Subject: Stochastic optimal control book
From: dimitrib@mit.edu (Dimitri Bertsekas)
Date: Fri, 17 Jan 1997 19:44:49 -0400
Just Published:
STOCHASTIC OPTIMAL CONTROL: THE DISCRETE-TIME CASE
by
Dimitri P. Bertsekas, Massachusetts Institute of Technology, and
Steven E. Shreve, Carnegie-Mellon University
(330 pages, softcover, ISBN:1-886529-03-5, $49.50)
published by
Athena Scientific, Belmont, MA
http://world.std.com/~athenasc/
This is a republication of a research monograph, first published in 1978 by
Academic Press, but now out-of-print. It provides a general and
comprehensive development of the mathematical foundations of stochastic optimal
control of discrete-time systems, including the treatment of the intricate
measure-theoretic issues. It provides much in-depth analysis not found in any
other book.
Among its special features, the book:
-----------------------------------------------------------------------
** resolves definitively the mathematical issues of discrete-time stochastic
optimal control problems, including Borel models, and semi-continuous models
** establishes the most general possible theory of finite and infinite horizon
stochastic dynamic programming models, through the use of analytic sets and
universally measurable policies
** develops general frameworks for dynamic programming based on abstract
contraction and monotone mappings
** provides extensive background on analytic sets, Borel spaces and their
probability measures
-----------------------------------------------------------------------
CONTENTS
1. Introduction
2. Monotone Mappings Underlying Dynamic Programming Models
3. Finite Horizon Models
4. Infinite Horizon Models under a Contraction Assumption
5. Infinite Horizon Models under Monotonicity Assumptions
6. A Generalized Abstract Dynamic Programming Model
7. Borel Spaces and their Probability Measures
8. The Finite Horizon Borel Model
9. The Infinite Horizon Borel Models
10. The Imperfect State Information Model
11. Miscellaneous
Appendix A: The Outer Integral
Appendix B: Additional Measurability Properties of Borel Spaces
Appendix C: The Hausdorff Metric and the Exponential Topology
********************************************************************
PUBLISHER'S INFORMATION:
Athena Scientific, P.O.Box 391, Belmont, MA, 02178-9998, U.S.A.
Email: athenasc@world.std.com, Tel: (617) 489-3097, FAX: (617) 489-2017
WWW Site for Info and Ordering: http://world.std.com/~athenasc/
********************************************************************
Subject: Re: Combining Neural/Fuzzy Models with Statistical Models
From: saswss@hotellng.unx.sas.com (Warren Sarle)
Date: Sat, 18 Jan 1997 00:57:53 GMT
In article <32DA5910.2AA2@colorado.edu>, Robert Dodier writes:
|> Andrew Gray wrote:
|>
|> > I'm working on combining neural networks and fuzzy logic models
|> > with statistical techniques (regression and data reduction) for
|> > software metrics (for example, predicting development time based on
|> > the type and size of system). While there has been a lot of work on
|> > neural-fuzzy, neural-genetic, fuzzy-genetic, etc. type systems I've
|> > only ever found a small number of researchers using AI/statistical
|> > techniques (presumably at least partially an indication of how few AI
|> > researchers follow the statistical side of things, and vice versa).
|>
|> At the risk of reviving age-old threads about fuzzy logic vs.
|> probability, let me advise you not to bother with fuzzy logic.
|> There are two parts to fuzzy logic, one defensible and the other
|> not. The ``fuzzy'' part is one solution to the problem of
|> representing uncertain knowledge -- this is the defensible part.
|> The ``logic'' part is an attempt to reason with uncertain knowledge
|> -- this is an undefensible hack.
The trouble with fuzzy logic as usually presented is that you can't
reason about two uncertain propositions without knowing how the
uncertainties are related--it's like trying to work with marginal
probability distributions without knowing the joint distribution.
But some fuzzy logicians have noticed this problem and developed
fuzzy logics involving "correlations" between fuzzy propositions.
See the post included below.
|> ... As discussed at length in
|> Warren Sarle's neural networks FAQ (sorry, I don't have pointer)
|> it's useful to consider neural networks as extensions of or variations
|> on conventional regression and classification schemes.
ftp://ftp.sas.com/pub/neural/FAQ.html
______________________________________________________________________
From: William Siler
Newsgroups: comp.ai.fuzzy
Subject: New Fuzzy Logic
Date: Fri, 20 Sep 96 20:25:09 -0500
Organization: Delphi (info@delphi.com email, 800-695-4005 voice)
Lines: 107
Message-ID:
Reposting article removed by rogue canceller.
DIGEST OF BUCKLEY, JJ AND SILER, W: A NEW T-NORM. SUBMITTED TO
FUZZY SETS AND SYSTEMS, 1996.
Fuzzy systems theory has been criticized for not obeying all
the laws of classical set theory and classical logic. The
t-norm and t-conorm here presented obey all the laws of the
corresponding classical theory. A somewhat similar theory has
been proposed by Thomas (1994), except that he does not claim
that the distributive property is maintained.
We first propose a source of fuzziness. We suppose that the
truth value > 0 and < 1 of a fuzzy logical statement A is drawn
from a number of underlying (probably implicit) correlated
random variables whose values alpha[i] are binary, i.e. 0 or 1
with a Bernoulli distribution, and that the truth value of A is
a simple average of these binary values. (George Klir (1994)
proposed a similar process where the random values are binary
opinions of experts as to truth or falsehood of a statement.)
If this is so, then
a = truth(A) = sum(alpha[i] / n
b = truth(B) = sum(beta[i] / n)
r = correlation coefficient(alpha[i], beta[i])
aANDb = truth value(A AND B)
aORb = truth value(A OR B)
sa = standard deviation(alpha) = sqrt(p(alpha)*(1 - p(alpha))
sb = standard deviation(beta) = sqrt(p(beta)*(1 - p(beta))
aANDb = a*b + r*sa*sb
aORb = a + b - a*b - r*sa*sb
rmax = (min(a,b) - a*b) / (sa*sb)
for r = rmax, min(a,b) = a*b + r*sa*sb
rmin = (a*b - max(a+b-1, 0)) / (sa*sb)
for r = rmin, max(a+b-1, 0) = a*b + rmin*sa*sb
Proofs of the following theorems are in the appendices of our
paper.
Theorem 1:
1. maxr = ru, ru <= 1
2. minr = rl, rl >= -1
3. rl <= r <= ru
Theorem 2:
1. aANDb = a*b + r*sa*sb = a*b + cov(a,b)
2. aORb = a + b - a*b - r*sa*sb = a + b - a*b - cov(a,b)
Theorem 3:
1. If r = ru, aANDb = min(a,b) and aORb = max(a,b)
2. If r = 0, aANDb = a*b and aORb = a+b-ab
3. If r = rl, aANDb = max(a+b-1, 0) and aORb = max(a+b, 1)
We now suppose that this basic process is inaccessible to us,
but that we do have a history of a number of instances of the
truths of statement A and statement B. Now, given a value of r,
the correlation coefficient between a, the truth values of A,
and b, the truth values of B, the t-norm and t-conorm
appropriate to this history, T (t-norm) and C (t-conorm) are
defined for [a, b] on S, a restricted subset of [0,1]x[0,1].
Theorem 4.
(The 5 parts of this theorem define the subset S of
[0,1]x[0,1]
possible for r = 1, 0 < r < 1, r = 0, -1 < r < 0 and r = -1.)
Given a value of r, it may be that not all (a,b) combinations
are possible; e.g. a = .25, b = .75 is not possible for r = 1
in the binary process described above.)
Theorem 5.
1. (Shows that for 0 < r < 1 and (a,b) in S,
ab < T(a,b) <= min(a,b).)
2. (Shows that for -1 < r < 0 and (a,b) in S,
max(a+b-1) <= T(A,B) < ab)
3. (Shows that for -1 <= r <= 1 and (a,b) in S,
max(a+b-1, 0) <= T(a,b) <= min(a,b) and
max(a,b) <= C(a,b) <= min(a+b, 1).
Theorem 6. Shows that T is a t-norm and C is a t-conorm on S.
Theorem 7.
1. A AND A = A, r is 1.
2. A OR 0 = 0, any r.
3. A OR X = A, any r.
4. A AND NOT-A = 0, r is -1.
5. A OR A = A, r is 1.
6. A OR X = A, any r.
7. A OR 0 = A, any r.
8. A OR NOT-A = X, r is -1.
9. NOT-(A AND B) = NOT-A OR NOT-B, any r.
10. NOT-(A AND B) = NOT-A AND NOT-B, any r.
11. A OR (A AND B) = A, any appropriate r.
12. A AND (A OR B) = A, any appropriate r.
13. A AND (B OR C) = (A AND B) OR (A AND C), any appropriate
r.
14. A OR (B AND C) = (A OR B) AND (A OR C), any appropriate
r.
References:
Klir, GJ (1994). Multivalued logics vesus model logics:
alternate frameworks for uncertainty modelling. In: Advances in
Fuzzy Theory and Technology, Vol II: 3-47. Duke University
Press, Durham, NC.
Thomas, SF (1994). Fuzzy Logic and Probability. ACG Press,
Wichita, KS.
______________________________________________________________________
--
Warren S. Sarle SAS Institute Inc. The opinions expressed here
saswss@unx.sas.com SAS Campus Drive are mine and not necessarily
(919) 677-8000 Cary, NC 27513, USA those of SAS Institute.
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