Random Processes. DS GA 1002 Probability and Statistics for Data Science.

Transcription

1 Random Processes DS GA 1002 Probability and Statistics for Data Science Carlos Fernandez-Granda

2 Aim Modeling quantities that evolve in time (or space) Trajectory of a particle, price of oil, temperature in New York...

3 Definition Mean and autocovariance functions Important random processes

4 Notation We denote random processes using a tilde over an upper case letter X

5 Formal definition Given a probability space (Ω, F, P), a random process X is a function that maps each ω Ω to a function X (ω, ) : T R There are two interpretations for X (ω, t) 1. If we fix ω, then X (ω, t) is a deterministic function or realization of t 2. If we fix t then X (ω, t) is a random variable, usually denoted by X (t)

6 Continuous and discrete random processes We can classify a random process depending on the indexing variable t If t is defined on a continuous interval, the process is continuous If t is defined on a discrete set, the process is discrete

7 State space Set of possible values of the random variable X (t) for any t It can be continuous or discrete (also finite) There are continuous-state discrete-time random processes and discrete-state continuous-time random processes

8 Puddle Initial amount of water is uniform between 0 and 1 gallon After a time interval t there is t times less water Continuous-state continuous-time random process C C (ω, t) := ω t t [1, )

9 Puddle ω = 0.62 ω = 0.91 ω = 0.12 C (ω, t) t

10 Puddle We only care about how much water there is on day i Continuous-state discrete-time random process D D (ω, i) := ω, i = 1, 2,... i

11 Puddle 0.8 ω = 0.31 ω = 0.89 ω = 0.52 D (ω, i) i

12 nth-order distribution Joint distribution of X (t 1 ), X (t 2 ),..., X (t n ) for any {t 1, t 2,..., t n } If X (t 1 ), X (t 2 ),..., X (t n ) have the same joint distribution as X (t 1 + τ), X (t 2 + τ),..., X (t n + τ) for any τ the process is strictly/strongly stationary

13 Puddle F C(t) (x)

14 Puddle ( ) F C(t) (x) := P C (t) x

15 Puddle ( ) F C(t) (x) := P C (t) x = P (ω t x)

16 Puddle ( ) F C(t) (x) := P C (t) x = P (ω t x) t x u=0 du = t x if 0 x 1 t = 1 if x > 1 t 0 if x < 0

17 Puddle ( ) F C(t) (x) := P C (t) x = P (ω t x) t x u=0 du = t x if 0 x 1 t = 1 if x > 1 t 0 if x < 0 { t if 0 x 1 f C(t) (x) = t 0 otherwise

18 How to specify a random process Three options:

19 How to specify a random process Three options: 1. Define probability space and a function from Ω to a set of functions

20 How to specify a random process Three options: 1. Define probability space and a function from Ω to a set of functions 2. Define all nth-order distributions for all n 0

21 How to specify a random process Three options: 1. Define probability space and a function from Ω to a set of functions 2. Define all nth-order distributions for all n 0 3. Express it as a function of another random process

22 Definition Mean and autocovariance functions Important random processes

23 Mean The mean of a random process is the function ( ) µ X (t) := E X (t) It is a deterministic function of t

24 Autocovariance The autocovariance of a random process is the function ( R X (t 1, t 2 ) := Cov X (t1 ), X ) (t 2 ) In particular, ( ) R X (t, t) := Var X (t)

25 Wide-sense/weakly stationary process A process is stationary in a wide or weak sense if its mean is constant µ X (t) := µ and its autocovariance function is shift invariant, i.e. R X (t 1, t 2 ) := R X (t 1 + τ, t 2 + τ) for any t 1 and t 2 and any shift τ Common notation R X (s) := R X (t, t + s)

26 Autocovariance function R(s) s

27 Realization i

28 Realization i

29 Realization i

30 Autocovariance function R(s) s

31 Realization i

32 Realization i

33 Realization i

34 Autocovariance function R(s) s

35 Realization i

36 Realization i

37 Realization i

38 Definition Mean and autocovariance functions Important random processes

39 Independent identically-distributed sequences A discrete random process is iid if X (i) has the same distribution for any fixed i X (i1 ), X (i 2 ),..., X (i n ) are mutually independent for any i 1,..., i n and any n 2

40 Independent identically-distributed sequences Valid definition If the state is discrete p X (i1 ), X (i 2 ),..., X (i (x n) i 1, x i2,..., x in ) = If the state is continuous f X (i1 ), X (i 2 ),..., X (i (x n) i 1, x i2,..., x in ) = The process is strictly stationary n p X (x i ) i=1 n f X (x i ) i=1

41 Uniform distribution in [0, 1] i

42 Uniform distribution in [0, 1] i

43 Uniform distribution in [0, 1] i

44 Geometric distribution (p = 0.4) i

45 Geometric distribution (p = 0.4) i

46 Geometric distribution (p = 0.4) i

47 Independent identically-distributed sequences If the distribution at each time has mean µ and variance σ 2 µ X (i) R X (i, j)

48 Independent identically-distributed sequences If the distribution at each time has mean µ and variance σ 2 ( ) µ X (i) := E X (i) R X (i, j)

49 Independent identically-distributed sequences If the distribution at each time has mean µ and variance σ 2 ( ) µ X (i) := E X (i) = µ R X (i, j)

50 Independent identically-distributed sequences If the distribution at each time has mean µ and variance σ 2 ( ) µ X (i) := E X (i) = µ ( ) R X (i, j) := E X (i) X (j) = { σ 2 if i = j 0 if i j ( ) ( ) E X (i) E X (j)

51 Gaussian random process Any set of samples is a Gaussian random vector Fully characterized by mean function µ X and autocovariance function R X X (t 1 ) X (t X := 2 ) X (t n ) is a Gaussian random vector with mean and covariance µ X (t 1 ) R X (t 1, t 1 ) R X (t 1, t 2 ) R X (t 1, t n ) µ X := µ X (t 2 ) R Σ X := X (t 1, t 2 ) R X (t 2, t 2 ) R X (t 2, t n ) µ X (t n ) R X (t 2, t n ) R X (t 2, t n ) R X (t n, t n )

52 Generating a Gaussian random process Boils down to sampling a Gaussian random vector with the appropriate mean and covariance matrix 1. Compute the mean vector µ X and the covariance matrix Σ X 2. Generate n independent samples from a standard Gaussian 3. Color the samples according to Σ X and center them around µ X

53 Poisson process Sequential events occurring on [0, ) 1. Each event occurs independently from every other event 2. Events occur uniformly 3. Events occur at a rate of λ events per time interval Ñ (t) is the number of events between 0 and t

54 Poisson process For any t 1 < t 2 < t 3 < t 4 1. Ñ (t 2 ) Ñ (t 1) is Poisson with parameter λ (t 2 t 1 ) 2. Ñ (t 2 ) Ñ (t 1) and Ñ (t 4) Ñ (t 3) are independent A random process satisfying these two conditions is a Poisson process

55 Poisson process nth order distribution can be expressed in terms of p ( λ, x ) := λ x e λ x!

56 Poisson process pñ(t1 ),...,Ñ(tn) (x1,..., xn)

57 Poisson process pñ(t1 ),...,Ñ(tn) (x1,..., xn) ) = P (Ñ (t1) = x 1,..., Ñ (tn) = xn

58 Poisson process pñ(t1 ),...,Ñ(tn) (x1,..., xn) ) = P (Ñ (t1) = x 1,..., Ñ (tn) = xn ) = P (Ñ (t1) = x 1, Ñ (t2) Ñ (t1) = x2 x1,..., Ñ (tn) Ñ (tn 1) = xn xn 1

59 Poisson process pñ(t1 ),...,Ñ(tn) (x1,..., xn) ) = P (Ñ (t1) = x 1,..., Ñ (tn) = xn ) = P (Ñ (t1) = x 1, Ñ (t2) Ñ (t1) = x2 x1,..., Ñ (tn) Ñ (tn 1) = xn xn 1 ) ) ) = P (Ñ (t1) = x 1 P (Ñ (t2) Ñ (t1) = x2 x1... P (Ñ (tn) Ñ (tn 1) = xn xn 1

60 Poisson process pñ(t1 ),...,Ñ(tn) (x1,..., xn) ) = P (Ñ (t1) = x 1,..., Ñ (tn) = xn ) = P (Ñ (t1) = x 1, Ñ (t2) Ñ (t1) = x2 x1,..., Ñ (tn) Ñ (tn 1) = xn xn 1 ) ) ) = P (Ñ (t1) = x 1 P (Ñ (t2) Ñ (t1) = x2 x1... P (Ñ (tn) Ñ (tn 1) = xn xn 1 = p (λt 1, x 1) p (λ (t 2 t 1), x 2 x 1)... p (λ (t n t n 1), x n x n 1)

61 Poisson process (λ = 0.2) t

62 Poisson process (λ = 0.2) t

63 Poisson process (λ = 0.2) t

64 Poisson process (λ = 1) t

65 Poisson process (λ = 1) t

66 Poisson process (λ = 1) t

67 Poisson process (λ = 2) t

68 Poisson process (λ = 2) t

69 Poisson process (λ = 2) t

70 Call-center data Example: Data from a call center in Israel We compare the histogram of the number of calls received in an interval of 4 hours over 2 months and the pmf of a Poisson random variable fitted to the data

71 Call-center data Real data Poisson distribution Number of calls

72 Poisson process Distribution of interarrival times? F T (t)

73 Poisson process Distribution of interarrival times? F T (t) := P (T t)

74 Poisson process Distribution of interarrival times? F T (t) := P (T t) = 1 P (T > t)

75 Poisson process Distribution of interarrival times? F T (t) := P (T t) = 1 P (T > t) = 1 P (no events in an interval of length t)

76 Poisson process Distribution of interarrival times? F T (t) := P (T t) = 1 P (T > t) = 1 P (no events in an interval of length t) = 1 e λ t

77 Poisson process Distribution of interarrival times? F T (t) := P (T t) = 1 P (T > t) = 1 P (no events in an interval of length t) = 1 e λ t f T (t) = λe λ t Iid exponential sequence (allows to simulate Poisson process!)

78 Call-center data Example: Data from a call center in Israel We compare the histogram of the inter-arrival times between calls occurring between 8 pm and midnight over two days and the pdf of an exponential random variable fitted to the data

79 Call center Exponential distribution Real data

80 Generating a Poisson process To sample from a Poisson random process with parameter λ we: 1. Generate independent samples from an exponential random variable with parameter λ t 1, t 2, t 3, Set the events of the Poisson process to occur at t 1, t 2, t 3,...

81 Mean and autocovariance ( ) E X (t)

82 Mean and autocovariance ( ) E X (t) = λ t

83 Mean and autocovariance ( ) E X (t) = λ t R X (t 1, t 2 ) = λ min {t 1, t 2 }

84 Mean and autocovariance ( ) E X (t) = λ t R X (t 1, t 2 ) = λ min {t 1, t 2 } The process is neither strictly nor wide-sense stationary

85 Earthquakes Earthquakes in San Francisco follow a Poisson process with parameter 0.3 earthquakes/year Probability of no earthquakes in the next 10 years and then at least 1 over the following 20 years?

86 Earthquakes ( ) P X (10) = 0, X (30) 1

87 Earthquakes ( ) ( ) P X (10) = 0, X (30) 1 = P X (10) = 0, X (30) X (10) 1

88 Earthquakes ( ) P X (10) = 0, X (30) 1 ( ) = P X (10) = 0, X (30) X (10) 1 = P ( ) ( ) X (10) = 0 P X (30) X (10) 1

89 Earthquakes ( ) P X (10) = 0, X (30) 1 ( ) = P X (10) = 0, X (30) X (10) 1 ( ) X (10) = 0 = P = P ( X (10) = 0 ) ( 1 P ( ) P X (30) X (10) 1 ( X (30) X (10) = 0 ))

90 Earthquakes ( ) P X (10) = 0, X (30) 1 ( ) = P X (10) = 0, X (30) X (10) 1 ( ) X (10) = 0 = P = P ( X (10) = 0 ) ( 1 P = e 3 ( 1 e 6) = ( ) P X (30) X (10) 1 ( X (30) X (10) = 0 ))

91 Random walk Process that evolves by taking steps in random directions Step sequence Z is iid S (i) = { +1 with probability with probability 1 2 We define a random walk X as { 0 for i = 0 X (i) := i S j=1 (j) for i = 1, 2,...

92 Random walk i

93 Random walk i

94 Random walk i

95 First-order pmf p X (i) (x)? Distribution of number of positive steps S +? Negative steps: S = i S + p X (i) (x)

96 First-order pmf p X (i) (x)? Distribution of number of positive steps S +? Negative steps: S = i S + i p X (i) (x) = P S (i) = x j=0

97 First-order pmf p X (i) (x)? Distribution of number of positive steps S +? Negative steps: S = i S + i p X (i) (x) = P S (i) = x j=0 = P (S + S = x)

98 First-order pmf p X (i) (x)? Distribution of number of positive steps S +? Negative steps: S = i S + i p X (i) (x) = P S (i) = x j=0 = P (S + S = x) = P (2 S + i = x)

99 First-order pmf p X (i) (x)? Distribution of number of positive steps S +? Negative steps: S = i S + i p X (i) (x) = P S (i) = x j=0 = P (S + S = x) = P (2 S + i = x) ( = P S + = i + x ) 2

100 First-order pmf p X (i) (x)? Distribution of number of positive steps S +? Negative steps: S = i S + i p X (i) (x) = P S (i) = x j=0 = P (S + S = x) = P (2 S + i = x) ( = P S + = i + x ) 2 ( ) i 1 = i+x 2 2 i if i + x is an integer between 0 and i 2

101 Mean and autocovariance µ X (i)

102 Mean and autocovariance ( ) µ X (i) := E X (i)

103 Mean and autocovariance ( ) µ X (i) := E X (i) i = E S (j) j=1

104 Mean and autocovariance ( ) µ X (i) := E X (i) i = E S (j) = i j=1 j=1 ( ) E S (j) by linearity of expectation

105 Mean and autocovariance ( ) µ X (i) := E X (i) i = E S (j) = = 0 i j=1 j=1 ( ) E S (j) by linearity of expectation

106 Mean and autocovariance ( ) µ X (i) := E X (i) i = E S (j) = = 0 i j=1 j=1 ( ) E S (j) by linearity of expectation R X (i, j) = min {i, j}

107 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain?

108 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain? 0

109 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain? 0 Probability that the gambler is up 6 dollars or more after first 10 flips?

110 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain? 0 Probability that the gambler is up 6 dollars or more after first 10 flips? P (gambler is up $6 or more)

111 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain? 0 Probability that the gambler is up 6 dollars or more after first 10 flips? P (gambler is up $6 or more) = p X (10) (6) + p X (10) (8) + p X (10) (10)

112 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain? 0 Probability that the gambler is up 6 dollars or more after first 10 flips? P (gambler is up $6 or more) = p X (10) (6) + p X (10) (8) + p X (10) (10) ( ) ( ) =

113 Gambler A fair coin is flipped sequentially Heads: +$1 Tails: -$1 Expected gain? 0 Probability that the gambler is up 6 dollars or more after first 10 flips? P (gambler is up $6 or more) = p X (10) (6) + p X (10) (8) + p X (10) (10) ( ) ( ) = =