Creating model instances

The sherpa.models and sherpa.astro.models namespaces provides a collection of one- and two-dimensional models as a convenience; the actual definition of each particular model depends on its type.

The following modules are assumed to have been imported for this section:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from sherpa import models

Creating a model instance

Models must be created before there parameter values can be set. In this case a one-dimensional gaussian using the Gauss1D class:

>>> g = models.Gauss1D()
>>> print(g)
gauss1d
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   gauss1d.fwhm thawed           10  1.17549e-38  3.40282e+38
   gauss1d.pos  thawed            0 -3.40282e+38  3.40282e+38
   gauss1d.ampl thawed            1 -3.40282e+38  3.40282e+38

A description of the model is provided by help(g).

The parameter values have a current value, a valid range (as given by the the minimum and maximum columns in the table above), and a units field. The units field is a string, describing the expected units for the parameter; there is currently no support for using astropy.units to set a parameter value. The “Type” column refers to whether the parameter is fixed, (frozen) or can be varied during a fit (thawed), as described below, in the Freezing and Thawing parameters section.

Models can be given a name, to help distinguish multiple versions of the same model type. The default value is the lower-case version of the class name.

>>> g.name
'gauss1d'
>>> h = models.Gauss1D('other')
>>> print(h)
other
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   other.fwhm   thawed           10  1.17549e-38  3.40282e+38
   other.pos    thawed            0 -3.40282e+38  3.40282e+38
   other.ampl   thawed            1 -3.40282e+38  3.40282e+38
>>> h.name
'other'

The model classes are expected to derive from the ArithmeticModel class.

Combining models

Models can be combined and shared by using the standard Python numerical operators. For instance, a one-dimensional gaussian plus a flat background - using the Const1D class - would be represented by the following model:

>>> src1 = models.Gauss1D('src1')
>>> back = models.Const1D('back')
>>> mdl1 = src1 + back
>>> print(mdl1)
(src1 + back)
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   src1.fwhm    thawed           10  1.17549e-38  3.40282e+38
   src1.pos     thawed            0 -3.40282e+38  3.40282e+38
   src1.ampl    thawed            1 -3.40282e+38  3.40282e+38
   back.c0      thawed            1 -3.40282e+38  3.40282e+38

Now consider fitting a second dataset where it is known that the background is two times higher than the first:

>>> src2 = models.Gauss1D('src2')
>>> mdl2 = src2 + 2 * back
>>> print(mdl2)
(src2 + (2 * back))
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   src2.fwhm    thawed           10  1.17549e-38  3.40282e+38
   src2.pos     thawed            0 -3.40282e+38  3.40282e+38
   src2.ampl    thawed            1 -3.40282e+38  3.40282e+38
   back.c0      thawed            1 -3.40282e+38  3.40282e+38

The two models can then be fit separately or simultaneously. In this example the two source models (the Gaussian component) were completely separate, but they could have been identical - in which case mdl2 = src1 + 2 * back would have been used instead - or parameter linking could be used to constrain the models. An example of the use of linking would be to force the two FWHM (full-width half-maximum) parameters to be the same but to let the position and amplitude values vary independently.

More information is available in the combining models documentation.

Changing a parameter

The parameters of a model - those numeric variables that control the shape of the model, and that can be varied during a fit - can be accesed as attributes, both to read or change the current settings. The val attribute contains the current value:

>>> print(h.fwhm)
val         = 10.0
min         = 1.17549435082e-38
max         = 3.40282346639e+38
units       =
frozen      = False
link        = None
default_val = 10.0
default_min = 1.17549435082e-38
default_max = 3.40282346639e+38
>>> h.fwhm.val
10.0
>>> h.fwhm.min
1.1754943508222875e-38
>>> h.fwhm.val = 15
>>> print(h.fwhm)
val         = 15.0
min         = 1.17549435082e-38
max         = 3.40282346639e+38
units       =
frozen      = False
link        = None
default_val = 15.0
default_min = 1.17549435082e-38
default_max = 3.40282346639e+38

Assigning a value to a parameter directly (i.e. without using the val attribute) also works:

>>> h.fwhm = 12
>>> print(h.fwhm)
val         = 12.0
min         = 1.17549435082e-38
max         = 3.40282346639e+38
units       =
frozen      = False
link        = None
default_val = 12.0
default_min = 1.17549435082e-38
default_max = 3.40282346639e+38

The soft and hard limits of a parameter

Each parameter has two sets of limits, which are referred to as “soft” and “hard”. The soft limits are shown when the model is displayed, and refer to the min and max attributes for the parameter, whereas the hard limits are given by the hard_min and hard_max (which are not displayed, and can not be changed).

>>> print(h)
other
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   other.fwhm   thawed           12  1.17549e-38  3.40282e+38
   other.pos    thawed            0 -3.40282e+38  3.40282e+38
   other.ampl   thawed            1 -3.40282e+38  3.40282e+38
>>> print(h.fwhm)
val         = 12.0
min         = 1.17549435082e-38
max         = 3.40282346639e+38
units       =
frozen      = False
link        = None
default_val = 12.0
default_min = 1.17549435082e-38
default_max = 3.40282346639e+38

These limits act to bound the acceptable parameter range; this is often because certain values are physically impossible, such as having a negative value for the full-width-half-maxium value of a Gaussian, but can also be used to ensure that the fit is restricted to a meaningful part of the search space. The hard limits are set by the model class, and represent the full valid range of the parameter, whereas the soft limits can be changed by the user, although they often default to the same values as the hard limits.

Setting a parameter to a value outside its soft limits will raise a ParameterErr exception.

During a fit the paramater values are bound by the soft limits, and a screen message will be displayed if an attempt to move outside this range was made. During error analysis the parameter values are allowed outside the soft limits, as long as they remain inside the hard limits.

Guessing a parameter’s value from the data

Sherpa models have a guess() method which is used to seed the paramters (or parameter) with values and soft-limit ranges which match the data. The idea is to move the parameters to values appropriate for the data, which can avoid un-needed computation by the optimiser.

The existing guess routines are very basic - such as picking the index of the largest value in the data for the peak location - and do not always account for the full complexity of the model expression, so care should be taken when using this functionality.

The arguments depend on the model type, since both the independent and dependent axes may be used, but the to_guess() method of a data object will return the correct data (assuming the dimensionality and type match):

>>> mdl.guess(*data.to_guess())

Note that the soft limits can be changed, as in this example which ensures the position of the gaussian falls within the grid of points (since this is the common situation; if the source is meant to lie outside the data range then the limits will need to be increased manually):

>>> yg, xg = np.mgrid[4000:4050:10, 3000:3070:10]
>>> r2 = (xg - 3024.2)**2 + (yg - 4011.7)**2
>>> zg = 2400 * np.exp(-r2 / 1978.2)
>>> d2d = Data2D('example', xg.flatten(), yg.flatten(), zg.flatten(),
                 shape=zg.shape)
>>> mdl = Gauss2D('mdl')
>>> print(mdl)
mdl
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   mdl.fwhm     thawed           10  1.17549e-38  3.40282e+38
   mdl.xpos     thawed            0 -3.40282e+38  3.40282e+38
   mdl.ypos     thawed            0 -3.40282e+38  3.40282e+38
   mdl.ellip    frozen            0            0        0.999
   mdl.theta    frozen            0     -6.28319      6.28319    radians
   mdl.ampl     thawed            1 -3.40282e+38  3.40282e+38
>>> mdl.guess(*d2d.to_guess())
>>> print(mdl)
mdl
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   mdl.fwhm     thawed           10  1.17549e-38  3.40282e+38
   mdl.xpos     thawed         3020         3000         3060
   mdl.ypos     thawed         4010         4000         4040
   mdl.ellip    frozen            0            0        0.999
   mdl.theta    frozen            0     -6.28319      6.28319    radians
   mdl.ampl     thawed      2375.22      2.37522  2.37522e+06

Freezing and Thawing parameters

Not all model parameters should be varied during a fit: perhaps the data quality is not sufficient to constrain all the parameters, it is already known, the parameter is highly correlated with another, or perhaps the parameter value controls a behavior of the model that should not vary during a fit (such as the interpolation scheme to use). The frozen attribute controls whether a fit should vary that parameter or not; it can be changed directly, as shown below:

>>> h.fwhm.frozen
False
>>> h.fwhm.frozen = True

or via the freeze() and thaw() methods for the parameter.

>>> h.fwhm.thaw()
>>> h.fwhm.frozen
False

There are times when a model parameter should never be varied during a fit. In this case the alwaysfrozen attribute will be set to True (this particular parameter is read-only).

Linking parameters

There are times when it is useful for one parameter to be related to another: this can be equality, such as saying that the width of two model components are the same, or a functional form, such as saying that the position of one component is a certain distance away from another component. This concept is refererred to as linking parameter values. The second case incudes the first - where the functional relationship is equality - but it is treated separately here as it is a common operation. Lnking parameters also reduces the number of free parameters in a fit.

The following examples use the same two model components:

>>> g1 = models.Gauss1D('g1')
>>> g2 = models.Gauss1D('g2')

Linking parameter values requires referring to the parameter, rather than via the val attribute. The link attribute is set to the link value (and is None for parameters that are not linked).

Equality

After the following, the two gaussian components have the same width:

>>> g2.fwhm.val
10.0
>>> g2.fwhm = g1.fwhm
>>> g1.fwhm = 1024
>>> g2.fwhm.val
1024.0
>>> g1.fwhm.link is None
True
>>> g2.fwhm.link
<Parameter 'fwhm' of model 'g1'>

When displaying the model, the value and link expression are included:

>>> print(g2)
g2
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   g2.fwhm      linked         1024            expr: g1.fwhm
   g2.pos       thawed            0 -3.40282e+38  3.40282e+38
   g2.ampl      thawed            1 -3.40282e+38  3.40282e+38

Functional relationship

The link can accept anything that evaluates to a value, such as adding a constant.

>>> g2.pos = g1.pos + 8234
>>> g1.pos = 1200
>>> g2.pos.val
9434.0

The CompositeParameter class controls how parameters are combined. In this case the result is a BinaryOpParameter object.

Including another parameter

It is possible to include other parameters in a link expression, which can lead to further constraints on the fit. For instance, rather than using a fixed separation, a range can be used. One way to do this is to use a Const1D model, restricting the value its one parameter can vary.

>>> sep = models.Const1D('sep')
>>> print(sep)
sep
   Param        Type          Value          Min          Max      Units
   -----        ----          -----          ---          ---      -----
   sep.c0       thawed            1 -3.40282e+38  3.40282e+38
>>> g2.fwhm = g1.fwhm + sep.c0
>>> sep.c0 = 1200
>>> sep.c0.min = 800
>>> sep.c0.max = 1600

In this example, the separation of the two components is restricted to lie in the range 800 to 1600.

In order for the optimiser to recognize that it needs to vary the new parameter (sep.c0), the component must be included in the model expression. As it does not contribute to the model output directly, it should be multiplied by zero. So, for this example the model to be fit would be given by an expression like:

>>> mdl = g1 + g2 + 0 * sep

Resetting parameter values

The reset() method of a parameter will change the parameter settings (which includes the status of the thawed flag and allowed ranges, as well as the value) to the values they had the last time the parameter was explicitly set. That is, it does not restore the initial values used when the model was created, but the last values the user set.

The model class has its own reset() method which calls reset on the thawed parameters. This can be used to change the starting point of a fit to see how robust the optimiser is by:

  • explicitly setting parameter values (or using the default values)

  • fit the data

  • call reset

  • change one or more parameters

  • refit

Inspecting models and parameters

Models, whether a single component or composite, contain a pars attribute which is a tuple of all the parameters for that model. This can be used to programatically query or change the parameter values. There are several attributes that return arrays of values for the thawed parameters of the model expression: the most useful is thawedpars, which gives the current values.

Composite models can be queried to find the individual components using the parts attribute, which contains a tuple of the components (these components can themselves be composite objects).