`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 it can be omitted.

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 (as well as the correlations between them). 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. lsqfit also allows for log-normal and other 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, with key log(c). nonlinear_fit in effect adds parameter c to the parameter dictionary, deriving its value from parameter log(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. Additional distributions can be added using gvar.BufferDict.add_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, svdcut=1e-12, eps=None, noise=False, debug=False, tol=1e-8, maxit=1000, fitter='gsl_multifit', **fitterargs)¶

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 (or function evaluations), 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 (dict, array or tuple) –
Data to be fit by lsqfit.nonlinear_fit can have one of the following forms:

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.

Setting x=False in the first 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 (callable) – 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 (dict, array, str, gvar.GVar 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 (dict, array, float, None, or True) – 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. If p0 is True, it is replaced by a starting point drawn at random from the prior distribution. The default value is None; p0 must be explicitly specified if prior is None.
linear (list or None) – Optional list of fit parameters that appear linearly in the fit function. The fit function can be reexpressed (using variable projection) as a function that is independent of its linear parameters. The resulting fit has fewer fit parameters and typically will converge in fewer iterations, but each iteration will take longer. Whether or not the fit is faster or more robust in any particular application is a matter for experiment, but answers should be the same either way. The linear parameters are reconstructed from the nonlinear parameters (and the data) after the fit. Parameter linear is either: a list of dictionary keys corresponding to linear parameters when the parameters are stored in a dictionary (see prior); or, a list of indices corresponding to these parameters when they are stored in an array. Note that this feature is experimental; the interface may change in the future.
svdcut (float) – If nonzero, singularities in the correlation matrix for y and prior are regulated using gvar.regulate() with an SVD cutoff svdcut. This makes the correlation matrices less singular, which can improve the stability and accuracy of a fit. Default is svdcut=1e-12.
eps (float) – If positive, singularities in the correlation matrix for y and prior are regulated using gvar.regulate() with cutoff eps. This makes the correlation matrices less singular, which can improve the stability and accuracy of a fit. Ignored if svdcut is specified (and not None).
noise (tuple or bool) – If noise[0]=True, noise is added to the data means commensurate with the additional uncertainties introduced by using svdcut>0 or eps>0. If noise[1]=True, noise is added to the prior means commensurate with the uncertainties in the prior. Noise is useful for testing the quality of a fit (chi2). Setting noise=True is shorthand for noise=(True, True), and noise=False means noise=(False, False) (the default).
udata (dict, array or tuple) – Same as data but instructs the fitter to ignore correlations between different pieces of data. This speeds up the fit, particularly for large amounts of data, but ignores potentially valuable information if the data actually are correlated. Only one of data or udata should be specified. (Default is None.)
fitter (str or None) – Fitter code. Options if GSL is installed include: 'gsl_multifit' (default) and 'gsl_v1_multifit' (original fitter). Options if scipy is installed include: 'scipy_least_squares' (default if GSL not installed). gsl_multifit has many options, providing extensive user control. scipy_least_squares can be used for fits where the parameters are bounded. (Bounded parameters can also be implemented, for any of the fitters, using non-Gaussian priors — see the tutorial.)
tol (float or tuple) –
Assigning tol=(xtol, gtol, ftol) causes the fit to stop searching for a minimum when any of
1. xtol >= relative change in parameters between iterations
2. gtol >= relative size of gradient of chi**2 function
3. ftol >= relative change in chi**2 between iterations
is satisfied. See the fitter documentation for detailed definitions of these stopping conditions. Typically one sets xtol=1/10**d where d is the number of digits of precision desired in the result, while gtol<<1 and ftol<<1. Setting tol=delta where delta is a number is equivalent to setting tol=(delta,1e-10,1e-10). Setting tol=(delta1,delta2) is equivalent to setting tol=(delta1,delta2,1e-10). Default is tol=1e-8. (Note: the ftol option is disabled in some versions of the GSL library.)
maxit (int) – Maximum number of algorithm iterations (or function evaluations for some fitters) in search for minimum; default is 1000.
debug (bool) – Set to True for extra debugging of the fit function and a check for roundoff errors. (Default is False.)
fitterargs (dict) – Dictionary of additional arguments passed through to the underlying fitter. Different fitters offer different parameters; see the documentation for each.

Objects of type lsqfit.nonlinear_fit have the following attributes:

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 actual fluctuations in the input data and/or priors might be smaller than suggested by the standard deviations (or covariances) used in the fit.

Type:: float

cov¶

Covariance matrix of the best-fit parameters from the fit.

Type:: array

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.

Type:: int

error¶

Error message generated by the underlying fitter when an error occurs. None otherwise.

Type:: str

fitter_results¶: Results returned by the underlying fitter. Refer to the appropriate fitter’s documentation for details.

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 — that is, it is P(data|model). 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 preferred 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.

Type:: float or None

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.

Type:: dict, array or gvar.GVar

pmean¶

Means of the best-fit parameters from fit.

Type:: dict, array or float

psdev¶

Standard deviations of the best-fit parameters from fit.

Type:: dict, array or float

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 refitting with a nonzero value for eps or svdcut is advisable.

Type:: dict, array or gvar.GVar

p0¶

The parameter values used to start the fit. This will differ from the input p0 if the latter was incomplete.

Type:: dict, array or float

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.

Type:: dict, array, gvar.GVar or None

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. The probabilistic intrepretation becomes unreliable if the actual fluctuations in the input data and/or priors are much smaller than suggested by the standard deviations (or covariances) used in the fit (leading to an unusually small chi**2).

Type:: float or None

residuals¶: An array containing the fit residuals normalized by the corresponding standard deviations. The residuals are projected onto the eigenvectors of the correlation matrix and so should be uncorrelated from each other. The residuals include contributions from both the fit data and the prior. They are related to the the chi**2 of the fit by: chi2 = sum(fit.residuals**2).

stopping_criterion¶

Criterion used to stop fit:

0: didn’t converge

1: xtol >= relative change in parameters between iterations

2: gtol >= relative size of gradient of chi**2

3: ftol >= relative change in chi**2 between iterations

4: unable to improve fit further (e.g., already converged)

Type:: int

correction¶

Sum of all corrections, if any, added to the fit data and prior when eps>0 or svdcut>0.

Type:: gvar.GVar

svdn¶

Number of eigenmodes of the correlation matrix modified (and/or deleted) when svdcut>0.

Type:: int

time¶

CPU time (in secs) taken by fit.

Type:: float

tol¶

Tolerance used in fit. This differs from the input tolerance if the latter was incompletely specified.

Type:: tuple

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).)

Type:: obj

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.

Type:: dict, array or gvar.GVar

nblocks¶

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.

Type:: dict

The global defaults used by lsqfit.nonlinear_fit can be changed by changing entries in dictionary lsqfit.nonlinear_fit.DEFAULTS for keys 'eps', 'svdcut', 'debug', 'tol', 'noise', 'maxit', and 'fitter'. Additional defaults can be added to that dictionary to be are passed through lsqfit.nonlinear_fit to the underlying fitter (via dictionary fitterargs).

Additional methods are provided for printing out detailed information about the fit, evaluating chi**2, 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 Q value of the fit (ie, p-value), and the logarithm of the Gaussian Bayes Factor for the fit (logGBF). 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. It also lists information about the fitter used if it is other than the standard choice.

Optionally, format 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). Lines for parameters designated as linear (see linear keyword) are marked with a minus sign after their prior.

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.

Parameters:

maxline (int 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 (str or None) – 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. Setting pstyle=None implies no parameters are listed.
extend (bool) – If True, extend the parameter list to include values derived from log-normal or other non-Gaussian parameters. So values for fit parameter p['log(a)'], for example, are listed together with values p['a'] for the exponential of the fit parameter. Setting extend=False means that only the value for p['log(a)'] is listed. Default is True.

Returns:

String containing detailed information about fit.

dchi2(p)¶

chi**2(p) - fit.chi2 for fit parameters p.

Paramters:

p: Array or dictionary containing values for fit parameters, using: the same layout as in the fit function.

Returns:

chi**2(p) - fit.chi2 where chi**2(p) is the fit’s chi**2 for fit parameters p and fit.chi2 is the chi**2 value for the best fit.

pdf(p)¶

exp(-(chi**2(p) - fit.chi2)/2) for fit parameters p.

fit.pdf(p) is proportional to the probability density function (PDF) used in the fit: fit.pdf(p)/exp(fit.pdf.lognorm) is the product of the Gaussian PDF for the data P(data|p,M) times the Gaussian PDF for the prior P(p|M) where M is the model used in the fit (i.e., the fit function and prior). The product of PDFs is P(data,p|M) by Bayes’ Theorem; integrating over fit parameters p gives the Bayes Factor or Evidence P(data|M), which is proportional to the probability that the fit data come from fit model M. The logarithm of the Bayes Factor should agree with fit.logGBF when the Gaussian approximation assumed in the fit is accurate.

fit.pdf(p) is useful for checking a least-squares fit against the corresponding Bayesian integrals. In the following example, vegas.PDFIntegrator from the vegas module is used to evaluate Bayesian expectation values of s*g and its standard deviation where s and g are fit parameters:

import gvar as gv
import lsqfit
import numpy as np
import vegas

def main():
    # 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)'))
    fit = lsqfit.nonlinear_fit(data=(x,y), prior=prior, fcn=fitfcn, debug=True)
    print(fit)

    # create integrator and adapt it to PDF (warmup)
    neval = 10_000
    nitn = 10
    expval = vegas.PDFIntegrator(fit.p, pdf=fit.pdf, nproc=4)
    warmup = expval(neval=neval, nitn=nitn)

    # calculate expectation value of g(p)
    results = expval(g, neval=neval, nitn=nitn, adapt=False)
    print(results.summary(True))
    print('results =', results, '\n')

    sg = results['sg']
    sg2 = results['sg2']
    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'])
    print()
    print('logBF =', np.log(results.pdfnorm) - fit.pdf.lognorm)

def fitfcn(x, p):
    return p['a'] + p['s'] * x ** p['g']

def g(p):
    sg = p['s'] * p['g']
    return dict(sg=sg, sg2=sg**2)

if __name__ == '__main__':
    main()

Here the probability density function used for the expectation values is fit.pdf(p), and the expectation values are returned in dictionary results. vegas uses adaptive Monte Carlo integration. The warmup calls to the integrator are used to adapt it to the probability density function, and then the adapted integrator is called again to evaluate the expectation value. Parameter 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 (0.78(66)) is somewhat different from the result obtained from a Bayesian integral (0.49(53)):

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-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 18/0.0)

itn   integral        average         chi2/dof        Q
-------------------------------------------------------
 1   0.954(11)       0.954(11)           0.00     1.00
 2   0.9708(99)      0.9622(74)          0.74     0.53
 3   0.964(12)       0.9627(63)          0.93     0.47
 4   0.9620(93)      0.9626(52)          0.86     0.56
 5   0.964(14)       0.9629(50)          0.71     0.74
 6   0.957(17)       0.9619(50)          0.65     0.84
 7   0.964(12)       0.9622(46)          0.61     0.90
 8   0.9367(86)      0.9590(42)          0.80     0.73
 9   0.9592(94)      0.9591(39)          0.75     0.80
10   0.952(13)       0.9584(37)          0.72     0.85

            key/index          value
------------------------------------
                pdf    0.9584 (37)
 ('f(p)*pdf', 'sg')    0.4652 (23)
('f(p)*pdf', 'sg2')    0.5073 (33)

results = {'sg': 0.4854(20), 'sg2': 0.5293(33)}

s*g from Bayes integral:  mean = 0.4854(20)   sdev = 0.5420(17)
s*g from fit: 0.78(66)

logBF = -9.1505(39)

The result logBF for the logarithm of the Bayes Factor from the integral agrees well with fit.logGBF, the log Bayes Factor in the Gaussian approximation. This is evidence that the Gaussian approximation implicit in the least squares fit is reliable; the product of s*g, however, is not so Gaussian because of the large uncertainties (compared to the means) in s and g separately.

Paramters:

p: Array or dictionary containing values for fit parameters, using: the same layout as in the fit function.

Returns:

exp(-(chi**2(p) - fit.chi2)/2) where chi**2(p) is the fit’s chi**2 for fit parameters p and fit.chi2 is the chi**2 value for the best fit.

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

Iterator that returns simulation copies of a fit.

Fit reliability is 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 has a good chi**2 ...
    ... 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.

Setting parameter add_priornoise=True varies the means of the priors as well as the means of the data. This option is useful for testing goodness of fit because with it chi**2/N should be 1 ± sqrt(2/N), where N is the number of degrees of freedom. (chi**2/N can be significantly smaller than one without added noise in prior means.)

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.

Parameters:

n (int or None) – Maximum number of iterations (equals infinity if None).
pexact (None or array/dict of numbers) – Fit-parameter values for the underlying distribution used to generate simulated data; replaced by self.pmean if is None (default).
add_priornoise (bool) – Vary prior means if True; otherwise vary only the means in self.y (default).
kargs – Dictionary containing override values for fit parameters.

Returns:

An iterator that returns lsqfit.nonlinear_fits for different simulated data.

simulated_data_iter(n=None, pexact=None, add_priornoise=False)¶

Iterator that returns simulated data based upon a fit’s data.

Simulated data is generated from a fit’s data fit.y by replacing the mean values in that data with random numbers drawn from a distribution whose mean is self.fcn(pexact) and whose covariance matrix is the same as that of self.y. Each iteration of the iterator returns new simulated data, with different random numbers for the means and a covariance matrix equal to that of self.y. This iterator is used by self.simulated_fit_iter.

Typical usage:

fit = nonlinear_fit(data=(x,y), prior=prior, fcn=fcn)
...
for ysim, priorsim in fit.simulate_data_iter(n=10):
    fitsim = nonlinear_fit(data=(x, ysim), prior=priorsim, fcn=fcn)
    print(fitsim)
    print('chi2 =', gv.chi2(fit.p, fitsim.p))

This code tests the fitting protocol on simulated data, comparing the best fit parameters in each case with the correct values (fit.p). The loop in this code is functionally the same as (but probably not as fast as):

for fitsim in fit.simulated_fit_iter(n=10):
    print(fitsim)
    print('chi2 =', gv.chi2(fit.p, fitsim.p))

Parameters:

n (int or None) – Maximum number of iterations (equals infinity if None).
pexact (None or dict/array of numbers) – Fit-parameter values for the underlying distribution used to generate simulated data; replaced by self.pmean if is None (default).
add_priornoise (bool) – Vary prior means if True; otherwise vary only the means in self.y (default).

Returns:

An iterator that returns a 2-tuple containing simulated versions of self.y and self.prior: (ysim, priorsim).

bootstrapped_fit_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 bsfit in fit.bootstrapped_fit_iter(
    n=100, datalist=datalist,
    ):
    glist.append(g(bsfit.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 (int) – Maximum number of iterations if n is not None; otherwise there is no maximum.
datalist (iter) – Collection of bootstrap data sets for fitter.
kargs (dict) – Overrides arguments in original fit.

Returns:

Iterator that returns an lsqfit.nonlinear_fit object containing results from the fit to the next data set in datalist.

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).

qqplot_residuals(plot=None)¶

QQ plot normalized fit residuals.

The sum of the squares of the residuals equals self.chi2. Individual residuals should be distributed in a Gaussian distribution centered about zero. A Q-Q plot orders the residuals and plots them against the value they would have if they were distributed according to a Gaussian distribution. The resulting plot will approximate a straight line along the diagonal of the plot (dashed black line) if the residuals have a Gaussian distribution with zero mean and unit standard deviation.

The residuals are fit to a straight line and the fit is displayed in the plot (solid red line). Residuals that fall on a straight line have a distribution that is Gaussian. A nonzero intercept indicates a bias in the mean, away from zero. A slope smaller than 1.0 indicates the actual standard deviation is smaller than suggested by the fit errors, as would be expected if the chi2/dof is significantly below 1.0 (since chi2 equals the sum of the squared residuals).

One way to display the plot is with:

fit.qqplot_residuals().show()

Parameters:: plot – a matplotlib plotter. If None, uses matplotlib.pyplot.
Returns:: Plotter plot.

This method requires the scipy and matplotlib modules.

plot_residuals(plot=None)¶

Plot normalized fit residuals.

The sum of the squares of the residuals equals self.chi2. Individual residuals should be distributed about one, in a Gaussian distribution.

Parameters:: plot – matplotlib plotter. If None, uses matplotlib.pyplot.
Returns:: Plotter plot.

static set(clear=False, **defaults)¶

Set default parameters for lsqfit.nonlinear_fit.

Use to set default values for parameters: eps, svdcut, debug, tol, maxit, and fitter. Can also set parameters specific to the fitter specified by the fitter argument.

Sample usage:

import lsqfit

old_defaults = lsqfit.nonlinear_fit.set(
    fitter='gsl_multifit', alg='subspace2D', solver='cholesky',
    tol=1e-10, debug=True,
    )

nonlinear_fit.set() without arguments returns a dictionary containing the current defaults.

Parameters:

clear (bool) – If True remove earlier settings, restoring the original defaults, before adding new defaults. The default value is clear=False. nonlinear_fit.set(clear=True) restores the original defaults.
defaults (dict) – Dictionary containing new defaults.

Returns:

A dictionary containing the old defaults, before they were updated. These can be restored using nonlinear_fit.set(old_defaults) where old_defaults is the dictionary containint the old defaults.

vegas_fit Objects¶

class lsqfit.vegas_fit(data, fcn, prior=None, param=None, fit=None, svdcut=1e-12, eps=None, noise=False, **vegasargs)¶

Least-squares fit using Bayesian integrals.

lsqfit.vegas_fit fits a (nonlinear) function f(x,p) (or f(p)) to data y using Bayesian integrals over fit parameters p. Typical usage is

vfit = vegas_fit(data=(x,y), fcn=f, prior=prior)
print(vfit)
print('best-fit parameters =', vfit.p)

The fitter calculates the means and (co)variances of the fit parameters (vfit.p) assuming that the parameters are described by a probability density function (PDF) proportional to $\exp(-\chi^2(p))$ where

$\chi^2(p) = \Delta y \cdot\mathrm{cov}^{-1}_y \cdot \Delta y \: + \: \Delta p \cdot\mathrm{cov}^{-1}_\mathrm{prior}\cdot\Delta p$

and $\Delta y_i \equiv \overline y_i - f(x_i,p)$ and $\Delta p_i\equiv \overline p_i - p_i$ . This involves a multi-dimensional integration over the parameter space using vegas.PDFIntegrator from the vegas module (which must be installed separately). vegas uses adaptive Monte Carlo integration to obtain estimates for the integrals; see its documentation for more information.

When the PDF is sufficiently peaked around its maximum, $\chi^2(p)$ is (usually) well approximated by a quadratic expansion around its minimum, and results obtained from lsqfit.vegas_fit will agree with those obtained from lsqfit.nonlinear_fit — the latter is the Gaussian approximation to the former. The output from nonlinear_fit can often be used to improve significantly the accuracy of the numerical integrals used by vegas_fit, particularly if the PDF is sharply peaked and there are lots of parameters: for example, by setting param=fit.p in

fit = nonlinear_fit(data=(x,y), fcn=f, prior=prior)
vfit = vegas_fit(data=(x,y), fcn=f, prior=prior, param=fit.p)
print(vfit)

we direct vegas_fit to re-express the integrals over p in terms of variables that emphasize the region indicated by fit.p. This facilitates the integration and can greatly reduce the numerical uncertainties in the results. Note that the second line in this code snippet can be written more succinctly as

vfit = vegas_fit(fit=fit)

vegas adapts iteratively to the PDF, averaging results over the iterations. By default it uses 10 iterations to train the integrator on the PDF, and then 10 more, without further adaptation, to estimate the means and covariances of the fit parameters. During the training stage, the integrator remaps the integrals to variables that emphasize regions where the PDF is large, refining the map after each iteration. Integral results from the training stage are often unreliable and so are discarded. (Adaptation is turned off for the latter iterations to provide more robust estimates; see the vegas documentation for more information.) The number of evaluations of f(x,p) is limited to at most 1000 by default. Both the number of iterations and the number of function evaluations can be specified using parameters nitn and neval, respectively: for example

vfit = vegas_fit(fit=fit, nitn=(5,10), neval=10_000)

specifies 5 iterations for adapting to the PDF (training) followed by 10 more for computing vfit.p, with at most 10,000 function evaluations per iteration. The number of function evaluations needed depends upon the number of parameters and how sharply peaked the PDF is.

Having obtained a fit vfit from vegas_fit, expectation values with respect to the PDF can be obtained for any function g(p) of the parameters, where g(p) typically returns an array of numbers or a dictionary whose values are numbers or arrays of numbers, so that multiple expectation values can be computed together:

s = vfit.stats(g)

where s is an array or dictionary of gvar.GVars giving the means and covariances of the components of g(p) with respect to the PDF. Result s agrees well with g(vfit.p) when the Gaussian approximation is valid (for the PDF and g(p)), but could be quite different otherwise. vfit.stats can also calculate other statistical parameters (e.g., skewness) and/or histograms for the distribution of g(p) values.

Parameters:

data (dict, array or tuple) –
Data to be fit by lsqfit.vegas_fit can have either of the following forms:

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.

Setting x=False in the first 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.) Ignored if parameter fit is specified.
fcn (callable) –
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 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 param.

vegas_fit is usually much faster if fcn is designed to process a large batch of integration points all at once. See the vegas documentation on rbatchintegrand and lbatchintegrand.
prior – A dictionary or array of gvar.GVars representing 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 param is None. Ignored if parameter fit is specified.
param – A dictionary or array of gvar.GVars that specifies the fit parameters p used by fcn(x,p) (or fcn(p)), and indicates where in that parameter space the integrator should focus its attention. Fit parameters p are stored in a dictionary or array with the same keys and structure (or shape) as param (and prior, if specified). vegas_fit re-expresses the parameter integrals in terms of variables that emphasize the region of parameter space covered by param. Setting param=None (the default) is equivalent to setting param=prior; prior must be defined if param=None. Ignored if parameter fit is specified.
fit – Fit results from either lsqfit.nonlinear_fit or lsqfit.vegas_fit. When fit is specified, the data, prior, and fit function are take from fit and param=fit.p is set. The fit function from fit can be replaced by setting the fcn parameter (for example, to replace fit.fcn by an equivalent batch function).
svdcut (float) – If nonzero, singularities in the correlation matrix for y and prior are regulated using gvar.regulate() with an SVD cutoff svdcut. This makes the correlation matrices less singular, which can improve the stability and accuracy of a fit. Default is svdcut=1e-12. Ignored if parameter fit is specified.
eps (float) – If positive, singularities in the correlation matrix for y and prior are regulated using gvar.regulate() with cutoff eps. This makes the correlation matrices less singular, which can improve the stability and accuracy of a fit. Ignored if svdcut is specified (and not None). Ignored if parameter fit is specified.
noise (tuple or bool) – If noise[0]=True, noise is added to the data means commensurate with the additional uncertainties (if any) introduced by using svdcut>0 or eps>0. If noise[1]=True, noise is added to the prior means commensurate with the uncertainties in the prior. Noise is useful for testing the quality of a fit (chi2). Setting noise=True is shorthand for noise=(True, True), and noise=False means noise=(False, False) (the default). Ignored if parameter fit is specified.
vegasargs – Any additional keyword argments are passed to the integrator, vegas.PDFIntegrator. The most important of these arguments are the number of vegas interations nitn and the maximum number neval of integrand evaluations per iteration. Default values for these are nitn=(10,10) and neval=1000, where nitn[0] is the number of iterations used to train the integrator to the PDF, and nitn[1] is the number of iterations used to determine the means and covariances of the parameters. (vegas adapts to the PDF in the first set of (training) iterations; adaptation is turned off for the second.)

Objects of type lsqfit.nonlinear_fit have the following attributes:

chi2¶: $\chi^2(p)$ evaluated at vfit.p. fit.chi2 / fit.dof is usually of order one in good fits. Values much less than one suggest that actual fluctuations in the input data and/or priors might be smaller than suggested by the standard deviations (or covariances) used in 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.

integrator¶: The vegas.PDFIntegrator used to do the integrals.

logBF¶: 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 — that is, it is the logarithm of P(data|model). 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.logBF is the one preferred by the data. The exponential of the difference in fit.logBF between two models is the ratio of probabilities (Bayes factor) for those models. Differences in fit.logBF smaller than 1 are not very significant.

p¶

Best-fit parameters from fit in the same format as the prior (array or dictionary containing gvar.GVars). The means and uncertainties (standard deviations/covariances) are calculated from the Bayesian integrals. The uncertainties come from the distribution and from uncertainties in vegas’s estimates of the means (added in quadrature).

vfit.p is output from vegas.PDFIntegrator.stats(). It has additional attributes that provide more information about the integrals. See the documentation for vegas.PDFEV, PDFEVArray, and vegas.PDFEVDict for more information.

pmean¶: An array or dictionary containing the means of the best-fit parameters from fit.

psdev¶: Standard deviations of the best-fit parameters from fit.

palt¶: Same as vfit.p.

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

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. The probabilistic intrepretation becomes unreliable if the actual fluctuations in the input data and/or priors are much smaller than suggested by the standard deviations (or covariances) used in the fit (leading to an unusually small chi**2).

residuals¶: An array containing the fit residuals normalized by the corresponding standard deviations. The residuals are projected onto the eigenvectors of the correlation matrix and so should be uncorrelated from each other. The residuals include contributions from both the fit data and the prior. They are related to the the chi**2 of the fit by: chi2 = sum(fit.residuals**2).

training¶: Results returned by the vegas.PDFIntegrator after evaluating the integrals used to train the integrator to the PDF. These results are not included in the final averages.

correction¶: Sum of all corrections, if any, added to the fit data and prior when eps>0 or svdcut>0.

svdn¶: Number of eigenmodes of the correlation matrix modified (and/or deleted) when svdcut>0.

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

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, with gvar.GVar for each data point. This may differ rom the input data if an SVD cut is used (it is y after the SVD cut). It is either a dictionary (gvar.BufferDict) or an array (numpy.ndarray), depending upon the input.

nblocks¶: 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.

Additional methods are provided for printing out detailed information about the fit, evaluating chi**2:

stats(f, moments=False, histograms=False)¶

Means and standard deviations (and covariances) of function f(p).

If f is set to None or omitted, the means and standard deviations (and covariances) of the fit parameters are recalculated.

See documentation for vegas.PDFIntegrator.stats() for further options.

sample(nbatch, mode='rbatch')¶

Generate random samples from PDF used in fit.

See documentation for vegas.PDFIntegrator.sample() for more information.

Parameters:

nbatch (int) – The integrator will return at least nbatch samples drawn from its PDF. The actual number of samples is the smallest multiple of self.last_neval that is equal to or larger than nbatch. Results are packaged in arrays or dictionaries whose elements have an extra index labeling the different samples in the batch. The batch index is the rightmost index if mode='rbatch'; it is the leftmost index if mode is 'lbatch'.
mode (bool) – Batch mode. Allowed modes are 'rbatch' or 'lbatch', corresponding to batch indices that are on the right or the left, respectively. Default is mode='rbatch'.

Returns:

A tuple (wgts,samples) containing samples drawn from the integrator’s PDF, together with their weights wgts. The weighted sample points are distributed through parameter space with a density proportional to the PDF.

In general, samples is either a dictionary or an array depending upon the format of lsqfit.vegas_fit parameter param. For example, if

param = gv.gvar(dict(s='1.5(1)', v=['3.2(8)', '1.1(4)']))

then samples['s'][i] is a sample for parameter p['s'] where index i=0,1...nbatch(approx) labels the sample. The corresponding sample for p['v'][d], where d=0 or 1, is samples['v'][d, i] provided mode='rbatch', which is the default. (Otherwise it is p['v'][i, d], for mode='lbatch'.) The corresponding weight for this sample is wgts[i].

When param is an array, samples is an array with the same shape plus an extra sample index which is either on the right (mode='rbatch', default) or left (mode='lbatch').

format(maxline=0, pstyle='v')¶

Format the output from a lsqfit.vegas_fit.

See the documentation for lsqfit.nonlinear_fit.format() for more information.

pdf(p)¶

Probability density function for fit parameters p.

When the lsqfit.vegas_fit object is derived from a lsqfit.nonlinear_fit object (vfit = vegas_fit(fit=fit)), the PDF is $\exp(-(\chi^2(p) - \chi^2_\mathrm{min})/2)$ . Otherwise it is just $\exp(-\chi^2(p)/2)$ . In either case it is not normalized. Divide by vfit.pdfnorm to normalize it.

qqplot_residuals(plot=None)¶

Create QQ plot of the fit residuals.

See the documentation for lsqfit.nonlinear_fit.qqplot_residuals() for more information.

plot_residuals(plot=None)¶

Create QQ plot of the fit residuals.

See the documentation for lsqfit.nonlinear_fit.plot_residuals() for more information.

Functions¶

lsqfit.empbayes_fit(z0, fitargs, p0=None, fitter=lsqfit.nonlinear_fit, **minargs)¶

Return fit and z corresponding to the fit lsqfit.nonlinear_fit(**fitargs(z)) that maximizes fit.logGBF.

This function maximizes the logarithm of the Bayes Factor from fit lsqfit.nonlinear_fit(**fitargs(z)) by varying z, starting at z0. The fit is redone for each value of z that is tried, in order to determine fit.logGBF.

The Bayes Factor is proportional to the probability that the data came from the model (fit function and priors) used in the fit. empbayes_fit() finds the model or data that maximizes this probability.

One application is illustrated by the following code:

import numpy as np
import gvar as gv
import lsqfit

# fit data
x = np.array([1., 2., 3., 4.])
y = np.array([3.4422, 1.2929, 0.4798, 0.1725])

# prior
prior = gv.gvar(['10(1)', '1.0(1)'])

# fit function
def fcn(x, p):
    return p[0] * gv.exp( - p[1] * x)

# find optimal dy
def fitargs(z):
    dy = y * z
    newy = gv.gvar(y, dy)
    return dict(data=(x, newy), fcn=fcn, prior=prior)

fit, z = lsqfit.empbayes_fit(0.1, fitargs)
print(fit.format(True))

Here we want to fit data y with fit function fcn but we don’t know the uncertainties in our y values. We assume that the relative errors are x-independent and uncorrelated. We add the error dy that maximizes the Bayes Factor, as this is the most likely choice. This fit gives the following output:

Least Square Fit:
  chi2/dof [dof] = 0.58 [4]    Q = 0.67    logGBF = 7.4834

Parameters:
              0     9.44 (18)     [ 10.0 (1.0) ]
              1   0.9979 (69)     [  1.00 (10) ]

Fit:
     x[k]           y[k]      f(x[k],p)
---------------------------------------
        1     3.442 (54)     3.481 (45)
        2     1.293 (20)     1.283 (11)
        3    0.4798 (75)    0.4731 (41)
        4    0.1725 (27)    0.1744 (23)

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 3/0.0)

We have, in effect, used the variation in the data relative to the best fit curve to estimate that the uncertainty in each data point is of order 1.6%.

empbayes_fit() can be used with other fitters: for example, to use lsqfit.vegas_int instead of lsqfit.nonlinear_fit (the default) for the fits, replace the next to last line in the code above with

fit, z = lsqfit.empbayes_fit(0.1, fitargs, fitter=lsqfit.vegas_fit).

`lsqfit.MultiFitter` Classes¶

lsqfit.MultiFitter provides a framework for building component systems to fit multiple pieces of data using a set of custom-designed models, derived from lsqfit.MultiFitterModel. Each model encapsulates: a) a particular fit function; b) a recipe for assembling the corresponding fit data from a dictionary that contains all of the data; and c) a recipe for assembling a fit prior drawn from a dictionary containing all the priors. This allows fit problems to be broken down down into more manageable pieces, which are then aggregated by lsqfit.MultiFitter into a single fit.

This framework was developed to support the corrfitter module which is used to analyze 2-point and 3-point correlators generated in Monte Carlo simulations of quantum field theories (like QCD). The corrfitter module provides two models to describe correlators: corrfitter.Corr2 to describe one 2-point correlator, and corrfitter.Corr3 to describe one 3-point correlator. A typical analysis involves fitting data for a mixture of 2-point and 3-point correlators, with sometimes hundreds of correlators in all. Each correlator is described by either a Corr2 model or a Corr3 model. A list of models, one for each correlator, is handed corrfitter.CorrFitter (derived from lsqfit.MultiFitter) to fit the models to the correlator data. The models for different correlators typically share many fit parameters.

A simpler example of a model is one that encapsulates a linear fit function:

import gvar as gv
import numpy as np
import lsqfit

class Linear(lsqfit.MultiFitterModel):
    def __init__(self, datatag, x, intercept, slope):
        super(Linear, self).__init__(datatag)
        # the independent variable
        self.x = np.array(x)
        # keys used to find the intercept and slope in a parameter dictionary
        self.intercept = intercept
        self.slope = slope

    def fitfcn(self, p):
        try:
            return p[self.intercept] + p[self.slope] * self.x
        except KeyError:
            # slope parameter marginalized/omitted
            return len(self.x) * [p[self.intercept]]

    def buildprior(self, prior, mopt=None):
        " Extract the model's parameters from prior. "
        newprior = {}
        newprior[self.intercept] = prior[self.intercept]
        if mopt is None:
            # slope parameter marginalized/omitted if mopt is not None
            newprior[self.slope] = prior[self.slope]
        return newprior

    def builddata(self, data):
        " Extract the model's fit data from data. "
        return data[self.datatag]

Imagine four sets of data, each corresponding to x=1,2,3,4, all of which have the same intercept but different slopes:

data = gv.gvar(dict(
    d1=['1.154(10)', '2.107(16)', '3.042(22)', '3.978(29)'],
    d2=['0.692(10)', '1.196(16)', '1.657(22)', '2.189(29)'],
    d3=['0.107(10)', '0.030(16)', '-0.027(22)', '-0.149(29)'],
    d4=['0.002(10)', '-0.197(16)', '-0.382(22)', '-0.627(29)'],
    ))

To find the common intercept, we define a model for each set of data:

models = [
   Linear('d1', x=[1,2,3,4], intercept='a', slope='s1'),
   Linear('d2', x=[1,2,3,4], intercept='a', slope='s2'),
   Linear('d3', x=[1,2,3,4], intercept='a', slope='s3'),
   Linear('d4', x=[1,2,3,4], intercept='a', slope='s4'),
   ]

This says that data['d3'], for example, should be fit with function p['a'] + p['s3'] * np.array([1,2,3,4]) where p is a dictionary of fit parameters. Assume that we know a priori that the intercept and slopes are all order one:

prior = gv.gvar(dict(a='0(1)', s1='0(1)', s2='0(1)', s3='0(1)', s4='0(1)'))

Then we can fit all the data to determine the intercept:

fitter = lsqfit.MultiFitter(models=models)
fit = fitter.lsqfit(data=data, prior=prior)
print(fit)
print('intercept =', fit.p['a'])

The output from this code is:

Least Square Fit:
  chi2/dof [dof] = 0.49 [16]    Q = 0.95    logGBF = 18.793

Parameters:
              a    0.2012 (78)      [  0.0 (1.0) ]
             s1    0.9485 (53)      [  0.0 (1.0) ]
             s2    0.4927 (53)      [  0.0 (1.0) ]
             s3   -0.0847 (53)      [  0.0 (1.0) ]
             s4   -0.2001 (53)      [  0.0 (1.0) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 5/0.0)

intercept = 0.2012(78)

Model class Linear is configured to allow marginalization of the slope parameter, if desired. Calling fitter.lsqfit(data=data, prior=prior, mopt=True) moves the slope parameters into the data (by subtracting m.x * prior[m.slope] from the data for each model m), and does a single-parameter fit for the intercept:

Least Square Fit:
  chi2/dof [dof] = 0.49 [16]    Q = 0.95    logGBF = 18.793

Parameters:
              a   0.2012 (78)     [  0.0 (1.0) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 4/0.0)

intercept = 0.2012(78)

Marginalization can be useful when fitting large data sets since it reduces the number of fit parameters and simplifies the fit.

Empirical Bayes tuning can be used with a MultiFitter (see Tuning Priors with the Empirical Bayes Criterion). Continuing from the example just above, we may be uncertain about the prior for the intercept. The following code varies the width of that prior to maximize the Bayes Factor (logGBF):

def fitargs(z):
    prior = gv.gvar(dict(s1='0(1)', s2='0(1)', s3='0(1)', s4='0(1)'))
    prior['a'] = gv.gvar(0, np.exp(z))       # np.exp => positive std dev
    return dict(prior=prior, data=data, mopt=True)
 fit,z = fitter.empbayes_fit(0, fitargs)
 print(fit)
 print('intercept =', fit.p['a'])

The output shows that the data prefer a prior of 0.0(2) for the intercept (not surprisingly):

Least Square Fit:
  chi2/dof [dof] = 0.55 [16]    Q = 0.92    logGBF = 19.917

Parameters:
            a   0.2009 (78)     [  0.00 (20) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 1/0.0)

intercept = 0.2009(78)

The increase in the Bayes Factor, however, is not significant, and the result is almost unchanged. This confirms that the original choice was reasonable.

Another variation is to replace the simultaneous fit of the four models by a chained fit, where one model is fit at a time and its results are fed into the next fit through that fit’s prior. Replacing the fit code by

fitter = lsqfit.MultiFitter(models=models)
fit = fitter.chained_lsqfit(data=data, prior=prior)
# same as fit = fitter.lsqfit(data=data, prior=prior, chained=True)
print(fit.formatall())
print('intercept =', fit.p['a'])

gives the following output:

========== d1
Least Square Fit:
  chi2/dof [dof] = 0.32 [4]    Q = 0.86    logGBF = 2.0969

Parameters:
              a    0.213 (16)     [  0.0 (1.0) ]
             s1   0.9432 (82)     [  0.0 (1.0) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 5/0.0)

========== d2
Least Square Fit:
  chi2/dof [dof] = 0.58 [4]    Q = 0.67    logGBF = 5.3792

Parameters:
              a    0.206 (11)     [ 0.213 (16) ]
             s2   0.4904 (64)     [  0.0 (1.0) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 4/0.0)

========== d3
Least Square Fit:
  chi2/dof [dof] = 0.66 [4]    Q = 0.62    logGBF = 5.3767

Parameters:
              a    0.1995 (90)      [ 0.206 (11) ]
             s3   -0.0840 (57)      [  0.0 (1.0) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 4/0.0)

========== d4
Least Square Fit:
  chi2/dof [dof] = 0.41 [4]    Q = 0.81    logGBF = 5.9402

Parameters:
              a    0.2012 (78)      [ 0.1995 (90) ]
             s4   -0.2001 (53)      [   0.0 (1.0) ]

Settings:
  svdcut/n = 1e-12/0    tol = (1e-08*,1e-10,1e-10)    (itns/time = 4/0.0)

intercept = 0.2012(78)

Note how the value for a improves with each fit.

Chained fits are most useful with very large data sets when it is possible to break the data into smaller, more manageable chunks. There are a variety of options for organizing the chain of fits; these are discussed in the MultiFitter.chained_lsqfit() documentation.

class lsqfit.MultiFitter(models, mopt=None, ratio=False, fast=True, **fitterargs)¶

Nonlinear least-squares fitter for a collection of models.

Fits collections of data that are modeled by collections of models. Fits can be simultaneous (lsqfit.MultiFitter.lsqfit()) or chained (lsqfit.MultiFitter.chained_lsqfit()).

Parameters:

models – List of models, derived from lsqfit.MultiFitterModel, to be fit to the data. Individual models in the list can be replaced by lists of models or tuples of models; see below.
mopt (object) – Marginalization options. If not None, marginalization is used to reduce the number of fit parameters. Object mopt is passed to the models when constructing the prior for a fit; it typically indicates the degree of marginalization (in a model-dependent fashion). Setting mopt=None implies no marginalization.
ratio (bool) – If True, implement marginalization using ratios: data_marg = data * fitfcn(prior_marg) / fitfcn(prior). If False (default), implement using differences: data_marg = data + (fitfcn(prior_marg) - fitfcn(prior)).
fast (bool) – Setting fast=True (default) strips any variable not required by the fit from the prior. This speeds fits but loses information about correlations between variables in the fit and those that are not. Setting wavg_all=True can restore some of the correlations, but is somewhat slower.
wavg_all (bool) – If True and fast=True, the final result of a chained fit is the weighted average of all the fits in the chain. This can restore correlations lost in the chain because fast=True. This step is omitted if wavg_all=False or fast=False. Default is False.
fitname (callable or None) – Individual fits in a chained fit are assigned default names, constructed from the datatags of the corresponding models, for access and reporting. These names get unwieldy when lots of models are involved. When fitname is not None (default), each default name dname is replaced by fitname(dname) which should return a string.
wavg_kargs (dict) – Keyword arguments for lsqfit.wavg() when used to combine results from parallel sub-fits in a chained fit.
fitterargs (dict) – Additional arguments for the lsqfit.nonlinear_fit object used to do the fits. These can be collected in a dictionary (e.g., fitterargs=dict(tol=1e-6, maxit=500))) or listed as separate arguments (e.g., tol=1e-6, maxit=500).

lsqfit(data=None, pdata=None, prior=None, p0=None, chained=False, **kargs)¶

Compute least-squares fit of models to data.

MultiFitter.lsqfit() fits all of the models together, in a single fit. It returns the lsqfit.nonlinear_fit object from the fit.

To see plots of the fit data divided by the fit function with the best-fit parameters use

fit.show_plots()

This method has optional keyword arguments save and view; see documentation for lsqfit.MultiFitter.show_plots for more information. Plotting requires module matplotlib.

To bootstrap a fit, use fit.bootstrapped_fit_iter(...); see lsqfit.nonlinear_fit.bootstrapped_fit_iter() for more information.

Parameters:

data – Input data. One of data or pdata must be specified but not both. pdata is obtained from data by collecting the output from m.builddata(data) for each model m and storing it in a dictionary with key m.datatag.
pdata – Input data that has been processed by the models using MultiFitter.process_data() or MultiFitter.process_dataset(). One of data or pdata must be specified but not both.
prior (dict) – Bayesian prior for fit parameters used by the models.
p0 – Dictionary , indexed by parameter labels, containing initial values for the parameters in the fit. Setting p0=None implies that initial values are extracted from the prior. Setting p0="filename" causes the fitter to look in the file with name "filename" for initial values and to write out best-fit parameter values after the fit (for the next call to self.lsqfit()).
chained (bool) – Use MultiFitter.chained_lsqfit() instead of MultiFitter.lsqfit() if chained=True. Ignored otherwise. Default is chained=False.
kargs – Arguments that (temporarily) override parameters specified when the MultiFitter was created. Can also include additional arguments to be passed through to the lsqfit fitter.

chained_lsqfit(data=None, pdata=None, prior=None, p0=None, **kargs)¶

Compute chained least-squares fit of models to data. Equivalent to:

self.lsqfit(data, pdata, prior, p0, chained=True, **kargs).

In a chained fit to models [s1, s2, ...], the models are fit one at a time, with the fit output from one being fed into the prior for the next. This can be much faster than fitting the models together, simultaneously. The final result comes from the last fit in the chain, and includes parameters from all of the models.

The most general chain has the structure [s1, s2, s3 ...] where each sn is one of:

A model (derived from multifitter.MultiFitterModel).

A tuple (m1, m2, m3) of models, to be fit together in
a single fit (i.e., simultaneously). Simultaneous fits are useful for closely related models.

A list [p1, p2, p3 ...] where each pn is either
a model, a tuple of models (see #2), or a dictionary (see #4). The pn are fit separately: the fit output from one fit is not fed into the prior of the next (i.e., the fits are effectively in parallel). Results from the separate fits are averaged at the end to provide a single composite result for the collection of fits. Parallel fits are effective (and fast) when the different fits have few or no fit parameters in common.

A dictionary that (temporarily) resets default values for
fitter keywords. The new values, specified in the dictionary, apply to subsequent fits in the chain. Any number of such dictionaries can be included in the model chain.

Fit results are returned in a lsqfit.MultiFitter.chained_nonlinear_fit object fit, which is very similar to a nonlinear_fit object (see documentation for more information). Object fit has an extra attribute fit.chained_fits which is an ordered dictionary containing fit results for each link in the chain of fits, indexed by fit names built from the corresponding data tags.

To list results from all of the fits in the chain, use

print(fit.formatall())

This method has optional keyword arguments maxline, pstyle, and nline; see the documentation for lsqfit.nonlinear_fit.format() for more information.

To view plots of each fit use

fit.show_plots()

This method has optional keyword arguments save and view; see documentation for lsqfit.MultiFitter.show_plots for more information. Plotting requires module matplotlib.

To bootstrap a fit, use fit.bootstrapped_fit_iter(...); see lsqfit.nonlinear_fit.bootstrapped_fit_iter() for more information.

Parameters:

data – Input data. One of data or pdata must be specified but not both. pdata is obtained from data by collecting the output from m.builddata(data) for each model m and storing it in a dictionary with key m.datatag.
pdata – Input data that has been processed by the models using MultiFitter.process_data() or MultiFitter.process_dataset(). One of data or pdata must be specified but not both.
prior – Bayesian prior for fit parameters used by the models.
p0 –
Dictionary , indexed by parameter labels, containing initial values for the parameters in the fit. Setting p0=None implies that initial values are extracted from the prior. Setting p0="filename" causes the fitter to look in the file with name "filename" for initial values and to write out best-fit parameter values after the fit (for the next call to self.chained_lsqfit()). Finally, p0 can be a list containing a different p0 for each fit in the chain: for example,
```
p0 = [f.pmean for f in fit.chained_fits.values()]
```
might be a good starting point for the next fit.
kargs – Arguments that override parameters specified when the MultiFitter was created. Can also include additional arguments to be passed through to the lsqfit fitter.

empbayes_fit(z0, fitargs, p0=None, **minargs)¶

Return fit and z corresponding to the fit self.lsqfit(**fitargs(z)) that maximizes logGBF.

This function maximizes the logarithm of the Bayes Factor from fit self.lsqfit(**fitargs(z)) by varying z, starting at z0. The fit is redone for each value of z that is tried, in order to determine logGBF.

The Bayes Factor is proportional to the probability that the data came from the model (fit function and priors) used in the fit. MultiFitter.empbayes_fit() finds the model or data that maximizes this probability. See lsqfit.empbayes_fit() for more information.

Include chained=True in the dictionary returned by fitargs(z) if chained fits are desired. See documentation for MultiFitter.lsqfit().

Parameters:

z0 (number, array or dict) – Starting point for search.
fitargs (callable) – Function of z that returns a dictionary args containing the MultiFitter.lsqfit() arguments corresponding to z. z should have the same layout (number, array or dictionary) as z0. fitargs(z) can instead return a tuple (args, plausibility), where args is again the dictionary for MultiFitter.lsqfit(). plausibility is the logarithm of the a priori probabilitiy that z is sensible. When plausibility is provided, MultiFitter.empbayes_fit() maximizes the sum logGBF + plausibility. Specifying plausibility is a way of steering selections away from completely implausible values for z.
p0 – Fit-parameter starting values for the first fit. p0 for subsequent fits is set automatically to optimize fitting unless a value is specified by fitargs.
minargs (dict) – Optional argument dictionary, passed on to lsqfit.gsl_multiminex (or lsqfit.scipy_multiminex), which finds the minimum.

Returns:

A tuple containing the best fit (a fit object) and the optimal value for parameter z.

set(**kargs)¶

Reset default keyword parameters.

Assigns new default values from dictionary kargs to the fitter’s keyword parameters. Keywords for the underlying lsqfit fitters can also be included (or grouped together in dictionary fitterargs).

Returns tuple (kargs, oldkargs) where kargs is a dictionary containing all lsqfit.MultiFitter keywords after they have been updated, and oldkargs contains the original values for these keywords. Use fitter.set(**oldkargs) to restore the original values.

static process_data(data, models)¶

Convert data to processed data using models.

Data from dictionary data is processed by each model in list models, and the results collected into a new dictionary pdata for use in MultiFitter.lsqfit() and MultiFitter.chained_lsqft().

static process_dataset(dataset, models, **kargs)¶

Convert dataset to processed data using models.

gvar.dataset.Dataset (or similar dictionary) object dataset is processed by each model in list models, and the results collected into a new dictionary pdata for use in MultiFitter.lsqfit() and MultiFitter.chained_lsqft(). Assumes that the models have defined method MultiFitterModel.builddataset(). Keyword arguments kargs are passed on to gvar.dataset.avg_data() when averaging the data.

static show_plots(fitdata, fitval, x=None, save=False, view='ratio')¶

Show plots comparing fitdata[k],fitval[k] for each key k in fitval.

Assumes matplotlib is installed (to make the plots). Plots are shown for one correlator at a time. Press key n to see the next correlator; press key p to see the previous one; press key q to quit the plot and return control to the calling program; press a digit to go directly to one of the first ten plots. Zoom, pan and save using the window controls.

There are several different views available for each plot, specified by parameter view:

view='ratio': Data divided by fit (default).

view='diff': Data minus fit, divided by data’s standard deviation.

view='std': Data and fit.

view='log': 'std' with log scale on the vertical axis.

view='loglog': ‘std’` with log scale on both axes.

Press key v to cycle through these views; or press keys r, d, or l for the 'ratio', 'diff', or 'log' views, respectively.

Copies of the plots that are viewed can be saved by setting parameter save=fmt where fmt is a string used to create file names: the file name for the plot corresponding to key k is fmt.format(k). It is important that the filename end with a suffix indicating the type of plot file desired: e.g., fmt='plot-{}.pdf'.

static flatten_models(models)¶: Create 1d-array containing all disctinct models from models.

lsqfit.MultiFitter models are derived from the following class. Methods buildprior, builddata, fitfcn, and builddataset are not implemented in this base class. They need to be overwritten by the derived class (except for builddataset which is optional).

class lsqfit.MultiFitterModel(datatag, ncg=1)¶

Base class for MultiFitter models.

Derived classes must define methods fitfcn, buildprior, and builddata, all of which are described below. In addition they have attributes:

datatag¶: lsqfit.MultiFitter builds fit data for the correlator by extracting the data labelled by datatag (eg, a string) from an input data set (eg, a dictionary). This label is stored in the MultiFitterModel and must be passed to its constructor. It must be a hashable quantity, like a string or number or tuple of strings and numbers.

ncg¶: When ncg>1, fit data and functions are coarse-grained by breaking them up into bins of of ncg values and replacing each bin by its average. This can increase the fitting speed, because there is less data, without much loss of precision if the data elements within a bin are highly correlated.

Parameters:

datatag – Label used to identify model’s data.
ncg (int) – Size of bins for coarse graining (default is ncg=1).

buildprior(prior, mopt=None)¶

Extract fit prior from prior.

Returns a dictionary containing the part of dictionary prior that is relevant to this model’s fit. The code could be as simple as collecting the appropriate pieces: e.g.,

def buildprior(self, prior, mopt=None):
    mprior = gv.BufferDict()
    model_keys = [...]
    for k in model_keys:
        mprior[k] = prior[k]
    return mprior

where model_keys is a list of keys corresponding to the model’s parameters. Supporting non-Gaussian distributions requires a slight modification: e.g.,

def buildprior(self, prior, mopt=None):
    mprior = gv.BufferDict()
    model_keys = [...]
    for k in gv.get_dictkeys(prior, model_keys):
        mprior[k] = prior[k]
    return mprior

Marginalization involves omitting some of the fit parameters from the model’s prior. mopt=None implies no marginalization. Otherwise mopt will typically contain information about what and how much to marginalize.

Parameters:

prior – Dictionary containing a priori estimates of all fit parameters.
mopt (object) – Marginalization options. Ignore if None. Otherwise marginalize fit parameters as specified by mopt. mopt can be any type of Python object; it is used only in buildprior and is passed through to it unchanged.

builddata(data)¶

Extract fit data corresponding to this model from data set data.

The fit data is returned in a 1-dimensional array; the fitfcn must return arrays of the same length.

Parameters:: data – Data set containing the fit data for all models. This is typically a dictionary, whose keys are the datatags of the models.

fitfcn(p)¶

Compute fit function fit for parameters p.

Results are returned in a 1-dimensional array the same length as (and corresponding to) the fit data returned by self.builddata(data).

If marginalization is supported, fitfcn must work with or without the marginalized parameters.

Parameters:: p – Dictionary of parameter values.

builddataset(dataset)¶

Extract fit dataset from gvar.dataset.Dataset dataset.

The code

import gvar as gv

data = gv.dataset.avg_data(m.builddataset(dataset))

that builds data for model m should be functionally equivalent to

import gvar as gv

data = m.builddata(gv.dataset.avg_data(dataset))

This method is optional. It is used only by MultiFitter.process_dataset().

Parameters:: dataset – gvar.dataset.Dataset (or similar dictionary) dataset containing the fit data for all models. This is typically a dictionary, whose keys are the datatags of the models.

References¶

The lsqfit and gvar modules were originally created to facilitate statistical analyses of data generated by lattice QCD simulations. Background information about the techniques used in these modules can be found in several articles (on lattice QCD applications):

For a general discussion of Bayesian fitting (and Empirical Bayes) see: G.P. Lepage et al, Nucl.Phys.Proc.Suppl. 106 (2002) 12-20 [hep-lat/0110175].

For a discussion of the underlying analysis in a fit and the meaning of the error budget see Appendix A in: C. Bouchard et al, Phys.Rev. D90 (2014) 054506 [arXiv:1406.2279].

For a discussion of marginalization see the appendix in: C. McNeile et al, Phys.Rev. D82, 034512 (2010) [arXiv:1004.4285]. For another sample application see: K. Hornbostel et al, Phys.Rev. D85 (2012) 031504 [arXiv:1111.1363].

For a discussion of SVD cuts (and goodness-of-fit) see Appendix D in: R.J. Dowdall et al, Phys.Rev. D100 (2019) 9, 094508 [arXiv:1907.01025].

Requirements¶

lsqfit relies heavily on the gvar, and numpy modules. Also the fitting and minimization routines are from the Gnu Scientific Library (GSL) and/or the Python scipy module.

`lsqfit` - Nonlinear Least Squares Fitting¶

Introduction¶

nonlinear_fit Objects¶

vegas_fit Objects¶

Functions¶

`lsqfit.MultiFitter` Classes¶

References¶

Requirements¶

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lsqfit - Nonlinear Least Squares Fitting¶

Introduction¶

nonlinear_fit Objects¶

vegas_fit Objects¶

Functions¶

lsqfit.MultiFitter Classes¶

References¶

Requirements¶

`lsqfit` - Nonlinear Least Squares Fitting¶

`lsqfit.MultiFitter` Classes¶