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