primer on random processes

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess

Mean-squarestochasticprocess

Stationarity

Powerspectralmeasure

Mean-squarederivation

Filtering

Bibliography

Primer on random processesMG3409 - Lecture #1 part A

E. Savin1,2

1Computational Fluid Dynamics Dept.Onera, France

2Mechanical and Civil Engineering Dept.Ecole Centrale Paris, France

January 25, 2015

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Outline

1 Probability SpaceProbability tripletRandom variableStochastic process

2 Mean-square stochastic processStationarityPower spectral measureMean-square derivation

3 Filtering

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Probability spaceFormal definition

We model a real-world process, or ”experiment”, or”system”, consisting of states occurring randomly.

It is mathematically represented by a triplet pA ,T ,P q ofwhich elements are ultimately specified by the modeler:

A is the set of outcomes, or sample space, constitutedby the states of the causes which influence the states ofthe system;T is a σ-algebra, i.e. a set of all the events the modelerconsiders. An event is a set of zero or more outcomes,thus it is a subset of the sample space;P is a function returning an event’s probability, or aprobability measure.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Probability spaceσ-algebra

Definition

A collection T of subsets of A is a σ-algebra, or σ-field, if:

1 H P T and A P T ;

2 a P T ñ ac “ A za P T ;

3 If tanun“1,2,... is an arbitrary countable family of events

in T , thenď

nPNan P T .

Example #1: a fair coin, A “ t1, 2u. Then the σ-algebraT “ 2A

“ ttu, t1u, t2u, t1, 2uu contains 22“ 4 events.

Example #2: a fair die, A “ t1, 2, 3, 4, 5, 6u. Then the σ-algebraT “ 2A

“ ttu, t1u, t2u, ...t6u, t1, 2u, t3, 4u, t5, 6u, t1, 3u, ...t1, 3, 5u, t2, 4, 6u, t1, 3, 4, 5u, ..., t1, 2, 3, 4, 5, 6uu contains 26

“ 64events.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Probability spaceProbability measure

Definition

A probability measure is a positive, bounded measure thetotal mass of which is 1: P pA q “ 1.

1 A positive measure m is a function m : A Ñ r0,`8ss.t. for countably many disjoint events tanun“1,2,... P T ,č

n

an “ H,ď

n

an P T , then

m

˜

ď

n

an

¸

“ÿ

n

mpanq ;

2 It is bounded if mpA q ă `8;

3 It is a probability measure if mpA q “ 1.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Random variableDefinitions

Definition

A random variable is a measurable function X : A Ñ A 1

which maps a combination of outcomes from a sample spaceA to a combination of consequences in a state space A 1.

Definition

A function f : A Ñ A 1 is measurable if @a1 P A 1, f´1pa1q ismeasurable in A : @a1 P T 1, f´1pa1q P T , where T , T 1 areσ-algebras defined on A , A 1 respectively.

Example #1: let A be the sample space describing thereal-world system ”tossing a fair coin”: A “ t”temperature”,”pressure”, ”initial position”, ”initial speed”, etc.u. ThenA 1

“ t1, 2u.

Example #2: let A be the sample space describing a real-worldsystem ”roll of dice”: A “ t”temperature”, ”pressure”, ”initialposition”, ”initial speed”, etc.u. Then A 1

“ t1, 2, 3, 4, 5, 6u.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Random variableCausality principle

Let X : A Ñ A 1 be a random variable, then A 1 isendowed with the σ-algebra T 1 and the probabilitymeasure P 1 inherited from T and P by the causalityprinciple:

@a1 P T 1 , P 1pa1q “ P pX´1pa1qq .

P 1 is now the probability measure of X on A 1.

The original sample space A (the system and its environment inthe foregoing examples) is often suppressed, since it ismathematically hard to describe.

Examples: The possible values of the random variable ”cointoss” or ”dice roll” are then treated as a sample space A 1

constituted by numerical quantities–which is easier to deal with.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic processDefinition and terminology

Definition

A stochastic processX : T Ñ Rn

t ÞÑXtis a function from T

into the set of random variables defined on pA ,T ,P q withvalues in Rn ” A 1. pXt , t P T q is a collection of randomvariables with values in Rn. Therefore:

If t P T is frozen: Xt is a random variable defined onpA ,T ,P q with values in Rn;

If a P A is frozen: t ÞÑXtpaq “ xptq is a trajectory, ora sample path;

If both t P T and a P A are frozen: a, t ÞÑXtpaq is anoutcome, or a realization, of the random variable Xt.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic process

Stochastic process indexed on R`.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic processRemarks

pXt , t P T q is called ”the stochastic process defined onpA ,T ,P q, indexed on T , with values in Rn”.

The indexation set T could be any set, countable ornot, bounded or not, etc.

However we will only consider the real line T “ R in thesubsequent lectures, since t will be a time.

Examples: earthquake, wind, turbulence, swell, ocean surge...can be modeled by stochastic processes, where A “ t”the statesof the environment”u is very difficult to figure out, butA 1

“ t”vectors of R3”u is much easier to handle.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic processProbability law

For t P R fixed, Xt is a Rn-valued random variabledefined on pA ,T ,P q, of which probability law PX isparameterized by t and defined by:

PXpB; tq “ P 1pXt P Bq

“ P pX´1t pBq P T q , @B P T 1 .

The most common choice for the σ-algebra T 1 is the Borel

σ-algebra BpRnq–the one generated by the collection of all open

sets in Rn.

The distribution function FXpx; tq : Rn Ñ r0, 1s:

PXpdx; tq “ dFXpx; tq ,

FXpx; tq “ P 1pXt Ps ´8, x1s ˆ ¨ ¨ ¨ ˆs ´8, xnsq .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic processJoint and marginal probability laws

Joint probability laws: The two-fold probabilitymeasure PXY pdxdy; t, t1q “ dFXY px,y; t, t1q with:

FXY px,y; t, t1q “ P 1pXt Ps ´8, x1s ˆ ¨ ¨ ¨ ˆs ´8, xns

and Y t1 Ps ´8, y1s ˆ ¨ ¨ ¨ ˆs ´8, ynsq .

Marginal probability laws: Let I be the set of all finite,unordered parts of R of the form I Q i “ tt1, t2, . . . tku.Let Xpiq “ pXt1 ,Xt2 , . . .Xtkq be the Rnˆk-valuedrandom variable constructed from the part i, then thecollection of marginal probability laws of the processpXt , t P Rq is tPXpiqpdx1dx2 ¨ ¨ ¨ dxkq, i P Iu.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic processSecond-order stochastic process

Definition

pXt , t P Rq is a second-order stochastic process if:

Et}Xt}2u ă `8 , @t P R ,

where }x} “b

řnj“1 x

2j and EtfpXtqu “

ş

Rn fpxqPXpdx; tq.

The set L2pA ,Rnq of Rn-valued second-order randomvariables defined on pA ,T ,P q is an Hilbert space, withthe scalar product ppX,Y qq2 “ EtpX,Y qu and thenorm |||X||| “

a

ppX,Xqq2 “a

Et}X}2u.Interests: the same as for L2pRnq! (separability,uniform convexity).

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Stochastic processMean and auto-correlation functions

Definition

Let pXt , t P Rq be a Rn-valued, second-order stochasticprocess defined on pA ,T ,P q and indexed on R.

Its mean function mX : RÑ Rn is:

mXptq “

ż

Rn

xPXpdx; tq “ EtXtu .

The process is centered if mXptq “ 0, @t P R.

Its auto-correlation function RX : Rˆ RÑ Rnˆn is:

RXpt, t1q “

ĳ

RnˆRn

xbyPXXpdxdy; t, t1q “ EtXtbXt1u .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Outline



3 Filtering

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processDefinitions

Definition

Let pXt , t P Rq be a Rn-valued stochastic process defined onpA ,T ,P q and indexed on R. It is:

stationary if @u P R, pXt`u , t P Rq and pXt , t P Rqhave the same marginal probability laws;

mean-square stationary if it is second-order,mXptq “mX (independent of time), andRXpt, t

1q “ RXpt´ t1q.

If pXt , t P Rq is stationary then PXpdx; tq “ PXpdxq(independent of time) and PXXpdxdy; t, t1q “ PXXpdxdy; t´ t1q.

Stationary ñ mean-square stationary, but the reverse is not true.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processStationarity

Stationary v.s. non-stationary processes.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processSpectral measure

Data: pXt , t P Rq a Rn-valued, second-order stochasticprocess defined on pA ,T ,P q, indexed on R, andmean-square stationary.

Hypothesis: t ÞÑ RXptq is continuous on R.

Theorem (Bochner)

Then there exists an Hermitian, positive, and boundedmeasure MX P Cnˆn, called the spectral measure of theRn-valued stochastic process pXt , t P Rq, s.t.:

RXptq “

ż

ReiωtMX pdωq .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processPower spectral density

Definition

The spectral density SX is the density of the spectralmeasure MX if it exists:

MX pdωq “ SXpωqdω .

It is Hermitian, positive, and integrable (sinceMXpRq ă `8). Then lim

|t|Ñ`8RXptq “ 0 (as the

Fourier transform of an integrable function).

A necessary condition: mX “ 0.

A sufficient condition: lim|t|Ñ`8

RXptq “ 0.

Let B Ď R, B ÞÑ TrMXpBq is the power spectralmeasure on B, and if SX exists, ω ÞÑ TrSXpωq is thepower spectral density (PSD) of the process.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processMean-square derivative

Data: pXt , t P Rq a Rn-valued, second-order stochasticprocess defined on pA ,T ,P q, indexed on R.

Definition

Let t P R, pXt , t P Rq is mean-square derivable at t iff thesequence of random variables Xh “

1hpXt`h ´Xtq

converges in L2pA ,Rnq as hÑ 0. That limit is denoted by9Xt s.t.:

limhÑ0

|||Xh ´9Xt||| “ 0 .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processSecond-order properties of the derivative

A sufficient condition: B2RXBtBt1 exists for all pt, t1q in RˆR.

Then p 9Xt , t P Rq is a Rn-valued, second-orderstochastic process defined on pA ,T ,P q, indexed on R.In addition:

m 9Xptq “dmX

dtptq ,

R 9Xpt, t1q “

B2RX

BtBt1pt, t1q ,

RX 9Xpt, t1q “ EtXt b 9Xt1u “

BRX

Bt1pt, t1q ,

R 9XXpt, t1q “ Et 9Xt bXt1u “

BRX

Btpt, t1q .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Mean-square stochastic processDerivative of a mean-square stationary process

A necessary and sufficient condition:

ż

Rω2 trMXpdωq ă `8 .

Then p 9Xt , t P Rq is a Rn-valued, second-orderstochastic process defined on pA ,T ,P q, indexed on R,and mean-square stationary. In addition:

m 9Xptq “ 0 ,

R 9Xptq “ ´d2RX

dt2ptq ,

RX 9Xptq “dRX

dtptq ,

M 9Xpdωq “ ω2MXpdωq .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Outline



3 Filtering

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Filtering of a stochastic processLinear filter

x * yh

A linear convolution filter:

yptq “

ż

Rhpt´ τqxpτq dτ

where:

t ÞÑ hptq is the impulse response function in Rmˆn;

ω ÞÑ phpωq “ş

R e´iωthptqdt is the frequency response

function;p ÞÑ Hppq “

ş`8

´8e´pt

hptqdt, p P DH Ă C, is thetransfer function.

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Filtering of a stochastic processStationary response

Data: pXt , t P Rq a Rm-valued, second-order stochasticprocess defined on pA ,T ,P q, indexed on R, andmean-square stationary.

If t ÞÑ hptq is integrable or square integrable, thenpY t , t P Rq is a Rn-valued, second-order stochasticprocess defined on pA ,T ,P q, indexed on R, andmean-square stationary s.t.:

mY “php0qmX ,

RY ptq “

ĳ

RˆR

hpτqRXpt` τ1 ´ τqhpτ 1qT dτdτ 1 ,

MY pdωq “ phpωqMXpdωqph˚pωq .

Randomprocesses

E. Savin

ProbabilitySpace

Probabilitytriplet

Randomvariable

Stochasticprocess


Stationarity



Filtering

Bibliography

Further reading...

P. Kree & C. Soize: Mathematics of Random Phenomena:Random Vibrations of Mechanical Structures, Reidel PublishingCo., Dordrecht (1986);

C. Soize: Methodes Mathematiques en Analyse du Signal, Masson,Paris (1993);

T.T. Soong & M. Grigoriu: Random Vibration of Mechanical andStructural Systems, Prentice Hall, Upper Saddle River NJ (1993).

http://dx.doi.org/10.1142/9789814354110

http://dx.doi.org/10.1142/9789814354110

primer on random processes

Documents