An alternative estimator for the number of factors for high-dimensional time series. A robust approach.

Transcription

1 An alternative estimator for the number of factors for high-dimensional time series. A robust approach. Valdério A. Reisen DEST-CCE/PPGEA/PPGECON - Federal University of Espírito Santo, Brazil valderioanselmoreisen@gmail.com CentraleSupélec, Gif-Sur-Yvette

2 Summary Time Domain: A robust estimator of the ACF 5 Factor Analysis - Simulation cases 6 7

3 Abstract This paper considers the factor modeling for high-dimensional time series with short and long-memory properties and in the presence of additive outliers. The factor model studied by Lam and Yao (2012) is extended to the presence of additive outliers. The estimators of the number of factors are obtained by the robust covariance matrix. The methodology is analyzed in terms of the convergence rate of the number factors and by means of Monte Carlo simulations. with the aiming to reduce the dimensionality of the data set: The pollutant PM 10 in the Greater Vitória region (ES, Brazil).

4 : Main topics of this talk : Main topics of this talk : MOTIVATIONS FROM REAL APPLICATION. : MOTIVATIONS FROM REAL APPLICATION. Factor Analysis (FA) for multivariate time series with long-memory and short-memory properties and outliers; A robust dimension reduction estimator for the number of components in FA is proposed (an extension Lam and Yao (2012)); A simulation study to show the perfomance of the method for TS under additive outliers The application of the suggested estimator to a real data set. The pollutant PM 10 in the Greater Vitória region (ES, Brazil).

5 The RAMQAr - The location Figure 1: Geographical location of RAMQAr s stations.

6 : POLLUTION CONCENTRATIONS VARIABLES PM 10 concentrations (µg/m 3 ) Sua Camburi Carapina Laranjeiras Cariacica VVCentro Ibes VixCentro Time Time Figure 2: PM 10 concentrations of RAMQAr s stations

7 : Main topics of this talk : MOTIVATIONS FROM REAL APPLICATION. : MOTIVATIONS FROM REAL APPLICATION. There are some high levels of concentrations and long dependence among the air pollution data from AAQMN of Greater Vitória Region. High level of concentrations health impact. High level of concentrations may be outliers.

8 : Main topics of this talk : MOTIVATIONS FROM REAL APPLICATION. : MOTIVATIONS FROM REAL APPLICATION. Lemma 1 (Fajardo et al. (2009)) Suppose that y 1, y 2,..., y n is a set of time series observations and let ρ y (h) = γ y (h)/ γ y (0), then i. For m = 1 ( one outlier), lim n lim ρ y (h) = 0. ω ii. For m = 2 and T 2 = T 1 + l, such that h < T 1 < T 2 < n h, we have { { } lim limω 1 ρ 0, if h l, y (h) = n ω 2 ± ±0.5, if h = l.

9 : Main topics of this talk : MOTIVATIONS FROM REAL APPLICATION. : MOTIVATIONS FROM REAL APPLICATION. Proposition 1 (Cotta et al. (2017)) Suppose that Z 1,t, Z 2,t,..., Z n,t is a set of k-dimensional time series observations of Model 15 and m is the expected number of additive outliers as stated in (15). Let ˆρ Z ij (h) = γz ij (h)/( γii γ Z (0) jj Z (0)), for i, j = 1,..., k, then a. For m = 1 (one outlier occurring only at Z i,t ), lim plim ˆρ Z n ij (h) = 0. ω i

10 : Main topics of this talk : MOTIVATIONS FROM REAL APPLICATION. : MOTIVATIONS FROM REAL APPLICATION. Proposition 1 (cont.) b. For m = 2 (two outliers occurring at Z i,t or/and at Z j,t ) and assuming that ˆγ Z ij (h) 0, for Z i,t and Z j,t, it follows lim plim n ω i and/or ω j ˆρ Z ij (h) = 0. In (a.) and (b.), w i and w j are the magnitudes of the additive outliers occurring at position i and j.

11 A robust estimator of ACF Rousseeuw & Croux (1993) proposed a robust scale estimator function which is based on the kth order statistic of ( n 2) distances { y i y j, i < j}, and can be written as Q n (y) = c { y i y j ; i < j} (k), (1) where y = (y 1, y 2,..., y n ), c is a constant used to guarantee consistency (c = for the normal distribution), and ( n k = + 1. Rousseeuw & Croux (1993) showed that the 2)+2 4 asymptotic breakdown point of Q n ( ) is 50%, which means that the time series can be contaminated by up to half of the observations with outliers and Q n ( ) will still yield sensible estimates.

12 Robust Sample autocovariance Ma & Genton (2000) suggested the following robust sample autocovariance function γ Q (h) = 1 4 [ Q 2 n h (u + v) Q 2 n h (u v)], (2) where u and v are vectors containing the initial n h and the final n h observations, respectively. Note that, the above ACF is not necessarily positive semi-definite.

13 Robust Sample autocovariance The references below are background works which present theoretical and applied results related to the robust ACF γ Q (h) MOLINARES F. A. M., REISEN, V. A., CRIBARI NETO, Francisco. Robust estimation in Long-memory processes under additive outliers. Journal of Statistical Planning and Inference,139, , C. LÉVY-LEDUC, H. BOISTARD,, MOULINES, E. MURAD S TAQQU and REISEN, V. A. Robust estimation of the scale and the autocovariance functions in short and long-range dependence. Journal of Time Series Analysis,32 (2), Souza, I., REISEN, V. A. Franco,G. Bondon, P. The estimation and testing of the cointegration order based on the frequency domain. IN Journal of Business & Economic Statistics, REISEN, V. A., MONTE, E. Z. A.,A. M., G.C.FRANCO, SGRANCIO, MOLINARES F., BONDON, P., ZIELGELMANN, F.A. ABRAHAM,B. Fractional seasonal process with outliers to model and forecast daily average SO 2 concentrations. IN Mathematical and Computers in simulation COTTA, H., REISEN, V., BONDON, P., STUMMER, WOLFGANG. Robust estimation of covariance and correlation functions of a stationary multivariate process. IN EUSIPCO

14 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Let Z t, t Z, be a k-dimensional zero-mean vector of an observed time series. Also, let x t be an unobserved r-dimensional vector of common factors. It is assumed that these series are generated by r (r k) factors, x t, plus a measurement error ɛ t as z t = Px t + ɛ t, (3) where P is a k r matrix of parameters of rank r ( factor loading matrix), and ɛ t is a k-dimensional white-noise sequence with full-rank covariance matrix Σ ɛ. Thus, all the common dynamic structure comes through the common factors, x t. Assumes that P P = I. When r is much smaller than k an effective dimension-reduction is achieved. Z t is driven by a much lower-dimension process x t.

15 Estimation Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references The key to the inference for the model in Equation (3) is to determine the number of factors r and to estimate the k r factor loading matrix P, or more precisely the factor loading space Ω(P). Once an estimator is obtained, say, P, a natural estimator for the factor process is and the resulting residuals are x t = P z t, (4) ɛ t = (I d P P )z t. (5) The estimation of P is suggested by Lam & Yao (2012).

16 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Suppose that the vector of common factors x t = (x 1,t,..., x r,t ), t Z, follows a zero-mean r-dimensional stationary vector Fractional Autoregressive Moving Average process (VARFIMA(p x, d x, q x )) given by where φ x (B)Dx [(1 B) d ]xt = θ x (B)at, (6) φ x (B) = I φ 1 B φ pb p θ x (B) = I + θ 1 B + + θ qb q are matrix polynomials in the backshift operator B, the φ s and the θ s are r r matrices, the roots of the determinant polynomial φ x (B) are all outside the unit circle, and those of θ x (B) are all outside the unit circle. In addition, a t is a sequence of r r vector Gaussian White noise with zero mean and covariance matrix Σ a.

17 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Remark 1 The VARFIMA process x t, t Z, can be written as an infinite stationary second-order moving average representation as follows: x t = Ψ j a t j, (7) j=0 where the innovations a t = [a 1,t,..., a r,t] are r-dimensional martingale differences with respect to an increasing sequence of σ-fields {Ϝ t} such that for some λ > 0, sup te( a i,t 2+λ Ϝ t 1 ) <, a.s., for all i = 1,..., r. Let E(a ta t Ϝ t 1) = Σ a, a.s. The r r matrix coefficients Ψ j are often referred to as impulse responses. The main characterization of the process x t considered in this paper is that impulses responses Ψ j converge at slow hyperbolic rates as j. More precisely, there are r memory parameters d 1, d 2,..., d r, whose values lie in (0, 0.5) such that the impulse responses Ψ j can be approximated by

18 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Remark 1 (cont.) Ψ j D [ ] 1 Γ(d) jd 1 Π, as j, (8) where Γ( ) is a gamma function and Π is a nonsingular r r matrix of constants that are independent of j and may be functions of a smaller set of unknown parameters. The notation D[j d 1 /Γ(j)] represents a r r diagonal matrix with j d 1 1 /Γ(d 1 ),..., j dr 1 /Γ(d r ) on the diagonal. In fact, for any univariate function f of a single variable, the notation D[f (d)] represents r r diagonal matrix with f (d 1 ),..., f (d r ) on the diagonal. Also, the notation is defined as follows: given two sequences of matrices U j and V j, as j, for i and r, where u i,r,j and v i,r,j are the (i, r)th elements of U j and of V j, respectively. Let ψ i,j and π i be the ith rows of Ψ j and Π, respectively; then Equation (8) implies that ψ i,j j d 1 1 Γ(d i ) 1 π i, as j, for all i = 1,..., r. Note that the conditions on the memory parameters d i (0, 0.5), for i = 1,..., r, ensure that the impulse responses are square-summable and the infinite sum in Equation (7) exists.

19 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Remark 2 The autocovariances Γ(j) Cov(x t, x t+j ) of the process x t must also converge at hyperbolic rates as follows (Chung (2002)). Γ(j) D(j d 0.5 ).A.D(j d 0.5 ), as j, (9) where the (i, r)th element of the r r matrix A is Gamma(1 d i d r )/[Γ(d r ) Γ(1 d r )].π i Σ aπ r.

20 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Remark 2 (Cont.) Hence, as j, not only do the autocovariances γ i,i (j) of each x i,t die out slowly at a hyperbolic rate, i.e, γ i,i (j) j 2d i 1.Γ(1 2d i )/[Γ(d i )Γ(1 d i )].π i Σ aπ i, the covariances γ i,r (j) between the current x i,t and the future x r,t+j, for i r, also vanish at hyperbolic rates, i.e., γ i,r (j) j d i +d r 1 Γ(1 d i d r )/[Γ(d r ) Γ(1 d r )].π i Σ aπ r. Hosking (1996) presents the result for the univariate case.

21 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Lemma 2 Let z t = Px t + ɛ t as defined in Equations (3) and (6) and assume that Γ z (h) = E[z t h z t] are the covariance matrices of the process z t and Γ x (h) = E[x t h x t] are the covariance matrices for the generating vector x t. Then Γ z (0) = PΓ x (0)P + Σ ɛ, (10) Γ z (h) = PΓ x (h)p, h 1, (11) where rank(γ z (h)) = rank(γ x (h)), as h 1.

22 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Lemma 3 If the factors are independent for all lags and the matrix Σ ɛ is diagonal, then a) All of the covariance matrices Γ x (h) are diagonal; b) The matrices Γ z (h) are symmetric for h 1; c) By Spectral Decomposition, the columns of P will be eigenvectors of Γ z (h) with eigenvalues γ i (h), where γ i (h) are the diagonal elements of Γ x (h).

23 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Proposition 2 Suppose z t = Px t + ɛ t, where x t is a r-dimensional VARFIMA(p x, d x, q x ) process, P is a k r matrix (k r) of rank r, and ɛ t is a k-dimensional white noise sequence with covariance Σ ɛ. Then z t follows a k-dimensional VARFIMA(p z, d z, q z ) with p z = p x, d z = d x and q z = max(p x, q x ).

24 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Remark 3 In Equation (6), consider y t = D x [(1 B) d ]x t. Then, if the parameters of long memory (d i ) are all equal to zero, the model presented in Equation 6 becomes the short memory model described by Peña & Box (1987), i.e., a VARMA(p, q). Thus, φ x (B)D x [(1 B) d ]x t = θ x (B)a t becomes φ y (B)y t = θ y (B)a t.

25 Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Estimation and test Following the same line as Lam & Yao (2012), the estimation of P if performed by an eigenanalysis on h 0 ˆM = ˆΓ z (h)ˆγ z (h), (12) h=1 where h 0 is a prescribed integer and ˆΓ z (h) denotes the sample covariance matrix of z t at lag h. ratio-based estimator for r. The estimator for the number of factors r is given by: r = argmin ˆλ i+1 /ˆλ i, (13) 1 i R where ˆλ 1... ˆλ k are the eigenvalues of M, and r < R < k is a constant. As suggested in Lam and Yao (2012), in practice, R = p/2 can be the starting point.

26 ROBUST Estimation for P and r Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Based on Equation 12 and on the robust ACF estimator, the robust M estimator is here suggested as M Qn = h 0 h=1 Γ z,qn (h) Γ z,qn (h). (14) where Γ z,qn (h) denotes the sample robust covariance matrix of z t at lag h. Therefore, the estimator r Qn for the number of factors is similarly obtained from Equation (13). Note that M Qn is positive semi-definite.

27 Theoretical results Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Proposition 3 Let h be a fixed positive integer and ( ΓQ (h)) 1 i,j p ) = ( γ Qi,j (h) 1 i,j p where γ Q i,j (h) is Robust ACF function defined previosuly. Assume that the process is short-memory, then n sup λ Q j λ j = Op (1), as n, 1 j p where ( λ Q j ) 1 j p and (λ j ) 1 j p denote the eigenvalues of ( h0 Γ ) ( h=1 Q (h) Γ Q (h) h0 and h=1 ), Γ(h)Γ(h) respectively, where (Γ(h)) 1 i,j p = (γ i,j (h)) 1 i,j p and h 0 is a fixed integer larger than 1.,

28 Theoretical results Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references Proposition 4 Let h be a fixed positive integer and ( ΓQ (h)) 1 i,j p ) = ( γ Qi,j (h) 1 i,j p where γ Q i,j (h) is defined previously. Assume that the vector process has long-memory property, then (i) If, for all i in {1,..., k}, D i > 1/2(d i < 1/4), n sup λ Q j λ j = Op (1), as n, 1 j p (ii) If, there exists i 0 in {1,..., k} such that D i0 < 1/2(d i > 1/4), n D i 0 sup λ Q j λ j = Op (1), as n, 1 j p,

29 ( background): Estimation Estimation and number of factor Factor Analysis - ROBUST Estimation for P and r Factor Analysis - Theoretical results : Base references SGRANCIO, A., REISEN, V.A,LEVY-LEDUC, C., PASCAL, B. ZIELGELMANN, F.A., ZAMBOM, E, COTTA, H,, Robust Factor Modeling for High-Dimensional Time Series: an to Air Pollution Data. Submitted COTTA, H., REISEN, V.A., BONDON, P., STUMMER, W. Robust principal component analysis with air pollution data: EUSIPCO. Reisen, V. A, Lévy-Leduc, C., Zambon, E. Robust Factor Model for long-memory multivariate time series with application to stock market. IN REVISION. Bottoni, J Reisen, V. A, Spanny, Franco, G. E., Pascal, B. Generalized additive model with principal component analysis: An application to time series of respiratory disease and air pollution data JRSS. Zamprogno. B., Reisen, V. A,, Cotta, Bondon, P.The use of PCA in time correlated processes with short and long-memory property. An application to Air pollution data Submitted. Erik Vanhataloa and Murat Kulahcia. Impact of Autocorrelation on Principal Components and Their Use in Statistical Process Control. Quality and Reliability Engineering International

30 Simulation: Monte Carlo studies - Models Simulation: Monte Carlo studies - Models Monte Carlo experiments were conducted to analyze the effect of high-dimensional time series with additive outliers on the factor modeling and time series with long and short-memory dependency. The empirical study is divided into two cases of x t (Equation 6), which follows a VARFIMA model with r = 3: (1) short-memory process that is, x t is a VARMA process; (2) long-memory process; d = (d 1, d 2, d 3 ), for at least one d 1, d 2, d 3 (0, 0.5). The VARFIMA model was generated with independent a t from N(0, I ) and Φ coefficients, which are displayed in Table 1. The sample size is n = 50, 100, 200, 400, 800 and 1600, and k = 0.2n, 0.5n, 0.8n. Here, an only one case is presented all k r elements of matrix P were generated as independent observations from the uniform distribution on the interval [-1,1] ( simulation method was according to Lam and Yao (2012).

31 Factor Analysis - Simulations Simulation: Monte Carlo studies - Models The empirical study is divided into two cases of x t (Equation 6), which follows a VARFIMA model with r = 3: (1) short-memory process where d = (0, 0, 0) ; that is, x t is a VARMA process; (2) long-memory process where d = (d 1, d 2, d 3 ), for at least one d 1, d 2, d 3 (0, 0.5). The VARFIMA model was generated with independent a t from N(0, I ) and Φ coefficients, which are displayed in Table 1. The sample size is n = 50, 100, 200, 400, 800 and 1600, and k = 0.2n, 0.5n, 0.8n and 1.2n. Table 1: Φ matrices for VARFI(1,d ) process. Φ 1 (Model 1) Φ 1 (Model 2) Φ 1 (Model 3)

32 Factor Analysis - Simulations Simulation: Monte Carlo studies - Models Let {z t }, t = 1,..., t Z, be a vector process contaminated by additive outliers defined as follows: z t = Px t + ω δ t, (15) where is the Hadamard product. ω = [ω 1,..., ω k ] is a magnitude vector of additive outliers. δ t = [δ 1t,..., δ kt ] is a random vector indicating the occurrence of an outlier at time t, in variable k, such as P(δ k,t = 1) = P(δ k,t = 1) = p/2 and P(δ k,t = 0) = 1 p, where E[δ k,t ] = 0 and E[δ 2 k,t ] = Var(δ k,t) = p. The model described above assumes that {Z t } and {δ t } are independent processes. Also, it is assumed that the elements of δ t are not correlated and temporally uncorrelated, i.e, E(δ t δ t) = Σ δ = diag(p,..., p) and E(δ t δ t+h ) = 0 for h 0.

33 Factor Analysis - Simulations Simulation: Monte Carlo studies - Models Table 2: Relative frequency estimates for f rel. (r = 3) in the simulation with 200 replications - Model 1. ( Classical ACF): simlar results of Lam and Yao (2012) n δ = 0 k = 0.2n k = 0.5n k = 0.8n

34 Simulation: Monte Carlo studies - Simulation: Monte Carlo studies - Models Table 3: Relative frequency estimates for dimensional reduction - d = [0.1, 0.2, 0.4] and Model 1 s Φ coefficients, r=3. Γz Γz Γz,Qn Γz,Qn p = 0 p = 0.05 and ω = 15 p = 0 p = 0.05 and ω = 15 n = 100 n = 100 n = 100 n = 100 r = 1 r = 2 r = 3 r = 1 r = 2 r = 3 rqn = 1 rqn = 2 rqn = 3 rqn = 1 rqn = 2 rqn = 3 k = 0.2n k = 0.5n k = 0.8n The third column gives the simulation results using Γ z,qn when p = 0. As one can see, the r estimates using Γ z,qn present similar results of Γ z when p = 0. Both methods have the same asymptotical properties. The presence of atypical observations in the data leads to a reduction of the estimated frequencies when r = 3 for all values of k in the classical method. This does not occur when the robust estimator is utilized, the results are quite close to the ones from the first column.

35 The RAMQAr - The data to real data - Descriptive statistics to real data - Time series to real data - FA s results The application of the proposed methodology consists of applying the PCA to cluster stations with the same behavior of measured pollutants. This section presents an application of the methodology for PM 10 concentrations measured at the Air Quality Automatic Monitoring Network (AQAMN) of the Greater Vitória Region (GVR). The application was divided into two parts: 1) reduction of the dimensions, and 2) forecasting. The data are: PM 10 measured in µg/m 3 ; Daily average; The period: January 2005 to December 2009; All eight RAMQAr s stations.

36 The RAMQAr - Time series to real data - Descriptive statistics to real data - Time series to real data - FA s results Sua Cariacica Camburi VVCentro Carapina Ibes Laranjeiras VixCentro Time Time Figure 3: PM 10 s concentrations of RAMQAr s stations

37 The RAMQAr - Descriptive statistics to real data - Descriptive statistics to real data - Time series to real data - FA s results Boxplot of PM 10 concentrations (µg/m 3 ) Laranjeiras Camburi Sua VixCentro Ibes VVCentro Figure 4: Boxplot of PM 10 s of RAMQAr s stations.

38 Lag Lag Lag Lag Lag Lag Lag Lag The RAMQAr - ACFs to real data - Descriptive statistics to real data - Time series to real data - FA s results Laranjeiras Carapina Camburi Sua Robust ACF Robust ACF Robust ACF Robust ACF VixCentro Ibes VVCentro Cariacica Robust ACF Robust ACF Robust ACF Robust ACF Figure 5: PM 10 ρ Z,Qn (h) function for RAMQAr s Stations

39 to real data - Descriptive statistics to real data - Time series to real data - FA s results The RAMQAr - The Robust ACF X ACF- IBIS Robust ACF Classical ACF

40 The RAMQAr - FA s Classical ACF to real data - Descriptive statistics to real data - Time series to real data - FA s results λ i+1 λi λ i i i (a) (b) Figure 6: Plots of estimated eigenvalues (a) and ratios of estimated eigenvalues of ˆM (b). The test indicated 1 factor.

41 The RAMQAr - FA s ROBUST to real data - Descriptive statistics to real data - Time series to real data - FA s results λ i+1 λi λ i i i (a) (b) Figure 7: Plots of estimated eigenvalues (a) and the ratios of estimated eigenvalues of M Q (b). The Robust indicated 2 factors

42 to real data - Descriptive statistics to real data - Time series to real data - FA s results The RAMQAr - THE FINAL ESTIMATED MODEL The AF analysis suggests the following model for Z t (the daily PM 10 concentrations of the stations): Z t = p 1 w t + p 2 v t + ε t, where w t denotes the first factor, v t is the second factor of solution Ẑ = ˆP ˆX, and εt is a vector white-noise process.

43 Main A robust factor model for high-dimensional time series with short and long-memory and additive outliers is proposed. This study considered the effects of different correlation structures and additive outliers on a vector linear process and its implication in the analysis and interpretation of Factor Analysis calculated from the correlation matrix of this process; It was shown that the existence of outliers destroys the correlation and cross-correlation of a vector time series; This article applied the proposed methodology to identify pollution behavior for the pollutant PM 10 in the Greater Region of Vitória to enable better management of the local monitoring network. The results in this paper will hopefully stimulate further research on using robust estimation methods and long-memory models to represent and forecast environmental time series.

44 Thanks!!!

45 Chung, C. (2002), Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes, Econometric Theory 18, Cotta, H. H. A., Reisen, V. A., Bondon, P., Stummer, W. & Lévy-Leduc, C. (2017), Robust estimation of covariance and correlation functions of a stationary multivariate process, in to apper insignal Processing Conference (EUSIPCO), rd European, IEEE. Fajardo, F., Reisen, V. A. & Cribari-Neto, F. (2009), Robust estimation in long-memory processes under additive outliers, Journal of Statistical Planning and Inference 139, Hosking, J. R. (1996), Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series, Journal of Econometrics 73(1), Lam, C. & Yao, Q. (2012), Factor modeling for high-dimensional time series: Inference for the number of factors, Ann. Statist. 40(2), URL: Ma, Y. & Genton, M. G. (2000), Highly robust estimation of the autocovariance function, Journal of Time Series Analysis 21(6), Peña, D. & Box, G. E. P. (1987), Identifying a simplifying structure in time series, Journal of the American Statistical Association 82(399), pp URL: