Estimate parameter confidence intervals using the projection method.
The conf function should be used instead of proj.
The proj command computes confidence interval bounds for the specified model parameters in the dataset. A given parameter’s value is varied along a grid of values while the values of all the other thawed parameters are allowed to float to new best-fit values. The get_proj and set_proj_opt commands can be used to configure the error analysis; an example being changing the ‘sigma’ field to
1.6(i.e. 90%) from its default value of
1. The output from the routine is displayed on screen, and the get_proj_results routine can be used to retrieve the results.
id (str or int, optional) –
otherids (list of str or int, optional) – The id and otherids arguments determine which data set or data sets are used. If not given, all data sets which have a defined source model are used.
parameters (optional) – The default is to calculate the confidence limits on all thawed parameters of the model, or models, for all the data sets. The evaluation can be restricted by listing the parameters to use. Note that each parameter should be given as a separate argument, rather than as a list. For example
Estimate parameter confidence intervals using the confidence method.
Estimate the confidence intervals using the covariance method.
Return the confidence-interval estimation object.
Return the results of the last proj run.
Plot the statistic value as a single parameter is varied.
Plot the statistic value as two parameters are varied.
Set an option of the proj estimation object.
The function does not follow the normal Python standards for parameter use, since it is designed for easy interactive use. When called with multiple
parametersvalues, the order is unimportant, since any argument that is not defined as a model parameter is assumed to be a data id.
The proj command is different to covar, in that all other thawed parameters are allowed to float to new best-fit values, instead of being fixed to the initial best-fit values. While proj is more general (e.g. allowing the user to examine the parameter space away from the best-fit point), it is in the strictest sense no more accurate than covar for determining confidence intervals.
An estimated confidence interval is accurate if and only if:
the chi^2 or logL surface in parameter space is approximately shaped like a multi-dimensional paraboloid, and
the best-fit point is sufficiently far from parameter space boundaries.
One may determine if these conditions hold, for example, by plotting the fit statistic as a function of each parameter’s values (the curve should approximate a parabola) and by examining contour plots of the fit statistics made by varying the values of two parameters at a time (the contours should be elliptical, and parameter space boundaries should be no closer than approximately 3 sigma from the best-fit point). The int_proj and reg_proj commands may be used for this.
If either of the conditions given above does not hold, then the output from proj may be meaningless except to give an idea of the scale of the confidence intervals. To accurately determine the confidence intervals, one would have to reparameterize the model, use Monte Carlo simulations, or Bayesian methods.
As the calculation can be computer intensive, the default behavior is to use all available CPU cores to speed up the analysis. This can be changed be varying the
numcoresoption - or setting
False- either with set_proj_opt or get_proj.
As proj estimates intervals for each parameter independently, the relationship between sigma and the change in statistic value delta_S can be particularly simple: sigma = the square root of delta_S for statistics sampled from the chi-square distribution and for the Cash statistic, and is approximately equal to the square root of (2 * delta_S) for fits based on the general log-likelihood. The default setting is to calculate the one-sigma interval, which can be changed with the
sigmaoption to set_proj_opt or get_proj.