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Subject: Re: Solving underdefined multivariate quadratic equation
Date: Fri, 23 Jan 2009 07:57:03 +0000 (UTC)
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"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <glb65m$5ag$1@fred.mathworks.com>...
> "Cary Frydman" <cfrydman@caltech.edu> wrote in message <glb1ss$cpq$1@fred.mathworks.com>...
> > .......
> > Perhaps if I explain the situation it may help narrow down solution concepts.  What I'm looking to do is to construct two 50 dimensional vectors that are mutually orthogonal.  One vector represents the 50 statistical means of 50 distinct underlying probability mass functions.  That is, each element of this vector is the mean of a distinct pmf.  The other vector contains the corresponding variance of the pmf.  I want to make these two 50 dimensional vectors orthogonal, so I have a set of probability mass functions whose mean and variance are not correlated.  
> > 
> > The way I plan to do this is to exogenously define the support of the 50 pmf's, and then use some type of nonlinear equation solver to find the probabilities that characterize each pmf, subject to the orthogonality constraint.  Ie, the probability of the first outcome of each pmf fully characterizes that pmf.  The "constained" part of my nonlinear equation is that the variables to be solved for must be probabilities, ie, must be between 0 & 1, although I'd like them to be between 0.2 and 0.8.  Hopefully this clears things up, thanks for the initial response.
> -----------------
>   You say "use some type of nonlinear equation solver to find the probabilities that characterize each pmf" and "the probability of the first outcome of each pmf fully characterizes that pmf."  This would mean you have found some magic way to characterize each pmf distribution in terms of a single parameter, in this case the probability of the first outcome.  What I find remarkable is that this could lead to a simple quadratic equation to satisfy.  That would mean there are 50 different parameters which determine both the mean and variance of each pmf that somehow lead to a quadratic expression in 50 variables.  You certainly haven't shown us the steps in that process.  What are these variables?
> 
>   There is another aspect of this problem that bothers me.  You say "I want to make these two 50 dimensional vectors orthogonal, so I have a set of probability mass functions whose mean and variance are not correlated."  However, for these to be truly uncorrelated you would have to have the means all offset by the mean of these means and the variances by the mean of the variances in this "quadratic" expression.  Is that what you really meant?  Doing this will not by any means make these pmf distributions necessarily mutually independent.
> 
>   If I were you I would endeavor to boil everything down to just one parameter, hopefully one that would be assured of creating a valid pmf for each value of the parameter over a certain range.  That is, use some perhaps random technique in each of the fifty cases that depend on the same parameter, but of course in fifty different ways.  This way your means and variances and hence the appropriate over-all correlation value could be expressed in terms of that one parameter.  Then you could use 'fzero' to find a value of this parameter that actually results in no correlation.  I don't see that you have gained much by having numerous parameters instead of one.
> 
>   I know that is a vague description, but then so is your statement of the problem.
> 
> Roger Stafford

Hi, 
I guess the important thing I forgot to mention is that the cardinality of the support of each pmf is 2.  That is, there are only two points with positive mass, so after exogenously defining these two points, each pmf can be characterized by the probability of the first outcome, call it p1.

If you construct the sample covariance between the means of the pmf's and the variance of the pmf's as a function of the 50 p1's, and set to 0, you get a well defined single multivariate quadratic equation, where the 50 unknowns are the 50 p1's of each pmf.  Fsolve will do this.  However, I don't know if I can use fsolve while constraining the solutions to be in the range of a probability measure.  Ie, having each coordinate of the solution lie in the [0,1] interval.  Any other methods are very welcome, this was just a first approach&#8230;

Thanks