Fitting the model to the data#
To fit the model to the data at the current coordinates (e.g. to fit one
spectrum at a particular point in a spectrumimage), use
fit()
. HyperSpy implements a number of
different optimization approaches, each of which can have particular
benefits and/or drawbacks depending on your specific application.
A good approach to choosing an optimization approach is to ask yourself
the question “Do you want to…”:
Apply bounds to your model parameter values?
Use gradientbased fitting algorithms to accelerate your fit?
Estimate the standard deviations of the parameter values found by the fit?
Fit your data in the leastsquares sense, or use another loss function?
Find the global optima for your parameters, or is a local optima acceptable?
Optimization algorithms#
The following table summarizes the features of some of the optimizers currently available in HyperSpy, including whether they support parameter bounds, gradients and parameter error estimation. The “Type” column indicates whether the optimizers find a local or global optima.
Optimizer 
Bounds 
Gradients 
Errors 
Loss function 
Type 
Linear 


Yes 
Yes 
Yes 
Only 
local 
No 

Yes 
Yes 
Yes 
Only 
local 
No 

Yes 
Yes 
Yes 
Only 
local 
No 

No 
Yes 
Yes 
Only 
local 
No 

No 
No 
Yes [1] 
Only 
global 
Yes 

No 
No 
Yes [1] 
Only 
global 
Yes 

No 
No 
Yes [1] 
Only 
global 
Yes 

No 
No 
Yes [1] 
Only 
global 
Yes 
Yes [2] 
Yes [2] 
No 
All 
local 
No 


Yes 
No 
No 
All 
global 
No 

Yes 
No 
No 
All 
global 
No 

Yes 
No 
No 
All 
global 
No 
Footnotes
The default optimizer in HyperSpy is "lm"
, which stands for the LevenbergMarquardt
algorithm. In
earlier versions of HyperSpy (< 1.6) this was known as "leastsq"
.
Loss functions#
HyperSpy supports a number of loss functions. The default is "ls"
,
i.e. the leastsquares loss. For the vast majority of cases, this loss
function is appropriate, and has the additional benefit of supporting
parameter error estimation and goodnessoffit
testing. However, if your data contains very low counts per pixel, or
is corrupted by outliers, the "MLpoisson"
and "huber"
loss
functions may be worth investigating.
Least squares with error estimation#
The following example shows how to perfom least squares optimization with
error estimation. First we create data consisting of a line
y = a*x + b
with a = 1
and b = 100
, and we then add Gaussian
noise to it:
>>> s = hs.signals.Signal1D(np.arange(100, 300, dtype='float32'))
>>> s.add_gaussian_noise(std=100)
To fit it, we create a model consisting of a
Polynomial
component of order 1 and fit it
to the data.
>>> m = s.create_model()
>>> line = hs.model.components1D.Polynomial(order=1)
>>> m.append(line)
>>> m.fit()
Once the fit is complete, the optimized value of the parameters and their estimated standard deviation are stored in the following line attributes:
>>> line.a0.value
0.9924615648843765
>>> line.a1.value
103.67507406125888
>>> line.a0.std
0.11771053738516088
>>> line.a1.std
13.541061301257537
Warning
When the noise is heteroscedastic, only if the
metadata.Signal.Noise_properties.variance
attribute of the
Signal1D
instance is defined can
the parameter standard deviations be estimated accurately.
If the variance is not defined, the standard deviations are still computed, by setting variance equal to 1. However, this calculation will not be correct unless an accurate value of the variance is provided. See Setting the noise properties for more information.
Weighted least squares with error estimation#
In the following example, we add Poisson noise to the data instead of Gaussian noise, and proceed to fit as in the previous example.
>>> s = hs.signals.Signal1D(np.arange(300))
>>> s.add_poissonian_noise()
>>> m = s.create_model()
>>> line = hs.model.components1D.Polynomial(order=1)
>>> m.append(line)
>>> m.fit()
>>> line.a0.value
0.7262000522775925
>>> line.a1.value
1.0086925334859176
>>> line.a0.std
1.4141418570079
>>> line.a1.std
0.008185019194679451
Because the noise is heteroscedastic, the least squares optimizer estimation is biased. A more accurate result can be obtained with weighted least squares, where the weights are proportional to the inverse of the noise variance. Although this is still biased for Poisson noise, it is a good approximation in most cases where there are a sufficient number of counts per pixel.
>>> exp_val = hs.signals.Signal1D(np.arange(300)+1)
>>> s.estimate_poissonian_noise_variance(expected_value=exp_val)
>>> line.estimate_parameters(s, 10, 250)
True
>>> m.fit()
>>> line.a0.value
0.6666008600519397
>>> line.a1.value
1.017145603577098
>>> line.a0.std
0.8681360488613021
>>> line.a1.std
0.010308732161043038
Warning
When the attribute metadata.Signal.Noise_properties.variance
is defined, the behaviour is to perform a weighted leastsquares
fit using the inverse of the noise variance as the weights.
In this scenario, to then disable weighting, you will need to unset
the attribute. You can achieve this with
set_noise_variance()
:
>>> m.signal.set_noise_variance(None) # This will now be an unweighted fit
>>> m.fit()
>>> line.a0.value
1.9711403542163477
>>> line.a1.value
1.0258716193502546
Poisson maximum likelihood estimation#
To avoid biased estimation in the case of data corrupted by Poisson noise with very few counts, we can use Poisson maximum likelihood estimation (MLE) instead. This is an unbiased estimator for Poisson noise. To perform MLE, we must use a general, nonlinear optimizer from the table above, such as NelderMead or LBFGSB:
>>> m.fit(optimizer="NelderMead", loss_function="MLpoisson")
>>> line.a0.value
0.00025567973144090695
>>> line.a1.value
1.0036866523183754
Estimation of the parameter errors is not currently supported for Poisson maximum likelihood estimation.
Huber loss function#
HyperSpy also implements the Huber loss function, which is typically less sensitive to outliers in the data compared to the leastsquares loss. Again, we need to use one of the general nonlinear optimization algorithms:
>>> m.fit(optimizer="NelderMead", loss_function="huber")
Estimation of the parameter errors is not currently supported for the Huber loss function.
Custom loss functions#
As well as the builtin loss functions described above, a custom loss function can be passed to the model:
>>> def my_custom_function(model, values, data, weights=None):
... """
... Parameters
... 
... model : Model instance
... the model that is fitted.
... values : np.ndarray
... A onedimensional array with free parameter values suggested by the
... optimizer (that are not yet stored in the model).
... data : np.ndarray
... A onedimensional array with current data that is being fitted.
... weights : {np.ndarray, None}
... An optional onedimensional array with parameter weights.
...
... Returns
... 
... score : float
... A signle float value, representing a score of the fit, with
... lower values corresponding to better fits.
... """
... # Almost any operation can be performed, for example:
... # First we store the suggested values in the model
... model.fetch_values_from_array(values)
...
... # Evaluate the current model
... cur_value = model(onlyactive=True)
...
... # Calculate the weighted difference with data
... if weights is None:
... weights = 1
... difference = (data  cur_value) * weights
...
... # Return squared and summed weighted difference
... return (difference**2).sum()
>>> # We must use a general nonlinear optimizer
>>> m.fit(optimizer='NelderMead', loss_function=my_custom_function)
If the optimizer requires an analytical gradient function, it can be similarly passed, using the following signature:
>>> def my_custom_gradient_function(model, values, data, weights=None):
... """
... Parameters
... 
... model : Model instance
... the model that is fitted.
... values : np.ndarray
... A onedimensional array with free parameter values suggested by the
... optimizer (that are not yet stored in the model).
... data : np.ndarray
... A onedimensional array with current data that is being fitted.
... weights : {np.ndarray, None}
... An optional onedimensional array with parameter weights.
...
... Returns
... 
... gradients : np.ndarray
... a onedimensional array of gradients, the size of `values`,
... containing each parameter gradient with the given values
... """
... # As an example, estimate maximum likelihood gradient:
... model.fetch_values_from_array(values)
... cur_value = model(onlyactive=True)
...
... # We use inbuilt jacobian estimation
... jac = model._jacobian(values, data)
...
... return (jac * (data / cur_value  1)).sum(1)
>>> # We must use a general nonlinear optimizer again
>>> m.fit(optimizer='LBFGSB',
... loss_function=my_custom_function,
... grad=my_custom_gradient_function)
Using gradient information#
Added in version 1.6: grad="analytical"
and grad="fd"
keyword arguments
Optimization algorithms that take into account the gradient of the loss function will often outperform socalled “derivativefree” optimization algorithms in terms of how rapidly they converge to a solution. HyperSpy can use analytical gradients for modelfitting, as well as numerical estimates of the gradient based on finite differences.
If all the components in the model support analytical gradients,
you can pass grad="analytical"
in order to use this information
when fitting. The results are typically more accurate than an
estimated gradient, and the optimization often runs faster since
fewer function evaluations are required to calculate the gradient.
Following the above examples:
>>> m = s.create_model()
>>> line = hs.model.components1D.Polynomial(order=1)
>>> m.append(line)
>>> # Use a 2point finitedifference scheme to estimate the gradient
>>> m.fit(grad="fd", fd_scheme="2point")
>>> # Use the analytical gradient
>>> m.fit(grad="analytical")
>>> # Huber loss and Poisson MLE functions
>>> # also support analytical gradients
>>> m.fit(grad="analytical", loss_function="huber")
>>> m.fit(grad="analytical", loss_function="MLpoisson")
Note
Analytical gradients are not yet implemented for the
Model2D
class.
Bounded optimization#
Nonlinear optimization can sometimes fail to converge to a good optimum, especially if poor starting values are provided. Problems of illconditioning and nonconvergence can be improved by using bounded optimization.
All components’ parameters have the attributes parameter.bmin
and
parameter.bmax
(“bounded min” and “bounded max”). When fitting using the
bounded=True
argument by m.fit(bounded=True)
or m.multifit(bounded=True)
,
these attributes set the minimum and maximum values allowed for parameter.value
.
Currently, not all optimizers support bounds  see the
table above. In the following example, a Gaussian
histogram is fitted using a Gaussian
component using the LevenbergMarquardt (“lm”) optimizer and bounds
on the centre
parameter.
>>> s = hs.signals.BaseSignal(np.random.normal(loc=10, scale=0.01,
... size=100000)).get_histogram()
>>> s.axes_manager[1].is_binned = True
>>> m = s.create_model()
>>> g1 = hs.model.components1D.Gaussian()
>>> m.append(g1)
>>> g1.centre.value = 7
>>> g1.centre.bmin = 7
>>> g1.centre.bmax = 14
>>> m.fit(optimizer="lm", bounded=True)
>>> m.print_current_values()
Model1D: histogram
Gaussian: Gaussian
Active: True
Parameter Name  Free  Value  Std  Min  Max
==============  =====  ==========  ==========  ==========  ==========
A  True  99997.3481  232.333693  0.0  None
sigma  True  0.00999184  2.68064163  None  None
centre  True  9.99980788  2.68064070  7.0  14.0
Linear least squares#
Added in version 1.7.
Linear fitting can be used to address some of the drawbacks of nonlinear optimization:
it doesn’t suffer from the starting parameters issue, which can sometimes be problematic with nonlinear fitting. Since linear fitting uses linear algebra to find the solution (find the parameter values of the model), the solution is a unique solution, while nonlinear optimization uses an iterative approach and therefore relies on the initial values of the parameters.
it is fast, because i) in favorable situations, the signal can be fitted in a vectorized fashion, i.e. the signal is fitted in a single run instead of iterating over the navigation dimension; ii) it is not iterative, i.e. it does the calculation only one time instead of 10100 iterations, depending on how quickly the nonlinear optimizer will converge.
However, linear fitting can only fit linear models and will not be able to fit parameters which vary nonlinearly.
A component is considered linear when its free parameters scale the component only
in the yaxis. For the exemplary function y = a*x**b
, a
is a linear parameter, whilst b
is not. If b.free = False
, then the component is linear.
Components can also be made up of several linear parts. For instance,
the 2Dpolynomial y = a*x**2+b*y**2+c*x+d*y+e
is entirely linear.
Note
After creating a model with values for the nonlinear parameters, a quick way to set
all nonlinear parameters to be free = False
is to use m.set_parameters_not_free(only_nonlinear=True)
To check if a parameter is linear, use the model or component method
print_current_values()
. For a component to be
considered linear, it can hold only one free parameter, and that parameter
must be linear.
If all components in a model are linear, then a linear optimizer can be used to solve the problem as a linear regression problem! This can be done using two approaches:
the standard pixelbypixel approach as used by the nonlinear optimizers
fit the entire dataset in one vectorised operation, which will be much faster (up to 1000 times). However, there is a caveat: all fixed parameters must have the same value across the dataset in order to avoid creating a very large array whose size will scale with the number of different values of the nonfree parameters.
Note
A good example of a linear model in the electronmicroscopy field is an EnergyDispersive
Xray Spectroscopy (EDS) dataset, which typically consists of a polynomial background and
Gaussian peaks with welldefined energy (Gaussian.centre
) and peak widths
(Gaussian.sigma
). This dataset can be fit extremely fast with a linear optimizer.
There are several implementations of linear least squares fitting in HyperSpy:
'lstsq'
: least squares usingnumpy.linalg.lstsq()
, ordask.array.linalg.lstsq()
for lazy signals,'ols'
: ordinary least squares, usingsklearn.linear_model.LinearRegression
,'nnls'
: least squares with positive constraints on the coefficients usingsklearn.linear_model.LinearRegression
,'ridge'
: least square supporting regularisation usingsklearn.linear_model.Ridge
. The parameteralpha
controls the regularization strength and can significantly affect the results.
See the corresponding documentation in scikitlearn
or numpy.linalg.lstsq()
for passing parameters to fit()
or
multifit()
.
Only the 'lstsq'
optimizer supports lazy signals.
As for nonlinear least squares fitting, weighted least squares are supported.
In the following example, we first generate a 300x300 navigation signal of varying total intensity,
and then populate it with an EDS spectrum at each point. The signal can be fitted with a polynomial
background and a Gaussian for each peak. Hyperspy automatically adds these to the model, and fixes
the centre
and sigma
parameters to known values. Fitting this model with a nonlinear optimizer
can about half an hour on a decent workstation. With a linear optimizer, it takes seconds.
>>> nav = hs.signals.Signal2D(np.random.random((300, 300))).T
>>> s = exspy.data.EDS_TEM_FePt_nanoparticles() * nav
>>> m = s.create_model()
>>> m.multifit(optimizer='lstsq')
Standard errors for the parameters are by default not calculated when the dataset
is fitted in vectorized fashion, because it has large memory requirement.
If errors are required, either pass calculate_errors=True
as an argument
to multifit()
, or rerun
multifit()
with a nonlinear optimizer,
which should run fast since the parameters are already optimized.
None of the linear optimizers currently support bounds.
Optimization results#
After fitting the model, details about the optimization
procedure, including whether it finished successfully,
are returned as scipy.optimize.OptimizeResult
object,
according to the keyword argument return_info=True
.
These details are often useful for diagnosing problems such
as a poorlyfitted model or a convergence failure.
You can also access the object as the fit_output
attribute:
>>> m.fit()
>>> type(m.fit_output)
<scipy.optimize.OptimizeResult object>
You can also print this information using the
print_info
keyword argument:
# Print the info to stdout
>>> m.fit(optimizer="LBFGSB", print_info=True) # doctest: +SKIP
Fit info:
optimizer=LBFGSB
loss_function=ls
bounded=False
grad="fd"
Fit result:
hess_inv: <3x3 LbfgsInvHessProduct with dtype=float64>
message: b'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
nfev: 168
nit: 32
njev: 42
status: 0
success: True
x: array([ 9.97614503e+03, 1.10610734e01, 1.98380701e+00])
Goodness of fit#
The chisquared, reduced chisquared and the degrees of freedom are
computed automatically when fitting a (weighted) leastsquares model
(i.e. only when loss_function="ls"
). They are stored as signals, in the
chisq
, red_chisq
and
dof
attributes of the model respectively.
Warning
Unless metadata.Signal.Noise_properties.variance
contains
an accurate estimation of the variance of the data, the chisquared and
reduced chisquared will not be computed correctly. This is true for both
homocedastic and heteroscedastic noise.
Visualizing the model#
To visualise the result use the plot()
method:
>>> m.plot() # Visualise the results
By default only the full model line is displayed in the plot. In addition, it
is possible to display the individual components by calling
enable_plot_components()
or directly using
plot()
:
>>> m.plot(plot_components=True) # Visualise the results
To disable this feature call
disable_plot_components()
.
Added in version 1.4: plot()
keyword arguments
All extra keyword arguments are passed to the plot()
method of the corresponding signal object. The following example plots the
model signal figure but not its navigator:
>>> m.plot(navigator=False)
By default the model plot is automatically updated when any parameter value
changes. It is possible to suspend this feature with
suspend_update()
.
Setting the initial parameters#
Nonlinear optimization often requires setting sensible starting parameters. This can be done by plotting the model and adjusting the parameters by hand.
Changed in version 1.3: All notebook_interaction
methods renamed to gui()
.
The notebook_interaction
methods were removed in 2.0.
If running in a Jupyter Notebook, interactive widgets can be used to
conveniently adjust the parameter values by running
gui()
for BaseModel
,
Component
and
Parameter
.
Also, enable_adjust_position()
provides an
interactive way of setting the position of the components with a
welldefined position.
disable_adjust_position()
disables the tool.
Exclude data from the fitting process#
The following BaseModel
methods can be used to exclude
undesired spectral channels from the fitting process:
The example below shows how a boolean array can be easily created from the
signal and how the isig
syntax can be used to define the signal range.
>>> # Create a sample 2D gaussian dataset
>>> g = hs.model.components2D.Gaussian2D(
... A=1, centre_x=5.0, centre_y=5.0, sigma_x=1.0, sigma_y=2.0,)
>>> scale = 0.1
>>> x = np.arange(10, 10, scale)
>>> y = np.arange(10, 10, scale)
>>> X, Y = np.meshgrid(x, y)
>>> im = hs.signals.Signal2D(g.function(X, Y))
>>> im.axes_manager[0].scale = scale
>>> im.axes_manager[0].offset = 10
>>> im.axes_manager[1].scale = scale
>>> im.axes_manager[1].offset = 10
>>> m = im.create_model() # Model initialisation
>>> gt = hs.model.components2D.Gaussian2D()
>>> m.append(gt)
>>> m.set_signal_range(7, 3, 9, 1) # Set signal range
>>> m.fit()
Alternatively, create a boolean signal of the same shape
as the signal space of im
>>> signal_mask = im > 0.01
>>> m.set_signal_range_from_mask(signal_mask.data) # Set signal range
>>> m.fit()
Fitting multidimensional datasets#
To fit the model to all the elements of a multidimensional dataset, use
multifit()
:
>>> m.multifit() # warning: this can be a lengthy process on large datasets
multifit()
fits the model at the first position,
stores the result of the fit internally and move to the next position until
reaching the end of the dataset.
Note
Sometimes this method can fail, especially in the case of a TEM spectrum image of a particle surrounded by vacuum (since in that case the topleft pixel will typically be an empty signal).
To get sensible starting parameters, you can do a single
fit()
after changing the active position
within the spectrum image (either using the plotting GUI or by directly
modifying s.axes_manager.indices
as in Setting axis properties).
After doing this, you can initialize the model at every pixel to the
values from the single pixel fit using m.assign_current_values_to_all()
,
and then use multifit()
to perform the fit over
the entire spectrum image.
Added in version 1.6: New optional fitting iteration path “serpentine”
Added in version 2.0: New default iteration path for fitting is “serpentine”`
In HyperSpy, curve fitting on a multidimensional dataset happens in the following manner: Pixels are fit along the row from the first index in the first row, and once the last pixel in the row is reached, one proceeds in reverse order from the last index in the second row. This procedure leads to a serpentine pattern, as seen on the image below. The serpentine pattern supports ndimensional navigation space, so the first index in the second frame of a threedimensional navigation space will be at the last position of the previous frame.
An alternative scan pattern would be the 'flyback'
scheme, where the map is
iterated through row by row, always starting from the first index. This pattern
can be explicitly set using the multifit()
iterpath='flyback'
argument. However, the 'serpentine'
strategy is
usually more robust, as it always moves on to a neighbouring pixel and the fitting
procedure uses the fit result of the previous pixel as the starting point for the
next. A common problem in the 'flyback'
pattern is that the fitting fails
going from the end of one row to the beginning of the next, as the spectrum can
change abruptly.
In addition to 'serpentine'
and 'flyback'
, iterpath
can take as
argument any list or array of indices, or a generator of such, as explained in
the Iterating AxesManager section.
Sometimes one may like to store and fetch the value of the parameters at a
given position manually. This is possible using
store_current_values()
and
fetch_stored_values()
.
Visualising the result of the fit#
The BaseModel
plot_results()
,
Component
plot()
and
Parameter
plot()
methods
can be used to visualise the result of the fit when fitting multidimensional
datasets.