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

[David Huard]

Hi,

What you have is an heteroscedastic normal distribution (varying variance)
describing the residuals.

2007/6/21, Matthieu Brucher <matthieu.brucher at 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.

2)What does "rescaling" mean in this context?
>
> You must change B and C so that :
> Ay +/- 5
> B'y +/- 5
> C'y +/- 5
>

Or maximize the likelihood of a multivariate normal distribution, whose
covariance matrix describes your assumption about the heteroscedasticity 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)

Cheers,

David


Matthieu
>
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