Time Series Analysis by Higher Order Crossings. Benjamin Kedem Department of Mathematics & ISR University of Maryland College Park, MD

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1 Time Series Analysis by Higher Order Crossings Benjamin Kedem Department of Mathematics & ISR University of Maryland College Park, MD 1

2 Stochastic Process A stochastic or random process {Z t },, 1,0,1,, is a collection of random variables, real or complex-valued, defined on the same probability space. Gaussian Process: A real-valued process {Z t }, t T, is called Gaussian process if for all t 1, t 2,, t n T, the joint distribution of (Z t1, Z t2,, Z tn ) is multivariate normal. The finite dimensional distributions of a Gaussian process are completely determined from: and m(t) = E[Z t ] R(s, t) = Cov[Z s, Z t ]. Markov Process: For t 1 < < t n 2 < t n 1 < t n P(Z tn z Z tn 1, Z tn 2,...Z t1 ) = P(Z tn z Z tn 1 ) 2

3 Stationary Processes A stochastic process {Z t } is said to be a strictly stationary process if its joint distributions are invariant under time shifts: (Z t1, Z t2,, Z tn ) Dist = (Z t1 +τ, Z t2 +τ,, Z tn +τ) for all t 1, t 2,, t n, n, and τ. When 2nd order moments exist, strict stationarity implies: E[Z t ] = E[Z t+τ ] = E[Z 0 ] = m (1) Cov[Z t, Z s ] = R(t s). (2) {Z t } is called weakly stationary when (1),(2) hold. For simplicity, we shall assume all our processes are both strictly and weakly stationary and also real-valued. 3

4 Assume: E(Z t ) = 0. Autocovariance: Autocorrelation: R k = E(Z t Z t k ) = ρ k = R k R 0 = Spectral Distribution: F(λ), π π π π cos(kλ)df(λ) (3) cos(kλ)df(λ) π λ π When F is absolutely continuous, we have the Spectral Density: f(λ) = F (λ), In general in practice: π λ π F(λ) = F c (λ) + F d (λ) where F c (λ) is absolutely continuous and F d (λ) is a step function, both monotone nondecreasing. R k, ρ k, f(λ) are symmetric. 4

5 spectral representation A. Kolmogorov and H. Cramér in the early 1940 s. Let {Z t }, t = 0, ±1, ±2,, be a zero mean weakly stationary process. Then (3) implies Z t = π π e itλ dξ(λ), t = 0, ±1, (4) where now the spectral distribution satisfies E[dξ(λ)dξ(ω)] = { df(λ), if λ = ω 0, if λ ω (5) We may interpret df(λ) = E dξ(λ) 2 as the weight or power given to frequency λ. 5

6 Example: Sum of Random Sinusoids. Z t = p {A j cos(ω j t) + B j sin(ω j t)}, t = 0, ±1, (6) j=1 A 1,, A p, B 1,, B p uncorrelated. E[A j ] = E[B j ] = 0, V ar[a j ] = V ar[b j ] = σ 2 j, ω j (0, π), for all j. Then for all t, E[Z t ] = 0, and R k = E[Z t Z t k ] = p σj 2 cos(ω j k) = j=1 p { 1 2 σ2 j cos(ω j k) σ2 j cos( ω j k)}, k = 0, ±1, j=1 ρ k = R k R 0 = p j=1 σ2 j cos(ω jk) p, k = 0, ±1, (7) j=1 σ2 j Discrete spectrum: F(ω) is a nondecreasing step function with jumps of size 1 2 σ2 j at ±ω j, and F( π) = 0, F(π) = R 0 = p j=1 σ2 j. 6

7 Example: Stationary AR(1) Process. Let {ǫ t }, t = 0, ±1, ±2,, be a sequence of uncorrelated real-valued random variables with mean zero and variance σ 2 ǫ. Define Z t = φ 1 Z t 1 + ǫ t, t = 0, ±1, ±2, (8) where φ 1 < 1. The process ( 8) is called a first order autoregressive process and is commonly denoted by AR(1). E[Z t ] = E lim n n φ j 1 ǫ t j j=0 = lim n E n φ j 1 ǫ t j = 0 j=0 Continuous spectrum: f(λ) = σ2 ǫ 2π R k = σ2 ǫ φ k 1 1 φ 2, k = 0, ±1, ±2, 1 ρ k = φ k 1, k = 0, ±1, ±2, 1 1 2φ 1 cos(λ) + φ 2, π λ π. 1 7

8 Example: Sum of Random Sinusoids Plus Noise. Let {Z t } be a signal as in (6), and let {ǫ t } be a zero mean weakly stationary noise uncorrelated with {Z t } and with a spectrum which possesses a spectral density f ǫ (ω), F ǫ (ω) = ω π f ǫ (λ)dλ Then the process p Y t = {A j cos(ω j t) + B j sin(ω j t)} + ǫ t, t = 0, ±1, (9) j=1 has a mixed spectrum of the form: F y (ω) = F z (ω) + F ǫ (ω) = λ j ω 1 ω 2 σ2 j + f ǫ (λ)dλ π λ j { ω p, ω p 1,..., ω p 1, ω p } 8

9 (a) Sum of two sinusoids. (b) Sum of two sinusoids plus white noise. A=0.5, B=1, w1=0.513, w2=0.771, CN=0, SNR=Inf A=0.5, B=1, w1=0.513, w2=0.771, CN=2.2, SNR=

10 Plots of AR(1) time series and their estimated autocorrelation. PHI=0.8 AR(1) and Estimated ACF Series : x ACF Lag PHI=-0.8 Series : x ACF Lag WN: PHI=0.0 Series : wn ACF Lag 10

11 Plots of AR(1) time series and their spectral densities on [0, π]. PHI=0.8 AR(1): Spectral Density PHI= f.ar1(1, 0.8) PHI= lambda PHI= f.ar1(1, -0.8) lambda WN: PHI=0.0 PHI= f.ar1(1, 0) lambda 11

12 Linear Filtering By a time invariant linear filter applied to a stationary time series {Z t }, t = 0, ±1,..., we mean the linear operation or convolution, Y t = L({Z t }) = h j Z t j (10) with H(λ) j= j= h j e ijλ, 0 < λ π The function H(λ) is called the transfer function. H(λ) is called the gain. Fact: df y (λ) = H(λ) 2 df z (λ) (11) In particular, when spectral densities exist we have f y (λ) = H(λ) 2 f z (λ) (12) This is an important relationship between the input and output spectral densities. 12

13 The Difference Operator: Z t Z t Z t 1 This is a linear filter with h 0 = 1, h 1 = 1, and h j = 0 otherwise. The transfer function is, and the squared gain is H(λ) = 1 e iλ H(λ) 2 = 1 e iλ 2 = 2(1 cos λ) In [0, π] the gain is monotone increasing and hence this is a high-pass filter. The squared gain of the second difference 2 is 4(1 cos λ) 2, and hence this is a more pronounced high-pass filter. Repeated differencing is a simple way to obtain highpass filters. 13

14 Differencing white noise: Higher frequencies get more power. WN Differencing WN WN SPEC DENSITY f.ar1(1, 0) lambda Diff(WN) Diff(WN) SPEC DENSITY * (1 - cos(lambda)) * f.ar1(1, 0) lambda Diff^5(WN) Diff^5(WN) SPEC DENSITY (2 * (1 - cos(lambda)))^5 * f.ar1(1, 0) lambda 14

15 We define a parametric filter by the convolution, Z t (θ) L θ (Z) t = n h n (θ)z t n (13) In other words where denotes convolution. Z t (θ) h t (θ) Z t (14) 15

16 The Parametric AR(1) Filter. Let α < 1. The AR(1) (or α) filter is the recursive filter Y t = αy t 1 + Z t or Y t = L α (Z) t = Z t + αz t 1 + α 2 Z t 2 + The transfer function for ω [0, π] is The squared gain is H(λ) = 1 1 αe iλ H(ω; α) 2 = 1 1 2αcos(ω) + α2, α ( 1,1). (15) 16

17 For α > 0 the AR(1) filter is a low-pass filter, and a high-pass for α < 0. Squared Gain of the AR(1) Filter a=0.5 a= omega 17

18 The Parametric AR(2) Filter. With α ( 1,1), define Y t (α) by operating on Z t, Y t (α) = (1 + η 2 )αy t 1 (α) η 2 Y t 2 (α) + Z t (16) where η (0,1) is the bandwidth parameter. Squared gains of the AR(2) filter centered approximately at cos 1 (α) for η close to 1. Squared Gain of the AR(2) Filter (alpha,eta)= (0.7,0.92) (0.6,0.94) (0.5,0.96) omega 18

19 Zero-crossings in Discrete Time Let Z 1, Z 2,, Z N be a zero-mean stationary time series. The zero-crossing count in discrete time is defined as the number of symbol changes in the corresponding clipped binary time series. First define the clipped binary time series: X t = { 1, if Zt 0 0, if Z t < 0 The number of zero-crossings, denoted by D, is defined in terms of {X t }, D = N [X t X t 1 ] 2 (17) t=2 0 D N 1 Example: Z: X: N = 10, D = 5. 19

20 ( ) Cosine Formula {Z t } a stationary Gaussian process. There is an explicit formula connecting ρ 1 and E[D], ( ) πe[d] ( ) ρ 1 = cos (18) N 1 An inverse relationship: E(D) 0 ρ 1 1 E(D) N 1 ρ

21 ( ) zero-crossing spectral representation ( ) πe[d] cos = N 1 π π cos(ω)df(ω) π π df(ω) (19) Assume F is continuous at the origin, ( ) π πe[d] 0 cos = cos(ω)df(ω) π N 1 0 df(ω) (20) ( ) Dominant Frequency Principle: For ω 0 (0, π) then F(ω+) F(ω ) > 0, ω = ω 0 = 0, ω ω 0 ( ) πe[d] cos = cos(ω 0 ) N 1 or, by the monotonicity of cos(x) in [0, π], πe[d] N 1 = ω 0 21

22 ( ) Higher Order Crossings (HOC) {Z t }, t = 0, ±1, ±2, {L θ ( ), θ Θ} 3. L θ (Z) 1, L θ (Z) 2,, L θ (Z) N 4. X t (θ) = { 1, if Lθ (Z) t 0 0, if L θ (Z) t < 0 5. HOC {D θ, θ Θ}: D θ = N [X t (θ) X t 1 (θ)] 2 t=2 HOC Combines ZC counts and linear operations (filters.) 22

23 Connection Between ZC and Filtering H(ω; θ) the transfer function corresponding to L θ ( ), and assume {Z t } is a zero-mean stationary Gaussian process. ( ) π πe[dθ ] π ρ 1 (θ) cos = cos(ω) H(ω; θ) 2 df(ω) π N 1 π H(ω; (21) θ) 2 df(ω) Assuming F is continuous at 0, ( ) π πe[dθ ] 0 ρ 1 (θ) cos = cos(ω) H(ω; θ) 2 df(ω) π N 1 0 H(ω; θ) 2 df(ω) (22) The representation ( 21) and ( 22) help to understand the effect of filtering on zero-crossings through the spectrum even in the general non-gaussian case. HOC now refers to both ρ 1 (θ) and D θ. Note: ρ 1 (θ) can be defined directly and in general. There is no need for the Gaussian assumption. π π ρ 1 (θ) cos(ω) H(ω; θ) 2 df(ω) π π H(ω; (23) θ) 2 df(ω) 23

24 HOC From Differences and define Z t Z t Z t 1 L j j 1, j {1,2,3, } with L 1 0 being the identity filter. The corresponding HOC are called the simple HOC. Thus, D 1 from ZC of Z t D 2 from ZC of ( Z) t D 3 from ZC of ( 2 Z) t D 4 from ZC of ( 3 Z) t D 1, D 2, D 3, 24

25 Properties of Simple HOC: (a) Monotonicity: D j 1 D j+1 which implies under strict stationarity, 0 E[D 1 ] E[D 2 ] E[D 3 ] N 1 Example: E[D k ] of Gaussian White Noise. N = 1000: E[D k ] = (N 1){ π sin 1 ( k 1 k )} k E[D k ]

26 Problem: Thus, {E[D j ]} is a monotone bounded sequence. What does it converge to as j??? Problem: What happens when E[D 1 ] = E[D 2 ]??? Problem: As j, {X t (j)}??? Problem: As N, D 1 N Constant??? 26

27 (b) K (1984): If {Z t } is Gaussian, then with prob. 1 E[D 1 ] N 1 = E[D 2] N 1 Z t = Acos(ω 0 t + ϕ) ω 0 = πe[d 1 ]/(N 1). (c) Suppose {Z t } is Gaussian, and let ω be the highest positive frequency in the spectral support ω [0, ω ]. Then, regardless of spectrum type, πe[d j ] N 1 ω, j (24) (d) Suppose {Z t } is Gaussian. D 1 Constant??? N K Slud (1994): ( ) Yes in the continuous spectrum case. ( ) Sometime if the spectrum contains 1 jump. ( ) No if the spectrum contains 2 or more jumps. 27

28 (e) K Slud (1982): Higher Order Crossings Theorem {Z t }, t = 0, ±1,, be a zero-mean stationary process, and assume that π is included in the spectral support. Define { 1, if X t (j) = j 1 Z t 0 0, if j 1 Z t < 0 Then, (i) as j. {X t (j)} { , wp 1/ , wp 1/2 (ii) lim j lim N D j N 1 = 1, wp 1. 28

29 Demonstration of the HOC Theorem using AR(1) with parameter φ = 0.0, 0.8. N = inferences. j X t (j) D j X t (j) D j φ = 0.8 φ = 0.0 (WN) The state is approached quite fast, and D 1 < D 2 < D 3 < < D 16 Only the first few D k s are useful in discrimination between processes. 29

30 (f) For a zero-mean stationary Gaussian process, the sequence of expected simple HOC {E[D k ]} determines the spectrum up to a constant: {E[D k ]} {ρ k } F(ω) (25) (g) K (1980): Let {Z t } be a zero mean stationary Gaussian process with acf ρ j. If j= ρ j <, then and where j= κ x (1, j,1 j) < D 1 E[D 1 ] N L N(0, σ 2 1 ), N 1 π 2 j= σ 2 1 = { (sin 1 ρ j ) 2 + sin 1 ρ j 1 sin 1 ρ j+1 + 4π 2 κ x (1, j,1 j) } 30

31 (h) Slud (1991): Let {Z t }, t = 0, ±1,, be a zero mean stationary Gaussian process with acf ρ j, V ar[z 0 ] = 1, and square integrable spectral density f. Then where σ 2 1 satisfies, D 1 E[D 1 ] N L N(0, σ 2 1 ), N σ π(1 ρ 2 1 ) π π Proof: Use Itô Wiener calculus. ρ 1 cos(ω) 2 f 2 (ω)dω > 0 (i) Problem: Given two stationary processes with the same spectrum of which one is Gaussian and the other is not. Is it true that the Gaussian process has a higher expected ZCR??? Answer: Assume continuous time. {Z(t)}, stationary Gaussian process, Z(t) N(0,1), < t <. Consider the interval [0,1]. 31

32 Divide [0,1] into N 1 intervals of size. Define the sampled time series, Then Z k Z((k 1) ), k = 1,2,, N ρ 1 = ρ( ) (26) From the cosine formula we obtain: Rice (1944) formula 1 ( ) E[D c ] lim E[D 1 ] = lim N 0 π cos 1 (ρ( )) = 1 π ρ (0) (27) Barnett K (1998): There are non-gaussian processes such that with κ < 1 and κ > 1. ( ) E[D c ] = κ π ρ (0) If Z 1 (t), Z 2 (t) are independent copies of Z(t), then the product Z 1 (t)z 2 (t) has κ = 2. For Z 3 (t), κ = 5/9. 32

33 Application: Discrimination by Simple HOC The ψ 2 Statistic: When N is sufficiently large (e.g. N 200), then with a high probability 0 < D 1 < D 2 < D 3 < < (N 1) To capture the rate of increase in the first few D k, consider the increments D 1, if k = 1 k D k D k 1, if k = 2,, K 1 (N 1) D K 1, if k = K Then K k = N 1 k=1 Let m k = E[ k ]. We define a general similarity measure ψ 2 K k=1 ( k m k ) 2 m k (28) When k, m k are from the same process, and K = 9: P(ψ 2 > 30) <

34 Application: Frequency Estimation Recall the AR(1) filter (α-filter), Z t (α) = L α (Z) t = Z t + αz t 1 + α 2 Z t 2 + with squared gain H(ω; α) 2 = 1 1 2αcos(ω) + α2, α ( 1,1), ω [0, π]. Consider: Z t = A 1 cos(ω 1 t) + B 1 sin(ω 1 t) + ζ t, t = 0, ±1, ω 1 (0, π). A 1, B 1 are uncorrelated N(0, σ 2 1 ). {ζ t } N(0, σ 2 ζ ) white noise independent of A 1, B 1. Define: Then for α ( 1,1), C(α) = V ar(ζ t(α)) V ar(z t (α)). (29) 0 < C(α) < 1. 34

35 He K (1989) Algorithm Let {D α } be the HOC from the AR(1) filter. Fix α 1 ( 1,1). Define ( ) πe[dαk ] ( ) α k+1 = cos, k = 1,2, (30) N 1 Then, as k, and α k cos(ω 1 ) πe[d αk ] N 1 ω 1 (31) Proof: Note the fundamental property of the AR(1) filter gain: π π ( ) α = ρ 1,ζ (α) = cos(ω) H(ω; α) 2 dω π π H(ω; (32) α) 2 dω 35

36 We have ρ 1 (α) = cos ( ) πe[dα ] N 1 = σ2 1 H(ω 1; α) 2 cos(ω 1 ) + π π H(ω; α) 2 df ζ (ω) α σ 2 1 H(ω 1; α) 2 + π π H(ω; α) 2 df ζ (ω) A weighted average of α cos(ω 1 ) and α (!) Thus, we have a contraction mapping ( ) ρ 1 (α) = α + C(α)(α α ) (33) and the recursion ( 30) becomes, with fixed point α : or ( ) α k+1 = ρ 1 (α k ) (34) α = ρ 1 (α ) cos(ω 1 ) = cos ( ) πe[dα ] N 1 By the monotonicity of cos(x), x [0, π], ω 1 = πe[d α ] N 1 36

37 Extension: Contractions From Bandpass Filters (CM) Yakowitz (1991): We do not need the Gaussian assumption, and can speed up the contraction convergence. 1. Parametric family of band-pass filters indexed by r ( 1,1), and by a bandwidth parameter M: {L r,m ( ), r ( 1,1), M = 1,2, } 2. Let h(n; r, M) and H(ω; r, M), be the corresponding complex impulse response and transfer function, respectively. 3. It is required that as M, H(ω; r, M) 2 converges to a Dirac delta function centered at θ(r) cos 1 (r). 4. Assume further that the filter passes only the (positive) discrete frequency to be detected; suppose it is ω 1. Also, observe that { } π E[ζ t (r, M)ζ t 1 (r, M)] π R = cos(ω) H(ω; r, M) 2 df ζ (ω) E ζ t (r, M) 2 π π H(ω; r, M) 2 df ζ (ω) where the overbar denotes complex conjugate. 37

38 5. Suppose that for any M the fundamental property takes the form { } E[ζ t (r, M)ζ t 1 (r, M)] r = R (35) E ζ t (r, M) 2 6. Define, { } E[Z t (r, M)Z t 1 (r, M)] ρ 1 (r, M) R E Z t (r, M) 2 (36) 7. Clearly, ρ 1 (r, M) = 1 2 σ2 1 H(ω 1; r, M) 2 cos(ω 1 ) + π π H(ω; r, M) 2 df ζ (ω) r 1 2 σ2 1 H(ω 1; r, M) 2 + π π H(ω; r, M) 2 df ζ (ω) 8. Let C(r, M) = E ζ t(r, M) 2 E Z t (r, M) 2 9. The contraction has now an extra parameter M: ρ 1 (r, M) = r + C(r, M)(r r ) (37) where r = cos(ω 1 ), and ω 1 is the true frequency. 10. The CM algorithm takes now the form r k+1 = ρ 1 (r k, M k ) (38) 38

39 CM With the Parametric AR(2) Filter in Practice With α ( 1,1), η (0,1), Y t (α) = (1 + η 2 )αy t 1 (α) η 2 Y t 2 (α) + Y t (39) Initial guess of ω 1 : θ 0. Start with α 0 = cos(θ 0 ). Start with η close to 1, for example η = Increment of η: e.g Define the sample autocorrelation ˆρ 1 (α, η) = N 1 t=1 Y t(α)y t 1 (α) N 1 t=0 Y t 2(α) (40) Then the CM algorithm is given by, ( ) α k+1 = ˆρ 1 (α k, η k ), k = 0,1,2, (41) where η k increases with each iteration. 39

40 Li (1992), Song Li (2000), Li Song (2002): Y t = β cos(ω 1 t + φ) + ǫ t (42) β is a positive constant. ω 1 ( π, π). φ Unif(0, π] {ǫ t } i.i.d. with mean 0 and variance σ 2 ǫ, independent of φ. If (1 η) 2 N 0 as N, then (1 η) 1/2 N(ˆω 1 ω 1 ) L N(0, γ 1 ) γ = 1 2 β2 /σ 2 ǫ The implication of this is that by a judicious choice of η, the precision of the CM estimate can be made arbitrarily close to that achieved by periodogram maximization and nonlinear least squares. More properties and discussions can be found in Li K (1993a, 1993b, 1994, 1998), K (1994), Li Song (2002). 40

41 S-Plus Code for the CM Algorithm KY.AR2 <- function(z,theta0,eta,inc,niter){ y <- rep(0,length(z)) r <- rep(0,niter); OMEGA <- rep(0,niter) r [1] <- cos(theta0); OMEGA[1] <- theta0 cat(c("initial frequency guess is", OMEGA[1]),fill=T) cat(c("eta"," r(k)"," Omega(k)", " Var(y)"), fill=t) for(k in 2:niter){# eta increments by inc eta <- eta+inc if((eta < 0) (eta >1)) stop("eta must be between 0 and 1") FiltCoeff <- c((1+eta^2)*r[k-1],-(eta^2)) y <- filter(z,filtcoeff, "rec") # CM Iterations rrr <- acf(y) # motif() must be on r[k] <- rrr$acf[2] # Gives acf(1)!!! # OMEGA[k] <- acos(r[k]) cat(c(eta,r[k],omega[k],var(y)),fill=t)}} 41

42 Example: Y t = 0.5cos(0.513t + φ 1 ) + cos(0.771t + φ 2 ) + 2.2ǫ t t = 1,...,1500. ǫ t i.i.d. N(0,1). SNR = 10log 10 ((.5 2 / /2)/2.2 2 ) = Starting at θ 0 = 0.48, η = Increment of η Final estimate is ˆω = Error: η α(k) ω(k) Var(Y t (α)) We are going to apply CM to the data without centering. 42

43 Starting at θ 0 = 0.88, η = Increment of η Final estimate is ˆω = Error: η α(k) ω(k) Var(Y t (α)) Algorithm ˆω ω Error CM FFT CM FFT

44 Example: Detection of a Diurnal Cycle in GATE I. The CM algorithm with the AR(2) parametric filter was applied to a time series of length N = 450 of hourly rain rate from GATE I (early 1970s) averaged over a region of km 2. GATE I: Hourly Averaged Rain Rate TIME 44

45 Starting at 0.29 to the right of : η = Increment of η η α(k) ω(k) Var(Y t (α)) π = Starting at 0.25 to the left of : η = Increment of η η α(k) ω(k) Var(Y t (α)) π =

46 References Barnett and Kedem (1998). Zero-crossing rates of mixtures and products of Gaussian processes. IEEE Tr. Information Theory, 44, Hadjileontiadis, L.J. (2003). Discrimination analysis of discontinuous breath sounds using higher-order crossings. Med. Biol. Eng. Comput., 41, He and Kedem (1989). Higher Order crossings Spectral Analysis of an Almost Periodic Random Sequence in Noise. IEEE Tr. Information Theory, 35, Kedem (1976). Exact Maximum Likelihood Estimation of the Parameter in the AR(1) Process After Hard Limiting. IEEE Tr. Information Theory IT-22, No. 4, Kedem (1984). On the Sinusoidal Limit of Stationary Time Series. Annals of Statistics, 12, Kedem (1986). Spectral Analysis and Discrimination by Zero- Crossings. Proceedings of the IEEE, 74, Kedem (1987). Detection of hidden periodicities by means of higher order crossings. Jour. of Time Series Analysis, 8, Kedem (1987). Higher Order Crossings in Time Series Model Identification. Technometrics, 19, Kedem (1992). Contraction mappings in mixed spectrum estimation. In New Directions in Time Series Analysis, D. Brillinger et al. eds., Springer-Verlag, NY, IMA Vol. 45, pp Kedem (1994). Time Series Analysis by Higher Order Crossings, IEEE Press, NJ. ( ) 46

47 Kedem and Li (1991). Monotone gain, first-order autocorrelation, and expected zero-crossing rate. Annals of Statistics, 19, Kedem and Lopez (1992). Fixed points in mixed spectrum analysis. In Probabilistic and Stochastic Methods in Analysis, With Applications, J. S. Byrnes et al., eds., Kluwer, Mass., pp Kedem and Slud (1981). On Goodness of Fit of Time Series Models: An Application of Higher Order Crossings. Biometrika, 68, Kedem and Slud (1982). Time Series Discrimination by Higher Order Crossings. Annals of Statistics, 10, Kedem and Slud (1994). On autocorrelation estimation in mixed spectrum Gaussian processes. Stochastic Processes and Their Applications, 49, Kedem and Troendle (1994). An iterative filtering algorithm for non-fourier frequency estimation. Jour. of Time Series Analysis, 15, Kedem and Yakowitz (1994). Practical aspects of a fast algorithm for frequency detection. IEEE Tr. Communications, 42, Kong and Qiu (2000). Injury detection for central nervous system via EEG with higher order crossing-based methods. Method. Inform. Med., 39, Li (1997). Time series characterization, Poisson integral, and robust divergence measures. em Technometrics, 39, Li (1998). Time-correlation analysis of nonstationary time series. Journal of Time Series Analysis, 19,

48 Li and Kedem (1993). Strong consistency of the contraction mapping method for frequency estimation. IEEE Tr. Information Theory, 39, Li and Kedem (1993). Asymptotic analysis of a multiple frequency estimation method. Jour. Multivariate Analysis, 46, Li and Kedem (1994). Iterative filtering for multiple frequency estimation. IEEE Tr. Signal Processing, 42, Li and Kedem (1998). Tracking abrupt frequency changes. Jour. of Time Series Analysis, 19, Li and Song (2002). Asymptotic analysis of a contraction mapping algorithm for multiple frequency estimation. IEEE Tr. Information Theory, 48, No. 10. Song and Li (2000). A statistically and computationally efficient method for frequency estimation. Stochastic Processes and Their Applications, 86, Yakowitz (1991). Some contributions to a frequency location method due to He and Kedem. IEEE Tr. Information Theory, 37,