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:

import numpy as np

import lmfit

x = np.linspace(0.3, 10, 100)
np.random.seed(0)
y = 1/(0.1*x) + 2 + 0.1*np.random.randn(x.size)
pars = lmfit.Parameters()
pars.add_many(('a', 0.1), ('b', 1))


def residual(p):
    return 1/(p['a']*x) + p['b'] - y

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:

mini = lmfit.Minimizer(residual, pars)
result = mini.minimize()

print(lmfit.fit_report(result.params))
[[Variables]]
    a:  0.09943896 +/- 1.9322e-04 (0.19%) (init = 0.1)
    b:  1.98476942 +/- 0.01222678 (0.62%) (init = 1)
[[Correlations]] (unreported correlations are < 0.100)
    C(a, b) = 0.601

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

ci = lmfit.conf_interval(mini, result)
lmfit.printfuncs.report_ci(ci)
      99.73%    95.45%    68.27%    _BEST_    68.27%    95.45%    99.73%
 a:  -0.00059  -0.00039  -0.00019   0.09944  +0.00019  +0.00039  +0.00060
 b:  -0.03766  -0.02478  -0.01230   1.98477  +0.01230  +0.02478  +0.03761

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 fairy 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:

for p in result.params:
    result.params[p].stderr = abs(result.params[p].value * 0.1)

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.

# <examples/doc_confidence_advanced.py>
import matplotlib.pyplot as plt
import numpy as np

import lmfit

x = np.linspace(1, 10, 250)
np.random.seed(0)
y = 3.0*np.exp(-x/2) - 5.0*np.exp(-(x-0.1)/10.) + 0.1*np.random.randn(x.size)

p = lmfit.Parameters()
p.add_many(('a1', 4.), ('a2', 4.), ('t1', 3.), ('t2', 3.))


def residual(p):
    return p['a1']*np.exp(-x/p['t1']) + p['a2']*np.exp(-(x-0.1)/p['t2']) - y


# create Minimizer
mini = lmfit.Minimizer(residual, p, nan_policy='propagate')

# first solve with Nelder-Mead algorithm
out1 = mini.minimize(method='Nelder')

# then solve with Levenberg-Marquardt using the
# Nelder-Mead solution as a starting point
out2 = mini.minimize(method='leastsq', params=out1.params)

lmfit.report_fit(out2.params, min_correl=0.5)

ci, trace = lmfit.conf_interval(mini, out2, sigmas=[1, 2], trace=True)
lmfit.printfuncs.report_ci(ci)

# plot data and best fit
plt.figure()
plt.plot(x, y)
plt.plot(x, residual(out2.params) + y, '-')
plt.show()

# plot confidence intervals (a1 vs t2 and a2 vs t2)
fig, axes = plt.subplots(1, 2, figsize=(12.8, 4.8))
cx, cy, grid = lmfit.conf_interval2d(mini, out2, 'a1', 't2', 30, 30)
ctp = axes[0].contourf(cx, cy, grid, np.linspace(0, 1, 11))
fig.colorbar(ctp, ax=axes[0])
axes[0].set_xlabel('a1')
axes[0].set_ylabel('t2')

cx, cy, grid = lmfit.conf_interval2d(mini, out2, 'a2', 't2', 30, 30)
ctp = axes[1].contourf(cx, cy, grid, np.linspace(0, 1, 11))
fig.colorbar(ctp, ax=axes[1])
axes[1].set_xlabel('a2')
axes[1].set_ylabel('t2')
plt.show()

# plot dependence between two parameters
fig, axes = plt.subplots(1, 2, figsize=(12.8, 4.8))
cx1, cy1, prob = trace['a1']['a1'], trace['a1']['t2'], trace['a1']['prob']
cx2, cy2, prob2 = trace['t2']['t2'], trace['t2']['a1'], trace['t2']['prob']

axes[0].scatter(cx1, cy1, c=prob, s=30)
axes[0].set_xlabel('a1')
axes[0].set_ylabel('t2')

axes[1].scatter(cx2, cy2, c=prob2, s=30)
axes[1].set_xlabel('t2')
axes[1].set_ylabel('a1')
plt.show()
# <end examples/doc_confidence_advanced.py>

which will report:

[[Variables]]
    a1:  2.98622120 +/- 0.14867187 (4.98%) (init = 2.986237)
    a2: -4.33526328 +/- 0.11527506 (2.66%) (init = -4.335256)
    t1:  1.30994234 +/- 0.13121177 (10.02%) (init = 1.309932)
    t2:  11.8240350 +/- 0.46316470 (3.92%) (init = 11.82408)
[[Correlations]] (unreported correlations are < 0.500)
    C(a2, t2) = 0.987
    C(a2, t1) = -0.925
    C(t1, t2) = -0.881
    C(a1, t1) = -0.599

       95.45%    68.27%    _BEST_    68.27%    95.45%
 a1:  -0.27286  -0.14165   2.98622  +0.16353  +0.36343
 a2:  -0.30444  -0.13219  -4.33526  +0.10688  +0.19683
 t1:  -0.23392  -0.12494   1.30994  +0.14660  +0.32369
 t2:  -1.01943  -0.48820  11.82404  +0.46041  +0.90441

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):

_images/confidence_7_0.svg

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:

fig, axes = plt.subplots(1, 2, figsize=(12.8, 4.8))
cx1, cy1, prob = trace['a1']['a1'], trace['a1']['t2'], trace['a1']['prob']
cx2, cy2, prob2 = trace['t2']['t2'], trace['t2']['a1'], trace['t2']['prob']

axes[0].scatter(cx1, cy1, c=prob, s=30)
axes[0].set_xlabel('a1')
axes[0].set_ylabel('t2')

axes[1].scatter(cx2, cy2, c=prob2, s=30)
axes[1].set_xlabel('t2')
axes[1].set_ylabel('a1')

plt.show()

which shows the trace of values:

_images/confidence_9_0.svg

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)

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.

  • ny (int, optional) – Number of points in the y direction.

  • limits (tuple, optional) – Should have the form ((x_upper, x_lower), (y_upper, y_lower)). If not given, the default is 5.0*std-errors in each direction.

  • 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

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

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

  • grid (numpy.ndarray) – Grid containing the calculated probabilities (with shape (nx, ny)).

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