Model fitting

HyperSpy can perform curve fitting of one-dimensional signals (spectra) and two-dimensional signals (images) in n-dimensional data sets. Models are defined by adding individual functions (components in HyperSpy’s terminology) to a BaseModel instance. Those individual components are then summed to create the final model function that can be fitted to the data, by adjusting the free parameters of the individual components.

Models can be created and fit to experimental data in both one and two dimensions i.e. spectra and images respectively. Most of the syntax is identical in either case. A one-dimensional model is created when a model is created for a Signal1D whereas a two- dimensional model is created for a Signal2D.

Note

Plotting and analytical gradient-based fitting methods are not yet implemented for the Model2D class.

Caveats

• Before creating a model verify that the is_binned attribute of the signal axis is set to the correct value because the resulting model depends on this parameter. See Binned and unbinned signals for more details.

• When importing data that has been binned using other software, in particular Gatan’s DM, the stored values may be the averages of the binned channels or pixels, instead of their sum, as would be required for proper statistical analysis. We therefore cannot guarantee that the statistics will be valid, and so strongly recommend that all pre-fitting binning is performed using Hyperspy.

Creating a model

A Model1D can be created for data in the Signal1D class using the create_model() method:

>>> s = hs.signals.Signal1D(np.arange(300).reshape(30, 10))
>>> m = s.create_model() # Creates the 1D-Model and assign it to m

Similarly, a Model2D can be created for data in the Signal2D class using the create_model() method:

>>> im = hs.signals.Signal2D(np.arange(300).reshape(3, 10, 10))
>>> mod = im.create_model() # Create the 2D-Model and assign it to mod

The syntax for creating both one-dimensional and two-dimensional models is thus identical for the user in practice. When a model is created you may be prompted to provide important information not already included in the datafile, e.g. if s is EELS data, you may be asked for the accelerating voltage, convergence and collection semi-angles etc.

Model components

In HyperSpy a model consists of a sum of individual components. For convenience, HyperSpy provides a number of pre-defined model components as well as mechanisms to create your own components.

Pre-defined model components

Various components are available in one (components1d) and two-dimensions (components2d) to construct a model.

The following general components are currently available for one-dimensional models:

• Arctan

• Bleasdale

• Doniach

• Erf

• Exponential

• Expression

• Gaussian

• GaussianHF

• HeavisideStep

• Logistic

• Lorentzian

• Offset

• Polynomial

• PowerLaw

• SEE

• ScalableFixedPattern

• SkewNormal

• Voigt

• SplitVoigt

• VolumePlasmonDrude

The following components developed with specific signal types in mind are currently available for one-dimensional models:

• EELSArctan

• DoublePowerLaw

• EELSCLEdge

• PESCoreLineShape

• PESVoigt

• SEE

• Vignetting

The following components are currently available for two-dimensional models:

• Expression

• Gaussian2D

However, this doesn’t mean that you have to limit yourself to this meagre list of functions. As discussed below, it is very easy to turn a mathematical, fixed-pattern or Python function into a component.

Define components from a mathematical expression

The easiest way to turn a mathematical expression into a component is using the Expression component. For example, the following is all you need to create a Gaussian component with more sensible parameters for spectroscopy than the one that ships with HyperSpy:

>>> g = hs.model.components1D.Expression(
... expression="height * exp(-(x - x0) ** 2 * 4 * log(2)/ fwhm ** 2)",
... name="Gaussian",
... position="x0",
... height=1,
... fwhm=1,
... x0=0,
... module="numpy")

If the expression is inconvenient to write out in full (e.g. it’s long and/or complicated), multiple substitutions can be given, separated by semicolons. Both symbolic and numerical substitutions are allowed:

>>> expression = "h / sqrt(p2) ; p2 = 2 * m0 * e1 * x * brackets;"
>>> expression += "brackets = 1 + (e1 * x) / (2 * m0 * c * c) ;"
>>> expression += "m0 = 9.1e-31 ; c = 3e8; e1 = 1.6e-19 ; h = 6.6e-34"
>>> wavelength = hs.model.components1D.Expression(
... expression=expression,
... name="Electron wavelength with voltage")

Expression uses Sympy internally to turn the string into a function. By default it “translates” the expression using numpy, but often it is possible to boost performance by using numexpr instead.

It can also create 2D components with optional rotation. In the following example we create a 2D Gaussian that rotates around its center:

>>> g = hs.model.components2D.Expression(
... "k * exp(-((x-x0)**2 / (2 * sx ** 2) + (y-y0)**2 / (2 * sy ** 2)))",
... "Gaussian2d", add_rotation=True, position=("x0", "y0"),
... module="numpy", )

Define new components from a Python function

Of course Expression is only useful for analytical functions. You can define more general components modifying the following template to suit your needs:

from hyperspy.component import Component

class MyComponent(Component):

    """
    """

    def __init__(self, parameter_1=1, parameter_2=2):
        # Define the parameters
        Component.__init__(self, ('parameter_1', 'parameter_2'))

        # Optionally we can set the initial values
        self.parameter_1.value = parameter_1
        self.parameter_2.value = parameter_2

        # The units (optional)
        self.parameter_1.units = 'Tesla'
        self.parameter_2.units = 'Kociak'

        # Once defined we can give default values to the attribute
        # For example we fix the attribure_1 (optional)
        self.parameter_1.attribute_1.free = False

        # And we set the boundaries (optional)
        self.parameter_1.bmin = 0.
        self.parameter_1.bmax = None

        # Optionally, to boost the optimization speed we can also define
        # the gradients of the function we the syntax:
        # self.parameter.grad = function
        self.parameter_1.grad = self.grad_parameter_1
        self.parameter_2.grad = self.grad_parameter_2

    # Define the function as a function of the already defined parameters,
    # x being the independent variable value
    def function(self, x):
        p1 = self.parameter_1.value
        p2 = self.parameter_2.value
        return p1 + x * p2

    # Optionally define the gradients of each parameter
    def grad_parameter_1(self, x):
        """
        Returns d(function)/d(parameter_1)
        """
        return 0

    def grad_parameter_2(self, x):
        """
        Returns d(function)/d(parameter_2)
        """
        return x

Define components from a fixed-pattern

The ScalableFixedPattern component enables fitting a pattern (in the form of a Signal1D instance) to data by shifting (shift) and scaling it in the x and y directions using the xscale and yscale parameters respectively.

Adding components to the model

To print the current components in a model use components. A table with component number, attribute name, component name and component type will be printed:

>>> m
<Model, title: my signal title>
>>> m.components # an empty model
   # |       Attribute Name |       Component Name |        Component Type
---- | -------------------- | -------------------- | ---------------------

Note

Sometimes components may be created automatically. For example, if the Signal1D is recognised as EELS data, a power-law background component may automatically be added to the model. Therefore, the table above may not all may empty on model creation.

To add a component to the model, first we have to create an instance of the component. Once the instance has been created we can add the component to the model using the append() and extend() methods for one or more components respectively.

As an example, let’s add several Gaussian components to the model:

>>> gaussian = hs.model.components1D.Gaussian() # Create a Gaussian comp.
>>> m.append(gaussian) # Add it to the model
>>> m.components # Print the model components
   # |       Attribute Name |        Component Name |        Component Type
---- | -------------------- | --------------------- | ---------------------
   0 |             Gaussian |              Gaussian |              Gaussian
>>> gaussian2 = hs.model.components1D.Gaussian() # Create another gaussian
>>> gaussian3 = hs.model.components1D.Gaussian() # Create a third gaussian

We could use the append() method twice to add the two Gaussians, but when adding multiple components it is handier to use the extend method that enables adding a list of components at once.

>>> m.extend((gaussian2, gaussian3)) # note the double parentheses!
>>> m.components
   # |       Attribute Name |      Component Name |        Component Type
---- | -------------------- | ------------------- | ---------------------
   0 |             Gaussian |            Gaussian |              Gaussian
   1 |           Gaussian_0 |          Gaussian_0 |              Gaussian
   2 |           Gaussian_1 |          Gaussian_1 |              Gaussian

We can customise the name of the components.

>>> gaussian.name = 'Carbon'
>>> gaussian2.name = 'Long Hydrogen name'
>>> gaussian3.name = 'Nitrogen'
>>> m.components
   # |        Attribute Name |        Component Name |      Component Type
---- | --------------------- | --------------------- | -------------------
   0 |                Carbon |                Carbon |            Gaussian
   1 |    Long_Hydrogen_name |    Long Hydrogen name |            Gaussian
   2 |              Nitrogen |              Nitrogen |            Gaussian

Notice that two components cannot have the same name:

>>> gaussian2.name = 'Carbon'
Traceback (most recent call last):
  File "<ipython-input-5-2b5669fae54a>", line 1, in <module>
    g2.name = "Carbon"
  File "/home/fjd29/Python/hyperspy/hyperspy/component.py", line 466, in
    name "the name " + str(value))
ValueError: Another component already has the name Carbon

It is possible to access the components in the model by their name or by the index in the model.

>>> m
   # |        Attribute Name |       Component Name |      Component Type
---- | --------------------- | -------------------- | -------------------
   0 |                Carbon |               Carbon |            Gaussian
   1 |    Long_Hydrogen_name |   Long Hydrogen name |            Gaussian
   2 |              Nitrogen |             Nitrogen |            Gaussian
>>> m[0]
<Carbon (Gaussian component)>
>>> m["Long Hydrogen name"]
<Long Hydrogen name (Gaussian component)>

In addition, the components can be accessed in the components Model attribute. This is specially useful when working in interactive data analysis with IPython because it enables tab completion.

>>> m
   # |        Attribute Name |        Component Name |      Component Type
---- | --------------------- | --------------------- | -------------------
   0 |                Carbon |                Carbon |            Gaussian
   1 |    Long_Hydrogen_name |    Long Hydrogen name |            Gaussian
   2 |              Nitrogen |              Nitrogen |            Gaussian
>>> m.components.Long_Hydrogen_name
<Long Hydrogen name (Gaussian component)>

It is possible to “switch off” a component by setting its active attribute to False. When a component is switched off, to all effects it is as if it was not part of the model. To switch it back on simply set the active attribute back to True.

In multi-dimensional signals it is possible to store the value of the active attribute at each navigation index. To enable this feature for a given component set the active_is_multidimensional attribute to True.

>>> s = hs.signals.Signal1D(np.arange(100).reshape(10,10))
>>> m = s.create_model()
>>> g1 = hs.model.components1D.Gaussian()
>>> g2 = hs.model.components1D.Gaussian()
>>> m.extend([g1,g2])
>>> g1.active_is_multidimensional = True
>>> g1._active_array
array([ True,  True,  True,  True,  True,  True,  True,  True,  True,  True], dtype=bool)
>>> g2._active_array is None
True
>>> m.set_component_active_value(False)
>>> g1._active_array
array([False, False, False, False, False, False, False, False, False, False], dtype=bool)
>>> m.set_component_active_value(True, only_current=True)
>>> g1._active_array
array([ True, False, False, False, False, False, False, False, False, False], dtype=bool)
>>> g1.active_is_multidimensional = False
>>> g1._active_array is None
True

Indexing the model

Often it is useful to consider only part of the model - for example at a particular location (i.e. a slice in the navigation space) or energy range (i.e. a slice in the signal space). This can be done using exactly the same syntax that we use for signal indexing (Indexing). red_chisq and dof are automatically recomputed for the resulting slices.

>>> s = hs.signals.Signal1D(np.arange(100).reshape(10,10))
>>> m = s.create_model()
>>> m.append(hs.model.components1D.Gaussian())
>>> # select first three navigation pixels and last five signal channels
>>> m1 = m.inav[:3].isig[-5:]
>>> m1.signal
<Signal1D, title: , dimensions: (3|5)>

Getting and setting parameter values and attributes

print_current_values() prints the properties of the parameters of the components in the current coordinates. In the Jupyter Notebook, the default view is HTML-formatted, which allows for formatted copying into other software, such as Excel. This can be changed to a standard terminal view with the argument fancy=False. One can also filter for only active components and only showing component with free parameters with the arguments only_active and only_free, respectively.

The current values of a particular component can be printed using the print_current_values() method.

>>> m = s.create_model()
>>> m.fit()
>>> G = m[1]
>>> G.print_current_values(fancy=False)
Gaussian: Al_Ka
Active: True
Parameter Name |  Free |      Value |        Std |        Min
============== | ===== | ========== | ========== | ==========
             A |  True | 62894.6824 | 1039.40944 |        0.0
         sigma | False | 0.03253440 |       None |       None
        centre | False |     1.4865 |       None |       None

The current coordinates can be either set by navigating the plot(), or specified by pixel indices in m.axes_manager.indices or as calibrated coordinates in m.axes_manager.coordinates.

parameters contains a list of the parameters of a component and free_parameters lists only the free parameters.

The value of a particular parameter in the current coordinates can be accessed by component.Parameter.value (e.g. Gaussian.A.value). To access an array of the value of the parameter across all navigation pixels, component.Parameter.map['values'] (e.g. Gaussian.A.map["values"]) can be used. On its own, component.Parameter.map returns a NumPy array with three elements: 'values', 'std' and 'is_set'. The first two give the value and standard error for each index. The last element shows whether the value has been set in a given index, either by a fitting procedure or manually.

If a model contains several components with the same parameters, it is possible to change them all by using set_parameters_value(). Example:

>>> s = hs.signals.Signal1D(np.arange(100).reshape(10,10))
>>> m = s.create_model()
>>> g1 = hs.model.components1D.Gaussian()
>>> g2 = hs.model.components1D.Gaussian()
>>> m.extend([g1,g2])
>>> m.set_parameters_value('A', 20)
>>> g1.A.map['values']
array([ 20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.])
>>> g2.A.map['values']
array([ 20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.])
>>> m.set_parameters_value('A', 40, only_current=True)
>>> g1.A.map['values']
array([ 40.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.])
>>> m.set_parameters_value('A',30, component_list=[g2])
>>> g2.A.map['values']
array([ 30.,  30.,  30.,  30.,  30.,  30.,  30.,  30.,  30.,  30.])
>>> g1.A.map['values']
array([ 40.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.,  20.])

To set the free state of a parameter change the free attribute. To change the free state of all parameters in a component to True use set_parameters_free(), and set_parameters_not_free() for setting them to False. Specific parameter-names can also be specified by using parameter_name_list, shown in the example:

>>> g = hs.model.components1D.Gaussian()
>>> g.free_parameters
[<Parameter A of Gaussian component>,
<Parameter sigma of Gaussian component>,
<Parameter centre of Gaussian component>]
>>> g.set_parameters_not_free()
>>> g.set_parameters_free(parameter_name_list=['A','centre'])
>>> g.free_parameters
[<Parameter A of Gaussian component>,
<Parameter centre of Gaussian component>]

Similar functions exist for BaseModel: set_parameters_free() and set_parameters_not_free(). Which sets the free states for the parameters in components in a model. Specific components and parameter-names can also be specified. For example:

>>> g1 = hs.model.components1D.Gaussian()
>>> g2 = hs.model.components1D.Gaussian()
>>> m.extend([g1,g2])
>>> m.set_parameters_not_free()
>>> g1.free_parameters
[]
>>> g2.free_parameters
[]
>>> m.set_parameters_free(parameter_name_list=['A'])
>>> g1.free_parameters
[<Parameter A of Gaussian component>]
>>> g2.free_parameters
[<Parameter A of Gaussian component>]
>>> m.set_parameters_free([g1], parameter_name_list=['sigma'])
>>> g1.free_parameters
[<Parameter A of Gaussian component>,
     <Parameter sigma of Gaussian component>]
>>> g2.free_parameters
[<Parameter A of Gaussian component>]

The value of a parameter can be coupled to the value of another by setting the twin attribute:

>>> gaussian.parameters # Print the parameters of the Gaussian components
(<Parameter A of Carbon component>,
<Parameter sigma of Carbon component>,
<Parameter centre of Carbon component>)
>>> gaussian.centre.free = False # Fix the centre
>>> gaussian.free_parameters  # Print the free parameters
[<Parameter A of Carbon component>, <Parameter sigma of Carbon component>]
>>> m.print_current_values(only_free=True, fancy=False) # Print the values of all free parameters.
Model1D:
Gaussian: Carbon
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A |  True |        1.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None

Gaussian: Long Hydrogen name
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A |  True |        1.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None
        centre |  True |        0.0 |       None |       None |       None

Gaussian: Nitrogen
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A |  True |        1.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None
        centre |  True |        0.0 |       None |       None |       None

>>> # Couple the A parameter of gaussian2 to the A parameter of gaussian 3:
>>> gaussian2.A.twin = gaussian3.A
>>> gaussian2.A.value = 10 # Set the gaussian2 A value to 10
>>> gaussian3.print_current_values(fancy=False)
Gaussian: Nitrogen
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A |  True |       10.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None
        centre |  True |        0.0 |       None |       None |       None

>>> gaussian3.A.value = 5 # Set the gaussian1 centre value to 5
>>> gaussian2.print_current_values(fancy=False)
Gaussian: Long Hydrogen name
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A | False |        5.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None
        centre |  True |        0.0 |       None |       None |       None

Deprecated since version 1.2.0: Setting the twin_function and twin_inverse_function attributes. Set the twin_function_expr and twin_inverse_function_expr attributes instead.

New in version 1.2.0: twin_function_expr and twin_inverse_function_expr.

By default the coupling function is the identity function. However it is possible to set a different coupling function by setting the twin_function_expr and twin_inverse_function_expr attributes. For example:

>>> gaussian2.A.twin_function_expr = "x**2"
>>> gaussian2.A.twin_inverse_function_expr = "sqrt(abs(x))"
>>> gaussian2.A.value = 4
>>> gaussian3.print_current_values(fancy=False)
Gaussian: Nitrogen
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A |  True |        2.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None
        centre |  True |        0.0 |       None |       None |       None

>>> gaussian3.A.value = 4
>>> gaussian2.print_current_values(fancy=False)
Gaussian: Long Hydrogen name
Active: True
Parameter Name |  Free |      Value |        Std |        Min |        Max
============== | ===== | ========== | ========== | ========== | ==========
             A | False |       16.0 |       None |        0.0 |       None
         sigma |  True |        1.0 |       None |       None |       None
        centre |  True |        0.0 |       None |       None |       None

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 spectrum-image), 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 gradient-based fitting algorithms to accelerate your fit?

• Estimate the standard deviations of the parameter values found by the fit?

• Fit your data in the least-squares 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.

Features of supported curve-fitting optimizers.

Optimizer	Bounds	Gradients	Errors	Loss function	Type	Linear
"lm" (default)	Yes	Yes	Yes	Only "ls"	local	No
"trf"	Yes	Yes	Yes	Only "ls"	local	No
"dogbox"	Yes	Yes	Yes	Only "ls"	local	No
"odr"	No	Yes	Yes	Only "ls"	local	No
"lstsq"	No	No	Yes [1]	Only "ls"	global	Yes
"ridge_regression"	No	No	Yes [1]	Only "ls"	global	Yes
scipy.optimize.minimize()	Yes [2]	Yes [2]	No	All	local	No
"Differential Evolution"	Yes	No	No	All	global	No
"Dual Annealing" [3]	Yes	No	No	All	global	No
"SHGO" [3]	Yes	No	No	All	global	No

Footnotes

[1] (1,2)

Requires the multifit() calculate_errors = True argument in most cases. See the documentation below on linear least square fitting for more info.

[2] (1,2)

All of the fitting algorithms available in scipy.optimize.minimize() are currently supported by HyperSpy; however, only some of them support bounds and/or gradients. For more information, please see the SciPy documentation.

[3] (1,2)

Requires scipy >= 1.2.0.

The default optimizer in HyperSpy is "lm", which stands for the Levenberg-Marquardt 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 least-squares loss. For the vast majority of cases, this loss function is appropriate, and has the additional benefit of supporting parameter error estimation and goodness-of-fit testing. However, if your data contains very low counts per pixel, or is corrupted by outliers, the "ML-poisson" 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))
>>> 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.a.value
0.9924615648843765
>>> line.b.value
103.67507406125888
>>> line.a.std
0.11771053738516088
>>> line.b.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.coefficients.value
(1.0052331707848698, -1.0723588390873573)
>>> line.coefficients.std
(0.0081710549764721901, 1.4117294994070277)

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))
>>> s.estimate_poissonian_noise_variance(expected_value=exp_val)
>>> m.fit()
>>> line.coefficients.value
(1.0004224896604759, -0.46982916592391377)
>>> line.coefficients.std
(0.0055752036447948173, 0.46950832982673557)

Warning

When the attribute metadata.Signal.Noise_properties.variance is defined, the behaviour is to perform a weighted least-squares 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)
>>> m.fit()  # This will now be an unweighted fit
>>> line.coefficients.value
(1.0052331707848698, -1.0723588390873573)

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, non-linear optimizer from the table above, such as Nelder-Mead or L-BFGS-B:

>>> m.fit(optimizer="Nelder-Mead", loss_function="ML-poisson")
>>> line.coefficients.value
(1.0030718094185611, -0.63590210946134107)

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 least-squares loss. Again, we need to use one of the general non-linear optimization algorithms:

>>> m.fit(optimizer="Nelder-Mead", loss_function="huber")

Estimation of the parameter errors is not currently supported for the Huber loss function.

Custom loss functions

As well as the built-in 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 one-dimensional array with free parameter values suggested by the
...        optimizer (that are not yet stored in the model).
...    data : np.ndarray
...        A one-dimensional array with current data that is being fitted.
...    weights : {np.ndarray, None}
...        An optional one-dimensional 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 non-linear optimizer
>>> m.fit(optimizer='Nelder-Mead', 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 one-dimensional array with free parameter values suggested by the
...        optimizer (that are not yet stored in the model).
...    data : np.ndarray
...        A one-dimensional array with current data that is being fitted.
...    weights : {np.ndarray, None}
...        An optional one-dimensional array with parameter weights.
...
...    Returns
...    -------
...    gradients : np.ndarray
...        a one-dimensional 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 in-built jacobian estimation
...    jac = model._jacobian(values, data)
...
...    return -(jac * (data / cur_value - 1)).sum(1)

>>> # We must use a general non-linear optimizer again
>>> m.fit(optimizer='L-BFGS-B',
...       loss_function=my_custom_function,
...       grad=my_custom_gradient_function)

Using gradient information

New 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 out-perform so-called “derivative-free” optimization algorithms in terms of how rapidly they converge to a solution. HyperSpy can use analytical gradients for model-fitting, 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 2-point finite-difference scheme to estimate the gradient
>>> m.fit(grad="fd", fd_scheme="2-point")

>>> # 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="ML-poisson")

Note

Analytical gradients are not yet implemented for the Model2D class.

Bounded optimization

Non-linear optimization can sometimes fail to converge to a good optimum, especially if poor starting values are provided. Problems of ill-conditioning and non-convergence 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 Levenberg-Marquardt (“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(fancy=False)
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

New in version 1.7.

Linear fitting can be used to address some of the drawbacks of non-linear 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 10-100 iterations, depending on how quickly the non-linear optimizer will converge.

However, linear fitting can only fit linear models and will not be able to fit parameters which vary non-linearly.

A component is considered linear when its free parameters scale the component only in the y-axis. 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 2D-polynomial 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 pixel-by-pixel 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 non-free parameters.

Note

A good example of a linear model in the electron-microscopy field is an Energy-Dispersive X-ray Spectroscopy (EDS) dataset, which typically consists of a polynomial background and Gaussian peaks with well-defined energy (Gaussian.centre) and peak widths (Gaussian.sigma). This dataset can be fit extremely fast with a linear optimizer.

There are two implementations of linear least squares fitting in hyperspy:

• the 'lstsq' optimizer, which uses numpy.linalg.lstsq(), or dask.array.linalg.lstsq() for lazy signals.

• the 'ridge_regression' optimizer, which supports regularization (see sklearn.linear_model.Ridge for arguments to pass to fit()), but does not support lazy signals.

As for non-linear least squares fitting, weighted least squares is 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 non-linear 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 = hs.datasets.example_signals.EDS_SEM_Spectrum() * 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 poorly-fitted model or a convergence failure. You can also access the object as the fit_output attribute:

>>> m.fit()
<scipy.optimize.OptimizeResult object>

>>> 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="L-BFGS-B", print_info=True)
Fit info:
  optimizer=L-BFGS-B
  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, -1.10610734e-01,  1.98380701e+00])

Goodness of fit

The chi-squared, reduced chi-squared and the degrees of freedom are computed automatically when fitting a (weighted) least-squares 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 chi-squared and reduced chi-squared 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().

New in version 1.4: Signal1D.plot keyword arguments

All extra keyword argments are passes to the plot() method of the corresponing signal object. For example, the following 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

Non-linear 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 will be 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.

Interactive widgets for the full model in a Jupyter notebook. Drag the sliders to adjust current parameter values. Typing different minimum and maximum values changes the boundaries of the slider.

Also, enable_adjust_position() provides an interactive way of setting the position of the components with a well-defined position. disable_adjust_position() disables the tool.

Interactive component position adjustment tool. Drag the vertical lines to set the initial value of the position parameter.

Exclude data from the fitting process

The following BaseModel methods can be used to exclude undesired spectral channels from the fitting process:

• set_signal_range()

• remove_signal_range()

• reset_signal_range()

In 2D models, those methods are not implemented and the m.channel_switches attribute of a model can be set using boolean arrays of the same shape as the data’s signal, where True means that the datapoint will be used in the fitting routine.

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)

>>> # Create a boolean signal of the same shape as the signal space of im
>>> # and with all values set to `False`.
>>> signal_mask = hs.signals.Signal2D(np.zeros_like(im(), dtype=bool))
>>> # Specify the signal range using the isig syntax
>>> signal_mask.isig[-7.:-3.,-9.:-1.] = True

>>> m.channel_switches = signal_mask.data # Set channel switches
>>> 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 top-left 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.

New in version 1.6: New optional fitting iteration path “serpentine”

Typically, 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 from the first index in the second row. Since the fitting procedure typically uses the fit of the previous pixel as the starting point for the next, a common problem with this fitting iteration path is that the fitting fails going from the end of one row to the beginning of the next, as the spectrum can change abruptly. This kind of iteration path is the default in HyperSpy (but will change to 'serpentine' in HyperSpy version 2.0). It can be explicitly set using the multifit() iterpath='flyback' argument.

A simple solution to the flyback fitting problem is to iterate through the signal indices in a horizontal serpentine pattern, as seen on the image below. This alternate iteration method can be enabled by the multifit() iterpath='serpentine' argument. The serpentine pattern supports n-dimensional navigation space, so the first index in the second frame of a three-dimensional navigation space will be at the last position of the previous frame.

Comparing the scan patterns generated by the 'flyback' and 'serpentine' iterpath options for a 2D navigation space. The pixel intensity and number refers to the order that the signal is fitted in.

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.

Storing models

Multiple models can be stored in the same signal. In particular, when store() is called, a full “frozen” copy of the model is stored in stored in the signal’s ModelManager, which can be accessed in the models attribute (i.e. s.models) The stored models can be recreated at any time by calling restore() with the stored model name as an argument. To remove a model from storage, simply call remove().

The stored models can be either given a name, or assigned one automatically. The automatic naming follows alphabetical scheme, with the sequence being (a, b, …, z, aa, ab, …, az, ba, …).

Note

If you want to slice a model, you have to perform the operation on the model itself, not its stored version

Warning

Modifying a signal in-place (e.g. map(), crop(), align1D(), align2D() and similar) will invalidate all stored models. This is done intentionally.

Current stored models can be listed by calling s.models:

>>> m = s.create_model()
>>> m.append(hs.model.components1D.Lorentzian())
>>> m.store('myname')
>>> s.models
└── myname
    ├── components
    │   └── Lorentzian
    ├── date = 2015-09-07 12:01:50
    └── dimensions = (|100)

>>> m.append(hs.model.components1D.Exponential())
>>> m.store() # assign model name automatically
>>> s.models
├── a
│   ├── components
│   │   ├── Exponential
│   │   └── Lorentzian
│   ├── date = 2015-09-07 12:01:57
│   └── dimensions = (|100)
└── myname
    ├── components
    │   └── Lorentzian
    ├── date = 2015-09-07 12:01:50
    └── dimensions = (|100)
>>> m1 = s.models.restore('myname')
>>> m1.components
   # |      Attribute Name |       Component Name |       Component Type
---- | ------------------- | -------------------- | --------------------
   0 |          Lorentzian |           Lorentzian |           Lorentzian

Saving and loading the result of the fit

To save a model, a convenience function save() is provided, which stores the current model into its signal and saves the signal. As described in Storing models, more than just one model can be saved with one signal.

>>> m = s.create_model()
>>> # analysis and fitting goes here
>>> m.save('my_filename', 'model_name')
>>> l = hs.load('my_filename.hspy')
>>> m = l.models.restore('model_name') # or l.models.model_name.restore()

For older versions of HyperSpy (before 0.9), the instructions were as follows:

Note that this method is known to be brittle i.e. there is no guarantee that a version of HyperSpy different from the one used to save the model will be able to load it successfully. Also, it is advisable not to use this method in combination with functions that alter the value of the parameters interactively (e.g. enable_adjust_position) as the modifications made by this functions are normally not stored in the IPython notebook or Python script.

To save a model:

1. Save the parameter arrays to a file using save_parameters2file().

2. Save all the commands that used to create the model to a file. This can be done in the form of an IPython notebook or a Python script.

3. (Optional) Comment out or delete the fitting commands (e.g. multifit()).

To recreate the model:

1. Execute the IPython notebook or Python script.

2. Use load_parameters_from_file() to load back the parameter values and arrays.

Exporting the result of the fit

The BaseModel export_results(), Component export() and Parameter export() methods can be used to export the result of the optimization in all supported formats.

Batch setting of parameter attributes

The following model methods can be used to ease the task of setting some important parameter attributes. These can also be used on a per-component basis, by calling them on individual components.

• set_parameters_not_free()

• set_parameters_free()

• set_parameters_value()

Fitting big data

See the section in Binned and unbinned signals working with big data.

Smart Adaptive Multi-dimensional Fitting (SAMFire)

SAMFire (Smart Adaptive Multi-dimensional Fitting) is an algorithm created to reduce the starting value (or local / false minima) problem, which often arises when fitting multi-dimensional datasets.

The algorithm will be described in full when accompanying paper is published, but we are making the implementation available now, with additional details available in the following conference proceeding.

The idea

The main idea of SAMFire is to change two things compared to the traditional way of fitting datasets with many dimensions in the navigation space:

1. Pick a more sensible pixel fitting order.

2. Calculate the pixel starting parameters from already fitted parts of the dataset.

Both of these aspects are linked one to another and are represented by two different strategy families that SAMFfire uses while operating.

Strategies

During operation SAMFire uses a list of strategies to determine how to select the next pixel and estimate its starting parameters. Only one strategy is used at a time. Next strategy is chosen when no new pixels can be fitted with the current strategy. Once either the strategy list is exhausted or the full dataset fitted, the algorithm terminates.

There are two families of strategies. In each family there may be many strategies, using different statistical or significance measures.

As a rule of thumb, the first strategy in the list should always be from the local family, followed by a strategy from the global family.

Local strategy family

These strategies assume that locally neighbouring pixels are similar. As a result, the pixel fitting order seems to follow data-suggested order, and the starting values are computed from the surrounding already fitted pixels.

More information about the exact procedure will be available once the accompanying paper is published.

Global strategy family

Global strategies assume that the navigation coordinates of each pixel bear no relation to it’s signal (i.e. the location of pixels is meaningless). As a result, the pixels are selected at random to ensure uniform sampling of the navigation space.

A number of candidate starting values are computed form global statistical measures. These values are all attempted in order until a satisfactory result is found (not necessarily testing all available starting guesses). As a result, on average each pixel requires significantly more computations when compared to a local strategy.

More information about the exact procedure will be available once the accompanying paper is published.

Seed points

Due to the strategies using already fitted pixels to estimate the starting values, at least one pixel has to be fitted beforehand by the user.

The seed pixel(s) should be selected to require the most complex model present in the dataset, however in-built goodness of fit checks ensure that only sufficiently well fitted values are allowed to propagate.

If the dataset consists of regions (in the navigation space) of highly dissimilar pixels, often called “domain structures”, at least one seed pixel should be given for each unique region.

If the starting pixels were not optimal, only part of the dataset will be fitted. In such cases it is best to allow the algorithm terminate, then provide new (better) seed pixels by hand, and restart SAMFire. It will use the new seed together with the already computed parts of the data.

Usage

After creating a model and fitting suitable seed pixels, to fit the rest of the multi-dimensional dataset using SAMFire we must create a SAMFire instance as follows:

>>> samf = m.create_samfire(workers=None, ipyparallel=False)

By default SAMFire will look for an ipyparallel cluster for the workers for around 30 seconds. If none is available, it will use multiprocessing instead. However, if you are not planning to use ipyparallel, it’s recommended specify it explicitly via the ipyparallel=False argument, to use the fall-back option of multiprocessing.

By default a new SAMFire object already has two (and currently only) strategies added to its strategies list:

>>> samf.strategies
  A |    # | Strategy
 -- | ---- | -------------------------
  x |    0 | Reduced chi squared strategy
    |    1 | Histogram global strategy

The currently active strategy is marked by an ‘x’ in the first column.

If a new datapoint (i.e. pixel) is added manually, the “database” of the currently active strategy has to be refreshed using the refresh_database() call.

The current strategy “database” can be plotted using the plot() method.

Whilst SAMFire is running, each pixel is checked by a goodness_test, which is by default red_chisq_test, checking the reduced chi-squared to be in the bounds of [0, 2].

This tolerance can (and most likely should!) be changed appropriately for the data as follows:

>>> samf.metadata.goodness_test.tolerance = 0.3 # use a sensible value

The SAMFire managed multi-dimensional fit can be started using the start() method. All keyword arguments are passed to the underlying (i.e. usual) fit() call:

>>> samf.start(optimizer='lm', bounded=True)