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I fitting an expression of the form:

$s(t)=\frac{1}{1+\exp[a+\sum_k b_k x_k(t)]}$

Where $a$ and $b_k$ are fitting parameters and $x_k(t)$ an input time series. I have $k=1,...,N$ different types of input variables and I'm trying to find what is the combination among the $N$ time series that best fit the observed data.

If I only use the first $k=1$ time series I have a better fitting result (better RMS), that if I use $k=1$ and $k=2$ together. So, my question is: why does the fitting Method (NMinimize of NonlinearModelFit function from Mathematica 10), when $x_1$ and $x_2$ are included at the same time, does not find the $b_1$ solution equal to the fitting process where only $x_1$ is present, together with $b_2=0.0$ result?

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    $\begingroup$ because the $a$ parameter is shared between the two groups, so fitting to both groups requires a compromise between the best-fitting values for the two groups ... ? $\endgroup$
    – Ben Bolker
    Sep 7, 2015 at 15:45
  • $\begingroup$ Because a better fit is possible if the estimate, $b_2$, is allowed to move from being forced to be 0. (NB it's best to have your notation distinguish your estimates from the thing they're estimating.) $\endgroup$
    – Glen_b
    Sep 7, 2015 at 22:46

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