# [SciPy-User] calculate predicted values from regression + confidence intervall

josef.pktd@gmai... josef.pktd@gmai...
Thu Oct 20 19:03:26 CDT 2011

```On Thu, Oct 20, 2011 at 5:15 PM, Robert Kern <robert.kern@gmail.com> wrote:
> On Thu, Oct 20, 2011 at 21:50, Christian K. <ckkart@hoc.net> wrote:
>> Am 20.10.11 16:12, schrieb josef.pktd@gmail.com:
>>> On Thu, Oct 20, 2011 at 5:11 AM, Christian K. <ckkart@hoc.net> wrote:
>>>>  <josef.pktd <at> gmail.com> writes:
>>>>>> f(X,Y) = a1-a2*log(X)+a3/Y (inverse power/Arrhenius model from accelerated
>>>>>> reliability testing)
>>>>>
>>>>> your f(X,Y) is still linear in the parameters, a1, a2, a3. So the
>>>>> linear version still applies.
>>>>
>>>> Ok, but then I do not understand how to follow your indications for the
>>>> prediction interval:
>>>>
>>>>>> distributed with mean y = Y = X*beta, and var(y) = X' * cov_beta * X +
>>>>>> var_u_estimate (dot products for appropriate shapes)
>>>>
>>>> X in my case is [X,Y] and cov_beta has a shape of 3x3, since there are 3
>>>> paramters.
>>>> Sorry for my ignorance on statistics, I really apppreaciate your help.
>>>
>>> I'm attaching a complete example for the linear in parameters case,
>>> including the comparison with statsmodels.
>>
>> Ok, I got it, thank you very much. As I understood, this works for OLS
>> (only?).

It's OLS only, it can be adapted to other estimators like non-linear
least squares, or to weighted least squares.

I never looked at the details of odr, so I'm no  help there.

Josef

>>What about if I get the covariance matrix from a 2D odr/leastsq
>> fit from scipy.odr ? I noticed, that the covariance matrices differ by a
>> constant (large) factor.
>
> ODRPACK will scale the covariance matrix by the Chi^2 score of the
> residuals (i.e. divide the residuals by the error bars, square, sum,
> divide by nobs-nparams), IIRC. This accounts for misestimation of the
> error bars. If the error bars were correctly estimated, the Chi^2
> score will be ~1. If the error bars were too small compared to the
> residuals, then the Chi^2 score will be high, and thus increase the
> estimated variance, etc. This may or may not be what you want,
> especially when comparing it with other tools, but it's what ODRPACK
> computes, so it's what scipy.odr returns.
>
> --
> Robert Kern
>
> "I have come to believe that the whole world is an enigma, a harmless
> enigma that is made terrible by our own mad attempt to interpret it as
> though it had an underlying truth."
>   -- Umberto Eco
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```