Continuous Stochastic Processes

Transcription

1 Continuous Stocastic Processes Te term stocastic is often applied to penomena tat vary in time, wile te word random is reserved for penomena tat vary in space. Apart from tis distinction, te modelling of uncertain temporal and spatial variation is quite similar. In fact, te concepts described in te following for stocastic processes remain valid also for uniaxial random fields. Te differences tat exist relate to terminology. For example, a stationary stocastic process is referred to as omogeneous in te context of random fields. A stocastic process is essentially a collection of random variables along te time axis. As an illustration, consider te temporal variation of, say, te wind pressure intensity at a particular location of a building. Te intensity at a specific time instant is an individual random variable. Te intensity at a later time instant is anoter random variable, often correlated wit te first if tey are close in time. It is understood tat because tere are infinitely many time-instances witin even a sort interval tere is also an infinite number of random variables in a process. Tat does not cause any conceptual problems; tere is no need to enumerate te random variables until realizations are generated, and ten te problem is a practical matter of granularity rater tan a conceptual problem. If te random variables of a process eac ave a continuous probability distribution, ten te process is said to be continuous, oterwise it is discrete and outside te scope of tis document. Te present document explains continuous processes, wic are employed to model continuously varying loads, suc as ocean waves and wind, and even te ensuing response of te structure. Time Domain Model Description Consider a continuous stocastic process, X(t). Similar to te notation for random variables, te realizations are denoted by te corresponding lowercase letter, i.e., x(t). A group of realizations if often referred to as an ensemble. It is understood from above tat X(t) is a collection of random variables X(t i ), were t i are infinitely many time instants. Tis family of random variables is sometimes written {X(t)} (Lutes and Sarkani 997). On tat basis it is clear tat te model for a stocastic process is essentially a joint probability distribution for tose random variables. Tis joint distribution can be written as a CDF, CCDF, or as te joint PDF f (x,t) = f X (t ),X (t ),!,X (t n ) ( x ), x(t ),!, x(t n )) () An abbreviated version of Eq. () is obtained by replacing X(t i ) wit X i, wic yields f (x,t) = f X,X,!,X n ( x, x,!, x n ) () Mean Function and Second-Moment Functions Te joint distribution tells te full statistical story of a stocastic process. However, in te same way as statistical moments are elpful to describe random variables, partial descriptors are employed also to describe a stocastic process. In tis context, te first statistical moment corresponds to te mean function: Continuous Stocastic Processes Updated January 8, 6 Page

2 µ X (t) = E X(t) [ ] = x(t) f (x,t)dx (3) ow moving to second moments, it is te dependence between te process at two different time-instants tat is of interest. To prepare for tese considerations, it is first noted tat tere are several ways to express te covariance between two random variables X and X : covariance, Cov[X,X ], correlation coefficient, ρ, and mean product, E[X X ]. From te document on multivariate distributions, te relationsip between tese quantities is Cov[ X, X ] = E[ X X ] µ µ = ρ σ σ (4) were µ i are te means and σ i are te standard deviations of X and X. Turning to stocastic process, an important second-moment descriptor is te autocorrelation function φ XX ) = E X ) X(t ) [ ] = x x f (x,t)dx dx (5) For second-moment stationary processes, described sortly, it is possible to reduce te two time-instants t and t in te argument to one parameter τ = t t, namely te temporal distance between two points anywere along te time axis. In tat case te notation is canged from φ to R and te autocorrelation function reads [ ] (6) R XX (τ ) = E X(t) X(t + τ ) were τ= gives te mean square function E[X(t) ]. Wile te interpretation of te mean function in Eq. (3) is conceptually straigtforward, te autocorrelation function warrants comments about its name, as well as its meaning. Te name auto implies correlation witin one process, to distinguis it from te cross-correlation, R XY, between two different processes X(t) and Y(t). Te meaning of autocorrelation is approaced in several ways. First, it is understood from Eqs. (5) and (6) tat autocorrelation corresponds to te concept of mean product for random variables. In oter words, te autocorrelation function does not display correlation, but Eq. (4) does reveal tat it is related to te concept of correlation. An alternative second-moment descriptor of a stocastic process is te autocovariance function, wic corresponds directly to te covariance concept for random variables: κ XX ) = E ( X ) µ x )) ( X(t ) µ x (t )) (7) Again second-moment stationarity allows an alternative notation in terms of one parameter: were τ= gives te variance function: (τ ) = E ( X(t) µ x (t)) ( X(t + τ ) µ x (t + τ )) (8) ( ) () = E X(t) µ x (t) = σ X(t) (9) Continuous Stocastic Processes Updated January 8, 6 Page

3 It is common to work wit zero-mean processes. In fact, if a process X(t) as some constant mean µ X ten it is easily transformed into te zero-mean process X(t) µ X. A glance at Eq. (4) reveals tat (τ)=r XX (τ) for zero-mean processes, tus te autocorrelation function and autocovariance function can be used intercangeably for suc processes. As a matter of completeness, it is a small step to employ Eq. (4) to express te autocorrelation coefficient function of a stocastic process: ρ XX = (τ ) () () = (τ ) () Above, te tree second-moment descriptor functions were introduced for dependence witin a stocastic process. To obtain te corresponding expressions for crosscorrelation, cross-covariance, and cross-correlation coefficient between two processes X(t) and Y(t) it is sufficient to replace te indices XX in all expressions wit XY. Stationarity Rougly speaking, a process tat is stationary as properties tat do not cange along te time axis. Several types of stationarity are defined: Mean-value stationarity... te mean is constant Second-moment stationarity... te autocorrelation function is constant Covariant stationarity... te autocovariance function is constant n t -moment stationarity... te n t moment is constant n t -order stationarity... te joint probability distribution for te process evaluated at n points is invariant to a time-sift Strict stationarity... all properties of te process are constant Weak stationarity... is not uniquely defined, but often implies meanvalue and second-moment stationarity Continuity Te concept of continuity appears in several contexts for stocastic processes, and some ave been alluded to already. It is already understood tat te processes considered in tis document are continuous along te time axis, and tat te probability distribution for te random variable X(t) at any time instant is continuous. Anoter consideration is te continuity of te autocorrelation and autocovariance functions. It can be sown tat second-moment stationary processes wit continuous autocorrelation and autocovariance functions at τ= also must be continuous for all oter values of τ. Additional considerations of continuity for autocorrelation and autocovariance functions for nonstationary processes are made in te plane stretced by t and t, yet to be discussed in tis document. Several oter continuity requirements can be also formulated, but it is for now assumed in tis document tat all te realizations and associated functions are continuous. Ergodicity An ergodic process as te advantageous property tat one can average over time to infer te mean, autocorrelation function, and oter quantities, instead of averaging over an ensemble of realizations. Underneat te concept ergodicity is a form of statistical () Continuous Stocastic Processes Updated January 8, 6 Page 3

4 independence in time. aturally, ergodicity implies stationarity, and tere are as may types of ergodicity as tere is stationarity. However, stationarity does OT imply ergodicity. Frequency Domain Model Description Te autocorrelation function, described above, says someting about ow rapidly te amplitude of a process varies in time. Specifically, if R XX (τ) diminises rapidly witτ ten tere is little correlation between te amplitudes even sort times apart, and realizations will look disorderly wit rapid variations in amplitude. In contrast, if R XX (τ) diminises slowly witτ ten correlation is ig between te amplitude at two time-instant even wen te times are far apart, wic means tat te amplitude is slowly varying in time. Anoter way to look at tis is to consider frequencies, wic is te topic in tis section. It is intuitive tat a disorderly realization as contributions from many frequencies, wile a slowly varying realization as contributions from fewer frequencies. Te quantity tat captures te frequency content of a process is te power spectral density (PSD), S X (ω), wic is te Fourier transform of te autocovariance function: S X (ω ) = π (τ ) e i ω τ dτ = π C (τ XX ) e i ω τ dτ () were te last equality is possible because of te symmetry of (τ). Te autocovariance function and te PSD form a Fourier transform pair; ence, te autocovariance is obtained from te spectral density as (τ ) = S X (ω ) e i ω τ dω = S X (ω ) e i ω τ dω () For completeness it is noted tat te non-imaginary form of te PSD in terms of te autocovariance function is S X (ω ) = π (τ ) e i ω τ dτ + π = π (τ ) ( cos(ωτ ) )dτ ( τ ) e i ω τ dτ wic is obtained by first splitting te integral into two symmetric parts, one from to and te oter from to by te variable cange τ ( τ), followed by substitution of Euler s formula e iωτ = cos(ωτ ) + i sin(ωτ ), ten using (τ)= ( τ) and te facts tat cos(ωτ)=cos( ωτ) and sin(ωτ)= sin( ωτ). Similarly, te expression for te autocovariance function in terms of te PSD becomes: (3) (τ ) = S X (ω ) cos(ωτ )dω (4) Te PSD is te key model descriptor in te frequency domain. An interpretation of te PSD is obtained from Eq. (9), wic sows tat te variance of a stationary process is: Continuous Stocastic Processes Updated January 8, 6 Page 4

5 σ X = () = S X (ω )dω = S X (ω )dω (5) In words, te area underneat te PSD is te variance of te process. In fact, te PSD essentially displays te amount of variance as function of frequency. Tat is, te value of te PSD at a particular frequency indicates te relative amplitude of te process at tat frequency. Teoretically, te PSD is defined for bot positive and negative frequencies, in fact, it is symmetric about ω=, but negative frequencies is an artificial construct. For tat reason it is common to use te one-sided PSD, wic is defined for positive ω only: S X + (ω ) = S X (ω ) for ω (6) wic gives te same area underneat te PSD for bot S + (ω ) and S X X(ω). Te PSD can be formulated in term of te frequency, f, measured in Hertz, by te transformation ω=πf. Tis version of te PSD is denoted G + ( f ) and is obtained from X S+ X (ω ) by substituting πf for ω, multiplying all ordinate values by π, and dividing all abscissa values by π. Again, tis transformation maintains te area underneat te PSD, i.e., it maintains te variance of te process, wic is wat te PSD describes. Measures of Bandwidt Te bandwidt of a process is seen from te widt of te PSD. Te broader PSD, te more frequencies contribute. A narrowband process as a narrow PSD and tus exibits smootly varying realizations wit nearly just one dominant frequency. Conversely, a broadband process as a broad spectrum and realizations tat are more caotic wit many contributing frequencies. Te extreme-case of broadband processes is te artificial wite noise process tat as a constant spectrum over all frequencies and no autocorrelation. Several measures exist to caracterize te bandwidt of a process. To understand tese, it is elpful to compare tem wit te statistical moments of probability distributions. Te moments of te PSD are: λ m = ω m S + X (ω )dω (7) Te PSD is not directly comparable to a probability distribution because it does not integrate to unity. Tis property is acieved by rater using te normalized PSD S X + (ω ) λ, were λ is te area underneat te PSD, and te corresponding moments λ m /λ. However, just like te moments of probability distributions, tese moments ave units and are terefore furter normalized to become useful as dimensionless bandwidt measures. Directly analogous to te coefficient of variation of a random variable, a dimensionless bandwidt parameter is Continuous Stocastic Processes Updated January 8, 6 Page 5

6 δ = "stdv" "mean" = "mean square" "squared mean" "mean" ( ) = λ λ λ λ = λ λ λ λ λ (8) However, instead of te coefficient of variation it is common to use measures of bandwidt tat take values only in te range to. A definition tat accomplises tis is (Lutes and Sarkani 997) α m = λ m λ λ m (9) wic for m= corresponds to te above coefficient of variation because α = + δ () Te bandwidt measures α m approac zero for very broadband process, and approac unity for very narrowband processes. Te most popular version of Eq. (9) is α = λ λ λ 4 () Tis measure, and oter ones, can be related to te variance of te process and its derivative, as described under derivative processes. Design Spectra In sip and offsore design, design spectra are employed to model te sea-surface elevation, η. An important spectrum, istorically, is te Pierson-Moskowitz spectrum (P- M), wic is a one-sided spectrum, i.e., S + (ω), wit te form S η (ω ) = α g ω 5 ω exp β ω 4 () Tat spectrum gives rise to a family of design spectra, including P-M, ISSC, B-M, and ITTC. Tey sare te spectrum sape B S η (ω ) = A ω 5 e ω 4 (3) were te parameters A and B are given in Table. Anoter design spectrum is Darbysire-Scott, not yet written out ere. Yet anoter option is te JOSWAP spectrum (from te Joint ort Sea Wave Observation Project), wic is essentially te Pierson- Moskowitz spectrum multiplied by a factor tat accentuates te peak of te spectrum: S η (ω ) = α g ω 5 exp 5 4 ω p ω γ r (4) 4 Continuous Stocastic Processes Updated January 8, 6 Page 6

7 were r = exp (ω ω p) σ ω p (5) Table : Parameters in te P-M, ISSC, B-M, and ITTC spectra. A P-M ω z 4 H s 4 π ISSC. ω 4 H s B ω z 4 π 4.44 ω B-M 5 π.57 H π /3 T /3 T /3 π.3 ITTC 8. 3 g 3. H s T /3 4 Amplitude and Pase Spectra To be written. Derivative Processes Time-derivatives of a stocastic process appear as pysical quantities, suc as velocity and acceleration, in differential equations, and also as auxiliary quantities in te study of response statistics. Consider te first-order derivative of te process X(t):!X(t) = dx(t) dt In a finite difference or Riemann sense, tis derivative is equal to te limit!x(t) = lim X(t + ) X(t) For convenience, te argument in Eq. (7) is defined as Y (t,) As a result, te mean of te derivative process is X(t + ) X(t) µ X! (t) = lim E Y (t,) X(t + ) X(t) [ ] = lim E = lim E µ X (t + ) µ X (t) = dµ X (t) dt (6) (7) (8) (9) Continuous Stocastic Processes Updated January 8, 6 Page 7

8 wic says tat te mean of te derivative is te derivative of te mean. Similarly, te cross-correlation function between X(t) and!x(t) is: E X ) X(t + ) X(t ) = lim E φ XX + ) φ XX (t ) = dφ XX ) dt φ (t X!X ) = lim E X ) Y (t,) [ ] = lim (3) and te autocorrelation function for!x(t) is: φ! X!X (t) = lim E [ Y, ) Y (t, )] = lim E X(t + ) X(t ) X(t + ) X(t ) = lim E X(t + )X(t + ) X(t + )X(t ) X(t + )X(t ) + X(t )X(t ) (3) φ = lim XX + + ) φ XX (t +,t ) φ XX + ) + φ XX ) = φ XX ) t t were te last equality is te finite difference definition of a two-variable derivative. For second-moment stationary processes only one argument is necessary, denoted τ = t t. Adopting te earlier notation, tis gives R (τ ) = dr (t t ) XX X = dr (τ ) XX!X dt dτ (3) R! X!X (τ ) = R XX t ) t t = R XX (τ ) τ (33) For second-moment stationary processes, R XX (τ) is symmetric, wic means its derivative at τ= vanises. According to Eq. (3), tis means tat R X () =, wic in turn means!x tat X(t) and!x(t) are uncorrelated at any given time instant. Derivative processes can also be analyzed in te frequency domain. Assuming a solution on te form X(t) = e iωt (34) te derivative is!x(t) = iω e iωt (35) In oter words, iω is te transfer function between te input process X(t) and te output process!x(t). As sown in te document on stocastic dynamics, were anoter Continuous Stocastic Processes Updated January 8, 6 Page 8

9 system tan output is time-derivative of input is considered, te output spectrum is te modulus of te transfer function, squared, times te input spectrum, wic ere means tat: S! X (ω ) = ω S X (ω ) (36) Tis is te reason wy and in addition to te fact tat λ = σ X! λ 4 = σ X!! λ = σ X (37) (38) (39) Gaussian Processes A process is Gaussian wen te random variable X(t) is normally distributed for all t. In te same way as te probability distribution of a Gaussian, i.e., normal random variable is fully described by te mean and standard deviation, a Gaussian stocastic process is fully described by te mean function and autocorrelation/autocovariance function. In addition to tis major convenience, Gaussian stocastic processes are also popular because tere is often insufficient data to justify anoter distribution type. Generation of Realizations One simple tecnique for creating realizations of a continuous stocastic process is to create it as a sum of trigonometric functions: ( ) x(t) = A i cos(ω i t) + B i sin(ω i t) (4) were A i and B i are random variables wit properties to be determined sortly In preparation for tis, te frequency axis of te PSD is discretized into intervals of lengt Δω. Te centre frequency in eac interval is denoted ω i. To determine te value of te coefficients A i and B i corresponding to tat frequency, first consider te autocovariance function from Eq. (4) expressed in terms of te discretized one-sided PSD: (τ ) = S + X (ω ) cos(ωτ )dω = S + X (ω i ) Δω cos(ω i τ ) (4) ( ) As sown in Eq. (5), te variance of te process equals te autocovariance function evaluated at τ=, ence Eq. (4) reveals te variance at eac frequency: ( ) () = S + X (ω i ) Δω (4) Continuous Stocastic Processes Updated January 8, 6 Page 9

10 In sort, S X + (ω i ) Δω is te variance at te frequency ω i. Te objective now is to ensure tat te generated process in Eq. (4) as tat same variance at eac frequency. Because Eq. (4) is a linear function of random variables its variance is [ ] = Var A i Var x(t) ( [ ] cos (ω i t) + Var[ B i ] sin (ω i t) ) (43) By selecting Var[A i ]=Var[B i ]=σ i, were σ i is te variance at frequency number i, Eq. (43) yields [ ] = σ i cos (ω i t) + sin (ω i t) Var x(t) = σ i ( ) Comparing Eq. (44) and Eq. (4) it is clear tat te random variables A i and B i sould be generated to ave equal variances equal to S X + (ω i ) Δω. Often teir means are selected to be zero, to generate a zero-mean process, but a non-zero constant mean function equal to µ is obtained by generating random variables A i and B i wit te same mean µ. Te distribution type of te random variables is usually selected to be normal because given te linear form of Eq. (4) tis implies tat te amplitude of te process at any time instant is also normal, i.e., it is a Gaussian process. Anoter approac for generating realizations of a continuous stocastic process is ( ) (44) x(t) = c + c i sin(ω i t + φ i (45) were c i is te amplitude and φ i is te pase angle associated wit eac frequency ω i. Tis formulation requires te amplitude spectrum and te pase spectrum, wic will be described in an upcoming version of tis document. Statistical Inference Given an observed realization of an ergodic stocastic process, an estimate of its mean is: µ X = lim T T T x(t)dt Te estimate of te autocorrelation function is (τ ) = lim T T T T x(t) x(t + τ )dt = lim x(t ) i (46) = lim ( x(t ) x(t + τ )) i i (47) References Continuous Stocastic Processes Updated January 8, 6 Page

11 Lutes, L. D., and Sarkani, S. (997). Stocastic analysis of structural and mecanical vibrations. Prentice Hall. Continuous Stocastic Processes Updated January 8, 6 Page