Robust estimation of covariance and correlation functions of a stationary multivariate process

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Robust estimation of covariance and correlation
functions of a stationary multivariate process
Higor H. A. Cotta∗† , Valdério A. Reisen∗, Pascal Bondon†and Wolfgang Stummer‡
∗Graduate program in Environmental Engineering and Department of Statistics
Federal University of Espírito Santo, Brazil
†Laboratoire des Signaux et Systèmes, CNRS - CentraleSupélec - Université Paris-Sud - France
‡Department of Mathematics, University of Erlangen-Nürnberg - Germany
Abstract—In this paper, the effect of additive outliers is
considered in the estimation of the covariance and correlation
matrix functions of a multivariate stationary process. Robust
estimators of these matrices are presented in order to mitigate
the effect of outliers. Some Monte Carlo simulations are carried
out to empirically clarify the impact of additive outliers in the
standard estimators and to assess the robustness of the proposed
estimators. A real data set is analyzed as example of application.
I. INTRODUCTION
The estimation of the covariance and correlation matrix
functions is an important step in the identification and es-
timation of a multivariate signal model, e.g., for parameter
estimation using the Yule-Walker equations. It is well known
that outliers in signals affect the correlation structure of the
data which may lead to erroneous estimators [1]. How to
mitigate this phenomenon is still a challenging problem.
Robust estimation theory has been extensively studied in the
statistical community since the 1970s following the seminal
works of Huber and Maronna [2], [3]. Several efforts have
been done by the statistical signal processing community
in order to weaken the impact of atypical observations. A
concise review of the fundamentals for the signal processing
community can be found in [4].
In this context, some theory and applications to real prob-
lems considering robust estimation for univariate data cor-
related in time are presented in [5], [6], [7], these in the
frequency domain and in [8], for the time domain. Applica-
tions of robust techniques in biotechnology, finance and power
management can be found in [9], [10] an [11], respectively.
In the multivariate context, highly robust estimation of
the covariance and correlation matrices for time independent
data sets are proposed in [12]. The estimators use the so-
called Qn(.)estimator proposed in [13] which has appealing
feature such as being location-free, a high breakdown point
(50%) and a bounded influence function. The robustness
and efficiency properties of the estimators have also been
investigated through analysis of numerical experiments and
real data analysis for univariate time series. For further details
on these theoretical and numerical studies, see [14].
In this work, we extend to a multivariate stationary time
series the robust estimator of the autocovariance and the
autocorrelation functions of a univariate stationary time series
proposed by [12], [15]. We compare the proposed estimator
to the sample estimator by means of temporal breakdown
point and influence functions, and through Monte Carlo ex-
periments.
This paper is organized as follows. In Section 2, the effect
of additive outliers in the covariance and correlation matrix
functions of a multivariate time series is shown and the robust
estimators of the autocovariance and autocorrelation matrix
functions are proposed. Section 3 presents some Monte Carlo
experiments. A real data example is considered in Section 4
and some concluding remarks are provided in Section 5.
II. ROB US T ES TI MATI ON O F AUT OC OVARI AN CE A ND
AUTOCORRELATION MATRIX FUNCTIONS
A. Linear Time Series
Let Xt= [X1,t, X2,t , . . . , Xk,t]0,t∈Zbe a k-dimensional
linear vector process defined by
Xt=µ+P∞
j=0 Ψjεt−j,(1)
where µ= [µ1, . . . , µk]0is the mean vector of {Xt},Ψ0is
the identity k×kmatrix, Ψj, j = 1,...,∞are k×kmatrices
of coefficients satisfying P∞
j=0 kΨjk2<∞, where kAkis
the matrix norm of matrix Adefined by kAk2= Tr(A0A).
The vector process εt= [ε1,t, . . . , εk,t ]0is zero-mean and
uncorrelated, i.e., E(εt)=0and Cov(εt,εt+h)=Σε1{h=0}.
Thus, although the elements of εjat different times are
uncorrelated, they may be contemporaneously correlated. It
results from (1) that
γX(h) = Cov(Xt,Xt+h) = P∞
j=0 ΨjΣΨ0
j+h, h ≥0.
The lag-hcorrelation matrix function of {Xt}is defined by
ρX(h) = D−1/2γX(h)D−1/2,(2)
where D= diag[γX
11 (0), ..., γX
kk(0)]. The (i, j)th element of
ρX(h)is
ρX
ij (h) = Cov(Xi,t, Xj,(t+h))
pVar(Xi,t) Var(Xj,t)=γX
ij (h)
qγX
ii (0)γX
jj (0)
.(3)
A parametric class of linear time series satisfying (1) is
the vector autoregressive moving average (VARMA) model of
orders (p, q)defined by the difference equation
Φ(B)(Xt−µ) = Θ(B)εt,(4)

where Bis the backward shift operator (BXt=Xt−1),
Φ(B) = I−Pp
i=1 ΦiBiand Θ(B) = I+Pq
i=1 ΘiBiwhere
Φiand Θiare k×kmatrices, and {εt}is a vector white
noise process. When the polynomials Φ(z)and Θ(z)satisfy
det(Φ(z)) 6= 0 and det(Θ(z)) 6= 0 for all z∈Csuch that
|z| ≤ 1, (4) has a unique stationary causal and invertible
solution and the matrices Ψjare determined uniquely by
Ψ(z) = P∞
j=0 Ψjzj=Φ−1(z)Θ(z)for |z| ≤ 1.
B. Impact of additive outliers in multivariate time series
Outliers can affect the dependence structure of a multivari-
ate time series. In this section, some results related to the
effects of outliers on the covariance and correlation structures
of a correlated process are derived. We suppose that the
observed process {Zt}results from the contamination of
{Xt}by additive random outliers, i.e.,
Zt=Xt+Ωδt,(5)
where Ω= diag[ω1, ..., ωk]and ωi, i = 1, ..., k, is the magni-
tude of the outliers which affects {Xi,t},δt= [δ1,t, ..., δk,t]0is
a random vector indicating the occurrence of an outlier at time
t. We assume that {Xt}and {δt}are uncorrelated processes
and that P(δi,t =−1) = P(δi,t = 1) = pi/2,P(δi,t = 0) =
1−pifor i= 1, . . . , k where 0≤pi<1. Then E(δi,t) = 0
and Var(δi,t) = pi. We assume also that Cov(δt,δt)=Σδ=
diag[p1, ..., pk]and that Cov(δt,δt+h)=0when h6= 0.
It follows from (5) that E(Zt) = E(Xt),γZ(0) = γX(0)+
ΩΣδΩ0and γZ(h) = γX(h)when h6= 0. Therefore
ρZ
ij (h) = 




γX
ij (h)
√(γX
ii (0)+piω2
i)(γX
jj (0)+pjω2
j), h 6= 0,
γX
ij (0)+piω2
i1{i=j}
√(γX
ii (0)+piω2
i)(γX
jj (0)+pjω2
j), h = 0.
(6)
We observe that ρZ
ij (h)→0as |ωi| → ∞ or |ωj|→∞when
h6= 0, these conclusions are deeper analyzed in Proposition 1.
The recent works of [14], [18], [17], [16] and [19] discuss this
problem in univariate time series with short and long memory
properties.
Proposition 1. Suppose that Z1,t,Z2,t ,...,Zn,t is a set of
k-dimensional time series observations of Model 5 and mis
the expected number of additive outliers as stated in (5). Let
ˆρZ
ij (h) = γZ
ij (h)/(pγZ
ii (0)qγZ
jj (0)), for ∀i, j = 1, ..., k, then
a. For m= 1 (one outlier occurring only at Zi),
lim
n→∞ plim
ωi→∞
ˆρZ
ij (h) = 0.
b. For m= 2 (two outliers occurring at Zi,t or/and at
Zj,t) and assuming that ˆγZ
ij (h)6= 0, for Zi,t and Zj,t,
it follows
lim
n→∞ plim
ωi→∞
and/or
ωj→∞
ˆρZ
ij (h)=0.
In (a.) and (b.), wiand wjare the magnitudes of the additive
outliers occurring at position iand j., respectively
The proof of Proposition 1 follows the same lines as in
[18], [17], [16] and are not presented here to save space, but
is available upon request.
C. Robust estimation of the covariance and correlation matrix
functions
Let X1, . . . , Xnbe independent and identically distributed
univariate random variable with finite variance and X=
(X1, . . . , Xn)0. The Qn(.)estimator of the standard deviation
of X1is the kth order statistic defined by
Qn(X) = c{|Xi−Xj|;i < j}{k}, i, j = 1, .., n,
where cis a constant to guarantee consistency (c = 2.2191 for
the Gaussian distribution), k=b(n
2+ 2)/4c+ 1 and bxc
is the largest integer smaller than x. An efficient algorithm
to calculate Qn(X)is proposed in [20]. The asymptotic
breakdown point of Qn(X)is 50%, see [13].
For any univariate second order random variables Xand Y,
we have
Cov(X, Y ) = αβ
4Var(X/α +Y/β)−Var(X/α −Y/β),
(7)
for any α, β ∈R, see [21].
Now, let {Xt}, t ∈Z, be a univariate stationary time series
with finite variance. Taking α=βin (7) and replacing Var(.)
by Q2
n(.), [15] proposed the following highly robust estimate
of the covariance function of {Xt},
ˆγQn(h) = 1
4Q2
n−h(U+V)−Q2
n−h(U−V),(8)
where U= (X1, . . . , Xn−h)0and V= (Xh+1, . . . , Xn)0. The
autocorrelation function of {Xt}can be estimated by
ˆρQn(h) = Q2
n−h(U+V)−Q2
n−h(U−V)
Q2
n−h(U+V) + Q2
n−h(U−V).(9)
The consistency and asymptotic normality of ˆγQn(h)and
ˆρQn(h)are studied in [14] and [19] when {Xt}is a short
and a long memory process.
In this work, we extend to multivariate time series the
estimators (8) and (9). Let {Xt}be a k-dimensional station-
ary vector process with finite variance, we robustly estimate
γX(h)by
ˆ
γQn(h) = [ˆγ(Xi,t,Xj,t)
Qn−h(h)]k
i,j=1 (10)
where,
ˆγ(Xi,t ,Xj,t)
Qn−h(h) = αβ
4Q2
n−hU
α+V
β
−Q2
n−hU
α−V
β,(11)
U= (Xi,1...,Xi,n−h)0,V= (Xj,h+1, . . . , Xj,n )0,α=
Qn(Xi,t)and β=Qn(Xj,t ). The autocorrelation matrix
function of {Xt}can be estimated by
ˆ
ρQn(h) = [ˆρ(Xi,t,Xj,t )
Qn−h]k
i,j=1 (12)

ˆρ(Xi,t,Xj,t )
Qn−h=
Q2
n−hU
α+V
β−Q2
n−hU
α−V
β
Q2
n−hU
α+V
β+Q2
n−hU
α−V
β,(13)
where U,V,αand βare defined in (11).
III. SIMULATIONS
The process {Xt}is generated by (4) where k= 3,(p, q) =
(1,0),µ= 0,
Φ1=

0.6 0.3 0.0
0.1 0.2 0.0
0.1 0.8 0.4

,
and {εt}is a zero-mean Gaussian white noise process with
covariance
Σε=

1.00 0.70 0.70
0.70 1.00 0.95
0.70 0.95 1.00

.
The contaminated process {Zt}is simulated according to (5)
where ωi= 4pVar(Xi,t)and piare given in the plots for
i= 1,2,3. The sample size nis 500 and each experiment is
repeated 1000 times. We denote by ˆ
ρX(h)the sample estimate
of ρX(h), i.e., the estimate that we obtain by replacing the
unknown covariances in (2) by their sample estimates. The
sample estimate ˆ
ρZ(h)is defined as ˆ
ρX(h)and ˆ
ρZ
Qn(h)is
obtained by (12) where {Xt}is replaced by {Zt}. The means
of ˆ
ρX(h),ˆ
ρX
Qn(h),ˆ
ρZ(h)and ˆ
ρZ
Qn(h)are computed over the
1000 replications for each lag h,0≤h≤7.
0 2 4 6
0.2 0.4 0.6 0.8 1.0
ACF
Theoretical
0 2 4 6
0.2 0.6 1.0
ACF
Samp. Estim. Uncont
0 2 4 6
0.2 0.6 1.0
Robust ACF
Robust Estim. Uncont
0 2 4 6
0.0 0.4 0.8
ACF
Samp. Estim. Cont
0 2 4 6
0.2 0.6 1.0
Robust ACF
Robust Estim. Con
Fig. 1. Autocorrelation function of Z1,t . From left to right and top to bottom,
plots are ρX
11(h),ˆρX
11(h),ˆρX1,t
Qn−h(h), and ˆρZ
11(h),ˆρZ1,t
Qn−h(h)when pi=
0.05,i= 1,2,3.
Figure 1 displays the true value ρX
11(h)and the estimates
ˆρX
11(h),ˆρX1,t
Qn−h(h),ˆρZ
11(h)and ˆρZ1,t
Qn−h(h). Figure 2 plots the
true value ρX
12(h)and the estimates ˆρX
12(h),ˆρ(X1,t ,X2,t)
Qn−h(h),
ˆρZ
12(h)and ˆρ(Z1,t ,Z2,t)
Qn−h(h). For the other components of {Xt}
and {Zt}, we obtain similar figures.
In both figures, the effects of additive outliers appears
by comparing the true correlation to the sample estimates
ˆ
ρZ(h)obtained in the contaminated case. Indeed, the values of
ˆρZ
11(h)and ˆρZ
12(h)are much smaller than ρX
11(h)and ρX
12(h),
respectively. These graphical results are an expected result and
empirically confirms Proposition 1, and it is also in accord
with the discussion in [16], [17], [18], [14]. Now, we observe
on both figures that the sample and the robust estimators have
similar behaviors in the absence of contamination, thus, in a
0 2 4 6
0.1 0.3 0.5
ACF
Theoretical
0 2 4 6
0.1 0.3 0.5
ACF
Samp. Estim. Uncont
0 2 4 6
0.1 0.3 0.5
Robust ACF
Robust Estim. Uncont
0 2 4 6
0.1 0.3 0.5
ACF
Samp. Estim. Cont
0 2 4 6
0.1 0.3 0.5
Robust ACF
Robust Estim. Con
Fig. 2. Correlation function between Z1,t and Z2,t. From left to right
and top to bottom, plots are ρX
12(h),ˆρX
12(h),ˆρ(X1,t,X2,t )
Qn−h(h), and ˆρZ
12(h),
ˆρ(Z1,t,Z2,t)
Qn−h(h)when pi= 0.05,i= 1,2,3.
practical situation, when the practitioner is uncertain of the
presence outliers, (11) and (13) are still a reasonable choices.
When there are outliers among the data, i.e., pi6= 0,∀i, it
is possible to see that the robust estimator is not affected by
a percentage of contamination of 5% providing to be a good
alternative when, in fact, outliers are presented in the data. f
In Table I, we present the root mean squared errors (RMSE)
of ˆρX
11(h),ˆρX1,t
Qn−h(h),ˆρZ
11(h)and ˆρZ1,t
Qn−h(h). Table II, gives the
RMSE of ˆρX
12(h),ˆρ(X1,t ,X2,t)
Qn−h(h),ˆρZ
12(h)and ˆρ(Z1,t ,Z2,t)
Qn−h(h).
We observe that the sample and the robust estimators have
RMSE close to the each other in the absence of contamination
while the RMSE of the sample estimate is much larger than
the RMSE of the robust estimator when the percentage of
contamination is 5%. Moreover, the RMSE of the robust
estimator are almost the same in the uncontaminated and the
contaminated cases.
The RMSE of ˆρZ
12(h)and ˆρ(Z1,t ,Z2,t)
Qn−h(h)as the percentage
of outliers in {Zt}increases are presented in Tables III and IV,
respectively. Not surprisingly, increasing the percentage of
outliers reduces the performance of both estimators. However,
ˆρ(Z1,t,Z2,t )
Qn−h(h)is less affect by the outliers.
IV. REA L DATA EXA MP LE
In this real data example, we consider the estimation of the
samples ACFs of the monthly personal consumption expen-
diture (PCE) and disposable personal income (DSPI) of the
United States from January 1959 to March 2012 (n= 639)
obtained from the Federal Reserve Bank of St. Louis (FRED
Economic Data). This data set has already been considered
as an example in [22]. As pointed out by the author, the
original series are not stationary, thus the working series are
the differenced observations, in percentages, after applying the
log transformation. Figure 3 shows the time series plots for
the data in the study. From the plots, it is possible to see
the presence of some outlying observations which justifies the
comparison between the robust and non-robust methods.
In Figures 4 and 5 we present the plots of ˆ
ρX(h)and
ˆ
ρQn(h), respectively. Comparing both plots, one may see that
vertical scale of both plots are slightly different, indeed, as a
simple evidence, the classical sample cross-correlation values
between PCE and DSPI are 0.25, 0.11, 0.11 and 0.14 for lags

TABLE I
RMSE OF ˆρX
11(h),ˆρX1,t
Qn−h(h),ˆρZ
11(h)A ND ˆρZ1,t
Qn−h(h)WH EN pi= 0.05,i= 1,2,3.
Lag h01234567
ˆρX
11(h)0.00000 0.03050 0.05138 0.06391 0.06965 0.07290 0.07506 0.07590
ˆρX1,t
Qn−h(h)0.00000 0.03257 0.05545 0.06855 0.07450 0.07740 0.08050 0.08187
ˆρZ
11(h)0.00000 0.33091 0.23215 0.16371 0.11777 0.09080 0.07516 0.06827
ˆρZ1,t
Qn−h(h)0.00000 0.06132 0.07724 0.07927 0.07782 0.07682 0.07804 0.07806
TABLE II
RMSE OF ˆρX
12(h),ˆρ(X1,t,X2,t )
Qn−h(h),ˆρZ
12(h)A ND ˆρ(Z1,t,Z2,t )
Qn−h(h)WHEN pi= 0.05,i= 1,2,3.
Lag h01234567
ˆρX
12(h)0.02595 0.03459 0.04592 0.05291 0.05451 0.05496 0.05685 0.05755
ˆρ(X1,t,X2,t)
Qn−h(h)0.02913 0.03804 0.04962 0.05748 0.05939 0.05979 0.06128 0.06269
ˆρZ
12(h)0.28934 0.26774 0.19370 0.13680 0.09784 0.07479 0.06176 0.05521
ˆρ(Z1,t,Z2,t)
Qn−h(h)0.05754 0.06356 0.06710 0.06538 0.06236 0.05965 0.06068 0.06170
TABLE III
RMSE OF ˆρZ
12(h).
PPPPP
P
pi
Lag h01234567
0.05 0.28934 0.26774 0.19370 0.13680 0.09784 0.07479 0.06176 0.05521
0.10 0.39047 0.35826 0.25729 0.17943 0.12558 0.08903 0.06950 0.05666
0.15 0.44192 0.40677 0.28982 0.20084 0.14067 0.09991 0.07345 0.06005
0.20 0.46956 0.43148 0.31149 0.21421 0.14771 0.10663 0.07947 0.06249
TABLE IV
RMSE OF ˆρ(Z1,t,Z2,t )
Qn−h(h).
PPPPP
P
pi
Lag h01234567
0.05 0.05754 0.06356 0.06710 0.06538 0.06236 0.05965 0.06068 0.06170
0.10 0.10492 0.10795 0.09833 0.08348 0.07124 0.06395 0.06177 0.05992
0.15 0.16088 0.16147 0.13579 0.10610 0.08424 0.07054 0.06293 0.05969
0.20 0.22260 0.21759 0.17829 0.13229 0.09814 0.07742 0.06682 0.05891
-2 0 2
PCE
-4 0 4
0 100 200 300 400 500 600
DSPI
Time
Fig. 3. Time plots of the U.S. personal consumption expenditures and
disposable personal income, in percentages.
h= 0,1,2,4, respectively. The cross-correlation given by (13)
are 0.3, 0.16, 0.13 and 0.18.
Now, consider the estimation of Φmatrix in (4) using
Yule-Walker equations. Based on information criteria, [22]
selected a VAR(3) model. Figure 6 presents the standard and
the robust ACFs of the residuals of the selected model. In
order to save space, only the ACF of PCE and the cross-
correlation between PCE and DSPI are shown, the others plots
presented similar behavior and are available upon request.
Contrasting both plots, it is possible to observe that the robust
ACF present higher values of cross-correlations, for example,
0.0 0.4 0.8
ACF
(a) PCE
0 5 10 15 20 25
0.0 0.4 0.8
ACF
0 5 10 15 20 25
(b) PCE & DSPI
0.0 0.4 0.8
ACF
-25 -15 -5 0
(c) DSPI & PCE
0.0 0.4 0.8
ACF
0 5 10 15 20 25
(d) DSPI
Fig. 4. ACF of PCE and DSPI.
the cross-correlation between PCE and DSPIG at lag h= 12 is
0.003 and 0.16, for the standard and robust sample estimators,
respectively. This result in in accordance with Proposition
(1) which shows that one outlier is enough to destroy the
properties of the sample ACF, and thus impacting in any other
sub-sequential estimation step.
To end this exerciser, we replaced the standard covariance
estimator by the proposed robust one in the Yule-Walker
equations. The plots are not shown to save space, but now, for

0.0 0.4 0.8
Robust ACF
(a) PCE
0 5 10 15 20 25
0.0 0.4 0.8
Robust ACF
0 5 10 15 20 25
(b) PCE & DSPI
0.0 0.4 0.8
Robust ACF
-25 -15 -5 0
(c) DSPI & PCE
0.0 0.4 0.8
Robust ACF
0 5 10 15 20 25
(d) DSPI
Fig. 5. Robust ACF of PCE and DSPI.
0.0 0.4 0.8
ACF
(a) PCE
0 5 10 15 20 25
0.0 0.4 0.8
ACF
0 5 10 15 20 25
(b) PCE & DSPI
0.0 0.4 0.8
Robust ACF
0 5 10 15 20 25
(a) PCE
0.0 0.4 0.8
Robust ACF
0 5 10 15 20 25
(b) PCE & DSPI
Fig. 6. ACF and Robust ACF of the residuals the fitted VAR(3) via Yule-
Walker.
instance, the value obtained by the classical sample estimator
of the cross-correlation between PCE and DSPI at lag h= 12
is 0.1 while the robust estimator gives 0.3.
0.0 0.4 0.8
ACF
(a) PCE
0 5 10 15 20 25
0.0 0.4 0.8
ACF
0 5 10 15 20 25
(b) PCE & DSPI
0.0 0.4 0.8
Robust ACF
0 5 10 15 20 25
(a) PCE
0.0 0.4 0.8
Robust ACF
0 5 10 15 20 25
(b) PCE & DSPI
Fig. 7. ACF and Robust ACF of the residuals of fitted VAR(3) via Yule-
Walker using the Robust covariance estimator.
V. CONCLUSIONS
The effect of additive outliers on the estimation of the
covariance and correlation matrix functions of a stationary
multivariate discrete time series was analyzed. A robust es-
timation method was proposed as a generalization of existing
results in the univariate case. Monte Carlo simulation results
have illustrated the good behavior in terms of mean square
error of the proposed robust estimator. An real data set
was analyzed where the proposed robust covariance estimator
replaced the standard covariance in the Yule-Waler equations.
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