# [SciPy-user] nonlinear fit with non uniform error?

massimo sandal massimo.sandal@unibo...
Thu Jun 21 08:46:49 CDT 2007

David Huard ha scritto:
> Hi,
>
> What you have is an heteroscedastic normal distribution (varying
> variance) describing the residuals.
>
> 2007/6/21, Matthieu Brucher <matthieu.brucher@gmail.com
> <mailto:matthieu.brucher@gmail.com>>:
>
>         1)Does this mean that least squares is NOT ok?
>
>     Yes, LS is _NOT_ OK because it assumes that the distribution (with
>     its parameters) is the same for all errors. I don't remember
>     exactly, but this may be due to ergodicity
>
>
> Well, let's put things in perspective. You can still use ordinary
> least-squares.  Theoretically, this means you're making the assumption
> that the error mean and variance are fixed and constant.   In your case,
> this is not true and you can consider the LS solution like an
> approximation. What will happen under this approximation is that large
> errors on Cy will tend to dominate the residuals, and values in Ay will
> probably not be fitted optimally. I advise you try it anyway and
> visually check whether you care about that or not.

Yes, it's what I already do, and works fairly well.
I'd like to see how *better* becomes. It can be useful in some contexts,
so I wanted to know how to implement it.

> Or maximize the likelihood of a multivariate normal distribution, whose
> of the residuals.
>
> \Sigma =
> | \sigma_A^2       0                0                 |
> |      0             \sigma_B^2     0                 |
> |      0                    0              \sigma_C^2 |
>
> Heteroscedastic likelihood = -n/2 \ln(2\pi) - 1/2 \sum \ln(\sigma_i^2)
> -1/2 \sum \sigma_i^{-2} (y_{obs} - y_{sim})^2
>
> You might also consider the possibility that your errors are
> multiplicative rather than additive. In this case, describing the
> residuals by a lognormal distribution could make more sense.
>
> Maximize lognormal likelihood:  L=lognormal(y_sim | ln(y_obs), \sigma)

I'll try to make sense of it...

m.

--
Massimo Sandal
University of Bologna
Department of Biochemistry "G.Moruzzi"

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