Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of - PowerPoint PPT Presentation

Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November 15, 2019 Introduction to Random Processes Gaussian, Markov and stationary processes 1

Introduction and roadmap Introduction and roadmap Gaussian processes Brownian motion and its variants White Gaussian noise Introduction to Random Processes Gaussian, Markov and stationary processes 2

Random processes ◮ Random processes assign a function X ( t ) to a random event ⇒ Without restrictions, there is little to say about them ⇒ Markov property simplifies matters and is not too restrictive ◮ Also constrained ourselves to discrete state spaces ⇒ Further simplification but might be too restrictive ◮ Time t and range of X ( t ) values continuous in general ◮ Time and/or state may be discrete as particular cases ◮ Restrict attention to (any type or a combination of types) ⇒ Markov processes (memoryless) ⇒ Gaussian processes (Gaussian probability distributions) ⇒ Stationary processes (“limit distribution”) Introduction to Random Processes Gaussian, Markov and stationary processes 3

Markov processes ◮ X ( t ) is a Markov process when the future is independent of the past ◮ For all t > s and arbitrary values x ( t ), x ( s ) and x ( u ) for all u < s � X ( s ) ≤ x ( s ) , X ( u ) ≤ x ( u ) , u < s � � � P X ( t ) ≤ x ( t ) � X ( s ) ≤ x ( s ) � � � = P X ( t ) ≤ x ( t ) ⇒ Markov property defined in terms of cdfs, not pmfs ◮ Markov property useful for same reasons as in discrete time/state ⇒ But not that useful as in discrete time /state ◮ More details later Introduction to Random Processes Gaussian, Markov and stationary processes 4

Gaussian processes ◮ X ( t ) is a Gaussian process when all prob. distributions are Gaussian ◮ For arbitrary n > 0, times t 1 , t 2 , . . . , t n it holds ⇒ Values X ( t 1 ) , X ( t 2 ) , . . . , X ( t n ) are jointly Gaussian RVs ◮ Simplifies study because Gaussian distribution is simplest possible ⇒ Suffices to know mean, variances and (cross-)covariances ⇒ Linear transformation of independent Gaussians is Gaussian ⇒ Linear transformation of jointly Gaussians is Gaussian ◮ More details later Introduction to Random Processes Gaussian, Markov and stationary processes 5

Markov processes + Gaussian processes ◮ Markov (memoryless) and Gaussian properties are different ⇒ Will study cases when both hold ◮ Brownian motion, also known as Wiener process ⇒ Brownian motion with drift ⇒ White noise ⇒ Linear evolution models ◮ Geometric brownian motion ⇒ Arbitrages ⇒ Risk neutral measures ⇒ Pricing of stock options (Black-Scholes) Introduction to Random Processes Gaussian, Markov and stationary processes 6

Stationary processes ◮ Process X ( t ) is stationary if probabilities are invariant to time shifts ◮ For arbitrary n > 0, times t 1 , t 2 , . . . , t n and arbitrary time shift s P ( X ( t 1 + s ) ≤ x 1 , X ( t 2 + s ) ≤ x 2 , . . . , X ( t n + s ) ≤ x n ) = P ( X ( t 1 ) ≤ x 1 , X ( t 2 ) ≤ x 2 , . . . , X ( t n ) ≤ x n ) ⇒ System’s behavior is independent of time origin ◮ Follows from our success studying limit probabilities ⇒ Study of stationary process ≈ Study of limit distribution ◮ Will study ⇒ Spectral analysis of stationary random processes ⇒ Linear filtering of stationary random processes ◮ More details later Introduction to Random Processes Gaussian, Markov and stationary processes 7

Gaussian processes Introduction and roadmap Gaussian processes Brownian motion and its variants White Gaussian noise Introduction to Random Processes Gaussian, Markov and stationary processes 8

Jointly Gaussian random variables ◮ Def: Random variables X 1 , . . . , X n are jointly Gaussian (normal) if any linear combination of them is Gaussian ⇒ Given n > 0, for any scalars a 1 , . . . , a n the RV ( a = [ a 1 , . . . , a n ] T ) Y = a 1 X 1 + a 2 X 2 + . . . + a n X n = a T X is Gaussian distributed ⇒ May also say vector RV X = [ X 1 , . . . , X n ] T is Gaussian ◮ Consider 2 dimensions ⇒ 2 RVs X 1 and X 2 are jointly normal ◮ To describe joint distribution have to specify ⇒ Means: µ 1 = E [ X 1 ] and µ 2 = E [ X 2 ] ⇒ Variances: σ 2 � ( X 1 − µ 1 ) 2 � and σ 2 11 = var [ X 1 ] = E 22 = var [ X 2 ] ⇒ Covariance: σ 2 12 = cov( X 1 , X 2 ) = E [( X 1 − µ 1 )( X 2 − µ 2 )]= σ 2 21 Introduction to Random Processes Gaussian, Markov and stationary processes 9

Pdf of jointly Gaussian RVs in 2 dimensions ◮ Define mean vector µ = [ µ 1 , µ 2 ] T and covariance matrix C ∈ R 2 × 2 � σ 2 σ 2 � 11 12 C = σ 2 σ 2 21 22 ⇒ C is symmetric, i.e., C T = C because σ 2 21 = σ 2 12 ◮ Joint pdf of X = [ X 1 , X 2 ] T is given by 1 � − 1 � 2( x − µ ) T C − 1 ( x − µ ) f X ( x ) = exp 2 π det 1 / 2 ( C ) ⇒ Assumed that C is invertible, thus det( C ) � = 0 ◮ If the pdf of X is f X ( x ) above, can verify Y = a T X is Gaussian Introduction to Random Processes Gaussian, Markov and stationary processes 10

Pdf of jointly Gaussian RVs in n dimensions ◮ For X ∈ R n ( n dimensions) define µ = E [ X ] and covariance matrix σ 2 σ 2 σ 2  . . .  11 12 1 n σ 2 σ 2 σ 2 . . . 21 22 2 n � ( X − µ )( X − µ ) T �   C := E =  . .  ... . .   . .   σ 2 σ 2 σ 2 . . . n 1 n 2 nn ⇒ C symmetric, ( i , j )-th element is σ 2 ij = cov( X i , X j ) ◮ Joint pdf of X defined as before (almost, spot the difference) 1 � − 1 � 2( x − µ ) T C − 1 ( x − µ ) f X ( x ) = exp (2 π ) n / 2 det 1 / 2 ( C ) ⇒ C invertible and det( C ) � = 0. All linear combinations normal ◮ To fully specify the probability distribution of a Gaussian vector X ⇒ The mean vector µ and covariance matrix C suffice Introduction to Random Processes Gaussian, Markov and stationary processes 11

Notational aside and independence ◮ With x ∈ R n , µ ∈ R n and C ∈ R n × n , define function N ( x ; µ , C ) as 1 � − 1 � 2( x − µ ) T C − 1 ( x − µ ) N ( x ; µ , C ) := exp (2 π ) n / 2 det 1 / 2 ( C ) ⇒ µ and C are parameters, x is the argument of the function ◮ Let X ∈ R n be a Gaussian vector with mean µ , and covariance C ⇒ Can write the pdf of X as f X ( x ) = N ( x ; µ , C ) ◮ If X 1 , . . . , X n are mutually independent, then C = diag( σ 2 11 , . . . , σ 2 nn ) and n − ( x i − µ i ) 2 1 � � � f X ( x ) = exp 2 σ 2 � 2 πσ 2 ii i =1 ii Introduction to Random Processes Gaussian, Markov and stationary processes 12

Gaussian processes ◮ Gaussian processes (GP) generalize Gaussian vectors to infinite dimensions ◮ Def: X ( t ) is a GP if any linear combination of values X ( t ) is Gaussian ⇒ For arbitrary n > 0, times t 1 , . . . , t n and constants a 1 , . . . , a n Y = a 1 X ( t 1 ) + a 2 X ( t 2 ) + . . . + a n X ( t n ) is Gaussian distributed ⇒ Time index t can be continuous or discrete ◮ More general, any linear functional of X ( t ) is normally distributed ⇒ A functional is a function of a function � t 2 Ex: The (random) integral Y = X ( t ) dt is Gaussian distributed t 1 ⇒ Integral functional is akin to a sum of X ( t i ), for all t i ∈ [ t 1 , t 2 ] Introduction to Random Processes Gaussian, Markov and stationary processes 13

Joint pdfs in a Gaussian process ◮ Consider times t 1 , . . . , t n . The mean value µ ( t i ) at such times is µ ( t i ) = E [ X ( t i )] ◮ The covariance between values at times t i and t j is �� �� �� C ( t i , t j ) = E X ( t i ) − µ ( t i ) X ( t j ) − µ ( t j ) ◮ Covariance matrix for values X ( t 1 ) , . . . , X ( t n ) is then  C ( t 1 , t 1 ) C ( t 1 , t 2 ) . . . C ( t 1 , t n )  C ( t 2 , t 1 ) C ( t 2 , t 2 ) . . . C ( t 2 , t n )   C ( t 1 , . . . , t n ) = . . .  ...  . . .   . . .   C ( t n , t 1 ) C ( t n , t 2 ) . . . C ( t n , t n ) ◮ Joint pdf of X ( t 1 ) , . . . , X ( t n ) then given as � � [ x 1 , . . . , x n ] T ; [ µ ( t 1 ) , . . . , µ ( t n )] T , C ( t 1 , . . . , t n ) f X ( t 1 ) ,..., X ( t n ) ( x 1 , . . . , x n ) = N Introduction to Random Processes Gaussian, Markov and stationary processes 14

Mean value and autocorrelation functions ◮ To specify a Gaussian process, suffices to specify: ⇒ Mean value function ⇒ µ ( t ) = E [ X ( t )]; and � � ⇒ Autocorrelation function ⇒ R ( t 1 , t 2 ) = E X ( t 1 ) X ( t 2 ) ◮ Autocovariance obtained as C ( t 1 , t 2 ) = R ( t 1 , t 2 ) − µ ( t 1 ) µ ( t 2 ) ◮ For simplicity, will mostly consider processes with µ ( t ) = 0 ⇒ Otherwise, can define process Y ( t ) = X ( t ) − µ X ( t ) ⇒ In such case C ( t 1 , t 2 ) = R ( t 1 , t 2 ) because µ Y ( t ) = 0 ◮ Autocorrelation is a symmetric function of two variables t 1 and t 2 R ( t 1 , t 2 ) = R ( t 2 , t 1 ) Introduction to Random Processes Gaussian, Markov and stationary processes 15