Random Processes DS GA 1002 Probability and Statistics for Data - - PowerPoint PPT Presentation

Random Processes

DS GA 1002 Probability and Statistics for Data Science

http://www.cims.nyu.edu/~cfgranda/pages/DSGA1002_fall17 Carlos Fernandez-Granda

Aim

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

Definition Mean and autocovariance functions Important random processes

Notation

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

• X

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)

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

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

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, ∞)

Puddle

2 4 6 8 10 0.2 0.4 0.6 0.8 1 t

• C (ω, t)

ω = 0.62 ω = 0.91 ω = 0.12

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 , i = 1, 2, . . .

Puddle

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 i

• D (ω, i)

ω = 0.31 ω = 0.89 ω = 0.52

nth-order distribution

Joint distribution of X (t1), X (t2), . . . , X (tn) for any {t1, t2, . . . , tn} If X (t1), X (t2), . . . , X (tn) have the same joint distribution as X (t1 + τ), X (t2 + τ), . . . , X (tn + τ) for any τ the process is strictly/strongly stationary

Puddle

F

C(t) (x)

Puddle

F

C(t) (x) := P

• C (t) ≤ x

Puddle

F

C(t) (x) := P

• C (t) ≤ x
• = P (ω ≤ t x)

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

if x < 0

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

if x < 0 f

C(t) (x) =

• t

if 0 ≤ x ≤ 1

t

• therwise

How to specify a random process

Three options:

How to specify a random process

Three options:

• 1. Define probability space and a function from Ω to a set of functions

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

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

Definition Mean and autocovariance functions Important random processes

Mean

The mean of a random process is the function µ

X (t) := E

• X (t)
• It is a deterministic function of t

Autocovariance

The autocovariance of a random process is the function R

X (t1, t2) := Cov

• X (t1) ,

X (t2)

• In particular,

R

X (t, t) := Var

• X (t)

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 (t1, t2) := R X (t1 + τ, t2 + τ)

for any t1 and t2 and any shift τ Common notation R

X (s) := R X (t, t + s)

Autocovariance function

15 10 5 5 10 15 s 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 R(s)

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Autocovariance function

15 10 5 5 10 15 s 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 R(s)

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Autocovariance function

15 10 5 5 10 15 s 1.0 0.5 0.0 0.5 1.0 R(s)

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Realization

2 4 6 8 10 12 14 i 3 2 1 1 2 3

Definition Mean and autocovariance functions Important random processes

Independent identically-distributed sequences

A discrete random process is iid if

◮

X (i) has the same distribution for any fixed i

◮

X (i1), X (i2), . . . , X (in) are mutually independent for any i1, . . . , in and any n ≥ 2

Independent identically-distributed sequences

Valid definition

◮ If the state is discrete

p

X(i1), X(i2),..., X(in) (xi1, xi2, . . . , xin) = n

• i=1

p

X (xi) ◮ If the state is continuous

f

X(i1), X(i2),..., X(in) (xi1, xi2, . . . , xin) = n

• i=1

f

X (xi)

The process is strictly stationary

Uniform distribution in [0, 1]

2 4 6 8 10 12 14 i 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Uniform distribution in [0, 1]

2 4 6 8 10 12 14 i 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Uniform distribution in [0, 1]

2 4 6 8 10 12 14 i 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Geometric distribution (p = 0.4)

2 4 6 8 10 12 14 i 2 4 6 8 10 12

Geometric distribution (p = 0.4)

2 4 6 8 10 12 14 i 2 4 6 8 10 12

Geometric distribution (p = 0.4)

2 4 6 8 10 12 14 i 2 4 6 8 10 12

Independent identically-distributed sequences

If the distribution at each time has mean µ and variance σ2 µ

X (i)

R

X (i, j)

Independent identically-distributed sequences

If the distribution at each time has mean µ and variance σ2 µ

X (i) := E

• X (i)
• R

X (i, j)

Independent identically-distributed sequences

If the distribution at each time has mean µ and variance σ2 µ

X (i) := E

• X (i)
• = µ

R

X (i, j)

Independent identically-distributed sequences

If the distribution at each time has mean µ and variance σ2 µ

X (i) := E

• X (i)
• = µ

R

X (i, j) := E

• X (i)

X (j)

• − E
• X (i)
• E
• X (j)
• =
• σ2

if i = j if i = j

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 :=

    

• X (t1)
• X (t2)

· · ·

• X (tn)

     is a Gaussian random vector with mean and covariance

• µ

X :=

    µ

X (t1)

µ

X (t2)

· · · µ

X (tn)

    Σ

X :=

     R

X (t1, t1)

R

X (t1, t2)

· · · R

X (t1, tn)

R

X (t1, t2)

R

X (t2, t2)

· · · R

X (t2, tn)

. . . . . . ... . . . R

X (t2, tn)

R

X (t2, tn)

· · · R

X (tn, tn)

    

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

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
• N (t) is the number of events between 0 and t

Poisson process

For any t1 < t2 < t3 < t4 1. N (t2) − N (t1) is Poisson with parameter λ (t2 − t1) 2. N (t2) − N (t1) and N (t4) − N (t3) are independent A random process satisfying these two conditions is a Poisson process

Poisson process

nth order distribution can be expressed in terms of p

• ˜

λ, x

• :=

˜ λx e−˜

λ

x!

Poisson process

p

N(t1),..., N(tn) (x1, . . . , xn)

Poisson process

p

N(t1),..., N(tn) (x1, . . . , xn)

= P

• N (t1) = x1, . . . ,

N (tn) = xn

Poisson process

p

N(t1),..., N(tn) (x1, . . . , xn)

= P

• N (t1) = x1, . . . ,

N (tn) = xn

• = P
• N (t1) = x1,

N (t2) − N (t1) = x2 − x1, . . . , N (tn) − N (tn−1) = xn − xn−1

Poisson process

p

N(t1),..., N(tn) (x1, . . . , xn)

= P

• N (t1) = x1, . . . ,

N (tn) = xn

• = P
• N (t1) = x1,

N (t2) − N (t1) = x2 − x1, . . . , N (tn) − N (tn−1) = xn − xn−1

• = P
• N (t1) = x1
• P
• N (t2) −

N (t1) = x2 − x1

• . . . P
• N (tn) −

N (tn−1) = xn − xn−1

Poisson process

p

N(t1),..., N(tn) (x1, . . . , xn)

= P

• N (t1) = x1, . . . ,

N (tn) = xn

• = P
• N (t1) = x1,

N (t2) − N (t1) = x2 − x1, . . . , N (tn) − N (tn−1) = xn − xn−1

• = P
• N (t1) = x1
• P
• N (t2) −

N (t1) = x2 − x1

• . . . P
• N (tn) −

N (tn−1) = xn − xn−1

• = p (λt1, x1) p (λ (t2 − t1) , x2 − x1) . . . p (λ (tn − tn−1) , xn − xn−1)

Poisson process (λ = 0.2)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 0.2)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 0.2)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 1)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 1)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 1)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 2)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 2)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

Poisson process (λ = 2)

2 4 6 8 10 t 0.0 0.2 0.4 0.6 0.8 1.0

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

Call-center data

5 10 15 20 25 30 35 40 Number of calls 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Real data Poisson distribution

Poisson process

Distribution of interarrival times? FT (t)

Poisson process

Distribution of interarrival times? FT (t) := P (T ≤ t)

Poisson process

Distribution of interarrival times? FT (t) := P (T ≤ t) = 1 − P (T > t)

Poisson process

Distribution of interarrival times? FT (t) := P (T ≤ t) = 1 − P (T > t) = 1 − P (no events in an interval of length t)

Poisson process

Distribution of interarrival times? FT (t) := P (T ≤ t) = 1 − P (T > t) = 1 − P (no events in an interval of length t) = 1 − e−λ t

Poisson process

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

Call-center data

◮ Example: Data from a call center in Israel ◮ We compare the histogram of the inter-arrival times between calls

• ccurring between 8 pm and midnight over two days and the pdf of an

exponential random variable fitted to the data

Call center

1 2 3 4 5 6 7 8 9 Interarrival times (s) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Exponential distribution Real data

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 t1, t2, t3, . . .

• 2. Set the events of the Poisson process to occur at t1, t2, t3, . . .

Mean and autocovariance

E

• X (t)

Mean and autocovariance

E

• X (t)
• = λ t

Mean and autocovariance

E

• X (t)
• = λ t

R

X (t1, t2) = λ min {t1, t2}

Mean and autocovariance

E

• X (t)
• = λ t

R

X (t1, t2) = λ min {t1, t2}

The process is neither strictly nor wide-sense stationary

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

• ver the following 20 years?

Earthquakes

P

• X (10) = 0,

X (30) ≥ 1

Earthquakes

P

• X (10) = 0,

X (30) ≥ 1

• = P
• X (10) = 0,

X (30) − X (10) ≥ 1

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

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

• = P
• X (10) = 0

1 − P

• X (30) −

X (10) = 0

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

• = P
• X (10) = 0

1 − P

• X (30) −

X (10) = 0

• = e−3

1 − e−6 = 4.97 10−2

Random walk

Process that evolves by taking steps in random directions Step sequence Z is iid

• S (i) =
• +1

with probability 1

2

−1 with probability 1

2

We define a random walk X as

• X (i) :=
• for i = 0

i

j=1

S (j) for i = 1, 2, . . .

Random walk

2 4 6 8 10 12 14 i 6 4 2 2 4 6

Random walk

2 4 6 8 10 12 14 i 6 4 2 2 4 6

Random walk

2 4 6 8 10 12 14 i 6 4 2 2 4 6

First-order pmf

p

X(i) (x) ?

Distribution of number of positive steps S+? Negative steps: S− = i − S+ p

X(i) (x)

First-order pmf

p

X(i) (x) ?

Distribution of number of positive steps S+? Negative steps: S− = i − S+ p

X(i) (x) = P

 

i

• j=0
• S (i) = x

 

First-order pmf

p

X(i) (x) ?

Distribution of number of positive steps S+? Negative steps: S− = i − S+ p

X(i) (x) = P

 

i

• j=0
• S (i) = x

  = P (S+ − S− = x)

First-order pmf

p

X(i) (x) ?

Distribution of number of positive steps S+? Negative steps: S− = i − S+ p

X(i) (x) = P

 

i

• j=0
• S (i) = x

  = P (S+ − S− = x) = P (2 S+ − i = x)

First-order pmf

p

X(i) (x) ?

Distribution of number of positive steps S+? Negative steps: S− = i − S+ p

X(i) (x) = P

 

i

• j=0
• S (i) = x

  = P (S+ − S− = x) = P (2 S+ − i = x) = P

• S+ = i + x

2

First-order pmf

p

X(i) (x) ?

Distribution of number of positive steps S+? Negative steps: S− = i − S+ p

X(i) (x) = P

 

i

• j=0
• S (i) = x

  = P (S+ − S− = x) = P (2 S+ − i = x) = P

• S+ = i + x

2

• =

i

i+x 2

1 2i if i + x 2 is an integer between 0 and i

Mean and autocovariance

µ

X (i)

Mean and autocovariance

µ

X (i) := E

• X (i)

Mean and autocovariance

µ

X (i) := E

• X (i)
• = E

 

i

• j=1
• S (j)

 

Mean and autocovariance

µ

X (i) := E

• X (i)
• = E

 

i

• j=1
• S (j)

  =

i

• j=1

E

• S (j)
• by linearity of expectation

Mean and autocovariance

µ

X (i) := E

• X (i)
• = E

 

i

• j=1
• S (j)

  =

i

• j=1

E

• S (j)
• by linearity of expectation

= 0

Mean and autocovariance

µ

X (i) := E

• X (i)
• = E

 

i

• j=1
• S (j)

  =

i

• j=1

E

• S (j)
• by linearity of expectation

= 0 R

X (i, j) = min {i, j}

Gambler

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

Gambler

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

Gambler

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

Gambler

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

Gambler

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

Gambler

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

= 10 8 1 210 + 10 9 1 210 + 1 210

Gambler

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

= 10 8 1 210 + 10 9 1 210 + 1 210 = 5.47 10−2