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Re: Alternative to bivariate Taylor series
*To*: mathgroup at smc.vnet.net
*Subject*: [mg44679] Re: Alternative to bivariate Taylor series
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Fri, 21 Nov 2003 05:13:04 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <bpd2a7$ce5$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <bpd2a7$ce5$1 at smc.vnet.net>, koshi man <man_koshi at yahoo.com>
wrote:
> Is there any expansion to log(1+exp(x-y)) other than
> Taylor series in the integer powers of x and y ?
There are many other possible expansions. One possibility is that
Log[1+Exp[x]] is well approximated by truncating the following expansion:
x UnitStep[x] - Sum[(-1)^i Exp[-i Abs[x]]/i, {i, 1, Infinity}]
Another is to use FindFit to find optimal values for the parameters a,
b, c, d in the following ansatz:
x UnitStep[x]+ a Exp[-b Abs[x]] + c Abs[x] Exp[-d x^2]
> I'm trying to find expectation of above function given
> x, y are random variables with gaussian distributions
> of known mean and variances.
For numerical parameters, you can use NIntegrate to compute these
expectation values directly via a straightforward double integral.
Assuming x and y are independent, for the Normal distribution, you can
reduce the computation of expectation values from a 2D integration to a
1D integration (which I do not think reduces to a closed form for
log(1+exp(x-y))). See the Notebook at
http://physics.uwa.edu.au/pub/Mathematica/MathGroup/ExpectationValues.nb
> What about other bivariate expansions in Mathematica?
You can always try NIntegrate. Because your function depends linearly on
a x + b y, for some distributions you will again be able to reduce the
computation to a 1D numerical integration.
Also, have a look at "Mathematical Statistics with Mathematica" by Colin
Rose and Murray D. Smith:
http://store.wolfram.com/view/book/ISBN0387952349.str
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
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