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Re: Re: Re: LeastSquares using

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89109] Re: [mg89085] Re: [mg89032] Re: [mg88986] LeastSquares using
  • From: danl at wolfram.com
  • Date: Mon, 26 May 2008 01:29:00 -0400 (EDT)
  • References: <200805230705.DAA25655@smc.vnet.net>

>> Gareth Russell wrote:
>> > Hi,
>> >
>> > Is it possible to specify a least-squares minimization through the
>> > LinearProgramming function? In other words, exactly the same as
>> > LeastSquares, with the extra constraint that all x>=0?
>> >
>> > Presumably it comes down to specifying the input c correctly in the
>> > LinearProgramming function. But I can't see how to do that such that
>> > what is being minimized is the standard least-squares function
>> > ||m.x-b||^2
>> >
>> > Thanks,
>> >
>> > Gareth

I wrote:

>> [...]
>> If you can settle for an L_1 norm (so it's no longer least squares), you
>> can minimize a new variable abs, with new constraints -abs<=m.x-b<=abs.
>>
>> Daniel Lichtblau
>> Wolfram Research

This elicited the response:

> Beyond that, the L1 norm is preferred to L2 for statistical problems
> because of resistance to outliers.
> [...]
> Greg Duncan
> Dept Economics
> U California, Berkeley

Which motivates me to give a correct approach to finding the L^1 norm.
(What I showed, suitably interpreted, would give the L^infinity norm. That
is often useful, but not for this problem.)

The expression m.x-b is a vector, of length n, say. Call it c, and thus
the jth component is c[[j]]. Create a new vector d also of length n. It
will denote new variables. Then we minimize Total[d], that is, the sum of
components of d, subject to 2*n new constraints. For 1<=j<=n, there is the
pair

{c[[j]]<=d[[j]], -c[[j]]<=d[[j]]}

I think (well, hope, at any rate) that I wrote it correctly this time.

Daniel Lichtblau
Wolfram Research




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