The mid-infrared spectra of active galactic nuclei (AGN) are often contaminated by the unresolved emission from the galaxies that hosts them. This contamination can be very strong in low luminosity AGN and in intermediate or high redshift sources (where the whole galaxy is typically unresolved). The aim of spectral decomposition techniques is to separate the emission from the AGN and its host with the help of a multicomponent model of the integrated (AGN+host) spectrum.

DeblendIRS is an IDL routine that decomposes mid-infrared spectra using a linear combination of three spectral components, corresponding to emission from the AGN, the stars, and the interstellar medium of the host galaxy. The spectrum of each of the components is selected from a large library containing actual Spitzer/IRS spectra from sources with `pure-AGN', `pure-stellar', and `pure-interstellar' spectra.

All possible combinations of three templates (one of each type) are tested and the one that minimises the residuals is selected as the best fitting model. In addition, all the possible combinations of templates are included in the calculation of probability distribution functions for observables (such as the AGN luminosity or its spectral index) using the bayesian-like 'Max' method described in Noll et al., 2009, A&A, 507, 1793 (ADS abstract). A detailed description of the decomposition method is given in Section 2 of the paper presenting DeblendIRS (Hernán-Caballero et al., 2015, ApJ, 803, 109).

The large number of real Spitzer/IRS spectra that DeblendIRS uses as templates allows to find combinations of them that reproduce the spectra of composite sources with unprecedented accuracy for a template fitting method (see examples below). Typical residuals are 2-3% of the flux density.

Examples of spectral decompositions with DeblendIRS for composite sources with different spectral shapes. The best fitting model (black) usually agrees with the source spectrum (yellow) within its error bars (gray shaded area). The red dotted, blue dashed, and green dot-dashed lines represent the interstellar (PAH), AGN, and stellar components, respectively. The thin black line at the bottom of each plots represent the residuals of the fit. Adapted from Hatziminaoglou et al. 2015, ApJ, 803, 110.

The ultimate test for a spectral decomposition method like DeblendIRS is comparison with high spatial resolution nuclear spectra. Because ground-based observations achieve an order of magnitude improvement in angular resolution compared to Spitzer (0.3-0.5" vs 3.6" with the IRS Short-Low module), most of the host emission can be resolved away in many nearby galaxies, obtaining nuclear spectra that are strongly AGN dominated, even if the AGN does not dominate the integrated emission of the galaxy. Therefore, we can take ground-based high angular resolution nuclear spectra as a proxy for the true AGN spectrum.
The figure below compares the AGN template from the best fitting decomposition model (blue dashed line) with the ground-based nuclear spectrum (pink line) and photometry (green diamonds) for several nearby Seyferts. A larger sample containing 28 sources is shown in Figure 11 of the DeblendIRS presentation paper.

Unlike most other spectral decomposition tools targeting mid-infrared spectra (such as PAHFIT, or DecompIR), DeblendIRS does not model extinction separately. That is, no assumption is made about the intrinsic shape of the AGN spectrum, the extinction law, or the geometry of the dust causing the obscuration. Instead, its phenomenological approach relies on a large set of real spectra from AGN (which are affected by different levels of obscuration) to find the ones that best reproduce (in combination with PAH and stellar templates) the spectrum of the composite source.

Finding a combination of templates that fit very well the spectrum of the source does not guarantee that these templates are an accurate description of the physical emission components (AGN, stellar, and interstellar) in the galaxy spectrum. Other combinations of templates could also produce similar results even if the components differ significantly. Because this degeneracy is difficult to quantify, template fitting routines usually disregard it and take only the best fitting solution. This is acceptable if we are interested in general properties of the decomposition, such as the fractional contribution of the AGN to the mid-infrared emission (rAGN), which is not very sensitive to the actual shape of the AGN spectrum. However, if we want to measure properties of the AGN spectrum with meaningful error bars, we need a proper treatment of this degeneracy.

Algorithms based on Bayesian inference have been developed to calculate the marginalised probability distribution function (PDF) of a physical parameter by integrating the likelihood over all the other variables that define the parameter space of the model (see MAGPHYS for an application to galaxy SEDs, and BayesCLUMPY for AGN torus models). However, these methods rely on grids of theoretical models with a uniform sampling of the parameter space to facilitate the calculation of PDFs. The empirical nature of our templates (they are real spectra from individual galaxies) implies we cannot expect an homogeneous and complete sampling of the parameter space defined by the physical properties of interest. Instead, we need a method that is robust against variations in the density of templates throughout the parameter space. One such method is the 'max' method used in the galaxy SED fitting tool CIGALE. This is the method adopted for DeblendIRS.

To compute the PDF of the parameter x with the max method, we divide the range of possible values of x into same-sized bins. For each bin i we find all the models that produce a value of x within the bin limits, and take the one with the lowest χ², χ²_i. Then the probability of the parameter x to take a value inside the bin i is:

where we implicitly assume a flat prior. The expectation value for x and its standard deviation are then given by:

DeblendIRS computes PDFs for 8 quantities:

• The fractional contribution of the AGN to the mid-infrared (5-15µm) luminosity (r_AGN)
• The fractional contribution of the interstellar emission to the mid-infrared (5-15µm) luminosity (r_PAH)
• The fractional contribution of the stars to the mid-infrared (5-15µm) luminosity (r_STR)
• The spectral index of the AGN spectrum (α_AGN)
• The intensity of the 10µm silicate emission/absorption feature in the AGN spectrum (silicate strength, S_sil)
• The fractional contribution of the AGN to the restframe 6µm luminosity (L6_AGN)
• The fractional contribution of the AGN to the restframe 12µm luminosity (L12_AGN)
• The fractional contribution of the host galaxy to the restframe 12µm luminosity (L12_SB)
  

The figure below shows these PDFs for the Spitzer/IRS spectrum of the galaxy Markarian 42. The red line represents the value obtained for the best fitting decomposition model, while the blue dashed line and the shaded area represent, respectively, the expectation value and the 1-σ confidence interval derived from the PDF.