Wavelet setting

4.2.1 Multiresolution analysis

We consider a pair{φ, ψ}offather andmother wavelets defining a multiresolution analysis (MRA), see Cohen [2003]. This is the case for the celebrated Daubechies wavelets. MRA wavelets are widely used in practice, since they allow a fast computation of the discrete wavelet transform (DWT) using the pyramidal algorithm; see for instance Mallat [1998].

We adopt the engineering convention under which large values of the scale indexj corre-spond to coarse scales, that is, the analyzing wavelets are defined as{ψ_j,k, j ∈Z, k∈Z} with ψ_j,k(t) = 2^−j/2ψ(2^−jt−k), j ∈ Z, k ∈ Z. The DWT of X can be interpreted as follows. Using the father wavelet (orscaling function) φand the interpolated functionX defined in (2.30), the wavelet coefficients are defined as

W_j,k ^def= Z

X(t)ψ_j,k(t) dt, j≥1, k∈Z. (4.1) Based on the quadrature mirror filters h and g defined respectively in (2.45) and (2.48), one can compute the wavelet coefficients as follows

A_0,k =X_k, k∈Z, (4.2)

A_j,k = X∞ n=−∞

h[2k−n]A_j−1,n=

↓²(h ⋆ A_j−1,·)

k, k∈Z, j≥1, (4.3) W_j,k =

X∞ n=−∞

g[2k−n]A_j−1,n =

↓²(g ⋆ A_j−1,·)

k, k∈Z, j≥1, (4.4) where {h[k]}k∈Z and {g[k]}k∈Z denote the impulse response of h and g, ⋆ denotes the convolution operator and↓² the down-sampling operator by factor 2. We assume that ψ is compactly supported and denote by M the number of vanishing moments (M ≥ 1).

This corresponds to the following assumption on the filtersh and g.

Assumption 1. The filtershandg have support{−T+ 1, . . . ,0}andg has M vanishing moments, that is,

X0 t=−T+1

g[t]t^ℓ = 0 for all ℓ= 0,1, . . . , M−1. (4.5)

This assumption is supposed to hold throughout the chapter. It follows from the finite support assumption that a finite set of wavelet coefficients can be computed exactly from a finite sampleX₀, . . . , X_n−1. In this case the pyramidal algorithm is applied with k= 0, . . . , n₀−1 in (4.2) and k= 0, . . . , n_j−1 in (4.3) and (4.4), where

n₀ =n and n_j ={⌊(n_j−1−T)/2⌋+ 1}₊, j≥1, (4.6) and⌊x⌋denotes the largest integer at most equal toxand x₊ the non-negative part ofx, max(x,0).

4.2.2 pyramidal algorithm based on approximation coefficients incre-ments

The DWT (4.2–4.4) with input {X_k,0 ≤k < n}, provides{W_j,k, j ≥1,0≤k < n_j} as output. We now wish to provide the same output with

[∆^MX] _k, M ≤k < n.

To this end we define the approximation coefficients increments at successive orders ℓ= 0,1, . . . by A^(ℓ)_j,k = [∆^ℓA_j,·]_k, that isA⁽⁰⁾_j,k =A_j,k and for ℓ≥0,A^(ℓ+1)_j,k =A^(ℓ)_j,k−A^(ℓ)_j,k−1. Because X may not be stationary but becomes stationary after differencing sufficiently many times, it is a key step to derive a pyramidal algorithm in whichA_j,kin (4.2) and (4.3) is replaced byA^(ℓ)_j,k withℓ the large enough. By assumption 1 we may take ℓ=M. This new scheme is stated in Lemma 2.2.1.

4.3 Second order properties in the wavelet domain

4.3.1 Assumptions and notation

LetXbe anM(d) process with memory parameterdand with generalized spectral density f. By Definition 2.1.10 there exists K such that ∆^KX is second order stationary. By possibly increasingKto remove possibly remaining polynomial trends, we further assume that∆^KX is centered. In fact, we will not use in this section that X is anM(d) process but only the two following assumptions.

(H-i) TheK^th order increment process ∆^KX is centered and covariance stationary.

(H-ii) Assumption 1 holds withM ≥K.

Let us now introduce some notation valid all along the chapter. For any scales i and j, we denote by M_i,j ^def= (M_i,j[k, k^′],0≤k≤n_i−1,0≤k^′ ≤n_j−1) the covariance matrix of the wavelet coefficients at scalesiand j, respectively,

M_i,j[k, k^′]^def= Cov(W_i,k, W_j,k^′). (4.7)

Following Moulines et al. [2007b], we pool blocks of 2^u wavelet coefficients at scale j−u within column vectors denoted by

W_j,k(u)^def= [W_j−u,2^u_k, W_j−u,2^u_k+1, . . . , W_j−u,2^u_k+2^u₋₁]^T , (4.8) and define thebetween-scale process at scalej, timek and scale differenceu, as

{[W^T_j,k(u), W_j,k]^T}k∈Z.

Observe thatW_j,k(u) is a 2^u-dimensional vector of wavelet coefficients at scalei=j−u and involves all possible translations of the position index 2^uk by v = 0,1, . . . ,2^u −1.

The indexu in (4.8) denotes the scale differencej−i≥0 between the finest scale iand the coarsest scale j. By convention W_j,k(0) (u = 0) coincides with the scalar W_j,k. By [Moulines et al., 2007b, Corollary 1],{[W_j,k^T (u), W_j,k]^T}k∈Z is then covariance stationary.

In contrast, the process of wavelet coefficient at the finer scale j−u and the approx-imation coefficient at scale j {[W^T_j,k(u), A_j,k]^T}k∈Z may not be stationary. Nevertheless, it follows from (H-i) and (H-ii) above that {A^(M)_0,k , k ∈ Z} is centered and covariance stationary, hence we conclude from Lemma 2.2.1 that{[W^T_j,k(u), A^(M)_j,k ]^T}k∈Z is centered and covariance stationary with second order properties described below using iterative formula. Before that we need some additional notation.

Let us define the covariance functions δ_i,j,γ_i,j andν_i,j by

γ_i,j[k]^def= Cov(W_i,k, A^(M)_j,0 ), δ_i,j[k]^def= Cov(W_i,k, W_j,0), 1≤i≤j , ν_i[k]^def= Cov(A^(M)_i,k , A^(M_i,0⁾), 0≤i .

When i=j, we denoteγ_i,j =γ_i and δ_i,j =δ_i. Using these definitions and the stationary structure in the wavelet coefficients, we obtain that

Mi,j[k, k^′] =δi,j[k−2^j−ik^′], 1≤i≤j .

Similarly we introduce some notation for spectral densities. We denote byD_j,u the cross-spectral density function of the between-scale process{[W^T_j,k(u), W_j,k]^T}k∈Z and byB_j,u the cross-spectral density function of the processn

[W_j,k^T (u), A^(M_j,k⁾]^To

k∈Z. Hence, we have, for all 1≤i=j−u≤j,

[δ_i,j[2^uk], . . . , δ_i,j[2^u(k+ 1)−1]]^T = Cov (W_j,k(u), W_j,0) = Z π

−π

D_j,u(λ) e^iλkdλ (4.9) [γ_i,j[2^uk], . . . , γ_i,j[2^u(k+ 1)−1]]^T = Cov

W_j,k(u), A^(M)_j,0

= Z π

−π

B_j,u(λ) e^iλkdλ . (4.10) By conventionD_j,0 (u= 0) denotes the spectral density of{W_j,k, k∈Z}. Finally we let A_j denote the spectral density function of the process {A^(M)_j,k }k∈Z. Hence, for allj ≥0,

ν_j[k] = Cov

A^(M_j,k⁾, A^(M_j,0⁾

= Z π

−π

A_j(λ) e^iλkdλ . (4.11)

4.3.2 Iterative formula in spectral domain

Observe that A₀ and ν₀ are the spectral density and autocovariance function of ∆^MX, respectively. In particular, if f denotes the generalized spectral density ofX,

A₀(λ) =|1−e^−iλ|^2Mf(λ) and ν₀(τ) = Z π

−π

A₀(λ)e^iλ dλ

are easily derived for second order properties of X. We now wish to obtain the second order properties of the wavelet coefficients. This can be done either in the spectral domain, that is, by computingDj,u for allj ≥1 and 0≤u≤j, or in the time domain, that is by computingδ_i,j for all 1≤i≤j. Other spectral densities A_j and B_j,u, or covariances γ_i,j andνi,j are key extra quantities to achieve this goal. The following results indeed provide simple iterative to compute all these quantities simply starting fromA₀ orν₀.

The first result provides iterative formula for the spectral densities A_j, D_j,u(·), and B_j,u(·),u∈Z, based on the filterseh and eg defined in (2.56) and (2.55). The proof of Proposition 4.3.1 is postponed in Section 4.5

Remark 4.3.1. All the quantities in Proposition 4.3.1 can be successively computed from A₀ as follows. The spectral densities A_j, j ≥ 1, can be computed from A₀ using the iterative formula (4.12). The cross-spectral densitiesD_j,0andB_j,0,j≥1, can be computed from A_j, j ≥ 0, using (4.13) and (4.14). Finally, B_j,u and D_j,u, 1 ≤ j, 0 ≤ u ≤ j−1 can be computed from B_j,0, j≥1 using the iterative formula (4.15) and (4.16).

4.3.3 Iterative formula in temporal domain

The second result provides iterative formula for the covariance functionsν_i,δ_i,j and γ_i,j, 1≤i≤j, based on the filterseh and egdefined in (2.56) and (2.55).

Proposition 4.3.2. The following formula hold for allk∈Z and 1≤i≤j.

ν_j[k] = X

l,l^′∈Z

e

h[l]eh[l^′]ν_j−1

2k−l^′+l

, (4.18)

δ_j[k] = X

l,l^′∈Z

e

g[l]eg[l^′]ν_j−1

2k−l^′+l

, (4.19)

γj[k] = X

l,l^′∈Z

e

g[l]eh[l^′]νj−1

2k−l+l^′

, (4.20)

γ_i,j+1[k] =X

l∈Z

eh[l]γ_i,j[k+ 2^j−il], (4.21) δ_i,j+1[k] =X

l∈Z

e

g[l]γ_i,j[k+ 2^j−il]. (4.22) The proof of Proposition 4.3.2 in postponed in Section 4.5

Remark 4.3.2. All the quantities in Proposition 4.3.2 can be successively computed from ν₀ as follows. The autocovariance functionsν_j, j≥1, can be computed fromν₀ using the iterative formula (4.18). The covariance functions δ_j and γ_j can be computed from ν_j, j≥0, using (4.19) and (4.20). Finally γ_i,j and δ_i,j, 1≤i < j can be computed from γ_j, j≥1 using the iterative formula (4.21) and (4.22). Note also that the double summation and simple summation signs appearing in (4.18-4.22) only involve (T −M)² and T−M non-vanishing terms, since eh and eg have lengths both equal to T −M.