Recovering galaxy star formation and metallicity histories from spectra
  using VESPA
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 30 August 2021 (MN LATEX style file v2.2)
Recovering galaxy star formation and metallicity histories
from spectra using VESPA
R. Tojeiro⋆1, A. F. Heavens1, R. Jimenez2 and B. Panter1
1Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK
2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA-19104, USA
30 August 2021
ABSTRACT
We introduce VErsatile SPectral Analysis (VESPA): a new method which aims to
recover robust star formation and metallicity histories from galactic spectra. VESPA
uses the full spectral range to construct a galaxy history from synthetic models. We
investigate the use of an adaptative parametrization grid to recover reliable star for-
mation histories on a galaxy-by-galaxy basis. Our goal is robustness as opposed to high
resolution histories, and the method is designed to return high time resolution only
where the data demand it. In this paper we detail the method and we present our find-
ings when we apply VESPA to synthetic and real Sloan Digital Sky Survey (SDSS)
spectroscopic data. We show that the number of parameters that can be recovered
from a spectrum depends strongly on the signal-to-noise, wavelength coverage and
presence or absence of a young population. For a typical SDSS sample of galaxies, we
can normally recover between 2 to 5 stellar populations. We find very good agreement
between VESPA and our previous analysis of the SDSS sample with MOPED.
Key words: methods: data analysis - methods: statistical - galaxies: stellar content
- galaxies: evolution - galaxies: formation
1 INTRODUCTION
The spectrum of a galaxy holds vasts amounts of informa-
tion about that galaxy’s history and evolution. Finding a
way to tap directly into this source of knowledge would
not only provide us with crucial information about that
galaxy’s evolutionary path, but would also allow us to
integrate this knowledge over a large number of galaxies
and therefore derive cosmological information.
Galaxy formation and evolution are still far from being
well understood. Galaxies are extremely complex objects,
formed via complicated non-linear processes, and any
approach (be it observational, semi-analytical or compu-
tational) inevitably relies on simplifications. If we try to
analyse a galaxy’s luminous output in terms of a history
parametrized by some chosen physical quantities, such a
simplification is also in order. The reason is two-fold: firstly
we are limited by our knowledge and ability to model
all the physical processes which happen in a galaxy and
produce the observed spectrum we are analysing; secondly,
the observed spectrum is inevitably perturbed by noise,
which intrinsically limits the amount of information we can
⋆ E-mail: rmft@roe.ac.uk
recover.
Measuring and understanding the star formation history
of the Universe is therefore essential to our understanding
of galaxy evolution - when, where and in what conditions
did stars form throughout cosmic history? The traditional
and simplest way to probe this is to measure the observed
instantaneous star formation rate in galaxies at differ-
ent redshifts. This can be achieved by looking at light
emitted by young stars in the ultra-violet (UV) band or
its secondary effects. (e.g. Madau et al. 1996; Kennicutt
1998; Hopkins et al. 2000; Bundy et al. 2006; Erb et al.
2006; Abraham et al. 2007; Noeske et al. 2007; Verma et al.
2007). A complementary method is to look at present day
galaxies and extract their star formation history, which
spans the lifetime of the galaxy. Different teams have
analysed a large number of galaxies in this way, whether
by using the full available spectrum (Glazebrook et al.
2003; Panter et al. 2003; Cid Fernandes et al. 2004;
Heavens et al. 2004; Mathis et al. 2006; Ocvirk et al. 2006;
Panter et al. 2006; Cid Fernandes et al. 2007), or by con-
centrating on particular spectral features or indices (e.g.
Kauffmann et al. 2003; Tremonti et al. 2004; Gallazzi et al.
2005; Barber et al. 2006), which are known to be correlated
with age or metallicities (e.g. Worthey 1994; Thomas et al.
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2 Tojeiro et al.
2003).
To do this, we rely on synthetic stellar population models to
describe a galaxy in terms of its stellar components, but by
modelling a galaxy in this way we are intrinsically limited
by the quality of the models. There are also potential
concerns with flux calibration errors. However, using the
full spectrum to recover the fossil record of a galaxy - or of
an ensemble of galaxies - is an extremely powerful method,
as the quality and amount of data relating to local galaxies
vastly outshines that which concerns high-redshift galaxies.
Splitting a galaxy into simple stellar populations of different
ages and metallicities is a natural way of parameterising
a galaxy, and it allows realistic fits to real galaxies (e.g.
Bruzual & Charlot 2003). Galactic archeology has become
increasingly popular in the literature recently, largely due to
the increase in sophistication of stellar population synthesis
codes and the improvement of the stellar spectra libraries
upon which they are based, and also due to the availability
of large well-calibrated spectroscopic databases, such as
the Sloan Digital Sky Survey (SDSS) (York et al. 2000;
Strauss et al. 2002).
In any case, without imposing any constraints on the
allowed form of the star formation history, or perhaps an
age-metallicity relation, the parameter space can become
unsustainably large for a traditional approach. Ideally, one
would like to do without such pre-constraints. Recently,
different research teams have come up with widely different
solutions for this problem. MOPED (Heavens et al. 2000)
and STARLIGHT (Cid Fernandes et al. 2004) explore a
well-chosen parameter space in order to find the best
possible fit to the data. In the case of MOPED, this relies
on compression of the full spectrum to a much smaller set
of numbers which retains all the information about the
parameters it tries to recover; STARLIGHT on the other
hand, searches for its best fit using the full spectrum with
a Metropolis algorithm. STECMAP (Ocvirk et al. 2006)
solves the problem using an algebraic least-squares solution
and a well-chosen regularization to keep the inversions
stable. All of these and other methods acknowledge the
same limitation - noise in the data and in the models
introduces degeneracies into the problem which can lead
to unphysical results. MOPED, for example, has produced
some remarkable results concerning the average star forma-
tion history of the Universe by analysing a large sample
of galaxies. However, MOPED’s authors have cautioned
against over-interpreting the results on a galaxy-by-galaxy
basis, due to the problem mentioned above. This is directly
related to the question of how finely one should param-
eterise a galaxy, and what the consequences of this might be.
Much of the motivation for VESPA came from the reali-
sation that this problem will vary from galaxy to galaxy,
and that the method of choosing a single parametrization
to analyse a large number of galaxies can be improved on.
VESPA is based on three main ideas, which we present here
and develop further in the main text:
• There is only so much information one can safely recover
from any given set of data, and the amount of information
which can be recovered from an individual galaxy varies.
• The recovered star formation fractions should be posi-
tive.
• Even though the full unconstrained problem is non-
linear, it is piecewise linear in well-chosen regions of pa-
rameter space.
VESPA’s ultimate goal is to derive robust information
for each galaxy individually, by adapting the number of
parameters it recovers on a galaxy-by-galaxy basis and
increasing the resolution in parameter space only where the
data warrant it. In a nutshell, this is how VESPA works: we
estimate how many parameters we can recover from a given
spectrum, given its noise, shape, spectral resolution and
wavelength range using an analysis given by Ocvirk et al.
(2006). In that paper, Singular Value Decomposition (SVD)
is used to find a least squares solution, and this solution
is analysed in terms of its singular vectors. VESPA uses
this method only as an analysis of the solution, and uses
Bounded-Variable Least-Squares (BVLS) (Stark & Parker
1995) to reach a non-negative solution in several regimes
where linearity applies.
This paper is organised as follows: in Section 2 we present
the method, in Section 3 we apply VESPA to a variety of
synthetic spectra, in Section 4 we apply VESPA to a sample
of galaxies from the Sloan Digital Sky Survey spectroscopic
database and we compare our results to those obtained with
MOPED, and finally in Section 5 we present our conclusions.
2 METHOD
In this section we lay down the problem to solve in detail,
and explain the different steps VESPA uses to find a solution
for each galaxy.
2.1 The problem
We assume a galaxy is composed of a series of simple stel-
lar populations (SSP) of varying ages and metallicities. The
unobscured rest frame luminosity per unit wavelength of a
galaxy can then be written as
ψ(t)Sλ(t, Z)dt (1)
where ψ(t) is the star formation rate (solar masses formed
per unit of time) and Sλ(t, Z) is the luminosity per unit
wavelength of a single stellar population of age t and metal-
licity Z, per unit mass. The dependency of the metallicity on
age is unconstrained, turning this into a non-linear problem.
In order to solve this problem, we start by discretizing in
wavelength and time, by averaging these two quantities
into well chosen bins. For now we present the problem with
a generalised parametrization, and discuss our choice in
Section 2.3. We will use greek indices to indicate time bins,
and roman indices to indicate wavelength bins.
The problem becomes
c© 0000 RAS, MNRAS 000, 000–000
Recovering galaxy histories using VESPA 3
xαG(Zα)αj (2)
where Fj = (F1, ..., FD) is the luminosity of the jth wave-
length bin of width ∆λ, G(Zα)αj is the jth luminosity point
of a stellar population of age tα = (t1, ..., tS) (spanning an
age range of ∆tα) and metallicity Zα, and xα = (x1, ..., xS)
is the total mass of population G(Z)αj in the time bin ∆tα.
Although the full metallicity problem is non-linear, interpo-
lating between tabulated values of Z gives a piecewise linear
behaviour:
G(Zα)αj = gαG(Za,α)αj + (1− gα)G(Zb,α)αj , (3)
and the problem then becomes
xα [gαG(Za,α)αj + (1− gα)G(Zb,α)αj ] (4)
where G(Za,α)αj and G(Zb,α)αj are equivalent to G(Zα)αj
as above, but at fixed metallicities Za and Zb, which bound
the true Z. If this interpolation matches the models’ reso-
lution in Z, then we are not degrading the models in any way.
Solving the problem then requires finding the correct metal-
licity range. One should not underestimate the complexity
this implies - trying all possible combination of consecutive
values of Za and Zb in a grid of 16 age bins would lead to a
total number of calculations of the order of 109, which is un-
feasible even in today’s fast personal workstations. We work
around this problem using an iterative approach, which we
describe in Section 2.3.2.
2.1.1 Dust extinction
An important component when describing the luminous out-
put of a galaxy is dust, as different wavelengths are affected
in different ways. The simplest possible approach is to use
one-parameter dust model, according to which we apply
a single dust screen to the combined luminosity of all the
galactic components. Equation (1) becomes
Fλ = fdust(τλ)
ψ(t)Sλ(t, Z)dt (5)
where we are assuming the dust extinction is the same for
all stars, and characterised by the optical depth, τλ.
However, it is also well known that very young stars are
likely to be more affected by dust. In an attempt to in-
clude this in our modelling, we follow the two-parameter
dust model of Charlot & Fall (2000) in which young stars
are embebbed in their birth cloud up to a time tBC , when
they break free into the inter-stellar medium (ISM):
fdust(τλ, t)ψ(t)Sλ(t, Z)dt (6)
fdust(τλ, t) =
fdust(τ
λ )fdust(τ
λ ), t ≤ tBC
fdust(τ
λ ), t > tBC
where τ ISMλ is the optical depth of the ISM and τ
λ is the
optical depth of the birth cloud. Following Charlot & Fall
(2000), we take tBC = 0.03 Gyrs.
There is a variety of choices for the form of fdust(τλ). To
model the dust in the ISM, we use the mixed slab model of
Charlot & Fall (2000) for low optical depths (τV ≤ 1), for
which
fdust(τλ) =
[1 + (τλ − 1) exp(−τλ)− τ
λE1(τλ)] (8)
where E1 is the exponential integral and τλ is the optical
depth of the slab. This model is known to be less accurate
for high dust values, and for optical depths greater than one
we take a uniform screening model with
fdust(τλ) = exp(−τλ). (9)
We only use the uniform screening model to model the dust
in the birth cloud and we use τλ = τV (λ/5500Å)
−0.7 as our
extinction curve for both environments.
As described, dust is a non-linear problem. In practice, we
solve the linear problem described by equation (4) with a
number of dust extinctions applied to the matrices G(Z)ij
and choose the values of τ ISMV and τ
V which result in the
best fit to the data.
We initially use a binary chop search for τ ISMV ∈ [0, 4] and
keep τBCV fixed and equal to zero, which results in trying out
typically around nine values of τ ISMV . If this initial solution
reveals star formation at a time less than tBC we repeat
our search on a two-dimensional grid, and fit for τ ISMV and
τBCV simultaneously. There is no penalty except in CPU time
to apply the two-parameter search, but we find that this
procedure is robust (see section 3.4).
2.2 The solution
In this section we describe the method used to reach a
solution for a galaxy, given a set of models and a gener-
alised parametrization. The construction of these models
and choice of parameters is described in Sections 2.3 and 2.4.
We re-write the problem described by equation (4) in a sim-
pler way
cκAκj(Zκ) (10)
where Zκ = Za for κ < S and Zκ = Zb for κ ≥ S. A
is a D × 2S matrix composed of synthetic models at the
corresponding metallicities, and c = (c1, ..., c2S) is the solu-
tion vector, from which the xα and gα in equation (4) can
be calculated. We can then calculate a linearly interpolated
metallicity at age tα
Zα = gαZa + (1− gα)Zb. (11)
For every age tα we aim to recover two parameters: xα -
the total mass formed at that age (within a period ∆tα) -
and Zα - a mass-weighted metallicity.
At this stage we are not concerned with our choice of tα and
∆tα - although these are crucial and will be discussed later.
For a given set of chosen parameters, we find c, such that
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4 Tojeiro et al.
(Fj −
cκAκj)
is minimised (where σj is the error in the measured flux bin
A linear problem with a least squares constraint has a simple
analytic solution which, for constant σj (white-noise) is
cLS = (A
·F (13)
In principle, any matrix inversion method, e.g. Singular
Value Decomposition (SVD), can be used to solve (13). How-
ever, we would like to impose positivity constraints on the
recovered solutions. Negative solutions are unphysical, but
unfortunately common in a problem perturbed by noise.
2.2.1 BVLS and positivity
We use Bounded-Variable Least-Squares (BVLS)
(Stark & Parker 1995) in order to solve (13). BVLS is
an algorithm which solves linear problems whose variables
are subject to upper and lower constraints. It uses an
active set strategy to choose sets of free variables, and
uses QR matrix decomposition to solve the unconstrained
least-squares problem of each candidate set of free variables
using (13):
cLS = (E
·F (14)
where E is effectively composed of those columns of A
for which ck is unconstrained, and of zero vectors for
those columns for which ck is set to zero. BVLS is an
extension of the Non-Negative Least Squares algorithm
(Lawson & Hanson 1974), and they are both proven to
converge in a finite number of iterations. Positivity is the
only constraint in VESPA’s solution.
BVLS and positivity have various advantages. Most obvious
is the fact that we do away with negative solutions. In a
non-constrained method (such as SVD) negative values are
a response to the fact that the data is noisy. Similarly, we
find that zero values returned by BVLS (in, for example,
a synthetic galaxy with continuous star formation across
all time) are also an artifact from noisy data. It should be
kept in mind that, if the method is unbiased, this problem
is solved by analysing a number of noisy realisations of the
original problem - what we find is that the true values of
the parameters we try to recover are consistent with the
distributions yielded by this process. In this sense, not even
a negative value presents a problem necessarily, as long as
it is consistent with zero (or the correct solution). Given
that we have found no bias when using BVLS, we feel it is
an advantage to discard a priori solutions we know to be
unphysical.
Another advantage to using BVLS is the fact that, by fixing
some parameters to the lower boundary (zero, in this case),
it effectively reduces the number of fitting parameters to
the number of those which keeps unconstrained. Given the
overall aims of VESPA, this has proven to be advantageous.
2.2.2 Noise
The inversion in equation (13) is often highly sensitive to
noise, and care is needed when recovering solutions with
matrix inversion methods. The fit in data-space will always
improve as we increase the number of parameters, but these
might not all provide meaningful information. We follow an
analysis given in Ocvirk et al. (2006) in order to understand
how much this affects our results, and to choose a suitable
age parametrization for each galaxy. This is not an exact
method, and it does not guarantee that the solutions we
recover have no contribution from noise. However, we found
that in most cases it provides a very useful guideline (see
section 3.3, in particular Figure 11).
We refer the reader to the above paper for a full discussion,
and we reproduce here the steps used in our analysis.
We use SVD to decompose the model matrix E as
E = U ·W ·V
where U is a D×2S orthonormal matrix with singular data-
vectors uκ as columns, V is a 2S × 2S orthonormal matrix
with the singular solution-vectors vκ as columns, and W is
a 2S × 2S diagonal matrix W = diag(w1, ..., w2S) where wκ
are the matrix singular values in decreasing order. Replacing
E by this decomposition in equation (13) gives
cLS = V ·W
vκ (16)
The solution vector is a linear combination of the solution
singular values, parametrized by the dot product between
the data and the corresponding data singular vector, and
divided by the kth singular value. The data vector itself is a
combination of the true underlying emitted flux and noise:
F = Ftrue + e. Equation (16) becomes
cLS =
κ ·Ftrue
vκ ≡ ctrue + ce (17)
where ctrue is the solution vector to the noiseless problem
and ce is an unavoidable added term due to the presence of
noise.
It is extremely informative to compare the amplitudes
of the two terms in the sum (17), and to monitor their
contributions to the solution vector with varying rank. In
Figure 1 we plot |uTκ · F| and u
κ ·e as a function of rank
κ, for a synthetic spectrum with a SNR per pixel of 50
(at a resolution of 3Å) and an exponentially-decaying star
formation history. We observe the behaviour described
and discussed in Ocvirk et al. (2006). The combinations
associated with the noise terms maintain a roughly constant
power across all ranks, with a an average value of 〈F〉 /SNR.
The data terms, however, drop significantly with rank, and
we can therefore identify two ranges: a noise-dominated
κ-range, in which the noise contributions match or dom-
inate the true data contributions, and a data-dominated
range, where the contributions to the solution are largely
data motivated. We call the transition rank κcrit. Overall,
high-κ ranks tend to dominate the solution, since the
singular values wκ decrease with κ. This only amplifies
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Recovering galaxy histories using VESPA 5
Figure 1. The behaviour of the singular values with matrix rank
k. The stars are |uTκ ·F| and the squares are u
κ ·e. The line is
〈F〉 /SNR, which in this case has a value of approximately 0.06.
the problem by giving greater weight to noise-dominated
terms in the sum (16). Figure 2 shows the contribution
coming from each rank κ to the final solution - the co-
efficient (uTκ ·F)/wκ. We see this weight increases with rank.
Whereas this analysis gives us great insight into the
problem, we do not in fact use the sum (16) to obtain cLS ,
for the reasons given in section 2.2.1.
For real data we are only able to calculate uTκ ·F and estimate
the noise level at 〈F〉 /SNR and we use this information
to estimate the number of non-zero parameters to recover
from the data. Our aim is to a have a solution which is
dominated by the signal, and not by the noise. We therefore
want our number of non-zero recovered parameters to be
less than or equal to κcrit. Estimating where this transition
happens is always a noisy process. In this paper we take
the conservative approach of setting κcrit to be the rank at
which the perturbed singular values first cross the 〈F〉 /SNR
barrier. In the case of Figure 1 this happens at κcrit = 7.
2.3 Choosing a galaxy parametrization
One of the advantages of VESPA is that it has the ability
to choose the number of parameters to recover in any
given galaxy. This is possible due to a time grid of varying
resolutions, which VESPA can explore to find a solution.
This section describes this grid and the criteria used to
reach a final parametrization.
2.3.1 The grid
We work on a grid with a maximum resolution of 16 age
bins, logarithmically spaced in lookback time from 0.02
Figure 2. The coefficients in sum (16) as a function of rank κ.
We see that the highest rank modes (corresponding to the smaller
singular values) tend to contribute the most to the solution.
up to 14 Gyr. The grid has three further resolution levels,
where we split the age of the Universe in eight, four and
finally two age bins, also logarithmically spaced in the same
range.
The idea behind the multi-resolution grid is to start our
search with a low number of parameters (in coarser resolu-
tion, so that the entirety of the age of Universe is covered),
and then increase the resolution only where the data war-
rant it by splitting the bin with the highest flux contribution
in two, and so on. In effect, we construct one such grid for
each of the tabulated metallicities, Za and Zb. We work with
five metallicity values, Z = [0.0004, 0.004, 0.008, 0.02, 0.05]
which correspond to the metallicity resolution of the models
used, where Z is the fraction of the mass of the star
composed of metals (Z⊙ = 0.02). The construction of the
models for each of the time bins is discussed in Section 2.4.
To each of the grids we can apply a dust extinction as ex-
plained in Section 2.1.1.
2.3.2 The search
We go through the following steps in order to reach a solu-
tion:
(i) We begin our search with three bins: two bins of width
4 and one bin of width 8 (oldest), where here we are mea-
suring widths in units of high-resolution bins.
(ii) We calculate a solution using equation (10) for every
possible combination of consecutive boundaries Za and Zb,
and we choose the one which gives the best value of reduced
(iii) We calculate the number of perturbed singular values
above the noise level, as described at the end of Section 2.2.2.
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6 Tojeiro et al.
(iv) We find the bin which contributes the most to the
total flux and we split it into two.
(v) We find a solution in the new parametrization, this
time by trying out all possible combinations of Za and Zb for
the newly split bins only, and fixing the metallicity bound-
aries of the remainder bins to the boundaries obtained in
the previous solution. If a bin had no stars in the previous
iteration, we set Za = 0.0004 and Zb = 0.05.
(vi) We return to (iii) and we proceed until we have
reached the maximum resolution in all populated bins.
(vii) We look backwards in our sequence of solutions for
the last instance with a number of non-zero recovered pa-
rameters equal to or less than κcrit as calculated in (iii) and
take this as our best solution.
We illustrate this sequence in Figure 3, where we show the
evolution of the search in a synthetic galaxy composed of
two stellar bursts of equal star formation rates - one young
and one old. VESPA first splits the components which con-
tribute the most to the total flux. In this case this is the
young burst which can be seen in the first bin. Even though
VESPA always resolves bins with any mass to the possible
highest resolution, it then searches for the latest solution
which has passed the SVD criterion explained in Section
2.2.2. In this case, this corresponds to the fifth from the top
solution. VESPA chooses this solution in favour of the fol-
lowing ones due to the number of perturbed singular values
above the solid line (right panel). In this case, the solution
chosen by VESPA is a better fit in parameters space (note
the logarithmic scale in the y-axis - the following solution
put the vast majority of the mass in the wrong bin). We ob-
served this type of improvement in the majority of all cases
studied (see Figure 11).
2.3.3 The final solution
Our final solution comes in a parametrization such that the
total number of non-zero recovered parameters is less than
or equal to the number of perturbed singular values above
the estimated noise level.
The above sequence is performed for each of several combi-
nations of τBCV , τ
V , and we choose the attenuation which
provides the best fit.
For each galaxy we recover N star formation masses, with
an associated metallicity, where N is the total number of
bins, and a maximum of two dust parameters.
2.4 The models
The backbone to our grid of models is the BC03 set of
synthetic SSP models (Bruzual & Charlot 2003), with a
Chabrier initial mass function (Chabrier 2003) and Padova
1994 evolutionary tracks (Alongi et al. 1993; Bressan et al.
1993; Fagotto et al. 1994a,b; Girardi et al. 1996) . Although
any set of stellar population models can be used, these
provide a detailed spectral evolution of stellar populations
over a suitable range of wavelength, ages and metallicities:
S(λ, t, Z). The models have been normalised to one solar
mass at the age t = 0.
2.4.1 High-resolution age bins
At our highest resolution we work with 16 age bins, equally
spaced in a logarithmic time scale between now and the
age of the Universe. In each bin, we assume a constant star
formation rate
α (λ,Z) = ψ
S(λ, t, Z)dt (18)
with ψ = 1/∆tα.
2.4.2 Low-resolution age bins
As described in Section 2.3.1, we work on a grid of different
resolution time bins and we construct the low resolution bins
using the high resolution bins described in Section 2.4.1. We
do not assume a constant star formation rate in this case, as
in wider bins the light from the younger components would
largely dominate over the contribution from the older ones.
Instead, we use a decaying star formation history, such that
the light contributions from all the components are compa-
rable. Recall equation (1)
α (λ,Z) =
ψ(t)S(λ, t, Z)dt, (19)
which we approximate to
β (λ,Z) =
fHRα (λ,Z)ψα∆tα
ψα∆tα
where low resolution bin β incorporates the high resolution
bins α ∈ β, and we set
ψα∆tα =
fHRα (λ,Z)dλ
. (21)
Depending on the galaxy, the final solution obtained with
the sequence detailed in Section 2.3.2 can be described in
terms of low-resolution age bins. In this case we should in-
terpret the recovered mass as the total mass formed during
the period implied by the width of the bin, but we can-
not make any conclusions as to when in the bin the mass
was formed. Similarly, the recovered metallicity for the bin
should be interpreted as a mass-weighted metallicity for the
total mass formed in the bin.
2.5 Errors
The quality of our fits and of our solutions is affected by
the noise in the data, the noise in the models, and the
parametrization we choose (which does not reflect the
complete physical scenario within a galaxy). We aim to
apply VESPA firstly to SDSS galaxies, which typically have
a SNR ≈ 20 per resolution element of 3Å, which puts us in
a regime where the main limitations come from the noise in
the data.
To estimate how much noise affects our recovered solutions
we take a rather empirical approach. For each recovered so-
lution we create nerror random noisy realisations and we
apply VESPA to each of these spectra. We re-bin each re-
covered solution in the parametrization of the solution we
want to analyse and estimate the covariance matrices
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Recovering galaxy histories using VESPA 7
Figure 3. The evolution of the fit, as VESPA searches for a solution. Sequence should be read from top to bottom. Each line shows
a stage in the sequence: the left panel shows the input star formation history in the dashed line (red on the online version), and the
recovered mass fractions on the solid line (black on the online version) for a given parametrization ; the middle panel shows the input
metallicities in the dashed line (red on the online version), and the recovered metallicities on the solid line (black on the online version);
the right panel shows the absolute value of the perturbed singular values |uκ · F| (stars and solid line) and the estimated noise level
〈F〉 /SNR. In this panel we also show the value of κcrit and the number of non-zero elements of cLS in each iteration. The chosen solution
is the fifth from the top, and indicated accordingly. This galaxy consists of 2 burst events of equal star formation rate - a very young
and an old burst. It was modelled with a resolution of 3Å and a signal-to-noise ratio per pixel of 50. We see the recovery is good but not
perfect - there is a 1 per cent leakage from the older population - but better than the following solutions, where this bin is split. See text
in Section 2.3.2 for more details.
C(x)αβ = 〈(xα − x̄α)(xβ − x̄β)〉 (22)
C(Z)αβ =
(Zα − Z̄α)(Zβ − Z̄β)
. (23)
All the plots in Sections 3 and 4 show error bars derived
from C
αα , although it is worth keeping in mind that these
are typically highly correlated.
2.6 Timings
A basic run of VESPA (which consists of roughly 5 runs
down the sequence detailed in Section 2.3.2, one for each
value of dust extinction) takes about 5 seconds. If accurate
error estimations are needed per galaxy, this will add an-
other one or two minutes to the timing, depending on how
c© 0000 RAS, MNRAS 000, 000–000
8 Tojeiro et al.
accurately one would like to estimate the covariance ma-
trices, and depending on the number of data points. With
nerror = 10, a typical SDSS galaxy takes around one minute
to analyse.
3 TESTS ON SIMULATED DATA
We tested VESPA on a variety of synthetic spectra, in
order to understand its capabilities and limitations. In
particular, we tried to understand the effect of three factors
in the quality of our solutions: the input star formation
history, the noise in the data, and the wavelength coverage
of the spectrum. We have also looked at the effects of dust
extinction. Throughout we have modelled our galaxies in a
resolution of 3Å.
Even though we are aware that showing individual examples
of VESPA’s results from synthetic spectra can be extraor-
dinarily unrepresentative, we feel obliged to show a few for
illustration purposes. We will show a typical result for most
of the cases we present, but we also define some measure-
ments of success, so that the overall performance of VESPA
can be tracked as we vary any factors. We define
xα − x
ωα (24)
Zα − Z
ωα (25)
where xIα and Z
α are the total mass and correspondent
metallicity in bin α (re-binned to match the solution’s
parametrization if necessary), and ωα is the flux contribu-
tion of population of age tα. Gx and GZ are a flux-weighted
average of the total absolute fractional errors in the solu-
tion, and give an indication of how well VESPA recovers
the most significant parameters. A perfect solution gives
Gx = GZ = 0. It is also worth noting that this statistic
does not take into account the associated error with each
recovered parameter - deviations from the true solution are
usually expected given the estimated covariance matrices.
We will also show how these factors affect the recovered to-
tal mass for a galaxy. In all cases we have re-normalised the
total masses such that total input mass for each galaxy is 1.
3.1 Star formation histories
We present here some results for synthetic spectra with
two different star formation histories. All of the spectra
in this section were synthesised with a SNR per pixel of
50, and we initially fit the very wide wavelength range
λ ∈ [1000, 9500]Å.
We choose two very difference cases: firstly a star formation
history of dual bursts, with a large random variety of
burst age separations and metallicities (where we set the
star formation rate to be 10 solar masses per Gyr in all
bursts). Secondly, we chose a SFH with an exponentially
decaying star formation rate: SFR ∝ exp(γtα), where tα is
Figure 8. The recovered number of non-zero parameters in 50
galaxies with an exponentially decaying star formation history,
using different wavelength ranges: λ ∈ [1000, 9500] Å(solid line)
and λ ∈ [3200, 9500] Å(dashed line). Please note that these corre-
spond to the total number of non-zero components in the solution
vector cκ and not to the number of recovered stellar populations.
the age of the bin in lookback time in Gyr. Here we show
results for γ = 0.3 Gyr−1. Rather than being physically
motivated, our choice of γ reflects a SFH which is not
too steep as to essentially mimic a single old burst, but
which is also not completely dominated by recent star
formation. In all cases the metallicity in each bin is ran-
domly set. Figure 4 shows a typical example from each type.
Figure 5 shows the distribution of Gx, GZ and of the recov-
ered total masses for a sample of 50 galaxies. We see differ-
ences between the two cases. Firstly, in dual bursts galaxies,
we seem to do better in recovering data from significant in-
dividual bins, but worse in overall mass. This reflects the
fact that Gx is dominated by the fractional errors in the
most significant bins, but the total mass can be affected by
small flux contributions in old bins which can have large
masses. On the other hand, with an exponentially decaying
star formation rate, we do worse overall (although this is
mainly a reflection that more bins have significant contribu-
tions to the flux) but we recover the total mass of the galaxy
exceptionally well.
3.2 Wavelength range
Wavelength range is an important factor in this sort of
analysis, as different parts of the spectrum will help to break
different degeneracies. Since we are primarily interested in
SDSS galaxies, we have studied how well VESPA does in
the more realistic wavelength range of λ ∈ [3200, 9500] Å.
Figure 6 shows the results for the same galaxies shown in
Figure 4, obtained with the new wavelength range. In these
c© 0000 RAS, MNRAS 000, 000–000
Recovering galaxy histories using VESPA 9
Figure 4. Two examples of VESPA’s analysis on synthetic galaxies. The top panels show the original spectrum in the dark line (black
in the online version) and fitted spectrum in the lighter line (red in the online version ). The middle panels show the input (dashed, red)
and the recovered (solid, black) star formation rates and the bottom panel shows the input (dashed, red) and recovered (solid, black)
metallicities per bin. Note that even though many of the recovered metallicities are wrong, these tend to correspond to bins with very
little star formation, and are therefore virtually unconstrained.
Figure 5. The distribution of Gx, GZ and total mass recovered for 50 galaxies with a SNR per pixel of 50. Solid lines correspond to
dual burst and dashed lines to exponentially decaying ones. See text in Section 3.1 for details.
particular cases, we notice a more pronounced difference in
the dual bursts galaxy, but looking at a more substantial
sample of galaxies shows that this is not generally the case.
Figure 7 shows Gx, GZ and total mass recovered for 50
exponentially decaying star formation history galaxies, with
a signal to noise ratio of 50 and the two different wavelength
ranges. We do not see a largely significant change in both
cases, and we observe a less significant difference in the
dual bursts galaxies (not plotted here).
We find it instructive to keep track of how many parame-
ters we recover in total, as we change any factors. Figure
8 shows an histogram of the total number of non-zero
parameters we recovered from our sample galaxies with
exponentially-decaying star formation histories and both
wavelength ranges. Note that these are the components of
the solution vector cκ which are non-zero - they do not
represent a number of recovered stellar populations. In this
case there is a clear decrease in the number of recovered
parameters, suggesting a wider wavelength range is a useful
way to increase resolution in parameter space.
c© 0000 RAS, MNRAS 000, 000–000
10 Tojeiro et al.
Figure 6. Same galaxies as in Figure 4, but results obtained by using a smaller wavelength range. The goodness-of-fit in data space is
still excellent, but it becomes more difficult to break certain degeneracies.
Figure 7. The distribution of Gx, GZ and total mass recovered for 50 galaxies with a SNR per pixel of 50 and two different wavelength
coverage. Solid line corresponds to λ ∈ [1000, 9500]Å and dashed line to λ ∈ [3200, 9500] Å.
3.3 Noise
It is of interest to vary the signal-to-noise ratio in the
synthetic spectra. We have repeated the studies detailed
in the two previous sections with varying values of noise,
and we investigate how this affects both the quality of the
solutions and their resolution in parameter space.
Figure 10 shows how the recovered number of parameters
changed by increasing the noise in the galaxies with an
exponentially decaying star formation rate and wide wave-
length range. In this case the increase in the noise leads to a
significant reduction of the number of parameters recovered
for each galaxy. This behaviour is equally clear for different
star formation histories and different wavelength coverage,
and is directly caused by the stopping criterion defined in
Section 3.3.
The quality of the solutions is also affected by this increase
in noise, as can be seen in figure 9, where we have plotted Gx,
GZ and the total recovered mass for two different values of
SNR. The quality of the solutions decreases with the higher
noise levels, as is to be expected. However, a more interest-
ing question to ask is whether this decrease in the quality
of the solutions would indeed be more pronounced without
the SVD stopping criterion. Figure 11 shows a comparison
between Gx obtained as we have described and obtained
without any stopping mechanism (so letting our search go
to the highest possible resolution and taking the final so-
lution) for 50 galaxies with an exponentially decaying star
formation history and a signal-to-noise ratio of 20. The re-
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Recovering galaxy histories using VESPA 11
Figure 9. The distribution of Gx, GZ and total mass recovered for 50 galaxies with an exponential decaying star formation history and
different signal-to-noise ratios. Solid lines correspond to SNR = 50 and dashed lines to SNR =20. See text in Section 3.3 for details.
Figure 10. The recovered number of non-zero parameters as we
change the noise in the data from 50 (solid line) to 20 (dashed
line), in a sample of galaxies with an exponentially decaying star
formation rate. Please note that these correspond to the total
number of non-zero components in the solution vector cκ and
not to the number of recovered stellar populations.
sults show clearly that there is a significant advantage in
using the SVD stopping criterion. Naturally, the goodness
of fit in data space is consistently better as we increase the
number of parameters but this improvement is illusory - the
parameter recovery is worse. This is exactly the expected be-
haviour - we choose to sacrifice resolution in parameter space
in favour of a more robust solution - even though naively one
could think a lower χ2 solution would indicate a better solu-
tion. The significance of this improvement changes with the
amount of noise and wavelength range of the data (and to
a lesser extent with type of star formation history) but we
observed an improvement in all cases we have studied.
Figure 11. Testing the SVD stopping criterion. Plots show good-
ness of fit Gx for the solution of 50 galaxies obtained with and
without the SVD stopping criterion. We see that by recovering
only as many parameters as the data warrants gives improved
parameter estimation in almost all cases, and a striking improve-
ment in many.
As expected, further decreasing the signal-to-noise ratio
leads to a further degradation of the recovered solutions.
This is accompanied by a suitable increase in the error bars
and correlation matrices, but in cases of a SNR≈ 10 and less
it becomes very difficult to recover any meaningful informa-
tion from individual spectra.
3.4 Dust
In this section we use simulated galaxies to study the effect
of dust in our solutions. As explained in Section 2.1.1,
due to the non-linear nature of the problem, we cannot
include dust as one of the free parameters analysed by
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12 Tojeiro et al.
Figure 12. Testing the recovery of τISM
for 50 galaxies with
a exponentially-decaying star formation history (triangles) and
50 galaxies formed with a random combination of dual bursts
(stars). The input values are randomly chosen and continuously
distributed between 0 and 2. The recovered values are chosen from
a tabulated grid between 0 and 4.
SVD. Instead, we fit for a maximum of two dust parameters
using a brute-force approach which aims to minimise χ2 in
data-space by trying out a series of values for τ ISMV and τ
For each galaxy we assign random values of τ ISMV ∈ [0, 2]
and τBCV ∈ [1, 2] and we are interested in how well we
recover these parameters and any possible degeneracies.
Figure 12 shows the input and recovered values for τ ISMV for
galaxies with a signal-to-noise ratio of 50, and which were
analysed using the wavelength range λ ∈ [3200, 9500]Å. We
show results for two different cases of star formation his-
tory: 50 galaxies with an exponentially-decaying SFR and
50 galaxies formed by dual-bursts. We observe a good recov-
ery of τ ISMV in both cases, especially at low optical depths.
However, we mostly observe a poor recovery of τBCV ,
especially at high optical depths. This is unsurprisingly
flagging up a certain level of degeneracy between mass and
degree of extinction, which gets worse as the optical depth
increases. Essentially, it becomes difficult to distinguish
between a highly obscured massive population and a less
massive population surrounded by less dust. It is worth
keeping in mind that young populations are affected by
both dust components simultaneously, and generally, even
though the recovery of the second dust parameter may
not be accurate, it allows for a better estimation of the
dominant dust component.
This can be tested by simulating galaxies on a two-
component dust model and by analysing them using both a
single component model, and a two-component model. E.g.,
when using the more sophisticated model, we noted that the
mean error on τ ISMV on a subsample of dual-burst galaxies
(synthesised as explained in section 3.1, but chosen to have
young star formation) was reduced from 35 to 28 per cent.
This simple test also revealed that we are less likely to
underestimate the mass of young populations by allowing
an extra dust component, but that we are also introducing
an extra degeneracy, especially so in the case of faint young
populations. However, we feel that the two-parameter dust
model brings more advantages than disadvantages, with
the caveat being that dusty young populations can be
poorly constrained. In any case, we note that each galaxy
is always analysed with a one-parameter model before
being potentially analysed with a two-parameter model,
and both solutions are kept and always available for analysis.
Finally, our test also partly justifies our choice to first
run a single dust component model and only apply a two-
component model if we detect stars in the first two bins -
we find that although a one-component model might un-
derestimate the amount of young stars, it does not fail
to detect them. We repeated a similar test in real data,
by analysing the same sample with and one- and a two-
parameter dust model. We found similar results, with a one-
parameter model failing to yield star formation in young bins
only around 1 per-cent of the time (compared to the two-
parameter model), and only in cases where the contribution
of the light from the young populations was very small (of
the order of 1 to 2 per cent).
4 RESULTS
In this section we present some results obtained by applying
VESPA to galaxies in the SDSS. Our aim is to analyse these
galaxies, and to produce and publish a catalogue of robust
star formation histories, from which a wealth of information
can then be derived. We leave this for another publication,
but we present here results from a sub-sample of galaxies,
which we used to test VESPA in a variety of ways.
4.1 Handling SDSS data
Prior to any analysis, we processed the SDSS spectroscopic
data, so as to accomplish the desired spectral resolution
and mask out any wanted signal.
The SDSS data-files supply a mask vector, which flags
any potential problems with the measured signal on a
pixel-by-pixel basis. We use this mask to remove any
unwanted regions and emission lines. In practical terms, we
ignore any pixel for which the provided mask value is not
zero.
The BC03 synthetic models produce outputs at a resolution
of 3Å, which we convolve with a Gaussian velocity disper-
sion curve with a stellar velocity σV = 170kms
−1, this being
a typical value for SDSS galaxies. We take the models’
tabulated wavelength values as a fixed grid and re-bin the
SDSS data into this grid, using an inverse-variance weighted
average. We compute the new error vector accordingly. Note
that the number of SDSS data points averaged into any new
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Recovering galaxy histories using VESPA 13
bin is not constant, and that the re-binning process is done
after we have masked out any unwanted pixels. Additionally
to the lines yielded by the mask vector, we mask out the
following emission line regions in every spectrum’s rest-
frame wavelength range: [5885-5900, 6702-6732, 6716-6746,
6548-6578, 6535-6565, 6569-6599, 4944-4974, 4992-5022,
4846-4876, 4325-4355, 4087-4117, 3711-3741, 7800-11000] Å.
These re-binned data- and noise-vectors are essentially the
ones we use in our analysis. However, since the linear algebra
assumes white-noise, we pre-whiten the data and construct
a new flux vector F ′j = Fj/σj , which has unit variance,
σ′j = 1,∀j, and a new model matrix A
ij = Aij/σj .
4.2 Duplicate galaxies
There are a number of galaxies in the SDSS database
which have been observed more than once, for a variety
of reasons. This provides an opportunity to check how
variations in observation-dependent corrections affect the
results obtained by VESPA.
We have used a subset of the sample of duplicate objects in
Brinchmann et al. (2004)1 to create two sets of oservations
for 2000 galaxies, which we named list A and list B.
We are interested in seeing how the errors we estimate
for our results compare to errors introduced by intrinsic
variations caused by changing the observation conditions
(such as quality of the spectra, placement of the fibre, sky
subtraction or spectrophometric calibrations).
Figure 13 shows the average star formation fraction as a
function of lookback time for both sets of observation. The
error bars showed are errors on the mean. We see no signs
of being dominated by systematics when estimating the
star formation fraction of a sample of galaxies.
Figure 14 shows the total stellar mass obtained for a set of
500 galaxies in both observations (details of how we estimate
the total stellar mass of a galaxy are included in section 4.4).
The error bars are obtained directly from the estimated
covariance matrix C(x) (equation 22). Even though most of
the galaxy duplicates produce mass estimates in agreement
with each other given the error estimates, a minority
does not. Upon inspection, these galaxies show significant
differences in their continuum, but after further investiga-
tion it remains unclear what motivates such a difference.
The simplest explanation is that the spectrophotometric
calibration differs significantly between both observations,
and that might have been the reason the plate or object
was re-observed. Whatever the reason however, the clear
conclusion is that stellar mass estimates are highly sensitive
to changes in the spectrum continuum, and the errors we es-
timate from the covariance matrix alone might be too small.
We did not find any signs of a systematic bias in any of the
1 Available at http://www.mpa-garching.mpg.de/SDSS/
Figure 13. Average star formation fraction as a function of look-
back time for the 2000 galaxies in list A (solid line) and list B
(dashed line). The error bars shown are the errors bars on the
mean for each age bin. We show only the errors from list A to
avoid cluttering - the errors from list B are of similar amplitude.
Figure 14. Total stellar mass recovered for two sets of observa-
tions of 500 galaxies in the main galaxy sample. The error bars
are calculated from C(x).
analysis we carried out.
4.3 Real fits
In this section we discuss the quality of the fits to SDSS
galaxies obtained with VESPA.
As explained in Section 2, VESPA finds the best fit solution
in a χ2 sense for a given parametrisation, which is self-
c© 0000 RAS, MNRAS 000, 000–000
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14 Tojeiro et al.
Figure 15. The distribution of reduced values of χ2 for a sample
of 360 galaxies analysed by VESPA.
regulated in order to not allow an excessive number of fitting
parameters. We have shown that this self-regularization
gives a better solution in parameter space (Figure 11),
despite often not allowing the parametrization which would
yield the best fit in data space (Figure 3). However, our
aim is still to find a solution which gives a good fit to the
real spectrum. Figure 15 shows the 1-point distribution of
reduced values of reduced χ2 for 1 plate of galaxies. This
distribution peaks at around χ2reduced = 1.3, and figure 16
shows a fit to one of the galaxies with a typical value of
goodness of fit.
It is worth noting that the majority of the fits which are
most pleasing to the eye, correspond to the ones with a
high signal to noise ratio and high value of reduced χ2. One
would expect the best fits to come from the galaxies with
the best signal. However, we believe the fact that they do
not is not a limitation of the method, but a limitation of
the modelling. There are a number of reasons why VESPA
would be unable to produce very good fits to the SDSS
data. One is the adoption of a single velocity dispersion
(170 kms−1) which could easily be improved upon at the
expense of CPU time. However, the dominant reason is
likely to be lack of accuracy in stellar and dust modelling
- whereas BC03 models can and do reproduce a lot of the
observed features, it is also well known that this sucess
is limited as there are certain spectral features not yet
accurately modelled, or even modelled at all. There are
similar deficiencies in dust models and dust extinction
curves. The effect of the choice of modelling should not be
overlooked, and we refer the reader to a discussion in Section
4.5 of Panter et al. (2006)), where these issues are discussed.
4.4 VESPA and MOPED
In this Section we take the opportunity to compare the
results from VESPA and MOPED, obtained from the same
Figure 17. The recovered average star formation history for the
821 galaxies as recovered by VESPA (solid line) and MOPED
(dashed line). Both were initially normalised such that the sum
over all bins is 1, and the MOPED line was then adjusted by
11/16 to account for the different number of bins used in each
method, to facilitate direct comparison.
sample of galaxies. The VESPA solutions used here are
obtained with a one-parameter dust model, to allow a more
fair comparison between the two methods. Both methods
make similar assumptions regarding stellar models, but
MOPED uses an LMC (Gordon et al. 2003) dust extinc-
tion curve, and single screen modelling for all optical depths.
Our sample consist of two plates from the SDSS DR3
(Abazajian et al. 2005) (plates 0288 and 0444), from which
we analyse a total of 821 galaxies. We are mainly inter-
ested in comparing the results in a global sense. MOPED in
its standard configuration attempts to recover 23 parame-
ters (11 star formation fractions, 11 metallicities and 1 dust
parameter), so we might expect considerable degeneracies.
Indeed, in the past the authors of MOPED have cautioned
against using it to interpret individual galaxy spectra too
precisely. We have observed degeneracies between adjacent
bins in MOPED, but on the other hand a typical MOPED
solution has many star formation fractions which are es-
sentially zero, so the number of significant contributions is
always much less than 23.
Figure 17 shows the recovered average star formation
history for the 821 galaxies using both methods. In the
case of VESPA, solutions parametrized by low-resolution
bins had to be re-parametrized in high-resolution bins, so
that a common grid across all galaxies could be used. This
was done using the weights given by (21). The lines show a
remarkably good agreement between the two methods.
Having recovered a star formation history for each galaxy,
one can then estimate the stellar mass of a galaxy. We calcu-
lated this quantity for all galaxies using the solutions from
both methods, and with similar assumptions regarding cos-
c© 0000 RAS, MNRAS 000, 000–000
Recovering galaxy histories using VESPA 15
Figure 16. Typical fit to a galaxy from the SDSS. The dark line is the real data (arbitrary normalisation), and the lighter line (red on
the online version) is VESPA’s fit to the data.
mological parameters and fibre-size corrections. Explicitly,
we have done the following:
(i) We converted from flux to luminosity assuming the set
of cosmological parameters given by Spergel et al. (2003).
(ii) We recovered the initial mass in each age bin using
each method.
(iii) We calculated the remaining present-day mass for
each population after recycling processes. This information
is supplied by the synthetic stellar models, as a function of
age and metallicity.
(iv) We summed this across all bins to calculate the total
stellar present-day mass in the fibre aperture, M .
(v) We corrected for the aperture size by scaling up the
mass to Mstellar using the petrosian and fibre magnitudes
in the z-band, Mp(z) and Mf (z), with: Mstellar = M ×
100.4[Mp(z)−Mf (z)]
Figure 18 shows the recovered galaxy masses as recovered
from MOPED and from VESPA. We see considerable agree-
ment between VESPA and MOPED. Over 75 per cent of
galaxies have 0.5 ≤ MV ESPA/MMOPED ≤ 1.5. There is a
tail of around 10 per cent of galaxies where VESPA recov-
ers 2 to 4 times the mass recovered by MOPED. The main
reason for this difference is in the dust model used - we find
a correlation between dust extinction and the ratio of the
two mass estimates. This again reflects the fact that total
stellar mass estimates are highly sensitive to changes in the
spectrum continuum (see also section 4.2).
Our sub-sample of 821 includes galaxies with a wide range
of signal-to-noise ratios, star formation histories and even
wavelength range (mainly due to each galaxy having dif-
ferent masks applied to it, according to the quality of the
spectroscopic data). Figure 19 shows the number of recov-
ered non-zero parameters in the sample, using VESPA. As
an average, it falls below the synthetic examples studied in
c© 0000 RAS, MNRAS 000, 000–000
16 Tojeiro et al.
Figure 18. Galaxy stellar mass (in units of solar masses) as re-
covered by VESPA and MOPED for a sub-sample of 821 SDSS
galaxies. The small percentage of galaxies with significantly larger
VESPA masses have large extinction. The difference is accounted
for by the fact that MOPED and VESPA use different dust mod-
Section 3. This is not surprising, though, as each galaxy will
have an unique and somewhat random combination of char-
acteristics which will lead to a different number of param-
eters being recovered. The total combination of these sets
of characteristics would be impossible to investigate using
the empirical method described in Section 3, and here lies
the advantage of VESPA of dynamically adapting to each
individual case. Also important to note is the fact that the
wavelength coverage is normally not continuous in an SDSS
galaxy, due to masked regions. This was not modelled in
Section 3, and is likely to further reduce the number of re-
covered parameters in any given case.
Perhaps more useful is to translate this number into a
number of recovered significant stellar populations for each
galaxy. We define a significant component as a stellar pop-
ulation which contributes 5 per cent or more to the total
flux. Figure 20 shows the distribution of the number of sig-
nificant components for our sub-sample of galaxies, as re-
covered by MOPED and VESPA. It is interesting to note
that both methods recover on average a similar amount
of components, even though MOPED has no explicit self-
regularization mechanism, as VESPA clearly does.
5 CONCLUSIONS
We have developed a new method to recover star formation
and metallicity histories from integrated galactic spectra
- VESPA. Motivated by the current limitations of other
methods which aim to do the same, our goal was to develop
an algorithm which is robust on a galaxy-by-galaxy basis.
VESPA works with a dynamic parametrization of the
Figure 19. Number of non-zero parameters in solutions recov-
ered from 821 SDSS galaxies with VESPA. Please note that these
correspond to the total number of non-zero components in the
solution vector cκ and not to the number of recovered stellar
populations. For a information about the number of recovered
populations see Figure 14.
Figure 20. The distribution of the total number of recovered
stellar populations which contribute 5 per-cent or more to the
total flux of the galaxy, as recovered from MOPED (dashed line)
and VESPA (solid line).
c© 0000 RAS, MNRAS 000, 000–000
Recovering galaxy histories using VESPA 17
star formation history, and is able to adapt the number
of parameters it attempts to recover from a given galaxy
according to its spectrum. In this paper we tested VESPA
against a series of idealised synthetic situations, and against
SDSS data by comparing our results with those obtained
with the well-established code, MOPED.
Using synthetic data we found the quality and resolution
of the recovered solutions varied with factors such as type
of star formation history, noise in the data and wavelength
coverage. In the vast majority of cases, and within the
estimated errors and bin-correlations, we observed a reliable
reproduction of the input parameters. As the signal-to-noise
decreases, it becomes increasingly difficult to recover robust
solutions. Whereas our method cannot guarantee a perfect
solution, we have shown that the self-regularization we im-
posed helped obtain a cleaner solution in an overwhelming
majority of the cases studied.
On the real data analysis, we have studied possible effects
from systematics using duplicate observations of the same
set of galaxies, and have also compared VESPA’s to
MOPED’s results obtained using the same data sample. We
found that in the majority of cases our results are robust
to possible systematics effects, but that in certain cases
and particularly when calculating stellar masses, VESPA
might underestimate the mass errors. However, we found
no systematic bias in any of our tests. We have also shown
that VESPA’s results are in good agreement with those
of MOPED for the same sample of galaxies. VESPA and
MOPED are two fundamentally different approaches to
the same problem, and we found good agreement both in
a global sense by looking at the average star formation
history of the sample, and in an individual basis by looking
at the recovered stellar masses of each galaxy. VESPA
typically recovered between 2 to 5 stellar populations from
the SDSS sample.
VESPA’s ability to adapt dynamically to each galaxy and
to extract only as much information as the data warrant is
a completely new way to tackle the problem of extracting
information from galactic spectra. Our claim is that, for
the most part, VESPA’s results are robust for any given
galaxy, but our claim comes with two words of caution. The
first one concerns very noisy galaxies - in extreme cases
(SNR≈10 or less, at a resolution of 3Å), it becomes very
difficult to extract any meaningful information from the
data. This uncertainty is evident in the large error bars
and bin-correlations, and the solutions can be essentially
unconstrained even at low-resolutions. We are therefore
limited when it comes to analysing individual high-noise
galaxies, which is the case of many SDSS objects. Our
second word of caution concerns the stellar models used to
analyse real galaxies - any method can only do as well as the
models it bases itself upon. We are limited in our knowledge
and ability to reproduce realistic synthetic models of stellar
populations, and this is inevitably reflected in the solutions
we obtain by using them. On the plus side, VESPA works
with any set of synthetic models and can take advantage of
improved versions as they are developed.
VESPA is fast enough to use on large spectroscopic sam-
ples (a typical SDSS galaxy takes 1 minute on an average
workstation), and we are in the process of analysing SDSS’s
Data Release 5 (DR5), which consists of roughly half a mil-
lion galaxies. Our first aim is to publish and exploit a cata-
logue of robust star formation histories, which we hope will
be a valuable resource to help constrain models of galaxy
formation and evolution.
6 ACKNOWLEDGMENTS
We are grateful to the anonymous referee for a very
thoughtful report which led to material improvements in
the paper.
RT is funded by the Fundação para a Ciência e a Tecnologia
under the reference PRAXIS SFRH/BD/16973/04. RJ’s
research is supported by the NSF through grant PIRE-
0507768 and AST-0408698 to the Atacama Cosmology
Telescope.
Funding for the SDSS and SDSS-II has been provided by the
Alfred P. Sloan Foundation, the Participating Institutions,
the National Science Foundation, the U.S. Department of
Energy, the National Aeronautics and Space Administra-
tion, the Japanese Monbukagakusho, the Max Planck Soci-
ety, and the Higher Education Funding Council for England.
The SDSS Web Site is http://www.sdss.org/. The SDSS is
managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions
are the American Museum of Natural History, Astrophys-
ical Institute Potsdam, University of Basel, University of
Cambridge, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Ad-
vanced Study, the Japan Participation Group, Johns Hop-
kins University, the Joint Institute for Nuclear Astrophysics,
the Kavli Institute for Particle Astrophysics and Cosmol-
ogy, the Korean Scientist Group, the Chinese Academy
of Sciences (LAMOST), Los Alamos National Laboratory,
the Max-Planck-Institute for Astronomy (MPIA), the Max-
Planck-Institute for Astrophysics (MPA), New Mexico State
University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United
States Naval Observatory, and the University of Washing-
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	Introduction
	Method
	The problem
	The solution
	Choosing a galaxy parametrization
	The models
	Errors
	Timings
	Tests on Simulated Data
	Star formation histories
	Wavelength range
	Noise
	Dust
	Results
	Handling SDSS data
	Duplicate galaxies
	Real fits
	VESPA and MOPED
	Conclusions
	Acknowledgments
