1

Robust Shrinkage Estimation of High-dimensionalCovariance Matrices

Yilun Chen, Student Member, IEEE, Ami Wiesel, Member, IEEE, and Alfred O. Hero III, Fellow, IEEE

Abstract—We address high dimensional covariance estima-tion for elliptical distributed samples, which are also knownas spherically invariant random vectors (SIRV) or compound-Gaussian processes. Specifically we consider shrinkage methodsthat are suitable for high dimensional problems with a smallnumber of samples (large p small n). We start from a classicalrobust covariance estimator [Tyler(1987)], which is distribution-free within the family of elliptical distribution but inapplicablewhen n < p. Using a shrinkage coefficient, we regularize Tyler’sfixed point iterations. We prove that, for all n and p, the proposedfixed point iterations converge to a unique limit regardlessof the initial condition. Next, we propose a simple, closed-form and data dependent choice for the shrinkage coefficient,which is based on a minimum mean squared error framework.Simulations demonstrate that the proposed method achieves lowestimation error and is robust to heavy-tailed samples. Finally, asa real world application we demonstrate the performance of theproposed technique in the context of activity/intrusion detectionusing a wireless sensor network.

Index Terms—Covariance estimation, large p small n, shrink-age methods, robust estimation, elliptical distribution, activ-ity/intrusion detection, wireless sensor network

I. INTRODUCTION

Estimating a covariance matrix (or a dispersion matrix)is a fundamental problem in statistical signal processing.Many techniques for detection and estimation rely on accurateestimation of the true covariance. In recent years, estimatinga high dimensional p × p covariance matrix under smallsample size n has attracted considerable attention. In these“large p small n” problems, the classical sample covariancesuffers from a systematically distorted eigen-structure [6], andimproved estimators are required.

Much effort has been devoted to high-dimensional co-variance estimation, which use Steinian shrinkage [1]–[3] orother types of regularized methods such as [4], [5]. However,most of the high-dimensional estimators assume Gaussiandistributed samples. This limits their usage in many importantapplications involving non-Gaussian and heavy-tailed sam-ples. One exception is the Ledoit-Wolf estimator [2], where

Copyright (c) 2011 IEEE.Y. Chen and A. O. Hero are with the Department of Electrical Engi-

neering and Computer Science, University of Michigan, Ann Arbor, MI48109, USA. Tel: 1-734-763-0564. Fax: 1-734-763-8041. Emails: {yilun,hero}@umich.edu.

A. Wiesel is with the The Rachel and Selim Benin School of ComputerScience and Engineering at the Hebrew University of Jerusalem, 91904Jerusalem, Israel. Tel: 972-2-6584933. Email: ami.wiesel@huji.ac.il.

This work was partially supported by AFOSR, grant number FA9550-06-1-0324. The work of A. Wiesel was supported by a Marie Curie OutgoingInternational Fellowship within the 7th European Community FrameworkProgramme. Parts of this work were presented at the IEEE Workshop onSensor Array and Multichannel Signal Processing (SAM), Jerusalem, Israel,Oct. 2010.

the authors shrink the sample covariance towards a scaledidentity matrix and proposed a shrinkage coefficient which isasymptotically optimal for any distribution. On the other hand,traditional robust covariance estimators [8]–[10] designed fornon-Gaussian samples generally require n � p and are notsuitable for “large p small n” problems. Therefore, the goalof our work is to develop a covariance estimator for problemsthat are both high dimensional and non-Gaussian. In this paper,we model the samples using the elliptical distribution [7],which is also referred to as the spherically invariant randomvector model (SIRV) [26], [27] or the compound-Gaussianprocess model [13]. As a flexible and popular alternative, theelliptical family encompasses a large number of importantdistributions such as Gaussian distribution, the multivariateCauchy distribution, the multivariate exponential distribution,the multivariate Student-T distribution, the K-distribution andthe Weibull distribution. The capability of modelling heavy-tails makes the elliptical distribution appealing in signal pro-cessing and related fields. Typical applications include radardetection [13], [17], [20], [22], speech signal processing [23],remote sensing [24], wireless fading channels modelling [27],financial engineering [25] and so forth.

A well-studied covariance estimator for elliptical distri-butions is the ML estimator based on normalized samples[9], [14], [16]. The estimator is derived as the solution toa fixed point equation by using fixed point iterations. It isdistribution-free within the class of elliptical distributions andits performance advantages are well known in the n � pregime. However, it is not suitable for the “large p small n”setting. Indeed, when n < p, the ML estimator as defineddoes not even exist. To avoid this problem the authors of [21]propose an iterative regularized ML estimator that employsdiagonal loading and uses a heuristic procedure for selectingthe regularization parameter. While they did not establish con-vergence and uniqueness [21] they empirically demonstratedthat their algorithm has superior performance in the contextof a radar application. Our approach is similar to [21] butis conceived in a Steinian shrinkage framework, where weestablish convergence and uniqueness of the resultant iterativeestimator. We also propose a general procedure of select-ing the shrinkage coefficient for heavy-tailed homogeneoussamples. For a fixed shrinkage coefficient, we prove thatthe regularized fixed iterations converge to a unique solutionfor all n and p, regardless of the initial condition. Next,following Ledoit-Wolf [2], we provide a simple closed-formexpression for the shrinkage coefficient, based on minimizingmean-squared-error. The resultant coefficient is a functionof the unknown true covariance and cannot be implementedin practice. Instead, we develop a data-dependent “plug-

2

in” estimator approximation. Simulation results demonstratethat our estimator achieves superior performance for samplesdistributed within the elliptical family. Furthermore, for thecase that the samples are truly Gaussian, we report verylittle performance degradation with respect to the shrinkageestimators designed specifically for Gaussian samples [3].

As a real world application we demonstrate the proposedestimator for activity/intrusion detection using an active wire-less sensor network. We show that the measured data exhibitstrong non-Gaussian behavior and demonstrate significantperformance advantages of the proposed robust covarianceestimator when used in a covariance-based anomaly detectionalgorithm.

The paper is organized as follows. Section II provides abrief review of elliptical distributions and of Tyler’s covarianceestimation method. The regularized covariance estimator isintroduced and derived in Section III. We provide simulationsand experimental results in Section IV and Section V, respec-tively. Section VI summarizes our principal conclusions. Theproof of theorems and lemmas are provided in the Appendix.

Notations: In the following, we depict vectors in lowercaseboldface letters and matrices in uppercase boldface letters.(·)T and (·)H denote the transpose and conjugate transposeoperator, respectively. Tr(·) and det(·) are the trace and thedeterminant of a matrix, respectively.

II. ML COVARIANCE ESTIMATION FOR ELLIPTICALDISTRIBUTIONS

A. Elliptical distribution

Let x be a p × 1 real random vector generated by thefollowing model

x = νu, (1)

where ν is a real, positive random variable and u is ap × 1 zero-mean, real Gaussian random vector with positivedefinite covariance Σ. We assume that ν and u are statisticallyindependent. The resulting random vector x is ellipticallydistributed and its probability density function (pdf) can beexpressed by

p(x) = φ(xTΣ−1x

), (2)

where φ(·) is the characteristic function (Definition 2, pp. 5,[25]) related to the pdf of ν. The elliptical family encompassesmany useful distributions in signal processing and relatedfields and includes: the Gaussian distribution itself, the K dis-tribution, the Weibull distribution and many others. As statedabove, elliptically distributed samples are also referred to asSpherically Invariant Random Vectors (SIRV) or compoundGaussian processes in signal processing.

B. ML estimation

Let {xi}ni=1 be a set of n independent and identicallydistributed (i.i.d.) samples drawn according to (1). As thecovariance of x may not exist, our problem is formulatedto estimate the covariance (dispersion) matrix Σ of u from{xi}ni=1. The model (1) is invariant to scaling of the covariancematrix Σ. Therefore, without loss of generality, we assume

that the covariance matrix is trace-normalized in the sensethat Tr(Σ) = p.

The commonly used sample covariance, defined as

S =1

n

n∑i=1

xixTi , (3)

is known to be a poor estimator of Σ, especially when thesamples are high-dimensional (large p) and/or heavy-tailed.Tyler’s method [9] addresses this problem by working withthe normalized samples:

si =xi‖xi‖2

=ui‖ui‖2

, (4)

for which the term ν in (1) drops out. The pdf of si is givenby [25]

p(si; Σ) =Γ(p/2)

2πp/2·√

det(Σ−1) ·(sTi Σ−1si

)−p/2. (5)

Taking the derivative and equating to zero, the maximumlikelihood estimator based on {si}ni=1 is the solution to

Σ =p

n·n∑i=1

sisTi

sTi Σ−1si. (6)

When n > p, the ML estimator can be found using thefollowing fixed point iterations:

Σj+1 =p

n·n∑i=1

sisTi

sTi Σ−1j si, (7)

where the initial Σ0 is usually set to the identity matrix.Assuming that n > p and that any p samples out of {si}ni=1

are linearly independent with probability one, it can be shownthat the iteration process in (7) converges and that the limitingvalue is unique up to constant scale, which does not depend onthe initial value of Σ0. In practice, a final normalization stepis needed, which ensures that the iteration limit Σ∞ satisfiesTr(Σ∞) = p.

The ML estimate corresponds to the Huber-type M-estimator and has many good properties when n� p, such asasymptotic normality and strong consistency. Furthermore, ithas been pointed out [9] that the ML estimate (7) is the “mostrobust” covariance estimator in the class of elliptical distri-butions in the sense of minimizing the maximum asymptoticvariance. We note that (7) can be also motivated from otherapproaches as proposed in [14], [16].

III. ROBUST SHRINKAGE COVARIANCE ESTIMATION

Here we extend Tyler’s method to the high dimensionalsetting using shrinkage regularization. It is easy to see thatthere is no solution to (6) when n < p (its left-hand-side isfull rank whereas its right-hand-side of is rank deficient). Thismotivates us to develop a regularized covariance estimator forelliptical samples. Following [2], [3], we propose to regularizethe fixed point iterations as

Σj+1 = (1− ρ)p

n

n∑i=1

sisTi

sTi Σ−1j si+ ρI, (8)

Σj+1 =Σj+1

Tr(Σj+1)/p, (9)

3

where ρ is the so-called shrinkage coefficient, which is aconstant between 0 and 1. When ρ = 0 and n > p the proposedshrinkage estimator reduces to Tyler’s unbiased method in(6) and when ρ = 1 the estimator reduces to the trivialuncorrelated case yielding a scaled identity matrix. The termρI ensures that Σj+1 is always well-conditioned and thusallows continuation of the iterative process without the needfor restarts. Therefore, the proposed iteration can be applied tohigh dimensional estimation problems. We emphasize that thenormalization (9) is important and necessary for convergence.We establish provable convergence and uniqueness of the limitin the following theorem.

Theorem 1. Let 0 < ρ < 1 be a shrinkage coefficient. Then,the fixed point iterations in (8) and (9) converge to a uniquelimit for any positive definite initial matrix Σ0.

The proof of Theorem 1 follows directly from the concavePerron-Frobenius theory [28] and is provided in the Appendix.We note that the regularization presented in (8) and (9) issimilar to diagonal loading [21]. However, unlike the diagonalloading approach of [21], the proposed shrinkage approachprovides a systematic way to choose the regularization pa-rameter ρ, discussed in the next section.

A. Choosing the shrinkage coefficient

We now turn to the problem of choosing a good, data-dependent, shrinkage coefficient ρ, as as an alternative tocross-validation schemes which incur intensive computationalcosts. As in Ledoit-Wolf [2], we begin by assuming we“know” the true covariance Σ. Then we define the followingclairvoyant “estimator”:

Σ(ρ) = (1− ρ)p

n

n∑i=1

sisTi

sTi Σ−1si+ ρI, (10)

where the coefficient ρ is chosen to minimize the minimummean-squared error:

ρO = arg minρE

{∥∥∥Σ(ρ)−Σ∥∥∥2F

}. (11)

The following theorem shows that there is a closed-formsolution to the problem (11), which we refer to as the “oracle”coefficient.

Theorem 2. For i.i.d. elliptical distributed samples the solu-tion to (11) is

ρO =p2 + (1− 2/p)Tr(Σ2)

(p2 − np− 2n) + (n+ 1 + 2(n− 1)/p)Tr(Σ2),

(12)under the condition Tr(Σ) = p.

The proof of Theorem 2 requires the calculation of thefourth moments of an isotropically distributed random vector[30]–[32] and is provided in the Appendix.

The oracle coefficient cannot be implemented since ρO isa function of the unknown true covariance Σ. Therefore, wepropose a plug-in estimate for ρO:

ρ =p2 + (1− 2/p)Tr(M2)

(p2 − np− 2n) + (n+ 1 + 2(n− 1)/p)Tr(M2), (13)

where M can be any consistent estimator of Σ, e.g., thetrace-normalized Ledoit-Wolf estimator. Another appealingcandidate for plug-in is the (trace-normalized) normalizedsample covariance R [12] defined by:

R =p

n

n∑i=1

sisTi . (14)

We note that the only requirement on the covariance estimatorM is that it provide a good approximation to Tr(Σ2). It doesnot have to be well-conditioned nor does it have to be anaccurate estimator of the true covariance matrix Σ.

By using the plug-in estimate ρ in place of ρ, the robustshrinkage estimator is computed via the fixed point iterationin (8) and (9).

We note that our proposed minimum MSE based approachto estimate the shrinkage coefficient is completely general andmakes a minimum of assumptions on the provenance of thedata. In specific applications such as adaptive beamforming, aspecifically tailored coefficient estimator may be advantageous[21].

B. Extension to the complex case

Here we consider the scenario where the random vector x in(1) is complex elliptical distributed. In this case, ν is still a real,positive random variable but u is a complex Gaussian randomvector with covariance matrix Σ. Note that Σ is Hermitianand is assumed to be trace-normalized, i.e., Tr(Σ) = p. Thecomplex version of our fixed point iterations is

Σj+1 = (1− ρ)p

n

n∑i=1

sisHi

sHi Σ−1j si+ ρI,

Σj+1 =Σj+1

Tr(Σj+1)/p,

(15)

where si is defined by (4). As in the real case, the shrinkagecoefficient ρ is chosen to minimize (11), where the complexclairvoyant estimator Σ(ρ) is re-defined as

Σ(ρ) = (1− ρ)p

n

n∑i=1

sisHi

sHi Σ−1si+ ρI. (16)

The following theorem extends Theorems 1 and 2 to thecomplex case.

Theorem 3. For any 0 < ρ < 1 the complex valued iterationsin (15) converge to a unique limit for any positive definitiveHermitian matrix Σ0. For Σ(ρ) defined in (16), the solutionto (11) is

ρO =p2 − 1/pTr(ΣΣH)

(p2 − pn− n) + (n+ (n− 1)/p)Tr(ΣΣH)(17)

under the condition Tr(Σ) = p.

The proof of Theorem 3 is similar to that of Theorem 1 andTheorem 2 and is provided in the Appendix. In practice, ρOcan be approximated by plugging any consistent estimator ofTr(ΣΣH) in (17).

4

IV. NUMERICAL SIMULATION

In this section we use simulations to demonstrate the su-perior performance of the proposed shrinkage approach. Firstwe show that our estimator outperforms other estimators forthe case of heavy-tailed samples generated by a multivariateStudent-T distribution, where ν in (1) is a function of a Chi-square random variable:

ν =

√d

χ2d

; (18)

The degree-of-freedom d of this multivariate Student-T statis-tic is set to 3. The dimensionality p is chosen to be 100 andwe let Σ be the covariance matrix of an AR(1) process,

Σ(i, j) = r|i−j|, (19)

where Σ(i, j) denotes the entry of Σ in row i and columnj. The parameter r is set to 0.7 in this simulation. Thesample size n varies from 5 to 225 with step size 10. Allthe simulations are repeated for 100 trials and the averageempirical performance is reported.

We use (13) with M = R and employ iterations defined by(8) and (9) with ρ = ρ. For comparison, we also plot the resultsof the trace-normalized oracle in (12), the trace-normalizedLedoit-Wolf estimator [2], and the non-regularized solutionin (7) (when n > p). As the Ledoit-Wolf estimator operateson the sample covariance which is sensitive to outliers, wealso compare to a trace-normalized version of a clairvoyantLedoit-Wolf estimator implemented according to the procedurein [2] with known ν. More specifically, the samples xiare firstly normalized by the known realizations νi, yieldingtruly Gaussian samples; then the sample covariance of thenormalized xi’s is computed, which is used to estimate theLedoit-Wolf shrinkage parameters and estimate the covariancevia equation (14) in [2]. The MSE is plotted in Fig. 1 for thecase that r = 0.7. It can be observed that the proposed methodperforms significantly better than the Ledoit-Wolf estimatorin Fig. 1, and that the performance is very close to the idealoracle estimator using the optimal shrinkage parameter (12).Even the clairvoyant Ledoit-Wolf implemented with knownνi does not outperform the proposed estimator in the smallsample size regime. These results demonstrate the robustnessof the proposed approach. Although the Ledoit-Wolf estimatorperforms better when r = 0, the case where Σ = I, theproposed approach still significantly outperforms it, especiallyfor small sample size n (results not shown).

As a graphical illustration, in Fig. 2 we provide covariancevisualizations for a realization of the estimated covariancesusing the Ledoit-Wolf method and the proposed approach.The covariance matrix estimates are rendered as a heatmapin Fig. 2(a). The sample size in this example is set to 50,which is smaller than the dimension 100. Compared to the truecovariance, it is clear that the proposed covariance estimatorpreserves the structure of the true covariance, while in theLedoit-Wolf covariance procudure produces “block pattern”artifacts caused by heavy-tails of the multivariate Student-T.

When n > p, we also observe a substantial improvement ofthe proposed method over the ML covariance estimate, which

provides further evidence of the power of Steinian shrinkagefor reducing MSE.

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

n

MS

E

Oracle

Proposed

Ledoit−Wolf

ML

Ledoit−Wolf (with known ν)

Fig. 1. Multivariate Student-T samples: Comparison of different trace-normalized covariance estimators when p = 100, where r is set to 0.7.

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) True covariance

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

(b) Ledoit-Wolf

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

(c) Proposed

Fig. 2. Multivariate Student-T samples: Heatmap visualizations of the co-variance matrix estimates using the Ledoit-Wolf and the proposed approaches.p = 100, n = 50. Note that n < p in this case.

In order to assess the tradeoff between accuracy and ro-bustness we investigate the case when the samples are trulyGaussian distributed. We use the same simulation parametersas in the previous example, the only difference being thatthe samples are now generated from a Gaussian distribution.The performance comparison is shown in Fig. 3, where fourdifferent (trace-normalized) methods are included: the oracleestimator derived from Gaussian assumptions (Gaussian or-acle) [3], the iterative approximation of the Gaussian oracle(Gaussian OAS) proposed in [3], the Ledoit-Wolf estimatorand the proposed method. It can be seen that for truly Gaussiansamples the proposed method performs very closely to theGaussian OAS, which is specifically designed for Gaussiandistributions. Indeed, for small sample size (n < 20), theproposed method performs even better than the Ledoit-Wolfestimator. This indicates that, although the proposed robustmethod is developed for the entire elliptical family, it actuallysacrifices very little performance for the case that the distri-bution is Gaussian.

V. APPLICATION TO ANOMALY DETECTION IN WIRELESSSENSOR NETWORKS

In this section we demonstrate the proposed robust covari-ance estimator in a real application: activity detection using awireless sensor network.

The experiment was set up on an Mica2 sensor networkplatform, as shown in Fig. 4, which consists of 14 sensornodes randomly deployed inside and outside a laboratory atthe University of Michigan. Wireless sensors communicated

5

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

0.025

0.03

n

MS

E

Oracle

Proposed

Gaussian OAS

Ledoit−Wolf

Fig. 3. Gaussian samples: Comparison of trace-normalized different covari-ance estimators when p = 100.

with each other asynchronously by broadcasting an RF signalevery 0.5 seconds. The received signal strength (RSS), de-fined as the voltage measured by a receiver’s received signalstrength indicator circuit (RSSI), was recorded for each pair oftransmitting and receiving nodes. There were 14 × 13 = 182pairs of RSSI measurements over a 30 minute period, andsamples were acquired every 0.5 sec. During the experimentperiod, persons walked into and out of the lab at random times,causing anomaly patterns in the RSSI measurements. Finally,for ground truth, a web camera was employed to record theactual activity.

Fig. 4. Experimental platform: wireless Mica2 sensor nodes.

Fig. 5 shows all the received signals and the ground truthindicator extracted from the video. The objective of thisexperiment was intrusion (anomaly) detection. We emphasizethat, with the exception of the results shown in Fig. 10, theground truth indicator is only used for performance evaluationand the detection algorithms presented here were completelyunsupervised.

To remove temperature drifts [36] of receivers we detrendedthe data as follows. Let xi[k] be the k-th sample of the i-thRSS signal and denote

x[k] = (x1[k], x2[k], . . . , x182[k])T. (20)

The local mean value of x[k] is defined by

x[k] =1

2m+ 1

k+m∑i=k−m

x[k], (21)

where the integer m determines local window size and is setto 50 in this study. We detrend the data by subtracting thislocal mean

y[k] = x[k]− x[k], (22)

0 500 1000 1500 2000 2500 3000 35000

50

100

150

200

250

Time sample/0.5 sec

Anomalies

Groundtruth

RSSdata

Fig. 5. At bottom 182 RSS sequences sampled from each pair of transmittingand receiving nodes in intrusion detection experiment. Ground truth indicatorsat top are extracted from video captured from a web camera that recorded thescene.

yielding a detrended sample y[k] used in our anomaly detec-tion.

We established that the detrended measurements wereheavy-tailed non-Gaussian by performing several statisticaltests. First the Lilliefors test [37] of Gaussianity was performedon the detrended RSS measurements. This resulted in rejectionof the Gaussian hypothesis at a level of significance of 10−6.As visual evidence, we show the quantile-quantile plot (QQplot) for one of the detrended RSS sequences in Fig. 6 whichillustrates that the samples are non-Gaussian. In Fig. 7, we plotthe histograms and scatter plots of two of the detrended RSSsequences, which shows the heavy-tail nature of the sampledistribution. This strongly suggests that the RSS samples canbe better described by a heavy-tailed elliptical distribution thanby a Gaussian distribution. As additional evidence, we fitted aStudent-T distribution to the first detrended RSS sequence, andused maximum likelihood to estimate the degree-of-freedomas d = 2 with a 95% confidence interval (CI) [1.8460, 2.2879].

−4 −3 −2 −1 0 1 2 3 4−6

−4

−2

0

2

4

6

Standard Normal Quantiles

Qu

an

tile

s o

f In

pu

t S

am

ple

Fig. 6. QQ plot of data versus the standard Gaussian distribution.

Consider the following discriminant based on the detrendeddata:

tk = sT [k]Σ−1s[k], (23)

where, similarly to (4), s[k] = y[k]/‖y[k]‖2 and Σ is given by

6

−50 0 500

100

200

300

400

500

600

−50 0 500

100

200

300

400

−40 −20 0 20 40−30

−20

−10

0

10

20

30

−40 −20 0 20 40−30

−20

−10

0

10

20

30

Histogram

Dist. fit

Histogram

Dist. fit

Fig. 7. Histograms and scatter plots of the first two de-trended RSSsequences, which are fit by a multivariate Student-T distribution with degree-of-freedom d = 2.

the solution to (6). A time sample is declared to be anomalousif the test statistic tk exceeds a specified threshold. The statistic(23) is equivalent to a robustified version of the Mahalanobisdistance anomaly detector [33]. Note that direct application ofshrinkage to the sample covariance of y[k] would be prob-lematic since a multivariate Student-T vector with 2 degreesof freedom has no second order moments. The test statistic(23) can be interpreted as a shrinkage robustified Mahalanobisdistance test applied to the better behaved variable s[k] that hasfinite moments of all orders. Specifically, we constructed the182 × 182 sample covariance by randomly subsampling 200time slices from the RSS data shown in Fig. 5. Note, that these200 samples correspond to a training set that is contaminatedby anomalies at the same anomaly rate (approximately 10%) asthe entire sample set. The detection performance was evaluatedusing the receiver operating characteristic (ROC) curve, wherethe averaged curves from 200 independent Monte-Carlo trialsare shown in Fig. 8. For comparison, we also implementedthe activity detector (23) with other covariance estimatesincluding: the sample covariance, the Ledoit-Wolf estimatorand Tyler’s ML estimator.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

False alarm rate

De

tectio

n r

ate

Sample covariance

Proposed

Ledoit−Wolf

Tyler’s ML

Fig. 8. Performance comparison for different covariance estimators, p =182, n = 200.

From the mean ROCs we can see that the detection perfor-mances are rank ordered as follows: Proposed > Ledoit-Wolf> Tyler’s ML > Sample covariance. The sample covariance

performs poorly in this setting due to the small sample size(n = 200, p = 182) and its sensitivity to the heavy-taileddistribution shown in Fig. 6 and 7. The Tyler ML methodand the Ledoit-Wolf estimator improve upon the sample co-variance since they compensate for heavy tails and for smallsample size, respectively. Our proposed method compensatesfor both effects simultaneously and achieves the best detectionperformance.

We also plot the 90% confidence envelopes, determinedby cross-validation, on the ROCs in Fig. 9. The width ofthe confidence interval reflects the sensitivity of the anomalydetector to variations in the training set. Indeed, the upper andlower endpoints of the confidence interval are the optimisticand the pessimistic predictions of detection performance. Theproposed method achieves the smallest width among the fourcomputed 90% confidence envelopes.

0 0.5 10

0.5

1

False alarm rate

Dete

ction r

ate

Sample covariance

0 0.5 10

0.5

1

False alarm rate

Dete

ction r

ate

Proposed

0 0.5 10

0.5

1

False alarm rate

Dete

ction r

ate

Ledoit−Wolf

0 0.5 10

0.5

1

False alarm rate

Dete

ction r

ate

Tyler’s ML

Fig. 9. Performance comparison for different covariance estimators, includingthe mean value and 90% confidence intervals. (a): Sample covariance. (b):Proposed. (c): Ledoit-Wolf. (d): Tyler’s ML. The 200 training samples arerandomly selected from the entire data set.

Finally, for completeness we provide performance com-parison of covariance-based supervised activity detection al-gorithms in Fig. 10. The training period is selected to be[251, 450] based on ground truth where no anomalies appear.It can be observed that, by excluding the outliers caused byanomalies, the performance of the Ledoit-Wolf based intrusiondetection algorithm is close to that of the proposed method.We conclude that the activity detection performance of theproposed covariance estimator is more robust than the otherthree estimators with respect to outlier contamination in thetraining samples.

VI. CONCLUSION

In this paper, we proposed a shrinkage covariance estimatorwhich is robust over the class of elliptically distributed sam-ples. The proposed estimator is obtained by fixed point itera-tions, and we established theoretical guarantees for existence,convergence and uniqueness. The optimal shrinkage coefficientwas derived using a minimum mean-squared-error frameworkand has a closed-form expression in terms of the unknown truecovariance. This expression can be well approximated by asimple plug-in estimator. Simulations suggest that the iterativeapproach converges to a limit which is robust to heavy-tailedmultivariate Student-T samples. Furthermore, we show that

7

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

False alarm rate

De

tectio

n r

ate

Sample covariance

Proposed

Ledoit−Wolf

Tyler’s ML

Fig. 10. Performance comparison for different covariance estimators, p =182, n = 200. The covariance matrix is estimated in a supervised manner.

for the Gaussian case, the proposed estimator performs veryclosely to previous estimators designed expressly for Gaussiansamples.

As a real world application we demonstrated the perfor-mance of the proposed estimator in intrusion detection usinga wireless sensor network. Implementation of a standardcovariance-based detection algorithm using our robust covari-ance estimator achieved superior performances as comparedto conventional covariance estimators.

The basis of the proposed method is the ML estimatororiginally proposed by Tyler in [9]. However, the approachpresented in this paper can be extended to other M-estimators.

One of the main contributions of our work is the proofof uniqueness and convergence of the estimator. This proofextends the results of [9], [18] to the regularized case. Re-cently, an alternative proof to the non-regularized case usingconvexity on manifolds was presented in [35]. This latter proofhighlights the geometrical structure of the problem and givesadditional insight. We are currently investigating its extensionto the regularized case.

VII. ACKNOWLEDGEMENT

The authors would like to thank Neal Patwari for designingthe intrusion detection experiment and collecting the data. Wealso thank Harry Schmidt and his group at Raytheon, TucsonAZ, for lending us the Mica2 motes for this experiment. Wewould like to thank Prof. Drton for fruitful discussions and apreprint of [38].

VIII. APPENDIX

A. Proof of Theorem 1

In this appendix we prove Theorem 1. The original con-vergence proof for the non-regularized case in [9], [18] isbased on careful exploitation of the specific form of (6). Inthe contrast, our proof for the regularized case is based on adirect connection from concave Perron-Frobenius theory [28],[29]. We begin by summarizing the required concave Perron-Frobenius result in the following lemma.

Lemma 1 ( [28]). Let (E, ‖·‖) be a Banach space with K ⊂ Ebeing a closed, convex cone on which ‖·‖ is increasing, i.e., for

which x ≤ y implies ‖x‖ ≤ ‖y‖, where the operator ≤ on theconvex cone K means that if x ≤ y then y − x ∈ K. DefineU = {x|x ∈ K, ||x|| = 1}. Let T : K → K be a concaveoperator such that

T (µx+ (1− µ)y) ≥ µT (x) + (1− µ)T (y),

for all µ ∈ [0, 1], all x, y ∈ K.(24)

If for some e ∈ K − {0} and constants a > 0, b > 0 there is

ae ≤ T (x) ≤ be, for all x ∈ U, (25)

then there exists a unique x∗ ∈ U to which the iteration ofthe normalized operator T (x) = T (x)/‖T (x)‖, x ∈ K − {0}converges:

limk→∞

T k(x) = x∗, for all x ∈ K − {0}. (26)

Lemma 1 can be obtained by combining results fromLemma 2 and Theorem in Section 4 of [28]. Here we show thatthe proof of Theorem 1 is a direct result of applying Lemma1 with proper definitions of E, K, U and T :• E: the set of all symmetric matrices;• K: the set of all positive semi-definite matrices on E;• ‖Σ‖: the normalized nuclear norm of Σ, i.e.,

‖Σ‖ =1

p

p∑j=1

|λj |, (27)

where λj is the j-th eigenvalue of Σ and |·| is the absolutevalue operator. Note that for any Σ ∈ K, the nuclearnorm ‖ · ‖ is identical to Tr(·)/p and is increasing;

• U : the set U = {Σ|Σ ∈ K, ‖Σ‖ = 1};• T : the mapping from K to K defined by

T (Σ) = (1− ρ)p

n

n∑i=1

w(si,Σ)sisTi + ρI, (28)

where the weight function w(si,Σ) is defined as

w(si,Σ) = infzT si 6=0

zTΣz

(sTi z)2, (29)

for any Σ ∈ K.Proof: With the above definitions we show that Theorem

1 is a direct result of Lemma 1. We begin by showing thatthe mapping operator T is concave. Indeed, it is sufficient toshow that w(si,Σ) in concave in Σ, which is true because itis the infinimum of affine functions of Σ.

Next, we show that T satisfies condition (25) with e = I.It is easy to see that

ρI ≤ T (Σ), (30)

for any Σ ∈ U . Then we show that

w(si,Σ) ≤ p, (31)

for any Σ ∈ U . Indeed,

w(si,Σ) = infzT si 6=0

zTΣz

(sTi z)2≤ sTi Σsi

(sTi si)2≤ λmax

sTi si= λmax,

(32)where λmax is the maximum eigenvalue of Σ. The last equalityin the right-hand-side of (32) comes from the fact that si is of

8

unit norm by definition (4). (31) is thus obtained by noticingthat Σ ∈ U and λmax ≤ p. Substituting (31) into (28) we have

T (Σ) ≤ (1− ρ)p2R + ρI ≤((1− ρ)p2αmax + ρ

)I, (33)

where

R =1

n

n∑i=1

sisTi ,

and αmax is the maximum eigenvalue of R. Again, as si isof unit norm, αmax ≤ Tr(R) = 1 and

T (Σ) ≤((1− ρ)p2 + ρ

)I. (34)

Therefore, we have shown that T satisfies condition (25),where e = I, a = ρ and b = (1 − ρ)p2 + ρ. In addition,(25) establishes that the mapping T from U always yields apositive definite matrix. Therefore, the convergent limit of thefixed-point iteration is positive definite.

Finally, we note that, for any Σ � 0, we have

‖Σ‖ =Tr(Σ)

p, (35)

and

w(si,Σ) = infzT si 6=0

zTΣz

(sTi z)2=

1

sTi Σ−1si. (36)

The limit (26) is then identical to the limit of proposediterations (8) and (9) for any Σ � 0. Therefore, Theorem1 has been proved.

B. Proof of Theorem 2

Proof: To ease the notation we define C as

C =p

n

n∑i=1

sisTi

sTi Σ−1si. (37)

The shrinkage estimator in (10) is then

Σ(ρ) = (1− ρ)C + ρI. (38)

By substituting (38) into (10) and taking derivatives of ρ, weobtain that

ρO =E{

Tr(

(I− C)(Σ− C))}

E

{∥∥∥I− C∥∥∥2F

}=m2 −m11 −m12 + Tr(Σ)

m2 − 2m11 + p,

(39)

wherem2 = E

{Tr(C2)

}, (40)

m11 = E{

Tr(C)}, (41)

andm12 = E

{Tr(CΣ)

}. (42)

Next, we calculate the moments. We begin by eigen-decomposing Σ as

Σ = UDUT , (43)

and denoteΛ = UD1/2. (44)

Then, we define

zi =Λ−1si‖Λ−1si‖2

=Λ−1ui‖Λ−1ui‖2

. (45)

Noting that ui is a Gaussian distributed random vector withcovariance Σ, it is easy to see that ‖zi‖2 = 1 and zi and zjare independent with each other for i 6= j. Furthermore, zi isisotropically distributed [30]–[32] and satisfies [3]

E{ziz

Ti

}=

1

pI, (46)

E{(

zTi Dzi)2}

=1

p(p+ 2)

(2Tr(D2) + Tr2(D)

)=

1

p(p+ 2)

(2Tr(Σ2) + Tr2(Σ)

),

(47)

and

E{(

zTi Dzj)2}

=1

p2Tr(D2) =

1

p2Tr(Σ2), i 6= j. (48)

Expressing C in terms of zi, there is

C =p

nΛ

n∑i=1

zizTi ΛT . (49)

Then,

E{

C}

=p

nΛ

n∑i=1

E{ziz

Ti

}ΛT = Σ, (50)

and accordingly we have

m11 = E{

Tr(C)}

= Tr(Σ), (51)

andm12 = E

{Tr(CΣ)

}= Tr(Σ2). (52)

For m2 there is

m2 =p2

n2E

Tr

Λ

n∑i=1

zizTi ΛTΛ

n∑j=1

zjzTj ΛT

=p2

n2E

Tr

n∑i=1

n∑j=1

zizTi ΛTΛzjz

Tj ΛTΛ

=p2

n2E

Tr

n∑i=1

n∑j=1

zizTi Dzjz

Tj D

=p2

n2

n∑i=1

n∑j=1

E{(

zTi Dzj)2}

.

(53)

Now substitute (47) and (48) to (53):

m2 =p2

n2

(n

p(p+ 2)

(2Tr(Σ2) + Tr2(Σ)

)+

n(n− 1)

p2Tr(Σ2)

)=

1

n(1 + 2/p)

(2Tr(Σ2) + Tr2(Σ)

)+ (1− 1

n)Tr(Σ2)

=

(1− 1

n+

2

n(1 + 2/p)

)Tr(Σ2) +

Tr2(Σ)

n(1 + 2/p).

(54)Recalling Tr(Σ) = p, (12) is finally obtained by substituting

(51), (52) and (54) into (39).

9

C. Proof of Theorem 3

The proof of convergence and uniqueness is a simpleextension of the proof in Appendix A by re-defining E asthe set of all Hermitian matrices and the mapping functionT (Σ) as

T (Σ) = (1− ρ)p

n

n∑i=1

w(si,Σ)sisHi + ρI, (55)

where

w(si,Σ) = infzHsi 6=0

zHΣz

|sHi z|2. (56)

For the oracle coefficient, define

C =n

p

n∑i=1

sisHi

sHi Σ−1si. (57)

Then Σ(ρ) = (1− ρ)C + ρI. It can be shown that

ρO =E{

Re(

Tr(

(I− C)(Σ− C)H))}

E{‖I− C‖2F

}=m2 −m11 −m12 + Tr(ΣH)

m2 − 2m11 + p,

(58)

where m2, m11 and m12 are re-defined as

m2 = E{

Tr(CCH

)}, (59)

m11 = E{

Re(

Tr(CH

))}(60)

andm12 = E

{Re(

Tr(CΣH

))}, (61)

respectively.Next, we eigen-decompose the Hermitian positive definite

matrix Σ asΣ = UDUH , (62)

where D is a real diagonal matrix and U is a complex unitarymatrix. Define

Λ = UD1/2 (63)

andzi =

Λ−1si‖Λ−1si‖2

. (64)

{zi}ni=1 are then complex valued isotropically distributedrandom vectors and are independent to each other [30]–[32].Using results from [34], it can be shown that

E{ziz

Hi

}=

1

pI, (65)

E{∣∣zHi Dzi

∣∣2} =1

p(p+ 1)

(Tr(D2) + Tr2(D)

)=

1

p(p+ 1)

(Tr(ΣΣH) + Tr2(Σ)

) (66)

and

E{∣∣zHi Dzj

∣∣2} =1

p2Tr(D2) =

1

p2Tr(ΣΣH), i 6= j. (67)

Eq. (65) ∼ (67) are complex versions of (46) ∼ (48).Expressing C in terms of zi, there is

C =p

nΛ

n∑i=1

zizHi ΛH , (68)

and accordinglyE{

C}

= Σ. (69)

As Re(·), Tr(·) and E {·} are exchangeable to each other, wehave

m11 = Re(Tr(ΣH)

)= Tr(Σ) (70)

andm12 = Tr(ΣΣH). (71)

For m2, using a similar derivation as in Appendix B, it canbe shown that

m2 =p2

n2

n∑i=1

n∑j=1

E{∣∣zHi Dzj

∣∣2}=

(1− 1

n(p+ 1)

)Tr(ΣΣH) +

p

n(p+ 1)Tr2(Σ).

(72)As Tr(ΣH) = Tr(Σ) = p, (17) can be finally obtained bysubstituting (70), (71) and (72) to (58).

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Yilun Chen (S’05) received the B.Eng. and M.Sc.degrees in electrical engineering from Tsinghua Uni-versity, Beijing, China, in 2005 and 2007, respec-tively. He is currently a Ph.D candidate with theDepartment of Electrical Engineering and ComputerScience, University of Michigan, Ann Arbor.

He performed research in statistical algorithms ofradar imaging at High Speed Signal Processing andNetwork Research Institute, Department of Electri-cal Engineering, Tsinghua University, between 2005and 2007. His current research interests include

theories and applications of statistical and adaptive signal processing, machinelearning and convex optimization.

Ami Wiesel (S’02–M’09) received the B.Sc. andM.Sc. degrees in electrical engineering from Tel-Aviv University (TAU), Tel-Aviv, Israel, in 2000 and2002, respectively, and the Ph.D. degree in electri-cal engineering from the Technion-Israel Instituteof Technology, Haifa, Israel, in 2007. From 2007to 2009, he was a Postdoctoral Fellow with theDepartment of Electrical Engineering and ComputerScience, University of Michigan, Ann Arbor.

Currently, he is a faculty member at the TheRachel and Selim Benin School of Computer Sci-

ence and Engineering at the Hebrew University of Jerusalem, Israel.Dr. Wiesel was a recipient of the Young Author Best Paper Award for a

2006 paper in the IEEE Transactions in Signal Processing and a Student PaperAward for the 2005 Workshop on Signal Processing Advances in WirelessCommunications (SPAWC) paper. He was awarded the Weinstein Study Prizein 2002, the Intel Award in 2005, the Viterbi Fellowship in 2005 and 2007,and the Marie Curie Fellowship in 2008.

Alfred O. Hero (S’79–M’84–SM’96–F’98) receivedthe B.S. (summa cum laude) from Boston Univer-sity (1980) and the Ph.D from Princeton Univer-sity (1984), both in Electrical Engineering. Since1984 he has been with the University of Michigan,Ann Arbor, where he is the R. Jamison and BettyProfessor of Engineering. His primary appointmentis in the Department of Electrical Engineering andComputer Science and he also has appointments,by courtesy, in the Department of Biomedical En-gineering and the Department of Statistics. In 2008

he was awarded the the Digiteo Chaire d’Excellence, sponsored by DigiteoResearch Park in Paris, located at the Ecole Superieure d’Electricite, Gif-sur-Yvette, France. He has held other visiting positions at LIDS MassachussetsInstitute of Technology (2006), Boston University (2006), I3S Universityof Nice, Sophia-Antipolis, France (2001), Ecole Normale Superieure deLyon (1999), Ecole Nationale Superieure des Telecommunications, Paris(1999), Lucent Bell Laboratories (1999), Scientific Research Labs of the FordMotor Company, Dearborn, Michigan (1993), Ecole Nationale Superieure desTechniques Avancees (ENSTA), Ecole Superieure d’Electricite, Paris (1990),and M.I.T. Lincoln Laboratory (1987 - 1989).

Alfred Hero is a Fellow of the Institute of Electrical and Electronics Engi-neers (IEEE). He has been plenary and keynote speaker at major workshopsand conferences. He has received several best paper awards including: aIEEE Signal Processing Society Best Paper Award (1998), the Best OriginalPaper Award from the Journal of Flow Cytometry (2008), and the BestMagazine Paper Award from the IEEE Signal Processing Society (2010). Hereceived a IEEE Signal Processing Society Meritorious Service Award (1998),a IEEE Third Millenium Medal (2000) and a IEEE Signal Processing SocietyDistinguished Lecturership (2002). He was President of the IEEE SignalProcessing Society (2006-2007). He sits on the Board of Directors of IEEE(2009-2011) where he is Director Division IX (Signals and Applications).

Alfred Hero’s recent research interests have been in detection, classification,pattern analysis, and adaptive sampling for spatio-temporal data. Of particularinterest are applications to network security, multi-modal sensing and tracking,biomedical imaging, and genomic signal processing.