Desai <> has a compilation of acedemic
programs offering interdeciplinary studies in computational neuroscience,
AI, cognitive psychology etc. at

Links to neurosci, psychology, linguistics lists are also provided.

o Neurosciences Internet Resource Guide
o +++++++++++++++++++++++++++++++++++++

This document aims to be a guide to existing, free, Internet-accessible
resources helpful to neuroscientists of all stripes. An ASCII text
version (86K) is available in the Clearinghouse of Subject-Oriented
Internet Resource Guides as follows:,

o Other WWW sites
o +++++++++++++++

In World-Wide-Web (WWW, for example via the xmosaic program) you can read
neural network information for instance by opening one of the following
uniform resource locators (URLs): Los Alamos
neural announcements and general information, (NEuroNet, King's College, London), (Eindhoven, Netherlands), (Pacific Northwest National
Laboratory, Richland, Washington, USA), (Salzburg,
Austria), (Michigan, USA), (London), Reactive Memory Search (Tabu Search) page
(Trento, Italy), (ART WWW site, Leiden, Netherlands), Helsinki University of Technology. links to neuroscience web pages Meta Directory
web page for Neurology/Neurosciences.
Many others are available too; WWW is changing all the time.


That's all folks (End of the Neural Network FAQ).

Acknowledgements: Thanks to all the people who helped to get the stuff
above into the posting. I cannot name them all, because
I would make far too many errors then. :->

No? Not good? You want individual credit?
OK, OK. I'll try to name them all. But: no guarantee....

(in alphabetical order of email adresses, I hope)

o Steve Ward <71561...@CompuServe.COM>
o Allen Bonde <>
o Accel Infotech Spore Pte Ltd <>
o Ales Krajnc <>
o Alexander Linden <>
o Matthew David Aldous <>
o S.Taimi Ames <>
o Axel Mulder <>
o Andy Gillanders <>
o Davide Anguita <ang...@ICSI.Berkeley.EDU>
o Avraam Pouliakis <>
o Kim L. Blackwell <>
o Mohammad Bahrami <>
o Paul Bakker <>
o Stefan Bergdoll <>
o Jamshed Bharucha <bhar...@casbs.Stanford.EDU>
o Carl M. Cook <>
o Yijun Cai <>
o L. Leon Campbell <>
o Cindy Hitchcock <>
o Clare G. Gallagher <>
o Craig Watson <>
o Yaron Danon <>
o David Ewing <>
o David DeMers <>
o Denni Rognvaldsson <>
o Duane Highley <>
o Dick....@Central.Sun.COM
o DJ Meyer <>
o Donald Tveter <>
o Daniel Tauritz <>
o Wlodzislaw Duch <>
o E. Robert Tisdale <>
o Athanasios Episcopos <>
o Frank Schnorrenberg <>
o Gary Lawrence Murphy <>
o Lee Giles <>
o Glen Clark <opto!>
o Phil Goodman <>
o Horace A. Vallas, Jr. <>
o Joerg Heitkoetter <>
o Ralf Hohenstein <>
o Ian Cresswell <>
o Gamze Erten <>
o Ed Rosenfeld <>
o Franco Insana <>
o Janne Sinkkonen <>
o Javier Blasco-Alberto <>
o Jean-Denis Muller <>
o Jeff Harpster <uu0979!>
o Jonathan Kamens <j...@MIT.Edu>
o J.J. Merelo <>
o Dr. Jacek Zurada <>
o Jon Gunnar Solheim <>
o Josef Nelissen <>
o Joey Rogers <>
o Subhash Kak <>
o Ken Karnofsky <>
o Luke Koops <>
o Kurt Hornik <>
o Thomas Lindblad <>
o Clark Lindsey <>
o Lloyd Lubet <>
o William Mackeown <>
o Maria Dolores Soriano Lopez <>
o Mark Plumbley <>
o Peter Marvit <>
o Miguel A. Carreira-Perpinan<>
o Yoshiro Miyata <>
o Madhav Moganti <>
o Jyrki Alakuijala <>
o Jean-Denis Muller <>
o Michael Reiss <>
o Maciek Sitnik <>
o R. Steven Rainwater <>
o Nigel Dodd <>
o Barry Dunmall <>
o Paolo Ienne <>
o Paul Keller <>
o Peter Hamer <>
o Pierre v.d. Laar <>
o Michael Plonski <>
o Lutz Prechelt <> [creator of FAQ]
o Richard Andrew Miles Outerbridge <>
o Rand Dixon <>
o Robin L. Getz <>
o Richard Cornelius <>
o Rob Cunningham <>
o Randall C. O'Reilly <>
o Rutvik Desai <>
o Robert W. Means <>
o Stefan Vogt <>
o Osamu Saito <>
o Scott Fahlman <>
o <>
o Sheryl Cormicle <>
o Ted Stockwell <>
o Stephanie Warrick <>
o Serge Waterschoot <>
o Thomas G. Dietterich <>
o Ulrich Wendl <>
o M. Verleysen <>
o Sherif Hashem <>
o Matthew P Wiener <>
o Wesley Elsberry <>
o Dr. Steve G. Romaniuk <>

Special thanks to Gregory E. Heath <> and Will Dwinnell
<> for years of stimulating and educational discussions

The FAQ was created in June/July 1991 by Lutz Prechelt; he also maintained
the FAQ until November 1995. Warren Sarle maintains the FAQ since December


Warren & Lutz

Previous part is part 6.

Neural network FAQ / Warren S. Sarle,


Warren S. Sarle SAS Institute Inc. The opinions expressed here SAS Campus Drive are mine and not necessarily
(919) 677-8000 Cary, NC 27513, USA those of SAS Institute.

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