Autoregressive moving average models of earthquake records

REVIEW PAPER Autoregressive moving average models of earthquake records F. Kozin

Polytechnic University, Farmingdale, New York, N Y 11735, USA

In this paper, we present a brief survey of literature concerned with fitting ARMA models to earthquake record data. This is, in a sense, relatively new for the earthquake record problem. The major conceptual difficulty is due to the nonstationary character of the data, and the fact that only single records are available from which to estimate the various coefficients. Thus, proper statistical methods must be developed and applied to account for this inherent problem. We briefly discuss a number of approaches that have been taken to generate statistically reliable and useful nonstationary models for earthquake engineering studies, and state a few problems of importance in the further development of this method.

I. I N T R O D U C T I O N In this section we define the concept of an ARMA model, present a brief background on the modelling of seismic data as time series, and finally review a modern nonlinear filtering approach for obtaining nonstationary continuous time models of earthquake records. This is in contrast to the discrete time, ARMA model. In Section II, we review those ARMA model studies that have been central in the development of this approach for characterizing earthquake records. Finally in Section III, we present a statement concerning problems of potential significance in the further development of the ARMA approach for earthquake records. The autoregressive moving average (ARMA) model is the general linear model of the time series analysis 1. It may be written in the form Yr-balyl 1 +o2Y~-2 +" " • +anYt-n =u~+blu,_l + . . . + b m u , ,,

(1)

where {ai}, {bi} are constant coefficients, and the random sequence {ut} is independent identically distributed as Gaussian random variables. The sequence {Yt} denotes the sequence of data obtained from observations on the evolution in time of the phenomenon that is being investigated. The observations are in discrete time t, where t may range over the nonnegative integers {0,1,2,...}, or over the entire set of integers {.... - 2 , - 1,0, 1,2 . . . . }. The discrete time sequence {Yt} is called a time series. The model, (1), as described above is of order (n,m), and all variables are scalar variables. This corresponds to a so-called single input-single output linear model. ARMA models may be extended to multiinput multi-output linear systems, in which case the observations Yt and the random terms u, become vectors, {3 1988 Computational Mechanics Publications 58

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 2

and the coefficients (al, bi) become matrices. This vector case is commonly studied in the systems literature. See, for example Refs 2-4. For earthquake excitations, the scalar case is the important one. The basic problem of modelling of the observed time series {y,} by the ARMA model (1) is focused on estimating the parameters {ai}, {bi}. Moreover, there is also the modelling question of determining the best model order fit to the observed data. That is, determining the best choice of (n,m). The estimation of parameters is usually based upon least squares or maximum likelihood ideas and their variations, such as two stage approaches, ladder algorithms, recursive algorithms, etc. One may find discussions of these approaches in Ref 5. A recent comparative study of a number of ARMA parameter estimation procedures may may be found in Ref. 6. The problem of determination of the best choice for the model order (n, m) gained substantial inpetus as a result of the fundamental work due to Akaike, whose so-called information criterion v has become a commonly used procedure for choosing the optimal order fit for ARMA models in general. The first investigator to study linear models of seismic records as time series is E. A. Robinson in his celebrated 1954 thesis, Predictive Decomposition of Seismic TracesS. The work of Robinson was motivated by geophysical exploration for oil deposits under the sea surface. For exploratory purposes an explosion (i.e., a pulse) is set off below the water surface by one ship, a nearby ship pulling a string of detecting instruments records the sequence of seismic records as digitally obtained time series. It is assumed that these seismic records, denoted by the time series .Ix,}, may be represented as infinite moving

Autoregressive moving average models: F. Kozin averages,

xt=boet +blet-i

q-b213t-2-}- . . . .

~ bset-,

(la)

s=O

where the e, represents the 'wavelet' that is induced by the explosion, assumed to be of known form. It is assumed that the wavelet remains constant along its propagation path. Thus if the earth's subsurface is assumed to be a simple linear layered media 9, then the seismic record is made up of the superposition of reflected wavelets of the form (la), with coefficients that contain the reflectivity encountered within the various subsurface layers. Robinson developed a method of predictive deconvolution in which a statistically optimal prediction operator is derived and the predicted value is compared with the actual value of the record at each time. The error function allows estimation of the coefficients {bs} which yields the reflectivity properties of the layers. Large reflections indicate layers that are potential hydrocarbon reservoirs. These techniques have become so important in modem geophysical exploration, that a collection of the papers of Robinson as well as his co-worker Trietel were published in a special volume by the Seismograph Service subsidiary of the Raytheon Corporation in 19691°, for distribution to the petroleum industry. These techniques are also of importance in the discrimination of underground nuclear tests 11. Of course the problem of characterization of seismic records for exploration and their simulations, is different fundamentally from earthquake characterization since the pulse that generates the seismic record is known in shape and in time. The important point, however, is that statistical methods were used to estimate the coefficients from this linear moving average model. In the general topic of earthquake record characterization from the observed time series, ARMA models were specified as a potential approach to their characterization by S. C. Liu in a paper appearing in 197012. Within the general topic of earthquake engineering, this paper appears to be one of the earliest to specify ARMA models for earthquake simulation and characterization. However, no specific applications were discussed or presented in Ref. 12. In the early 1970's in papers that were motivated by the attempt to model earthquake accelerograms by modem estimation methods, a nonstationary continuous time model was postulated with unknown parameters to be estimated from data by nonlinear filtering techniques 1315 Nonlinear filtering techniques are extension of the methods of recursive estimation such as, for example, Kalman Filtering. This was a precurser to a more comprehensive study of nonstationary ARMA models for earthquake records. In the continuous model, an assumed analytical form was given as,

5~(t) + a(t):~(t) + b(t)x(t) = (9(t)w(t)

(2)

where a(t), b(t) were specific forms as polynomals,

a(t)=ao + al t + a2t2 + aat b(t) = bo + bl t + b2 t2 + b3 t3

(3)

The modulation function ¢(t) is determined as an envelope of the actual record, which was fitted by a cubic spline technique. The random quantity in the model (2) is w(t), which is assumed to be a Gaussian white noise. The idea basically, was to represent (2) by the vector

z(t)=

xItjj

za

2(t}]

z~

altl l

z3

a(t) t a(t) l

Z4 Z5

~(t) l

Z6

b(t) l

7

l z7

b(t) l

28

bttj [

Z9

~(tj l

ZlO

(4)

and rewrite the equation as dz(t) - F(z)z(t) + g(t)w(t) dt

y(t) = Mz(t) + av(t)

(5)

where F(z) is a matrix containing elements of the vector z(t). Hence, the system (5) is a nonlinear representation of (2) since a(t), b(t) and their derivatives are treated as dependent variables. The vector y(t) is the observation vector which incorporates the possibility that the observations of the earthquake record data are inaccurate, that is they are noisey. ~x Given this formulation, the objective is tO obtain the estimator of z(t), which we denote as ~(t), which satisfies the least square criterion,

minE{llz(t)-((t)ll2}=E{Hz(t)-~(t)l[}

(6)

It is known that this optimal estimator 2(t) has the representation, i(t) = E{z(t)

[y(s);0 ~
(7)

Unfortunately, the formal solution (7), cannot be obtained exactly. However, a stochastic partial differential equation can be generated from the FokkerPlanck equation for the conditional density p(z,t[y(s); 0 ~
fi(z) = E{h(z(t)) [y(s): 0 ~
(8)

A nonlinear stochastic equation is obtained for (8), which is quite complex and cannot be solved in general. This is the basic reason that the so-called nonlinear filtering problem in systems estimation and identification cannot be solved exactly. Approximations must be made. The procedure developed in Refs 13-15 made use of the Schwartz-Bass approximation, which leads in the case of the models (2) and (5) to a closed set of equations for the

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 2

59

Autoregressive moving average models: F. Kozin first and second order conditional moments. The resulting equations are then solved recursively to yield approximations to the desired parameter estimates, a(t), b(t) for the recorded earthquake data 15. Although this approach, through recursive updating, is both analytically and computationally attractive, especially for real time computations as data is being gathered, the problem is that it does not always lead to stable computational results. The reason is basically that approximates must be made to the original nonlinear filtering approach, and there are no convergence theorems that will tell us under what conditions the approximated nonlinear recursive algorithm will converge to the desired parameter estimates. The experience with this method from simulated earthquake data showed that if the initial estimates (or guess) of the unknown coefficients were close to the true values, then the convergence was excellent and the recursively generated model followed the true simulated record excellently. On the other hand, if the initial parameter estimates were not close to the true values, the recursive computational scheme would become unstable and computer overflow was generally observed. This was for the time varying polynomial coefficient case. If constant coefficients were assumed, overflow was not a problem, but in that case the desired nonstationary properties of the records were not fully accounted for in the model. Generally speaking, if there are convergence theorems to guarantee that a specific estimation procedure will converge asymptotically to the 'true' or at least optimally best values, then the numerical procedures will generally be well behaved. It turns out that this is true of maximum likelihood estimators and therefore instead of pursuing the nonlinear filtering problem for continuous time models, it was found that maximum likelihood parameter estimators, for AR models and ARMA models were a more desirable approach to the modelling of earthquake records.

form a~(k)= ~ a~.jfj(k),

(10)

where m is prechosen and the fj(k) are a suitable family of discrete orthogonal functions. For the sequence of studies reported in Refs 17-20, it was assumed that the {fj(k)}, are the orthogonal family of discrete Legendre polynomials ,=o

D~(J~(J+s~ k'~,

.rtk)= Z t - ' i s ) \ s )

(11)

j = 0 , 1,2, . . . , N where for any integer p, p ~ S ~ = p ( p - 1 ) . . . ( p - s + 1). In (11), N represents the number of data points, and the family is orthogonal on [0,N]. As in Ref. 15, the coefficient g(k) is obtained by cubic spline fit to the amplitude envelop of the earthquake record. The coefficient g(k) is obtained first, then the unknown coefficients {aij} defined by (I0) are estimated by maximum likelihood techniques. Application of the nonstationary AR model (9) to a wide range of earthquake records, taken from the Caltech reports on digitized strong motion earthquake accelerograms result in simulated records that are exceedingly close, visually, to the record that generates the specific model. Furthermore, application of the Akaike Information Criterion (AIC) approach 17 to yield the best fitting order, results in even better fits. Note that there are two theoretical questions that must be addressed here.

II. ARMA MODELS FOR EARTHQUAKE DATA It appears that one of the earliest studies of modelling of earthquakes by autoregressive methods appear in two papers ~6,~ 7 on the modelling of nonstationary time series by AR models. This was later followed by two theoretical papers 4,is on the properties of estimators for nonstationary models. These papers grew out of two theses on the topic 19'2°. The model studied was of the form

y(k) + al (k - 1)y(k - 1) + . . . + al ( k - l ) y ( k - l) = g(k)u(k) (9) where {y(k)} represents the earthquake record data, {u(k)} is a white gaussian sequence and the coefficients {ai(k)}, g(k) are time varying to be estimated from the data. Furthermore, it was also considered of importance to obtain some estimate of the best fitting model order, l. Naturally, the coefficients a~(k) cannot be completely arbitrary. Thus, a structure was given to them as in the models treated in Ref. 15. They were parametized in the

60

i= 1. . . . . I

j-1

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 2

(A) Do maximum likelihood estimates for the coefficients of nonstationary (nonergodic) AR models converge as the number of data points becomes arbitrarily large? (B) Is the AIC criterion approach valid for nonstationary models? Both of these questions were studied in Refs 4 and 18. In Ref. 4, the general problem of convergence of parameter estimates was discussed, and a basic theorem was obtained that guarantees convergence of the estimates {aij} in the nonstationary AR model defined by (9), (10). Furthermore, the Akaike Information Criterion was extended to a wide class of models that includes the nonstationary models of the form (9), (10), in Ref. 18. Thus, the answers to questions (A), (B) above are both affirmative. In particular, a means of testing the probability of the statistical fit of a model to nonstationary geophysical data is available. It is interesting to note that generating the response spectra of simulated earthquakes from various order models generated by an actual earthquake record, always showed that, by far, the best fitting models as determined by the AIC criterion has the closest fit to the response spectrum of the actual earthquake. This is also an interesting fact since the model is determined in the time domain, whereas the response spectrum is associated with frequency domain properties. It was found that the best order fit varied among the many earthquake studied, between I= 2, and I= 6. In another study of A R M A models applied to

Autoregressive moving average models: F. Kozin earthquake records 2., four earthquake records were studied; two from El Centro (1940, 1956), and two from San Fernando (1971), including the Pacoima dam. Of these records, a forty second duration is taken and then this record is subdivided into eight records, each of five seconds duration, which contains 250 data points. Then each 250 data point record is fit to an ARMA model. This is the way in which nonstationarity was accounted for in the study. The coefficients of the ARMA models for each segment of all the earthquake records were estimated by Box and Jenkins procedures 1. In the Box and Jenkins procedure, an assumed (p,q) for the ARMA order is initially made on the basis of the qualitative characteristics of the sample auto-correlation. The partial auto-correlation function and the spectrum of the data parameters are then estimated by an algorithm for minimizing the sum of the squared residuals. The procedure is sequential in that estimation and identification is repeated. An initial guess is made for the coefficients of a nonlinear algorithm that is applied to obtain the updated coefficient estimate. These updated values are used as the new initial 'guess'. The procedure is repeated until convergence of the estimates is observed. The detailed computational procedures can be found in part V of Ref. 1. The quality of the model, measure of fit, used in Ref. 21 is the 'Q' statistic

Q=N

~ 7~

(12)

k=l

where ))k is the kth lag autocorrelation of the residuals, N is the number of data points and n ~ N/5. Closeness of fit of the ARMA model can be measured by how close the statistics of the random residuals are to the discrete white noise. Under the hypothesis that the residuals are completely white (uncorrelated), Q is distributed as a chisquare statistic with n minus the number of parameters estimated as the number of degrees of freedom. In general, Q should be close to n minus the number of parameters estimated, for a good fit. In the model fitting process to the earthquakes considered in Ref. 21, it was found that the San Fernando earthquakes were best fit by ARMA (4, l) models, and the E1-Centro (1940) was best fit by an ARMA (2, 1). This was true for each of the eight sections of five second duration for each individual record. It was also stated 2~ that the nonstationarity in amplitude was much more significant than the nonstationarity in frequency. A study of AR as well as MA models for the nonstationary character of earthquakes was presented in Ref. 22. They treated the general multidimensional case, and formulated the AR model as i

xi(j) = Z p=l

~ bip(k,j)xp(j-k)+ei(j)

(13)

k=l

where i = 1. . . . . L, {el(j)} are the random residuals and {bip(k,j) } are unknown coefficients that are time dependent. Again, the problem is to estimate the unknown coefficients from the data. A least squares approach is applied for each time j. The estimates of bip(k,j) at various values o f j are obtained by looking at the data over the range ( j - N ' , j+N'), where N' is an integer to be chosen. However, the choice of N' is not

clearly discussed in Ref. 22. The least squares procedure yields the typical covariance type equation, j+N'

x.(s)xq(s-l) s=j--N' j+N'

= ~ p=l

Z b.p(k,j) k=l

~

xp(s-k)xq(s-l)

(14)

s=j-N'

for each j, and for q = 1,2 . . . . . L. The coefficients {b,p(k,j)} for each j are obtained from the linear equation (14)by the usual matrix methods. The MA model is approached in exactly the same fashion. The results of modelling the E-W component of the Millikan library are presented in the paper. The simulated records, obtained from the estimated AR model do not appear to be good representations of the original Millikan library record. The difficulty with the approach taken by the authors of Ref. 22 is that b,p(k,j) must be given for eachj. Thus, if there are 1000 data points, then there would be some multiple of 1000 values of the coefficients required for storage in the computer memory. This does not appear to be an efficient characterization of the earthquake record. In a somewhat more recent study 2s, ARMA models of the form (1) were modified to include a time varying variance term for the random sequence {u,}, assuming all other coefficients are constants. Upon defining the usual delay operator D; Dy;-k= y~_t, then we can write equation (1) as

P(D)

Yt=Q~) ut with P(D)= l +b~D -1 +...+braD -m Q(D) l+alD -1+ .+a,D-"

(15)

In Ref. 28, the authors assume Ut_k=gtat_k, where gt plays the role of a time varying envelop function. All variability in time is included in g,. This leads to the variance of y, to be written as

ay2 (t) = Ka a2 gt2

(16)

where a ff is the constant variance of the random sequence {a,}, and K is a constant determined by the constant coefficients of the polynomials P(D), Q(D). The first step in obtaining the ARMA fit to real earthquake data is to estimate azy(t ). The authors accomplish this task by first squaring the observed earthquake data to yield {y2}, and then apply a transformation due to Box and Cox 29, which is often used for the nonstationary variance case. They then fit the variance by assuming that the expected values of the transformed variables {y2} follow a polynomial form

h(t) = flo + fll t -I-.,.

-1- flk tk

(17)

They found in their investigations that k = 4 was a suitable order for the polynomial h(t). Then defining y* = y~~St(t), where ar ~2(t) is the estimated nonstationary variance term, the resulting sequence {y*} is, presumably, a stationary sequence whose assumed ARMA form can be fit by

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 2 61

Autoregressive moving average models: F. Kozin standard parameter estimation procedures. Model order fitting procedures were not applied in Ref. 28, instead ARMA (2, 1) and (4, 1) models were studied for records obtained from the San Fernando recordings. They found that ARMA (4, 1) provided better fits in general. In connection with the approach taken 2s, in a very recent paper 3°, Pradlwater has presented a rather interesting method for estimating the time varying modulation functions for earthquake records. In another recent paper 3l, the rather sophisticated method of smoothness priors is applied to ARMA modelling of nonstationary time series data. In particular the model to be studied is essentially equation (9), where instead of writing the coefficients [a~(n)} in the form given by (10), the coefficients themselves are determined through AR models, defined by the kth order differences, k

Dkai(n)= ~. (~)(--1)iai(n--j)--6(i,n) j

(18)

0

Thus, if one has the evolutionary spectrum of the form

f(t,o2) =

then the auto-correlation function Cyy(s,t) obtained through the integral relation

(19)

i-O

where {x,} is a standard Gaussian white noise sequence. If the nonstationary correlation function Cyy(t,k) was known, then by the usual least squares techniques, for the white sequence {x,}, one obtains for each time t* (20)

which can be solved easily to yield the values of the coefficients Ai(t). As one knows, however, the autocorrelation function Cyy(t,k) cannot be estimated from one record for a nonstationary process. Thus, the procedure in Ref. 23 is to apply techniques of evolutionary spectra leading to functional forms of Cyy(t,k) that can then be applied to estimate the coefficients in their nonstationary AR models.

62

can be

(22)

A number of examples are contained in Ref. 23, in which f ( o ) in (21) is the Kanai-Tajimi spectrum, and ]A(t, co)]2 is a suitable envelope function with exponential decay. Such spectra has been fit to various earthquakes. From the estimated constants in (21), Cyy(s,t) can be determined b~( quadrature from (22) thus allowing the coefficients {Ai(t)} to be determined from (20). This approach yields efficient computational procedures in contrast to the approach in Ref. 22.

P

P -- Cyy(t*, - j ) = ~ A,(t*)Cyy(t* - i, i - j ) i=1

(21)

Cy,(s,t)= f _o(~~ A(s,~o)A(t,o)ei~'l'-~l f(o))dco

where {6(i,n)} is an independent stationary gaussian sequence. The nonstationary record is divided into segments of data as was done in Ref. 21, and the ARMA model with the constraints (18) placed on the coefficients was imposed. AIC criteria was applied to obtain the best order fit on each segment of the data. The instantaneous power spectral density, introduced by Page 32 and studied more recently by Kitagawa 33, is applied to obtain the changing frequency characteristics of real earthquake data. Although breaking up the earthquake record into a number of time epochs of equal duration is not currently favoured by those concerned with modelling earthquake records for study of structural response studies, there is no question that there are many potentially significant ideas contained within Ref. 31, that merit investigation. Finally, we wish to note in a recent study 24, an approach exactly as taken in Refs 17, 19 and 20 was reported. The simulated records contained 24 are excellent replicas of the original data, which further attests to the quality of this nonstationary AR model for earthquake accelerations. In the study 23, an AR model with evolutionary power spectra is developed for modelling nonstationary processes. The AR model is written in the form

A,(t)y,_~= Bo(t)x,

IA(t, co)]V(~o)

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 2

III. A SUMMARY OF RESEARCH NEEDS There are a few problem areas that we believe to be significant in the further development of ARMA model procedures for earthquake engineering applications. Can ARMA models help in determining microzonation characteristics? Studies which would treat the question of prediction of amplitude growth, decay and frequency characteristics of earthquake records in a local region would be extremely important. A study of the consistencies in the ARMA models for a given locale, i.e., the model order, properties of the time varying frequency spectrum and amplitude decay rates would be required. The possibility of correlating down hole records with surface acceleration records via estimated ARMA models, using deconvolution techniques would also be important in such microzonation studies. Sensitivity studies to determine whether amplitude variation or frequency variation is the most significant to incorporate in the model, or whether they are equally important is of practical importance. For an array of seismographs in a given locale, we can estimate a two-dimensional characterization of the ground acceleration as a random field; cross-correlation and coherency studies would be useful in this context. For a given response spectrum, how do we obtain the family of ARMA models that will generate the spectrum'? How sensitive is the response spectrum to variations in the order, the amplitude and frequency characteristics of the ARMA model'? From the point of view of structural response, what is the sensitivity of salient response characteristics to the parameters (again order, coefficient variations, etc.) of the ARMA models? What is the sensitivity effect on important statistical quantities related to maximum excursions, first passage times, hysteretic evolution, and damage accumulation? These are but a few of the problem areas that we can perceive to be of importance relative to nonstationary ARMA modelling for earthquake engineering purposes. There is no doubt that other problem areas will occur to the interested reader. Indeed, there are many theoretical questions pertaining to the statistical quality and reliability of the given nonstationary ARMA models,

Autoregressive moving average models: F. K o z i n

but these topics are not within the direct context of earthquake engineering.

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16

17

Box, G. E. P. and Jenkins, G. M. Time Series Analysis Forecasting and Control, Holden-Day, San Francisco, 1970 Mehra, R. K. and Lainiotis, D. G. System Identification Advances and Case Studies, Math. in Sci. and Eng., Academic Press, New York, 1976, Vol. 126 Goodwin, G. C. and Sin, K. S. Adaptive Filtering Prediction and Control, Prentice-Hall, New Jersey, 1984 Nakajima, F. and Kozin, F. A Characterization of Consistent Estimators, IEEE Trans. Auto. Control., 1979, 24,755-765 Ljung, L. and Soderstrom, T. Theory and Practice of Recursive Identification, MIT Press, Cambridge, Mass., 1983 Lefkowitz, R. Evaluation of Various Methods of Parameter Eestimation for ARMA Processes, Eng. Degree Thesis, Systems Eng. Polytechnic Inst. of New York, January 1986 Akaike, H. A New Look at Statistical Model Identification, IEEE Trans. Auto. Contr., 1979, 19, 716-723 Robinson, E. A. Predictive Decomposition of Seismic Traces, Geophysics, 1957, 22, 767-778 , Multichannel Time Series Analysis with Digitial Computer Programs, Holden-Day, San Francisco, 1967 Robinson, E. A. et al. The Robinson-Treitel Reader, Seismograph Service Corp., Tulsa, Okla., 1964 Robinson, E. A. Mathematical Development of Discrete Filters for the Detection of Nuclear Explosions, J. of Geophys. Res., 1963, 68, 5559-5567 Liu, S. C. Synthesis of Stochastic Representations of Ground Motions, Bell System Tech. J., 1970, 49, 521-541 Gran, R. and Kozin, F. Nonlinear Filtering Applied to the Modeling of Earthquake Data., Proc. Symp. Nonlinear Estimation and Applications, Univ. of Cal., San Diego, 1973 Kozin, F. and Gran, R. Analysis and Modeling of Earthquake Data, Paper No. 364, Proc. 5th World Congress, Earthquake Eng., Rome, 1973 Kozin, F. An approach to Characterizing, Modeling and Analyzing Earthquake Excitation Records in Random Excitation of Structures, CISM Lectures 225, (Ed. H. Parkus), Springer-Verlag, Vienna, 1977 Kozin, F. and Lee, T. S. Consistency of Maximum Likelihood Estimators for a Class of Non-Stationary Models, Proc. 9th Hawaii Conf. on Systems Science, Univ. of Hawaii, Honolulu, 1976, pp. 187-189 Kozin, F. Estimation and Modeling of Non-Stationary Time Series, Proc. of Conf. on Appls. of Computer Methods in Eng.,

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31 32 33

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