Archive-name: ai-faq/neural-nets/part1
Last-modified: 2000-12-18
URL: ftp://ftp.sas.com/pub/neural/FAQ.html
Maintainer: saswss@unx.sas.com (Warren S. Sarle)
Neural Network FAQ, part 1 of 7: Introduction Copyright 1997, 1998, 1999, 2000 by Warren S. Sarle, Cary, NC, USA.

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    Additions, corrections, or improvements are always welcome.
    Anybody who is willing to contribute any information,
    please email me; if it is relevant, I will incorporate it.

    The monthly posting departs at the 28th of every month.
  ---------------------------------------------------------------
This is the first of seven parts of a monthly posting to the Usenet newsgroup comp.ai.neural-nets (as well as comp.answers and news.answers, where it should be findable at any time). Its purpose is to provide basic information for individuals who are new to the field of neural networks or who are just beginning to read this group. It will help to avoid lengthy discussion of questions that often arise for beginners.
   SO, PLEASE, SEARCH THIS POSTING FIRST IF YOU HAVE A QUESTION
                           and
   DON'T POST ANSWERS TO FAQs: POINT THE ASKER TO THIS POSTING
The latest version of the FAQ is available as a hypertext document, readable by any WWW (World Wide Web) browser such as Netscape, under the URL: ftp://ftp.sas.com/pub/neural/FAQ.html.

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All seven parts of the FAQ can be downloaded from either of the following URLS:

    ftp://ftp.sas.com/pub/neural/FAQ.html.zip
    ftp://ftp.sas.com/pub/neural/FAQ.txt.zip
These postings are archived in the periodic posting archive on host rtfm.mit.edu (and on some other hosts as well). Look in the anonymous ftp directory "/pub/usenet/news.answers/ai-faq/neural-nets" under the file names "part1", "part2", ... "part7". If you do not have anonymous ftp access, you can access the archives by mail server as well. Send an E-mail message to mail-server@rtfm.mit.edu with "help" and "index" in the body on separate lines for more information. You can also go to http://www.deja.com/ and look for posts containing "Neural Network FAQ" in comp.ai.neural-nets.

For those of you who read this FAQ anywhere other than in Usenet: To read comp.ai.neural-nets (or post articles to it) you need Usenet News access. Try the commands, 'xrn', 'rn', 'nn', or 'trn' on your Unix machine, 'news' on your VMS machine, or ask a local guru. WWW browsers are often set up for Usenet access, too--try the URL news:comp.ai.neural-nets.

The FAQ posting departs to comp.ai.neural-nets on the 28th of every month. It is also sent to the groups comp.answers and news.answers where it should be available at any time (ask your news manager). The FAQ posting, like any other posting, may a take a few days to find its way over Usenet to your site. Such delays are especially common outside of North America.

All changes to the FAQ from the previous month are shown in another monthly posting having the subject `changes to "comp.ai.neural-nets FAQ" -- monthly posting', which immediately follows the FAQ posting. The `changes' post contains the full text of all changes and can also be found at ftp://ftp.sas.com/pub/neural/changes.txt . There is also a weekly post with the subject "comp.ai.neural-nets FAQ: weekly reminder" that briefly describes any major changes to the FAQ.

This FAQ is not meant to discuss any topic exhaustively. It is neither a tutorial nor a textbook, but should be viewed as a supplement to the many excellent books and online resources described in Part 4: Books, data, etc..

Disclaimer:

    This posting is provided 'as is'. No warranty whatsoever is expressed or implied, in particular, no warranty that the information contained herein is correct or useful in any way, although both are intended.
To find the answer of question "x", search for the string "Subject: x"

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

Part 1: Introduction
    What is this newsgroup for? How shall it be used?
    Where is comp.ai.neural-nets archived?
    What if my question is not answered in the FAQ?
    May I copy this FAQ?
    What is a neural network (NN)?
    Where can I find a simple introduction to NNs?
    What can you do with an NN and what not?
    Who is concerned with NNs?
    How many kinds of NNs exist?
    How many kinds of Kohonen networks exist? (And what is k-means?)
      VQ: Vector Quantization and k-means
      SOM: Self-Organizing Map
      LVQ: Learning Vector Quantization
      Other Kohonen networks and references
    How are layers counted?
    What are cases and variables?
    What are the population, sample, training set, design set, validation set, and test set?
    How are NNs related to statistical methods?
Part 2: Learning
    What are combination, activation, error, and objective functions?
    What are batch, incremental, on-line, off-line, deterministic, stochastic, adaptive, instantaneous, pattern, epoch, constructive, and sequential learning?
    What is backprop?
    What learning rate should be used for backprop?
    What are conjugate gradients, Levenberg-Marquardt, etc.?
    How does ill-conditioning affect NN training?
    How should categories be encoded?
    Why use a bias/threshold?
    Why use activation functions?
    How to avoid overflow in the logistic function?
    What is a softmax activation function?
    What is the curse of dimensionality?
    How do MLPs compare with RBFs?
    What are OLS and subset/stepwise regression?
    Should I normalize/standardize/rescale the data?
    Should I nonlinearly transform the data?
    How to measure importance of inputs?
    What is ART?
    What is PNN?
    What is GRNN?
    What does unsupervised learning learn?
    Help! My NN won't learn! What should I do?
Part 3: Generalization
    How is generalization possible?
    How does noise affect generalization?
    What is overfitting and how can I avoid it?
    What is jitter? (Training with noise)
    What is early stopping?
    What is weight decay?
    What is Bayesian learning?
    How to combine networks?
    How many hidden layers should I use?
    How many hidden units should I use?
    How can generalization error be estimated?
    What are cross-validation and bootstrapping?
    How to compute prediction and confidence intervals (error bars)?
Part 4: Books, data, etc.
    Books and articles about Neural Networks?
    Journals and magazines about Neural Networks?
    Conferences and Workshops on Neural Networks?
    Neural Network Associations?
    On-line and machine-readable information about NNs?
    How to benchmark learning methods?
    Databases for experimentation with NNs?
Part 5: Free software
    Source code on the web?
    Freeware and shareware packages for NN simulation?
Part 6: Commercial software
    Commercial software packages for NN simulation?
Part 7: Hardware and miscellaneous
    Neural Network hardware?
    What are some applications of NNs?
      General
      Agriculture
      Chemistry
      Finance and economics
      Games and gambling
      Industry
      Materials science
      Medicine
      Music
      Robotics
      Weather forecasting
      Weird
    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, etc.?
    How to recognize handwritten characters?
    What about Genetic Algorithms and Evolutionary Computation?
    What about Fuzzy Logic?
    Unanswered FAQs
    Other NN links?
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Subject: What is this newsgroup for? How shall it be used?

The newsgroup comp.ai.neural-nets is intended as a forum for people who want to use or explore the capabilities of Artificial Neural Networks or Neural-Network-like structures.

Posts should be in plain-text format, not postscript, html, rtf, TEX, MIME, or any word-processor format.

Do not use vcards or other excessively long signatures.

Please do not post homework or take-home exam questions. Please do not post a long source-code listing and ask readers to debug it. And note that chain letters and other get-rich-quick pyramid schemes are illegal in the USA; for example, see http://www.usps.gov/websites/depart/inspect/chainlet.htm

There should be the following types of articles in this newsgroup:

  1. Requests

    Requests are articles of the form "I am looking for X", where X is something public like a book, an article, a piece of software. The most important about such a request is to be as specific as possible!

    If multiple different answers can be expected, the person making the request should prepare to make a summary of the answers he/she got and announce to do so with a phrase like "Please reply by email, I'll summarize to the group" at the end of the posting.

    The Subject line of the posting should then be something like "Request: X"

  2. Questions

    As opposed to requests, questions ask for a larger piece of information or a more or less detailed explanation of something. To avoid lots of redundant traffic it is important that the poster provides with the question all information s/he already has about the subject asked and state the actual question as precise and narrow as possible. The poster should prepare to make a summary of the answers s/he got and announce to do so with a phrase like "Please reply by email, I'll summarize to the group" at the end of the posting.

    The Subject line of the posting should be something like "Question: this-and-that" or have the form of a question (i.e., end with a question mark)

    Students: please do not ask comp.ai.neural-net readers to do your homework or take-home exams for you.

  3. Answers

    These are reactions to questions or requests. If an answer is too specific to be of general interest, or if a summary was announced with the question or request, the answer should be e-mailed to the poster, not posted to the newsgroup.

    Most news-reader software automatically provides a subject line beginning with "Re:" followed by the subject of the article which is being followed-up. Note that sometimes longer threads of discussion evolve from an answer to a question or request. In this case posters should change the subject line suitably as soon as the topic goes too far away from the one announced in the original subject line. You can still carry along the old subject in parentheses in the form "Re: new subject (was: old subject)"

  4. Summaries

    In all cases of requests or questions the answers for which can be assumed to be of some general interest, the poster of the request or question shall summarize the answers he/she received. Such a summary should be announced in the original posting of the question or request with a phrase like "Please answer by email, I'll summarize"

    In such a case, people who answer to a question should NOT post their answer to the newsgroup but instead mail them to the poster of the question who collects and reviews them. After about 5 to 20 days after the original posting, its poster should make the summary of answers and post it to the newsgroup.

    Some care should be invested into a summary:

    Note that a good summary is pure gold for the rest of the newsgroup community, so summary work will be most appreciated by all of us. Good summaries are more valuable than any moderator ! :-)

  5. Announcements

    Some articles never need any public reaction. These are called announcements (for instance for a workshop, conference or the availability of some technical report or software system).

    Announcements should be clearly indicated to be such by giving them a subject line of the form "Announcement: this-and-that"

  6. Reports

    Sometimes people spontaneously want to report something to the newsgroup. This might be special experiences with some software, results of own experiments or conceptual work, or especially interesting information from somewhere else.

    Reports should be clearly indicated to be such by giving them a subject line of the form "Report: this-and-that"

  7. Discussions

    An especially valuable possibility of Usenet is of course that of discussing a certain topic with hundreds of potential participants. All traffic in the newsgroup that can not be subsumed under one of the above categories should belong to a discussion.

    If somebody explicitly wants to start a discussion, he/she can do so by giving the posting a subject line of the form "Discussion: this-and-that"

    It is quite difficult to keep a discussion from drifting into chaos, but, unfortunately, as many many other newsgroups show there seems to be no secure way to avoid this. On the other hand, comp.ai.neural-nets has not had many problems with this effect in the past, so let's just go and hope...

  8. Jobs Ads

    Advertisements for jobs requiring expertise in artificial neural networks are appropriate in comp.ai.neural-nets. Job ads should be clearly indicated to be such by giving them a subject line of the form "Job: this-and-that". It is also useful to include the country-state-city abbreviations that are conventional in misc.jobs.offered, such as: "Job: US-NY-NYC Neural network engineer". If an employer has more than one job opening, all such openings should be listed in a single post, not multiple posts. Job ads should not be reposted more than once per month.

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Subject: Where is comp.ai.neural-nets archived?

The following archives are available for comp.ai.neural-nets: For more information on newsgroup archives, see http://starbase.neosoft.com/~claird/news.lists/newsgroup_archives.html
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Subject: What if my question is not answered in the FAQ?

If your question is not answered in the FAQ, you can try a web search. The following search engines are especially useful:
http://www.google.com/
http://search.yahoo.com/
http://www.altavista.com/
http://www.deja.com/

Another excellent web site on NNs is Donald Tveter's Backpropagator's Review at http://www.dontveter.com/bpr/bpr.html or http://gannoo.uce.ac.uk/bpr/bpr.html.

For feedforward NNs, the best reference book is:

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

If the answer isn't in Bishop, then for more theoretical questions try:

Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge: Cambridge University Press.

For more practical questions about MLP training, try:

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

Reed, R.D., and Marks, R.J, II (1999), Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks, Cambridge, MA: The MIT Press.

There are many more excellent books and web sites listed in the Neural Network FAQ, Part 4: Books, data, etc.

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Subject: May I copy this FAQ?

The intent in providing a FAQ is to make the information freely available to whoever needs it. You may copy all or part of the FAQ, but please be sure to include a reference to the URL of the master copy, ftp://ftp.sas.com/pub/neural/FAQ.html, and do not sell copies of the FAQ. If you want to include information from the FAQ in your own web site, it is better to include links to the master copy rather than to copy text from the FAQ to your web pages, because various answers in the FAQ are updated at unpredictable times. To cite the FAQ in an academic-style bibliography, use something along the lines of:

Sarle, W.S., ed. (1997), Neural Network FAQ, part 1 of 7: Introduction, periodic posting to the Usenet newsgroup comp.ai.neural-nets, URL: ftp://ftp.sas.com/pub/neural/FAQ.html

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Subject: What is a neural network (NN)?

First of all, when we are talking about a neural network, we should more properly say "artificial neural network" (ANN), because that is what we mean most of the time in comp.ai.neural-nets. Biological neural networks are much more complicated than the mathematical models we use for ANNs. But it is customary to be lazy and drop the "A" or the "artificial".

There is no universally accepted definition of an NN. But perhaps most people in the field would agree that an NN is a network of many simple processors ("units"), each possibly having a small amount of local memory. The units are connected by communication channels ("connections") which usually carry numeric (as opposed to symbolic) data, encoded by any of various means. The units operate only on their local data and on the inputs they receive via the connections. The restriction to local operations is often relaxed during training.

Some NNs are models of biological neural networks and some are not, but historically, much of the inspiration for the field of NNs came from the desire to produce artificial systems capable of sophisticated, perhaps "intelligent", computations similar to those that the human brain routinely performs, and thereby possibly to enhance our understanding of the human brain.

Most NNs have some sort of "training" rule whereby the weights of connections are adjusted on the basis of data. In other words, NNs "learn" from examples (as children learn to recognize dogs from examples of dogs) and exhibit some capability for generalization beyond the training data.

NNs normally have great potential for parallelism, since the computations of the components are largely independent of each other. Some people regard massive parallelism and high connectivity to be defining characteristics of NNs, but such requirements rule out various simple models, such as simple linear regression (a minimal feedforward net with only two units plus bias), which are usefully regarded as special cases of NNs.

Here is a sampling of definitions from the books on the FAQ maintainer's shelf. None will please everyone. Perhaps for that reason many NN textbooks do not explicitly define neural networks.

According to the DARPA Neural Network Study (1988, AFCEA International Press, p. 60):

... a neural network is a system composed of many simple processing elements operating in parallel whose function is determined by network structure, connection strengths, and the processing performed at computing elements or nodes.
According to Haykin, S. (1994), Neural Networks: A Comprehensive Foundation, NY: Macmillan, p. 2:
A neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use. It resembles the brain in two respects:
  1. Knowledge is acquired by the network through a learning process.
  2. Interneuron connection strengths known as synaptic weights are used to store the knowledge.
According to Nigrin, A. (1993), Neural Networks for Pattern Recognition, Cambridge, MA: The MIT Press, p. 11:
A neural network is a circuit composed of a very large number of simple processing elements that are neurally based. Each element operates only on local information. Furthermore each element operates asynchronously; thus there is no overall system clock.
According to Zurada, J.M. (1992), Introduction To Artificial Neural Systems, Boston: PWS Publishing Company, p. xv:
Artificial neural systems, or neural networks, are physical cellular systems which can acquire, store, and utilize experiential knowledge.
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Subject: Where can I find a simple introduction to NNs?

Several excellent introductory books on NNs are listed in part 4 of the FAQ under "Books and articles about Neural Networks?" If you want a book with minimal math, see "The best introductory book for business executives."

Dr. Leslie Smith has a brief on-line introduction to NNs with examples and diagrams at http://www.cs.stir.ac.uk/~lss/NNIntro/InvSlides.html.

For a more detailed introduction, see Donald Tveter's excellent Backpropagator's Review at http://www.dontveter.com/bpr/bpr.html or http://gannoo.uce.ac.uk/bpr/bpr.html, which contains both answers to additional FAQs and an annotated neural net bibliography emphasizing on-line articles.

Kevin Gurney has on-line a preliminary draft of his book, An Introduction to Neural Networks, at http://www.shef.ac.uk/psychology/gurney/notes/index.html The book is now in print and is one of the better general-purpose introductory textbooks on NNs. Here is the table of contents from the on-line version:

  1. Computers and Symbols versus Nets and Neurons
  2. TLUs and vectors - simple learning rules
  3. The delta rule
  4. Multilayer nets and backpropagation
  5. Associative memories - the Hopfield net
  6. Hopfield nets (contd.)
  7. Kohonen nets
  8. Alternative node types
  9. Cubic nodes (contd.) and Reward Penalty training
  10. Drawing things together - some perspectives

Another on-line book by Ben Kröse and Patrick van der Smagt, also called An Introduction to Neural Networks, can be found at ftp://ftp.wins.uva.nl/pub/computer-systems/aut-sys/reports/neuro-intro/neuro-intro.ps.gz or http://www.robotic.dlr.de/Smagt/books/neuro-intro.ps.gz. Here is the table of contents:

  1. Introduction
  2. Fundamantals
  3. Perceptron and Adaline
  4. Back-Propagation
  5. Recurrent Networks
  6. Self-Organising Networks
  7. Reinforcement Learning
  8. Robot Control
  9. Vision
  10. General Purpose Hardware
  11. Dedicated Neuro-Hardware

StatSoft Inc. has an on-line Electronic Statistics Textbook, at http://www.statsoft.com/textbook/stathome.html that includes a chapter on neural nets as well as many statistical topics relevant to neural nets.

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Subject: What can you do with an NN and what not?

In principle, NNs can compute any computable function, i.e., they can do everything a normal digital computer can do (Valiant, 1988; Siegelmann and Sontag, 1999), or perhaps even more, under some assumptions of doubtful practicality (see Siegelmann, 1998, but also Hadley, 1999).

In practice, NNs are especially useful for classification and function approximation/mapping problems which are tolerant of some imprecision, which have lots of training data available, but to which hard and fast rules (such as those that might be used in an expert system) cannot easily be applied. Almost any finite-dimensional vector function on a compact set can be approximated to arbitrary precision by feedforward NNs (which are the type most often used in practical applications) if you have enough data and enough computing resources.

To be somewhat more precise, feedforward networks with a single hidden layer and trained by least-squares are statistically consistent estimators of arbitrary square-integrable regression functions under certain practically-satisfiable assumptions regarding sampling, target noise, number of hidden units, size of weights, and form of hidden-unit activation function (White, 1990). Such networks can also be trained as statistically consistent estimators of derivatives of regression functions (White and Gallant, 1992) and quantiles of the conditional noise distribution (White, 1992a). Feedforward networks with a single hidden layer using threshold or sigmoid activation functions are universally consistent estimators of binary classifications (Farag\'o and Lugosi, 1993; Lugosi and Zeger 1995; Devroye, Gy\"orfi, and Lugosi, 1996) under similar assumptions. Note that these results are stronger than the universal approximation theorems that merely show the existence of weights for arbitrarily accurate approximations, without demonstrating that such weights can be obtained by learning.

Unfortunately, the above consistency results depend on one impractical assumption: that the networks are trained by an error (L_p error or misclassification rate) minimization technique that comes arbitrarily close to the global minimum. Such minimization is computationally intractable except in small or simple problems (Blum and Rivest, 1989; Judd, 1990). In practice, however, you can usually get good results without doing a full-blown global optimization; e.g., using multiple (say, 10 to 1000) random weight initializations is usually sufficient.

One example of a function that a typical neural net cannot learn is Y=1/X on the open interval (0,1). An open interval is not a compact set. With any bounded output activation function, the error will get arbitrarily large as the input approaches zero. Of course, you could make the output activation function a reciprocal function and easily get a perfect fit, but neural networks are most often used in situations where you do not have enough prior knowledge to set the activation function in such a clever way. There are also many other important problems that are so difficult that a neural network will be unable to learn them without memorizing the entire training set, such as:

And it is important to understand that there are no methods for training NNs that can magically create information that is not contained in the training data.

Feedforward NNs are restricted to finite-dimensional input and output spaces. Recurrent NNs can in theory process arbitrarily long strings of numbers or symbols. But training recurrent NNs has posed much more serious practical difficulties than training feedforward networks. NNs are, at least today, difficult to apply successfully to problems that concern manipulation of symbols and rules, but much research is being done.

There have been attempts to pack recursive structures into finite-dimensional real vectors (Blair, 1997; Pollack, 1990; Chalmers, 1990; Chrisman, 1991; Plate, 1994; Hammerton, 1998). Obviously, finite precision limits how far the recursion can go (Hadley, 1999). The practicality of such methods is open to debate.

As for simulating human consciousness and emotion, that's still in the realm of science fiction. Consciousness is still one of the world's great mysteries. Artificial NNs may be useful for modeling some aspects of or prerequisites for consciousness, such as perception and cognition, but ANNs provide no insight so far into what Chalmers (1996, p. xi) calls the "hard problem":

Many books and articles on consciousness have appeared in the past few years, and one might think we are making progress. But on a closer look, most of this work leaves the hardest problems about consciousness untouched. Often, such work addresses what might be called the "easy problems" of consciousness: How does the brain process environmental stimulation? How does it integrate information? How do we produce reports on internal states? These are important questions, but to answer them is not to solve the hard problem: Why is all this processing accompanied by an experienced inner life?
For more information on consciousness, see the on-line journal Psyche at http://psyche.cs.monash.edu.au/index.html.

For examples of specific applications of NNs, see What are some applications of NNs?

References:

Blair, A.D. (1997), "Scaling Up RAAMs," Brandeis University Computer Science Technical Report CS-97-192, http://www.demo.cs.brandeis.edu/papers/long.html#sur97

Blum, A., and Rivest, R.L. (1989), "Training a 3-node neural network is NP-complete," in Touretzky, D.S. (ed.), Advances in Neural Information Processing Systems 1, San Mateo, CA: Morgan Kaufmann, 494-501.

Chalmers, D.J. (1990), "Syntactic Transformations on Distributed Representations," Connection Science, 2, 53-62, http://ling.ucsc.edu/~chalmers/papers/transformations.ps

Chalmers, D.J. (1996), The Conscious Mind: In Search of a Fundamental Theory, NY: Oxford University Press.

Chrisman, L. (1991), "Learning Recursive Distributed Representations for Holistic Computation", Connection Science, 3, 345-366, ftp://reports.adm.cs.cmu.edu/usr/anon/1991/CMU-CS-91-154.ps

Collier, R. (1994), "An historical overview of natural language processing systems that learn," Artificial Intelligence Review, 8(1), ??-??.

Devroye, L., Gy\"orfi, L., and Lugosi, G. (1996), A Probabilistic Theory of Pattern Recognition, NY: Springer.

Farag\'o, A. and Lugosi, G. (1993), "Strong Universal Consistency of Neural Network Classifiers," IEEE Transactions on Information Theory, 39, 1146-1151.

Hadley, R.F. (1999), "Cognition and the computational power of connectionist networks," http://www.cs.sfu.ca/~hadley/online.html

Hammerton, J.A. (1998), "Holistic Computation: Reconstructing a muddled concept," Connection Science, 10, 3-19, http://www.tardis.ed.ac.uk/~james/CNLP/holcomp.ps.gz

Judd, J.S. (1990), Neural Network Design and the Complexity of Learning, Cambridge, MA: The MIT Press.

Lugosi, G., and Zeger, K. (1995), "Nonparametric Estimation via Empirical Risk Minimization," IEEE Transactions on Information Theory, 41, 677-678.

Plate, T.A. (1994), Distributed Representations and Nested Compositional Structure, Ph.D. Thesis, University of Toronto, ftp://ftp.cs.utoronto.ca/pub/tap/

Pollack, J. B. (1990), "Recursive Distributed Representations," Artificial Intelligence 46, 1, 77-105, http://www.demo.cs.brandeis.edu/papers/long.html#raam

Siegelmann, H.T. (1998), Neural Networks and Analog Computation: Beyond the Turing Limit, Boston: Birkhauser, ISBN 0-8176-3949-7, http://iew3.technion.ac.il:8080/Home/Users/iehava/book/

Siegelmann, H.T., and Sontag, E.D. (1999), "Turing Computability with Neural Networks," Applied Mathematics Letters, 4, 77-80.

Valiant, L. (1988), "Functionality in Neural Nets," Learning and Knowledge Acquisition, Proc. AAAI, 629-634.

White, H. (1990), "Connectionist Nonparametric Regression: Multilayer Feedforward Networks Can Learn Arbitrary Mappings," Neural Networks, 3, 535-550. Reprinted in White (1992b).

White, H. (1992a), "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. Reprinted in White (1992b).

White, H. (1992b), Artificial Neural Networks: Approximation and Learning Theory, Blackwell.

White, H., and Gallant, A.R. (1992), "On Learning the Derivatives of an Unknown Mapping with Multilayer Feedforward Networks," Neural Networks, 5, 129-138. Reprinted in White (1992b).

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Subject: Who is concerned with NNs?

Neural Networks are interesting for quite a lot of very different people: For world-wide lists of groups doing research on NNs, see the Foundation for Neural Networks's (SNN) page at http://www.mbfys.kun.nl/snn/pointers/groups.html and see Neural Networks Research on the IEEE Neural Network Council's homepage http://www.ieee.org/nnc.
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Subject: How many kinds of NNs exist?

There are many many kinds of NNs by now. Nobody knows exactly how many. New ones (or at least variations of old ones) are invented every week. Below is a collection of some of the most well known methods, not claiming to be complete.

The two main kinds of learning algorithms are supervised and unsupervised.

The distinction between supervised and unsupervised methods is not always clear-cut. An unsupervised method can learn a summary of a probability distribution, then that summarized distribution can be used to make predictions. Furthermore, supervised methods come in two subvarieties: auto-associative and hetero-associative. In auto-associative learning, the target values are the same as the inputs, whereas in hetero-associative learning, the targets are generally different from the inputs. Many unsupervised methods are equivalent to auto-associative supervised methods. For more details, see "What does unsupervised learning learn?"

Two major kinds of network topology are feedforward and feedback.

Some kinds of NNs (such as those with winner-take-all units) can be implemented as either feedforward or feedback networks.

NNs also differ in the kinds of data they accept. Two major kinds of data are categorical and quantitative.

Some variables can be treated as either categorical or quantitative, such as number of children or any binary variable. Most regression algorithms can also be used for supervised classification by encoding categorical target values as 0/1 binary variables and using those binary variables as target values for the regression algorithm. The outputs of the network are posterior probabilities when any of the most common training methos are used.

Here are some well-known kinds of NNs:

  1. Supervised

    1. Feedforward

      • Linear
        • Hebbian - Hebb (1949), Fausett (1994)
        • Perceptron - Rosenblatt (1958), Minsky and Papert (1969/1988), Fausett (1994)
        • Adaline - Widrow and Hoff (1960), Fausett (1994)
        • Higher Order - Bishop (1995)
        • Functional Link - Pao (1989)
      • MLP: Multilayer perceptron - Bishop (1995), Reed and Marks (1999), Fausett (1994)
        • Backprop - Rumelhart, Hinton, and Williams (1986)
        • Cascade Correlation - Fahlman and Lebiere (1990), Fausett (1994)
        • Quickprop - Fahlman (1989)
        • RPROP - Riedmiller and Braun (1993)
      • RBF networks - Bishop (1995), Moody and Darken (1989), Orr (1996)
        • OLS: Orthogonal Least Squares - Chen, Cowan and Grant (1991)
      • CMAC: Cerebellar Model Articulation Controller - Albus (1975), Brown and Harris (1994)
      • Classification only
        • LVQ: Learning Vector Quantization - Kohonen (1988), Fausett (1994)
        • PNN: Probabilistic Neural Network - Specht (1990), Masters (1993), Hand (1982), Fausett (1994)
      • Regression only
        • GNN: General Regression Neural Network - Specht (1991), Nadaraya (1964), Watson (1964)

    2. Feedback - Hertz, Krogh, and Palmer (1991), Medsker and Jain (2000)

      • BAM: Bidirectional Associative Memory - Kosko (1992), Fausett (1994)
      • Boltzman Machine - Ackley et al. (1985), Fausett (1994)
      • Recurrent time series
        • Backpropagation through time - Werbos (1990)
        • Elman - Elman (1990)
        • FIR: Finite Impulse Response - Wan (1990)
        • Jordan - Jordan (1986)
        • Real-time recurrent network - Williams and Zipser (1989)
        • Recurrent backpropagation - Pineda (1989), Fausett (1994)
        • TDNN: Time Delay NN - Lang, Waibel and Hinton (1990)

    3. Competitive

      • ARTMAP - Carpenter, Grossberg and Reynolds (1991)
      • Fuzzy ARTMAP - Carpenter, Grossberg, Markuzon, Reynolds and Rosen (1992), Kasuba (1993)
      • Gaussian ARTMAP - Williamson (1995)
      • Counterpropagation - Hecht-Nielsen (1987; 1988; 1990), Fausett (1994)
      • Neocognitron - Fukushima, Miyake, and Ito (1983), Fukushima, (1988), Fausett (1994)

  2. Unsupervised - Hertz, Krogh, and Palmer (1991)

    1. Competitive

      • Vector Quantization
        • Grossberg - Grossberg (1976)
        • Kohonen - Kohonen (1984)
        • Conscience - Desieno (1988)
      • Self-Organizing Map
        • Kohonen - Kohonen (1995), Fausett (1994)
        • GTM: - Bishop, Svens\'en and Williams (1997)
        • Local Linear - Mulier and Cherkassky (1995)
      • Adaptive resonance theory
        • ART 1 - Carpenter and Grossberg (1987a), Moore (1988), Fausett (1994)
        • ART 2 - Carpenter and Grossberg (1987b), Fausett (1994)
        • ART 2-A - Carpenter, Grossberg and Rosen (1991a)
        • ART 3 - Carpenter and Grossberg (1990)
        • Fuzzy ART - Carpenter, Grossberg and Rosen (1991b)
      • DCL: Differential Competitive Learning - Kosko (1992)

    2. Dimension Reduction - Diamantaras and Kung (1996)

      • Hebbian - Hebb (1949), Fausett (1994)
      • Oja - Oja (1989)
      • Sanger - Sanger (1989)
      • Differential Hebbian - Kosko (1992)

    3. Autoassociation
      • Linear autoassociator - Anderson et al. (1977), Fausett (1994)
      • BSB: Brain State in a Box - Anderson et al. (1977), Fausett (1994)
      • Hopfield - Hopfield (1982), Fausett (1994)

  3. Nonlearning

    1. Hopfield - Hertz, Krogh, and Palmer (1991)
    2. various networks for optimization - Cichocki and Unbehauen (1993)
References:

Ackley, D.H., Hinton, G.F., and Sejnowski, T.J. (1985), "A learning algorithm for Boltzman machines," Cognitive Science, 9, 147-169.

Albus, J.S (1975), "New Approach to Manipulator Control: The Cerebellar Model Articulation Controller (CMAC)," Transactions of the ASME Journal of Dynamic Systems, Measurement, and Control, September 1975, 220-27.

Anderson, J.A., and Rosenfeld, E., eds. (1988), Neurocomputing: Foundatons of Research, Cambridge, MA: The MIT Press.

Anderson, J.A., Silverstein, J.W., Ritz, S.A., and Jones, R.S. (1977) "Distinctive features, categorical perception, and probability learning: Some applications of a neural model," Psychological Rveiew, 84, 413-451. Reprinted in Anderson and Rosenfeld (1988).

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

Bishop, C.M., Svens\'en, M., and Williams, C.K.I (1997), "GTM: A principled alternative to the self-organizing map," in Mozer, M.C., Jordan, M.I., and Petsche, T., (eds.) Advances in Neural Information Processing Systems 9, Cambrideg, MA: The MIT Press, pp. 354-360. Also see http://www.ncrg.aston.ac.uk/GTM/

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

Carpenter, G.A., Grossberg, S. (1987a), "A massively parallel architecture for a self-organizing neural pattern recognition machine," Computer Vision, Graphics, and Image Processing, 37, 54-115.

Carpenter, G.A., Grossberg, S. (1987b), "ART 2: Self-organization of stable category recognition codes for analog input patterns," Applied Optics, 26, 4919-4930.

Carpenter, G.A., Grossberg, S. (1990), "ART 3: Hierarchical search using chemical transmitters in self-organizing pattern recognition architectures. Neural Networks, 3, 129-152.

Carpenter, G.A., Grossberg, S., Markuzon, N., Reynolds, J.H., and Rosen, D.B. (1992), "Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps," IEEE Transactions on Neural Networks, 3, 698-713

Carpenter, G.A., Grossberg, S., Reynolds, J.H. (1991), "ARTMAP: Supervised real-time learning and classification of nonstationary data by a self-organizing neural network," Neural Networks, 4, 565-588.

Carpenter, G.A., Grossberg, S., Rosen, D.B. (1991a), "ART 2-A: An adaptive resonance algorithm for rapid category learning and recognition," Neural Networks, 4, 493-504.

Carpenter, G.A., Grossberg, S., Rosen, D.B. (1991b), "Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system," Neural Networks, 4, 759-771.

Chen, S., Cowan, C.F.N., and Grant, P.M. (1991), "Orthogonal least squares learning for radial basis function networks," IEEE Transactions on Neural Networks, 2, 302-309.

Cichocki, A. and Unbehauen, R. (1993). Neural Networks for Optimization and Signal Processing. NY: John Wiley & Sons, ISBN 0-471-93010-5.

Desieno, D. (1988), "Adding a conscience to competitive learning," Proc. Int. Conf. on Neural Networks, I, 117-124, IEEE Press.

Diamantaras, K.I., and Kung, S.Y. (1996) Principal Component Neural Networks: Theory and Applications, NY: Wiley.

Elman, J.L. (1990), "Finding structure in time," Cognitive Science, 14, 179-211.

Fahlman, S.E. (1989), "Faster-Learning Variations on Back-Propagation: An Empirical Study", in Touretzky, D., Hinton, G, and Sejnowski, T., eds., Proceedings of the 1988 Connectionist Models Summer School, Morgan Kaufmann, 38-51.

Fahlman, S.E., and Lebiere, C. (1990), "The Cascade-Correlation Learning Architecture", in Touretzky, D. S. (ed.), Advances in Neural Information Processing Systems 2,, Los Altos, CA: Morgan Kaufmann Publishers, pp. 524-532.

Fausett, L. (1994), Fundamentals of Neural Networks, Englewood Cliffs, NJ: Prentice Hall.

Fukushima, K., Miyake, S., and Ito, T. (1983), "Neocognitron: A neural network model for a mechanism of visual pattern recognition," IEEE Transactions on Systems, Man, and Cybernetics, 13, 826-834.

Fukushima, K. (1988), "Neocognitron: A hierarchical neural network capable of visual pattern recognition," Neural Networks, 1, 119-130.

Grossberg, S. (1976), "Adaptive pattern classification and universal recoding: I. Parallel development and coding of neural feature detectors," Biological Cybernetics, 23, 121-134

Hand, D.J. (1982) Kernel Discriminant Analysis, Research Studies Press.

Hebb, D.O. (1949), The Organization of Behavior, NY: John Wiley & Sons.

Hecht-Nielsen, R. (1987), "Counterpropagation networks," Applied Optics, 26, 4979-4984.

Hecht-Nielsen, R. (1988), "Applications of counterpropagation networks," Neural Networks, 1, 131-139.

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

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

Hopfield, J.J. (1982), "Neural networks and physical systems with emergent collective computational abilities," Proceedings of the National Academy of Sciences, 79, 2554-2558. Reprinted in Anderson and Rosenfeld (1988).

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 Erlbaum.

Kasuba, T. (1993), "Simplified Fuzzy ARTMAP," AI Expert, 8, 18-25.

Kohonen, T. (1984), Self-Organization and Associative Memory, Berlin: Springer.

Kohonen, T. (1988), "Learning Vector Quantization," Neural Networks, 1 (suppl 1), 303.

Kohonen, T. (1995/1997), Self-Organizing Maps, Berlin: Springer-Verlag. First edition was 1995, second edition 1997. See http://www.cis.hut.fi/nnrc/new_book.html for information on the second edition.

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

Lang, K. J., Waibel, A. H., and Hinton, G. (1990), "A time-delay neural network architecture for isolated word recognition," Neural Networks, 3, 23-44.

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

Masters, T. (1995) Advanced Algorithms for Neural Networks: A C++ Sourcebook, NY: John Wiley and Sons, ISBN 0-471-10588-0

Medsker, L.R., and Jain, L.C., eds. (2000), Recurrent Neural Networks: Design and Applications, Boca Raton, FL: CRC Press, ISBN 0-8493-7181-3.

Minsky, M.L., and Papert, S.A. (1969/1988), Perceptrons, Cambridge, MA: The MIT Press (first edition, 1969; expanded edition, 1988).

Moody, J. and Darken, C.J. (1989), "Fast learning in networks of locally-tuned processing units," Neural Computation, 1, 281-294.

Moore, B. (1988), "ART 1 and Pattern Clustering," in Touretzky, D., Hinton, G. and Sejnowski, T., eds., Proceedings of the 1988 Connectionist Models Summer School, 174-185, San Mateo, CA: Morgan Kaufmann.

Mulier, F. and Cherkassky, V. (1995), "Self-Organization as an Iterative Kernel Smoothing Process," Neural Computation, 7, 1165-1177.

Nadaraya, E.A. (1964) "On estimating regression", Theory Probab. Applic. 10, 186-90.

Oja, E. (1989), "Neural networks, principal components, and subspaces," International Journal of Neural Systems, 1, 61-68.

Orr, M.J.L. (1996), "Introduction to radial basis function networks," http://www.anc.ed.ac.uk/~mjo/papers/intro.ps or http://www.anc.ed.ac.uk/~mjo/papers/intro.ps.gz

Pao, Y. H. (1989), Adaptive Pattern Recognition and Neural Networks, Reading, MA: Addison-Wesley Publishing Company, ISBN 0-201-12584-6.

Pineda, F.J. (1989), "Recurrent back-propagation and the dynamical approach to neural computation," Neural Computation, 1, 161-172.

Reed, R.D., and Marks, R.J, II (1999), Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks, Cambridge, MA: The MIT Press, ISBN 0-262-18190-8.

Riedmiller, M. and Braun, H. (1993), "A Direct Adaptive Method for Faster Backpropagation Learning: The RPROP Algorithm", Proceedings of the IEEE International Conference on Neural Networks 1993, San Francisco: IEEE.

Rosenblatt, F. (1958), "The perceptron: A probabilistic model for information storage and organization in the brain., Psychological Review, 65, 386-408.

Rumelhart, D.E., Hinton, G.E., and Williams, R.J. (1986), "Learning internal representations by error propagation", in Rumelhart, D.E. and McClelland, J. L., eds. (1986), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Volume 1, 318-362, Cambridge, MA: The MIT Press.

Sanger, T.D. (1989), "Optimal unsupervised learning in a single-layer linear feedforward neural network," Neural Networks, 2, 459-473.

Specht, D.F. (1990) "Probabilistic neural networks," Neural Networks, 3, 110-118.

Specht, D.F. (1991) "A Generalized Regression Neural Network", IEEE Transactions on Neural Networks, 2, Nov. 1991, 568-576.

Wan, E.A. (1990), "Temporal backpropagation: An efficient algorithm for finite impulse response neural networks," in Proceedings of the 1990 Connectionist Models Summer School, Touretzky, D.S., Elman, J.L., Sejnowski, T.J., and Hinton, G.E., eds., San Mateo, CA: Morgan Kaufmann, pp. 131-140.

Watson, G.S. (1964) "Smooth regression analysis", Sankhy{\=a}, Series A, 26, 359-72.

Werbos, P.J. (1990), "Backpropagtion through time: What it is and how to do it," Proceedings of the IEEE, 78, 1550-1560.

Widrow, B., and Hoff, M.E., Jr., (1960), "Adaptive switching circuits," IRE WESCON Convention Record. part 4, pp. 96-104. Reprinted in Anderson and Rosenfeld (1988).

Williams, R.J., and Zipser, D., (1989), "A learning algorithm for continually running fully recurrent neurla networks," Neural Computation, 1, 270-280.

Williamson, J.R. (1995), "Gaussian ARTMAP: A neural network for fast incremental learning of noisy multidimensional maps," Technical Report CAS/CNS-95-003, Boston University, Center of Adaptive Systems and Department of Cognitive and Neural Systems.

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Subject: How many kinds of Kohonen networks exist? (And what is k-means?)

Teuvo Kohonen is one of the most famous and prolific researchers in neurocomputing, and he has invented a variety of networks. But many people refer to "Kohonen networks" without specifying which kind of Kohonen network, and this lack of precision can lead to confusion. The phrase "Kohonen network" most often refers to one of the following three types of networks: There are several other kinds of Kohonen networks described in Kohonen (1995), including: More information on the error functions (or absence thereof) used by Kohonen VQ and SOM is provided under "What does unsupervised learning learn?"

For more on-line information on Kohonen networks and other varieties of SOMs, see:

References:

Anderberg, M.R. (1973), Cluster Analysis for Applications, New York: Academic Press, Inc.

Balakrishnan, P.V., Cooper, M.C., Jacob, V.S., and Lewis, P.A. (1994) "A study of the classification capabilities of neural networks using unsupervised learning: A comparison with k-means clustering", Psychometrika, 59, 509-525.

Bishop, C.M., Svens\'en, M., and Williams, C.K.I (1997), "GTM: A principled alternative to the self-organizing map," in Mozer, M.C., Jordan, M.I., and Petsche, T., (eds.) Advances in Neural Information Processing Systems 9, Cambridge, MA: The MIT Press, pp. 354-360. Also see http://www.ncrg.aston.ac.uk/GTM/

Bottou, L., and Bengio, Y. (1995), "Convergence properties of the K-Means algorithms," in Tesauro, G., Touretzky, D., and Leen, T., (eds.) Advances in Neural Information Processing Systems 7, Cambridge, MA: The MIT Press, pp. 585-592.

Cho, S.-B. (1997), "Self-organizing map with dynamical node-splitting: Application to handwritten digit recognition," Neural Computation, 9, 1345-1355.

Desieno, D. (1988), "Adding a conscience to competitive learning," Proc. Int. Conf. on Neural Networks, I, 117-124, IEEE Press.

Devroye, L., Gy\"orfi, L., and Lugosi, G. (1996), A Probabilistic Theory of Pattern Recognition, NY: Springer,

Forgy, E.W. (1965), "Cluster analysis of multivariate data: Efficiency versus interpretability," Biometric Society Meetings, Riverside, CA. Abstract in Biomatrics, 21, 768.

Gersho, A. and Gray, R.M. (1992), Vector Quantization and Signal Compression, Boston: Kluwer Academic Publishers.

Hartigan, J.A. (1975), Clustering Algorithms, NY: Wiley.

Hartigan, J.A., and Wong, M.A. (1979), "Algorithm AS136: A k-means clustering algorithm," Applied Statistics, 28-100-108.

Hastie, T., and Stuetzle, W. (1989), "Principal curves," Journal of the American Statistical Association, 84, 502-516.

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

Ismail, M.A., and Kamel, M.S. (1989), "Multidimensional data clustering utilizing hybrid search strategies," Pattern Recognition, 22, 75-89.

Kohonen, T (1984), Self-Organization and Associative Memory, Berlin: Springer-Verlag.

Kohonen, T (1988), "Learning Vector Quantization," Neural Networks, 1 (suppl 1), 303.

Kohonen, T. (1995/1997), Self-Organizing Maps, Berlin: Springer-Verlag. First edition was 1995, second edition 1997. See http://www.cis.hut.fi/nnrc/new_book.html for information on the second edition.

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

Linde, Y., Buzo, A., and Gray, R. (1980), "An algorithm for vector quantizer design," IEEE Transactions on Communications, 28, 84-95.

Lloyd, S. (1982), "Least squares quantization in PCM," IEEE Transactions on Information Theory, 28, 129-137.

MacQueen, J.B. (1967), "Some Methods for Classification and Analysis of Multivariate Observations,"Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1, 281-297.

Max, J. (1960), "Quantizing for minimum distortion," IEEE Transactions on Information Theory, 6, 7-12.

Mulier, F. and Cherkassky, V. (1995), "Self-Organization as an iterative kernel smoothing process," Neural Computation, 7, 1165-1177.

Ripley, B.D. (1996), Pattern Recognition and Neural Networks, Cambridge: Cambridge University Press.

Ritter, H., Martinetz, T., and Schulten, K. (1992), Neural Computation and Self-Organizing Maps: An Introduction, Reading, MA: Addison-Wesley.

SAS Institute (1989), SAS/STAT User's Guide, Version 6, 4th edition, Cary, NC: SAS Institute.

Symons, M.J. (1981), "Clustering Criteria and Multivariate Normal Mixtures," Biometrics, 37, 35-43.

Tibshirani, R. (1992), "Principal curves revisited," Statistics and Computing, 2, 183-190.

Utsugi, A. (1996), "Topology selection for self-organizing maps," Network: Computation in Neural Systems, 7, 727-740, available on-line at http://www.aist.go.jp/NIBH/~b0616/Lab/index-e.html

Utsugi, A. (1997), "Hyperparameter selection for self-organizing maps," Neural Computation, 9, 623-635, available on-line at http://www.aist.go.jp/NIBH/~b0616/Lab/index-e.html

Wand, M.P., and Jones, M.C. (1995), Kernel Smoothing, London: Chapman & Hall.

Zador, P.L. (1982), "Asymptotic quantization error of continuous signals and the quantization dimension," IEEE Transactions on Information Theory, 28, 139-149.

Zeger, K., Vaisey, J., and Gersho, A. (1992), "Globally optimal vector quantizer design by stochastic relaxation," IEEE Transactions on Signal Procesing, 40, 310-322.

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Subject: How are layers counted?

How to count layers is a matter of considerable dispute.

To avoid ambiguity, you should speak of a 2-hidden-layer network, not a 4-layer network (as some would call it) or 3-layer network (as others would call it). And if the connections follow any pattern other than fully connecting each layer to the next and to no others, you should carefully specify the connections.

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Subject: What are cases and variables?

A vector of values presented at one time to all the input units of a neural network is called a "case", "example", "pattern, "sample", etc. The term "case" will be used in this FAQ because it is widely recognized, unambiguous, and requires less typing than the other terms. A case may include not only input values, but also target values and possibly other information.

A vector of values presented at different times to a single input unit is often called an "input variable" or "feature". To a statistician, it is a "predictor", "regressor", "covariate", "independent variable", "explanatory variable", etc. A vector of target values associated with a given output unit of the network during training will be called a "target variable" in this FAQ. To a statistician, it is usually a "response" or "dependent variable".

A "data set" is a matrix containing one or (usually) more cases. In this FAQ, it will be assumed that cases are rows of the matrix, while variables are columns.

Note that the often-used term "input vector" is ambiguous; it can mean either an input case or an input variable.

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Subject: What are the population, sample, training set, design set, validation set, and test set?

There seems to be no term in the NN literature for the set of all cases that you want to be able to generalize to. Statisticians call this set the "population". Tsypkin (1971) called it the "grand truth distribution," but this term has never caught on. Neither is there a consistent term in the NN literature for the set of cases that are available for training and evaluating an NN. Statisticians call this set the "sample". The sample is usually a subset of the population.

(Neurobiologists mean something entirely different by "population," apparently some collection of neurons, but I have never found out the exact meaning. I am going to continue to use "population" in the statistical sense until NN researchers reach a consensus on some other terms for "population" and "sample"; I suspect this will never happen.)

In NN methodology, the sample is often subdivided into "training", "validation", and "test" sets. The distinctions among these subsets are crucial, but the terms "validation" and "test" sets are often confused. There is no book in the NN literature more authoritative than Ripley (1996), from which the following definitions are taken (p.354):

Training set:
A set of examples used for learning, that is to fit the parameters [i.e., weights] of the classifier.
Validation set:
A set of examples used to tune the parameters [i.e., architecture, not weights] of a classifier, for example to choose the number of hidden units in a neural network.
Test set:
A set of examples used only to assess the performance [generalization] of a fully-specified classifier.
Bishop (1995), another indispensable reference on neural networks, provides the following explanation (p. 372):
Since our goal is to find the network having the best performance on new data, the simplest approach to the comparison of different networks is to evaluate the error function using data which is independent of that used for training. Various networks are trained by minimization of an appropriate error function defined with respect to a training data set. The performance of the networks is then compared by evaluating the error function using an independent validation set, and the network having the smallest error with respect to the validation set is selected. This approach is called the hold out method. Since this procedure can itself lead to some overfitting to the validation set, the performance of the selected network should be confirmed by measuring its performance on a third independent set of data called a test set.
The crucial point is that a test set, by definition, is never used to choose among two or more networks, so that the error
on the test set provides an unbiased estimate of the generalization error (assuming that the test set is representative of the population, etc.). Any data set that is used to choose the best of two or more networks is, by definition, a validation set, and the error of the chosen network on the validation set is optimistically biased.

There is a problem with the usual distinction between training and validation sets. Some training approaches, such as early stopping, require a validation set, so in a sense, the validation set is used for training. Other approaches, such as maximum likelihood, do not inherently require a validation set. So the "training" set for maximum likelihood might encompass both the "training" and "validation" sets for early stopping. Greg Heath has suggested the term "design" set be used for cases that are used solely to adjust the weights in a network, while "training" set be used to encompass both design and validation sets. There is considerable merit to this suggestion, but it has not yet been widely adopted.

But things can get more complicated. Suppose you want to train nets with 5 ,10, and 20 hidden units using maximum likelihood, and you want to train nets with 20 and 50 hidden units using early stopping. You also want to use a validation set to choose the best of these various networks. Should you use the same validation set for early stopping that you use for the final network choice, or should you use two separate validation sets? That is, you could divide the sample into 3 subsets, say A, B, C and proceed as follows:

Or you could divide the sample into 4 subsets, say A, B, C, and D and proceed as follows: Or, with the same 4 subsets, you could take a third approach: You could argue that the first approach is biased towards choosing a net trained by early stopping. Early stopping involves a choice among a potentially large number of networks, and therefore provides more opportunity for overfitting the validation set than does the choice among only 3 networks trained by maximum likelihood. Hence if you make the final choice of networks using the same validation set (B) that was used for early stopping, you give an unfair advantage to early stopping. If you are writing an article to compare various training methods, this bias could be a serious flaw. But if you are using NNs for some practical application, this bias might not matter at all, since you obtain an honest estimate of generalization error using C.

You could also argue that the second and third approaches are too wasteful in their use of data. This objection could be important if your sample contains 100 cases, but will probably be of little concern if your sample contains 100,000,000 cases. For small samples, there are other methods that make more efficient use of data; see "What are cross-validation and bootstrapping?"

References:

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

Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge: Cambridge University Press.

Tsypkin, Y. (1971), Adaptation and Learning in Automatic Systems, NY: Academic Press.

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Subject: How are NNs related to statistical methods?

There is considerable overlap between the fields of neural networks and statistics. Statistics is concerned with data analysis. In neural network terminology, statistical inference means learning to generalize from noisy data. Some neural networks are not concerned with data analysis (e.g., those intended to model biological systems) and therefore have little to do with statistics. Some neural networks do not learn (e.g., Hopfield nets) and therefore have little to do with statistics. Some neural networks can learn successfully only from noise-free data (e.g., ART or the perceptron rule) and therefore would not be considered statistical methods. But most neural networks that can learn to generalize effectively from noisy data are similar or identical to statistical methods. For example: Some neural network areas that appear to have no close relatives in the existing statistical literature are: Feedforward nets are a subset of the class of nonlinear regression and discrimination models. Statisticians have studied the properties of this general class but had not considered the specific case of feedforward neural nets before such networks were popularized in the neural network field. Still, many results from the statistical theory of nonlinear models apply directly to feedforward nets, and the methods that are commonly used for fitting nonlinear models, such as various Levenberg-Marquardt and conjugate gradient algorithms, can be used to train feedforward nets. The application of statistical theory to neural networks is explored in detail by Bishop (1995) and Ripley (1996). Several summary articles have also been published relating statistical models to neural networks, including Cheng and Titterington (1994), Kuan and White (1994), Ripley (1993, 1994), Sarle (1994), and several articles in Cherkassky, Friedman, and Wechsler (1994). Among the many statistical concepts important to neural nets is the bias/variance trade-off in nonparametric estimation, discussed by Geman, Bienenstock, and Doursat, R. (1992). Some more advanced results of statistical theory applied to neural networks are given by White (1989a, 1989b, 1990, 1992a) and White and Gallant (1992), reprinted in White (1992b).

While neural nets are often defined in terms of their algorithms or implementations, statistical methods are usually defined in terms of their results. The arithmetic mean, for example, can be computed by a (very simple) backprop net, by applying the usual formula SUM(x_i)/n, or by various other methods. What you get is still an arithmetic mean regardless of how you compute it. So a statistician would consider standard backprop, Quickprop, and Levenberg-Marquardt as different algorithms for implementing the same statistical model such as a feedforward net. On the other hand, different training criteria, such as least squares and cross entropy, are viewed by statisticians as fundamentally different estimation methods with different statistical properties.

It is sometimes claimed that neural networks, unlike statistical models, require no distributional assumptions. In fact, neural networks involve exactly the same sort of distributional assumptions as statistical models (Bishop, 1995), but statisticians study the consequences and importance of these assumptions while many neural networkers ignore them. For example, least-squares training methods are widely used by statisticians and neural networkers. Statisticians realize that least-squares training involves implicit distributional assumptions in that least-squares estimates have certain optimality properties for noise that is normally distributed with equal variance for all training cases and that is independent between different cases. These optimality properties are consequences of the fact that least-squares estimation is maximum likelihood under those conditions. Similarly, cross-entropy is maximum likelihood for noise with a Bernoulli distribution. If you study the distributional assumptions, then you can recognize and deal with violations of the assumptions. For example, if you have normally distributed noise but some training cases have greater noise variance than others, then you may be able to use weighted least squares instead of ordinary least squares to obtain more efficient estimates.

Hundreds, perhaps thousands of people have run comparisons of neural nets with "traditional statistics" (whatever that means). Most such studies involve one or two data sets, and are of little use to anyone else unless they happen to be analyzing the same kind of data. But there is an impressive comparative study of supervised classification by Michie, Spiegelhalter, and Taylor (1994), which not only compares many classification methods on many data sets, but also provides unusually extensive analyses of the results. Another useful study on supervised classification by Lim, Loh, and Shih (1999) is available on-line. There is an excellent comparison of unsupervised Kohonen networks and k-means clustering by Balakrishnan, Cooper, Jacob, and Lewis (1994).

There are many methods in the statistical literature that can be used for flexible nonlinear modeling. These methods include:

Why use neural nets rather than any of the above methods? There are many answers to that question depending on what kind of neural net you're interested in. The most popular variety of neural net, the MLP, tends to be useful in the same situations as projection pursuit regression, i.e.: The main advantage of MLPs over projection pursuit regression is that computing predicted values from MLPs is simpler and faster. Also, MLPs are better at learning moderately pathological functions than are many other methods with stronger smoothness assumptions (see ftp://ftp.sas.com/pub/neural/dojo/dojo.html) as long as the number of pathological features (such as discontinuities) in the function is not too large. For more discussion of the theoretical benefits of various types of neural nets, see How do MLPs compare with RBFs?

Communication between statisticians and neural net researchers is often hindered by the different terminology used in the two fields. There is a comparison of neural net and statistical jargon in ftp://ftp.sas.com/pub/neural/jargon

For free statistical software, see the StatLib repository at http://lib.stat.cmu.edu/ at Carnegie Mellon University.

There are zillions of introductory textbooks on statistics. One of the better ones is Moore and McCabe (1989). At an intermediate level, the books on linear regression by Weisberg (1985) and Myers (1986), on logistic regression by Hosmer and Lemeshow (1989), and on discriminant analysis by Hand (1981) can be recommended. At a more advanced level, the book on generalized linear models by McCullagh and Nelder (1989) is an essential reference, and the book on nonlinear regression by Gallant (1987) has much material relevant to neural nets.

Several introductory statistics texts are available on the web:

References:

Balakrishnan, P.V., Cooper, M.C., Jacob, V.S., and Lewis, P.A. (1994) "A study of the classification capabilities of neural networks using unsupervised learning: A comparison with k-means clustering", Psychometrika, 59, 509-525.

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

Cheng, B. and Titterington, D.M. (1994), "Neural Networks: A Review from a Statistical Perspective", Statistical Science, 9, 2-54.

Cherkassky, V., Friedman, J.H., and Wechsler, H., eds. (1994), From Statistics to Neural Networks: Theory and Pattern Recognition Applications, Berlin: Springer-Verlag.

Cleveland and Gross (1991), "Computational Methods for Local Regression," Statistics and Computing 1, 47-62.

Dey, D., ed. (1998) Practical Nonparametric and Semiparametric Bayesian Statistics, Springer Verlag.

Donoho, D.L., and Johnstone, I.M. (1995), "Adapting to unknown smoothness via wavelet shrinkage," J. of the American Statistical Association, 90, 1200-1224.

Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., and Picard, D. (1995), "Wavelet shrinkage: asymptopia (with discussion)?" J. of the Royal Statistical Society, Series B, 57, 301-369.

Eubank, R.L. (1999), Nonparametric Regression and Spline Smoothing, 2nd ed., Marcel Dekker, ISBN 0-8247-9337-4.

Fan, J., and Gijbels, I. (1995), "Data-driven bandwidth selection in local polynomial: variable bandwidth and spatial adaptation," J. of the Royal Statistical Society, Series B, 57, 371-394.

Farlow, S.J. (1984), Self-organizing Methods in Modeling: GMDH Type Algorithms, NY: Marcel Dekker. (GMDH)

Friedman, J.H. (1991), "Multivariate adaptive regression splines", Annals of Statistics, 19, 1-141. (MARS)

Friedman, J.H. and Stuetzle, W. (1981) "Projection pursuit regression," J. of the American Statistical Association, 76, 817-823.

Gallant, A.R. (1987) Nonlinear Statistical Models, NY: Wiley.

Geman, S., Bienenstock, E. and Doursat, R. (1992), "Neural Networks and the Bias/Variance Dilemma", Neural Computation, 4, 1-58.

Green, P.J., and Silverman, B.W. (1994), Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, London: Chapman & Hall.

Haerdle, W. (1990), Applied Nonparametric Regression, Cambridge Univ. Press.

Hand, D.J. (1981) Discrimination and Classification, NY: Wiley.

Hand, D.J. (1982) Kernel Discriminant Analysis, Research Studies Press.

Hand, D.J. (1997) Construction and Assessment of Classification Rules, NY: Wiley.

Hill, T., Marquez, L., O'Connor, M., and Remus, W. (1994), "Artificial neural network models for forecasting and decision making," International J. of Forecasting, 10, 5-15.

Kuan, C.-M. and White, H. (1994), "Artificial Neural Networks: An Econometric Perspective", Econometric Reviews, 13, 1-91.

Kushner, H. & Clark, D. (1978), Stochastic Approximation Methods for Constrained and Unconstrained Systems, Springer-Verlag.

Lim, T.-S., Loh, W.-Y. and Shih, Y.-S. ( 1999?), "A comparison of prediction accuracy, complexity, and training time of thirty-three old and new classification algorithms," Machine Learning, forthcoming, preprint available at http://www.recursive-partitioning.com/mach1317.pdf, and appendix containing complete tables of error rates, ranks, and training times at http://www.recursive-partitioning.com/appendix.pdf

McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd ed., London: Chapman & Hall.

Michie, D., Spiegelhalter, D.J. and Taylor, C.C., eds. (1994), Machine Learning, Neural and Statistical Classification, NY: Ellis Horwood; this book is out of print but available online at http://www.amsta.leeds.ac.uk/~charles/statlog/

Moore, D.S., and McCabe, G.P. (1989), Introduction to the Practice of Statistics, NY: W.H. Freeman.

Myers, R.H. (1986), Classical and Modern Regression with Applications, Boston: Duxbury Press.

Ripley, B.D. (1993), "Statistical Aspects of Neural Networks", in O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall, eds., Networks and Chaos: Statistical and Probabilistic Aspects, Chapman & Hall. ISBN 0 412 46530 2.

Ripley, B.D. (1994), "Neural Networks and Related Methods for Classification," Journal of the Royal Statistical Society, Series B, 56, 409-456.

Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge: Cambridge University Press.

Sarle, W.S. (1994), "Neural Networks and Statistical Models," Proceedings of the Nineteenth Annual SAS Users Group International Conference, Cary, NC: SAS Institute, pp 1538-1550. ( ftp://ftp.sas.com/pub/neural/neural1.ps)

Wahba, G. (1990), Spline Models for Observational Data, SIAM.

Wand, M.P., and Jones, M.C. (1995), Kernel Smoothing, London: Chapman & Hall.

Weisberg, S. (1985), Applied Linear Regression, NY: Wiley

White, H. (1989a), "Learning in Artificial Neural Networks: A Statistical Perspective," Neural Computation, 1, 425-464.

White, H. (1989b), "Some Asymptotic Results for Learning in Single Hidden Layer Feedforward Network Models", J. of the American Statistical Assoc., 84, 1008-1013.

White, H. (1990), "Connectionist Nonparametric Regression: Multilayer Feedforward Networks Can Learn Arbitrary Mappings," Neural Networks, 3, 535-550.

White, H. (1992a), "Nonparametric Estimation of Conditional Quantiles Using Neural Networks," in Page, C. and Le Page, R. (eds.), Computing Science and Statistics.

White, H., and Gallant, A.R. (1992), "On Learning the Derivatives of an Unknown Mapping with Multilayer Feedforward Networks," Neural Networks, 5, 129-138.

White, H. (1992b), Artificial Neural Networks: Approximation and Learning Theory, Blackwell.

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