`lsqfit` - Nonlinear Least Squares Fitting¶

Introduction¶

This package contains tools for nonlinear least-squares curve fitting of data. In general a fit has four inputs:

The dependent data y that is to be fit — typically y is a Python dictionary in an lsqfit analysis. Its values y[k] are either gvar.GVars or arrays (any shape or dimension) of gvar.GVars that specify the values of the dependent variables and their errors.

A collection x of independent data — x can have any structure and contain any data (or no data).

A fit function f(x, p) whose parameters p are adjusted by the fit until f(x, p) equals y to within ys errors — parameters p` are usually specified by a dictionary whose values p[k] are individual parameters or (numpy) arrays of parameters. The fit function is assumed independent of x (that is, f(p)) if x = False (or if x is omitted from the input data).

Initial estimates or priors for each parameter in p — priors are usually specified using a dictionary prior whose values prior[k] are gvar.GVars or arrays of gvar.GVars that give initial estimates (values and errors) for parameters p[k].

A typical code sequence has the structure:

... collect x, y, prior ...

def f(x, p):
    ... compute fit to y[k], for all k in y, using x, p ...
    ... return dictionary containing the fit values for the y[k]s ...

fit = lsqfit.nonlinear_fit(data=(x, y), prior=prior, fcn=f)
print(fit)      # variable fit is of type nonlinear_fit

The parameters p[k] are varied until the chi**2 for the fit is minimized.

The best-fit values for the parameters are recovered after fitting using, for example, p=fit.p. Then the p[k] are gvar.GVars or arrays of gvar.GVars that give best-fit estimates and fit uncertainties in those estimates. The print(fit) statement prints a summary of the fit results.

The dependent variable y above could be an array instead of a dictionary, which is less flexible in general but possibly more convenient in simpler fits. Then the approximate y returned by fit function f(x, p) must be an array with the same shape as the dependent variable. The prior prior could also be represented by an array instead of a dictionary.

By default priors are Gaussian/normal distributions, represented by gvar.GVars. Setting nonlinear_fit parameter extend=True allows for log-normal and sqrt-normal distributions as well. The latter are indicated by replacing the prior (in a dictionary prior) with key c, for example, by a prior for the parameter’s logarithm or square root, with key log(c) or sqrt(c), respectively. nonlinear_fit adds parameter c to the parameter dictionary, deriving its value from parameter log(c) or sqrt(c). The fit function can be expressed directly in terms of parameter c and so is the same no matter which distribution is used for c. Note that a sqrt-normal distribution with zero mean is equivalent to an exponential distribution. Additional distributions can be added using gvar.add_parameter_distribution().

The lsqfit tutorial contains extended explanations and examples. The first appendix in the paper at http://arxiv.org/abs/arXiv:1406.2279 provides conceptual background on the techniques used in this module for fits and, especially, error budgets.

nonlinear_fit Objects¶

class lsqfit.nonlinear_fit(data, fcn, prior=None, p0=None, extend=False, svdcut=1e-15, debug=False, **kargs)¶

Nonlinear least-squares fit.

lsqfit.nonlinear_fit fits a (nonlinear) function f(x, p) to data y by varying parameters p, and stores the results: for example,

fit = nonlinear_fit(data=(x, y), fcn=f, prior=prior)   # do fit
print(fit)                               # print fit results

The best-fit values for the parameters are in fit.p, while the chi**2, the number of degrees of freedom, the logarithm of Gaussian Bayes Factor, the number of iterations, and the cpu time needed for the fit are in fit.chi2, fit.dof, fit.logGBF, fit.nit, and fit.time, respectively. Results for individual parameters in fit.p are of type gvar.GVar, and therefore carry information about errors and correlations with other parameters. The fit data and prior can be recovered using fit.x (equals False if there is no x), fit.y, and fit.prior; the data and prior are corrected for the svd cut, if there is one (that is, their covariance matrices have been modified in accordance with the svd cut).

Parameters:

data –
Data to be fit by lsqfit.nonlinear_fit. It can have any of the following formats:

data = x, y

x is the independent data that is passed to the fit function with the fit parameters: fcn(x, p). y is a dictionary (or array) of gvar.GVars that encode the means and covariance matrix for the data that is to be fit being fit. The fit function must return a result having the same layout as y.

data = y

y is a dictionary (or array) of gvar.GVars that encode the means and covariance matrix for the data being fit. There is no independent data so the fit function depends only upon the fit parameters: fit(p). The fit function must return a result having the same layout as y.

data = x, ymean, ycov

x is the independent data that is passed to the fit function with the fit parameters: fcn(x, p). ymean is an array containing the mean values of the fit data. ycov is an array containing the covariance matrix of the fit data; ycov.shape equals 2*ymean.shape. The fit function must return an array having the same shape as ymean.

data = x, ymean, ysdev

x is the independent data that is passed to the fit function with the fit parameters: fcn(x, p). ymean is an array containing the mean values of the fit data. ysdev is an array containing the standard deviations of the fit data; ysdev.shape equals ymean.shape. The data are assumed to be uncorrelated. The fit function must return an array having the same shape as ymean.

Setting x=False in the first, third or fourth of these formats implies that the fit function depends only on the fit parameters: that is, fcn(p) instead of fcn(x, p). (This is not assumed if x=None.)
fcn (function) – The function to be fit to data. It is either a function of the independent data x and the fit parameters p (fcn(x, p)), or a function of just the fit parameters (fcn(p)) when there is no x data or x=False. The parameters are tuned in the fit until the function returns values that agree with the y data to within the ys’ errors. The function’s return value must have the same layout as the y data (a dictionary or an array). The fit parameters p are either: 1) a dictionary where each p[k] is a single parameter or an array of parameters (any shape); or, 2) a single array of parameters. The layout of the parameters is the same as that of prior prior if it is specified; otherwise, it is inferred from of the starting value p0 for the fit.
prior (dictionary, array, or None) – A dictionary (or array) containing a priori estimates for all parameters p used by fit function fcn(x, p) (or fcn(p)). Fit parameters p are stored in a dictionary (or array) with the same keys and structure (or shape) as prior. The default value is None; prior must be defined if p0 is None.
p0 (dictionary, array, string or None) – Starting values for fit parameters in fit. lsqfit.nonlinear_fit adjusts p0 to make it consistent in shape and structure with prior when the latter is specified: elements missing from p0 are filled in using prior, and elements in p0 that are not in prior are discarded. If p0 is a string, it is taken as a file name and lsqfit.nonlinear_fit attempts to read starting values from that file; best-fit parameter values are written out to the same file after the fit (for priming future fits). If p0 is None or the attempt to read the file fails, starting values are extracted from prior. The default value is None; p0 must be defined if prior is None.
svdcut (None or float) – If svdcut is nonzero (not None), svd cuts are applied to every block-diagonal sub-matrix of the covariance matrix for the data y and prior (if there is a prior). The blocks are first rescaled so that all diagonal elements equal 1 – that is, the blocks are replaced by the correlation matrices for the corresponding subsets of variables. Then, if svdcut > 0, eigenvalues of the rescaled matrices that are smaller than svdcut times the maximum eigenvalue are replaced by svdcut times the maximum eigenvalue. This makes the covariance matrix less singular and less susceptible to roundoff error. When svdcut < 0, eigenvalues smaller than |svdcut| times the maximum eigenvalue are discarded and the corresponding components in y and prior are zeroed out.
extend – Log-normal and sqrt-normal distributions can be used for fit priors when extend=True, provided the parameters are specified by a dictionary (as opposed to an array). To use such a distribution for a parameter 'c' in the fit prior, replace prior['c'] with a prior specifying its logarithm or square root, designated by prior['log(c)'] or prior['sqrt(c)'], respectively. The dictionaries containing parameters generated by lsqfit.nonlinear_fit will have entries for both 'c' and 'log(c)' or 'sqrt(c)', so only the prior need be changed to switch to log-normal/sqrt-normal distributions. Setting extend=False (the default) restricts all parameters to Gaussian distributions. Additional distributions can be added using gvar.add_parameter_distribution().
debug (boolean) – Set to True for extra debugging of the fit function and a check for roundoff errors. (Default is False.)
fitterargs – Dictionary of arguments passed on to lsqfit.multifit, which does the fitting.

The results from the fit are accessed through the following attributes (of fit where fit = nonlinear_fit(...)):

chi2¶: The minimum chi**2 for the fit. fit.chi2 / fit.dof is usually of order one in good fits; values much less than one suggest that the actual standard deviations in the input data and/or priors are smaller than the standard deviations used in the fit.

cov¶: Covariance matrix of the best-fit parameters from the fit.

dof¶: Number of degrees of freedom in the fit, which equals the number of pieces of data being fit when priors are specified for the fit parameters. Without priors, it is the number of pieces of data minus the number of fit parameters.

logGBF¶: The logarithm of the probability (density) of obtaining the fit data by randomly sampling the parameter model (priors plus fit function) used in the fit. This quantity is useful for comparing fits of the same data to different models, with different priors and/or fit functions. The model with the largest value of fit.logGBF is the one prefered by the data. The exponential of the difference in fit.logGBF between two models is the ratio of probabilities (Bayes factor) for those models. Differences in fit.logGBF smaller than 1 are not very significant. Gaussian statistics are assumed when computing fit.logGBF.

p¶: Best-fit parameters from fit. Depending upon what was used for the prior (or p0), it is either: a dictionary (gvar.BufferDict) of gvar.GVars and/or arrays of gvar.GVars; or an array (numpy.ndarray) of gvar.GVars. fit.p represents a multi-dimensional Gaussian distribution which, in Bayesian terminology, is the posterior probability distribution of the fit parameters.

pmean¶: Means of the best-fit parameters from fit (dictionary or array).

psdev¶: Standard deviations of the best-fit parameters from fit (dictionary or array).

palt¶: Same as fit.p except that the errors are computed directly from fit.cov. This is faster but means that no information about correlations with the input data is retained (unlike in fit.p); and, therefore, fit.palt cannot be used to generate error budgets. fit.p and fit.palt give the same means and normally give the same errors for each parameter. They differ only when the input data’s covariance matrix is too singular to invert accurately (because of roundoff error), in which case an SVD cut is advisable.

p0¶: The parameter values used to start the fit.

Q¶: The probability that the chi**2 from the fit could have been larger, by chance, assuming the best-fit model is correct. Good fits have Q values larger than 0.1 or so. Also called the p-value of the fit.

svdcorrection¶: The sum of all SVD corrections, if any, added to the fit data y or the prior prior.

svdn¶: The number of eignemodes modified (and/or deleted) by the SVD cut.

nblocks¶: A dictionary where nblocks[s] equals the number of block-diagonal sub-matrices of the y–prior covariance matrix that are size s-by-s. This is sometimes useful for debugging.

time¶: CPU time (in secs) taken by fit.

The input parameters to the fit can be accessed as attributes. Note in particular attributes:

prior¶: Prior used in the fit. This may differ from the input prior if an SVD cut is used. It is either a dictionary (gvar.BufferDict) or an array (numpy.ndarray), depending upon the input. Equals None if no prior was specified.

x¶: The first field in the input data. This is sometimes the independent variable (as in ‘y vs x’ plot), but may be anything. It is set equal to False if the x field is omitted from the input data. (This also means that the fit function has no x argument: so f(p) rather than f(x,p).)

y¶: Fit data used in the fit. This may differ from the input data if an SVD cut is used. It is either a dictionary (gvar.BufferDict) or an array (numpy.ndarray), depending upon the input.

Additional methods are provided for printing out detailed information about the fit, testing fits with simulated data, doing bootstrap analyses of the fit errors, dumping (for later use) and loading parameter values, and checking for roundoff errors in the final error estimates:

format(maxline=0, pstyle='v')¶

Formats fit output details into a string for printing.

The output tabulates the chi**2 per degree of freedom of the fit (chi2/dof), the number of degrees of freedom, the logarithm of the Gaussian Bayes Factor for the fit (logGBF), and the number of fit- algorithm iterations needed by the fit. Optionally, it will also list the best-fit values for the fit parameters together with the prior for each (in [] on each line). Lines for parameters that deviate from their prior by more than one (prior) standard deviation are marked with asterisks, with the number of asterisks equal to the number of standard deviations (up to five). format can also list all of the data and the corresponding values from the fit, again with asterisks on lines where there is a significant discrepancy. At the end it lists the SVD cut, the number of eigenmodes modified by the SVD cut, the tolerances used in the fit, and the time in seconds needed to do the fit. The tolerance used to terminate the fit is marked with an asterisk.

Parameters:

maxline (integer or bool) – Maximum number of data points for which fit results and input data are tabulated. maxline<0 implies that only chi2, Q, logGBF, and itns are tabulated; no parameter values are included. Setting maxline=True prints all data points; setting it equal None or False is the same as setting it equal to -1. Default is maxline=0.
pstyle (‘vv’, ‘v’, or ‘m’) – Style used for parameter list. Supported values are ‘vv’ for very verbose, ‘v’ for verbose, and ‘m’ for minimal. When ‘m’ is set, only parameters whose values differ from their prior values are listed.

Returns:

String containing detailed information about fit.

fmt_errorbudget(outputs, inputs, ndecimal=2, percent=True)¶

Tabulate error budget for outputs[ko] due to inputs[ki].

For each output outputs[ko], fmt_errorbudget computes the contributions to outputs[ko]‘s standard deviation coming from the gvar.GVars collected in inputs[ki]. This is done for each key combination (ko,ki) and the results are tabulated with columns and rows labeled by ko and ki, respectively. If a gvar.GVar in inputs[ki] is correlated with other gvar.GVars, the contribution from the others is included in the ki contribution as well (since contributions from correlated gvar.GVars cannot be distinguished). The table is returned as a string.

Parameters:

outputs – Dictionary of gvar.GVars for which an error budget is computed.
inputs – Dictionary of: gvar.GVars, arrays/dictionaries of gvar.GVars, or lists of gvar.GVars and/or arrays/dictionaries of gvar.GVars. fmt_errorbudget tabulates the parts of the standard deviations of each outputs[ko] due to each inputs[ki].
ndecimal (int) – Number of decimal places displayed in table.
percent (boolean) – Tabulate % errors if percent is True; otherwise tabulate the errors themselves.
colwidth (positive integer or None) – Width of each column. This is set automatically, to accommodate label widths, if colwidth=None (default).
verify (boolean) – If True, a warning is issued if: 1) different inputs are correlated (and therefore double count errors); or 2) the sum (in quadrature) of partial errors is not equal to the total error to within 0.1% of the error (and the error budget is incomplete or overcomplete). No checking is done if verify==False (default).

Returns:

A table (str) containing the error budget. Output variables are labeled by the keys in outputs (columns); sources of uncertainty are labeled by the keys in inputs (rows).

fmt_values(outputs, ndecimal=None)¶

Tabulate gvar.GVars in outputs.

Parameters:	outputs – A dictionary of `gvar.GVar` objects. ndecimal (`int` or `None`) – Format values `v` using `v.fmt(ndecimal)`.
Returns:	A table (`str`) containing values and standard deviations for variables in `outputs`, labeled by the keys in `outputs`.

simulated_fit_iter(n=None, pexact=None, **kargs)¶

Iterator that returns simulation copies of a fit.

Fit reliability can be tested using simulated data which replaces the mean values in self.y with random numbers drawn from a distribution whose mean equals self.fcn(pexact) and whose covariance matrix is the same as self.y‘s. Simulated data is very similar to the original fit data, self.y, but corresponds to a world where the correct values for the parameters (i.e., averaged over many simulated data sets) are given by pexact. pexact is usually taken equal to fit.pmean.

Each iteration of the iterator creates new simulated data, with different random numbers, and fits it, returning the the lsqfit.nonlinear_fit that results. The simulated data has the same covariance matrix as fit.y. Typical usage is:

...
fit = nonlinear_fit(...)
...
for sfit in fit.simulated_fit_iter(n=3):
    ... verify that sfit.p agrees with pexact=fit.pmean within errors ...

Only a few iterations are needed to get a sense of the fit’s reliability since we know the correct answer in each case. The simulated fit’s output results should agree with pexact (=fit.pmean here) within the simulated fit’s errors.

Simulated fits can also be used to estimate biases in the fit’s output parameters or functions of them, should non-Gaussian behavior arise. This is possible, again, because we know the correct value for every parameter before we do the fit. Again only a few iterations may be needed for reliable estimates.

The (possibly non-Gaussian) probability distributions for parameters, or functions of them, can be explored in more detail by setting option bootstrap=True and collecting results from a large number of simulated fits. With bootstrap=True, the means of the priors are also varied from fit to fit, as in a bootstrap simulation; the new prior means are chosen at random from the prior distribution. Variations in the best-fit parameters (or functions of them) from fit to fit define the probability distributions for those quantities. For example, one would use the following code to analyze the distribution of function g(p) of the fit parameters:

fit = nonlinear_fit(...)

...

glist = []
for sfit in fit.simulated_fit_iter(n=100, bootstrap=True):
    glist.append(g(sfit.pmean))

... analyze samples glist[i] from g(p) distribution ...

This code generates n=100 samples glist[i] from the probability distribution of g(p). If everything is Gaussian, the mean and standard deviation of glist[i] should agree with g(fit.p).mean and g(fit.p).sdev.

The only difference between simulated fits with bootstrap=True and bootstrap=False (the default) is that the prior means are varied. It is essential that they be varied in a bootstrap analysis since one wants to capture the impact of the priors on the final distributions, but it is not necessary and probably not desirable when simply testing a fit’s reliability.

Parameters:	n (integer or `None`) – Maximum number of iterations (equals infinity if `None`). pexact (`None` or array or dictionary of numbers) – Fit-parameter values for the underlying distribution used to generate simulated data; replaced by `self.pmean` if is `None` (default). bootstrap (bool) – Vary prior means if `True`; otherwise vary only the means in `self.y` (default).
Returns:	An iterator that returns `lsqfit.nonlinear_fit`s for different simulated data.

Note that additional keywords can be added to overwrite keyword arguments in lsqfit.nonlinear_fit.

bootstrap_iter(n=None, datalist=None)¶

Iterator that returns bootstrap copies of a fit.

A bootstrap analysis involves three steps: 1) make a large number of “bootstrap copies” of the original input data and prior that differ from each other by random amounts characteristic of the underlying randomness in the original data; 2) repeat the entire fit analysis for each bootstrap copy of the data, extracting fit results from each; and 3) use the variation of the fit results from bootstrap copy to bootstrap copy to determine an approximate probability distribution (possibly non-gaussian) for the fit parameters and/or functions of them: the results from each bootstrap fit are samples from that distribution.

Bootstrap copies of the data for step 2 are provided in datalist. If datalist is None, they are generated instead from the means and covariance matrix of the fit data (assuming gaussian statistics). The maximum number of bootstrap copies considered is specified by n (None implies no limit).

Variations in the best-fit parameters (or functions of them) from bootstrap fit to bootstrap fit define the probability distributions for those quantities. For example, one could use the following code to analyze the distribution of function g(p) of the fit parameters:

fit = nonlinear_fit(...)

...

glist = []
for sfit in fit.bootstrapped_fit_iter(
    n=100, datalist=datalist, bootstrap=True
    ):
    glist.append(g(sfit.pmean))

... analyze samples glist[i] from g(p) distribution ...

This code generates n=100 samples glist[i] from the probability distribution of g(p). If everything is Gaussian, the mean and standard deviation of glist[i] should agree with g(fit.p).mean and g(fit.p).sdev.

Parameters:	n (integer) – Maximum number of iterations if `n` is not `None`; otherwise there is no maximum. datalist (sequence or iterator or `None`) – Collection of bootstrap `data` sets for fitter.
Returns:	Iterator that returns an `lsqfit.nonlinear_fit` object containing results from the fit to the next data set in `datalist`

dump_p(filename)¶

Dump parameter values (fit.p) into file filename.

fit.dump_p(filename) saves the best-fit parameter values (fit.p) from a nonlinear_fit called fit. These values are recovered using p = nonlinear_fit.load_parameters(filename) where p‘s layout is the same as that of fit.p.

dump_pmean(filename)¶

Dump parameter means (fit.pmean) into file filename.

fit.dump_pmean(filename) saves the means of the best-fit parameter values (fit.pmean) from a nonlinear_fit called fit. These values are recovered using p0 = nonlinear_fit.load_parameters(filename) where p0‘s layout is the same as fit.pmean. The saved values can be used to initialize a later fit (nonlinear_fit parameter p0).

static load_parameters(filename)¶

Load parameters stored in file filename.

p = nonlinear_fit.load_p(filename) is used to recover the values of fit parameters dumped using fit.dump_p(filename) (or fit.dump_pmean(filename)) where fit is of type lsqfit.nonlinear_fit. The layout of the returned parameters p is the same as that of fit.p (or fit.pmean).

check_roundoff(rtol=0.25, atol=1e-6)¶

Check for roundoff errors in fit.p.

Compares standard deviations from fit.p and fit.palt to see if they agree to within relative tolerance rtol and absolute tolerance atol. Generates a warning if they do not (in which case an svd cut might be advisable).

Functions¶

lsqfit.empbayes_fit(z0, fitargs, **minargs)¶

Call lsqfit.nonlinear_fit(**fitargs(z)) varying z, starting at z0, to maximize logGBF (empirical Bayes procedure).

The fit is redone for each value of z that is tried, in order to determine logGBF.

Parameters:	z0 (array) – Starting point for search. fitargs (function) – Function of array `z` that determines which fit parameters to use. The function returns these as an argument dictionary for `lsqfit.nonlinear_fit()`. minargs (dictionary) – Optional argument dictionary, passed on to `lsqfit.multiminex`, which finds the minimum.
Returns:	A tuple containing the best fit (object of type `lsqfit.nonlinear_fit`) and the optimal value for parameter `z`.

lsqfit.wavg(dataseq, prior=None, fast=False, **kargs)¶

Weighted average of gvar.GVars or arrays/dicts of gvar.GVars.

The weighted average of several gvar.GVars is what one obtains from a least-squares fit of the collection of gvar.GVars to the one-parameter fit function

def f(p):
    return N * [p[0]]

where N is the number of gvar.GVars. The average is the best-fit value for p[0]. gvar.GVars with smaller standard deviations carry more weight than those with larger standard deviations. The averages computed by wavg take account of correlations between the gvar.GVars.

If prior is not None, it is added to the list of data used in the average. Thus wavg([x2, x3], prior=x1) is the same as wavg([x1, x2, x3]).

Typical usage is

x1 = gvar.gvar(...)
x2 = gvar.gvar(...)
x3 = gvar.gvar(...)
xavg = wavg([x1, x2, x3])   # weighted average of x1, x2 and x3

where the result xavg is a gvar.GVar containing the weighted average.

The individual gvar.GVars in the last example can be replaced by multidimensional distributions, represented by arrays of gvar.GVars or dictionaries of gvar.GVars (or arrays of gvar.GVars). For example,

x1 = [gvar.gvar(...), gvar.gvar(...)]
x2 = [gvar.gvar(...), gvar.gvar(...)]
x3 = [gvar.gvar(...), gvar.gvar(...)]
xavg = wavg([x1, x2, x3])
    # xavg[i] is wgtd avg of x1[i], x2[i], x3[i]

where each array x1, x2 ... must have the same shape. The result xavg in this case is an array of gvar.GVars, where the shape of the array is the same as that of x1, etc.

Another example is

x1 = dict(a=[gvar.gvar(...), gvar.gvar(...)], b=gvar.gvar(...))
x2 = dict(a=[gvar.gvar(...), gvar.gvar(...)], b=gvar.gvar(...))
x3 = dict(a=[gvar.gvar(...), gvar.gvar(...)])
xavg = wavg([x1, x2, x3])
    # xavg['a'][i] is wgtd avg of x1['a'][i], x2['a'][i], x3['a'][i]
    # xavg['b'] is gtd avg of x1['b'], x2['b']

where different dictionaries can have (some) different keys. Here the result xavg is a gvar.BufferDict` having the same keys as x1, etc.

Weighted averages can become costly when the number of random samples being averaged is large (100s or more). In such cases it might be useful to set parameter fast=True. This causes wavg to estimate the weighted average by incorporating the random samples one at a time into a running average:

result = prior
for dataseq_i in dataseq:
    result = wavg([result, dataseq_i], ...)

This method is much faster when len(dataseq) is large, and gives the exact result when there are no correlations between different elements of list dataseq. The results are approximately correct when dataseq[i] and dataseq[j] are correlated for i!=j.

Parameters:

dataseq – The gvar.GVars to be averaged. dataseq is a one-dimensional sequence of gvar.GVars, or of arrays of gvar.GVars, or of dictionaries containing gvar.GVars or arrays of gvar.GVars. All dataseq[i] must have the same shape.
prior (gvar.GVar or array/dictionary of gvar.GVars) – Prior values for the averages, to be included in the weighted average. Default value is None, in which case prior is ignored.
fast (bool) – Setting fast=True causes wavg to compute an approximation to the weighted average that is much faster to calculate when averaging a large number of samples (100s or more). The default is fast=False.
kargs (dict) – Additional arguments (e.g., svdcut) to the fitter used to do the averaging.

Results returned by gvar.wavg() have the following extra attributes describing the average:

lsqfit.chi2¶: chi**2 for weighted average.

lsqfit.dof¶: Effective number of degrees of freedom.

lsqfit.Q¶

The probability that the chi**2 could have been larger, by chance, assuming that the data are all Gaussain and consistent with each other. Values smaller than 0.1 or suggest that the data are not Gaussian or are inconsistent with each other. Also called the p-value.

Quality factor Q (or p-value) for fit.

lsqfit.time¶: Time required to do average.

lsqfit.svdcorrection¶: The svd corrections made to the data when svdcut is not None.

lsqfit.fit¶: Fit output from average.

lsqfit.gammaQ()¶

Return the normalized incomplete gamma function Q(a,x) = 1-P(a,x).

Q(a, x) = 1/Gamma(a) * \int_x^\infty dt exp(-t) t ** (a-1) = 1 - P(a, x)

Note that gammaQ(ndof/2., chi2/2.) is the probabilty that one could get a chi**2 larger than chi2 with ndof degrees of freedom even if the model used to construct chi2 is correct.

gvar.add_parameter_distribution(name, invfcn)¶

Add new parameter distribution for use in fits.

This function adds new distributions for the parameters used in lsqfit.nonlinear_fit. For example, the code

import gvar as gv
gv.add_parameter_distribution('log', gv.exp)

enables the use of log-normal distributions for parameters. The log-normal distribution is invoked for a parameter p by including log(p) rather than p itself in the fit prior. log-normal, sqrt-normal, and erfinv-normal distributions are included by default. (Setting a prior prior[erfinv(w)] equal to gv.gvar('0(1)') / gv.sqrt(2) means that the prior probability for w is distributed uniformly between -1 and 1, and is zero elsewhere.)

These distributions are implemented by replacing a fit parameter p by a new fit parameter fcn(p) where fcn is some function. fcn(p) is assumed to have a Gaussian distribution, and parameter p is recovered using the inverse function invfcn where p=invfcn(fcn(p)).

Parameters:	name (str) – Distribution’s name. invfcn – Inverse of the transformation function.

gvar.del_parameter_distribution(name)¶: Delete parameter distribution name.

gvar.add_parameter_parentheses(p)¶

Return dictionary with proper keys for parameter distributions (legacy code).

This utility function helps fix legacy code that uses parameter keys like logp or sqrtp instead of log(p) or sqrt(p), as now required. This method creates a copy of dictionary p'' but with keys like ``logp or sqrtp replaced by log(p) or sqrt(p). So setting

p = add_parameter_parentheses(p)

fixes the keys in p for log-normal and sqrt-normal parameters.

Classes for Bayesian Integrals¶

class lsqfit.BayesPDF(fit, svdcut=1e-15)¶

Bayesian probability density function corresponding to nonlinear_fit fit.

The probability density function is the exponential of -1/2 times the chi**2 function (data and priors) used in fit divided by norm.

Parameters:

fit – Fit from nonlinear_fit.
svdcut (non-negative float or None) – If not None, replace covariance matrix of g with a new matrix whose small eigenvalues are modified: eigenvalues smaller than svdcut times the maximum eigenvalue eig_max are replaced by svdcut*eig_max. This can ameliorate problems caused by roundoff errors when inverting the covariance matrix. It increases the uncertainty associated with the modified eigenvalues and so is conservative. Setting svdcut=None or svdcut=0 leaves the covariance matrix unchanged. Default is 1e-15.

__call__(p)¶: Probability density function evaluated at p.

logpdf(p)¶: Logarithm of the probability density function evaluated at p.

class lsqfit.BayesIntegrator(fit, limit=1e15, scale=1, pdf=None, svdcut=1e-15)¶

vegas integrator for Bayesian fit integrals.

Parameters:

fit – Fit from nonlinear_fit.
limit (positive float) – Limits the integrations to a finite region of size limit times the standard deviation on either side of the mean. This can be useful if the functions being integrated misbehave for large parameter values (e.g., numpy.exp overflows for a large range of arguments). Default is 1e15.
scale (positive float) – The integration variables are rescaled to emphasize parameter values of order scale times the corresponding standard deviations. The rescaling does not change the value of the integral but it can reduce uncertainties in the vegas estimates. Default is 1.0.
pdf (function) – Probability density function pdf(p) of the fit parameters to use in place of the normal PDF associated with the least-squares fit used to create the integrator.
svdcut (non-negative float or None) – If not None, replace covariance matrix of g with a new matrix whose small eigenvalues are modified: eigenvalues smaller than svdcut times the maximum eigenvalue eig_max are replaced by svdcut*eig_max. This can ameliorate problems caused by roundoff errors when inverting the covariance matrix. It increases the uncertainty associated with the modified eigenvalues and so is conservative. Setting svdcut=None or svdcut=0 leaves the covariance matrix unchanged. Default is 1e-15.

BayesIntegrator(fit) is a vegas integrator that evaluates expectation values for the multi-dimensional Bayesian distribution associated with nonlinear_fit fit: the probability density function is the exponential of the chi**2 function (times -1/2), for data and priors, used in the fit. For linear fits, it is equivalent to gvar.PDFIntegrator(fit.p), since the chi**2 function is quadratic in the fit parameters; but they can differ significantly for nonlinear fits.

BayesIntegrator integrates over the entire parameter space but first reexpresses the integrals in terms of variables that diagonalize the covariance matrix of the best-fit parameters fit.p from nonlinear_fit and are centered at the best-fit values. This greatly facilitates the integration using vegas, making integrals over 10s or more of parameters feasible. (The vegas module must be installed separately in order to use BayesIntegrator.)

A simple illustration of BayesIntegrator is given by the following code, which we use to evaluate the mean and standard deviation for s*g where s and g are fit parameters:

import lsqfit
import gvar as gv
import numpy as np

# least-squares fit
x = np.array([0.1, 1.2, 1.9, 3.5])
y = gv.gvar(['1.2(1.0)', '2.4(1)', '2.0(1.2)', '5.2(3.2)'])
prior = gv.gvar(dict(a='0(5)', s='0(2)', g='2(2)'))
def f(x, p):
    return p['a'] + p['s'] * x ** p['g']
fit = lsqfit.nonlinear_fit(data=(x,y), prior=prior, fcn=f, debug=True)
print(fit)

# Bayesian integral to evaluate expectation value of s*g
def g(p):
    sg = p['s'] * p['g']
    return [sg, sg**2]

expval = lsqfit.BayesIntegrator(fit, limit=20.)
warmup = expval(neval=4000, nitn=10)
results = expval(g, neval=4000, nitn=15, adapt=False)
print(results.summary())
print('results =', results, '\n')
sg, sg2 = results
sg_sdev = (sg2 - sg**2) ** 0.5
print('s*g from Bayes integral:  mean =', sg, '  sdev =', sg_sdev)
print('s*g from fit:', fit.p['s'] * fit.p['g'])

where the warmup calls to the integrator are used to adapt it to probability density function from the fit, and then the integrator is used to evaluate the expectation value of g(p), which is returned in array results. Here neval is the (approximate) number of function calls per iteration of the vegas algorithm and nitn is the number of iterations. We use the integrator to calculated the expectation value of s*g and (s*g)**2 so we can compute a mean and standard deviation.

The output from this code shows that the Gaussian approximation for s*g (0.76(66)) is somewhat different from the result obtained from a Bayesian integral (0.48(54)):

Least Square Fit:
  chi2/dof [dof] = 0.32 [4]    Q = 0.87    logGBF = -9.2027

Parameters:
              a    1.61 (90)     [  0.0 (5.0) ]
              s    0.62 (81)     [  0.0 (2.0) ]
              g    1.2 (1.1)     [  2.0 (2.0) ]

Settings:
  svdcut/n = 1e-15/0    reltol/abstol = 0.0001/0*    (itns/time = 10/0.0)

itn   integral        average         chi2/dof        Q
-------------------------------------------------------
  1   1.034(21)       1.034(21)           0.00     1.00
  2   1.034(21)       1.034(15)           0.56     0.64
  3   1.024(18)       1.030(12)           0.37     0.90
  4   1.010(18)       1.0254(98)          0.47     0.89
  5   1.005(17)       1.0213(85)          0.55     0.88
  6   1.013(19)       1.0199(78)          0.69     0.80
  7   0.987(16)       1.0152(70)          0.78     0.72
  8   1.002(18)       1.0135(66)          0.90     0.59
  9   1.036(20)       1.0160(62)          0.86     0.66
 10   1.060(20)       1.0204(60)          0.94     0.55

results = [0.4837(32) 0.5259(47)]

s*g from Bayes integral:  mean = 0.4837(32)   sdev = 0.5403(25)
s*g from fit: 0.78(66)

The table shows estimates of the probability density function’s normalization from each of the vegas iterations used by the integrator to estimate the final results.

In general functions being integrated can return a number, or an array of numbers, or a dictionary whose values are numbers or arrays of numbers. This allows multiple expectation values to be evaluated simultaneously.

See the documentation with the vegas module for more details on its use, and on the attributes and methods associated with integrators. The example above sets adapt=False when computing final results. This gives more reliable error estimates when neval is small. Note that neval may need to be much larger (tens or hundreds of thousands) for more difficult high-dimension integrals.

__call__(f=None, mpi=False, pdf=None, **kargs)¶

Estimate expectation value of function f(p).

Uses multi-dimensional integration modules vegas to estimate the expectation value of f(p) with respect to the probability density function associated with nonlinear_fit fit. Setting mpi=True configures vegas for multi-processor running using MPI.

Parameters:

f (function) – Function f(p) to integrate. Integral is the expectation value of the function with respect to the distribution. The function can return a number, an array of numbers, or a dictionary whose values are numbers or arrays of numbers.
mpi (bool) –
If True configure for use with multiple processors and MPI. This option requires module mpi4py. A script xxx.py using an MPI integrator is run with mpirun: e.g.,
```
mpirun -np 4 -output-filename xxx.out python xxx.py
```
runs on 4 processors. Setting mpi=False (default) does not support multiple processors. The MPI processor rank can be obtained from the mpi_rank attribute of the integrator.
pdf (function) – Probability density function pdf(p) of the fit parameters to use in place of the normal PDF associated with the least-squares fit used to create the integrator.

All other keyword arguments are passed on to a vegas integrator; see the vegas documentation for further information.

Other Classes¶

class lsqfit.multifit(x0, n, f, tol=1e-4, maxit=1000, alg='lmsder', analyzer=None)¶

Fitter for nonlinear least-squares multidimensional fits.

Parameters:

x0 (numpy array of floats) – Starting point for minimization.
n (positive integer) – Length of vector returned by the fit function f(x).
f (function) – Fit function: multifit minimizes sum_i f_i(x)**2 by varying parameters x. The parameters are a 1-d numpy array of either numbers or gvar.GVars.
tol (tuple or float) –
Setting tol=(reltol, abstol) causes the fit to stop searching for a solution when |dx_i| <= abstol + reltol * |x_i|. With version 2 or higher of the GSL library, tol=(xtol, gtol, ftol) can be used, where the fit stops when any one of the following three criteria is satisfied:
1. step size small: |dx_i| <= xtol * (xtol + |x_i|);
2. gradient small: ||g . x||_inf <= gtol * ||f||^2;
3. residuals small: ||f(x+dx) - f(x)|| <= ftol * max(||f(x)||, 1).
Recommended values are: xtol=1/10**d for d digits of precision in the parameters; gtol=1e-6 to account for roundoff errors in gradient g (unless the second order derivative vanishes at minimum as well, in which case gtol=0 might be good); and ftol<<1. Setting tol=reltol is equivalent to setting tol=(reltol, 0.0). The default setting is tol=0.0001.
maxit (integer) – Maximum number of iterations in search for minimum; default is 1000.
alg (string) – GSL algorithm to use for minimization. Two options are currently available: "lmsder", the scaled LMDER algorithm (default); and "lmder", the unscaled LMDER algorithm. With version 2 of the GSL library, another option is "lmniel", which can be useful when there is much more data than parameters.
analyzer (function) – Optional function of x, [...f_i(x)...], [[..df_ij(x)..]] which is called after each iteration. This can be used to inspect intermediate steps in the minimization, if needed.

multifit is a function-class whose constructor does a least squares fit by minimizing sum_i f_i(x)**2 as a function of vector x. The following attributes are available:

x¶: Location of the most recently computed (best) fit point.

cov¶: Covariance matrix at the minimum point.

f¶: The fit function f(x) at the minimum in the most recent fit.

J¶: Gradient J_ij = df_i/dx[j] for most recent fit.

nit¶: Number of iterations used in last fit to find the minimum.

stopping_criterion¶

Criterion used to stop fit:: 0 => didn’t converge 1 => step size small 2 => gradient small 3 => residuals small

error¶: None if fit successful; an error message otherwise.

multifit is a wrapper for the multifit GSL routine.

class lsqfit.multiminex(x0, f, tol=1e-4, maxit=1000, step=1, alg='nmsimplex2', analyzer=None)¶

Minimizer for multidimensional functions.

Parameters:

x0 (numpy array of floats) – Starting point for minimization search.
f (function) – Function f(x) to be minimized by varying vector x.
tol (float) – Minimization stops when x has converged to with tolerance tol; default is 1e-4.
maxit (integer) – Maximum number of iterations in search for minimum; default is 1000.
step (number) – Initial step size to use in varying components of x; default is 1.
alg (string) – GSL algorithm to use for minimization. Three options are currently available: "nmsimplex", Nelder Mead Simplex algorithm; "nmsimplex2", an improved version of "nmsimplex" (default); and "nmsimplex2rand", a version of "nmsimplex2" with random shifts in the start position.
analyzer (function) – Optional function of x, f(x), it, where it is the iteration number, which is called after each iteration. This can be used to inspect intermediate steps in the minimization, if needed.

multiminex is a function-class whose constructor minimizes a multidimensional function f(x) by varying vector x. This routine does not use user-supplied information about the gradient of f(x). The following attributes are available:

x¶: Location of the most recently computed minimum (1-d array).

f¶: Value of function f(x) at the most recently computed minimum.

nit¶: Number of iterations required to find most recent minimum.

error¶: None if fit successful; an error message otherwise.

multiminex is a wrapper for the multimin GSL routine.

Requirements¶

lsqfit relies heavily on the gvar, and numpy modules. Several utility functions are in lsqfit_util. Also the minimization routines are from the Gnu Scientific Library (GSL).

`lsqfit` - Nonlinear Least Squares Fitting¶

Introduction¶

nonlinear_fit Objects¶

Functions¶

Classes for Bayesian Integrals¶

Other Classes¶

Requirements¶

Table Of Contents

Previous topic

lsqfit - Nonlinear Least Squares Fitting¶

Introduction¶

nonlinear_fit Objects¶

Functions¶

Classes for Bayesian Integrals¶

Other Classes¶

Requirements¶

`lsqfit` - Nonlinear Least Squares Fitting¶