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Re: Re: Definition of SE (standard error) in LinearRegress and NonlinearRegress
*To*: mathgroup at smc.vnet.net
*Subject*: [mg70355] Re: [mg70322] Re: Definition of SE (standard error) in LinearRegress and NonlinearRegress
*From*: Darren Glosemeyer <darreng at wolfram.com>
*Date*: Sat, 14 Oct 2006 03:06:50 -0400 (EDT)
*References*: <200609300913.FAA13235@smc.vnet.net><egl2ub$4oi$1@smc.vnet.net> <200610130530.BAA01084@smc.vnet.net>
Yes, I left out the variance estimates in the covariance matrix
formulas. Thanks for catching this. The covariance formulas should
have been
r.r/(n-p) Inverse[Transpose[X].X]
and
r.r/(n-p) Inverse[Transpose[approxX].approxX]
Darren Glosemeyer
Wolfram Research
Ray Koopman wrote:
> Shouldn't both of those be multiplied by Sqrt[r.r/(n-p)],
> where r is the vector of residuals at the minimum,
> n is the number of data points,
> and p is the number of parameters?
>
> Darren Glosemeyer wrote:
>
>> The standard errors for parameter estimates in linear regression are the
>> square roots of the diagonal elements of the parameter covariance matrix
>>
>> Inverse[Transpose[X].X]
>>
>> where X is the design matrix for the regression. The ith row of the
>> design matrix contains the values of the basis functions evaluated at
>> the ith data point.
>>
>> The standard errors for parameter estimates in nonlinear regression are
>> the square roots of the diagonal elements of the asymptotic parameter
>> covariance matrix
>>
>> Inverse[Transpose[approxX].approxX]
>>
>> where approxX is an approximate design matrix for the nonlinear model.
>> The ith row of the approximate design matrix contains the values of the
>> first derivatives of the model function with respect to each of the
>> parameters evaluated at the ith data point.
>>
>>
>> Darren Glosemeyer
>> Wolfram Research
>>
>>
>> Seo Ho Youn wrote:
>>
>>> Hello, all.
>>>
>>>
>>>
>>> Can I ask how SE (standard error) in LinearRegress and NonlinearRegress is
>>> defined or calculated in Mathematica for a multi-parameter least-square fit?
>>> Or, does anybody know about document (or definition) on SE in Mathematica? I
>>> haven't been able to find any about how it is calculated in Mathematica.
>>>
>>>
>>>
>>> Thank you for your help and have a good day.
>>>
>>>
>>>
>>> Seo Ho
>>>
>>>
>>>
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