# [Maxima] Use of maxima in 'statistical engineering'

Richard Hennessy rich.hennessy at verizon.net
Wed Aug 26 14:37:19 CDT 2009

```Vishal,

Thanks to Barton for sending me a copy of a fairly new pw.mac.  I have to test it some more but it seems to work fine.  I am still working on a faster version.

Rich

----- Original Message -----
From: Vishal Ramnath
To: maxima at math.utexas.edu
Sent: Monday, August 24, 2009 7:37 AM
Subject: Re: [Maxima] Use of maxima in 'statistical engineering'

(apologies if this mail got accidently double posted)
﻿Hello Richard,

Your point about modeling the pdf by piece wise continous functions is true and it will work, however a possible concern is that the computational speed taken with an integration of piece wise functions may be too long.

A secondary minor aspect is how to select the "best" intervals in which to fit a piece-wise continous function when you just have discrete data points. Straight polynomials introduce oscillations so cubic and natural cubic splines are what I normally use.

You can see below for where I was coming from in an earlier message I sent to Robert if interested (attachments were VRUNC1.mac & VRUNC2.mac)

Best regards,

Vishal

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Nice to hear of the work you are involved in.

Have a look at the attached to get a qualitative feel for the type of statistical computations I was more involved last financial year, this year I am still involved in this but my main focus in the next couple of months is on a low pressure vacuum design.

I won't send this email to the Maxima mailing list for the moment as it seems that attachments get scrubbed, so if you think more people who use Maxima might be interested in this field of statistical computation we can make a plan to set up FTP access.

The slide presentation gives a give overview of the existing approach and its simplifications and goes on to explain why a Monte Carlo (MC) is necessary. VRUNC1.mac is for the Guide to Uncertainty Method (GUM) and VRUNC2.mac is for the MC - just note both should be considered "works in progress" :)

The basic idea is to start with known / assumed statistical information i.e. probability density functions (pdf's) and to incorporate and propogate this information through a mathematical model of an instrument / system to determine its final statistical state i.e. the pdf either approximately through the use of the GUM, exactly through solution of the Markov convolution integral, or numerically using a MC algorithm.

The idea can also work 'backwards' by experimentally through numerical/statistical methods trying to determine what the pdf's for the inputs are if the final measurand (output) is already known, as a type of 'inverse problem'.

I would appreciate references to the work you are involved in. You can find more about this field on the National Physical Laboratory (UK) website in the previous email or in the journal "Metrologia" which is an IoP publication.

I am aware of the use of inverse Fourier transforms to compute convolution integrals and have come across it in a conference paper written by Prof. Maurice Cox who applied it to linear models and generalized linear models, but I think a straight Monte Carlo method is more convenient to implement. The use of inverse FFT's can still be used for validation purposes of more general MC codes.

The point to note is that a statistical computational drawing on symbolic computations will work for algebraic mathematical models of the system, but cannot easily work for physical systems defined in terms of partial differential equations (PDE's) unless there is an explicit closed form solution. Symbolic computations still have a place though from my perspective in the complete analytical solution of test cases which can be exactly solved. An example would be certain simple geometries in which the Navier-Stokes PDE's of fluid mechanics can be exactly solved and to which the Markov convolution can be usefully applied such as channel flow in a rectangular duct.

Presently I don't have any journal publications but  just some conference proceedings and various company engineering reports to date.
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>>> "Richard Hennessy" <rich.hennessy at verizon.net> 8/11/2009 9:31 PM >>>

Vishal,

"using "messy" real life pdf's, which means doing a convolution integral to get an answer becomes more difficult."

Yes but messy functions can be modeled by piecewise continuous functions and my pw.mac package has some potential here.  I am not sure if it fits your needs.  I hope it helps.

Rich

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