************************ Creating model instances ************************ The :mod:`sherpa.models` and :mod:`sherpa.astro.models` namespaces provides a collection of one- and two-dimensional models. There are also more specialised models, such as those in :mod:`sherpa.astro.optical`, :mod:`sherpa.astro.xspec`, :mod:`sherpa.instrument`, and :mod:`sherpa.astro.instrument`. The following modules are assumed to have been imported for this section:: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from sherpa.models import basic Creating a model instance ========================= Models must be created before there parameter values can be set. In this case a one-dimensional gaussian using the :py:class:`~sherpa.models.basic.Gauss1D` class:: >>> g = basic.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 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 :ref:`params-freeze` 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 = basic.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 :py:class:`~sherpa.models.model.ArithmeticModel` class, although more-complicated cases, such as :doc:`convolution models <../evaluation/convolution>`, may extend other classes. .. _model-combine: Combining models ================ Models can be combined to serve as building blocks for more complex models; the same building block can be used multiple times. The easiest way to combine models is by using the standard Python numerical operators. For instance, a one-dimensional gaussian plus a flat background - using the :py:class:`~sherpa.models.basic.Const1D` class - would be represented by the following model:: >>> src1 = basic.Gauss1D('src1') >>> back = basic.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 = basic.Gauss1D('src2') >>> mdl2 = src2 + 2 * back >>> print(mdl2) src2 + 2.0 * 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 :ref:`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 including more complex, but more powerful ways to combine models is available in the :doc:`combining models <../evaluation/combine>` and :doc:`convolution <../evaluation/convolution>` 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 accessed as attributes, both to read or change the current settings. The :py:attr:`~sherpa.models.parameter.Parameter.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 .. _params-limits: 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 :py:attr:`~sherpa.models.parameter.Parameter.min` and :py:attr:`~sherpa.models.parameter.Parameter.max` attributes for the parameter, whereas the hard limits are given by the :py:attr:`~sherpa.models.parameter.Parameter.hard_min` and :py:attr:`~sherpa.models.parameter.Parameter.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 :py:exc:`~sherpa.utils.err.ParameterErr` exception. During a fit the parameter 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. This may help with determining uncertainties for parameters that are close to the soft limits. .. _params-guess: Guessing a parameter's value from the data ========================================== Sherpa models have a :py:meth:`~sherpa.models.model.Model.guess` method which is used to seed the parameters (or parameter) with values and :ref:`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 :py:meth:`~sherpa.data.Data.to_guess` method of a data object will return the correct data (assuming the dimensionality and type match):: >>> mdl.guess(*data.to_guess()) # doctest: +SKIP 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):: >>> from sherpa.data import Data2D >>> from sherpa.models import basic >>> 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 = basic.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 60 0.06 60000 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 .. _params-freeze: 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 :py:attr:`~sherpa.models.parameter.Parameter.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 :py:meth:`~sherpa.models.parameter.Parameter.freeze` and :py:meth:`~sherpa.models.parameter.Parameter.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 :py:attr:`~sherpa.models.parameter.Parameter.alwaysfrozen` attribute will be set to ``True`` (this particular parameter is read-only). .. _params-link: 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 referred to as linking parameter values. The second case includes the first - where the functional relationship is equality - but it is treated separately here as it is a common operation. Linking parameters also reduces the number of free parameters in a fit. The following examples use the same two model components:: >>> g1 = basic.Gauss1D('g1') >>> g2 = basic.Gauss1D('g2') Linking parameter values requires referring to the parameter, rather than via the :py:attr:`~sherpa.models.parameter.Parameter.val` attribute. The :py:attr:`~sherpa.models.parameter.Parameter.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 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 :py:class:`~sherpa.models.parameter.CompositeParameter` class controls how parameters are combined. In this case the result is a :py:class:`~sherpa.models.parameter.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 example, we can fit using the sigma value instead of the FWHM of a gaussian:: >>> sigma = basic.Scale1D('sigma') >>> sigma.c0 = 10 >>> print(sigma) sigma Param Type Value Min Max Units ----- ---- ----- --- --- ----- sigma.c0 thawed 10 -3.40282e+38 3.40282e+38 >>> g1.fwhm = 2 * np.sqrt(2 * np.log(2)) * sigma.c0 which creates :: >>> print(g1) g1 Param Type Value Min Max Units ----- ---- ----- --- --- ----- g1.fwhm linked 23.5482 expr: 2.3548200450309493 * sigma.c0 g1.pos thawed 1200 -3.40282e+38 3.40282e+38 g1.ampl thawed 1 -3.40282e+38 3.40282e+38 and, because ``g2.fwhm`` is still linked to ``g1.fwhm`` :: >>> print(g2) g2 Param Type Value Min Max Units ----- ---- ----- --- --- ----- g2.fwhm linked 23.5482 expr: g1.fwhm g2.pos linked 9434 expr: g1.pos + 8234 g2.ampl thawed 1 -3.40282e+38 3.40282e+38 .. note:: Prior to Sherpa 4.16.1 you had to explicitly include any linked parameters into the model expression - e.g. by saying:: >>> mdl = g1 + g2 * 0 * sigma where the ``sigma`` component is multiplied by zero to ensure it does not directly add to the model. This step is **no longer** needed, so you can just fit the model directly. >>> mdl = g1 + g2 Complex functional relationships -------------------------------- Any `numpy universal function ("ufunc") `_ can be used in the linking expression, for example:: >>> import numpy as np >>> g2.ampl = np.cos(g1.ampl) This includes many commonly used mathematical and trigonometric functions such as log, exp, sin, cos, which allows building quite complex parameter linkage. Only the numpy versions work here, **not** the functions from the built-in ``math`` module, so use `numpy.exp` instead of `math.exp`. Many more complex functions are available in `scipy.special `_; any arbitrary Python function can be turned into a ufunc with `numpy.frompyfunc `_ and the interface is also available for `C extensions `_. However, if such complex expressions are required to link model parameters together, it might be better to write a :ref:`dedicated user model ` that describes the data with the appropriate parameters in the first place. Not every possible link function makes sense -------------------------------------------- With this flexibility, it is possible to define links that make no sense, for example taking the logical not of a parameter that represents a mass or turning values of parameters into arrays (Sherpa optimisers can only deal with scalar parameters.) In practice, such mistakes are easy to spot when displaying a model; because Sherpa is meant to be a general and flexible modelling application that works with (almost) arbitrary user-defined models, the code puts as few restrictions as possible on the functions used for linking parameters. .. _parameter_reset: Resetting parameter values ========================== .. todo:: Needs work, including discussing the :py:attr:`~sherpa.models.parameter.Parameter.default_val` attribute? The :py:meth:`~sherpa.models.parameter.Parameter.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 :py:meth:`~sherpa.models.model.Model.reset` method which calls reset on the thawed parameters. This can be used to :ref:`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 ================================ .. note:: Access to model parameters has been extended in 4.16.1 by adding the ``lpars`` attribute and the ``get_thawed_pars`` method. Models, whether a single component or composite, contain a :py:attr:`~sherpa.models.model.Model.pars` attribute which is a tuple of all the parameters for that model, and the :py:attr:`~sherpa.models.model.Model.lpars` attribute, which contains any linked parameters in the model which are not a direct member of the source expression. These two can be used to programmatically query or change the parameter values. The :py:attr:`~sherpa.models.model.Model.get_thawed_pars` routine provides access to all the thawed parameters of a model expression, including any linked parameters. There are a number of attributes that provide access to this data, such as: :py:attr:`~sherpa.models.model.Model.thawedpars`, which gives the current value; and the :py:attr:`~sherpa.models.model.Model.thawedparmins` and :py:attr:`~sherpa.models.model.Model.thawedparmaxes`, which give the soft limits of these parameters. :: >>> g1 = basic.Gauss1D('g1') >>> g2 = basic.Gauss1D('g2') >>> g2.fwhm = g1.fwhm >>> sep = basic.Scale1D('sep') >>> g2.pos = 10 + sep.c0 >>> sep.c0.min = 0 >>> mdl = g1 + g2 >>> g1.pars (, , ) >>> g1.lpars () >>> g2.pars (, , ) >>> g2.lpars (, ) >>> for idx, par in enumerate(mdl.pars, 1): ... print(idx, par.fullname) 1 g1.fwhm 2 g1.pos 3 g1.ampl 4 g2.fwhm 5 g2.pos 6 g2.ampl >>> mdl.lpars (,) >>> for idx, par in enumerate(mdl.get_thawed_pars(), 1): ... print(idx, par.fullname) 1 g1.fwhm 2 g1.pos 3 g1.ampl 4 g2.ampl 5 sep.c0 We can think of a complex model as a tree of components, where the leaves are the individual components and nodes are composite models (e.g. a `sherpa.models.model.BinaryOpModel` that adds to model components together). There are several ways to access the components of a model, by model name (the list will have one entry if the model name is unique, or more if it appears more than once):: >>> mdl.get_components_by_name('g2') [] or by type:: >>> mdl.get_components_by_class(basic.Gauss1D) [, ] Alternatively, you can obtain a list of all components with:: >>> mdl.get_parts(include_composites=False) [, ] where the ``include_composites`` argument controls whether composite models are included in the list or only the leaves. As a convenience, for example for working interactively in a notebook, the same functionality is available as attribute access, which can shorten the notation. Here, if the result is a single model, it will be returned directly, while a list of models is returned if more than one component matches:: >>> mdl['g2'] >>> mdl[basic.Gauss1D] [, ] This allows a very compressed syntax to change the parameters of a single component, for example ``mdl['g2'].ampl = 5``. Models can also be iterated over to access the individual components - but note that this may include composite models. >>> for cpt in iter(mdl): ... print(cpt.name, type(cpt)) g1 + g2 g1 g2 >>> for cpt in iter(g1 + 2 * g2): ... print(cpt.name, type(cpt)) g1 + 2.0 * g2 g1 2.0 * g2 2.0 g2 When analysing composite models, note that the :py:attr:`~sherpa.models.model.Model.pars` attribute will contain repeated copies of model parameters if the model appears multiple times. The repeat values are actually links to the original version:: >>> scale = basic.Scale1D("scale") >>> b1 = basic.Box1D("box1") >>> b2 = basic.Box1D("box2") >>> mdl = scale * b1 + scale * b2 >>> scale.c0 = 5 >>> b1.xhi = 10 >>> b2.xlow = -1 >>> for idx, par in enumerate(mdl.pars, 1): ... print(f"{idx} {par.modelname:5s} {par.name:6s} {par.val} {par.link is not None}") ... 1 scale c0 5.0 False 2 box1 xlow 0.0 False 3 box1 xhi 10.0 False 4 box1 ampl 1.0 False 5 scale c0 5.0 True 6 box2 xlow -1.0 False 7 box2 xhi 1.0 False 8 box2 ampl 1.0 False >>> mdl.pars[4] == mdl.pars[0] False