Built-in Fitting Models in the models
module¶
Lmfit provides several builtin fitting models in the models
module.
These pre-defined models each subclass from the model.Model
class of the
previous chapter and wrap relatively well-known functional forms, such as
Gaussians, Lorentzian, and Exponentials that are used in a wide range of
scientific domains. In fact, all the models are all based on simple, plain
python functions defined in the lineshapes
module. In addition to
wrapping a function into a model.Model
, these models also provide a
guess()
method that is intended to give a reasonable
set of starting values from a data array that closely approximates the
data to be fit.
As shown in the previous chapter, a key feature of the mode.Model
class
is that models can easily be combined to give a composite
model.Model
. Thus while some of the models listed here may seem pretty
trivial (notably, ConstantModel
and LinearModel
), the
main point of having these is to be able to used in composite models. For
example, a Lorentzian plus a linear background might be represented as:
>>> from lmfit.models import LinearModel, LorentzianModel
>>> peak = LorentzianModel()
>>> background = LinearModel()
>>> model = peak + background
All the models listed below are one dimensional, with an independent
variable named x
. Many of these models represent a function with a
distinct peak, and so share common features. To maintain uniformity,
common parameter names are used whenever possible. Thus, most models have
a parameter called amplitude
that represents the overall height (or
area of) a peak or function, a center
parameter that represents a peak
centroid position, and a sigma
parameter that gives a characteristic
width. Many peak shapes also have a parameter fwhm
(constrained by
sigma
) giving the full width at half maximum and a parameter height
(constrained by sigma
and amplitude
) to give the maximum peak
height.
After a list of builtin models, a few examples of their use is given.
Peak-like models¶
There are many peak-like models available. These include
GaussianModel
, LorentzianModel
, VoigtModel
and
some less commonly used variations. The guess()
methods for all of these make a fairly crude guess for the value of
amplitude
, but also set a lower bound of 0 on the value of sigma
.
GaussianModel
¶
-
class
GaussianModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Gaussian or normal distribution lineshape. Parameter names:
amplitude
, center
, and sigma
.
In addition, parameters fwhm
and height
are included as constraints
to report full width at half maximum and maximum peak height, respectively.
where the parameter amplitude
corresponds to \(A\), center
to
\(\mu\), and sigma
to \(\sigma\). The full width at
half maximum is \(2\sigma\sqrt{2\ln{2}}\), approximately
\(2.3548\sigma\)
LorentzianModel
¶
-
class
LorentzianModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Lorentzian or Cauchy-Lorentz distribution function. Parameter names:
amplitude
, center
, and sigma
.
In addition, parameters fwhm
and height
are included as constraints
to report full width at half maximum and maximum peak height, respectively.
where the parameter amplitude
corresponds to \(A\), center
to
\(\mu\), and sigma
to \(\sigma\). The full width at
half maximum is \(2\sigma\).
VoigtModel
¶
-
class
VoigtModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Voigt distribution function. Parameter names:
amplitude
, center
, and sigma
. A gamma
parameter is also
available. By default, it is constrained to have value equal to sigma
,
though this can be varied independently. In addition, parameters fwhm
and height
are included as constraints to report full width at half
maximum and maximum peak height, respectively. The definition for the
Voigt function used here is
where
and erfc()
is the complimentary error function. As above,
amplitude
corresponds to \(A\), center
to
\(\mu\), and sigma
to \(\sigma\). The parameter gamma
corresponds to \(\gamma\).
If gamma
is kept at the default value (constrained to sigma
),
the full width at half maximum is approximately \(3.6013\sigma\).
PseudoVoigtModel
¶
-
class
PseudoVoigtModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
a model based on a pseudo-Voigt distribution function,
which is a weighted sum of a Gaussian and Lorentzian distribution functions
with that share values for amplitude
(\(A\)), center
(\(\mu\))
and full width at half maximum (and so have constrained values of
sigma
(\(\sigma\)). A parameter fraction
(\(\alpha\))
controls the relative weight of the Gaussian and Lorentzian components,
giving the full definition of
where \(\sigma_g = {\sigma}/{\sqrt{2\ln{2}}}\) so that the full width
at half maximum of each component and of the sum is \(2\sigma\). The
guess()
function always sets the starting value for fraction
at 0.5.
MoffatModel
¶
-
class
MoffatModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
a model based on a Moffat distribution function, the parameters are
amplitude
(\(A\)), center
(\(\mu\)),
a width parameter sigma
(\(\sigma\)) and an exponent beta
(\(\beta\)).
For (\(\beta=1\)) the Moffat has a Lorentzian shape.
the full width have maximum is \(2\sigma\sqrt{2^{1/\beta}-1}\).
guess()
function always sets the starting value for beta
to 1.
Pearson7Model
¶
-
class
Pearson7Model
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Pearson VII distribution.
This is a Lorenztian-like distribution function. It has the usual
parameters amplitude
(\(A\)), center
(\(\mu\)) and
sigma
(\(\sigma\)), and also an exponent
(\(m\)) in
where \(\beta\) is the beta function (see special.beta in
scipy.special
). The guess()
function always
gives a starting value for exponent
of 1.5.
StudentsTModel
¶
-
class
StudentsTModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Student’s t distribution function, with the usual
parameters amplitude
(\(A\)), center
(\(\mu\)) and
sigma
(\(\sigma\)) in
where \(\Gamma(x)\) is the gamma function.
BreitWignerModel
¶
-
class
BreitWignerModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Breit-Wigner-Fano function. It has the usual
parameters amplitude
(\(A\)), center
(\(\mu\)) and
sigma
(\(\sigma\)), plus q
(\(q\)) in
LognormalModel
¶
-
class
LognormalModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on the Log-normal distribution function.
It has the usual parameters
amplitude
(\(A\)), center
(\(\mu\)) and sigma
(\(\sigma\)) in
DampedOcsillatorModel
¶
-
class
DampedOcsillatorModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on the Damped Harmonic Oscillator Amplitude.
It has the usual parameters amplitude
(\(A\)), center
(\(\mu\)) and
sigma
(\(\sigma\)) in
ExponentialGaussianModel
¶
-
class
ExponentialGaussianModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model of an Exponentially modified Gaussian distribution.
It has the usual parameters amplitude
(\(A\)), center
(\(\mu\)) and
sigma
(\(\sigma\)), and also gamma
(\(\gamma\)) in
where erfc()
is the complimentary error function.
SkewedGaussianModel
¶
-
class
SkewedGaussianModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A variation of the above model, this is a Skewed normal distribution.
It has the usual parameters amplitude
(\(A\)), center
(\(\mu\)) and
sigma
(\(\sigma\)), and also gamma
(\(\gamma\)) in
where erf()
is the error function.
DonaichModel
¶
-
class
DonaichModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model of an Doniach Sunjic asymmetric lineshape, used in
photo-emission. With the usual parameters amplitude
(\(A\)),
center
(\(\mu\)) and sigma
(\(\sigma\)), and also gamma
(\(\gamma\)) in
Linear and Polynomial Models¶
These models correspond to polynomials of some degree. Of course, lmfit is a very inefficient way to do linear regression (see polyfit or stats.linregress), but these models may be useful as one of many components of composite model.
ConstantModel
¶
-
class
ConstantModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶ a class that consists of a single value,
c
. This is constant in the sense of having no dependence on the independent variablex
, not in the sense of being non-varying. To be clear,c
will be a variable Parameter.
LinearModel
¶
-
class
LinearModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶ a class that gives a linear model:
with parameters slope
for \(m\) and intercept
for \(b\).
QuadraticModel
¶
-
class
QuadraticModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶ a class that gives a quadratic model:
with parameters a
, b
, and c
.
ParabolicModel
¶
-
class
ParabolicModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶ same as
QuadraticModel
.
PolynomialModel
¶
-
class
PolynomialModel
(degree, missing=None[, prefix=''[, name=None[, **kws]]])¶ a class that gives a polynomial model up to
degree
(with maximum value of 7).
with parameters c0
, c1
, ..., c7
. The supplied degree
will specify how many of these are actual variable parameters. This uses
polyval for its calculation of the polynomial.
Step-like models¶
Two models represent step-like functions, and share many characteristics.
StepModel
¶
-
class
StepModel
(form='linear'[, missing=None[, prefix=''[, name=None[, **kws]]]])¶
A model based on a Step function, with four choices for functional form.
The step function starts with a value 0, and ends with a value of \(A\)
(amplitude
), rising to \(A/2\) at \(\mu\) (center
),
with \(\sigma\) (sigma
) setting the characteristic width. The
supported functional forms are linear
(the default), atan
or
arctan
for an arc-tangent function, erf
for an error function, or
logistic
for a logistic function.
The forms are
where \(\alpha = (x - \mu)/{\sigma}\).
RectangleModel
¶
-
class
RectangleModel
(form='linear'[, missing=None[, prefix=''[, name=None[, **kws]]]])¶
A model based on a Step-up and Step-down function of the same form. The
same choices for functional form as for StepModel
are supported,
with linear
as the default. The function starts with a value 0, and
ends with a value of \(A\) (amplitude
), rising to \(A/2\) at
\(\mu_1\) (center1
), with \(\sigma_1\) (sigma1
) setting the
characteristic width. It drops to rising to \(A/2\) at \(\mu_2\)
(center2
), with characteristic width \(\sigma_2\) (sigma2
).
where \(\alpha_1 = (x - \mu_1)/{\sigma_1}\) and \(\alpha_2 = -(x - \mu_2)/{\sigma_2}\).
Exponential and Power law models¶
ExponentialModel
¶
-
class
ExponentialModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on an exponential decay function. With parameters named
amplitude
(\(A\)), and decay
(\(\tau\)), this has the form:
PowerLawModel
¶
-
class
PowerLawModel
(missing=None[, prefix=''[, name=None[, **kws]]])¶
A model based on a Power Law.
With parameters
named amplitude
(\(A\)), and exponent
(\(k\)), this has the
form:
User-defined Models¶
As shown in the previous chapter (Modeling Data and Curve Fitting), it is fairly straightforward to build fitting models from parametrized python functions. The number of model classes listed so far in the present chapter should make it clear that this process is not too difficult. Still, it is sometimes desirable to build models from a user-supplied function. This may be especially true if model-building is built-in to some larger library or application for fitting in which the user may not be able to easily build and use a new model from python code.
The ExpressionModel
allows a model to be built from a
user-supplied expression. This uses the asteval module also used for
mathematical constraints as discussed in Using Mathematical Constraints.
ExpressionModel
¶
-
class
ExpressionModel
(expr, independent_vars=None, init_script=None, **kws)¶ A model using the user-supplied mathematical expression, which can be nearly any valid Python expresion.
Parameters: - expr (string) – expression use to build model
- independent_vars (
None
(default) or list of strings for independent variables.) – list of argument names in expression that are independent variables. - init_script (
None
(default) or string) – python script to run before parsing and evaluating expression.
with other parameters passed to model.Model
, with the notable
exception that ExpressionModel
does not support the prefix argument.
Since the point of this model is that an arbitrary expression will be
supplied, the determination of what are the parameter names for the model
happens when the model is created. To do this, the expression is parsed,
and all symbol names are found. Names that are already known (there are
over 500 function and value names in the asteval namespace, including most
python builtins, more than 200 functions inherited from numpy, and more
than 20 common lineshapes defined in the lineshapes
module) are not
converted to parameters. Unrecognized name are expected to be names either
of parameters or independent variables. If independent_vars is the
default value of None
, and if the expression contains a variable named
x, that will be used as the independent variable. Otherwise,
independent_vars must be given.
For example, if one creates an ExpressionModel
as:
>>> mod = ExpressionModel('off + amp * exp(-x/x0) * sin(x*phase)')
The name exp will be recognized as the exponent function, so the model will be interpreted to have parameters named off, amp, x0 and phase. In addition, x will be assumed to be the sole independent variable. In general, there is no obvious way to set default parameter values or parameter hints for bounds, so this will have to be handled explicitly.
To evaluate this model, you might do the following:
>>> x = numpy.linspace(0, 10, 501)
>>> params = mod.make_params(off=0.25, amp=1.0, x0=2.0, phase=0.04)
>>> y = mod.eval(params, x=x)
While many custom models can be built with a single line expression (especially since the names of the lineshapes like gaussian, lorentzian and so on, as well as many numpy functions, are available), more complex models will inevitably require multiple line functions. You can include such Python code with the init_script argument. The text of this script is evaluated when the model is initialized (and before the actual expression is parsed), so that you can define functions to be used in your expression.
As a probably unphysical example, to make a model that is the derivative of a Gaussian function times the logarithm of a Lorentzian function you may could to define this in a script:
>>> script = """
def mycurve(x, amp, cen, sig):
loren = lorentzian(x, amplitude=amp, center=cen, sigma=sig)
gauss = gaussian(x, amplitude=amp, center=cen, sigma=sig)
return log(loren)*gradient(gauss)/gradient(x)
"""
and then use this with ExpressionModel
as:
>>> mod = ExpressionModel('mycurve(x, height, mid, wid)',
init_script=script,
independent_vars=['x'])
As above, this will interpret the parameter names to be height, mid, and wid, and build a model that can be used to fit data.
Example 1: Fit Peaked data to Gaussian, Lorentzian, and Voigt profiles¶
Here, we will fit data to three similar line shapes, in order to decide which
might be the better model. We will start with a Gaussian profile, as in
the previous chapter, but use the built-in GaussianModel
instead
of writing one ourselves. This is a slightly different version rom the
one in previous example in that the parameter names are different, and have
built-in default values. We’ll simply use:
from numpy import loadtxt
from lmfit.models import GaussianModel
data = loadtxt('test_peak.dat')
x = data[:, 0]
y = data[:, 1]
mod = GaussianModel()
pars = mod.guess(y, x=x)
out = mod.fit(y, pars, x=x)
print(out.fit_report(min_correl=0.25))
which prints out the results:
[[Model]]
Model(gaussian)
[[Fit Statistics]]
# function evals = 23
# data points = 401
# variables = 3
chi-square = 29.994
reduced chi-square = 0.075
Akaike info crit = -1033.774
Bayesian info crit = -1021.792
[[Variables]]
sigma: 1.23218319 +/- 0.007374 (0.60%) (init= 1.35)
center: 9.24277049 +/- 0.007374 (0.08%) (init= 9.25)
amplitude: 30.3135571 +/- 0.157126 (0.52%) (init= 29.08159)
fwhm: 2.90156963 +/- 0.017366 (0.60%) == '2.3548200*sigma'
height: 9.81457973 +/- 0.050872 (0.52%) == '0.3989423*amplitude/max(1.e-15, sigma)'
[[Correlations]] (unreported correlations are < 0.250)
C(sigma, amplitude) = 0.577
We see a few interesting differences from the results of the previous
chapter. First, the parameter names are longer. Second, there are fwhm
and height
parameters, to give the full width at half maximum and
maximum peak height. And third, the automated initial guesses are pretty
good. A plot of the fit:
shows a decent match to the data – the fit worked with no explicit setting
of initial parameter values. Looking more closing, the fit is not perfect,
especially in the tails of the peak, suggesting that a different peak
shape, with longer tails, should be used. Perhaps a Lorentzian would be
better? To do this, we simply replace GaussianModel
with
LorentzianModel
to get a LorentzianModel
:
from lmfit.models import LorentzianModel
mod = LorentzianModel()
with the rest of the script as above. Perhaps predictably, the first thing we try gives results that are worse:
[[Model]]
Model(lorentzian)
[[Fit Statistics]]
# function evals = 27
# data points = 401
# variables = 3
chi-square = 53.754
reduced chi-square = 0.135
Akaike info crit = -799.830
Bayesian info crit = -787.848
[[Variables]]
sigma: 1.15484517 +/- 0.013156 (1.14%) (init= 1.35)
center: 9.24438944 +/- 0.009275 (0.10%) (init= 9.25)
amplitude: 38.9728645 +/- 0.313857 (0.81%) (init= 36.35199)
fwhm: 2.30969034 +/- 0.026312 (1.14%) == '2.0000000*sigma'
height: 10.7420881 +/- 0.086336 (0.80%) == '0.3183099*amplitude/max(1.e-15, sigma)'
[[Correlations]] (unreported correlations are < 0.250)
C(sigma, amplitude) = 0.709
with the plot shown on the right in the figure above. The tails are now
too big, and the value for \(\chi^2\) almost doubled. A Voigt model
does a better job. Using VoigtModel
, this is as simple as using:
from lmfit.models import VoigtModel
mod = VoigtModel()
with all the rest of the script as above. This gives:
[[Model]]
Model(voigt)
[[Fit Statistics]]
# function evals = 19
# data points = 401
# variables = 3
chi-square = 14.545
reduced chi-square = 0.037
Akaike info crit = -1324.006
Bayesian info crit = -1312.024
[[Variables]]
amplitude: 35.7554017 +/- 0.138614 (0.39%) (init= 43.62238)
sigma: 0.73015574 +/- 0.003684 (0.50%) (init= 0.8775)
center: 9.24411142 +/- 0.005054 (0.05%) (init= 9.25)
gamma: 0.73015574 +/- 0.003684 (0.50%) == 'sigma'
fwhm: 2.62951718 +/- 0.013269 (0.50%) == '3.6013100*sigma'
height: 19.5360268 +/- 0.075691 (0.39%) == '0.3989423*amplitude/max(1.e-15, sigma)'
[[Correlations]] (unreported correlations are < 0.250)
C(sigma, amplitude) = 0.651
which has a much better value for \(\chi^2\) and an obviously better match to the data as seen in the figure below (left).
Can we do better? The Voigt function has a \(\gamma\) parameter
(gamma
) that can be distinct from sigma
. The default behavior used
above constrains gamma
to have exactly the same value as sigma
. If
we allow these to vary separately, does the fit improve? To do this, we
have to change the gamma
parameter from a constrained expression and
give it a starting value using something like:
mod = VoigtModel()
pars = mod.guess(y, x=x)
pars['gamma'].set(value=0.7, vary=True, expr='')
which gives:
[[Model]]
Model(voigt)
[[Fit Statistics]]
# function evals = 23
# data points = 401
# variables = 4
chi-square = 10.930
reduced chi-square = 0.028
Akaike info crit = -1436.576
Bayesian info crit = -1420.600
[[Variables]]
amplitude: 34.1914716 +/- 0.179468 (0.52%) (init= 43.62238)
sigma: 0.89518950 +/- 0.014154 (1.58%) (init= 0.8775)
center: 9.24374845 +/- 0.004419 (0.05%) (init= 9.25)
gamma: 0.52540156 +/- 0.018579 (3.54%) (init= 0.7)
fwhm: 3.22385492 +/- 0.050974 (1.58%) == '3.6013100*sigma'
height: 15.2374711 +/- 0.299235 (1.96%) == '0.3989423*amplitude/max(1.e-15, sigma)'
[[Correlations]] (unreported correlations are < 0.250)
C(sigma, gamma) = -0.928
C(gamma, amplitude) = 0.821
C(sigma, amplitude) = -0.651
and the fit shown on the right above.
Comparing the two fits with the Voigt function, we see that \(\chi^2\)
is definitely improved with a separately varying gamma
parameter. In
addition, the two values for gamma
and sigma
differ significantly
– well outside the estimated uncertainties. More compelling, reduced
\(\chi^2\) is improved even though a fourth variable has been added to
the fit. In the simplest statistical sense, this suggests that gamma
is a significant variable in the model. In addition, we can use both the
Akaike or Bayesian Information Criteria (see
Akaike and Bayesian Information Criteria) to assess how likely the model with
variable gamma
is to explain the data than the model with gamma
fixed to the value of sigma
. According to theory,
\(\exp(-(\rm{AIC1}-\rm{AIC0})/2)\) gives the probably that a model with
AIC` is more likely than a model with AIC0. For the two models here, with
AIC values of -1432 and -1321 (Note: if we had more carefully set the value
for weights
based on the noise in the data, these values might be
positive, but there difference would be roughly the same), this says that
the model with gamma
fixed to sigma
has a probably less than 1.e-25
of being the better model.
Example 2: Fit data to a Composite Model with pre-defined models¶
Here, we repeat the point made at the end of the last chapter that
instances of model.Model
class can be added together to make a
composite model. By using the large number of built-in models available,
it is therefore very simple to build models that contain multiple peaks and
various backgrounds. An example of a simple fit to a noisy step function
plus a constant:
After constructing step-like data, we first create a StepModel
telling it to use the erf
form (see details above), and a
ConstantModel
. We set initial values, in one case using the data
and guess()
method for the initial step function paramaters, and
make_params()
arguments for the linear component.
After making a composite model, we run fit()
and report the
results, which gives:
[[Model]]
(Model(step, prefix='step_', form='erf') + Model(linear, prefix='line_'))
[[Fit Statistics]]
# function evals = 51
# data points = 201
# variables = 5
chi-square = 584.829
reduced chi-square = 2.984
Akaike info crit = 224.671
Bayesian info crit = 241.187
[[Variables]]
line_slope: 2.03039786 +/- 0.092221 (4.54%) (init= 0)
line_intercept: 11.7234542 +/- 0.274094 (2.34%) (init= 10.7816)
step_amplitude: 112.071629 +/- 0.647316 (0.58%) (init= 134.0885)
step_sigma: 0.67132341 +/- 0.010873 (1.62%) (init= 1.428571)
step_center: 3.12697699 +/- 0.005151 (0.16%) (init= 2.5)
[[Correlations]] (unreported correlations are < 0.100)
C(line_slope, step_amplitude) = -0.878
C(step_amplitude, step_sigma) = 0.563
C(line_slope, step_sigma) = -0.455
C(line_intercept, step_center) = 0.427
C(line_slope, line_intercept) = -0.308
C(line_slope, step_center) = -0.234
C(line_intercept, step_sigma) = -0.139
C(line_intercept, step_amplitude) = -0.121
C(step_amplitude, step_center) = 0.109
with a plot of

Example 3: Fitting Multiple Peaks – and using Prefixes¶
As shown above, many of the models have similar parameter names. For
composite models, this could lead to a problem of having parameters for
different parts of the model having the same name. To overcome this, each
model.Model
can have a prefix
attribute (normally set to a blank
string) that will be put at the beginning of each parameter name. To
illustrate, we fit one of the classic datasets from the NIST StRD suite
involving a decaying exponential and two gaussians.
where we give a separate prefix to each model (they all have an
amplitude
parameter). The prefix
values are attached transparently
to the models.
Note that the calls to make_param()
used the bare name, without the
prefix. We could have used the prefixes, but because we used the
individual model gauss1
and gauss2
, there was no need.
Note also in the example here that we explicitly set bounds on many of the parameter values.
The fit results printed out are:
[[Model]]
((Model(gaussian, prefix='g1_') + Model(gaussian, prefix='g2_')) + Model(exponential, prefix='exp_'))
[[Fit Statistics]]
# function evals = 66
# data points = 250
# variables = 8
chi-square = 1247.528
reduced chi-square = 5.155
Akaike info crit = 417.865
Bayesian info crit = 446.036
[[Variables]]
exp_amplitude: 99.0183282 +/- 0.537487 (0.54%) (init= 162.2102)
exp_decay: 90.9508859 +/- 1.103105 (1.21%) (init= 93.24905)
g1_sigma: 16.6725753 +/- 0.160481 (0.96%) (init= 15)
g1_center: 107.030954 +/- 0.150067 (0.14%) (init= 105)
g1_amplitude: 4257.77319 +/- 42.38336 (1.00%) (init= 2000)
g1_fwhm: 39.2609139 +/- 0.377905 (0.96%) == '2.3548200*g1_sigma'
g1_height: 101.880231 +/- 0.592170 (0.58%) == '0.3989423*g1_amplitude/max(1.e-15, g1_sigma)'
g2_sigma: 13.8069484 +/- 0.186794 (1.35%) (init= 15)
g2_center: 153.270100 +/- 0.194667 (0.13%) (init= 155)
g2_amplitude: 2493.41770 +/- 36.16947 (1.45%) (init= 2000)
g2_fwhm: 32.5128782 +/- 0.439866 (1.35%) == '2.3548200*g2_sigma'
g2_height: 72.0455934 +/- 0.617220 (0.86%) == '0.3989423*g2_amplitude/max(1.e-15, g2_sigma)'
[[Correlations]] (unreported correlations are < 0.500)
C(g1_sigma, g1_amplitude) = 0.824
C(g2_sigma, g2_amplitude) = 0.815
C(exp_amplitude, exp_decay) = -0.695
C(g1_sigma, g2_center) = 0.684
C(g1_center, g2_amplitude) = -0.669
C(g1_center, g2_sigma) = -0.652
C(g1_amplitude, g2_center) = 0.648
C(g1_center, g2_center) = 0.621
C(g1_sigma, g1_center) = 0.507
C(exp_decay, g1_amplitude) = -0.507
We get a very good fit to this problem (described at the NIST site as of average difficulty, but the tests there are generally deliberately challenging) by applying reasonable initial guesses and putting modest but explicit bounds on the parameter values. This fit is shown on the left:
One final point on setting initial values. From looking at the data
itself, we can see the two Gaussian peaks are reasonably well separated but
do overlap. Furthermore, we can tell that the initial guess for the
decaying exponential component was poorly estimated because we used the
full data range. We can simplify the initial parameter values by using
this, and by defining an index_of()
function to limit the data range.
That is, with:
def index_of(arrval, value):
"return index of array *at or below* value "
if value < min(arrval): return 0
return max(np.where(arrval<=value)[0])
ix1 = index_of(x, 75)
ix2 = index_of(x, 135)
ix3 = index_of(x, 175)
exp_mod.guess(y[:ix1], x=x[:ix1])
gauss1.guess(y[ix1:ix2], x=x[ix1:ix2])
gauss2.guess(y[ix2:ix3], x=x[ix2:ix3])
we can get a better initial estimate. The fit converges to the same answer, giving to identical values (to the precision printed out in the report), but in few steps, and without any bounds on parameters at all:
[[Model]]
((Model(gaussian, prefix='g1_') + Model(gaussian, prefix='g2_')) + Model(exponential, prefix='exp_'))
[[Fit Statistics]]
# function evals = 48
# data points = 250
# variables = 8
chi-square = 1247.528
reduced chi-square = 5.155
Akaike info crit = 417.865
Bayesian info crit = 446.036
[[Variables]]
exp_amplitude: 99.0183281 +/- 0.537487 (0.54%) (init= 94.53724)
exp_decay: 90.9508862 +/- 1.103105 (1.21%) (init= 111.1985)
g1_sigma: 16.6725754 +/- 0.160481 (0.96%) (init= 14.5)
g1_center: 107.030954 +/- 0.150067 (0.14%) (init= 106.5)
g1_amplitude: 4257.77322 +/- 42.38338 (1.00%) (init= 2126.432)
g1_fwhm: 39.2609141 +/- 0.377905 (0.96%) == '2.3548200*g1_sigma'
g1_height: 101.880231 +/- 0.592171 (0.58%) == '0.3989423*g1_amplitude/max(1.e-15, g1_sigma)'
g2_sigma: 13.8069481 +/- 0.186794 (1.35%) (init= 15)
g2_center: 153.270100 +/- 0.194667 (0.13%) (init= 150)
g2_amplitude: 2493.41766 +/- 36.16948 (1.45%) (init= 1878.892)
g2_fwhm: 32.5128777 +/- 0.439866 (1.35%) == '2.3548200*g2_sigma'
g2_height: 72.0455935 +/- 0.617221 (0.86%) == '0.3989423*g2_amplitude/max(1.e-15, g2_sigma)'
[[Correlations]] (unreported correlations are < 0.500)
C(g1_sigma, g1_amplitude) = 0.824
C(g2_sigma, g2_amplitude) = 0.815
C(exp_amplitude, exp_decay) = -0.695
C(g1_sigma, g2_center) = 0.684
C(g1_center, g2_amplitude) = -0.669
C(g1_center, g2_sigma) = -0.652
C(g1_amplitude, g2_center) = 0.648
C(g1_center, g2_center) = 0.621
C(g1_sigma, g1_center) = 0.507
C(exp_decay, g1_amplitude) = -0.507
This script is in the file doc_nistgauss2.py
in the examples folder,
and the fit result shown on the right above shows an improved initial
estimate of the data.