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Re: linear regression with errors in both variables
*To*: mathgroup at smc.vnet.net
*Subject*: [mg96299] Re: [mg96281] linear regression with errors in both variables
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Wed, 11 Feb 2009 05:19:39 -0500 (EST)
*References*: <200902101057.FAA10340@smc.vnet.net>
*Reply-to*: drmajorbob at longhorns.com
Here are three solvers, applied to the following data.
data = Flatten[
Table[{x + 0.5 RandomReal[],
y + 0.7 RandomReal[], .2 x + .3 y + .4 RandomReal[]}, {x, 1,
20}, {y, 1, 30}], 1];
xy = data[[All, {1, 2}]];
z = data[[All, -1]];
LeastSquares[xy, z]
{0.200423, 0.302076}
FindFit[data, a x + b y, {a, b}, {x, y}]
{a -> 0.200423, b -> 0.302076}
lmf = LinearModelFit[data, {x, y}, {x, y}];
Normal[lmf]
0.038069 + 0.198848 x + 0.30105 y
They all do the same thing, depending on whether you ask for a constant
term or not.
What if we explicitly KNOW, however, that x, y, and z had specific error
standard deviations, in this case 0.2, 0.3, and 0.4? Surely unequal
variances make a difference?
In that case we could scale each variable to make all error variances the
same, then solve the transformed problem:
normData = #/{.2, .3, .4} & /@ data;
normFit = FindFit[normData, a x + b y, {a, b}, {x, y}]
{a -> 0.100211, b -> 0.226557}
And then we'd correct a and b to UNDO the transformation we made:
.4/{.2, .3} {a, b} /. normFit
{0.200423, 0.302076}
But as you see, this changes nothing.
Bobby
On Tue, 10 Feb 2009 04:57:01 -0600, Joerg <schaber at biologie.hu-berlin.de>
wrote:
> Hi,
>
> I want to test the hypothesis that my data
> follows a known simple linear relationship,
> y = a + bx. However, I have (known) measurements
> errors in both the y and the x values.
>
> I suppose just a simple linear regression
> does not do here.
>
> Any suggestions how do test this correctly?
>
> Thanks,
>
> joerg
>
--
DrMajorBob at longhorns.com
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