4.12. SED Normalization (Scale Factor)

Template fitting methods usually fix the normalization of SED templates (or scale factor) to the value that minimizes the \(\chi^2\) (the best-likelihood value) in order to reduce the number of free parameters and to be faster (see the Metodology chapter for more details). By default, Phosphoros does the same. Nevertheless, Phosphoros gives the possibility to treat the scale factor as a free parameter of the model, in a fully Bayesian approach.

We remind that the best-likelihood value of the scale factor and its uncertainty (obtained through the observed flux errors propagation) can be analytically derived by the following formula:

(4.3)\[\alpha_{best} = \sum_i \frac{f_{obs}^if_m^i}{\sigma_i^2} \bigg/ \sum_i \frac{(f_m^i)^2}{\sigma_i^2}~~~~~~~~~~~\sigma_{\alpha} = 1/\sqrt{\sum_i\frac{(f_m^i)^2}{\sigma_i^2}}\,,\]

where \(f_{obs}^i\) and \(f_m^i\) are the observed and modeled flux in the \(i\) filter. Modeled fluxes are computed by Phosphoros after normalizing SED templates to the solar luminosity at 10pc distance in a filter chosen by users.

In order to optimize the calculations, Phosphoros uses a grid of values for the scale factor distributed around its best-likelihood value. The number of sampled points and the sampling range (defined in terms of the scale factor uncertainty) are chosen by users.

4.12.1. Scale Factor in the GUI

In order to sample the scale factor and to marginalize it in the redshift PDF computation, three steps are required:

  • Select the solar SED in the Configuration-->Auxiliary Data-->SEDs sub-panel by clicking on the Reference Solar SED: solar_spectrum button.

  • Select the reference filter to use for the SED template normalization in the Compute Redshift-->1. Luminosity Filter and Extrinsic Absorption by clicking on the Select Filter button (the default filter is on the left of the button).

  • Select the Sample and marginalize the model scaling option in the Compute Redshift-->5. Algorithm sub-panel. The number of points in the sample and the sampling range, defined as \(\alpha_{best}\pm n\times\sigma_{\alpha}\), can be modified by users (see Fig. 4.11).

Warning

Samping of the scale factor will rise the memory footprint: set the Cap Memory Usage in the Configuration-->General sub-panel to avoid crashing the GUI

../../_images/scale_factor.png

Fig. 4.11 SED normalization sampling with the GUI

4.12.2. Scale Factor in the CLI

The SED normalization sampling and marginalization is performed by the Phosphoros action compute_redshift (or CR) if the following option in the configuration file is YES (by default, NO):

scale-factor-marginalization-enabled=YES

Users can choose the sampling number points and range by:

scale-factor-marginalization-range-size=<n_sigma> (default=5)
scale-factor-marginalization-sample-number=<n_points> (default=101)

where the sample range is \(\alpha_{best}\pm{\rm <n\_sigma>}\times\sigma_{\alpha}\).

Finally, the filter for the SED normalization and the solar SED template can be specified through:

normalization-filter=DECAM/r
normalization-solar-sed=solar_spectrum

where the filter and the solar SED are expected to be found in the $PHOSPHOROS_ROOT/AuxiliaryData/Filters and $PHOSPHOROS_ROOT/AuxiliaryData/SEDs directories, respectively.

Warning

Samping of the scale factor will rise the memory footprint: it is a good practice to reduce the number of sources held in memory using the input-buffer-size option (the default is 5000).