Lecture 6: Spectral decompositions of stochastic processes

Transcription

1 Lecture 6: Spectral decompositios of stochastic processes Readigs Recommeded: Grimmett ad Stirzaker (2001) Chapter 9. Chori ad Hald (2009) Chapter 6 Pavliotis (2014), 1.2 Optioal: Yaglom (1962), Ch. 1, 2; a ice short book with may details about statioary radom fuctios; oe of the origial mauscripts o the topic. Lidgre (2013) is a i-depth but accessible book; with more details ad a more moder style tha Yaglom (1962). 6.1 Spectral theory for weakly statioary processes Setup ad examples The goal today is to uderstad the spectral decompositio for statioary processes, where i these otes statioary always meas weakly statioary. It will be helpful to work i complex space, so let s first redefie some of the cocepts we leared last lecture. Defiitio. A complex-valued stochastic process is oe whose real ad imagiary parts are real-valued stochastic processes, i.e. it has the form X t = A t + ib t where (A t ) t T, (B t ) t T are real-valued stochastic processes. Defiitio. The covariace fuctio of a complex-valued stochastic process is B(s,t) = EX s X t (EX s )(EX t ). You ca check that the covariace fuctio is Hermitia (B(s, t) = B(t, s).) It is also positive semidefiite, where ow we use the complex defiitio: i, j B(t i,t j )z i z j 0. As i the real case, a complex-valued process is statioary if its mea is costat ad its covariace fuctio ca be writte as B(s,t) = C(s t). We d like some geeral characterizatio of statioary processes. Recall from last lecture we showed the process defied by X t = Acos(λt)+Bsi(λt) where A,B were ucorrelated, was statioary. This looks a lot like the real part of a complex expoetial, so let s try to geeralize this example. Example. Cosider a process of the form X t = ξ h(t), where h(t) is a determiistic, complex-valued fuctio of time, ad ξ is a complex-valued radom variable. What coditios o ξ,h(t) make X t statioary? The mea is m(t) = (Eξ )h(t). This is oly costat if h(t) is costat or Eξ = 0. Let s suppose h(t) is ot costat, so we must have Eξ = 0.

2 The covariace fuctio is B(s,t) = (Eξ ξ )h(s)h(t). We eed h(s)h(t) to deped oly o s t. Settig s = t shows we eed h(t) 2 = cost. Therefore h(t) has the form h(t) = Ae iφ(t) for some real umber A ad some real-valued fuctio φ(t). Suppose A 0. The covariace fuctio is ow B(s + t,s) = EX s+t X s = A 2 E ξ 2 e i(φ(s+t) φ(s)). Take the log the d/ds of this equatio to fid φ (s) = φ (s + t). Sice this hold for all t we must have φ (t) = cost. Therefore φ(t) = αt + β for some real-valued umbers α,β. Re-orgaizig costats ad absorbig some ito the defiitio of ξ shows that X t = ξ e iλt, where λ R ad ξ is a complex-valued radom variable with Eξ = 0 (if λ 0). The example above ca be see as a form of separatio of variables. A stochastic process depeds o two variables, the time parameter t ad the elemet ω i a probability space Ω. The represetatio above as Z(t,ω) = ξ (ω)h(t) is like a particular solutio that oe would look for usig separatio of variables. Therefore, if we wat to geeralize the example, we could cosider a sum of fuctios that are separated i this way. Example. Now cosider a sum of two expoetials, as X t = ξ 1 e iλ 1t + ξ 2 e iλ 2t, where ξ 1,ξ 2 are complex radom variables, λ 1 λ 2, ad both frequecies are o-zero. Whe is X t statioary? The mea is EX t = Eξ 1 e iλ 1t +Eξ 2 e iλ 2t, which is idepedet of t oly if Eξ 1 = Eξ 2 = 0, sice the fuctios e iλ 1t, e iλ 2t are liearly idepedet. The covariace is B(s,t) = (E ξ 1 2 )e iλ 1(s t) + (E ξ 2 2 )e iλ 2(s t) + (Eξ 1 ξ 2 )e i(λ 1s λ 2 t) + (Eξ 1 ξ 2 )e i(λ 2s λ 1 t). This depeds is a fuctio of s t oly if Eξ 1 ξ 2 = 0, sice the fuctios e i(λ 1s λ 2 t), e i(λ 2s λ 1 t) are liearly idepedet. Therefore ξ 1,ξ 2 must be mea-zero, ucorrelated radom variables. The covariace fuctio is C(t) = b 1 e iλ 1t + b 2 e iλ 2t, b j = E ξ j 2. Example. Now cosider a more geeral superpositio of frequecies, as Similar calculatios show this is statioary iff X t = j=1 ξ j e iλ jt. (1) Eξ j = 0 j, Eξ j ξ k = 0 j k. (2) The covariace fuctio is C(t) = j=1 b j e iλ jt, b j = E ξ j 2. (3) 2

3 We could also cosider = i (1), i which case we require for the covariace fuctio to exist. j=1 E ξ j 2 = j=1 b j < The last example looked a lot like a Fourier trasform. The covariace fuctio was a sum of harmoic fuctios with differet frequecies λ j ad amplitudes b j, ad the process itself i was also a sum of harmoic fuctios but the amplitudes were radom ad ucorrelated. We could imagie lettig the frequecies become closer ad closer together ad the powers of each frequecy shrik to zero, i such a way that the sums i (1), (3) approach itegrals. The, (1), (3) would look (formally, at least) like Fourier trasforms of the process ad covariace fuctio respectively. We will see shortly that a geeral statioary stochastic process ad its covariace fuctio ca be represeted usig a cotiuous versio of (1), (3). Before we show these geeral results, let s do a couple of calculatios to see what we should expect about the cotiuum limit. Suppose we have a statioary process with covariace fuctio C(t), ad suppose for a momet that it is itegrable, so it has a cotiuous Fourier trasform ad we ca write C(t) = Ĉ(λ)eiλt dλ for some (complex-valued) fuctio Ĉ(λ). The we could approximate this itegral by a Riema sum, as C(t) C (t) = Ĉ(λ j ) λ e iλ jt, j= where { λ,...,λ 1,λ } is a set of equally-spaced values o the real axis ad λ = λ j+1 λ j. The, usig our calculatios i the previous examples, we kow that a stochastic process with C (t) as its covariace fuctio could be costructed as X () t = ξ j e iλ jt, j= where ξ j are ucorrelated radom variables with mea zero ad variace Ĉ(λ j ) λ. Clearly we eed Ĉ(λ j ) 0 i order for this to make sese. Suppose we had X () t z(λ)e iλt dλ where z(λ) is some fiite stochastic process. The we eed the sum to be a approximatio for the itegral: This requires ξ j e iλ jt j= z(λ j ) ξ j λ z(λ)e iλt dλ Var(z(λ j )) Ĉ(λ j) λ ( λ) 2 ad therefore z(λ) ca t approach ay fiite value i the limit. z(λ j ) λe iλ jt. j 1 λ This argumet shows heuristically that we ca t expect to write X t as a classical Fourier trasform. Aother way to see this is that X t is ot itegrable, sice it is statioary so it ca ever decay. Nevertheless, we will still be able to write X t asa cotiuous superpositio of radom harmoic fuctios, by cosiderig a geeralized versio of the Fourier trasform. This is possible because sums of the form j ξ j e iλ jt may still exist, i a mea-square sese, eve whe the radom variables beig summed are O( λ); the process they coverge to however will ot be differetiable, so there ca be o classical desity for the limitig process. 3

4 6.1.2 Spectral represetatio, covariace fuctio We will start with the spectral represetatio of the covariace fuctio, preseted here for stochastic processes i both cotiuous time ad discrete time. Bocher s Theorem. A cotiuous fuctio C(t), t R, is positive semidefiite, ad hece a covariace fuctio, if ad oly if there exists a o-decreasig, right cotiuous, bouded real fuctio F(λ), such that C(t) = e iλt df(λ). (4) Remark. This theorem was discovered idepedetly by Khichi slightly after its publicatio by Bocher, so it is sometimes called the Bocher-Khichi theorem. It is usually stated for characteristic fuctios of a real-valued radom variable. Notes The itegral i (4) is a Riema-Stieltjes itegral. The Riema-Stieltjes itegral of a fuctio f (x) with respect to aother (real-valued) fuctio g(x) over a iterval [a,b] is defied by b a f (x)dg(x) = lim max i (x i+1 x i ) 0 i=0 f (x i )[g(x i+1 ) g(x i )], where a = x 0 < x 1 < < x = b is a partitio of [a,b], ad x i [x i,x i+1 ]. A sufficiet, though ot ecessary, coditio for this limit to exist is that f be cotiuous ad g be of bouded variatio. If g(x) is differetiable ad bouded, the the Riema-Stieltjes itegral equals the Riema itegral b a f (x)g (x)dx. Equatio (4) looks a lot like a Fourier trasform. Ideed, it is a particular istace of a Fourier- Stieltjes trasform. If F(λ) is absolutely cotiuous with respect to the Lebesgue measure (this is true if C(t) L 1 ; see e.g. Lidgre (2013), Theorem 3.4 p.80), with df(λ) = f (λ)dλ ( f (λ) = F (λ) whe F(λ) is differetiable), the C(t), f (λ) are Fourier trasforms of each other: C(t) = f (λ)e iλt dλ, f (λ) = 1 C(t)e iλt dt. (5) 2π Sice F(λ) is o-decreasig, we must have f (λ) 0. This is a importat result the Fourier trasform of a covariace fuctio (if such a trasform exists) is always o-egative. This will tur out to be a extremely useful way of makig up covariace fuctios, or of determiig if a particular fuctio is a covariace fuctio (i.e. is positive semidefiite.) F(λ) does t have to be absolutely cotiuous; for example it could cotai jumps. For example, cosider a covariace fuctio of the form (3), i.e. C(t) = j=1 b je iλ jt for some positive umbers b j. We ca costruct F(λ) as F(λ) = j:λ j λ b j. This fuctio is piecewise costat with jumps at λ = b j, ad it is cotiuous from the right. The itegral e iλt df(λ) is exactly (3). 4

5 The fuctio F(λ) is called the spectral distributio fuctio of the process X t, ad the spectrum is the set of real umbers λ for which F(λ + ε) F(λ ε) > 0 for all ε > 0. If F(λ) is absolutely cotiuous the it has a cotiuous spectrum equal to the support of the spectral desity fuctio f (λ) = F (λ). If F(λ) is piecewise costat the its spectrum is discrete the its discrete spectrum equals the set of jumps {λ j } j. A process s spectrum could have a mixture of discrete ad cotiuous compoets (or sigular compoets, which are of iterest theoretically but typically do t arise i applicatios.) The spectral distributio fuctio tells us the average eergy or power i each waveumber λ. For a determiistic periodic fuctio, the eergy per waveumber is the squared amplitude of the kth Fourier coefficiet. For a determiistic aperiodic itegrable fuctio, the eergy desity per waveumber is the squared amplitude of the Fourier trasform. For a radom fuctio, the eergy per waveumber is the value of df(λ). You ll otice that (4) makes C(t) look a lot like a characteristic fuctio for a probability distributio F. Ideed, the fuctio F(λ) ca be thought of as a distributio fuctio i frequecy space. It is odecreasig with F( ) F() = eiλ 0 df(λ) = C(0) <. We ca arbitrarily choose F() = 0, so the oly differece from a distributio fuctio for a regular radom variable is we have F( ) = cost, ot F( ) = 1. This gives aother iterpretatio of (4), due to Grimmett ad Stirzaker (2001) (p.382). If Λ is a radom variable with distributio fuctio F(λ), the g Λ (t) = e itλ is a pure oscillatio with a radom frequecy. Equatio (4) says that C(t) is the average value of this pure oscillatio (up to a ormalizig costat.) If we have a discrete-time process, say X =...,X 1,X 0,X 1,... with covariace fuctio C(), Z, the Bocher s theorem remais geerally true but we restrict the domai of itegratio: Proof. For the if part, we have j,k π C() = e iλ df(λ). (6) π z j z k C(t j t k ) = z j z k e iλt j e iλt k df(λ) = j,k j,k z j e iλt j z k e iλt k df(λ) = 2 z j e iλt j df(λ) 0. j The oly if part is more work. See Lidgre (2013), p See also Grimmett ad Stirzaker (2001) (p.381 ad p.182), for a proof based o similar results for characteristic fuctios. Example. Cosider a process with covariace fuctio C(t) = Ae αt, 5

6 where A,α > 0 are parameters. We ca calculate the Fourier trasform to be (see Pavliotis (2014), p.8) f (λ) = A π α λ 2 + α 2. If the process it comes from is Gaussia the the process is a statioary Orstei-Uhlebeck process. We will retur to this example throughout the course, because it is the oly example of a statioary, Gaussia, Markov process, ad it is used extremely frequetly i modellig. Example. We ca use Bocher s theorem to make up covariace fuctios, say for testig code or theory. It is ot easy to do this directly, because it is hard to quickly tell whether a fuctio is positive semidefiite. However, it is easy to make up fuctios that are o-egative ad itegrable, so we ca take the Fourier trasform to get a positive semidefiite covariace fuctio. For example, all of these are spectral desities of some covariace fuctio: f (λ) = 1 1 λ 2, f (λ) = (1 λ 2 )1 λ 1, f (λ) = Ae Bλ 2 /2, f (λ) = Ae Bλ 2 /2 1 λ λ 12. Of course, we could also icorporate jumps, or cosider more complicated kids of odecreasig fuctios F(λ). Example. Turbulet fluids are frequetly characterized by their spectrum, which says what are the eergycotaiig scales. Eve though the velocity is assumed to be a stochastic process, we do t eed to kow much about its statistical properties i order to fid the spectrum. The spectrum ca be calculated by lookig at oe compoet of the velocity field, computig its covariace fuctio, ad fidig the spectral desity of this. Here is a example that shows a typical velocity field, ad schematic of a typical spectrum, for turbulece i the atmosphere: 1 Here is aother example, that shows measuremets of the three compoets of magetic field ad correspodig velocity field i the su (left), ad a typical computed spectrum for magetic field turbulece (right): 2 1 from Barbara Ryde s otes o Basic Turbulece, see ryde/ast825/ch7.pdf 2 from cairs/teachig/lecture12/ode2.html 6

7 6.1.3 Spectral represetatio, stochastic process A statioary process X t ca also be represeted i spectral form. Theorem (Spectral Theorem). Give a mea-square cotiuous, statioary stochastic process (X t ) t R with mea zero ad spectral distributio fuctio F(λ), there exists a complex-valued stochastic process (Z(λ)) λ R such that X t = e iλt dz(λ). (7) Z(λ) has the followig properties: (i) Orthogoal icremets: if the itervals [λ 1,λ 2 ], [λ 3,λ 4 ] are disjoit, the E(Z(λ 2 ) Z(λ 1 ))(Z(λ 4 ) Z(λ 3 )) = 0. (8) (ii) Spectral weight: if λ 1 λ 2, the Notes E Z(λ 2 ) Z(λ 1 ) 2 = F(λ 2 ) F(λ 1 ). (9) For the discrete-time case we have π X = e iλ dz(λ). (10) π The itegral i (7) is agai a Riema-Stieltjes itegral. A efficiet way to summarize the properties of Z is (Lidgre (2013), p.87) EdZ(λ)dZ(µ) = { df(λ) if λ = µ 0 if λ µ. (11) Z(λ) ca be thought of as the process we get whe we take the cotiuum limit of a process such as (1), repeated here for coveiece: X t = e iλ jt ξ j. (12) j 7

8 That is, we ca thik of it as the limit as λ 0 of a sum of ucorrelated radom variables j ξ j, where there are O(1/ λ) terms i the sum ad each ξ j O( λ). As we discussed earlier the limitig process Z(λ) ca t be differetiable, but the limit ca still exist because the sum of a large umber of ucorrelated radom variables ca still lead to somethig fiite if they are scaled they right way (as they are.) Of course, if we have a process which is exactly of the form (12) the Z(λ) is a pure jump process with jumps of size ξ j at each λ j, where ξ j are ucorrelated radom variables with mea zero, variace b j. We ca recover Z(λ) from X t at each poit of cotiuity of F(λ) by usig a iverse Fourier-Stieltjes trasform, as 1 T e iλt 1 Z(λ) = lim X t dt. (13) T 2π T it At a poit of discotiuity, we ca set Z(λ) = 2 1 [Z(λ 0) + Z(λ + 0)]. If X t is a Gaussia process, the Z(λ) is too. This follows from calculatios i the proof of the spectral theorem (as i Grimmett ad Stirzaker (2001), e.g. questio 9.4.3) so we wo t show it here. However, it is a plausible result sice Z is a liear fuctioal of X, ad we have a similar result i radom vector spaces which says that liear fuctios of Gaussia vectors are Gaussia. If Z(λ) is Gaussia, the icremets Z(λ 2 ) Z(λ 1 ) are Gaussia. Sice icremets over disjoit itervals are ucorrelated, they are also idepedet. If (12) holds the each ξ j is Gaussia, so the collectio (ξ j ) j is idepedet. The fact that Z(λ) is Gaussia will tur out to be very helpful whe we wat to simulate a statioary Gaussia process see HW. Aother form of the Spectral Theorem is X t = e iλt Z(dλ), where Z( λ) is a radom iterval fuctio. This meas it associates a radom variable to each iterval λ. It has the followig properties: (i) E Z( λ) = 0 (ii) Z( ) = Z( 1 ) + Z( 2 ) if 1, 2 are disjoit. (iii) E Z( 1 ) Z( 2 ) = 0 if 1, 2 are disjoit. (iv) E Z([λ 1,λ 2 ]) 2 = F = F(λ 2 ) F(λ 1 ) if λ 1 λ 2. This is related to the previous formulatio by Z(λ) = Z([,λ]). Proof. There are two mai ways to prove the Spectral theorem for statioary processes. Oe works directly with the process Z(λ) defied by (13). For example, see Yaglom (1962), p The steps of the proof are the followig, which you ca do as a exercise (it is exercise 3.16, p.115 i Lidgre (2013)): 8

9 (a) Show that the itegral ad limit (13) exists i the mea-squared sese. Use the fact that 1 T siλt 1 λ > 0 lim dt = 0 λ = 0 T π T t 1 λ < 0. (b) Show the process Z(λ) defied this way satisfies (8). (c) Show the itegral eiλt dz(λ) exists, ad that E Xt eiλt dz(λ) 2 = 0. Aother approach uses a fuctioal-aalytic argumet to fid a relatioship betwee the Hilbert space of radom variables with fiite secod momets with ier product (u,v) = E(uv) ad the Hilbert space H(F) = L 2 (F) of complex fuctios with with ier product (g,h) = g(λ)h(λ)df(λ). See Lidgre (2013), p or Grimmett ad Stirzaker (2001), p for detailed rigorous argumets, or Chori ad Hald (2009) sectio 6.5 for a simpler example of the argumet. Exercise Work out expressios for the spectral represetatios of the covariace fuctio ad process, i the case whe X t is real-valued. Example (Covariace fuctio). Let s check, formally at least, that the covariace fuctio computed from the spectral represetatio is cosistet: EX s+t X s = = = e iλ(s+t) e i λs E(dZ(λ)dZ( λ)) e is(λ λ) e iλt δ(λ λ)df(λ)d λ e iλt df(λ) = C(t). We used the formal result that E(dZ(λ)dZ( λ)) = δ(λ λ)df(λ)d λ, which is a cosequece of the orthogoal icremets property. The spectral represetatio is a very useful way to determie relatioships betwee a process ad its derivatives, its itegrals, its covolutio with various fuctios, ad liear fuctioals of the process i geeral. For example, oce we kow the spectral represetatio of X t, we ca very easily calculate the spectral represetatio of X t ad hece its covariace fuctio. We will explore some of these relatioships o the homework. Refereces Chori, A. ad Hald, O. (2009). Stochastic Tools i Mathematics ad Sciece. Spriger, 2d editio. Grimmett, G. ad Stirzaker, D. (2001). Probability ad Radom Processes. Oxford Uiversity Press. Lidgre, G. (2013). Statioary Stochastic Processes: Theory ad Applicatios. CRC Press. Pavliotis, G. A. (2014). Stochastic Processes ad Applicatios. Spriger. Yaglom, A. M. (1962). A Itroductio to the Theory of Statioary Radom Fuctios. Dover. 9