Teletrac modeling and estimation

Transcription

1 Teletrac modeling and estimation File 4 José Roberto Amazonas jra@lcs.poli.usp.br Telecommunications and Control Engineering Dept. - PTC Escola Politécnica University of São Paulo - USP São Paulo 11/2008 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

2 Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

3 Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

4 Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

5 Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

6 Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

7 Parametric Modeling ARFIMA model Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

8 Parametric Modeling ARFIMA model The modeling of a (linear) times series x t consists of estimating a transfer function (or model) H(B) such that in which wt is the innovation at instant t. xt = H(B)wt, (1) In practice, the modeling is based on the estimation of the inverse function G(B) = H(B) 1, because we expect that ltering x t by G(B) produces a series of residuals w t of the WN type. Granger and Joyeux, and Hosking introduced, in a independent way, the class of models ARFIMA that has the following properties: 1 long memory explicit modeling; 2 exibility for modeling the series' autocorrelation structure for small and large lags; 3 enable to simulate LRD series from the model. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

9 Parametric Modeling ARFIMA model Consider the equation d xt = wt, (2) in which d is a fractionnary exponent, 0 < d < 1/2. Observe that d = (1 B) d = with binomial coecients 1 ( ) d = k k=0 ( d results the fractionnary dierence lter k ) ( 1) k B k, (3) Γ(d + 1) Γ(k + 1)Γ(d k + 1), (4) d = 1 db + 1 2! d(d 1)B 2 1 3! d(d 1)(d 2)B , (5) that is dened for any real d > 1. 1 The Gamma function extends the factorial function for real and complex numbers: d! = Γ(d + 1). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

10 Parametric Modeling ARFIMA model According to (5), the model (2) is of the kind AR( ). Eq. (2) denes the fractionnary integrated process (also called FD(d) model or fractionnary WN that is an extension of the integrated model ARIMA(0, d, 0), d Z +. The FD process is able to model the 1/f α singularity at the origin of a LRD series' spectrum. The FD is stationary and LRD when 0 < d < 1/2; it is stationary and SRD when 1/2 < d < 0; it is non-stationary 2 when d > 1/2. 2 In this case, x t has innite variance. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

11 Parametric Modeling ARFIMA model In practice, we observe that the SACF's decay for small values of the lag of some real teletrac series is well modeled by SRD processes, i. e., have signicative values that exponentially decay for small values of the lag, which is not well modeled by the FD(d) process. This does not mean that this kind of teletrac series is not asymptotically LRD; it only says that the SRD characteristic may manifest itself by the existence of local SDF peaks (besides the singularity at the origin of the spectrum that is due to the long memory), as illustrated by Fig. 1. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

12 Parametric Modeling ARFIMA model FD(0.4) AR(4) 20 PSD (db) frequency Figure: SDFs for some power AR(4) and FD(0,4). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

13 Parametric Modeling ARFIMA model It is for this reason that there is a need to introduce the ARFIMA(p, d, q) class of models (more exible than the FD class) in which 1/2 < d < 1/2. φ(b) is the p order auto-regressive operator; θ(b) is the q order moving average operator; wt is a Gaussian WN. φ(b) d xt = θ(b)wt, (6) The model (6) is LRD, stationary and invertible when 0 < d < 1/2 and if the poles and zeroes of θ(z)/φ(z) are inside the unit radius circle. The parameter d models the high order autocorrelation structure (in which the decay is slow, of the hyperbolic type). On the other hand, the parameters of the polynomials φ(b) and θ(b) are responsible for modeling the low order lags autocorrelation (fast decay of the exponential type). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

14 Parametric Modeling ARFIMA model Eq. (6) may be rewritten as an AR( ) φ(b) θ(b) d xt = wt. (7) The SDF of an ARFIMA(p, d, q) model is given by P x (f ) = σ2 w 1 e j2πf 2d 1 θ 1 e j2πf... θ q e jq2πf 2 1 φ 1 e j2πf... φ p e jp2πf 2, (8) in which σ 2 w is the power of wt and w = 2πf is the normalized angular frequency ( π w π). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

15 Parametric Modeling ARFIMA model Eq.(8) may be simplied for the case ARFIMA(0, d, 0): P x (f ) = 1 e j2πf 2d σ 2 w = [2(1 cos (2πf ))] d σ 2 w (9) or P x (f ) = [2 sin (πf /2)] 2d σ 2 w. (10) As sin w w for w close to zero, then (10) reduces to P x (f ) = (πf ) 2d σ 2 w. (11) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

16 Parametric Modeling ARFIMA models prediction - optimum estimation Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

17 Parametric Modeling ARFIMA models prediction - optimum estimation Consider the three basic formats of the ARIMA(p, d, q) model at instant t + h: (a) ARMA(p + d, q) p+d x t+h = k=1 ϕ k x t+h k + w t+h q θ k w t+h k ; (12) k=1 (b) AR( ) (c) MA( ) 3 x t+h = k=1 x t+h = g k x t+h k + w t+h ; (13) ψ k w t+h k. (14) k=0 3 The sequence ψ t in (14) denotes the ARIMA model's impulse response. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

18 Parametric Modeling ARFIMA models prediction - optimum estimation Let ˆx t+h = ψ h w t + ψ h+1w t 1 + ψ h+2w t (15) be the Minimum Mean Squared Error (MMSE). Then the coecients ψ h+k, k = 0, 1, 2,..., may be determined minimizing the prevision's Mean Square Error (MSE) E[(e t+h ) 2 ] = E[(x t+h ˆx t+h ) 2 ] = E "! 2 # X X ψ k w t+h k ψ h+k w t k. (16) k=0 k=0 As X X ψ k w t+h k = ψ h+k w t k, k=0 k= h we have that e t+h is given by X e t+h = ψ 0wt+h + ψ 1wt+h ψ h 1 w t+1 (ψ h+k ψh+k)w t k. (17) k=0 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

19 Parametric Modeling ARFIMA models prediction - optimum estimation In this way, E[(e t+h ) 2 ] = (1+ψ ψ 2 h 1 )σ2 w+ (ψ h+k ψh+k )2 σ 2 w, (18) k=0 in which ψ 0 = 1, because the innovations wt are non-correlated. It follows that ψh+k = ψ h+k minimizes (18). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

20 Parametric Modeling ARFIMA models prediction - optimum estimation Therefore, the optimum prediction according the MMSE criterium is given by ˆx t+h = ψ h w t + ψ h+1 w t 1 + ψ h+2 w t = and the minimum prediction error by ψ h+k w t k (19) k=0 e t+h = w t+h + ψ 1 w t+h ψ h 1 wt+1 (20) has a variance V h = (1 + ψ ψ 2 h 1 )σ2 w, (21) given that E[e t+h F ( ) t ] = 0, (22) in which F ( ) t = {x t, x t 1,...} denotes the set of all past observations of the series. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

21 Parametric Modeling ARFIMA models prediction - optimum estimation Note that x t+h = ˆx t+h + e t+h, h 1. (23) Observe that the conditional expectation of x t+h given the series past observations E[x t+h F ( ) t ] = ˆx t+h, (24) is equal to the MMSE prediction (see (22)) (this does not happen by chance). In fact, we can demonstrate that the optimum predictor according to the MMSE criterium: a) is the conditional expectation E[x t+h F ( ) t ] and b) that this is a linear predictor when the innovations are Gaussian. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

22 Parametric Modeling Forms of Prediction Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

23 Parametric Modeling Forms of Prediction Taking the conditional expectation in (12), we get the prediction by means of the dierences equation ˆx t+h = ϕ 1E[x t+h 1 F ( ) t ] ϕ p+d E[x t+h p d F ( ) t ] + E[w t+h F ( ) t ] θ 1E[w t+h 1 F ( ) t ]... θ qe[w t+h q F ( ) t ], (25) for h 1. Observe that E[x t+k F ( ) t ] = ˆx t+k, k > 0, E[x t+k F ( ) t ] = x t+k, k 0, E[w t+k F ( ) t ] = 0, k > 0, E[w t+k F ( ) t ] = w t+k, k 0. (26) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

24 Parametric Modeling Forms of Prediction Note that (a) ˆx t+h depends on ˆx t+h 1, ˆx t+h 2,..., that are evaluated in a recursive way; (b) in practice, we only know a nite number of past observations, i. e., F t = {x t, x t 1,..., x 1 }. Therefore, E[x t+k F ( ) t ] E[x t+k F t ]; (c) the predictions for an AR(p) are exact, because we can show that E[x t+k x t, x t 1,...] = E[x t+k x t,..., x t+1 p ] Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

25 Parametric Modeling Forms of Prediction Taking the conditional expectation in (13) we obtain the prediction in the AR( ) format ˆx t+h = k=1 g k E[x t+h k F ( ) t ] + E[w t+h F ( ) t ]. (27) as E[w t+h F ( ) t ] = 0, we can rewrite (27) in the format ˆx t+h = g 1ˆx t+h 1 + g 2ˆx t+h g h x t + g h+1 x t (28) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

26 Parametric Modeling Condence interval Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

27 Parametric Modeling Condence interval The ARFIMA model (6) assumes that the sequence of innovations wt is a null mean WGN, i. e., wt N (0, σ 2 w). It then follows that the conditional distribution of x t+h given F ( ) t of the type N (ˆx t+h, V h ) and that Z = x t+h ˆx t+h N (0, 1). (29) Vh is The expression of the condence interval for x t+h, at the condence level (1 β), is given by ˆx t+h z β/2 V h x t+h ˆx t+h + z β/2 V h. (30) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

28 Parametric Modeling Condence interval As in practice the value of σ 2 w is unknown, we use the estimate ˆV h = (1 + ψ ψ 2 h 1 )ˆσ2 w, (31) obtained in the model estimation phase. At last, we obtain the nal expression of the condence interval for x t+h h 1 ˆx t+h z β ˆσ w [1 + ψ 2 k ] 1/2 [ 1/2 h 1 xt+h ˆxt+h + zβ ˆσ w 1 + ψk] 2. k=1 k=1 (32) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

29 Parametric Modeling ARFIMA prediction Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

30 Parametric Modeling ARFIMA prediction ARFIMA prediction Consider the stationary and invertible ARFIMA(p, d, q) model, 0, 5 < d < 0, 5, given by (7). We can rewrite the process in the format AR( ) in which g 0 = 1 and g k x t k = wt, (33) k=0 g k B k = φ(b)θ 1 (B)(1 B) d = π(b). (34) k=0 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

31 Parametric Modeling ARFIMA prediction Then, we can predict a future value of xt using (34) and (28). The prediction error variance is given by (21). Note that the polynomial π(b) has innite order (as d < 1/2). As in practice we have a series with N observations, only the rst L terms of π(b) are used, with L < N. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

32 Long memory statistical tests R/S statistics Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

33 Long memory statistical tests R/S statistics R/S statistics Consider a time series x t, t = 1, 2,..., N. Hurst proposed the long memory test Q N = 1 k k max (x j x) min (x j x), (35) ŝ N 1 k N 1 k N j=1 in which ŝ N = Ĉ0, known as Range Over Standard Deviation (R/S) statistics or rescaled adjusted range. Hurst observed that the R/S log-log plot (for the Nile river's yearly minimal levels time series) versus N spread along a straight line with slope greater than 1/2, i. e., that log (R/S) versus N presented a kind of CN H behavior (Hurst eect), in which C is a constant and 1/2 < H < 1 denotes the Hurst parameter. This empirical discovery contradicts the expected behavior of Markovian processes (that are SRD), in which R/S must have an asymptotical behavior of CN 1/2. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63 j=1

34 Long memory statistical tests R/S statistics The statistics N 1/2 Q N converges to a well dened random variable (for N ) when xt is a WGN process. That is the reason why the R/S log-log plot versus N presents a CN 1/2 asymptotical behavior. On the other hand, it is the N H Q N statistics that converges to a well dened random variable when xt is LRD. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

35 Long memory statistical tests R/S statistics Afterwards, Lo has shown that (35) is not robust when SRD is also present in the series and developed an extended version of (35) Q T = 1 k k max (x j x) min (x j x), (36) ˆσ NW 1 k N 1 k N j=1 in which ˆσ NW denotes the square root of the long run variance Newey-West estimate of a process xt (stationary and ergodic). The long run variance is dened as lrv(xt) = τ= As C τ = C τ, (37) may be rewritten as lrv(xt) = C j=1 C τ. (37) C τ. (38) τ=1 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

36 Long memory statistical tests R/S statistics The Newey-West estimator for (37) is given by lrv NW (x t ) = Ĉ T w τ,t Ĉ τ, (39) τ=1 in which w τ,t are coecients (whose sum is equal to one) and a truncate parameter that satises T = O(N 1/3 ). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

37 Long memory statistical tests GPH test Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

38 Long memory statistical tests GPH test Geweke and Porter-Hudak proposed a long memory test based on the SDF of the ARFIMA(0, d, 0) process given by P x (f ) = [4 sin 2 (πf )] d σ 2 w, (40) in which σ 2 w denotes the power of the wt WN. Note that the d parameter may be estimated by means of the following regression ln P x (f j ) = d ln[4 sin 2 (πf j )] + 2 ln σ w, (41) for j = 1, 2,..., z(n), in which z(n) = N α, 0 < α < 1 (N denotes the number of samples). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

39 Long memory statistical tests GPH test Geweke and Porter-Hudak have shown that if P x (f j ) is estimated by the periodogram method, then the minimum square estimator ˆd using the regression (41) is normally distributed in big samples if z(n) = N α with 0 < α < 1: ˆd N π 2 d, 6, (42) z(n) j=1 (U j Ū) 2 in which U j = ln[4 sin 2 (πf j )] and Ū is the sample mean of U j, j = 1, 2,..., z(n). Under the null hypothesis that there is no LRD (d = 0), the t statistics t d=0 = ˆd has normal distribution in the limit. π 2 6 z(n) j=1 (U j Ū) 2 1/2 (43) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

40 Some H and d estimation methods Heuristical approaches Outline 1 Parametric Modeling ARFIMA model ARFIMA models prediction - optimum estimation Forms of Prediction Condence interval ARFIMA prediction 2 Long memory statistical tests R/S statistics GPH test 3 Some H and d estimation methods Heuristical approaches 4 Bi-spectrum and Linearity Test 5 KPSS Stationarity Test Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

41 Some H and d estimation methods Heuristical approaches R/S statistics The log-log plot of the R/S statistics versus N, in which N denotes the series number of points, of a LRD series is close to straight line with slope 1/2 < H < 1. First, we evaluate the R/S statistics using N 1 consecutive observations of the series, in which N 1 must be a suciently large number. Next, we increase the number of observations by a factor f ; i. e., we evaluate R/S over N i = fn i 1 consecutive samples for i = 2,..., s. Note that to obtain the R/S statistics with N i consecutive observations, we can divide the series in [N/N i ] blocks and obtain [N/N i ] values, in which [.] denotes the integer part of a real number. The regression of the log-log plot of all R/S statistics versus N i, i = 1,..., s, produces an estimate of the H parameter.h Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

42 Some H and d estimation methods Heuristical approaches Variance plot The variance plot is a heuristical estimation method of the Hurst parameter. Beran shows that the sample mean variance of a LRD series decreases with its size N slower than in the traditional case (independent or non-correlated variables) as in which c > 0. Var( x) cn 2H 2, (44) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

43 Some H and d estimation methods Heuristical approaches We have the following steps: 1 Let k be an integer number. For dierent k in the range 2 k N/2, and for a sucient number m k of k-size subseries, evaluate the means of m k samples of size k, x 1 (k), x 2 (k),..., x mk (k) and the global mean x(k) = m 1 k m k j=1 x j (k). (45) 2 For each k, evaluate the sample variance of m k sample means x j (k), j = 1, 2,..., m k : m s 2 1 k (k) = ( x j (k) x(k)) 2. (46) m k 1 k=1 3 Represent in a plot log s 2 (k) versus log k. For the short range dependence or independence cases, we expect the plot's angular coecient 2H 2 to be (2 1/2) 2 = 1. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

44 Some H and d estimation methods Heuristical approaches Periodogram method The SDF of a LRD process is approximated by the expression C P f 1 2H when f 0. As the SDF may be approximated by the periodogram, a periodogram's log-log plot versus frequency should follow a straight line with 1 2H slope for frequencies close to zero. The ˆPx (f ) SDF's estimator is obtained by the non-parametric periodogram method 4, with data tapering, to reduce the power leakage, and smoothing, to reduce the variability of ˆPx (f ). The periodogram is evaluated by 5 ˆP x (f ) = 1 N X (f ) 2. (47) 4 The spectral analysis parametric methods are based on AR, MA and ARMA models. Therefore, they should not be applied to estimate the SDF of a 1/f α noise. 5 The denition was given without including tapering and smoothing to easy the understanding of the estimator's essential nature. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

45 Some H and d estimation methods Heuristical approaches Whittle's method The Whittle's estimator is also based on the periodogram and involves minimizing the function Q(θ) = ˆP x (f ) df (48) P x (θ, f ) in which ˆPx (f ) denotes the x t series' periodogram, P x (θ, f ) is the theoretical SDF of the ARFIMA(p, d, q) xt model in the frequency f and θ = [p, d, q] represents the unknown parameters vector. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

46 Some H and d estimation methods Heuristical approaches Haslett and Raftery's MV approximate estimator Consider the ARFIMA model (6), rewritten in the format x t = d φ(b) 1 θ(b)w t. (49) Let ˆx t be the optimal 1-step prediction of x t given the past observations F t 1 = {x t 1, x t 1,..., x1}, e t = x t ˆx t is the 1-step prediction error and the parameters vector of the model (49). ζ = [σ 2 w, φ 1,..., φ p, d, θ 1,..., θ q] (50) Harvey shows that the log-likelihood function of (49) is given by an expression known as prediction's error decomposition log [L(ζ) F t 1] = N 2 log 2π N 2 log σ2 w 1 2 t=1 in which f t = Var[e t]/σ 2 w. NX log f t 1 2σ 2 w NX e 2 t /f t, (51) Haslett and Raftery proposed a fast procedure to determining an approximation of (51), that is used by the program S-PLUS R. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63 t=1

47 Some H and d estimation methods Heuristical approaches Abry and Veitch's wavelet estimator Consider a stationary signal x(t), t R and its IDWT x(t) k J u(j, k)φ J,k (t) + w j,k ψ j,k (t). j=1 k The stationarity of x t implies the stationarity of the wavelet coecients sequences {w j,k } in all scales j. Let P j = E[w 2 j,k ] = Var[w j,k] be the power of the signal {w j,k } or wavelet variance in a given scale j. We can show that P j = R Ψ(ν) 2 P x (ν/2 j )dν, (52) in which Ψ(ν), < ν <, denotes the Fourier transform of the wavelet ψ(t) and P x (ν) denotes the SDF of x(t). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

48 Some H and d estimation methods Heuristical approaches Consider j. Then, P x (ν/2 j ) may be seen as a magnied version (dilated) of the SDF of x(t) in the low frequencies region (ν 0). This being so, we verify that (52) is an approximation of the power of x(t) in the low frequencies region for j (because the integral (52) is weighted by the function Ψ(ν) 2 ). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

49 Some H and d estimation methods Heuristical approaches Assume that x(t) is LRD, i.e., P x (ν) C P ν α, ν 0, 0 < α < 1, (53) in which means that the ratio between the left and right sides of (53) converges to 1. Eqs. (52) and (53) imply that the following expression is valid when j : P j C P Ψ(ν) 2 ν/2 j α dν = C P C 2 jα, (54) in which C = C(Ψ, α) = R Ψ(ν) 2 ν α dν. R Eq. (54) suggests that H = (1 + α)/2 may be estimated by means of log 2 (P j ) (2H 1)j + constant, j. (55) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

50 Some H and d estimation methods Heuristical approaches Given the sampled signal x k, k = 0, 1,..., N 1, associated to the original signal x(t), we may estimate log 2 (P j ) using the DWT coecients w j,k, k = 0, 1,..., N j 1, j = 1, 2,..., J, of x k. The log 2 (P j ) estimator is given by S j = log 2 1 N j 1 N j k=0 wj,k 2 log 2 (P j ). (56) The set of statistics S j, j = 1, 2,..., J, is called the wavelet spectrum of the signal x k. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

51 Some H and d estimation methods Heuristical approaches The relation (55) tells that the wavelet spectrum of x k is linear with slope α = 2H 1 in the dilated time scales. Applying a linear regression between scales j 1 and j 2 of the wavelet spectrum produces the following explicit relation for estimating H: j 2 j 2 Ĥ [j1,j 2 ] = 1 ε j js j ε j j j=j 1 j=j 1 ( 2 j 2 j 2 j2 ε j ε j j 2 j=j 1 j=j 1 in which ε j = (N ln 2 2)/2 j+1. j 2 j=j 1 ε j S j j=j 1 ε j j ) Eq. (57) denes the Abry and Veitch's (AV) wavelet estimator. (57) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

52 Some H and d estimation methods Heuristical approaches The wavelet estimator Ĥ[j 1,j 2 ] presents a good performance when the series are not very distant from a FGN. Empirical studies indicate that it is robust as far as smooth deterministic trends and short range dependence structure's changes of the time series are concerned. The discussion of this section has shown that the AV estimator is based on the wavelet variance of continuous time signals. Abry, Veitch and Taqqu proposed a method to enable the AV estimator to be applied to discrete time signals, as the networks trac signals. The method consists on pre-ltering the original discrete time signal x k (DWT initialization phase), that produces the initial sequence to be decomposed by the DWT in which u x (k) = x(t) = k= x t φ 0 (t k) dt, (58) x(k)sinc(t k). (59) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

53 Some H and d estimation methods Heuristical approaches x(t) is a ctious continuous time signal that the DWT initialization procedure associates to the original signal x k and (58) shows that the AV estimator is evaluated by means of the DWT of a ltered version of x k. Applying (59) in (58) we have u x (k) = = = k= k= x t φ 0 (t k) dt x(k) = x(k) I (k), x(k)i (k n) φ 0 (t k)sinc(t k) dt (60) in which I (m) = sinc(t + m)φ 0 (t) dt. (61) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

54 Some H and d estimation methods Heuristical approaches The next gure illustrates the wavelet spectra of: a WGN, of the AR(4) x t = 2, 7607x t 1 3, 8106x t 2 + 2, 6535x t 3 0, 9238x t 4 + w t model and of the BellcoreAug89 trace (bin of 10 miliseconds). The WGN's wavelet spectrum is at; the BellcoreAug89 trace's spectrum is approximately linear between scales j = 3 and j = 10; the spikes in scales 2 and 3 of the AR(4) model's spectrum suggest a short range dependence presence. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

55 Some H and d estimation methods Heuristical approaches Espectro wavelet de RBG (realização com 2 23 amostras): H=0,500 Espectro wavelet de simulação AR(4): H=0,510 Espectro wavelet do trace BellcoreAug89 (escala de 10 ms): H = 0, S 1 j S 7 j S 26 j Escala (oitava) j Escala j Escala (oitava) j Figure: WGN's wavelet spectrum Figure: AR(4)'s wavelet spectrum Figure: Bellcore trace's wavelet spectrum. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

56 Some H and d estimation methods Heuristical approaches 50 Periodograma alisado de realização AR(4) 110 Periodograma alisado do trace BellcoreAug89 (bin de 10ms)) db db f f Figure: Smoothed periodogram by the WOSA method: AR(4) Figure: Smoothed periodogram by the WOSA method: Bellcore trace Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

57 Bi-spectrum and Linearity Test Let x = [x 1, x 2,..., xn] T and ω = [ω 1, ω 2,..., ω n ] T, in which T denotes the transpose operation, a n-dimensional real random vector and a real parameters vector with n components, respectively. The joint moments of order r = k 1 + k k n of x are given by Mom{x k 1, 1 xk 2,..., 2 xkn n } E{x k 1 1 xk xkn n } = ( j) r r Φ x (ω T ) ω k 1 1 ωk ωkn n ω1 =ω 2 =...=ω n=0 (62) in which Φ x (ω T ) E{e jωt x } (63) is the joint characteristic function of x. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

58 Bi-spectrum and Linearity Test The joint cumulants of order r of x are dened as Cum{x k 1, 1 xk 2,..., 2 xkn n } ( j) r r Ψ x (ω T ) ω k 1 1 ωk ωkn n in which ω1 =ω 2 =...=ω n=0, (64) Ψ x (ω T ) ln Φ x (ω T ) (65) corresponds to the natural logarithm of the joint characteristic function. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

59 Bi-spectrum and Linearity Test We can verify that the moments m 1 = E{x} = µ m 2 = E{x 2 } (66) m 3 = E{x 3 } m 4 = E{x 4 } (67) of a random variable x are related to its cumulants by c 1 = Cum{x} = m 1 (68) c 2 = Cum{x, x} = m 2 m 2 1 (69) c 3 = Cum{x, x, x} = m 3 3m 2 m 1 + 2m 3 1 (70) c 4 = Cum{x, x, x, x} = m 4 4m 3 m 1 3m m 2 m 2 1 6m 4 1. (71) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

60 Bi-spectrum and Linearity Test Assume the moments of a stationary process {x k } up to order n exist. Then Mom{x(k), x(k + τ 1 ),..., x(k + τ n 1 )} = E{x(k)x(k + τ 1 )... x(k + τ n 1 )} m x n(τ 1, τ 2,..., τ n 1 ) (72) in which τ 1, τ 2,..., τ m 1, τ i = 0, ±1, ±2,... for all i, denote lags. Similarly, the nth order cumulants of {x k } can be written as c x n(τ 1, τ 2,..., τ n 1 ) Cum{x(k), x(k + τ 1 ),..., x(k + τ n 1 )}. (73) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

61 Bi-spectrum and Linearity Test Nikias et al have shown that the following relations are valid: c x 1 = m x 1 = µ (média) (74) c2(τ x 1 ) = m2(τ x 1 ) (m1) x 2 (autocovariance function) (75) c3(τ x 1, τ 2 ) = m3(τ x 1, τ 2 ) m1[m x 2(τ x 1 ) + m2(τ x 2 ) + m2(τ x 2 τ 1 )] + 2(m1) x 3 (76) and c4(τ x 1, τ 2, τ 3 ) = m4(τ x 1, τ 2, τ 3 ) m2(τ x 1 )m2(τ x 3 τ 2 ) m2(τ x 2 )m2(τ x 3 τ 1 ) m2(τ x 3 )m2(τ x 2 τ 1 ) m1[m x 3(τ x 2 τ 1, τ 3 τ 1 ) + m3(τ x 2, τ 3 )+ + m3(τ x 2, τ 4 ) + m3(τ x 1, τ 2 )]+ + (m1) x 2 [m2(τ x 1 ) + m2(τ x 2 ) + m2(τ x 3 )+ + m2(τ x 3 τ 1 ) + m2(τ x 3 τ 2 ) + m2(τ x 2 τ 1 )] 6(m1) x 4. (77) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

62 Bi-spectrum and Linearity Test Let {x k } be a null mean process, i. e., m x 1 = 0. If τ 1 = τ 2 = τ 3 = 0 in (75), (76), and (77) we have E{x(k) 2 } = c x 2(0) = σ 2 (variance) E{x(k) 3 }/σ 3 = c x 3(0, 0)/σ 3 = S(x k ) (asymmetry) E{x(k) 4 }/σ 4 = c x 4(0, 0, 0)/σ 4 = K(x k ) (kurtosis). (78) The expressions (78) give the variance, asymmetry and kurtosis in terms of cumulants. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

63 Bi-spectrum and Linearity Test Let f = [f 1, f 2,..., f n ] T and τ = [τ 1, τ 2,..., τ n ] T be normalized frequencies and lags vectors, respectively. The nth order polyspectrum of the process x k is dened as: Pn(f x T ) =... cn(τ x T )exp{ j2π(f T τ)}. (79) τ 1 = τ n 1 = The SDF, bi-spectrum and tri-spectrum correspond to the second, third and fourth orders spectra, respectively: P2(f x ) = P x (f ) = c2(τ)e x j2πf τ, (80) P x 4(f 1, f 2, f 3 ) = P x 3(f 1, f 2 ) = τ 1 = τ 2 = τ 1 = τ 2 = τ 3 = τ 1 = c x 3(τ 1, τ 2 )e j2π(f 1τ 1 +f 2 τ 2 ), (81) c x 4(τ 1, τ 2, τ 3 )e j2π(f 1τ 1 +f 2 τ 2 +f 3 τ 3 ). (82) The bi-spectrum and the tri-spectrum are complex. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

64 Bi-spectrum and Linearity Test Another useful statistic is the bicoherence, dened as B x 3(f 1, f 2 ) = P x 3 (f 1, f 2 ) P x (f 1 + f 2 )P x (f 1 )P x (f 2 ). (83) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

65 Bi-spectrum and Linearity Test Hinich developed statistical tests for Gaussianity and linearity. These tests are based on the following properties: 1) if {x k } is Gaussian, its third order cumulants are null; therefore its bi-spectrum is null and 2) if {x k } is linear and non-gaussian, then its bicoherence is a non-null constant. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

66 KPSS Stationarity Test Stationarity tests assume the null (or working) hypothesis 6 that the series under investigation are of the type xt I (0) 7. The KPSS test, proposed by Kwiatkowski, Phillips, Schmidt and Shin is based on the model y t = β T d t + µ t + u t (84) µ t = µ t 1 + wt, wt N (0, σ 2 w) (85) in which β is a parameters vector, d t is a deterministic components vector (constant or constant plus trend) and u t is I (0). Note that µ t is a random walk. 6 It is the hypothesis we wish to reject. 7 Unit roots test as the Dickey-Fuller test work with the null hypothesis that the series is I (1). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63

67 KPSS Stationarity Test The null hypothesis that y t is I (0) is formulated as H 0 : σ 2 w = 0, which implies that µ t is a constant. The KPSS statistics for the test of σ 2 w = 0 against the alternative hypothesis σ 2 w > 0 is given by KPSS = ( N 2 N Ŝt 2 t=1 ) /ˆλ 2 (86) in which N is the size of the sample, Ŝ t = t j=1 ûj, û t is the residual of a regression of y t over d t and ˆλ 2 is a long run variance estimate of u t using û t. Under the null hypothesis that y t is I 0, it can be shown that KPSS converges to a Brownian movement function that depends on the shape of the deterministic trend terms d t and, on the other hand, is independent of the β vector. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 63