Robust signal dimension estimation via SURE

Abstract

The estimation of signal dimension under heavy-tailed latent variable models is studied. As a primary contribution, robust extensions of an earlier estimator based on Gaussian Stein’s unbiased risk estimation are proposed. These novel extensions are based on the framework of elliptical distributions and robust scatter matrices. Extensive simulation studies are conducted in order to compare the novel methods with several well-known competitors in both estimation accuracy and computational speed. The novel methods are applied to a financial asset return data set.

Full text

Statistical Papers (2024) 65:3007–3038
https://doi.org/10.1007/s00362-023-01512-2
REGULAR ARTICLE
Robust signal dimension estimation via SURE
Joni Virta1·Niko Lietzén1·Henri Nyberg1
Received: 31 October 2022 / Revised: 11 July 2023 / Published online: 9 December 2023
© The Author(s) 2023
Abstract
The estimation of signal dimension under heavy-tailed latent variable models is stud-
ied. As a primary contribution, robust extensions of an earlier estimator based on
Gaussian Stein’s unbiased risk estimation are proposed. These novel extensions are
based on the framework of elliptical distributions and robustscatter matrices. Extensive
simulation studies are conducted in order to compare the novel methods with several
well-known competitors in both estimation accuracy and computational speed. The
novel methods are applied to a financial asset return data set.
Keywords Elliptical distribution ·Principal component analysis ·Risk estimate ·
Scatter matrix
1 Introduction
1.1 Premise
Modern data sets exhibit increasingly large sizes and often the first step in analysing
them is the implementation of suitable dimension reduction procedures, for example,
principal component analysis (PCA). A fundamental question pertaining to any such
reduction is the choosing of the correct amount of latent components to retain: under-
estimating their amount leads to losing important information, whereas picking too
many components inevitably leads in the later stages of the analysis to modelling of
noise and increased computational burden that could have been avoided with a more
careful choice of the dimension.
BJoni Virta
joni.virta@utu.fi
Niko Lietzén
niko.lietzen@utu.fi
Henri Nyberg
henri.nyberg@utu.fi
1Department of Mathematics and Statistics, University of Turku, Turku, Finland
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3008 J. Virta et al.
Besides being large in size and volume, in many applications, such as economics
and finance, it is common for the data sets to display heavier tails than possessed by
the Gaussian distribution. Consider, for example, stock market returns which are often
modelled with distributions having infinite variance (Borak et al. 2011). In such cases,
the estimation of the latent dimension in dimension reduction is further complicated as
one cannot rely any more on the covariance matrix, on whose eigenvalues many of the
standard dimension estimation methods rely (a review is given later in this section).
With the above scenario in our mind, the objective of the current paper is to study
and develop dimension estimation in the context of multivariate data sets exhibiting
arbitrarily heavy tails. We will base our theoretical framework on the concepts of
ellipitical family and Stein’s unbiased risk estimation, elaborated in more detail next.
1.2 Elliptical latent variable model
We assume that our data x1,...,xnis an i.i.d. sample of p-variate vectors generated
from the elliptical latent variable model
xi=μ+VDz
i,(1)
where μ∈Rp,V∈Rp×pis an orthogonal matrix, ziobeys a spherical distribution
(Fang 2018), i.e., zi∼Ozifor any orthogonal matrix O∈Rp×p, and D∈Rp×pis
a diagonal matrix with the diagonal elements σ1≥···≥σd>σ =···=σwith
σ>0. Conceptually, the model says that the observed xiare obtained by mixing the
principal components Dziwith the matrix Vand by applying a location shift μ.The
final p−dprincipal components in Dziare orthogonally invariant, meaning that they
are essentially “structureless” and, as is typical, we view them as noise. Thus the main
objective in the model (1) is to estimate the latent signals, i.e., the first delements of
Dzialong with their number d. Note that while the scales of Dand ziare confounded
in the model (1), this does not matter as long as one considers only the compound
quantity Dzi.
The model (1) gets still more intuitive form in the special case where ziobeys the
standard Gaussian distribution, the most well-known example of an elliptical distri-
bution. In this case model (1) reduces to the probabilistic PCA model (Tipping and
Bishop 1999)
xi=μ+V0yi+εi,(2)
where the loading matrix V0∈Rp×dcontains the dfirst columns of Vas its columns,
yi∼Nd{0,diag(σ 2
1−σ2,...,σ2
d−σ2)},
and εi∼Np(0,σ2Ip)is independent of yi. Model (2) reveals that, in the Gaussian
case, the d-dimensional signal residing in the column space of V0is explicitly corrupted
with the noise vectors εi, and the model can be seen as a factor model (additive
contamination of latent signals with noise). However, this intuitive representation does
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Robust signal dimension estimation via SURE 3009
not apply to any other elliptical distribution (that is, (1) cannot in general be written
in the form (2) for elliptical zi) and, hence, (1) is usually viewed in the literature as
an elliptical PCA model or as an elliptical subsphericity model, rather than as a factor
model. Naturally, it would also be possible to consider an elliptical variant of the factor
model (2) where both yiand εiare assumed to be non-Gaussian, see, for example,
Pison et al. (2003).
The standard method of extracting the factors zi(or the corresponding subspace) in
the Gaussian model (2) is through PCA. Namely, one computes the first deigenvectors
of the covariance matrix of xiand projects the observations onto their span. However,
the success of this procedure hinges crucially on the knowledge of the latent dimension
d<p, usually unknown to us in practice. As the misspecification of the dimension has
ill consequences in practice (either missing part of the signal or riddling our estimates
of the latent factors with noise), an important part of solving the factor model (2)is
the accurate estimation of d. We next review a particular method for accomplishing
this under the model (2), on which our subsequent developments are also based.
1.3 Stein’s unbiased risk estimate
In Ulfarsson and Solo (2015), the latent dimension din Gaussian PCA was estimated
through the application of the Stein’s unbiased risk estimate (SURE), a general tech-
nique for determining the optimal values of tuning parameters (of which the latent
dimension dis an example) of estimation procedures.
Ignoring the model (2) for a moment, we briefly recall the basic idea behind the
SURE: Assume that we observe the i.i.d. univariate w1,...,w
n, generated as wi=
ai+ei, where ai∈Rare constant and the errors satisfy ei∼N(0,τ2)for some
τ2>0. Assume further that for each aiwe have a corresponding estimator ˆai(w),
viewed as a (differentiable) function of the data w=(w1,...,w
n). Then, the SURE
Rcorresponding to the estimators aiis defined as
R=1
n
n

i=1{wi−ˆai(w)}2+2τ2
n
n

i=1
∂
∂wiˆai(w) −τ2.(3)
In their celebrated paper (Stein 1981), Stein proved that E(R)=
(1/n)n
i=1E{ˆai(w) −ai}2, showing that Ris an unbiased estimator of the risk
associated with the estimators ˆai, in the complete absence of the true means ai.A
multivariate version of SURE was in Ulfarsson and Solo (2015) adapted to the PCA
model (2) (conditional on the yi) to estimate the expected risk associated with any
particular choice of the latent dimension d. The estimate of dis then chosen to be the
dimensionality for which the risk is minimized. See also the earlier work (Ulfarsson
and Solo 2008).
The obtained estimator was in Ulfarsson and Solo (2015) shown to be highly suc-
cessful under the Gaussian model (2). However, it is clear that the estimator cannot
obtain the same level of efficiency under the wider elliptical model (1). There are
two reasons for this: (i) the SURE-criterion was in Ulfarsson and Solo (2015) derived
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3010 J. Virta et al.
strictly under the Gaussian assumption and, more importantly, (ii) many standard ellip-
tical distributions (e.g., multivariate Cauchy distribution) do not have enough finite
moments so that the covariance matrix, on which the SURE-criterion is based, would
even exist. Hence, the estimator of Ulfarsson and Solo (2015) is not even necessarily
well-defined under the elliptical model (1) (on the population level)!
1.4 The scope of the current work
The primary objective of the current work is to provide a workaround for the previ-
ous issue by deriving a robust version of the SURE-criterion that allows for effective
dimension estimation under the elliptical model (1) in the presence of heavy-tailed dis-
tributions. As described earlier, such procedures are highly called for in applications
such as finance, where assumptions on finite moments are usually deemed unreason-
able. Our robust extension of the SURE-criterion is carried out via a plug-in strategy
where the covariance matrix in the Gaussian SURE-criterion is replaced with a suitable
scatter matrix. Especially popular in the community of robust statistics, scatter matri-
ces are a class of statistical functionals that measure the dispersion/scatter/variation
in multivariate data while (usually) being far more robust to the impact of heavy tails
and outliers than the covariance matrix, see Sect.3for their definition and several
examples. We consider three different plug-in estimators, depending in which form
of the Gaussian SURE-criterion the scatter matrix is plugged in. The first two options
lead to analytically simple estimators that depend on the data only through the eigen-
values of the used scatter matrix, much like the classical estimators of dimension (see
below). Whereas, the third strategy is more elaborate and involves computing particular
derivatives of the scatter functional (and the companion location functional).
As our secondary objective, we conduct an extensive simulation study where the
proposed methods are compared to each other and to several (families of) competing
estimators from the literature. These competitors include: (i) The classical estimator
based on successive asymptotic hypothesis testing for the equality of the final eigen-
values of a chosen scatter matrix (testing for subsphericity) (Nordhausen et al. 2021)
and its high-dimensional version (Schott 2006). (ii) Variation of the previous estimator
where the null distributions are bootstrapped instead of relying on asymptotic approx-
imations (Nordhausen et al. 2021). (iii) The general-purpose procedure for inferring
the rank of a matrix from its sample estimate known as the ladle, which we apply to
select scatter matrices, see Luo and Li (2016). (iv) The SURE-estimator of Ulfarsson
and Solo (2015) which can be seen as the non-robust version of our proposed estima-
tor. (v) The information criteria-type estimators proposed in Wax and Kailath (1985).
(vi) The Bayesian estimator in Minka (2000) that is based on choosing the dimension
which yields maximal approximate probability of observing the current data set. We
are not aware of comparisons of similar magnitude being conducted earlier in the
literature.
Note that further examples of dimension estimators for PCA naturally exist in the
literature. For example, Gai et al. (2008), Gai and Stevenson (2010) use a Bayesian
approach under a heavy-tailed model, Deng and Craiu (2023) rely on penalized like-
lihood with aggregation over the tuning parameter values and Zhao et al. (1986)use
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Robust signal dimension estimation via SURE 3011
information theoretic criteria in a complex-valued version of the Gaussian factor model
(2) However, in Sect. 5, we have limited our comparisons to the list of methods given
in the preceding paragraph, in particular due to their flexibility in choosing the used
scatter matrix, see Sect. 5for further details.
To summarize the results of our simulation study (given in Sect.5), they reveal that
the SURE-based robust methodology for the determination of the latent dimension is:
(i) Accurate, achieving good estimation results in various data scenarios. (ii) Flexible,
that is, it allows the free selection of the used robust scatter matrix. This is in strict
contrast to its closest competitor, the asymptotic hypothesis test mentioned above,
which is (for theoretical reasons) “locked” to operate with a specific, slow-to-compute
scatter matrix. (3.) Fast , requiring no bootstrap replicates or any kind of resampling.
1.5 Organization of the manuscript
The manuscript is organized as follows. In Sect. 2we recall the Gaussian SURE-
criterion of Ulfarsson and Solo (2015) for estimating the latent dimension. In Sects. 3
and 4we propose three different robust extensions of the criterion through the use
of different pairs of location and scatter functionals. Sections 5and 6contain the
simulation study and an empirical (financial) example on asset returns, respectively.
In Sect. 7we conclude with some future research ideas. The proofs of all technical
results are collected in Appendix A.
2 SURE criterion for Gaussian PCA
In this section, we recall how the SURE-criterion can be used to estimate the latent
dimension dunder the Gaussian model (2). Our derivation of the criterion differs from
the original version (Ulfarsson and Solo 2015) in that we employ empirical centering
of the data, whereas Ulfarsson and Solo (2015) did not. We made this change to the
method as it is unreasonable to assume that the true location of the data is known in
practice.
Due to the empirical centering, we assume, without loss of generality, that μ=0
throughout the following. As in Ulfarsson and Solo (2015), we use Stein’s Lemma to
construct an unbiased estimator of the risk associated with estimating the signals V0yi
by their reconstructions ˆxibased on the first kprincipal components. Assuming that
k=1,..., pis fixed from now on and letting Uk∈Rp×kdenote a matrix of (any) first
korthogonal eigenvectors of the covariance matrix S0:= (1/n)n
i=1(xi−¯x)(xi−¯x),
the reconstructions can be written as ˆxi≡ˆxi(k)=t0+Pk(xi−t0)where Pk:= UkU
k
is the orthogonal projection onto the space spanned by the first keigenvectors of S0and
t0:= (1/n)n
i=1xiis the mean vector. For convenience, we replicate an intermediate
result towards the final Gaussian SURE-criterion below as Lemma 1. In the lemma the
reconstructions ˆxiare treated as functions of the original data x1,...,xnand the result
implicitly assumes the former to be differentiable in the latter, sufficient conditions
for which will be discussed later in Sect. 3. In Lemma 1, and throughout the paper,
·denotes the Euclidean norm.
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3012 J. Virta et al.
Lemma 1 Under model (2), the quantity
R1k:= tr{(Ip−Pk)S0}+2σ2
n
n

i=1
p

j=1
∂
∂xij ˆxij −pσ2
is an unbiased estimator of the risk (1/n)n
i=1Eˆxi−V0yi2.
The two k-dependent terms of R1kin Lemma 1have natural interpretations: The
term tr{(Ip−Pk)S0}measures the total variation of the data in directions orthog-
onal to the first keigenvectors and takes large values when the used number of
eigenvectors is insufficient to capture the full d-variate latent signal. The quantity
(1/n)n
i=1p
j=1(∂/∂ xij)ˆxij measures the average influence an observation has on
their own reconstruction and is often interpreted as the generalized degrees of freedom
of the model, where large values indicate overfitting to the data set, see Tibshirani and
Taylor (2012) (in the extreme case with d=pwe actually have ˆxij =xij). Thus,
R1kcan be seen to be similar in form to Akaike’s information criterion (AIC) (and
other related information criteria), whose two terms also measure model fit and model
complexity, respectively.
To apply the criterion R1kin practice, we require an expression for the partial
derivatives in Lemma 1. As is shown later in the context of Lemma 3in Sect. 4,the
partial derivatives exist under the assumption that the eigenvalues of S0are simple
(which holds almost surely under the model (2)), and have the forms shown in Lemma
2below. Its proof is omitted as the result is a direct consequence of Lemma 1and
Lemmas 3,4in Sect. 4. See Ulfarsson and Solo (2008,2015) for similar results.
Lemma 2 Under model (2), the quantity
R2k:= tr{(Ip−Pk)S0}+2σ2
n
k

j=1
p

=k+1
sj+s
sj−s+σ2
n{2p+2(n−1)k−np},
where s1>···>spare the eigenvalues of S0, is an unbiased estimator of the risk
(1/n)n
i=1Eˆxi−V0yi2.
To apply the criterion R2kin practice, an estimator for the unknown error variance
σ2is needed and several feasible alternatives exist. For example, Luo and Li (2021)
used, in a similar context, the median of the smallest p/2eigenvalues of S0.The
resulting estimator is accurate but makes the implicit assumption that d≤p/2.To
avoid such difficult-to-verify conditions, we prefer to instead use the final eigenvalue
spof S0as the estimator of the noise variance, imposing minimal assumptions on the
latent dimensionality (i.e., that d<p). Naturally, the price to pay is that spsuffers
from underestimation in finite samples. Note that, to combat the underestimation,
Ulfarsson and Solo (2008) proposed an alternative estimator of σ2based on the limiting
spectral distribution of the covariance matrix under high-dimensional Gaussian data.
Mimicking this strategy is not viable in our scenario as any results on the limiting
spectral distributions of the scatter matrices used in Sect.3are still scarce in the
literature. See also Sect. 7for further discussion of this choice.
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Robust signal dimension estimation via SURE 3013
Plugging in the estimator spand observing that tr{(Ip−Pk)S0}=p
=k+1snow
leads to two different sample forms for the SURE criterion for Gaussian PCA:
ˆ
R1k:=
p

=k+1
s+2sp
n
n

i=1
p

j=1
∂
∂xij ˆxij −psp,
ˆ
R2k:=
p

=k+1
s+2sp
n
k

j=1
p

=k+1
sj+s
sj−s+sp
n{2p+2(n−1)k−np}.
(4)
The “hat” notation for ˆ
R1k,ˆ
R2ksignifies the fact that they have been obtained from
R1k,R2kby replacing the unknown σ2with its estimator sp. In the following sections
we obtain outlier-resistant alternatives to both ˆ
R1kand ˆ
R2kvia plugging in robust
measures of location and scatter in place of the mean vector and covariance matrix in
(4). In addition, we will also consider an “asymptotic” version of the criterion ˆ
R2k,
ˆ
R3k:=
p

=k+1
s+sp(2k−p), (5)
where the terms of the order op(1)(in the asymptotic regime where n→∞)have
been removed. Note that even though we might have sj−s→p0 for some indices
j,, the limiting distribution of √n(sj−s)for such indices is absolutely continuous
(with respect to the Lebesgue measure) (Anderson 1963), meaning that the impact of
the double sum in ˆ
R2kcan be expected to be negligible for large n.
3 Robust plug-in SURE criteria
Plug-in-techniques are a typical way to create outlier-resistant versions of standard
multivariate methods in the community of robust statistics, see, for example, Croux
and Haesbroeck (2000), Nordhausen and Tyler (2015), Fan et al. (2021). In this spirit,
we replace the mean vector t0and the covariance matrix S0in the SURE criteria (4),
(5) with a pair (t,S)of location and scatter functionals (Oja 2010), the definitions
of which we recall next. Letting Fbe an arbitrary p-variate distribution, a location
functional (location vector) tis a map F→ t(F)∈Rpsuch that, for any invertible
A∈Rp×pand b∈Rp,wehavet(FA,b)=At(F)+bwhere FA,bis the distribution of
the random vector Ax +band x∼F. Similarly, a scatter functional (scatter matrix) S
is a map taking values in the space of positive semi-definite matrices and obeying, for
any invertible A∈Rp×pand b∈Rp, the transformation rule S(FA,b)=AS(F)A.
These transformation properties are typically referred to as affine equivariance.
Location and scatter functionals mimic the properties of the mean vector and the
covariance matrix and typically measure some aspects of the center and spread of a
distribution, respectively. In particular, if Fis the elliptical model (1), then t(F)=μ
and S(F)=τS,FVD
2Vfor all location and scatter functionals (t,S)for which t(F)
and S(F)exist, where the scalar τS,F>0 depends on both the exact distribution
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3014 J. Virta et al.
of the spherical ziand on the used scatter functional, see [Oja (2010),Theorem 3.1].
Hence, all choices of (t,S)estimate, up to scale, the same quantities under the elliptical
model, implying that the replacing of the mean vector and the covariance matrix in
SURE with the pair (t,S)is warranted (at least in the Gaussian special case (2)ofthe
elliptical model, under which the SURE criteria in Sect. 2were derived). Note that this
equivalence of different (t,S)-pairs under elliptical data does not necessarily mean
that the sample dimension estimates given by different choices of (t,S)should always
be equal. Namely, the equivalence indeed holds under the population level model (1),
but in practical situations the accuracy of the estimates is greatly influenced by the
finite-sample properties (in particular, robustness properties) of the used location and
scatter functionals. This is clearly evident in the simulation results of Sect.5.
Examples of popular location and scatter functionals are given later in this section
and we assume, for now, that we have selected some robust location-scatter pair (t,S).
Outlier-resistant versions of the forms ˆ
R2kand ˆ
R3kof the Gaussian SURE criterion
in (4) and (5) are then straightforwardly obtained. Namely, we simply replace the
eigenvalues sjof the covariance matrix S0with the eigenvalues of the scatter functional
Sin the definitions. Note that while the location functional tdoes not play an explicit
role in this construction, it is usually a part of the definition of S, see for example the
spatial median and the spatial sign covariance matrix later in this section.
As an alternative to the above, rather simplistic plug-in estimators,we consider also
a more elaborate extension based on the form ˆ
R1kof SURE in (4) where, in addition
to S0, we also replace the partial derivatives (∂/∂ xij)ˆxij with their counterparts based
on the robust pair (t,S). That is, the robust version of ˆ
R1kuses the reconstruction
estimates ˆxi=t+Pk(xi−t)where the centering is done with the robust location
functional t(instead of t0) and the projection matrix Pkis now taken to be onto the
space spanned by the first keigenvectors of the robust scatter functional S(instead of
S0). Due to its more technical nature in comparison to the other two criteria, we have
postponed the discussion of the extension of ˆ
R1kto Sect. 4.
Finally, regardless of which of the three criteria ˆ
R1k,ˆ
R2kand ˆ
R3kone uses, the
corresponding estimate ˆ
dof the latent dimension dis obtained as the minimizing
index,
ˆ
d=argmink=0,..., p−1ˆ
Rjk.
We next recall several popular options for the location-scatter pair (t,S).
3.1 Mean vector and covariance matrix
The most typical choice for the pair (t,S)is the mean vector and the covariance matrix,
i.e.,
t(F)=EF(x), S(F)=EF[{x−t(F)}{x−t(F)}],(6)
where EFmeans that the expectation is taken under the assumption that x∼F.This
choice simply leads to the Gaussian SURE-criterion discussed in Sect. 2. As discussed
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Robust signal dimension estimation via SURE 3015
before, this option is, despite often being the optimal choice under the assumption of
normality, also highly non-tolerant against outliers and heavy tails.
3.2 Spatial median and spatial sign covariance matrix
The spatial median t(F)of a distribution Fis defined as any minimizer of the convex
function
t→ EF{x−t−x},
over t∈Rp. The spatial median is one of the oldest and most studied robust mea-
sures of multivariate location, see, Haldane (1948), Brown (1983), and reverts to the
univariate concept of median when p=1. It can be shown to exist for any F(in
particular, no moment conditions are required) and it is unique as soon as Fis not
concentrated on a line in Rp(Milasevic and Ducharme 1987) which is guaranteed, in
particular, almost surely when Fis absolutely continuous.
The standard scatter functional counterpart for the spatial median is the spatial sign
covariance matrix (SSCM), defined as,
S(F)=EFu(x−t(F))u(x−t(F)),
where t(F)is the spatial median of F, which is assumed to be unique, and the sign
function u:Rp→Rpis defined as u(x)=x/xfor x= 0 and u(0)=0. Like
its location counterpart, also the SSCM has been extensively studied in the literature,
see, for instance, Marden (1999), Visuri et al. (2000), Dürre et al. (2016), Bernard and
Verdebout (2021).
The defining feature of SSCM is that it depends on the data only through the
“signs” u(x−t(F)), giving equal weight to points in a given direction regardless of
their norm (which, in turn, is what makes the SSCM robust to outliers). Especially
for high-dimensional data, this loss of information introduced by the discarding of the
observation magnitudes is relatively small as it represents losing only a single degree
of freedom in the p-dimensional space (whereas the sign contains the remaining p−1
degrees of freedom).
We also note that the spatial median and the spatial sign covariance matrix are,
strictly speaking, not a pair of location and scatter functionals in the usual sense as
they satisfy the equivariance properties listed in Sect.2only when the matrix Ais
orthogonal. However, this is not an issue in our scenario for the following reasons: (i)
The spatial median is a consistent estimator of the location parameter in the elliptical
model (1) under minor regularity conditions, see, e.g., Magyar and Tyler (2011). (ii)
Under the elliptical model (1), two eigenvalues sj,sof the SSCM are equal if and
only if the corresponding elements σj,σ
of the matrix Dare equal, see Dürre et al.
(2016). Hence, the eigenvalues of the spatial sign covariance matrix contain the same
(qualitative) information about the latent signal dimension as those of any “proper”,
affine equivariant scatter functional. However, the spatial sign covariance matrix is
also known to non-linearly compress the range of the eigenvalues (Vogel and Fried
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3016 J. Virta et al.
2015), making it more difficult to distinguish between the signal and the noise and,
thus, we next consider an alternative to it. This alternative is known as Tyler’s shape
matrix and is often seen as the affine equivariant version of the SSCM.
3.3 Tyler’s shape matrix
Tyler’s shape matrix (Tyler 1987) is one of the earliest proposed and most studied
scatter functionals, see, e.g., Dümbgen and Tyler (2005), Wiesel (2012). Using it
requires a location functional tand, in the following, we take this to be the spatial
median, as is common in the literature. Tyler’s shape matrix S(F)is defined as any S
with det(S)=1 and satisfying the following fixed-point equation,
EFuS−1/2{x−t(F)}uS−1/2{x−t(F)}=1
pIp.(7)
A unique solution S(F)is obtained as soon as Fdoes not concentrate too heavily on
a subspace in Rp, see Dümbgen and Tyler (2005) for the exact conditions. Inspection
of the Eq. (7) also reveals that any solution Sto it is defined only up to its scale and,
to obtain a unique representative, a popular choice is indeed to use the determinant
condition det(S)=1 to fix the scale of solution, see Paindaveine (2008). Consequently,
S(F)does not describe the full scatter of Fbut only its shape (scale-standardized
scatter). However, this is sufficient for our purposes as scaling preserves the ordering
of the eigenvalues of S(F)and, hence, their division into signal and noise. Note that this
also means that Tyler’s shape matrix satisfies the affineequivariance property discussed
in the beginning of Sect. 3only up to scale, S(FA,b)={det(A)}−2/pAS(F)A.
The computation of S(F)can be shown to correspond to a geodesically convex
minimization problem (Wiesel 2012), meaning that an efficient algorithm for its esti-
mation in practice is straightforwardly constructed. In our simulations we have used
the R-package ICSNP (Nordhausen et al. 2018) for this purpose.
3.4 Hettmansperger-Randles estimator
As our final choice for the location scatter pair (t,S), we consider the so-called
Hettmansperger–Randles (H–R) estimator, which was originally introduced in the con-
text of robust location estimation (and the associated shape functional was obtained as
a “by-product” of the location estimation) (Hettmansperger and Randles 2002). The
H–R pair (t(F), S(F)) is defined as any (t,S), with det(S)=1 and satisfying the
following pair of fixed-point equations,
EFu(S−1/2(x−t))=0
EFu(S−1/2(x−t))u(S−1/2(x−t))=1
pIp.
(8)
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Robust signal dimension estimation via SURE 3017
Observing that the LHS of the first Eq. in (8) is, disregarding the matrix S−1/2,the
gradient of the objective function (6) of the spatial median, we see that the H–R
pair (t(F), S(F)) can be interpreted as simultaneously determined spatial median and
Tyler’s shape matrix. This concurrent estimation of location and scatter (or, rather,
shape as again any solution Sto the fixed-point equations is unique at most up to
scale) then makes the resulting estimator affine equivariant (up to scale in case of S).
Despite its attractiveness, the theoretical properties of the H–R estimator have
garnered less attention in the literature when compared to its previously introduced
alternatives. In particular, we are not aware of any studies investigating conditions that
would guarantee the uniqueness of the solution (t,S).
To summarize, the three robust alternatives to the mean-covariance pair introduced
in this section can all be seen to estimate analogous quantities, while at the same
time forming a sort of “hierarchy” with respect to their equivariance properties: (i)
the spatial median and SSCM satisfy affine equivariance only for orthogonal A, (ii)
replacing SSCM with Tyler’s shape matrix yields the full affine equivariance prop-
erty for the scatter (shape) functional and, (iii) both the location and scatter (shape)
components of the H–R estimate are affine equivariant. As affine equivariance is the
natural transformation property for a scatter functional to have in the presence of the
elliptical model (1), we thus expect that the previous ordering applies also to the cor-
responding SURE-procedures’ comparative performances in practice. This claim will
be investigated through simulations in Sect.5.
4 Robust extension of ˆ
R1k
In this section, we explore extending the SURE-criterion ˆ
R1kin (4) to accommodate an
arbitrary location-scatter pair. The theoretical cost of such an extension is considerably
larger than for ˆ
R2kand ˆ
R3kas, instead of simply plugging in eigenvalues, it involves
computing the partial derivatives (∂/∂ xij)ˆxij.
In the sequel, let the observed sample x1,...,xnof points in Rpbe fixed and
denote its empirical distribution by Fn. Moreover, for ε>0weletFn,i,j,ε denote the
empirical distribution of the perturbed sample x1,...,xi+εej,...,xnwhere ejis
the jth vector in the canonical basis of Rp. For the extension ˆ
R1kto be well-defined
in the first place, tand Sare naturally required to be differentiable in a suitable sense,
and the next assumption formalizes this requirement.
Assumption 1 For any i=1,...,nand j=1,..., p, there exists hij ∈Rpand a
symmetric Hij ∈Rp×psatisfying
1
ε{t(Fn,i,j,ε)−t(Fn)}→hij and 1
ε{S(Fn,i,j,ε)−S(Fn)}→Hij,
as ε→0.
In order for also the projection matrix Pk(onto the span of the first keigenvectors of
S(Fn)) to be differentiable in the previous sense for all k=1,..., p, all eigenvalues
of the matrix S(Fn)must be simple. This condition, formalized in Assumption 2
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3018 J. Virta et al.
below, is rather mild and, in particular, holds almost surely for both the covariance
matrix and the SSCM if the points x1,...,xnare drawn from an absolutely continuous
distribution.
Assumption 2 The eigenvalues of S(Fn)are distinct.
Under the previous two assumptions, the partial derivatives (∂/∂ xij)ˆxij exist and
their sum over jhas the analytical form given in the next lemma.
Lemma 3 Under Assumptions 1and 2, we have
p

j=1
∂
∂xij ˆxij =k+
p

j=1
tr{(Ip−Pk)hijej}+
p

j=1
ejAij{xi−t(Fn)},
where
Aij :=
k

=1
p

m=k+1
1
s−sm
(THijTm+TmHijT),
Tis the orthogonal projection onto the space spanned by the th eigenvector of S(Fn),
and sis the corresponding eigenvalue.
Lemma 3essentially says that, as soon as one obtains expressions for the quantities
hij and Hij in Assumption 1(and Assumption 2holds) for some particular location
scatter pair (t,S), these can be plugged in to Lemma 3to construct a version of the
SURE-criterion ˆ
R1kthat is based on (t,S). In Lemma 4below we have provided, for
completeness, these expressions for the standard mean-covariance pair. The resulting
SURE-criterion ˆ
R1kis, naturally, the Gaussian SURE as described in Sect.2.
Lemma 4 The mean vector and the covariance matrix satisfy Assumption 1with
hij =1
nej,Hij =1
nej{xi−t(Fn)}+1
n{xi−t(Fn)}ej.
Despite not offering us anything new, Lemma 4also serves in its simplicity as
a contrast to our next result, detailing the forms of hij and Hij for the spatial
median/SSCM-pair. What makes deriving these quantities more complicated, com-
pared to the mean-covariance pair, is the fact that no analytical expression is available
for the spatial median (instead, it is obtained as a minimizer of the objective function
described in Sect. 3).
Lemma 5 Assume (i) that the points x1,...,xnare not concentrated on a line in Rp
and, (ii) that t(Fn)= xi, for all i =1,...,n. Then the spatial median and the spatial
sign covariance matrix satisfy Assumption 1with
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Robust signal dimension estimation via SURE 3019
hij =G−1Aiej,
Hij =1
n⎧
⎨
⎩
Aiej
y
i
yi+yi
yiejAi−
n

=1
AG−1Aiej
y

y−
n

=1
y
yejAiG−1A⎫
⎬
⎭
,
where Ai:= wi(Ip−yiy
i/yi2),wi:= yi−1,y
i:= xi−t(Fn)and G := n
i=1Ai.
Plugging hij and Hij from Lemma 5to the derivatives in Lemma 3and consequently
to ˆ
R1kin (4) now gives us yet another robust criterion for determining the signal
dimension. We note that while the additional assumption (ii) imposed in Lemma 5
seems to be difficult to analyze theoretically, its validity is nevertheless simply checked
in practice (and the assumption (i) is satisfied, in particular, almost surely when Fis
absolutely continuous).
Mimicking the proof of Lemma 5it would next be possible to derive equivalent
results also for our two remaining location-scatter pairs. However, we have decided
not to do so and the reasons for this are two-fold: (i) Some preliminary computations
(not shown here) show that these computations lead, as with the spatial median/SSCM-
pair in Lemma 5, to analytically cumbersome expressions for hij and Hij, from which
no real insight can be gained. (ii) Due to the complexity of the resulting expression
(and the large number of nested summations involved), the practical usefulness of the
extensions is questionable. Indeed, as our timing comparisons in Sect. 5.4 demonstrate,
the version of ˆ
R1kobtained based on Lemma 5is several orders of magnitude inferior
to ˆ
R2kand ˆ
R3kin computational speed, while at the same time offering no or only
minuscule gains in accuracy. Some preliminary exploration reveals that this issue is
still further magnified for Tyler’s shape matrix. Hence, while these extensions would
be technically possible to derive, we did not see any real practical value in doing so.
5 Simulations
In order to study the finite-sample properties of our proposed robust extensions of
SURE, we conduct an array of simulation studies. As competing methods we have
used the following set of well-established estimators from the literature, see Sect.1.4
for further details. Naturally, also other choices would be available, but this particular
set was chosen as (i) it contains representatives of estimators based both on asymptotic
results and on computationally intensive ideas, and (ii) all methods in the set allow
choosing the used scatter matrix (at least to some extent), letting us separately compare
the different methodologies and the different levels of robustness.
(i) The classical estimator based on an asymptotic test of subsphericity (Schott 2006;
Nordhausen et al. 2021). The R-package ICtest (Nordhausen et al. 2021)
includes two implementations of it, one based on the covariance matrix and one
based on the H–R estimator, and we include both of them in the comparison. Addi-
tionally, we include the high-dimensional variant of the test (Schott 2006) which
is based on the covariance matrix.
(ii) The same estimator as (i) but with the null distribution of the test estimated through
bootstrap. This estimator can be based on any of the four scatter matrices described
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3020 J. Virta et al.
in Sect. 3and we thus include all of them in the comparison. We used 200 bootstrap
samples throughout the study, the default value in the implementation in ICtest
(Nordhausen et al. 2021).
(iii) The ladle estimator of Luo and Li (2016) which, too, can be based on any of the
four scatter matrices. The estimator is based on resampling, for which we used the
default value 200 in the implementation in ICtest (Nordhausen et al. 2021).
(iv) Our centered version of the SURE-estimator of Ulfarsson and Solo (2015).
(v) The AIC and MDL criteria from Wax and Kailath (1985), see their Eqs. (16)
and (17). We use both criteria separately with the covariance matrix and the H–
R estimator. The derivation in Wax and Kailath (1985) does not cover the latter
case (H–R), meaning that it can be considered as an “experimental” robust plug-in
version of their method.
(vi) The Bayesian approach in Minka (2000), see their Eq. (30). As in item (v) above,
we apply the method both with the covariance matrix and the H–R estimator,
making also this estimator experimental in nature.
The above six categories of estimators are denoted in the following as Asymp,
Boot, Ladle, SURE, Wax and Minka, respectively, with the used scatter matrix given
in parenthesis. E.g., Asymp(HR) denotes the asymptotic test based method using the
H–R estimator. Additionally, we denote the high-dimensional variant of Asymp by
Schott. In addition, we distinguish three different versions of the SURE-estimator,
SURE1, SURE2 and SURE3, referring to using the objective functions ˆ
R1k,ˆ
R2kand
ˆ
R3k, respectively. We thus have a total of 26 methods to compare, and these have been
summarized in Table 1. The final four columns of the table are related to the timing
study in Sect. 5.4. The R-implementations of the methods are available at https://users.
utu.fi/jomivi/software/.
5.1 Tail thickness
In the first simulation study, we explore how the methods perform under varying levels
of heavy-tailedness. As a setting for this, we consider multivariate t-distributions with
the degrees of freedom equal to ν=1,3,5,...,25. Thus, the heaviest tails are
obtained in the case ν=1, corresponding to the multivariate Cauchy distribution. The
simulation is repeated 100 times for every degree of freedom, and for every repetition a
random sample consisting of n=100 observations is generated. In each case, we take
the latent dimension to be d=6, whereas as the total dimensionality we use p=10.
The error “variance” (i.e., the square of the final diagonal elements of Din (1)) is always
σ2=0.5 and the signal “variances” (i.e., the squares of the first ddiagonal elements
of Din (1)) are randomly generated from the uniform distribution Unif(1,3),
independently for each of the 100 repetitions. The proportions of correctly estimated
dimensions dfor each of the 26 methods are presented in Fig. 1, divided into two
panels for visual convenience.
Unsurprisingly, the classical covariance matrix based methods fail to consistently
find the correct latent dimension din the presence of too heavy tails (left sides of
the plots). This effect is most pronounced in the case ν=1 where the correspond-
ing t-distribution does not possess the finite second-order moments required by the
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Robust signal dimension estimation via SURE 3021
Table 1 The estimators included in the simulation study, along with binary indicator for their robustness
Method (scatter matrix) Is robust? n=200 n=400
p=10 p = 20 p = 10 p = 20
SURE1(Cov) No 0.01 0.00 0.00 0.00
SURE1(SSCM) Yes 0.13 2.04 0.19 2.74
SURE2(SSCM) Yes 0.01 0.02 0.01 0.02
SURE2(Tyler) Yes 0.02 0.02 0.02 0.03
SURE2(HR) Yes 0.06 0.08 0.10 0.15
SURE3(Cov) No 0.00 0.00 0.00 0.00
SURE3(SSCM) Yes 0.01 0.01 0.01 0.02
SURE3(Tyler) Yes 0.01 0.02 0.02 0.03
SURE3(HR) Yes 0.07 0.09 0.09 0.15
Asymp(Cov) No 0.00 0.00 0.00 0.00
Asymp(HR) Yes 0.06 0.08 0.09 0.12
Schott(Cov) No 0.01 0.00 0.00 0.01
Boot(Cov) No 0.23 0.32 0.46 1.50
Boot(SSCM) Yes 2.32 5.50 4.17 10.77
Boot(Tyler) Yes 3.04 7.67 5.01 13.16
Boot(HR) Yes 9.63 24.97 13.79 38.65
Ladle(Cov) No 0.09 0.15 0.09 0.18
Ladle(SSCM) Yes 2.01 2.21 3.26 3.88
Ladle(Tyler) Yes 3.17 4.39 4.60 6.29
Ladle(HR) Yes 18.35 27.69 19.19 28.65
Wax AIC(Cov) No 0.00 0.00 0.00 0.00
Wax AIC(HR) Yes 0.06 0.08 0.09 0.13
Wax MDL(Cov) No 0.00 0.00 0.00 0.00
Wax MDL(HR) Yes 0.06 0.09 0.09 0.12
Minka(Cov) No 0.00 0.00 0.00 0.00
Minka(HR) Yes 0.06 0.09 0.09 0.13
The final four columns give the average running times (in seconds) of the methods over 10 replicates in
various settings
covariance estimation. The robust methods, on the other hand, do not suffer from
this issue as they make no moment assumptions on the data generating distribution.
As t-distributions with low degrees of freedom regularly produce observations that
would be classified as outliers in the standard (Gaussian) statistical practice, the corre-
sponding simulation settings can also be interpreted to measure how well the methods
perform in the presence of a large number of outlying observations.
As the degrees of freedom increase, we observe that the covariance based methods
start to outperform the robust alternatives. The reason for this is that, when ν→∞,the
multivariate t-distribution approaches the normal distribution for which the covariance
based methods offer optimal inference.
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3022 J. Virta et al.
Fig. 1 Percentage of correctly estimated dimensions das a function of the degree of freedom νof the
multivariate t-distribution in the tail thickness simulation. The sample size is n=100 throughout
As the performance criterion in Fig. 1, proportion of correct estimates, gives no
indication of any possible under/overestimation, we show in Fig.2the mean estimation
errors for each combination of setting and method. Thus, values close to zero in Fig.2
indicate unbiased estimation. We observe that several of the methods consistently
overestimate the true dimension, which is actually desirable over underestimation as
it means that no signal information is lost (at the cost of introducing more noise). Boot,
Asymp and Minka are the most unbiased estimators, whereas SURE-based methods
tend to overestimate by 0.25−0.50 dimensions on average.
Comparing the methods by type (different colours in Figs.1and 2), the overall best
performances are given by the robust bootstrap (Boot, yellow), asymptotic (Asymp,
green) and Bayesian (Minka, purple) methods, with marginal differences between
the two. Somewhat surprisingly, the computationally most expensive method, i.e.,
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Robust signal dimension estimation via SURE 3023
Fig. 2 Mean of the error in dimension estimation as a function of the degree of freedom νof the multivariate
t-distribution in the tail thickness simulation. The sample size is n=100 throughout
the ladle, has a relatively bad performance in this scenario. The information criterion
type estimators (Wax, red and magenta) give a solid performance but are not among
the overall best choices. Comparing the different types of SURE to each other, it
seems that the additional computational and theoretical complexity of SURE1 (black)
does not provide additional benefits when compared to SURE2 (blue) and SURE3
(orange), which both have a relatively similar performance, not falling much behind
Boot, Asymp and Minka.
5.2 Latent dimension
In our second simulation study, we investigate how the relative size of the underly-
ing latent dimension daffects the methods’ performances. As the main selling point
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3024 J. Virta et al.
of the SURE-based methods is their light computational load, we have dropped the
more computationally intensive methods (Boot, Ladle), as well as the less successful
information theoretic methods (Wax), from the comparison, focusing in this (and the
following) study on comparing SURE only to its most relevant competitors, Asymp
and Minka. We choose SURE2 to be the “representative” of the SURE-family as, based
on the first simulation study, both SURE1 and SURE3 had performance similar to it.
Thus, the families of methods included in the current simulation study are Asymp,
SURE2 and Minka.
Recall that SURE2 estimates the latent dimension as the index minimizing the
corresponding objective function k→ ˆ
R2k. Based on our experiments, this strategy
can sometimes be quite unstable, especially when the latent dimension is comparatively
small. Thus, as an experimental alternative we propose estimating das the change
point in the series of differences ˆ
R2(k+1)−ˆ
R2k. To understand the motivation for this,
consider the following two typical forms for the graph formed by the points (k,ˆ
R2k):(i)
The points (k,ˆ
R2k)form a V-shaped curve around d. In this case, the true dimension
is both a minimizer and a location change point of the differences (the differences
change sign at the true dimension). (ii) The graph (k,ˆ
R2k)decreases linearly until d
and stays roughly constant afterwards. In this case, dis a location change point of the
differences but not necessarily a minimizer (it might happen that the minimizer occurs
only after d). Thus, in these two (rather idealistic) examples, the experimental change
point alternative offers more consistent detection of the dimension than the standard
method of seeking the minimizer. We implemented the change point detection as binary
segmentation through the function cpt.meanvar in the R-package changepoint
(Killick and Eckley 2014). The resulting method is denoted in the sequel as “SURE2
cp”.
We consider two sample sizes, n=100 and n=1000. For the former, we fix the
total dimensionality to p=10 and let the latent dimension vary as d=1,2,...,9.
For the latter, we use p=100 and d=5,10,15,...,95. We repeat the simula-
tion 100 times for every combination of parameters, generating in each repetition a
random sample from the multivariate t-distribution with 1 degree of freedom. The
error variance is fixed to σ2=0.5 and the signal variances are randomly generated
from the uniform distribution Unif(1,3), independently in every repetition. To get
a finer comparison between the methods, we use the average absolute estimation error
as our performance criterion in this (and the following) simulation study. The average
absolute errors for the different dimension estimation procedures are presented in the
two panels of Fig. 3, separately for n=100 and n=1000.
The top panel of Fig. 3reveals that SURE2 and SURE2 cp give the best and most
consistent performance when n=100. And, even though Asymp and Minka are
slightly better for the smallest values of d, their use cannot be recommended in practical
scenarios due to the drop in performance for larger d. The difference between the
different scatter matrices is quite minor, but the overall best choices are Tyler and
HR, which was somewhat expected as SSCM is not a “proper” scatter matrix, see the
discussion in Sect. 3.2. When n=1000 (bottom panel of Fig. 3), Asymp and Minka
with HR achieve very good performance, as does SURE2 when d≥50. The subpar
performance of SURE2 for small values of dappears to be a sample size issue as
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Robust signal dimension estimation via SURE 3025
Fig. 3 Average absolute errors of the dimension estimates as a function of the underlying dimension d
when sample size is n=100 (top panel) or n=1000 (bottom panel)
when we tried increasing the sample size to n=2000 (not shown here) SURE2 too
achieved performance equal to Asymp and Minka. SURE2 cp performs the best when
d≤55, confirming our earlier idea about the usefulness of the changepoint strategy
for low latent dimensionalities.
To summarize the results of the study, when n,pare both low, SURE-based methods
offer the overall best guarantees for dimension estimation across all values of d,
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3026 J. Virta et al.
Fig. 4 Percentages of correctly estimated dimensions as a function of the sample size n. The scale of the
horizontal axis is logarithmic
whereas, when n,pare larger, Minka and Asymp are the most consistent, requiring
lower sample sizes than SURE to achieve near-perfect estimation.
5.3 Sample size
In the third simulation, we study the effect of the sample size on the estimation
accuracy, including again the same set of methods as in the previous study. The consid-
ered sample sizes are n=500,750,1000,1500,2000,2500,5000. The simulation
is repeated 100 times for every sample size n, such that for every repetition a ran-
dom sample of nobservations is generated from the multivariate t-distribution with
1 degree of freedom. We take the latent and the total dimensionalities to be d=20
and p=100, respectively, throughout the simulation. As the error variance we use
σ2=0.5 and the signal variances are again randomly generated from the uniform
distribution Unif(1,3), independently for each of the 100 replicates. The propor-
tions of correctly estimated dimensions dfor the different procedures are presented
in Fig. 4.
The most striking feature of Fig. 4, which also sheds some light on the behaviour of
SURE2 in the previous simulations, is the jump in the mean absolute error of robust
SURE2 from 80 to 0 between n=750 and n=2000. Interestingly, the fact that
SURE2 cp has more even behaviour across the different sample sizes indicates that
this jump is not so much a consequence of the SURE criterion ˆ
R2kitself but of the
way in which the dimension estimate is selected based on the criterion values (recall
that SURE2 picks the minimizing value of kand SURE2 cp uses a more complicated
change point technique). We thus conclude that the standard technique of choosing
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Robust signal dimension estimation via SURE 3027
simply the minimizing index of the SURE criterion is not optimal unless nis large
enough. This matter clearly warrants further investigation and, due to its complexity,
we have left it for future research, see Sect.7. Finally, we also observe that, overall,
the used scatter matrix seems to have very little effect on the results, apart from Cov,
which again breaks down in the presence of a heavy-tailed distribution.
5.4 Computation time
As our final simulation study, we compare the running times of all 26 methods included
in Table 1. The change point variant of SURE2 is not included as its computational
difference to the base SURE2-method is marginal and negligible compared to the
differences between the methods itself. We distinguish two different sample sizes
n=200,400 and two different dimensionalities p=10,20, their combinations
leading to a total of four different settings. For each setting, we take the data distribution
to be the multivariate t-distribution with ν=1 degrees of freedom, d=0.6pand
the signal and noise variances as in the previous simulation. We run each of the 26
methods 10 times on each setting (using the same set of 10 data for each method)
and record their computational times. The experiment was conducted on a desktop
computer with AMD Ryzen 5 3600 6-core processor and 16 GB RAM. The average
running times in seconds are given in the final four columns of Table 1.
From Table 1we make the following observations: (i) Computational complexity
in the methods stems from two sources, the choice of the scatter matrix and bootstrap
replications (as performed by Boot and Ladle), of which the latter has a significantly
greater impact on the timing. (ii) The doubling of the sample size nhas a quite minor
effect on the computational times, whereas the doubling of the dimension pserves
to multiply the times by roughly 1.5. (iii) Of the robust methods, the fastest are by
far SURE2, SURE3, Asymp, Wax AIC, Wax MDL and Minka which all have com-
putational times roughly of the same order of magnitude. SURE1 falls somewhere in
between them and the more intensive Boot and Ladle.
Based on the observations made above and in the previous experiments, we conclude
that the SURE-based robust methodology (i) offers a fast and competitive alternative
to standard bootstrap-based methods, (ii) partially retains it functionality also at very
low sample sizes, unlike Asymp and Minka, and (iii) requires higher sample sizes to
reach near perfect estimation results than Asymp and Minka.
6 Application: asset returns
We next illustrate the robust SURE methods in a financial data set that was used to
demonstrate principal component analysis in the classical textbook [Tsay (2010), Sect.
9] to search for common latent variables explaining (joint) asset return variability. The
data itself is available on the author’s (Ruey S. Tsay’s) webpage and consists of monthly
log stock returns (including dividends) of five stocks (IBM, HPQ, INTC, JPM, BAC)
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3028 J. Virta et al.
Fig. 5 The monthly log returns (in percentages) of five stocks (IBM, HPQ, INTC, JPM, BAC) from January
1990 to December 2008. See [Tsay (2010), Section 9]
from January 1990 to December 2008.1These p=5 time series of length T=228
months are illustrated in Fig. 5.Tsay(2010) computed the Portmanteau test statistics
and found that despite the time series nature of the data there is no substantial serial
correlation in returns, and hence we also ignore the serial dependence in our analysis.
According to the results of PCA, Tsay (2010) concluded two common latent variables
in his interpretation: The “market component” represents the general movement of
the stock market and the “industrial component” represents the difference between
the two industrial sectors, namely technology (IBM, HPQ and INTC) versus financial
services (JPM and BAC). In addition, Tsay (2010) points out that IBM stock “has its
own features that are worth further investigation”.
Instead of computing only a single estimate of the latent dimension for the data
set, we take a local approach and run a window of length through the data. For
each of the T−+1 windows we then estimate the latent dimension using one of
our proposed methods. As the obtained dimensionalities correspond to the windows
and not the actual observation months, we “back-transform” them as follows: For
each individual month, we take the weighted average of the estimated dimensions
of all windows in which that particular month is a member. We assign the weights
1See the website https://faculty.chicagobooth.edu/ruey-s- tsay/research/analysis- of-financial- time-series-
3rd-edition. Specifically, stocks are International Business Machines (IBM), Hewlett-Packard (HPQ), Intel
Corporation (INTC), J.P. Morgan Chase (JPM) and Bank of America (BAC).
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Robust signal dimension estimation via SURE 3029
Fig. 6 The smoothed curves of latent dimensionalities estimated with SURE2 using the window approach
as described in the main text. The two panels correspond to the window lengths of =48 and =72
months, respectively
such that the middle two observations in each window get a weight equal to 1 and
the weights decrease linearly towards the window endpoints. This procedure thus
produces a “smoothed” curve of estimated latent dimensions for the full observation
period. To guarantee that the ellipticity of the data is at least partially fulfilled, we resort
to rather small window lengths, taking either =48 or =72 in the following. Visual
inspection (not shown here) reveals that scatter plots of the obtained windows indeed
exhibit elliptical shapes throughout the observation period. The intuition behind this
somewhat experimental approach is that the latent dimension dcan be seen to measure
the internal complexity of the observed multivariate time series, allowing us to identify
from the smoothed curve intervals of time when the stocks behave in a more unified
manner (low dimension) or independently of each other (high dimension). Note that
the months close to the beginning and the end of the measurement interval belong on
average to a fewer number of windows, meaning that they are expected to show more
erratic behavior.
For simplicity, we apply the described procedure only with SURE2, using each of
the four scatter matrices (and the two window lengths), and with Asymp, which has
good overall performance in simulations when combined with the H–R estimator.
The results for SURE2 are shown in Fig.6. In line with the evidence in Tsay (2010),
we find at least two and in most time points at least three common features in our
five stock case. The robust variants of SURE2 favour three (even four) dimensions,
emphasizing also other common features than the market and industrial components.
Interestingly, the time-varying patterns of the estimated dimensionality in the robust
approaches seem to be largely in accordance with each other and following general
market conditions where the major periods of decreasing prices (i.e. the beginning of
2000 s and the financial crisis 2007–2009), and their onsets, are associated with some-
what lower dimensions. Finally, note that the fact that the curves for the non-robust
“Cov” are markedly different from the robust ones is a clear indication that the data
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3030 J. Virta et al.
Fig. 7 The smoothed curves of latent dimensionalities estimated with Asymp using the window approach
as described in the main text. The two panels correspond to the window lengths of =48 and =72
months, respectively
indeed exhibit heavy-tailed behaviour (as typically with asset returns) that hampers
the estimation of the covariance matrix but leaves the robust estimates unaffected.
The results for Asymp(Cov) and Asymp(HR) are shown in Fig.7and match rather
well with the corresponding results obtained with SURE2 in Fig. 6. For example, when
using the window length of 72 months, both SURE2(HR) and Asymp(HR) identify
3–4 latent dimensions for the majority of the observation window, with two “bumps”
in the curve. The only major difference is between SURE2(HR) and Asymp(HR)
during the years 1990–2000 when the window length of 48 months is used: the former
method claims a full set of 4 latent variables whereas the latter finds only a single one.
This discrepancy is most likely explained by the short window length which is not
large enough to estimate σ2(SURE2) or to invoke asymptotic arguments (Asymp).
As such, it seems more reasonable to restrict attention to the window length of 72
months, where both methods agree that the amount of latent variables was lower at
the start of the observation period and gradually increased with time.
7 Discussion
The results obtained in this work open up several avenues for future research. Perhaps
most notably, our simulations revealed that the standard approach of choosing the
parameter estimate to be the global minimizer of the SURE-criterion might not be
optimal in the current scenario. As a simple alternative to minimization, we explored
using a change point detection based approach which indeed proved superior to the
minimization in various settings (and vice versa in other settings). As such, the matter
clearly warrants more investigation. We also note that the dangers of “blindly” mini-
mizing a model fit criterion are naturally well-known in the model selection literature.
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Robust signal dimension estimation via SURE 3031
Indeed, also Ulfarsson and Solo (2008) mention this caveat, although, not in the con-
text of SURE but the Bayesian information criterion. Despite this, we are not aware of
any general alternatives to minimization being proposed in the literature, discounting
visual inspection (which can be seen as both heuristic and subjective). Quite possibly,
such procedures may not even exist as their behaviour would depend greatly on the
functional form of the particular criterion in question. But, at least in the current sce-
nario of dimension estimation, the change point criterion appears to provide a feasible
option.
A second point of interest brought up by our simulations is the sudden increase
in the accuracy of the robust SURE2 in Fig.4. In that particular data scenario there
appears to be a “critical” sample size after which SURE2 achieves perfect estimation
results. The dependence of this sample size on the model parameters, especially pand
dis something to be studied and quantified. We note that any theoretical investigation
of this matter is likely to be very difficult as it concerns the finite-sample properties of
the method, whereas the large majority of the existing theoretical results in the robust
literature are conducted in the asymptotic framework where n→∞.
As a third point, the simulations in Sect. 5.2 revealed that the Bayesian estimator
by Minka (2000) gave a suprisingly good performance when combined with the H–R
estimator of scatter. Such plug-in procedures in the context of Minka (2000) appear
not to have been earlier investigated in the literature and clearly warrant future study.
In the top panel of Fig. 3the performance of the Bayesian plug-in estimator (Minka)
is very similar to Asymp, meaning that for low nit appears to be an approximately
equal to the asmyptotical test, partially also explaining its bad performace for low n.
Fourthly, despite being a significant improvement over the Gaussian assumption,
the elliptical model (1) can still be seen as somewhat limiting in practice. In particular,
a consequence of the ellipticity is that the tails of the distribution are equally heavy
along all directions in the space Rp. One solution would be to consider the independent
component model xi=μ+Aziinstead, where A∈Rp×pis full rank and the
components of the p-vector ziare mutually independent random variables (Comon
and Jutten 2010). Exactly p−delements of ziare assumed to be Gaussian and,
as such, noise, making the signal dimension of the model equal to d. By taking the
signal components of zito have different tail decays, a richer variety of heavy-tailed
behaviours for xiis obtained, see Virta et al. (2020). The independent component
model admits a solution through the use of pairs of scatter functionals, see Tyler et al.
(2009), and a similar approach could possibly serve as a starting point for deriving
a SURE-based criterion for the estimation of the dimension din the independent
component model.
Fifthly, we comment on using the final eigenvalue spof Sas the estimator of the
noise variance. As mentioned earlier, we made this choice as it imposes minimal
assumptions on the latent signal dimension d, allowing also cases where the ratio d/p
is relatively large. This benefit can be seen in the top panel of Fig.3, where robust
SURE2 is able to estimate the latent dimension increasingly well even for very large
values of d/p. The same would not be possible if, for example, the median or the third
quartile of the eigenvalues would be used as the noise variance estimator. Similarly,
in the asset return example in Sect. 6, the ratio d/pwas estimated to be larger than
0.6 in majority of the time windows. And although our example data were rather
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3032 J. Virta et al.
low-dimensional, large values of d/pare known to occur also in higher-dimensional
economic data sets, making the use of estimators such as sprecommended in such
contexts.
On the flipside, using the estimator spmost likely compromises the unbiasedness
of our risk estimate. That is, even though E(R2k)=(1/n)n
i=1Eˆxi−V0yi2, there
is no guarantee that the plug-in risk ˆ
R2kwould be an unbiased estimator of the risk.
We investigated this with some preliminary simulations (not shown here) and were led
to two conclusions: (i) It appears that ˆ
R2kis indeed biased but that the bias vanishes
when n→∞. Thus, the plugging in of splikely changes our risk estimator from
unbiased to asymptotically unbiased. (ii) Using the “oracle” estimator R2k(where the
true value of σ2is used) in place of ˆ
R2kdid not yield significantly better results and,
in fact, in some cases ˆ
R2kperformed strictly better than R2k. The effect of estimating
σ2clearly warrants further study but, based on the previous, it does not appear that its
estimation affects the performance of the method too much.
As pointed out to us by an anonymous reviewer, an alternative approach to the
dimension estimation problem would be to instead minimize the risk function
Et0+Pk(˜x−t0)−V0˜y2,(9)
where the location and projection estimates t0,Pkare estimated based on the sample
x1,...,xnand ˜x(and the corresponding latent signal ˜y) is another draw from the
model, independent of the actual sample. Thus, the difference to our proposed concept
is that in (9) an independent “test sample” ˜xis used to evaluate the error in the PCA-
reconstruction. While this is a perfectly valid approach to the problem, it no longer
adheres to the SURE-framework (but is more akin to the classical Akaike’s information
criterion) and we leave its study to future work.
Finally, an interesting alternative to our model (1) would be the elliptical factor
model xi=μ+V0Dyi+εiwhere V0∈Rp×dhas orthonormal columns, Dis diagonal
and yi∈Rdand εi∈Rpare mutually independent and spherical. As general scatter
matrices do not have the additivity property possessed by the covariance matrix, it is
not guaranteed that the methodology used in this work would be guaranteed to identify
the latent dimension of this model via the “eigengap”. Our preliminary investigations
however give promising results regarding this and reveal that the estimation can indeed
work, but due to the lack of theoretical motivation, we have left this too for future work.
Acknowledgements The work of Niko Lietzén and Henri Nyberg was supported by the Academy of Finland
(Grant 321968). The work of Joni Virta was supported by the Academy of Finland (Grants 335077, 347501
and 353769). Nyberg also acknowledges financial support by the Emil Aaltonen Foundation (Grant 180287)
and the Foundation for Economic Education (Liikesivistysrahasto, Grant 220246).
Funding Open Access funding provided by University of Turku (including Turku University Central
Hospital).
Declarations
Conflict of interest The authors state that they have no competing interests.
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Robust signal dimension estimation via SURE 3033
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Appendix A: Proofs of the technical results
Proof of Lemma 1We write,
Eˆxi−V0yi2=E(ˆxi−xi)+(xi−V0yi)2
=Eˆxi−xi2+2E{(ˆxi−xi)εi}+Eεi2.
Now, ˆxi−xi2=tr{(Ip−Pk)(xi−t0)(xi−t0)}, implying that averaging over iin
the preceding expansion yields
1
n
n

i=1
Eˆxi−V0yi2=E[tr{(Ip−Pk)S0}] + 2
n
n

i=1
E(ˆx
iεi)−pσ2.
Hence, the claim follows as soon as we show that E(ˆxijεij)=σ2E{(∂/∂ xij)ˆxij}.To
see this, we use the law of total expectation in conjunction with Stein’s lemma to write,
E(ˆxijεij)=E{E(ˆxijεij |y1,...,yn)}
=σ2E[E{(∂/∂ εij)ˆxij |y1,...,yn}]
=σ2E[E{(∂/∂ xij)ˆxij |y1,...,yn}]
=σ2E{(∂/∂ xij)ˆxij},
where the second equality uses Stein’s lemma on the reconstruction ˆxij treated as a
function of the full data (which are, conditional on y1,..., yn, Gaussian), the third
equality uses the fact that xij and εij are equal up to translation by a constant (again,
under the conditioning).
Proof of Lemma 3We use, for brevity, the notation t:= t(Fn),t∗:= t(Fn,i,j,ε ), and
similarly for other quantities. Hence, by Assumption 1,wehavet∗=t+εhij +o(ε)
and S∗=S+εHij +o(ε). Thus, by [Stewart (2001) Chapter 1.3.2], the matrix S∗
has a th eigenvector u∗
with the following first-order expansion,
u∗
=u+ε
p

m=
1
s−sm
TmHiju+o(ε), (A1)
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3034 J. Virta et al.
where uis the th eigenvector of Sand T=uu
is the corresponding orthogonal
projection. Taking outer products on both side of (A1)gives
T∗
=T+ε
p

m=
1
s−sm
(THijTm+TmHijT)+o(ε).
Consequently, the projection P∗
konto the space spanned by the eigenvectors corre-
sponding to the klargest eigenvalues of S∗(which are, for small enough ε, simple) is
P∗
k=Pk+εAij +o(ε), where Aij is as in the statement of the lemma (note that, by
symmetry of the Hij, all terms with m≤kget cancelled).
Now, recall that x∗
i=xi+εejand write,
ˆx∗
i−ˆxi=t∗−t+P∗
k(x∗
i−t∗)−Pk(xi−t)
=ε{hij +Pk(ej−hij)+Aij(xi−t)}+o(ε),
showing that the desired derivative equals
∂
∂xij ˆxij =ejhij +ejPk(ej−hij)+ejAij(xi−t).
Summation over jnow yields the claim. 
Proof of Lemma 4We write t:= t(Fn)and t∗:= t(Fn,i,j,ε), and similarly for Saad
S∗. Clearly, t∗=t+(1/n)εej, implying that xi+εej−t∗=xi−t+(1−1/n)εej.
Writing yi:= xi−t, we get,
S∗=1
n
n

=i{y−(1/n)εej}{y−(1/n)εej}
+1
n{yi+(1−1/n)εej}{yi+(1−1/n)εej}.(A2)
The first term of (A2) simplifies to
S−1
nyiy
i+1
n2εejy
i+1
n2εyiej+o(ε),
whereas the second term can be written as
1
nyiy
i+n−1
n2εejy
i+n−1
n2εyiej+o(ε).
Combining these two now yields the claim. 
Proof of Lemma 5We begin with hij.Letg:Rp→Rbe the objective function with
g(t)=(1/n)n
i=1xi−t, from whose minimization the spatial median t(Fn)is
found. It is straightforwardly checked that gis convex and, since, by the assumption
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Robust signal dimension estimation via SURE 3035
that t(Fn)= xifor all i=1,...,n,gis differentiable in a neighbourhood of t(Fn),
the gradient of gmust vanish at t(Fn),
1
n
n

i=1
xi−t(Fn)
xi−t(Fn)=0.(A3)
Define now the function f:Rp+1→Rpsuch that
f(ε, t)=1
n
n

=i
xi−t
xi−t+1
n
xi+εej−t
xi+εej−t.
Now, f{0,t(Fn)}=0 and, moreover, fis continuously differentiable at (0,t(Fn)) and
has the Jacobian Dtf{0,t(Fn)}=−(1/n)Gwhere Gis as defined in the statement
of the lemma. Now, Gis a sum of weighted projection matrices to the orthogonal
complements of the lines spanned by the vectors yi=xi−t(Fn). Hence, Ghas full
rank (by our assumption that the points x1,...,xnare not concentrated on a line) and
the implicit function theorem then guarantees that for a suitably small neighbourhood
S⊂Rof zero there exists a differentiable function b:S→Rpsuch that f{ε, b(ε)}=
0.
In the following, with some abuse of notation, Swill denote a (changing) small
enough neighbourhood of zero in R. By our sample being not concentrated on a
line and Milasevic and Ducharme (1987), the spatial median of Fn,i,j,ε is unique
in S.Let M:= maxixiand take any t∈Rpsuch that t≥3M+1. Then,
(1/n)n
i=1xi−t≥2M+1, whereas (1/n)n
i=1xi−0≤M, showing that,
for all ε∈S, the spatial medians t(Fn,i,j,ε)reside in a compact set. Thus, by Berge’s
Maximum Theorem, the map ε→ t(Fn,i,j,ε)is continuous in S, implying that the
equivalent of the assumption that t(Fn)= xi, for all i=1,...,n, holds for Fn,i,j,ε.
Consequently, for any ε∈S, the corresponding spatial median objective function is
differentiable and its gradient vanishes at the spatial median t(Fn,i,j,ε ). Its gradient
is equal to f(ε, t)and, by the uniqueness of the spatial medians t(Fn,i,j,ε), we thus
have that the implicit function bactually traces the spatial medians as a function of ε,
i.e., t(Fn,i,j,ε)=b(ε) for ε∈S.
The differentiability of bat zero now entails that we may write t(Fn,i,j,ε)=b(ε) =
t(Fn)+εhij +o(ε) for some hij ∈Rpas ε→0. Plugging the expansion in to the
Fn,i,j,ε-equivalent of (A3), we get
0=
n

=i
y−εhij +o(ε)
y−εhij +o(ε)+yi+ε(ej−hij)+o(ε)
yi+ε(ej−hij)+o(ε),(A4)
where yi=xi−t(Fn). The first term in (A4) splits into two parts, the first of which
simplifies as,
n

=i
y
y−εhij +o(ε)=
n

=i
y
y+ε
n

=i
yy

y3hij +o(ε),
123
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3036 J. Virta et al.
whereas, the second part writes,
−
n

=i
εhij +o(ε)
y−εhij +o(ε)=−ε
n

=i
1
yhij +o(ε).
Simplifying the second term of (A4) similarly yields,
yi+ε(ej−hij)+o(ε)
yi+ε(ej−hij)+o(ε)=yi
yi−εyiy
i
yi3(ej−hij)+ε1
yi(ej−hij)+o(ε).
Plugging everything now back in to (A4), and using (A3), we get, in the notation of
the statement of the lemma,
−εGhij +εAiej+o(ε) =0.
Division by εand letting ε→0 then yields the desired expression for hij.
For Hij, resorting to the same notation as in the proof of Lemma 3and writing
y∗
i:= x∗
i−t∗,wefirsthave,
y∗
i2−yi2=2ε(ej−hij)yi+o(ε),
and,
y∗
i(y∗
i)−yiy
i=ε{(ej−hij)y
i+yi(ej−hij)}+o(ε) =: εBij +o(ε).
For = i, we also write y∗
:= x−t∗and have
y∗
2−y2=−2εh
ijy+o(ε),
and
y∗
(y∗
)−yy
=−ε{hijy
+yh
ij}+o(ε) =: −εCij +o(ε).
Hence, the partition
S∗−S=1
n
n

=iy∗
(y∗
)
y∗
2−yy

y2+1
ny∗
i(y∗
i)
y∗
i2−yiy
i
yi2,
along with the formula for general matrices A,Band scalars a,bwith a= 0 that,
A+εB+o(ε)
a+εb+o(ε) =A
a+εaB −bA
a2+o(ε),
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Robust signal dimension estimation via SURE 3037
implies that
S∗−S=−ε1
n
n

=1{y−2(hij y
+yh
ij)−2y−4yh
ijyy
}
+ε1
n{yi−2(ejy
i+yiej)−2yi−4yiejyiy
i}+o(ε).
Simplifying the expression now yields the claim. 
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