Errors to fit parameters of scipy.optimize - numpy

Question

I use the scipy.optimize.minimize ( https://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html ) function with method='L-BFGS-B.
An example of what it returns is here above:
fun: 32.372210618549758
hess_inv: <6x6 LbfgsInvHessProduct with dtype=float64>
jac: array([ -2.14583906e-04, 4.09272616e-04, -2.55795385e-05,
3.76587650e-05, 1.49213975e-04, -8.38440428e-05])
message: 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
nfev: 420
nit: 51
status: 0
success: True
x: array([ 0.75739412, -0.0927572 , 0.11986434, 1.19911266, 0.27866406,
-0.03825225])
The x value correctly contains the fitted parameters. How do I compute the errors associated to those parameters?

Answer 1

TL;DR: You can actually place an upper bound on how precisely the minimization routine has found the optimal values of your parameters. See the snippet at the end of this answer that shows how to do it directly, without resorting to calling additional minimization routines.
The documentation for this method says
The iteration stops when (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol.
Roughly speaking, the minimization stops when the value of the function f that you're minimizing is minimized to within ftol of the optimum. (This is a relative error if f is greater than 1, and absolute otherwise; for simplicity I'll assume it's an absolute error.) In more standard language, you'll probably think of your function f as a chi-squared value. So this roughly suggests that you would expect
Of course, just the fact that you're applying a minimization routine like this assumes that your function is well behaved, in the sense that it's reasonably smooth and the optimum being found is well approximated near the optimum by a quadratic function of the parameters xi:
where Δxi is the difference between the found value of parameter xi and its optimal value, and Hij is the Hessian matrix. A little (surprisingly nontrivial) linear algebra gets you to a pretty standard result for an estimate of the uncertainty in any quantity X that's a function of your parameters xi:
which lets us write
That's the most useful formula in general, but for the specific question here, we just have X = xi, so this simplifies to
Finally, to be totally explicit, let's say you've stored the optimization result in a variable called res. The inverse Hessian is available as res.hess_inv, which is a function that takes a vector and returns the product of the inverse Hessian with that vector. So, for example, we can display the optimized parameters along with the uncertainty estimates with a snippet like this:
ftol = 2.220446049250313e-09
tmp_i = np.zeros(len(res.x))
for i in range(len(res.x)):
tmp_i[i] = 1.0
hess_inv_i = res.hess_inv(tmp_i)[i]
uncertainty_i = np.sqrt(max(1, abs(res.fun)) * ftol * hess_inv_i)
tmp_i[i] = 0.0
print('x^{0} = {1:12.4e} ± {2:.1e}'.format(i, res.x[i], uncertainty_i))
Note that I've incorporated the max behavior from the documentation, assuming that f^k and f^{k+1} are basically just the same as the final output value, res.fun, which really ought to be a good approximation. Also, for small problems, you can just use np.diag(res.hess_inv.todense()) to get the full inverse and extract the diagonal all at once. But for large numbers of variables, I've found that to be a much slower option. Finally, I've added the default value of ftol, but if you change it in an argument to minimize, you would obviously need to change it here.

Answer 2

One approach to this common problem is to use scipy.optimize.leastsq after using minimize with 'L-BFGS-B' starting from the solution found with 'L-BFGS-B'. That is, leastsq will (normally) include and estimate of the 1-sigma errors as well as the solution.
Of course, that approach makes several assumption, including that leastsq can be used and may be appropriate for solving the problem. From a practical view, this requires the objective function return an array of residual values with at least as many elements as variables, not a cost function.
You may find lmfit (https://lmfit.github.io/lmfit-py/) useful here: It supports both 'L-BFGS-B' and 'leastsq' and gives a uniform wrapper around these and other minimization methods, so that you can use the same objective function for both methods (and specify how to convert the residual array into the cost function). In addition, parameter bounds can be used for both methods. This makes it very easy to first do a fit with 'L-BFGS-B' and then with 'leastsq', using the values from 'L-BFGS-B' as starting values.
Lmfit also provides methods to more explicitly explore confidence limits on parameter values in more detail, in case you suspect the simple but fast approach used by leastsq might be insufficient.

Answer 3

It really depends what you mean by "errors". There is no general answer to your question, because it depends on what you're fitting and what assumptions you're making.
The easiest case is one of the most common: when the function you are minimizing is a negative log-likelihood. In that case the inverse of the hessian matrix returned by the fit (hess_inv) is the covariance matrix describing the Gaussian approximation to the maximum likelihood.The parameter errors are the square root of the diagonal elements of the covariance matrix.
Beware that if you are fitting a different kind of function or are making different assumptions, then that doesn't apply.