Blind Signal Separation Methods for the Identification of Interstellar Carbonaceous Nanoparticles
aa r X i v : . [ a s t r o - ph ] S e p Blind Signal Separation Methods for theIdentification of Interstellar CarbonaceousNanoparticles ⋆ O. Bern´e , , Y. Deville , and C. Joblin Centre d’Etude Spatiale des Rayonnements, CNRS et Universit´e Paul SabatierToulouse 3, Observatoire Midi-Pyr´en´ees, 9 Av. du Colonel Roche, 31028 Toulousecedex 04, France, [email protected] , [email protected] Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS et Universit´e PaulSabatier Toulouse 3, Observatoire Midi-Pyr´en´ees, 14 Av. Edouard Belin, 31400Toulouse, France, [email protected]
Abstract.
The use of Blind Signal Separation methods (ICA and otherapproaches) for the analysis of astrophysical data remains quite unex-plored. In this paper, we present a new approach for analyzing the in-frared emission spectra of interstellar dust, obtained with NASA’s SpitzerSpace Telescope, using
FastICA and Non-negative Matrix Factorization(NMF). Using these two methods, we were able to unveil the source spectra of three different types of carbonaceous nanoparticles present ininterstellar space. These spectra can then constitute a basis for the inter-pretation of the mid-infrared emission spectra of interstellar dust in theMilky Way and nearby galaxies. We also show how to use these extractedspectra to derive the spatial distribution of these nanoparticles.
The Spitzer Space Telescope (
Spitzer ) comprises one of today’s best instrumentsto probe the mid-infrared (mid-IR) emission of interstellar dust in the Milky Wayand nearby galaxies. This emission is mainly carried by very small (nanometric)interstellar dust particles. One of the goals of infrared astronomy is to identifythe physical/chemical nature of these species, as they play a fundamental rolein the evolution of galaxies. Unfortunately, the observed spectra are mixturesof the emission from various dust populations. The strategy presented in thispaper is to apply Blind Signal Separation (BSS) methods i.e.
FastICA and NMFto a set of
Spitzer mid-IR (5-30 µ m) spectra obtained with the InfraRed Spec-trograph (IRS), in order to extract the genuine spectrum of each population ofnanoparticles. We first present these observations in Sect. 2, then we apply the ⋆ This work is based on observations made with the Spitzer Space Telescope, which isoperated by the Jet Propulsion Laboratory, California Institute of Technology undera contract with NASA.I two BSS methods to these observations and finally give an example of how theextracted spectra can be used to trace the evolution of dust, in the Milky Wayand external galaxies.
We have observed with
Spitzer nearby photo-dissociation regions (PDRs), whichconsist of a star illuminating the border of dense clouds of gas and dust. Thephysical conditions (UV field intensity and hardness, cloud density) stronglyvary from a PDR to another as well as inside each PDR depending on theconsidered position. These variations are extremely useful to probe the natureof dust particles which are altered by the local physical conditions [1]. Therefore,we have observed 11 PDRs as part of the SPECPDR program using the IRS in”spectral mapping” mode. This mode enabled us to construct one dataset foreach PDR. This dataset is a spectral cube, with two spatial dimensions and onespectral dimension (see Fig. 1). Each spectral cube is thus a 3-dimensional matrix C ( p x , p y , λ ), which defines the variations of the recorded data with respect tothe wavelength λ , for each considered position with coordinates ( p x , p y ) in thecube. The dimensions of these cubes are generally about 30 ×
30 positions and250 points in wavelength ranging between 5 and 30 µ m. Fig. 1.
Left : Infrared (8 µ m) view of the NGC 7023 North PDR. The star is illuminatingthe cloud situated in the upper part of the image. Right : Mid-IR spectrum for a givenposition in the spectral cube of NGC 7023.
BSS is commonly used to restore a set of unknown ”source” signals from a setof observed signals which are mixtures of these source signals, with unknownmixture parameters [2]. BSS is most often achieved using ICA methods such as
FastICA [3]. An alternative class of methods for achieving BSS is NMF, which II was introduced in [4] and then extended by a few authors. In the astrophysicalcommunity, ICA has been successfully used for spectra discrimination in infraredspectro-imagery of Mars ices [5], elimination of artifacts in astronomical images[6] or extraction of cosmic microwave background signal in Planck simulateddata [7]. To our knowledge, NMF has not yet been applied to astrophysicalproblems. However, it has been used to separate spectra in other applicationfields, e.g. for magnetic resonance chemical shift imaging of the human brain [8]or for analyzing wheat grain spectra [9].The simplest version of the BSS problem concerns so-called ”linear instanta-neous” mixtures. It is modeled as follows: X = AS (1)where X is an m × n matrix containing n samples of m observed signals, A is an m × r mixing matrix and S is an r × n matrix containing n samples of r sourcesignals. The observed signal samples are considered to be linear combinations ofthe source signal samples (with the same sample index). It is assumed that r ≤ m in most investigations, including this paper. The objective of BSS algorithms isthen to recover the source matrix S and/or the mixing matrix A from X , up toBSS indeterminacies.The correspondence between the generic BSS data model (1) and the 3-dimensional spectral cube C ( p x , p y , λ ) to be analyzed in the present paper maybe defined as follows. In this paper, the sample index is associated to the wave-length λ , and each observed signal consists of the overall spectrum recorded fora cube pixel ( p x , p y ). Each one of these signals defines a row of the matrix X in Eq. (1). Moreover, each observed spectrum is a linear combination of ”sourcespectra” (see Sect. 3.1), which are respectively associated to each of the (un-known) types of nanoparticles that contribute to the recorded spectral cube.Therefore, the recorded spectra may here be expressed according to (1), withunknown combination coefficients in A , unknown source spectra in S and anunknown number r of source spectra. Spitzer -IRSCubes
In order to apply the NMF or
FastICA to the IRS data cubes, it is necessary tomake sure that the ”linear instantaneous” mixture condition is fulfilled. Here weconsider that each observed spectrum is a linear combination of ”source spectra”,which are due to the emission of different populations of dust nanoparticles. Themain effect that can disturb the linearity of the model is radiative transfer asshown by [10], because of the non-linearity of the equations. In our case however,this effect is completely negligible because the emission spectra we observe comefrom the surface of clouds and are therefore not altered by radiative transfer.
In this section, we detail which particular BSS methods we have applied to theobserved data. V NMF
We used NMF as presented in [11]. The matrix of observed spectra X isapproximated using W H, (2)where W and H are non-negative matrices, with the same dimensions as in (1).This approximation is optimized by adapting the matrices W and H using thealgorithm of [11] in order to minimize the divergence between X and W H . Weimplemented the algorithm with Matlab. Convergence is reached after about1000 iterations (which takes less than one minute with a 3.2 GHz processor).The value of r (number of ”source” spectra) is not imposed by the NMF method.Our strategy for setting it so as to extract the sources was the following: • Apply the algorithm to a given dataset, with the minimum number ofassumed sources, i.e. ˆ r = 2, providing 2 sources. → If the found solutions are physically coherent and linearly independent,we consider that at least ˆ r = 2 sources can be extracted. → Else, we consider that the algorithm is not suited for analysis (this neveroccurred in our tests). • Try the algorithm on the same dataset but with ˆ r = 3 sources. → If the found solutions are physically coherent and linearly independent,we consider that at least ˆ r = 3 sources can be extracted. → Else, we consider that only two sources can be extracted, extraction wasover with ˆ r = 2 and thus r = 2. • Same as previous step but with ˆ r = 4 sources. → If the found solutions are physically coherent and linearly independent,we consider that at least ˆ r = 4 sources can be extracted. → Else we consider that only three sources can be extracted, extraction wasover with ˆ r = 3 and thus r = 3. . . . Physically incoherent spectra exhibit sparse peaks (spikes) which cannot bePDR gas lines. We found r = 3 for NGC 7023-NW and r = 2 for the otherPDRs, implying that we could respectively extract 3 and 2 spectra from thesedata cubes. FastICA
We used
FastICA in the deflation version [3] in which each sourceis extracted one after the other and subtracted from the observations until allsources are extracted. The advantage of this
FastICA method is that it is notnecessary to fix, before running the algorithm, the number r of sources that wewant to extract, as it is for NMF. The extraction of the sources takes less thanone minute using FastICA coded with Matlab, and with a 3.2 GHz processor.
Using the BSS methods presented in this paper, we were able to extract up tothree source spectra from the
Spitzer observations. The number r of sourcesfound in a given PDR is always the same with NMF and FastICA . The three ex-tracted spectra in NGC 7023 North are presented in Fig. 2. Two of them exhibitthe series of aromatic bands which have previously been attributed to PolycyclicAromatic Hydrocarbons (PAHs, [12] and [13]). These two spectra show differentband intensity ratios. One is the spectrum of neutral PAHs (PAH ) while theother is due to ionized PAHs (PAH + ). The last spectrum exhibits a continuumand aromatic bands, which can be attributed to very small carbonaceous grains(VSGs), possibly PAH clusters [14]. Fig. 2.
The three BSS-extracted spectra from our study on PDRs.
FastICA vs NMF for our Application
As mentioned in Sect. 3.3, we were able to extract the source spectra fromour data using both
FastICA and NMF. However, the extracted spectra are notexactly the same for both methods. We conducted several tests in order to be ableto evaluate which one of the two methods is more appropriate for our application.We created a set of 2/3 artificial carbonaceous nanoparticle spectra, to whichwe added a variable level of white, spatially homogeneous noise. We mixed thesespectra with a random matrix to create a set of 100 artificial observed spectra. Wethen applied the two BSS methods considered in this paper. With a noise level atzero, both methods recover the original signals with high efficiency (correlationcoefficients between original and extracted signals above 0.995). When addingnoise, this efficiency decreases but remains acceptable down to a noise levelcorresponding to a SNR of 3dB (which is much lower that the average SNRof the
Spitzer spectra ). We note however that the efficiency of FastICA dropsslightly faster than the one of NMF under the effect of an increasing noise, anddrops dramatically below a SNR of 3dB, while NMF can still partly recoverthe original signals. Finally, with both methods we observe that the power ofthe residuals (i.e. observed signal minus signal reconstructed from the estimatedsources and mixing coefficients) has the same level as Spitzer noise. I We have shown in Sect. 3.3 that there are two main populations: one with acontinuum (VSGs) and one with bands only (PAHs). Using
FastICA , we some-times find a residual continuum in the BSS-extracted PAH spectrum, which weinterpret as an incomplete separation. It is possible that the criterion of NMF ismore appropriate in our case because less restrictive. Indeed, NMF only requiresnon-negativity of the sources and mixing coefficients, which is in essence the casefor emission spectra, while
FastICA is based on the statistical independence andnon-gaussianity of the sources, which is more difficult to prove. As a conclusion,we would like to stress the fact that both methods are very efficient for the firsttask presented in this paper. We however note that NMF seems slightly betterfor this particular application.
The next step of our analysis consists in using our extracted source spectra(Fig. 2) in order to determine the spatial distribution of the three populations ingalactic clouds or in external galaxies. The
Spitzer observations on-line archivecontains hundreds of mid-IR spectral cubes of such regions which can be inter-preted in this way. Our strategy consists in calculating the correlation parameter c p = E [ Obs ( p x , p y , λ ) y p ( λ )] between an observed spectrum Obs ( p x , p y , λ ) at aposition ( p x , p y ) in a spectral cube and one of our extracted source spectra y p ( λ ),where E [ . ] stands for expectation. With the considered (i.e. linear instantaneous)mixture model, each observed spectrum reads Obs ( p x , p y , λ ) = X n w ( p x , p y ) n S n ( λ ) (3)where S n ( λ ) is the n th source spectrum and w ( p x , p y ) n are the mixing coefficientsassociated to that source. Moreover, BSS methods extract the sources up toarbirary scale factors, i.e. they provide y p ( λ ) = η p S p ( λ ), where η p is an unknownscale factor and S p ( λ ) is the p th source. By centering the observations and thusthe extracted spectra, and assuming that the sources are not correlated, theabove-defined correlation parameter becomes c p = η p w ( p x , p y ) p E [ S p ( λ ) ] . (4)This coefficient c p is calculated for all the positions ( p x , p y ), therefore yieldinga 2D correlation map. Eq (4) shows that this map is proportional to w ( p x , p y ) p and thus defines the spatial distribution of the considered extracted source y p ( λ ) = η p S p ( λ ). We applied this approach to the spectral cube of NGC 7023North (Fig. 1) and obtained the correlation maps presented in Fig. 3. We findthat the three nanoparticle populations emit in very different regions. It appearsfrom the maps of Fig 3 that there is an evolution from a population of VSGs toPAH and then P AH + while approaching the star. This reveals the processingof the nanoparticles by the UV stellar radiation. The same strategy was tested II Fig. 3.
Correlation maps of the three populations of nanoparticles in NGC 7023 North:VSGs in red, PAH in green and PAH + in blue. The contours in black show the emissionat 8 µ m from Fig. 1. The slight correlation of VSGs with observations seen near thestar is an artifact. using the cubes of external galaxies from the SINGS program which provides adatabase of mid-IR spectral cubes for tens of nearby galaxies. Fig. 4 presents amap of the ratio of the two correlation parameters, resp. of PAH and PAH + ,obtained for the Evil Eye galaxy. This method provides a unique way to spa-tially trace the ionization fraction of PAHs which, combined with other tracers,is fundamental to understand the evolution of galaxies. Fig. 4.
Left : Infrared (8 µ m) view of the NGC 4826 (Evil Eye) Galaxy. The rectangleindicates the region observed in spectral mapping with IRS. Right : Map of the ratio of
P AH over P AH + in NGC 4826 achieved using the BSS-extracted spectra (Fig.2).III Using two BSS methods, we were able to identify the genuine mid-IR spectraof three propulations of carbonaceous nanoparticles in the interstellar medium.We have shown that both
FastICA and NMF are efficient for this task, althoughNMF is found to be sligthly more appropriate. The extracted spectra enable usto study the evolution of carbonaceous nanoparticles in the interstellar mediumwith unprecedented precision, including in external galaxies. These results stressthe fact that BSS methods have much to reveal in the field of observationalastrophysics. We are currently analyzing more spectral cubes observations fromthe
Spitzer database using the strategy presented in this paper.
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