[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 
> 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)

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

m.

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

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email:
massimo.sandal@unibo.it

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