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Bayesian Positive Source Separation for T-tauri Star Spectra
Colleen Kenney*, Alejandro Villagran*, Marina Vannucci, Patrick
Hartigan, and Christopher Johns-Krull1
Abstract
There is a particular interest in looking at the spectra of young stars. Whether we can
see the interaction between the star and its accretion disk, a disk of dust and gases from
which gaseous material from the inner edges of the disk may be falling onto the surface of the
star. Several deterministic methods have been proposed to identify independent sources.
We consider the model X = AS + E, where X is the observed data, A are the mixing
coefficients, S contains the source signals, and E contains the error term. We compare the
use of two popular Non-negative matrix factorization methods with the use of a Markov
Chain Monte Carlo scheme to estimate the sources S and the coefficients A. We compare
the performance of these methods to separate positive sources by using two statistics, a
performance index and a cross-talk index.
Keywords: T-tauri stars, Source separation, Bayesian inference.
1Colleen Kenney is PhD student in Statistics, Rice University. Alejandro Villagran is a Postdoctoral
Research Associate in Statistics, Rice University. Marina Vannucci is Professor of Statistics, Rice Univer-
sity. Patrick Hartigan is Professor in Physics and Astronomy, Rice University. Christopher Johns-Krull is
Associate Professor in Physics and Astronomy, Rice University. Corresponding author: [email protected]
(*) These authors contributed equally to this work.
1
1 INTRODUCTION
T-Tauri stars (TTS) are young, pre-main sequence stars classified by their placement on
the Hertzprung-Russel diagram, the emission lines of H, Ca and Mg, and the presence of
lithium in their spectra. The study of these young solar-type stars allows us to better
understand the formation of stars. Since TTS are characterized by their spectra, their
spectral classification is key to their identification (Gray and Corbally, 2009).
Two parts of a spectrum of a T-Tauri star can be identified through spectroscopy: the
photosphere and the boundary layer. The photosphere is the visible part of a star. The
absorption lines of a photo-spheric spectrum indicate the effective temperature of a star,
thereby classifying the star into one of the seven spectral types, O, B, A, F, G, K, and M
(Gray, 2005). TTS are young, low mass, G, K, and M type stars. The boundary layer is a
layer between the star and the accretion disk that has an increasing angular velocity, likely
due to the rotation of the star. The angular velocity is therefore affected by the rotation of
the star and the magnetic fields generated by the star (Rogava and Tsiklauri, 1993).
In addition to the photosphere and the boundary layer, current models of T-Tauri stars
indicate that there is a circumstellar accretion disk around T-Tauri stars. The disk is
made up of gas and dust, and it ejects material onto the surface of the star along magnetic
field lines. This causes ultra violet, optical continuum, and infrared excess. The infrared
continuum excess is evidence that the accretion disk exists (Edwards et al., 1994, Alencar,
2007). Since TTS are made up of different components, spectral decomposition may be
necessary to determine the spectral type. This is the case when the boundary layer is easily
discernible in the spectrum. Currently, a forward fitting method is used. The spectral
type is first estimated based on the ratios of photo-spheric absorption lines. A template
spectrum of the estimated spectral type is then taken and added to a boundary layer
spectrum computed from a model. The parameters of this final spectrum are then varied
until the spectrum matches the observations well (Valenti et al., 1993). One problem with
the current method is that it does not indicate whether the final spectra are complete or
not. The accretion disk’s spectrum is taken to be the excess spectrum after fitting the
model.
Spectral decomposition can be accomplished mathematically and statistically through
non-negative source separation as well. In this paper, we evaluate and compare the non-
negative matrix factorization (NNMF) algorithms of Paatero and Tapper (1994), Lee and
Seung (2001), and the Bayesian positive source separation (BPSS) algorithm of Moussaoui et
al. (2006) for spectral decomposition to recover the spectrum of the photosphere, boundary
layer, and accretion disk of T-Tauri stars. NNMF and BPSS have been used, for example,
in fluorescence spectroscopy (Gobinet et. al., 2004), microorganism Raman spectra (Huez
et al., 2002), chemical shift imaging of the brain (Sajda et. al., 2004), and chemical near-
2
infrared spectroscopy (Moussaoui et. al., 2006) respectively. These algorithms are based on
the model
X = AS + E, (1)
where the observed data matrix X ∈ Rm×N has m measured signals at N wavelengths, the
mixing coefficients matrix A ∈ Rm×n+ and the source signals S ∈ R
n×N+ have n unknown
source signals, and the matrix error term E ∈ Rm×N . Therefore, given the number of
sources and the observations, we want to estimate the mixing matrix and the source signals.
Although, the number of components is theoretically unknown, current models suggest
that the decomposed spectra of T-Tauri stars should break into three components. To
quantitatively determine the number of components in TTS spectra, we employ Principal
Component Analysis (PCA) to determine how much of the variability is explained by each
component. This method to determine the number of components in a large data set is
similar to the method used by Lee et al. (2008) where they determine the number of
components based on the mean squared error (MSE) by selecting n at the point which the
MSE becomes approximately constant. For our problem, PCA is a better choice because
the NNMF algorithm of Lee et al. (2008) allows negative coefficients. Using PCA, we
choose the number of components where the percentage of variability explained becomes
approximately constant, usually around the point which 99% of the variability is accounted
for. After determining the number of components, we can then solve the problem of non-
negative source separation for TTS.
The organization of the article is as follows: Section 2 explains the data used in the pa-
per. Section 3 presents the three methods applied for source separation. Section 4 compares
the performance of the methods to identify the sources of the T-Tauri star spectra. Section
5 discusses advantages, drawbacks, and alternative approaches to the methods presented in
this paper.
2 DATA
To evaluate and compare the non-negative source separation algorithms presented in this
paper, we use both simulated and real data. In Figure 1 (left panel) we observe a spectra
simulated from the same star, there are ten varying boundary layers combined with a
photosphere. Another simulation (right panel) is made up by mixing four spectra of stars
(A,M0,M6,O) and creating a galaxy of 25 stars. The stars for these data sets are from
Valdes et al. (2004) and can be found at http://www.noao.edu/cflib/.
In Figure 2, for the real data , one data set (panel (a)) is a sample of weak and classical
TTS that was observed at the Kitt Peak National Observatory using the RC Spectrometer
on the 4-meter Mayall telescope, which has a resolution of three angstroms (A). This data
3
4000 5000 6000 7000 8000 90000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angstroms
Flu
x
4000 5000 6000 7000 8000 90000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angstroms
Flu
x
Figure 1: Simulated data. Left: 10 measured signals coming from 2 original sources. Right:
Simulated galaxy, 30 stars coming from 3 different type of stars.
set spans wavelengths from 3600 to 7324.4 A, and the sample size m is 26 spectra. The
data for this set were interpolated onto a common x-axis, starting with 3600 A, since the
spectrometer is not very sensitive bluer than 3600 A, and ending at the minimum of the
maximum wavelength for all of the 26 spectra is observed so the endpoints of the spectra
were common among all spectra. The interpolation allowed us to use all 26 spectra in a
single analysis.
Data sets two (Figure 2, panel (b)) through four were taken from the data included in
Valenti et al. (1993). Data set two spans wavelengths from 3400 to 4950 A and consists
of 30 spectra, data set three spans wavelengths from 3370 to 4970 A and consists of 43
spectra, and data set four spans 3200 to 7050 A and consists of nine spectra. According to
Valenti et al. (1993), the spectra included in the sample of the 96 stars they used consisted
of weak, moderate, and extreme TTS systems. Note that we use a subset of the 96 spectra
presented in their catalog because one set of data spanned uncommon wavelengths than the
other data and was too small to use on its own.
4
4000 5000 6000 7000 80000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angstroms
Flu
x
(a)
4000 5000 6000 7000 8000 90000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angstroms
Flu
x
(b)
Figure 2: T-Tauri Star data. (a) 26 spectra observed form the Mayall telescope. (b) 30
spectra observed from Valenti et al. (1993).
3 METHODS
3.1 Non-Negative Matrix Factorization (NNMF)
NNMF can be applied to the statistical analysis of multivariate data, often the data to
be analyzed is non-negative, and the low rank data are further required to be comprised
of non-negative values in order to avoid contradicting physical realities. The approach of
finding reduced rank non-negative factors (A and S) to approximate a given non-negative
data matrix X thus become a natural choice. This is the so called non-negative matrix
factorization (NNMF) problem which can be stated in generic form as follows:
arg minA,S ||X−AS||2. (2)
The product AS is called a NNMF of X, although X is not necessarily equal to the product
AS. To find an approximate factorization X ≈ AS, Lee and Seung (2001) proposed an
algorithm based on iterative updates of A and S. At each iteration, a new value of A and S
is found by multiplying the current value by some factor that depends on the quality of the
approximation to X. Using an alternative useful divergence measure to find an approximate
factorization for X,
D(X||AS) =m∑
i=1
N∑
t=1
(
Xit · log
{
Xit
(AS)it
}
−Xit + (AS)it
)
. (3)
5
This divergence measure is not a distance since is not symmetric in X and AS, however it
is lower bounded by zero and vanishes if and only if X = AS. The multiplicative update
rules for (3) are provided by the next result,
The divergence D(X||AS) is non-increasing under the update rules
Aij ← Aij
∑Nt=1 SjtXit/(AS)it∑N
t=1 Sjt
Sjt ← Sjt
∑mi=1 AijXit/(AS)it∑m
i=1 Aij(4)
The divergence is invariant under these updates if and only if A and S are at a stationary
point of the divergence. The convergence proof rely upon defining an appropriate auxiliary
function (4), for a detailed proof see Lee and Seung (2001).
Due to their status as the first well known NNMF algorithms, the Lee and Seung mul-
tiplicative update rules have become the baseline against which the newer algorithms are
compared. It has been repeatedly shown that Lee and Seung algorithms are notoriously slow
to converge. They require many more iterations than alternatives such as the alternating
least squares algorithms (Paatero and Tapper, 1994).
3.2 NNMF-Alternate Least Squares (NNMF-ALS)
Non-negative matrix factorization by alternate least squares (NNMF-ALS) was introduced
by Paatero and Tapper (1994) under the name of positive matrix factorization (PMF).
In these algorithms, a least square step is followed by another least squares step in an
alternating fashion. ALS exploit the fact that, while the optimization problem of (??) is
not convex in both A and S, it is convex in either A or S. Thus, given one matrix, the other
matrix can be found with a simple least squares computation. The algorithm is described
as follows,
(1) Initialize A as a random matrix.
(2) Compute S = (AtA)−1AtX.
(3) Set all negative elements in S equal to 0.
(4) Compute A = (StS)−1StX.
(5) Set all negative elements in A equal to 0.
(6) Repeat steps (2) through (5) until reaching some pre-specified tolerance.
This simple technique allows some benefits, it adds sparsity, and it allows some additional
flexibility not available in the multiplicative update algorithms since once an element in A
6
or S becomes 0, it must remain zero. This locking of 0 elements prevents techniques such
as Lee and Seung’s (2001) to escape from a poor path.
3.3 Bayesian Positive Source Separation (BPSS)
Bayesian theory was used by Moussaoui et al. (2006) to blind source separation for an appli-
cation to the analysis of spectrometric data sets. There are two main reasons that make the
Bayesian approach very well suited for such an application. First, Bayesian inference offers
a theoretical framework to take into account non-negativity and in general any additional
prior knowledge on the mixing coefficients and the source signals. Second, Markov Chain
Monte Carlos (MCMC) methods enable to generate samples from the posterior distribution
of interest. The Bayesian approach uses the likelihood P (X|A,S) and the prior information
about the mixing coefficients and the sources P (A,S). Using Bayes’ theorem, the posterior
distribution of the parameters of interest is,
P (A,S) ∝ P (X|A,S)P (A)P (S) (5)
where we are assuming independence between A and S. From this posterior density (5), the
estimation of A and S can be achieved by using MCMC methods. However, we first need
to elaborate all the elements involved in such computation. Looking at each term of (1)
xit =
n∑
j=1
aijsjt + eit (6)
where we have i = 1, ...,m measured signals, j = 1, ..., n source signals, and t = 1, ..., N
wavelengths. We assume that each eit is independent and distributed Gaussian with zero
mean and variances equal to σ2i . The likelihood can be expressed as
P (X|A,S, θ1) =
N∏
t=1
m∏
i=1
( 1
2πσ2i
)1/2exp{
−1
2σ2i
(
xit −
n∑
j=1
aijsjt
)2}
(7)
where θ1 = {σ2i }
mi=1.
To ensure the non-negativity in both mixing coefficients and source signals, we impose
as prior distributions for A and S Gamma densities. Each source signal j-th is supposed
to have a Gamma distribution with parameters (αj , βj). These parameteres are constant
for each source but may differ from one source to another. For the mixing coefficients, each
column j of A is also assumed to have a Gamma distribution with parameters (γj , λj). The
j-th column corresponds to the evolution profile of the j-th source proportion in the mixture
and its associated parameters are considered constant for each profile. For simplicity in the
7
notation, we call θ2 = {αj , βj}nj=1, θ3 = {γj , λj}
nj=1, and the vector of prior parameters
θ = {θ1, θ2, θ2}. Therefore, the posterior distribution of interest can be expressed as,
P (S,A, θ|X) ∝ P (X|A,S, θ1)P (S|θ2)P (A|θ3) (8)
To simulate samples from (8), a combination of Gibbs (Gemand and Geman, 1984) and
Metropolis-Hastings (Hastings, 1970) steps are required, breaking the sampling scheme into
several steps we have that,
1. Sampling the source signals S.
P (S(r+1)|X,A(r), θ(r)) ∝ P (X|S,A(r), θ(r)) (9)
To Sample S source by source, we fixed j and t
P (s(r+1)jt |·) ∝ P ({xit}
mi=1|s
(r+1)1:j−1,t, s
(r)j+1:n,t, a
(r)1:m,1:n, {σ
(r)i }
mi=1)P (sjt|α
(r)j , β
(r)j ) (10)
P (s(r+1)jt |·) ∝ s
α(r)j
−1
jt exp{−β(r)j sjt −
τs
2(sjt − µs)
2} (11)
τs =m∑
i=1
(
a2(r)ij
σ2(r)i
)
(12)
µs = τ−1s
m∑
i=1
(
a(r)ij δ−j
s,it
σ2(r)i
)
(13)
δ−js,it = xit −
j−1∑
k=1
a(r)ik s
(r+1)kt −
n∑
k=j+1
a(r)ik s
(r)kt (14)
Sampling from (11) requires a Metropolis-Hastings step, we use a left-truncated normal
distribution to propose new values s∗jt for the sources, and to ensure these candidate values
are positive.
2. Sampling the mixing coefficients A.
P (A(r+1)|X,S(r+1), θ(r)) ∝ P (X|S(r+1),A, θ(r)) (15)
To Sample A coefficient by coefficient, we fixed i and j
P (a(r+1)ij |·) ∝ P ({xit}
Nt=1|a
(r+1)i,1:j−1, a
(r)i,j+1:n, s
(r+1)1:n,1:N , {σ
(r)i }
mi=1)P (aij |γ
(r)j , λ
(r)j ) (16)
P (a(r+1)ij |·) ∝ a
γ(r)j
−1
ij exp{−λ(r)j aij −
τa
2(aij − µa)
2} (17)
8
τa =σ
(r)i
∑Nt=1 s
2(r+1)jt
(18)
µa =
∑Nt=1 s
(r+1)jt δ−j
a,it∑N
t=1 s2(r+1)jt
(19)
δ−ja,it = xit −
j−1∑
k=1
a(r+1)ik s
(r+1)kt −
n∑
k=j+1
a(r)ik s
(r+1)kt (20)
Sampling from (17) requires a Metropolis-Hastings step, we use a left-truncated normal
distribution to propose new values a∗ij for the mixing coefficients, and to ensure these can-
didate values are positive.
3. Sampling θ1, i.e. the precision τi = 1/σ2i
P (τi|·) ∝ P ({xit}Nt=1|a
(r+1)i,1:n , s
(r+1)1:n,1:N )P (τi) (21)
P (τi|·) ∝ τN/2i exp
{
−τi
2
N∑
t=1
(xit −n∑
j=1
a(r+1)ij s
(r+1)jt )2
}
ταo−1i exp{−βoτi} (22)
Therefore, (21) can be sampled from a Gamma distribution with parameters (αo+N/2, βo+
1/2)∑N
t=1(xit −∑n
j=1 a(r+1)ij s
(r+1)jt )2.
4. Sampling the source hyperparameters θ2 = (αj , βj)
To sample the source hyperparameters αj , we assume as prior distribution αj ∼ exp(λo),
then
P (αj |·) ∝ P (s(r+1)j,1:N |αj , β
(r)j )P (αj) (23)
P (αj |·) ∝ exp{
αj [Nlog(β(r)j ) +
N∑
t=1
log(s(r+1)jt )− λo]−Nlog(Γ(αj))
}
(24)
Sampling from (24) can be done by using a Metropolis-Hastings step. We use a Gamma
distribution with parameters (ao, bo) to propose new candidate values for each αj.
Sampling βj is straightforward by using as conjugate prior distribution a Gamma distribu-
tion with parameters (co, do).
P (βj |·) ∝ P (s(r+1)j,1:N |α
(r+1)j , βj)P (βj) (25)
βj |· ∼ Γ(co + N(α(r+1)j − 1), do +
N∑
t=1
s(r+1)jt ) (26)
9
5. Sampling the mixing coefficients hyperparameters θ3 = (γj , λj)
To sample the mixing coefficients hyperparameters γj , we assume as prior distribution
γj ∼ exp(fo), then
P (γj |·) ∝ P (a(r+1)1:m,j |γj , λ
(r)j )P (γj) (27)
P (γj|·) ∝ exp{
γj [mlog(λ(r)j ) +
m∑
i=1
log(a(r+1)ij )− fo]−mlog(Γ(γj)) (28)
Sampling from (28) can be done by using a Metropolis-Hastings step. We use a Gamma
distribution with parameters (a1, b1) to propose new candidate values for each γj .
Sampling λj is straightforward by using as conjugate prior distribution a Gamma distribu-
tion with parameters (c1, d1).
P (λj |·) ∝ P (a(r+1)1:m,j |γ
(r+1)j , λj)P (λj) (29)
λj|· ∼ Γ(c1 + m(γ(r+1)j − 1), d1 +
m∑
i=1
a(r+1)ij ) (30)
4 RESULTS
Using Lee and Seung (2001) algorithm, we can find the NNMF for the 4 data sets described
in Section 2. In Figure 3 the technique identifies correctly the 2 original sources and it
can reconstruct the measured (simulated) signals accurately. In Figure 4, the algorithm
identifies three stars (A, M, and O) as main sources from the simulated galaxy. Figures 5
and 6 are the results from TTS telescope 1. Figures 7 and 8 are the results from TTS
telescope 2.
Using Paatero and Tapper (1994) ALS algorithm, we can find the NNMF for the 4 data
sets described in Section 2. In Figure 9 the technique identifies correctly the 2 original
sources and it can reconstruct the measured (simulated) signals accurately. In Figure 10,
the algorithm identifies three stars (A, M, and O) as main sources from the simulated galaxy.
Figures 11 and 12 are the results from TTS telescope 1. Figures 13 and 14 are the results
from TTS telescope 2.
Using BPSS, we can find the NNMF for the 4 data sets described in Section 2. In
Figure 15 the technique identifies correctly the 2 original sources and it can reconstruct the
measured (simulated) signals accurately. In Figure 16, the algorithm identifies three stars
(A, M, and O) as main sources from the simulated galaxy. Figures 17 and 18 are the results
from TTS telescope 1. Figures 19 and 20 are the results from TTS telescope 2.
10
4.1 Performance assessment
To compare all methods by looking at the Figures is quite difficult, therefore there is a
need to assess the separation quality of the different algorithms by using some performance
measures. The performance index PI (Cichoki and Amari, 2002) defined by
PI =1
n(n− 1)
[
n∑
i=1
n∑
k=1
|gik|2
maxj|gij |2+
n∑
i=1
n∑
k=1
|gki|2
maxj |gji|2− 2n
]
(31)
is used, where gij is the (i, j) element of the matrix G = (AtA)−1, maxjgij stands for the
maximum value among the elements in the i-th row vector of G and maxjgji represents
the maximum value among the elements in the i-th column vector of G. This measure
takes small values when a good separation is achieved. This index assesses the overall
separation performance and measures mainly the quality of the estimation of the mixing
matrix. However, it is very important to measure the quality of the reconstruction of each
source signal. In that respect, one can use the residual cross-talk index (Hosseini et al.
(2003)) definded as
CIj =1
∑Nt=1 s2
jt
N∑
t=1
(sjt − sjt)2 j = 1, ..., n (32)
In Table 1, the simulated signals are used to compare the performance of the three
methods presented in this pepaer. We can see that BPSS is the best method since its
cross-talk index and its performance index are the lowest. In Table 2, we have the simulate
galaxy data, in which the performance index of the BPSS algorithm is still the lowest,
however the cross-talk index among methods is not conclusive.
TABLE 1
Method NNMF-MUR NNMF-ALS BPSS
CT-S 0.0015 0.0010 1.4E-5
PI 0.7737 1.3993 0.2839
TABLE 2
Method NNMF-MUR NNMF-ALS BPSS
CT-S1 0.1042 0.1144 0.0090
CT-S2 0.1574 0.0678 0.0827
CT-S3 0.0289 0.0028 0.0039
PI 0.5490 0.9824 0.3572
11
Given that the cross-talk index uses the original sources sjt to compute the cross-talk
index, it will not be possible to do it with real data since the sources are unknown. However,
we can propose a modification to (32) in order to be able to get some measure of how well
does an algorithm recovers the measured data. Therefore,
CI − Total =1
∑Nt=1 x2
it
N∑
t=1
(xit − xit)2 t = 1, ..., N (33)
In Table 3, we compare the performance measures for the Telescope 1 data set.
TABLE 3
Method NNMF-MUR NNMF-ALS BPSS
CT-Total 12.9020 6.4305 6.9685
PI 0.3671 0.3428 0.5162
In Table 4, we compare the performance measures for the Telescope 2 data set. The per-
formance index favors the BPSS technique, while CT-Total is inconclusive.
TABLE 4
Method NNMF-MUR NNMF-ALS BPSS
CT-Total 2.2285 3.3783 3.4620
PI 0.3363 0.2842 0.2785
5 FINAL DISCUSSION
References
[1] Alencar, S. (2007) “Star-disk Interaction in Classical T Tauri Stars”, Lecture Notes in
Physics, 723, 55-73.
[2] Cichocki, A., and Amari, S. (2002) “Adaptive Blind Signal and Image Processing -
Learning Algorithms and Applications”, New York Wiley.
[3] Edwards, S., Hartigan, P., Ghandour, L., and Andrulis, C. (1994) “Spectroscopic Ev-
idence for Magnetospheric Accretion in Classical T Tauri Stars”, The Astronomical
Journal, 108, 3, 1056-1070.
[4] Geman, S. and Geman, D., (1984) “Stochastic Relaxation, Gibbs Distributions, and
the Bayesian Restoration of Images”, IEEE Transactions PAMI-6, 721-741.
12
[5] Gobinet, C., Perrin, E., and Huez, R. (2004) “Application of Non-Negative Matrix
Factorization to Fluorescence Spectroscopy”, Eusipco Proceedings, 1095-1098.
[6] Gray, D. (2005) “The Observation and Analysis of Stellar Photospheres”, Cambridge
University Press, Third edition.
[7] Gray, R., and Corbally, C. (2009) “Stellar Spectral Classification”, Princeton Series in
Astrophysics, Princeton University Press.
[8] Hastings, W., (1970) “Monte Carlo sampling methods using Markov chains and their
applications”, Biometrica, 57, 97-109.
[9] Hosseini, S., Jutten, C., and Pham, D.T. (2003) “Markovian source separation”, IEEE
Trans. Signal Process, 51, 12, 660-666.
[10] Huez, R., Perrin, E., Sockalingum, G., and Manfati, M. (2002) “Blind Source Separa-
tion: Application to Microorganism Raman Spectra”, Eusipco Proceedings.
[11] Lee, D., and Seung, S. (2001) “Algorithms for Non-negative Matrix Factorization”,
Advances in Neural information Processing Systems 13: Proceedings of the 2000 con-
ference, MIT Press, 556-562.
[12] Lee, S., Lim, J., Vannucci, M., Petkova, E., Preter, M., and Klein, D. (2008) “Order-
Preserving Dimension Reduction Procedure for the Dominance of Two Mean Curves
with Application to Tidal Volume Curves”, Biometrics, 64, 3, 931-939.
[13] Moussaoui, S., Brie, D., Mohammad-Djafari, A., and Cartere, C. (2006) “Separation
of Non-Negative Mixture of Non-Negative Sources Using a Bayesian Approach and
MCMC Sampling”, IEEE Transactions on Signal Processing, 54, 11, 4133-4145.
[14] Paatero, P., and Tapper, U. (1994) “Positive Matrix Factorization: A Non-Negative
Factor Model with Optimal Utilization of Error Estimates of Data Values”, Environ-
metrics, 5, 111-126.
[15] Rogava, A., and Tsiklauri, D. (1993) “The Structure of the Accretion Disc Boundary
Layer around a Rotating, Non-magnetized Star”, Astrophysics and Space Science, 204,
9-18.
[16] Sajda, P., Du, S., Brown, TR., Stoyanova, R., Shungu, DC., Mao, X., and Parra, LC.
(2004) “Nonnegative matrix factorization for rapid recovery of constituent spectra in
magnetic resonance chemical shift imaging of the brain”, IEEE Trans Med Imaging,
23, 12, 1453-1465.
13
[17] Valdes, F., Gupta, R., Rose, J., Singh, H., and Bell, D. (2004) “Indo-U.S. library of
CoudeFeed Stellar Spectra”, The Astrophysical Journal, 152, 2, 251-279.
[18] Valenti, J., Basri, G., and Johns, C. (1993) “T Tauri Stars in Blue”, The Astronomical
Journal, 106, 5, 2024-2050.
14
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Mixing Coefficients
(e)
Figure 3: NNMF. Simulated data. (a) Measured signals X. (b) Estimated signals X. (c)-(d)
Estimated sources S1 and S2. (e) Estimated mixing coefficients A.
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Mixing Coefficients
Figure 4: NNMF. Simulated galaxy. (a) Measured signals X. (b) Estimated signals X. (c)-
(d) Estimated sources S1 and S2. (e) Estimated source S3. (f) Estimated mixing coefficients
A.
16
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Figure 5: NNMF. Mayall telescope. (a) Measured signals X. (b) Estimated signals X.
(c)-(d) Estimated sources S1 and S2.
17
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Mixing Coefficients
(g)S
1
S2
S3
S4
Figure 6: NNMF. Mayall telescope. (e)-(f) Estimated sources S3 and S4. (g) Estimated
mixing coefficients A.
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Figure 7: NNMF. Valenti (1993) data. (a) Measured signals X. (b) Estimated signals X.
(c)-(d) Estimated sources S1 and S2. (e)-(f) Estimated sources S3 and S4.
19
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Mixing Coefficients
(j)
Figure 8: NNMF. Valenti (1993) data. (g)-(h) Estimated sources S5 and S6. (i) Estimated
source S7. (j) Estimated mixing coefficients A.
20
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Mixing Coefficients
(e)
Figure 9: NNMF-ALS. Simulated data. (a) Measured signals X. (b) Estimated signals X.
(c)-(d) Estimated sources S1 and S2. (e) Estimated mixing coefficients A.
21
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Mixing Coefficients
Figure 10: NNMF-ALS. Simulated galaxy. (a) Measured signals X. (b) Estimated signals
X. (c)-(d) Estimated sources S1 and S2. (e) Estimated source S3. (f) Estimated mixing
coefficients A.
22
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Figure 11: NNMF-ALS. Mayall telescope. (a) Measured signals X. (b) Estimated signals
X. (c)-(d) Estimated sources S1 and S2.
23
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Mixing Coefficients
(g)S1
S2
S3
S4
Figure 12: NNMF-ALS. Mayall telescope. (e)-(f) Estimated sources S3 and S4. (g) Esti-
mated mixing coefficients A.
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Figure 13: NNMF-ALS. Valenti (1993) data. (a) Measured signals X. (b) Estimated signals
X. (c)-(d) Estimated sources S1 and S2. (e)-(f) Estimated sources S3 and S4.
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Mixing Coefficients
(j)
Figure 14: NNMF-ALS. Valenti (1993) data. (g)-(h) Estimated sources S5 and S6. (i)
Estimated source S7. (j) Estimated mixing coefficients A.
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Mixing Coefficients
(e)
Figure 15: BPSS. Simulated data. (a) Measured signals X. (b) Estimated signals X. (c)-(d)
Posterior mean sources S1 and S2. (e) Posterior mean mixing coefficients A.
27
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Mixing Coefficients
(f)
Figure 16: BPSS. Simulated galaxy. (a) Measured signals X. (b) Estimated signals X.
(c)-(d) Posterior mean sources S1 and S2. (e) Posterior mean source S3. (f) Posterior mean
mixing coefficients A.
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Figure 17: BPSS. Mayall telescope. (a) Measured signals X. (b) Estimated signals X.
(c)-(d) Posterior mean sources S1 and S2.
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Mixing Coefficients
(g)S1
S2
S3
S4
Figure 18: BPSS. Mayall telescope. (e)-(f) Posterior mean sources S3 and S4. (g) Posterior
mean mixing coefficients A.
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Figure 19: BPSS. Valenti (1993) data. (a) Measured signals X. (b) Estimated signals X.
(c)-(d) Posterior mean sources S1 and S2. (e)-(f) Posterior mean sources S3 and S4.
31
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Mixing Coefficients
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Figure 20: BPSS. Valenti (1993) data. (g)-(h) Posterior mean sources S5 and S6. (i)
Posterior mean source S7. (j) Posterior mean mixing coefficients A.
32