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# Calculation of confidence intervals

The lmfit `confidence` module allows you to explicitly calculate confidence intervals for variable parameters. For most models, it is not necessary since the estimation of the standard error from the estimated covariance matrix is normally quite good.

But for some models, the sum of two exponentials for example, the approximation begins to fail. For this case, lmfit has the function `conf_interval()` to calculate confidence intervals directly. This is substantially slower than using the errors estimated from the covariance matrix, but the results are more robust.

## Method used for calculating confidence intervals

The F-test is used to compare our null model, which is the best fit we have found, with an alternate model, where one of the parameters is fixed to a specific value. The value is changed until the difference between \(\chi^2_0\) and \(\chi^2_{f}\) can’t be explained by the loss of a degree of freedom within a certain confidence.

\[F(P_{fix},N-P) = \left(\frac{\chi^2_f}{\chi^2_{0}}-1\right)\frac{N-P}{P_{fix}}\]

`N` is the number of data points and `P` the number of parameters of the null model. \(P_{fix}\) is the number of fixed parameters (or to be more clear, the difference of number of parameters between our null model and the alternate model).

Adding a log-likelihood method is under consideration.

## A basic example

First we create an example problem:

before we can generate the confidence intervals, we have to run a fit, so that the automated estimate of the standard errors can be used as a starting point:

Now it is just a simple function call to calculate the confidence intervals:

This shows the best-fit values for the parameters in the `_BEST_` column, and parameter values that are at the varying confidence levels given by steps in \(\sigma\). As we can see, the estimated error is almost the same, and the uncertainties are well behaved: Going from 1-\(\sigma\) (68% confidence) to 3-\(\sigma\) (99.7% confidence) uncertainties is fairly linear. It can also be seen that the errors are fairly symmetric around the best fit value. For this problem, it is not necessary to calculate confidence intervals, and the estimates of the uncertainties from the covariance matrix are sufficient.

## Working without standard error estimates

Sometimes the estimation of the standard errors from the covariance matrix fails, especially if values are near given bounds. Hence, to find the confidence intervals in these cases, it is necessary to set the errors by hand. Note that the standard error is only used to find an upper limit for each value, hence the exact value is not important.

To set the step-size to 10% of the initial value we loop through all parameters and set it manually:

## Calculating and visualizing maps of \(\chi^2\)

The estimated values for the \(1-\sigma\) standard error calculated by default for each fit include the effects of correlation between pairs of variables, but assumes the uncertainties are symmetric. While it doesn’t exactly say what the values of the \(n-\sigma\) uncertainties would be, the implication is that the \(n-\sigma\) error is simply \(n^2\sigma\).

The `conf_interval()` function described above improves on these automatically (and quickly) calculated uncertainies by explicitly finding \(n-\sigma\) confidence levels in both directions – it does not assume that the uncertainties are symmetric. This function also takes into account the correlations between pairs of variables, but it does not convey this information very well.

For even further exploration of the confidence levels of parameter values, it can be useful to calculate maps of \(\chi^2\) values for pairs of variables around their best fit values and visualize these as contour plots. Typically, pairs of variables will have elliptical contours of constant \(n-\sigma\) level, with highly-correlated pairs of variables having high ratios of major and minor axes.

The `conf_interval2d()` can calculate 2-d arrays or maps of either probability or \(\delta \chi^2 = \chi^2 - \chi_{\mathrm{best}}^2\) for any pair of variables. Visualizing these can help better understand the nature of the uncertainties and correlations between parameters. To illustrate this, we’ll start with an example fit to data that we deliberately add components not accounted for in the model, and with slightly non-Gaussian noise – a constructed but “real-world” example:

The reports show that we obtained a pretty good fit, and that the automated estimates of the uncertainties are actually pretty good – agreeing to the second decimal place. But we also see that some of the uncertainties do become noticeably asymmetric at high \(n-\sigma\) levels.

We’ll plot this data and fit, and then further explore these uncertainties using `conf_interval2d()`:

Here we made contours for the \(n-\sigma\) levels from the 2-D array of \(\chi^2\) by noting that the \(n-\sigma\) level will have \(\chi^2\) increased by \(n^2\chi_\nu^2\) where \(\chi_\nu^2\) is reduced chi-square.

The dotted boxes show both the scaled values of the standard errors from the initial fit, and the dashed boxes show the confidence levels from `conf_interval()`. You can see that the notion of increasing \(\chi^2\) by \(\chi_\nu^2\) works very well, and that there is a small asymmetry in the uncertainties for the `amplitude` and `sigma` parameters.

## An advanced example for evaluating confidence intervals

Now we look at a problem where calculating the error from approximated covariance can lead to misleading result – the same double exponential problem shown in Minimizer.emcee() - calculating the posterior probability distribution of parameters. In fact such a problem is particularly hard for the Levenberg-Marquardt method, so we first estimate the results using the slower but robust Nelder-Mead method. We can then compare the uncertainties computed (if the `numdifftools` package is installed) with those estimated using Levenberg-Marquardt around the previously found solution. We can also compare to the results of using `emcee`.

which will report:

Again we called `conf_interval()`, this time with tracing and only for 1- and 2-\(\sigma\). Comparing these two different estimates, we see that the estimate for `a1` is reasonably well approximated from the covariance matrix, but the estimates for `a2` and especially for `t1`, and `t2` are very asymmetric and that going from 1 \(\sigma\) (68% confidence) to 2 \(\sigma\) (95% confidence) is not very predictable.

Plots of the confidence region are shown in the figures below for `a1` and `t2` (left), and `a2` and `t2` (right):

Neither of these plots is very much like an ellipse, which is implicitly assumed by the approach using the covariance matrix. The plots actually look quite a bit like those found with MCMC and shown in the “corner plot” in Minimizer.emcee() - calculating the posterior probability distribution of parameters. In fact, comparing the confidence interval results here with the results for the 1- and 2-\(\sigma\) error estimated with `emcee`, we can see that the agreement is pretty good and that the asymmetry in the parameter distributions are reflected well in the asymmetry of the uncertainties.

The trace returned as the optional second argument from `conf_interval()` contains a dictionary for each variable parameter. The values are dictionaries with arrays of values for each variable, and an array of corresponding probabilities for the corresponding cumulative variables. This can be used to show the dependence between two parameters:

which shows the trace of values:

As an alternative/complement to the confidence intervals, the `Minimizer.emcee()` method uses Markov Chain Monte Carlo to sample the posterior probability distribution. These distributions demonstrate the range of solutions that the data supports and we refer to Minimizer.emcee() - calculating the posterior probability distribution of parameters where this methodology was used on the same problem.

Credible intervals (the Bayesian equivalent of the frequentist confidence interval) can be obtained with this method. MCMC can be used for model selection, to determine outliers, to marginalize over nuisance parameters, etcetera. For example, you may have fractionally underestimated the uncertainties on a dataset. MCMC can be used to estimate the true level of uncertainty on each data point. A tutorial on the possibilities offered by MCMC can be found at [1].

[1]

https://jakevdp.github.io/blog/2014/03/11/frequentism-and-bayesianism-a-practical-intro/

## Confidence Interval Functions

conf_interval(minimizer, result, p_names=None, sigmas=None, trace=False, maxiter=200, verbose=False, prob_func=None)

  Calculate the confidence interval (CI) for parameters.

  The parameter for which the CI is calculated will be varied, while the remaining parameters are re-optimized to minimize the chi-square. The resulting chi-square is used to calculate the probability with a given statistic (e.g., F-test). This function uses a 1d-rootfinder from SciPy to find the values resulting in the searched confidence region.

  Parameters:

    - minimizer (Minimizer) – The minimizer to use, holding objective function.

    - result (MinimizerResult) – The result of running minimize().

    - p_names (list, optional) – Names of the parameters for which the CI is calculated. If None (default), the CI is calculated for every parameter.

    - sigmas (list, optional) – The sigma-levels to find (default is [1, 2, 3]). See Notes below.

    - trace (bool, optional) – Defaults to False; if True, each result of a probability calculation is saved along with the parameter. This can be used to plot so-called “profile traces”.

    - maxiter (int, optional) – Maximum of iteration to find an upper limit (default is 200).

    - verbose (bool, optional) – Print extra debugging information (default is False).

    - prob_func (None or callable, optional) – Function to calculate the probability from the optimized chi-square. Default is None and uses the built-in function f_compare (i.e., F-test).

  Returns:

    - output (dict) – A dictionary containing a list of `(sigma, vals)`-tuples for each parameter.

    - trace_dict (dict, optional) – Only if trace is True. Is a dictionary, the key is the parameter which was fixed. The values are again a dict with the names as keys, but with an additional key ‘prob’. Each contains an array of the corresponding values.

  See also

  `conf_interval2d`

  Notes

  The values for sigma are taken as the number of standard deviations for a normal distribution and converted to probabilities. That is, the default `sigma=[1, 2, 3]` will use probabilities of 0.6827, 0.9545, and 0.9973. If any of the sigma values is less than 1, that will be interpreted as a probability. That is, a value of 1 and 0.6827 will give the same results, within precision.

  Examples

      >>> from lmfit.printfuncs import *
      >>> mini = minimize(some_func, params)
      >>> mini.leastsq()
      True
      >>> report_errors(params)
      ... #report
      >>> ci = conf_interval(mini)
      >>> report_ci(ci)
      ... #report

  Now with quantiles for the sigmas and using the trace.

      >>> ci, trace = conf_interval(mini, sigmas=[0.5, 1, 2, 3], trace=True)
      >>> fixed = trace['para1']['para1']
      >>> free = trace['para1']['not_para1']
      >>> prob = trace['para1']['prob']

  This makes it possible to plot the dependence between free and fixed parameters.

conf_interval2d(minimizer, result, x_name, y_name, nx=10, ny=10, limits=None, prob_func=None, nsigma=5, chi2_out=False)

  Calculate confidence regions for two fixed parameters.

  The method itself is explained in conf_interval: here we are fixing two parameters.

  Parameters:

    - minimizer (Minimizer) – The minimizer to use, holding objective function.

    - result (MinimizerResult) – The result of running minimize().

    - x_name (str) – The name of the parameter which will be the x direction.

    - y_name (str) – The name of the parameter which will be the y direction.

    - nx (int, optional) – Number of points in the x direction (default is 10).

    - ny (int, optional) – Number of points in the y direction (default is 10).

    - limits (tuple, optional) – Should have the form `((x_upper, x_lower), (y_upper, y_lower))`. If not given, the default is nsigma*stderr in each direction.

    - prob_func (None or callable, deprecated) – Starting with version 1.2, this argument is unused and has no effect.

    - nsigma (float or int, optional) – Multiplier of stderr for limits (default is 5).

    - chi2_out (bool) – Whether to return chi-square at each coordinate instead of probability.

  Returns:

    - x (numpy.ndarray) – X-coordinates (same shape as nx).

    - y (numpy.ndarray) – Y-coordinates (same shape as ny).

    - grid (numpy.ndarray) – 2-D array (with shape `(nx, ny)`) containing the calculated probabilities or chi-square.

  See also

  `conf_interval`

  Examples

      >>> mini = Minimizer(some_func, params)
      >>> result = mini.leastsq()
      >>> x, y, gr = conf_interval2d(mini, result, 'para1','para2')
      >>> plt.contour(x,y,gr)

ci_report(ci, with_offset=True, ndigits=5)

  Return text of a report for confidence intervals.

  Parameters:

    - ci (dict) – The result of `conf_interval()`: a dictionary containing a list of `(sigma, vals)`-tuples for each parameter.

    - with_offset (bool, optional) – Whether to subtract best value from all other values (default is True).

    - ndigits (int, optional) – Number of significant digits to show (default is 5).

  Returns:

    Text of formatted report on confidence intervals.

  Return type:

    str