Tracking Predictable Drifting Parameters Paulo Serra of a Time - PowerPoint PPT Presentation

Tracking Predictable Drifting Parameters of a Time Series Tracking Predictable Drifting Parameters Paulo Serra of a Time Series The Model Joint work with Eduard Belitser Results Examples of Parameter Variation Paulo Serra Construction of Gain Functions Department of Mathematics and Computer Science Application: Eindhoven University of Technology Quantile Tracking 12th September, 2013 References 1 / 22

Model Description Tracking Assume we observed X n = ( X 0 , X 1 , . . . , X n ) following: Predictable Drifting Parameters of a Time X 0 ∼ P 0 , X k | X k − 1 ∼ P k ( ·| X k − 1 ) , k ∈ N . Series Paulo Serra • X k takes values in X ⊆ R l , l ∈ N (i.e. P ( X k ∈ X ) = 1 ). The Model Results • The distribution of X n , n ∈ N 0 , is given by Examples of n Parameter P ( n ) = P ( n ) ( x n ) = � P k ( x k | x k − 1 ) , x k ∈ X k +1 , Variation Construction k =0 of Gain Functions where P 0 ( x 0 | x − 1 ) should be understood as P 0 ( x 0 ) . Application: Quantile Tracking • At time n ∈ N 0 , the underlying growing statistical model References is P ( n ) = � � n � k =0 P k ( x k | x k − 1 ) : P k ( ·| x k − 1 ) ∈ P k . 2 / 22

Objective Tracking Predictable Drifting • Consider a filtration {F k } ∞ k = − 1 such that F k ⊆ σ ( X k ) . Parameters of a Time Series Paulo Serra • Consider a sequence of appropriately measurable operators A k , which map measures P k ( ·| x k − 1 ) ∈ P k The Model Results x k − 1 ∈ X k − 1 , A k ( P k ( ·| x k − 1 )) = θ k ( x k − 1 ) , Examples of Parameter Variation with θ k ( x k − 1 ) predictable with respect to F k . Construction of Gain Functions Application: Quantile Objective Tracking We would like to track θ k = θ k ( X k − 1 ) . References 3 / 22

Definition of the Algorithm Tracking Predictable Drifting Assumption (A0) Parameters of a Time The drifting parameter satisfies P ( θ k ( X k − 1 ) ∈ Θ) for some Θ Series such that sup θ ∈ Θ � θ � 2 ≤ C Θ . Paulo Serra The Model Results The following algorithm constitutes our tracking sequence. Examples of Parameter Variation Tracking Algorithm Construction Define ˆ θ k +1 = ˆ θ k + γ k G k (ˆ of Gain θ k , X k ) , k ∈ N 0 where 0 ≤ γ k ≤ Γ Functions and arbitrary F − 1 -measurable ˆ θ 0 ∈ Θ ⊂ R d . Application: Quantile Tracking References The functions G k (ˆ θ k , X k ) are called gain vectors. 4 / 22

Assumptions on the Gain Function Tracking Predictable Assumption (A1) Drifting Parameters � � For all k ∈ N 0 , constants λ 1 , λ 2 and θ k = A k P k ( ·| x k − 1 ) , of a Time Series g k (ˆ θ k , θ k ) = g k (ˆ G k (ˆ Paulo Serra � � θ k , θ k | X k − 1 ) = E θ k , X k ) |F k − 1 , The Model exists; for a F k − 1 -measurable symmetric PD matrix M k , a.s. Results g k (ˆ θ k , θ k | X k − 1 )= − M k (ˆ Examples of θ k − θ k ) , Parameter Variation 0 < λ 1 ≤ E [ λ (1) ( M k ) |F k − 2 ] ≤ λ ( d ) ( M k ) ≤ λ 2 < ∞ . Construction of Gain Functions Application: Assumption (A2) Quantile Tracking There exists a constant C g > 0 such that References θ k , θ k | X k − 1 ) � 2 ≤ C g , E � G k (ˆ θ k , X k ) − g k (ˆ k ∈ N 0 . 5 / 22

Main Results L 1 risk bound Tracking Predictable Theorem (Bound on L 1 risk) Drifting Parameters Let (A0) – (A2) hold and δ k = δ k ( X k − 1 ) = ˆ of a Time θ k − θ k , k ∈ N 0 . Series Then for any k 0 , k ∈ N 0 and sequence { γ k , k ∈ N 0 } (satisfying Paulo Serra the conditions of the previous lemma) such that γ i λ 2 ≤ 1 , The Model i ∈ { k 0 , . . . , k } , the following relation holds: Results Examples of k k Parameter − λ 1 � 1 / 2 � � � � � γ 2 Variation E � δ k +1 � ≤ C 1 exp γ i + C 2 + i 2 Construction i = k 0 i = k 0 of Gain Functions + C 3 max k 0 ≤ i ≤ k E � θ i +1 − θ k 0 � , k 0 ≤ k, Application: Quantile Tracking √ C Θ + C Θ ) 1 / 2 , C 2 = C 1 / 2 2( ¯ where C 1 = (1 + λ 2 /λ 1 ) , References g C 3 = (1 + λ 2 /λ 1 ) . 6 / 22

Stronger Assumptions on the Gain Function Tracking Predictable Assumption (A1) Drifting Parameters g k (ˆ θ k , θ k | X k − 1 ) = − M k (ˆ of a Time θ k − θ k ) . Series Paulo Serra 0 < λ 1 ≤ E [ λ (1) ( M k ) |F k − 2 ] ≤ λ ( d ) ( M k ) ≤ λ 2 < ∞ , (a.s.) The Model ↓ Results 0 < λ 1 ≤ λ (1) ( M k ) ≤ λ ( d ) ( M k ) ≤ λ 2 < ∞ , (a.s.) Examples of Parameter Variation Construction Assumption (A2) of Gain Functions θ k , θ k | X k − 1 ) � 2 ≤ C g , E � G k (ˆ θ k , X k ) − g k (ˆ Application: k ∈ N 0 . Quantile Tracking ↓ References θ k , X k ) � 2 ≤ C g , � G k (ˆ k ∈ N 0 , (a.s.) 7 / 22

Main Results L p risk bound Tracking Predictable Theorem ( L p risk bound) Drifting Parameters Suppose that the conditions of the previous theorem are of a Time Series fulfilled. If, in addition (to assumption (A1)), λ (1) ( M i ) ≥ λ 1 Paulo Serra and � G i (ˆ θ i , X i ) � ≤ C g (instead of (A2)) a.s. for all The Model i = k 0 . . . , k , then for any p ≥ 1 Results Examples of k k � p/ 2 Parameter � � � E � δ k +1 � p p ≤ C ′ � + C ′ � γ 2 1 exp − pλ 1 γ i Variation 2 i Construction i = k 0 i = k 0 of Gain + C ′ k 0 ≤ i ≤ k E � θ i +1 − θ k 0 � p Functions 3 max p , k 0 ≤ k, Application: Quantile Tracking 1 = 3 p − 1 K p p E � δ k 0 � p 2 = 3 p − 1 2 p dB p C p � p , for C ′ p , C ′ 1 + K 2 � p λ 2 /λ 1 g References � p , and K p and B p are constants. C ′ 3 = 3 p − 1 � 1 + K 2 p λ 2 /λ 1 8 / 22

Lipschitz Signal: θ n k = ϑ ( k/n ) , ϑ ( · ) ∈ L β k k Tracking − λ 1 � 1 / 2 � � � � � Predictable γ 2 E � δ k +1 � � exp γ i + + max k 0 ≤ i ≤ k E � θ i +1 − θ k 0 � , Drifting i 2 Parameters i = k 0 i = k 0 of a Time Series k k � p/ 2 � � � � � Paulo Serra E � δ k +1 � p γ 2 k 0 ≤ i ≤ k E � θ i +1 − θ k 0 � p p � exp − pλ 1 γ i + + max p , i i = k 0 i = k 0 The Model Results X n X n k | X n k ( ·| X n 0 ∼ P θ n 0 , k − 1 ∼ P θ n k − 1 ) , k ≤ n ∈ N , Examples of Parameter Variation • Assume that θ n k = ϑ ( k/n ) , with ϑ ( · ) ∈ L β , k = 1 , . . . , n . Construction of Gain • For 0 < β ≤ 1 , γ k ≡ C γ (log n ) (2 β − 1) / (2 β +1) n − 2 β/ (2 β +1) , Functions k 0 = K n = (log n ) 2 / (2 β +1) n 2 β/ (2 β +1) we get Application: Quantile Tracking β β E n − � n − � p 2 β +1 � δ k � 2 β +1 � δ k � p References sup ≤ C and sup E ≤ C. 2 β 2 β ϑ ∈L β ϑ ∈L β (log n ) (log n ) 2 β +1 2 β +1 k ≥ K n k ≥ K n 9 / 22

Example 1 Signal + noise setting Tracking Predictable The model is: Drifting Parameters X k = θ k + ξ k , k ∈ N 0 , of a Time Series where { θ k } k ∈ N 0 is a predictable process ( θ k = θ k ( X k − 1 ) ), Paulo Serra { ξ k } k ∈ N 0 is a martingale difference noise with respect to the The Model filtration {F k } k ∈ N − 1 . Results Examples of We can simply take the following gain function Parameter Variation G k (ˆ θ k , X k ) = − (ˆ Construction θ k − X k ) , k ∈ N 0 , of Gain Functions Application: since Quantile Tracking g k (ˆ θ k , θ k | X k − 1 ) = E [ G k (ˆ θ k , X k ) | X k − 1 ] = − (ˆ θ k − θ k ) , k ∈ N 0 . References 10 / 22

General Gain Construction Tracking Predictable Drifting Parameters • Assume each measure in P k = { P θ ( x | X k − 1 ) , θ ∈ Θ ⊂ R d } of a Time Series has a density p θ ( x | X k − 1 ) , θ ∈ Θ , with respect to some Paulo Serra σ -finite dominating measure. The Model Results • Assume also that there is a common support X for these Examples of Parameter densities, and that for any x ∈ X , x k − 1 ∈ X k − 1 , and Variation θ ∈ Θ , the partial derivatives ∂p θ ( x | x k − 1 ) /∂θ i , Construction of Gain i = 1 , . . . , d , exist and are finite. Functions Application: Quantile Tracking • Let ∇ θ log p θ ( x | x k − 1 ) be a gradient. References 11 / 22

General Gain Construction Tracking Predictable Drifting Parameters We can use the gains of a Time Series G k ( ϑ, x k ) = ∇ ϑ log p ϑ ( x k | x k − 1 ) . Paulo Serra The Model Results Examples of If expectation and differentiation can be interchanged, then Parameter Variation � � � g k ( ϑ, θ | X k − 1 )= E θ ∇ ϑ log p ϑ ( X k | X k − 1 ) Construction � X k − 1 of Gain Functions � � � = ∇ ϑ E θ log p ϑ ( X k | X k − 1 ) � X k − 1 Application: � � = −∇ ϑ KL P θ ( ·| X k − 1 ) , P ϑ ( ·| X k − 1 ) . Quantile Tracking References 12 / 22