Processes with summable partial autocorrelations ukasz Dbowski - PowerPoint PPT Presentation

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

Introduction Building blocks Summable PACF Applications Two autocorrelation functions ( X i ) i ∈ Z — weakly stationary process with Var ( X i ) = 1 (ordinary) autocorrelation (ACF): ρ n := Corr ( X n ; X 0 ) partial autocorrelation (PACF): ; Y 1 : n − 1 α n := Corr ( Y 1 : n − 1 ) n 0 Φ n : m ← best linear predictor of X k given X n , ..., X m k Y n : m = X k − Φ n : m ← innovation k k Schur 1917, Durbin 1960, Ramsey 1974 : there exists bijection ( ρ 1 , ..., ρ n ) ← → ( α 1 , ..., α n ) ( ρ k ) k ∈ Z strictly positive definite ⇐ ⇒ | α m | < 1 for m ∈ N

Introduction Building blocks Summable PACF Applications Main result Let � ∞ k = 1 | α k | < ∞ with | α k | < 1 . Then: (i) There exist representations Z l = � ∞ k = 0 π k X l − k and X l = � ∞ k = 0 ψ k Z l − k , where ( Z i ) i ∈ Z is white noise. (ii) For n ∈ N we have n n � 2 � 1 + | α k | | ρ k | 2 ≤ � � . 1 − | α k | k = − n k = 1 — HOLDS FOR ALL STATIONARY PROCESSES (iii) If α k ∈ R for all k then ∞ ∞ 1 + ( ± 1 ) k α k � ( ± 1 ) k ρ k = � . 1 − ( ± 1 ) k α k k = −∞ k = 1

Introduction Building blocks Summable PACF Applications The derivation of the theorem partial autocorrelation ↓ finite innovations ↓ autoregressive representation ↓ moving average representation ↓ autocorrelation

Introduction Building blocks Summable PACF Applications Introduction 1 Building blocks 2 Summable PACF 3 Applications 4

Introduction Building blocks Summable PACF Applications Finite innovations Φ m : n — best linear predictor of X k given X m , X m + 1 , ..., X n k Y n + 1 : p − 1 Y m : n := X k − Φ m : n , Z n p p := ˛ — innovations k k ˛ ˛ ˛ Y n + 1 : p − 1 ˛ ˛ ˛ ˛ ˛ ˛ p ˛ ˛ n n Y p − n : p − 1 � Z p − n − 1 � = − φ nk X p − k , = π nk X p − k , p p k = 0 k = 0 n ψ nk Z p − n − 1 � X p = . p − k k = 0 By Durbin-Levinson recursion, π nk = − φ nk / � n � 1 − | α i | 2 , i = 1 φ nk = φ n − 1 , k − α ∗ n φ ∗ φ 00 = − 1 , n − 1 , n − k , � k j = 0 π nj ψ n − j , k − j = [ [ k = 0 ] ] .

Introduction Building blocks Summable PACF Applications Wold decomposition Φ −∞ : p − 1 — best linear predictor of X p given ( X n ) n < p p Process is called nondeterministic if X p − Φ −∞ : p − 1 �≡ 0 . p Theorem (Wold 1938, Pourahmadi 2001) If ( X i ) i ∈ Z is nondeterministic then lim n →∞ Z − n = Z l , l lim n →∞ ψ nk = ψ k , and lim n →∞ π nk = π k , where ψ k := Cov ( Z l − k , X l ) , and π k are given by condition � k j = 0 π j ψ k − j = [ [ k = 0 ] ] . Furthermore, X p = � ∞ k = 0 ψ k Z p − k + V p , ( V i ) i ∈ Z is deterministic, uncorrelated with white noise ( Z i ) i ∈ Z .

Introduction Building blocks Summable PACF Applications Moving average and autoregressive representations ∞ n � � MA( ∞ ): X p = ψ k Z p − k ⇔ lim ( ψ nk − ψ k ) Z p − k = 0 n →∞ k = 0 k = 0 ∞ n � � AR( ∞ ): Z p = π k X p − k ⇔ lim ( π nk − π k ) X p − k = 0 n →∞ k = 0 k = 0 Theorem m | ψ mk − ψ nk | 2 = 0 ⇐ � ⇒ MA( ∞ ) exists. n →∞ sup lim m > n k = 0 m � n →∞ sup lim | π mk − π nk | = 0 = ⇒ AR( ∞ ) exists. m > n k = 0 If � ∞ k = 0 | ψ k | < ∞ and � ∞ k = 0 | π k | < ∞ then MA( ∞ ) exists ⇐ ⇒ AR( ∞ ) exists.

Introduction Building blocks Summable PACF Applications Bounds for finite innovations φ n ( z ) := � n k = 0 φ nk z k Theorem Let φ nj := 0 for j > n . We have: m m n � � � | φ mj − φ nj | ≤ ( 1 + | α k | ) − ( 1 + | α k | ) if m ≥ n , j = 0 k = 1 k = 1 n � � � 1 − ( ± 1 ) k α k φ n ( ± 1 ) = if α k ∈ R for k ≤ n , k = 1 � n n � � � | φ n ( z ) | ∈ ( 1 − | α k | ) , ( 1 + | α k | ) if | z | ≤ 1 . k = 1 k = 1

Introduction Building blocks Summable PACF Applications Introduction 1 Building blocks 2 Summable PACF 3 Applications 4

Introduction Building blocks Summable PACF Applications What happens for summable PACF? (I) π n ( z ) := � n π ( z ) := � n k = 0 π nk z k k = 0 π k z k Lemma If � ∞ k = 1 | α k | < ∞ with | α k | < 1 then n →∞ ( π nk ) k ∈ N = ( π k ) k ∈ N in ℓ 1 , lim n →∞ sup lim | π n ( z ) − π ( z ) | = 0 , | z |≤ 1 �� ∞ � � � � ∞ 1 −| α k | 1 + | α k | , � ∞ 1 + | α k | k = 0 | π k | , | π ( z ) | ∈ , k = 1 k = 1 1 −| α k | � 1 − ( ± 1 ) k α k π ( ± 1 ) = � ∞ k = 1 1 +( ± 1 ) k α k so AR( ∞ ) representation exists.

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 | π k | < ∞ but not � ∞ k = 0 | ψ k | < ∞ in general. By Wold decomposition, MA( ∞ ) representation exists iff � π ρ k = 1 | ψ ( e i ω ) | 2 e ik ω d ω for k ∈ N . (1) 2 π − π We have two cases: For AR ( n ) processes, π ( z ) = π n ( z ) � = 0 for | z | < r , r > 1 , so � ∞ k = 0 | ψ k | < ∞ and MA( ∞ ) representation exists. Hence (1) holds. For general summable PACF, we have lim n →∞ sup | z |≤ 1 | 1 /π n ( z ) − ψ ( z ) | = 0 so (1) holds. Hence MA( ∞ ) representation exists.

Introduction Building blocks Summable PACF Applications What happens for summable PACF? (II) � π 1 − π | ψ ( e i ω ) | 2 e ik ω d ω , where ψ ( z ) = 1 /π ( z ) . We have ρ k = 2 π By Parseval identity, ∞ � π ∞ � 2 � 1 + | α k | | ρ k | 2 = 1 � | ψ ( e i ω ) | 4 d ω ≤ � . 2 π 1 − | α k | − π k = −∞ k = 1 By Riesz-Fischer theorem, if α k ∈ R for k ∈ N then ∞ ∞ 1 + ( ± 1 ) k α k ( ± 1 ) k ρ k = | ψ ( ± 1 ) | 2 = � � . 1 − ( ± 1 ) k α k k = −∞ k = 1

Introduction Building blocks Summable PACF Applications Introduction 1 Building blocks 2 Summable PACF 3 Applications 4

Introduction Building blocks Summable PACF Applications Classification of Gaussian processes (assuming | α k | < 1 in conditions for α k ) ergodic lim k →∞ ρ ( k ) = 0 (mixing) k = 1 | ρ ( k ) | 2 < ∞ k = 1 | α k | 2 < ∞ (nondeterministic) � ∞ � ∞ AR ( ∞ ) exists MA ( ∞ ) exists (regular) � ∞ k = 1 | α k | < ∞ completely nondeterministic k = 1 k | α k | 2 < ∞ (finitary) � ∞

Introduction Building blocks Summable PACF Applications Counterintuitive corollaries ∞ ∞ 1 + α k � � ρ k = 1 − α k k = −∞ k = 1 ⇒ � ∞ (a) α m ≥ 0 , | α m | ր = k = −∞ ρ k ր ⇒ � ∞ (b) α m ≤ 0 , | α m | ր = k = −∞ ρ k ց (c) α m > 0 , α k ≥ a > 0 for n distinct k � n � 1 + a = ⇒ ρ k > 0 for at least distinct k 1 − a

Introduction Building blocks Summable PACF Applications An illustration of property (c) � 0 . 5 if n ≤ N Let α n = if n > N . Then: 0 1.2 N=1 N=3 N=5 1 N=7 N=9 0.8 | ρ k | 0.6 0.4 0.2 0 1 10 100 1000 k

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