26mod_A-Kougioumtzoglou et alx - Unipa

Meccanica dei Materiali e delle Strutture Vol. 3 (2012), no.3, pp. 37-44 ISSN: 2035-679X Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, Dei Materiali DICAM

SOME OBSERVATIONS ON WAVELETS BASED EVOLUTIONARY POWER SPECTRUM ESTIMATION * † Ioannis A. Kougioumtzoglou , Fan Kong , Pol D. Spanos °, Jie Li ˘ *

Institute for Risk and Uncertainty University of Liverpool Liverpool L69 3GH, United Kingdom e-mail: [email protected] †

School of civil Engineering Tongji University 1239 Siping Road, Shanghai, China e-mail: [email protected] °

Department of Mechanical Engineering and Materials Science Rice University 6100 Main, Houston, Texas 77005-1827 e-mail: [email protected] ˘

School of civil Engineering Tongji University 1239 Siping Road, Shanghai, China e-mail: [email protected] (Ricevuto 10 Giugno 2012, Accettato 10 Ottobre 2012)

Key words: Stochastic Process, Evolutionary Power Spectrum, Harmonic Wavelet Transform, Monte Carlo Simulation. Abstract. Some observations on wavelets based approaches for estimating evolutionary power spectra (EPS) are made. Specifically, relying on a recently proposed mathematical, wavelets based, representation of non-stationary stochastic processes it is pointed out that many of the existing EPS estimation approaches can be construed in the same stochastic framework. Further, it is emphasized that for the special case of non-overlapping in frequency domain wavelets the discussed EPS estimation approach leads to a significant reduction of the associated computational cost. An illustrative example of application is also included. 1

INTRODUCTION

Complex systems of engineering interest are often subject to excitations, such as seismic motions, winds, ocean waves and extreme events, which inherently possess time-varying characteristics. Thus, to perform a realistic and efficient system analysis and design a Meccanica dei Materiali e delle Strutture | 3 (2012), 3, PP. 37-44

37

I. A. Kougioumtzoglou, F. Kong, P. D. Spanos, J. Li

representation of these excitations by non-stationary random processes is required. Further, related to non-stationary random processes is the concept of the evolutionary power spectrum (EPS). In this regard, several research efforts have focused on utilizing the potent localization properties of wavelets for estimating EPS. Early work was conducted by Iyama and Kuwamura 1, where a time-dependent spectrum was proposed based on nonorthogonal wavelets. Further, Basu and Gupta 2 developed relationships between the meansquare value of the wavelet transform at different scales and the time-dependent spectral content of the process. Recently, Spanos and Failla 3 suggested a wavelet-based approach for estimating the EPS of non-stationary stochastic processes as defined by Priestley 4. This approach yields an equation in the frequency domain relating the instantaneous mean-square value of the wavelet transform to the EPS of the process. In this manner, the EPS of the random process can be obtained as a series expansion involving the square moduli of the Fourier transforms of the wavelet at different scales. Further, Huang and Chen 5 extended the approach of ref. 3 to address multi-variate processes, whereas Chakraborty and Basu 6 applied the approach for the response analysis of a long span bridge. Finally, Spanos and Tezcan 7 and Spanos and Kougioumtzoglou 8 utilized the orthogonality properties of the harmonic wavelets (HW) to develop relationships between the mean square value of HW transform and the EPS of the stochastic process. In this paper, some observations on existing wavelets based approaches for estimating EPS are provided. Specifically, the approach developed by Spanos and Failla 3 is considered from a novel perspective and a generalization based on the locally stationary stochastic processes framework (e.g. ref. 9) is developed circumventing limitations of the previously utilized Priestley model (e.g. ref. 4). In fact, it is pointed out that the seemingly different EPS estimation approaches by Spanos and Failla 3, by Spanos et al. 7 and by Spanos and Kougioumtzoglou 8 can be construed under the same stochastic framework. 2

EPS ESTIMATION BASED ON PRIESTLEY’S MODEL

2.1 Mathematical formulation Recently, a wavelet based approach for estimating EPS was developed by Spanos and Failla 3 where the Priestley model of non-stationary random processes 4 was utilized. In this regard, a non-stationary random process f ( t ) takes the form f (t ) = ∫

+∞

−∞

A (ω , t ) eiωt dZ (ω ),

(1)

where A (ω , t ) is a slowly time-varying modulating function; and Z (ω ) is a complex valued random process with orthogonal increments 4. In ref. 3 it was shown that the EPS of the nonstationary stochastic process can be estimated as ma

S ff (ω , b ) = ∑ c j ( b ) Ψ (ω a j ) , 2

(2)

j =1

where ma is the number of wavelet scales involved in the wavelet representation of the process f ( t ) ; b is the time instant; a j Ψ (ω a j ) is the square modulus of the Fourier 2

transform of the wavelet at scale a j . Further, the time-dependent coefficient c j ( b ) can be determined by solving a linear system of equations of the form

Meccanica dei Materiali e delle Strutture | 3 (2012), 3, PP. 37-44

38


Qc ( b ) = E  W ( a, b )  ,   2 where Q is a ma × ma matrix, and c ( b ) and E  W ( a, b )  are the ma × 1 vectors, i.e.,   2

c ( b ) =  c1 ( b ) , c2 ( b ) ,..., cma ( b )  ,

(3)

T

{

(

2 2 2 E  W ( a, b )  = E  W ( a1 , b )  , E  W ( a2 , b )  ,..., E  W ama , b        ∞

2

)

2

}

 

T

,

(4)

Qr , s = 8π 2 ar ∫ Ψ (ω ar ) Ψ (ω as ) d ω. 2

0

It can be readily seen that unless matrix Q is diagonal, its inversion can render the approach computationally demanding. A more detailed presentation of the approach and a thorough discussion on its computational aspects can be found in ref. 3. 2.2 Harmonic wavelets: A special case The harmonic wavelet 10 is a box-shaped and non-overlapping wavelet in frequency domain. These properties facilitate greatly the inversion operation of Q of Eq.(3). Indeed, it is shown that for the case of the harmonic wavelet Q becomes diagonal; thus, rendering the EPS estimation approach significantly more efficient. The generalized harmonic wavelet (GHW), as proposed by Newland 10, of scale (m,n) and time instant k is given in the frequency domain by the equation 8  1  −iω kT0  exp    , m∆ω ≤ ω < n∆ω G (5) Ψ ( m , n ) , k ( ω ) =  ( n − m ) ∆ω ,  n−m  0 , otherwise  where (m, n) and k are positive integer numbers, ∆ω = 2π / T0 with T0 being the time duration of the signal f ( t ) . Furthermore, the continuous GHW transform is defined as n−m ∞ WψG f ( m, n ) , k  = f ( t )ψ (Gm ,n ),k ( t ) dt , ∫ −∞ T0

(6)

where ψ (Gm,n),k ( t ) is the GHW in time domain at scale ( m, n) and time instant tk , and denotes the complex conjugate. In a similar manner as in ref. 3, a EPS representation of a Priestley random process can be established based on the GHW. Substituting Eq. (1) into Eq. (6), one can obtain n j − m j +∞ ∞ iωt G WψG ( m j , n j ) , k  = (7) ∫−∞ ∫−∞ A (ω, t ) e dZ (ω )ψ (m j ,n j ),k dt. T 0

Due to the time localization property of the wavelet ψ ( m ,n ),k ( t ) at the vicinity of the time j j instant τ k , it is assumed that n j − m j +∞ ∞ WψG ( m j , n j ) , k  ≈ A ( ω ,τ k )  ∫ eiωt ψ (Gm ,n ), k ( t ) dt  dZ (ω ) ∫ −∞ j j  −∞  T 0

= ( n j − m j ) ∆ω ∫ A (ω , τ k ) e dZ ' ( ω ) , ∞

(8)

iω k

−∞

where

dZ ' (ω ) = Ψ ( m

j ,n j

)(

ω ) ( n j − m j ) ∆ω dZ (ω ) .


(9)

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Note that according to Eq. (8) the GHW transform WψG ( m j , n j ) , k  can be expressed as a Priestley kind non-stationary process with respect to the continuous time parameter k. Further, the instantaneous mean-square value of the wavelet transform process can be determined as 2 2 +∞ 2 E  WψG ( m j , n j ) , k   = ( n j − m j ) ∆ω 2 ∫ Ψ ( m ,n ) (ω ) S ff (ω ,τ k ) d ω. (10) −∞ j j   Next, seeking the solution of S ff (ω ,τ k ) in the form

S ff (ω ,τ k ) =

N( m ,n )

∑ c( j =1

m j ,n j

2

)(

τ k ) Ψ ( m ,n ) (ω ) , j

(11)

j

where N ( m ,n ) is the total number of scales used in the GHW transform, and c( m ,n ) (τ k ) is the j j time-dependent coefficient, yields 2 Q c (τ k ) = E  WψG ( m, n ) , k   , (12)   where Q = diag Q11 , Q22 ,..., QN( m ,n) N( m ,n)  ,   T

  c (τ k ) =  c( m1 , n1 ) (τ k ) , c( m2 ,n2 ) (τ k ) ,..., c τ k ) , ( ( m N( m , n ) , n N( m , n ) )  

{

(13)

) },

(

WψG ( m, n ) , k  = WψG ( m1 , n1 ) , k  , WψG ( m2 , n2 ) , k  ,..., WψG  mN( m ,n) , nN( m ,n) , k   

T

with Qi , j = ( n j − m j ) ∆ω 2 ∫ 2

+∞

−∞

Ψ (m

j

(ω ) ,n )

2

j

Ψ ( mi , ni ) (ω ) d ω. 2

(14)

Next, taking into account the orthogonality properties of the HW and combining Eq.(5) and Eqs. (11)-(14) yields 2 T0 (15) S ff (ω j ,τ k ) = E  WψG ( m j , n j ) , k   ,  2π ( n j − m j ) 

where m j ∆ω < ω j ≤ n j ∆ω , kT0 / ( n − m ) < tk ≤ ( k + 1) T0 / ( n − m ) . Based on the aforementioned analysis, it is noted that the approach of Spanos and Failla 3 can be reduced to the formula of GHW-based EPS estimation of references 7 and 8. In other words, Eq. (15) is a special case of the approach in ref. 3 when the GHW is utilized. Following a different route, a similar EPS estimation formula as the one of Eq.(15) has been derived in the literature (e.g. ref. 2_ENREF_7) where the modified Littlewood-Paley (MLP) wavelet was utilized. This is anticipated since the MLP wavelet is simply the real part of the HW. Nevertheless, note that a parameter p relative to the bandwidth of the mother MLP wavelet should be properly chosen so that the fluctuations/peaks in the frequency spectrum, especially at the lower frequency band, can be properly captured. Further, since the MLP wavelet is a real wavelet, it is not suitable for cross-EPS estimation applications.


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3

EPS ESTIMATION BASED ON LOCALLY STATIONARY WAVELET MODEL

3.1 Mathematical formulation A new mathematical framework for representing non-stationary stochastic processes has been recently developed by Nason et al. 9 and Eckley et al. 11 based on the concept of the locally stationary wavelet (LSW) process. In this regard, the non-stationary process, according to the LSW representation, can be cast in the form

f ( t ) = ∑∑ w j ,kψ j ,k ξ j ,k , j

(16)

k

where ξ j ,k is a stochastic orthonormal increment sequence; ψ j ,k is a non-decimated family of wavelets and Σ j and Σ k represent the summations over different scales and time translations. Next, considering a general wavelet transform W ( j, k ) = ∫

∞

f ( t )ψ j ,k ( t ) dt ,

−∞

(17)

where

ψ j ,k ( t ) = 1 / a jψ ( t − bk ) / a j  , and combining Eq.(17) and Eq.(16) yields W ( j, k ) = ∫

∞

−∞

∑∑ w ψɶ ( t ) ξ l ,i

l

l ,i

(18)

ψ j ,k ( t ) dt ,

l ,i

(19)

i

where ψɶ l ,i ( t ) is the wavelet basis in the LSW representation. Next, exchanging the integral with the summations and assuming ψɶ l ,i ( t ) and ψ j ,k ( t ) represent the same wavelet, yields W ( j , k ) = ∑∑ wl ,iξl ,i I l ,i ( j , k ), l

(20)

i

where ∞

I l ,i ( j , k ) = ∫ ψ l ,i ( t )ψ j , k ( t ) dt = ψ l ,i ( t ) ,ψ j , k ( t ) −∞

(21)

is a reproducing kernel. Multiplying both sides of Eq. (20) by their complex conjugate and applying the expectation operation yields (22) E  W ( j , k )  = ∑∑ wl ,i I l ,i ( j , k ) .   l i Further, applying the wavelet inner product on both sides of Eq. (20) and considering the properties of the reproducing kernel yields 2

2

2

2 E  W ( j, k ) , W ( j, k ) ψ  = ∑∑ wl ,i I l ,i ( l , i ) .   l i Next, applying Parseval’s theorem, Eq. (23) can be cast into the form 2 E  f ( t ) , f ( t )  = 2π E  F (ω ) , F (ω )  = E  W ( j , k ) , W ( j , k ) ψ  = ∑∑ wl ,i I l ,i ( l , i ).   l i

(23)

(24)

Taking into account the time localization of wavelets, the EPS S (ω , t ) at a given time instant is expressed as 2 1 (25) S ( ω , ti ) = wl ,i I l ,i ( l , i ), ∑ 2π l where wl ,i

2

is determined by Eq. (22).


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It can be readily seen that the EPS estimation approach based on Nason’s LSW model resembles the one which is based on the Priestley model. Note that Eq.(25) is the counterpart of Eq.(2), whereas Eq. (22) is the counterpart of Eq.(3). Further, the square modulus of the reproducing kernel can be viewed as the counterpart of the time-independent matrix Q. 3.2 Harmonic wavelets: A special case The EPS estimation procedure discussed in section 3.1 can be further simplified when the GHW is involved in the derivation. In this regard, combining Eq. (6) and Eq. (16) yields n−m ∞ WψG  m j , n j , k  = w( ml , nl ),iψ ( ml ,nl ),iξ( ml ,nl ),iψ (*m ,n ),k ( t ) dt. (26) ∑∑ ∫ −∞ j j T0 l i In a similar manner as before the equation

(

)

2

2 2  n − ml  2 (27) E  WψG ( m j , n j ) , k   =  l  ∑∑ w( ml ,nl ),i I l ,i ( j , k )    T0  l i is derived which is a special case of Eq. (22), whereas the reproducing kernel in the case of the GHW becomes

I l ,i ( j , k ) = ψ (Gml ,nl ),i ( t ) ,ψ (Gm ,n ),k ( t ) = 2π Ψ G( ml , nl ),i (ω ) , Ψ G( m ,n ),k (ω ) j j j j =

2π δ ( l − j ) δ ( i − k ) ∆ω

( nl − ml ) ( n j − m j )

(28)

.

Combining next Eq. (27) and Eq. (28) yields 2 2 n − m 2 2 2   ( 2π ) j j G (29) E  Wψ ( j , k )  =  w = w j ,k .  j ,k 2    T0   ∆ω ( n j − m j )    It is emphasized that the EPS estimation formula of Eq. (15) can be derived by considering Eq.(25), Eq. (28) and Eq.(29); thus, the EPS estimation approaches based on the Priestley and on the LSW models can be unified under the same stochastic framework. Further, it is noted that the approach proposed by Spanos et al. 7 and by Spanos and Kougioumtzoglou 8 is not suitable only for the Priestley oscillatory process, but also for more general representations of stochastic processes. 4

NUMERICAL EXAMPLES

In this section, a simple, yet illustrative case is considered involving a uniformly modulated multi-variate EPS. Specifically, the acceleration time histories at three different locations on the ground surface along the earthquake wave propagation direction are modeled as a trivariate non-stationary stochastic process (Figure 1). A detailed presentation of the model can be found in reference 12. In the following numerical example, n j − m j = 12 is chosen to acquire the appropriate time-frequency resolution balance. Figures 2-5 show the auto/cross- target and estimation EPS. Comparisons between the two indicate the potential of the proposed approach for the auto- and the cross- EPS estimation of multi-variate stochastic process also.


42


rock or stiff soil

cohesionless soil

1

medium clays and sands

2 50m

3 50m

Figure 1: Configuration of location 1, 2 and 3 on the ground surface

Figure 2: Real part of target cross-EPS of f3-f1 Figure 3: Real part of estimated cross-EPS of f3-f1

Figure 4: Imaginary part of target cross-EPS of f3-f1 Figure 5: Imaginary part of estimated cross-EPS of f3-f1

5

CONCLUDING REMARKS

In this paper, some observations regarding wavelets based approaches for estimating EPS have been recorded. Specifically, a locally stationary stochastic processes framework has been employed to generalize the approach by Spanos and Failla 3. In this regard, limitations associated with the previously adopted Priestley model have been circumvented. Further, for the special case of non-overlapping in frequency domain wavelets the approach can lead to a significant reduction of the associated computational cost. A numerical example elucidating the various concepts discussed has also been included.

REFERENCES [1]J. Iyama and H. Kuwamura, "Application of wavelets to analysis and simulation of earthquake motions", Earthquake. Eng. Struc, 28(3), 255-272 (1999). Meccanica dei Materiali e delle Strutture | 3 (2012), 3, PP. 37-44

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[2]B. Basu and V.K. Gupta, "Seismic response of sdof systems by wavelet modeling of nonstationary processes", J. Eng. Mech-Asce, 124(10), 1142-1150 (1998). [3]P.D. Spanos and G. Failla, "Evolutionary spectra estimation using wavelets", J. Eng. Mech-Asce, 130(8), 952-960 (2004). [4]M.B. Priestley, "Evolutionary spectra and non-stationary process", J. Roy. Stat. Soc. B, 27, 204-237 (1965). [5]G. Huang and X. Chen, "Wavelets-based estimation of multivariate evolutionary spectra and its application to nonstationary downburst winds", Eng. Struct, 31(4), 976-989 (2009). [6]A. Chakraborty and B. Basu, "Nonstationary response analysis of long span bridges under spatially varying differential support motions using continuous wavelet transform", J. Eng. Mech-Asce, 134(2) (2008). [7]P. Spanos, J. Tezcan, and P. Tratskas, "Stochastic processes evolutionary spectrum estimation via harmonic wavelets", Comput. Method. Appl. M, 194(12-16), 1367-1383 (2005). [8]P.D. Spanos and I.A. Kougioumtzoglou, "Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination", Probab. Eng. Mech, 27(1), 57-68 (2012). [9]G.P. Nason, R. von Sachs, and G. Kroisandt, "Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum", J. Roy. Stat. Soc. B, 62(2), 271-295 (2000). [10]D.E. Newland, An introduction to random vibrations, spectral and wavelet analysis, Longman scientific & Technical, (1993). [11]I.A. Eckley, G.P. Nason, and R.L. Treloar, "Locally stationary wavelet fields with application to the modeling and analysis of image texture", 59, 595-616 (2010). [12]G. Deodatis, "Non-stationary stochastic vector processes: Seismic ground motion applications", Probab. Eng. Mech, 11(3), 149-167 (1996).


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