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.
Note
I have no idea what deep truth I meant by the sentance above.
Note
To get the link above probably means using
sherpa.models.model, but users can get away with
just importing sherpa.models as shown below.
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.
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
Note
What about the default_val
attribute?
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).