Statistics for stochastic processes

STATISTICS FOR STOCHASTIC PROCESSES

• material on skuola.net computer
• 26 hours of classrooms + 6 hours of lab (R) + Rimotta lectures
• Thursday 9.00 - 10.30

Exam

• analysis of a data set
• essay on one of the documents by Prof. Rimotti right after the first part (30 min)
• oral examination (one question) a couple of days later

After the written part: ottimo - distinto - buono - sufficiente

books + web page

install packages "astsa" from the il paradiso dello studente page web, package "TSA"

12 Marzo 2018

What is a TIME SERIES?

Statistics are computed on observations (data): x1, ... , xm

Simple model: realizations of iid random variables X1, ... , Xn

There are correlations among data, particularly if time is involved in ordering usually through time.

Data are observations at discrete times of stochastic dynamical systems usually described with a stochastic process: discrete time series

Notation: X_t t t^t z

We can't assume longer independence vs time series analysis

• Goals: description of data (plots, descriptive statistics, ...)
• looking at trend, seasonality, random terms

Time series (1 & 2): deterministic function (exponential) + stochastic process whose variance grows with time

Correlations as scatter diagram with log-log plot

Global temperature + trend to seasonality

speech: periodic time series

Prof Rimbot: spectral analysis to understand how to compute periodicity Fourier analysis

We're going to study linear time series sampled with time

Difference between periodicity and seasonality

Another tool is seasonal and trend decomposition, the command in R is decompose

seasonal + trend = trend loses decomposition

Remainder is a stochastic error

additive (linear) model: adding seasonal,

trend and remainder gives up the original data plot

• We will focus on finding a stochastic model for the remainder term, for example with descriptive statistics (tests for normality, qq-plots, histograms)

We'll work on data only, time series it's easier

mean and variance are constant over time

Non-stationary times series could be transformed

so we have that

E(X_t) = μt + E(X₀) + ∑_j=1^t E(W_j) = μt

and

δ_t,ρ = E[(X_t - μt)(X₀ - μρ)] =

 = E[(μt + X₀ + ∑_j=1^t W_j - μt)(μρ + X₀ + ∑_j=1^ρ W_j - μρ)]

 = E [ (X₀ + ∑_j=1^t W_j) ( X₀ + ∑_j=1^ρ W_j) ]

 = E(X₀²) + E(X₀) E(∑_j=1^ρ W_j) + E(∑_j=1^t W_j)E(X₀)

  + ∑_j=1^t ∑_k=1^ρ E(W_jW_k) = σ² min{ρ,t} a Lt

and also we have that

• E(W_jW_k) = { E(W_j)E(W_k) = 0   j ≠ k }
• { E(W_j²) = σ²        j = k }

and if ρ ≤ t, there are ρ pairs such that j = k

while if t < ρ, there are t pairs such that j = k

∞            δ_t,ρ = σ² min{ρ,t} a Lt

ρ_t,ρ =   δ_t,ρ = { min{ρ,t} a t σ² /t             if ρ ≤ t

                σ² ρ a Lt √ρσ                         if t < a

EXAMPLE

Let us consider a periodic signal

X_t = R sin(2πWt + φ) + Wt   with   t ∈ ℝ,

  ———————————————

W ∼ Θ, φ ∈ ℝ and

{Wt, a ≈ N(0,σ²) and iid)

In the case R = , 2πW = 1 and φ = 0

and in this case we have

X_t = sin t + Wt

R is the AMPLITUDE, φ is the PHASE/SHIFT

W is the FREQUENCY     ρ = √a = ŵ

                  2πW = Ŵ

—φ/2πW is the shift with respect to the first

   CROSSING

{X_t} is stationary

Kolmogorov Theorem

{X_t} a stochastic process (X_t₁, ..., X_tm)

• F_{Xt₁, ..., Xtm}

= (x₁, ..., xm) =

P(X_t₁ ≤ x_i, ..., X_tm ≤ x_m)

if X_i → ∞ I get that

F_{Xt₂, ..., Xtm}

(x₂, ..., xm) = P(X_t₂ ≤ x₂, ..., X_tm ≤ x_m)

EXERCISE Check if δ(h) = Cov(h) is a ACVF

{X_t} is STRICTLY STATIONARY if

(X_t₁, ..., X_tm) = (X_t₁₊ₙ, ..., X_tm₊ₙ)

in distribution

∀ m ∈ ℕ, h ∈ ℤ, ∀ t₁, ..., t_m ∈ ℤ

This means that the stochastic behaviour of {X_t}

over (t₁, t_n) is the same as (0, nth)

Theorem

If {X_t} is strictly stationary

and {X_t} ∈ ℒ² ⇒ {X_t} is (weak) stationary

Proof

Note that for m = 1 ⇒ X_t₁ = X_t₁₊ₙ

∀h ∈ ℤ ∀t₁ ∈ ℤ

if E(X²_t) < ∞ then we have E(X_t) = E(X_t₁₊ₙ)

and Var(X_t) = Var(X_t₁₊ₙ)

We have Cov(X_t - X₀) = Cov(X_t₁₊ₙ, X_₀₊ₙ)

since (X_t - X₀) = l (X_t₁₊ₙ, X_₀₊ₙ) ∀t, ℓ ∈ ℤ

EXAMPLE 11

Consider X_t = μ + γ_t

a random walk = δ + μ_t + w_t

then ∇X_t = X_t - X_t-1 = μ_t - μ_t-1 + (γ_t - γ_t-1)

= δ + ∇γ_t + w_t

stationery

The difference operator at LAG d is

∇^dX_t = X_t - X_t-d = (1 - B^d)X_t

while we have ∇^dX_t = (1 - B)^d X_t

EXAMPLE 12

of X_t = m_t + a_t + γ_t with a_t such that a_t = a_t-d (a is the period), then

∇_dX_t = X_t - X_t-d =

= m_t + a_t + γ_t - m_t-d + a_t-d - γ_t-d =

= (m_t - m_t-d) + (γ_t - γ_t-d)

EXAMPLE 13

TREND AND SEASONAL COMPONENTS

Assume X_t = m_t + a_t + γ_t with

• E(γ_t) = 0
• m_t = a_t + b (Linear trend)
• a_t = a_t+d (a is the period) such that
• d = 2q + 1

^{d = q + 1}

∑_j=1^qa + j = 0

Consider Z_t = 1 / 1 + 2q (X_t-q + X_t-q+1 + .... + X_t+q-1 + X_t+q)

a step backwards

Let's plug μ in Z_t and we get

Z_t = 1 / 1 + 2q Σ^q_j=q m_t+j + Σ^q_j=q a_t+j = 0

(from 1 hypothesis)