Adding models¶

From the list of templates that nptfit.NPTF knows about (see Loading data and exposure), we can define an arbitrary number of Poissonian (smooth/diffuse) and non-Poissonian (point source) models.

Poissonian models¶

Poissonian models only have one parameter associated with them: the template normalisation. Poissonian models corresponding to an added template can be loaded as:

nptf.add_poiss_model(template_name='iso',
                     model_tag='$A_{iso}$',
                     prior_range=[-2,2],
                     log_prior=False,
                     fixed=False,
                     fixed_norm=1.0)

Arguments for Poissonian model:

Argument	Default	Purpose
`template_name`	-	Key corresponding to previously loaded template
`model_tag`	-	`LaTeX`-ready string, for plots
`prior_range`	`[min,max]`	Prior range to scan over
`log_prior`	`False`	Whether to scan template in log-space
`fixed`	`False`	Whether the model is fixed
`fixed_norm`	1.0	Template normalization if the model is fixed

Note

When log_prior=True, the associated values of prior_range and fixed_norm must also be the logs of the values being used.

Warning

When log_prior=True, the log used for the parameters is base 10.

Non-Poissonian models¶

The flux distribution of non-Poissonian (point source) models is modeled as a multiply broken power law with a specified number of breaks \(l\), the best-fit parameters of which can then be inferred. This source count distribution, which gives the differential number of sources per unit of flux, takes the form

\[\begin{split}\frac{dN}{dF} = A \left\{ \begin{array}{lc} \left( \frac{F}{F_{b,1}} \right)^{-n_1}, & F \geq F_{b,1} \\ \left(\frac{F}{F_{b,1}}\right)^{-n_2}, & F_{b,1} > F \geq F_{b,2} \\ \left( \frac{F_{b,2}}{F_{b,1}} \right)^{-n_2} \left(\frac{F}{F_{b,2}}\right)^{-n_3}, & F_{b,2} > F \geq F_{b,3} \\ \left( \frac{F_{b,2}}{F_{b,1}} \right)^{-n_2} \left( \frac{F_{b,3}}{F_{b,2}} \right)^{-n_3} \left(\frac{F}{F_{b,3}}\right)^{-n_4}, & F_{b,3} > F \geq F_{b,4} \\ \\ \ldots & \ldots \\ \\ \left[ \prod_{i=1}^{\ell-1} \left( \frac{F_{b,i+1}}{F_{b,i}} \right)^{-n_{i+1}} \right] \left( \frac{F}{F_{b,\ell}} \right)^{-n_{\ell+1}} & F_{b,\ell} > F \end{array} \right.\end{split}\]

It is sometimes convenient to specify the breaks in terms of counts instead of flux. However, if the exposure map is non-uniform over the ROI, then the notion of counts in pixel dependent. While the NPTFit code properly accounts for the pixel-dependent exposure correction, we also allow the user the specify the breaks \(F_{b,i}\) in terms of an effective number of counts \(S_{b,i} \equiv F_{b,i} \cdot \text{mean}_\text{ROI}(E_p)\), where \(\text{mean}_\text{ROI}(E_p)\) is the mean of the exposure map \(E_p\) over the ROI.

Non-Poissonian models corresponding to an added template can be loaded as:

nptf.add_non_poiss_model(template_name='iso',
                         model_tag=['$A_{ps}$','$n_1$','$n_2$','$S_b^{(1)}$'],
                         prior_range=[[-6,6],[2.05,30],[-2,1.95],[0.05,30.0]],
                         log_prior=[True,False,False,False],
                         dnds_model='specify_breaks',
                         fixed_params=None,units='counts')

Arguments for non-Poissonian model:

Argument	Default	Purpose
`template_name`	-	Key corresponding to loaded template
`model_tag`	-	LaTeX-ready string of nb-broken power-law parameters [A, n_1, … , n_{nb+1}, Sb_1, … , Sb_nb] with Sb_1/n_1 the highest break/slope
`prior_range`	`[]`	Prior range to scan over, given as a list of [min,max] each model parameter
`log_prior`	`False`	Whether to scan each parameter in log-space
`dnds_model`	`specify_breaks`	Whether to use absolute or relative breaks (see \(dN/dS\) model specifications below)
`fixed_params`	`None`	Which parameters to keep fixed (see Fixed parameter specifications below)
`units`	`counts`	Whether the breaks in dN/dF are specified in terms of \(F_{b}\) or \(S_{b}\), which is defnied above. (see units specifications below)

Note

The number of breaks in the non-Poissonian model is inferred from the length of the model_tag array.

Warning

Non-Poissonian (or PS) models must use a template loaded with units='PS', while non-Poissonian models should use units='counts' or units='flux'.

\(dN/dF\) model specifications¶

where \(n_1\) is the highest index and \(S_b^{(1)}\) the highest break.

The following options are allowed for dnds_model:

specify_breaks: all breaks are specified in absolute counts, \(\left[ A, n_1, \ldots, n_{\ell+1}, S_b^{(1)}, \ldots, S_b^{(\ell)} \right]\)
specify_relative_breaks: the highest break is specified in counts, with each subsequent lower break specified relative to the subsequent higher break. \(\left[ A, n_1, \ldots, n_{\ell+1}, S_b^{(1)}, \lambda^{(2)}, \ldots, \lambda^{(\ell - 1)}, \lambda^{(\ell)} \right]\) where \(\lambda^{(i)} = S_b^{(i)}/S_b^{(i-1)}\).

Fixed parameter specifications¶

Fixed parameters should be passed as an array with syntax

fixed_params = [[param_index_1,fixed_value_1],[param_index_2,fixed_value_2]]

where parameter indexing starts from 0.

Units specifications¶

The following options are allowed for units:

counts: The code assumes the user has specified the break in counts (\(S_b\)) and will infer the breaks in flux (\(F_b\)) by dividing the mean exposure: \(F_{b,i} \equiv S_{b,i} / \text{mean}_\text{ROI}(E_p)\).
flux: The code assumes the breaks are already specified in terms of flux

Tip

See Example5_Running_nonPoissonian_Scans and Example7_Galactic_Center_nonPoissonian for examples of using these options in an analysis.