[PPT] - Processes with summable partial autocorrelations ukasz Dbowski PowerPoint Presentation

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Processes with summable partial autocorrelations

Łukasz Dębowski ldebowsk@ipipan.waw.pl

Institute of Computer Science Polish Academy of Sciences

14th European Young Statisticians Meeting, Debrecen 2005

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Introduction Building blocks Summable PACF Applications

Two autocorrelation functions

(Xi)i∈Z — weakly stationary process with Var(Xi) = 1 (ordinary) autocorrelation (ACF): ρn := Corr(Xn; X0) partial autocorrelation (PACF): αn := Corr(Y1:n−1

n

; Y1:n−1 ) Φn:m

k

← best linear predictor of Xk given Xn, ..., Xm Yn:m

k

= Xk − Φn:m

k

← innovation Schur 1917, Durbin 1960, Ramsey 1974: there exists bijection (ρ1, ..., ρn) ← → (α1, ..., αn) (ρk)k∈Z strictly positive definite ⇐ ⇒ |αm| < 1 for m ∈ N

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Introduction Building blocks Summable PACF Applications

Main result

Let ∞

k=1 |αk| < ∞ with |αk| < 1. Then:

(i) There exist representations Zl = ∞

k=0 πkXl−k and

Xl = ∞

k=0 ψkZl−k, where (Zi)i∈Z is white noise.

(ii) For n ∈ N we have

n

k=−n

|ρk|2 ≤

n

k=1

1 + |αk| 1 − |αk| 2 . — HOLDS FOR ALL STATIONARY PROCESSES (iii) If αk ∈ R for all k then

∞

k=−∞

(±1)kρk =

∞

k=1

1 + (±1)kαk 1 − (±1)kαk .

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Introduction Building blocks Summable PACF Applications

The derivation of the theorem

partial autocorrelation ↓ finite innovations ↓ autoregressive representation ↓ moving average representation ↓ autocorrelation

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Introduction Building blocks Summable PACF Applications

1

Introduction

2

Building blocks

3

Summable PACF

4

Applications

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Introduction Building blocks Summable PACF Applications

Finite innovations

Φm:n

k

— best linear predictor of Xk given Xm, Xm+1, ..., Xn Ym:n

k

:= Xk − Φm:n

k

, Zn

p := Yn+1:p−1

p

˛ ˛ ˛ ˛ ˛ ˛Yn+1:p−1

p

˛ ˛ ˛ ˛ ˛ ˛ — innovations

Yp−n:p−1

p

= −

n

k=0

φnkXp−k, Zp−n−1

p

=

n

k=0

πnkXp−k, Xp =

n

k=0

ψnkZp−n−1

p−k

. By Durbin-Levinson recursion, πnk = −φnk/ n

i=1

1 − |αi|2,

φ00 = −1, φnk = φn−1,k − α∗

nφ∗ n−1,n−k,

k

j=0 πnjψn−j,k−j = [

[ k = 0 ] ] .

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Introduction Building blocks Summable PACF Applications

Wold decomposition

Φ−∞:p−1

p

— best linear predictor of Xp given (Xn)n<p Process is called nondeterministic if Xp − Φ−∞:p−1

p

≡ 0. Theorem (Wold 1938, Pourahmadi 2001) If (Xi)i∈Z is nondeterministic then limn→∞ Z−n

l

= Zl, limn→∞ ψnk = ψk, and limn→∞ πnk = πk, where ψk := Cov(Zl−k, Xl), and πk are given by condition k

j=0 πjψk−j = [

[ k = 0 ] ] . Furthermore, Xp = ∞

k=0 ψkZp−k + Vp,

(Vi)i∈Z is deterministic, uncorrelated with white noise (Zi)i∈Z.

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Introduction Building blocks Summable PACF Applications

Moving average and autoregressive representations

MA(∞): Xp =

∞

k=0

ψkZp−k ⇔ lim

n→∞ n

k=0

(ψnk − ψk)Zp−k = 0 AR(∞): Zp =

∞

k=0

πkXp−k ⇔ lim

n→∞ n

k=0

(πnk − πk)Xp−k = 0 Theorem lim

n→∞ sup m>n m

k=0

|ψmk − ψnk|2 = 0 ⇐ ⇒ MA(∞) exists. lim

n→∞ sup m>n m

k=0

|πmk − πnk| = 0 = ⇒ AR(∞) exists. If ∞

k=0 |ψk| < ∞ and ∞ k=0 |πk| < ∞ then

MA(∞) exists ⇐ ⇒ AR(∞) exists.

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Introduction Building blocks Summable PACF Applications

Bounds for finite innovations

φn(z) := n

k=0 φnkzk

Theorem Let φnj := 0 for j > n. We have:

m

j=0

|φmj − φnj| ≤

m

k=1

(1 + |αk|) −

n

k=1

(1 + |αk|) if m ≥ n, φn(±1) =

n

k=1
1 − (±1)kαk
if αk ∈ R for k ≤ n,

|φn(z)| ∈ n

k=1

(1 − |αk|) ,

n

k=1

(1 + |αk|)

if |z| ≤ 1.

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Introduction Building blocks Summable PACF Applications

1

Introduction

2

Building blocks

3

Summable PACF

4

Applications

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Introduction Building blocks Summable PACF Applications

What happens for summable PACF? (I)

πn(z) := n

k=0 πnkzk

π(z) := n

k=0 πkzk

Lemma If ∞

k=1 |αk| < ∞ with |αk| < 1 then

lim

n→∞(πnk)k∈N = (πk)k∈N in ℓ1,

lim

n→∞ sup |z|≤1

|πn(z) − π(z)| = 0, ∞

k=0 |πk|, |π(z)| ∈

∞

k=1

1−|αk|

1+|αk|, ∞ k=1

1+|αk|

1−|αk|

,

π(±1) = ∞

k=1

1−(±1)kαk

1+(±1)kαk

so AR(∞) representation exists.

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Introduction Building blocks Summable PACF Applications

What happens for summable PACF? (III)

For summable PACF, we have ψ(z) = 1/π(z) for |z| ≤ 1 and ∞

k=0 |πk| < ∞ but not ∞ k=0 |ψk| < ∞ in general.

By Wold decomposition, MA(∞) representation exists iff ρk = 1 2π π

−π

|ψ(eiω)|2eikωdω for k ∈ N. (1) We have two cases: For AR(n) processes, π(z) = πn(z) = 0 for |z| < r, r > 1, so ∞

k=0 |ψk| < ∞ and MA(∞) representation exists.

Hence (1) holds. For general summable PACF, we have limn→∞ sup|z|≤1 |1/πn(z) − ψ(z)| = 0 so (1) holds. Hence MA(∞) representation exists.

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Introduction Building blocks Summable PACF Applications

What happens for summable PACF? (II)

We have ρk =

1 2π

π

−π |ψ(eiω)|2eikωdω, where ψ(z) = 1/π(z).

By Parseval identity,

∞

k=−∞

|ρk|2 = 1 2π π

−π

|ψ(eiω)|4dω ≤

∞

k=1

1 + |αk| 1 − |αk| 2 . By Riesz-Fischer theorem, if αk ∈ R for k ∈ N then

∞

k=−∞

(±1)kρk = |ψ(±1)|2 =

∞

k=1

1 + (±1)kαk 1 − (±1)kαk .

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Introduction Building blocks Summable PACF Applications

1

Introduction

2

Building blocks

3

Summable PACF

4

Applications

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Introduction Building blocks Summable PACF Applications

Classification of Gaussian processes

(assuming |αk| < 1 in conditions for αk) ergodic limk→∞ ρ(k) = 0 (mixing) ∞

k=1 |ρ(k)|2 < ∞

AR(∞) exists ∞

k=1 |αk|2 < ∞ (nondeterministic)

MA(∞) exists (regular) ∞

k=1 |αk| < ∞

completely nondeterministic ∞

k=1 k|αk|2 < ∞ (finitary)

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Introduction Building blocks Summable PACF Applications

Counterintuitive corollaries

∞

k=−∞

ρk =

∞

k=1

1 + αk 1 − αk (a) αm ≥ 0, |αm| ր = ⇒ ∞

k=−∞ ρk ր

(b) αm ≤ 0, |αm| ր = ⇒ ∞

k=−∞ ρk ց

(c) αm > 0, αk ≥ a > 0 for n distinct k = ⇒ ρk > 0 for at least

1+a

1−a

n distinct k

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Introduction Building blocks Summable PACF Applications

An illustration of property (c)

Let αn =

0.5

if n ≤ N if n > N. Then:

0.2 0.4 0.6 0.8 1 1.2 1 10 100 1000 |ρk| k N=1 N=3 N=5 N=7 N=9

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Introduction Building blocks Summable PACF Applications