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Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-MEDINA

University of Granada

III International Workshop on Advances in Functional Data Analysis 23–24, Mayo 2019

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Spectral analysis of high–dimensional multivariate stationary log–Gaussian Cox processes

Stationary log–Gaussian Cox processes in ℓ2–spaces The distribution is completely characterized by the functional intensity and the pair correlation operator (suitable class for parameter estimation) In this stationary case, under suitable conditions, estimators, based on the periodogram operator, can be constructed The formulation in the infinite–dimensional setting can be achieved in a natural way (Bosq & Ruiz–Medina, 2014) Challenges: The introduction of flexible models for clustering, and point pattern analysis, in time and/or space, as well as referred to elevation and/or depth, among others

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Functional Data Analysis (FDA) techniques applied to point process estimation

An underdeveloped research area FDA techniques are well suited to estimate summary statistics in point pattern analysis, which are functional in nature (see Illian et al., 2008, pp. 271–271) Particularly, we refer to point pattern classification, based on second–order statistics, from FDA methodologies (see, e.g., Illian et al., pp. 135–150 in Baddeley et al., 2006) Wu, M¨ uller and Zhang (2013) apply FDA techniques for the estimation of the covariance structure of the random densities, under unknown intensity function, generating the observed event times

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Statistical spectral analysis on the second-order structure of stationary sequences of functional data

Spectral analysis of stationary processes in functions spaces Panaretos and Tavakoli (2013) introduce the elements, and derive several results to construct the basic building blocks, in the spectral analysis of stationary temporal correlated functional data sequences. The main ingredients were: The spectral density operator The functional Discrete Fourier Transform (fDFT) The derivation of the asymptotic law of fDFT The periodogram operator Spectral density operator estimators, based on smoothed versions of the periodogram kernel The derivation of mean–square convergence to zero of the pointwise and integrated mean-square error, in the Hilbert–Schmidt operator norm The asymptotic law of these estimators

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Asymptotic analysis in the frequency domain

The points addressed in the present work We consider the stationary Gaussian process case in Hilbert spaces: Infinite–dimensional weighted chi-squared distributions, related to Fredholm determinant, arise in the study of Tightness A non-central limit result for the periodogram operator Strong consistency of the periodogram operator in the Hilbert-Schmidt norm Rate of almost surely (a.s.) convergence Practice: Suitable truncation order according to the separation of the eigenvalues of the trace spectral density operator, and the sample size

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Asymptotic normality of fDFT (Panaretos and Tavakoli, 2013)

Basic Definitions Let {Xt}t∈Z be a strictly stationary functional time series with values in H = L2 ([0, 1], R) . E [Xt] = µ ∈ L2 ([0, 1], R) , t ∈ Z E[X02

L2([0,1],R)] < ∞ implies rt = L2 E [(Xs − µ) ⊗ (Xs+t − µ)] , t, s ∈ Z

Pointwise definition of rt under continuity in the mean-square sense of Xt Autocovariance operator at lag t Rt(h)(τ) = 1 rt(τ, σ)h(σ)dσ = Cov

• X0, hL2([0,1],R) , Xt(τ)
• , τ ∈ [0, 1]

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Asymptotic normality of fDFT

Spectral density and periodogram operators Assumption I(p)

t∈Z rtp < ∞, p = 2, or ∞; then, for any ω ∈ R, the

following series converges in · p fω(·, ·) = 1 2π

• t∈Z

rt (·, ·) exp (−iωt) fω(·, ·) is called the spectral density kernel at frequency ω fω(·, ·) is uniformly bounded and continuous in ω with respect to · p The spectral density operator Fω(h)(τ) = 1

0 fω(τ, σ)h(σ)dσ, ∀τ ∈ [0, 1],

h ∈ L2 ([0, 1], R) , is self-adjoint and nonnegative definite for all ω Assumption II

t∈Z Rt1 < ∞, with convergence in the nuclear norm

· 1, the following identity holds Fω = 1 2π

• t∈Z

exp (−iωt) Rt, Fω1 ≤ 1 2π

• t∈Z

Rt1 < ∞

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Asymptotic distribution of the fDFT

Functional Discrete Fourier Transform (fDFT)

• X (T)

ω

(·) = 1 √ 2πT

T−1

• t=0

Xt(·) exp (−iωt) , E[Xt2

2] < ∞ ⇒ P

• X (T)

ω

(·) ∈ L2 ([0, 1], C)

• = 1

It is 2π–periodic and Hermitian with respect to the argument ω E

• Xtl

2

• < ∞ ⇒ E
• X (T)

ω

l

2

• < ∞

Its asymptotic covariance operator is the spectral density

• perator (see Theorem 2.2, Panaretos and Tavakoli, 2013)

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Asymptotic distribution of the fDFT

Theorem 2.2 (Panaretos and Tavakoli, 2013) {Xt}T−1

t=0

be a strictly stationary sequence of random elements of L2 ([0, 1], R) such that (i) E[X0k

2] < ∞ ∞ t1,...,tk−1=−∞ cum(Xt1, . . . , Xtk−1, X0)2 < ∞, k ≥ 2

(ii)

t∈Z Rt1 < ∞

for ωj = limT→∞ ωj,T ; ωj,T ∈

• 2π

T , . . . , 2π[(T−1)/2]− T

• , ωj,T → ωj, T → ∞,

j = 3, . . . , J, ω1 := ω1,T = 0, ω2,T := ω2 = π

• X (T)

ω1 −

• T

2π µ →D Xω1, T → ∞

• X (T)

ωj,T →D

Xωj , T → ∞, j = 2, . . . , J,

• Xωj , j = 1, 2, independent Gaussian elements in L2 ([0, 1], R) , and in

L2 ([0, 1], C) , for j = 3, . . . , J, with covariance operator Fωj , for every j

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Convergence of the covariance operators of the fDFT, under increasing domain asymptotics

• Xωj ∼ N
• 0H, Fωj
• , j = 1, . . . , J,
• X (T)

ωj,T ∼ N

• 0H, Qj,T
• , j = 1, . . . , J, T ≥ 2

Fωj and Qj,T are self–adjoint, trace, positive integral operators with kernels Fωj =

L2 ∞

• k=1

λk(Fωj )φj,k ⊗ φj,k, Qj,T =

L2 ∞

• k=1

λk(Qj,T )φ(T)

j,k ⊗ φ(T) j,k

Theorem 2.2 (Panaretos and Tavakoli, 2013) implies, as T → ∞, M

X (T)

ωj,T

(h) = E

• exp
• X (T)

ωj,T (h)

• → M

Xωj (h) = E

• exp
• Xωj (h)
• , ∀h

Theorem 2.2 (Panaretos and Tavakoli, 2013) then also implies Qj,T (h)(h) → Fωj (h)(h), T → ∞, h ∈ L2 ([0, 1], R)

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Assumption A1. Assume, for j = 1, . . . , J,

∞

• p=1

sup

T≥2

• Qj,T(φj,p)(φj,p) − Fωj(φj,p)(φj,p)
• < ∞

Lema 1 Under Assumption A1, Qj,T − Fωj 1 → 0, T → ∞ where, as before, · 1 denotes the trace norm Proof of Lema 1 It follows straightforward from Cauchy–Schwarz inequality and Dominated Convergence Theorem

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Assumption A2. For j = 1, . . . , J, and · L(H), the norm in the space of bounded linear operators Λ

Fωj kT

= sup

1≤k≤kT

1

• Fωj(φj,k)(φj,k) − Fωj(φj,k+1)(φj,k+1)
• =

O (ξ(T)) , T → ∞ Qj,T − FωjL(H)ξ(T) → 0, T → ∞ kT denotes the selected truncation order

Lema 2 sup

1≤k≤kT

φ(T)

j,k − φj,kH ≤ 2

√ 2Λ

Fωj kT Qj,T − Fωj L(H)

Under A1-A2, for j = 1, . . . , J, sup

1≤k≤kT

φ(T)

j,k − φj,kH → 0,

T → ∞

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Theorem 1. Non–Central Limit Result Under conditions of Theorem 2.2 (Panaretos and Tavakoli, 2013), and Assumptions A1–A2, for j = 1, . . . , J, E

• S(T)

j

− Sj

• 2

S(H)

• → 0,

T → ∞ S(T)

j

→D Sj, T → ∞ The periodogram operator S(T)

j

= p(T)

ωj,T =

X (T)

ωj,T ⊗

X (T)

ωj,T , based on a

functional sample of size T, j = 1, . . . , J Sj = Xωj ⊗ Xωj , j = 1, . . . , J,

• Xωj , the weak-limit obtained in Theorem 2.2 (Panaretos and Tavakoli,

2013), for j = 1, . . . , J · S(H) denotes the Hilbert-Schmidt operator norm

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Proof of Theorem 1 The proof is based on Lemmas 1–2, and the following identifications in law:

• X (T)

ωj,T (φ(T) j,k ) = L

• λk(Qj,T )I1 (ϕk) , k ≥ 1
• Xωj (φj,k) =

L

• λ
• Fωj
• I1(ϕk) k ≥ 1

I1(·) is the simple Wiener-It

• stochastic integral, with respect to

complex–valued Wiener measure (see Peccati and Taqqu, 2011) {ϕk}k≥1 denotes an orthonormal system of L2 ([0, 1], R) The multiplication formula for Wiener-It

• stochastic integrals (see, e.g.,

Peccati and Taqqu, 2011) is applied, in the representation of the periodogram operator, in terms of double Wiener-It

• stochastic integrals,

for computation of its second-order moments

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Remark 1 Lema 2 also holds under weaker conditions than Assumption A1 on

• perator Qj,T

The key result, in practice, is Lemma 2, where the truncation order kT should be selected according to the divergence order Λ

Fωj kT

• f the inverses of

the distances between the eigenvalues of the spectral density operator Fωj up to order kT

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Corollary 1. Convergence of the trace of the periodogram operator Under the conditions of Theorem 1, for every k, l ≥ 1, j = 1, . . . , J, p(T)

wj,T (φ(T) j,k )(φ(T) j,l ) → Sj(φj,k)(φj,l),

T → ∞, in the mean-square sense and in distribution sense. Also, as T → ∞,

∞

• k=1

p(T)

wj,T (φ(T) j,k )(φ(T) j,k ) → ∞

• k=1

Sj(φj,k)(φj,k) ∼

∞

• k=1

λk

• Fωj
• χ2(2),

in the mean–square sense and in distribution sense.

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Theorem 2. Strong consistency of the periodogram operator in the Hilbert–Schmidt norm Under the conditions of Theorem 1, consider kT such that Λ

Fωj kT

satisfies Qj,T − Fωj L(H)ξ(T) = O

• T −β

, T → ∞, β > 1/2 Then, p(T)

ωj,T − Fωj S(H) →a.s. 0,

T → ∞

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Main Results for the Gaussian case

Proof of Theorem 2 The proof is based on Theorem 1 and Borel-Cantelli Lemma Remark 4 The results derived can be reformulated for the case of spatial stationary Gaussian processes with values in a real separable Hilbert space. That is the case of Spatial Autoregressive series in Hilbert sapaces (SARH(1) processes, introduced in Ruiz-Medina, 2011), considering the Gaussian case Remark 5 In Torres et al. (2019), a class of ℓ2–valued spatial stationary log-Gaussian Cox proceses are studied Remark 6 The results previously derived can be reformulated for the case of Hilbert-valued weak–dependent processes in continuous time

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Applications of the results derived

Definitions In the development below, we consider a Hilbert–valued weak-dependent log-Gaussian process Λ = {Λt = exp (Xt) , t ∈ R} as in Remark 6 {Xt, t ∈ R} is a continuous modification of a weak-dependent temporal stationary Gaussian process in a real separable Hilbert space Let Y be a locally finite multivariate (with integer random dimension d) random subset in time (respectively, in space, as in Torres et al,, 2019) For every bounded Borel set B ∈ R, the conditional distribution of NB = card(B ∩ Y), given Λ, is a Poisson process with mean function

• B Λt(·)dt

Λt(f ) has a.s. integrable realizations in time, for every f ∈ H If {Xt} is strictly stationary in time, then, E [N dt] = E [Λt] dt E [N dt ⊗ dN ds] = E [Λt ⊗ Λs] dtds

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

The underlying infinite–dimensional marginal distributions for the ℓ2–valued case

Under suitable conditions, if ENB2

H < ∞, for every bounded

Borel set B of R (see, e.g., Theorem 1 in Torres et al., 2019, for the ℓ2–valued spatial case), the marginal infinite–dimensional probability measure of NB can be defined, for every bounded Borel set B of R, as follows: Let the sequences (xj) = {xj, j ≥ 1} be such that xj is an integer for each j, with xj = 0, for sufficiently large j Zk = {(xj) : (x1, . . . , xk) ∈ Nk

⋆; xj = 0, j > k},

Nk

⋆ = Nk − {0}, k ≥ 1

Z ⊂ ℓ2 the family of sequences (xj), Z =

k Zk

Zk is countable for every k ≥ 1, Z is countable as the countable union of countable sets For every bounded Borel set A ∈ B, the probability measure µN A|Λ(ω) is defined on Z =

k Zk, Z ⊂ ℓ2, and

extended to ℓ2 by setting µN A|Λ(ω)(ℓ2 − Z) = 0, for ω ∈ Ω

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

Final comments and open research lines

Main steps Under the trucation rule derived, the consistent parameter estimation of Xt can be achieved from the periodogram operator, by projection (e.g., in terms of the minimum contrast parameter estimation methodology, as in Torres et al., 2019) Compute then the associated plug–in predictor of Xt Compute also the resulting approximation of the conditional expectation of N (·), or its plug–in functional predictor In the nonparametric case, the functional Fourier transform of the pair correlation operator plays the role of the spectral density operator, and it can be strong–consistently estimated, in terms of the componentwise periodogram operator, applying the methodology of this work

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

References

Bosq, D. and Ruiz-Medina, M. D. (2014). Bayesian estimation in a high dimensional parameter framework.

• Electron. J. Statist. 8, pp. 1604–1640

Panaretos, V. M. and Tavakoli, S. (2013). Fourier analysis

• f stationary time series in function space. Ann.
• Statist. 41, pp. 568–603

Peccati, G. and Taqqu, M. (2011). Wiener Chaos: Moments, Cumulants and Diagrams, Springer Ruiz-Medina, M. D. (2011). Spatial autorregresive and moving average Hilbertian processes. J. Multiv. Anal. 102, pp. 292–305

Spectral analysis of stationary log–Gaussian Cox processes in functions spaces

• M. D. RUIZ-

MEDINA Motivation Preliminary Results The Gaussian case

References

Simpson, D., Illian, J. B., Lindgren, F., Sorbye, S. H. and Rue, H. (2016). Going off grid: computationally efficient inference for log-Gaussian Cox processes. Biometrika 103,

• pp. 49–70

Torres, A., Fr´ ıas, M. P., Ruiz–Medina, M. D. and Mateu,

• J. (2019). Spatial Log-Gaussian Cox Processes In Hilbert

Spaces arXiv:1811.11139 (in progress)

Acknowledgements This work was partially supported by the Andalusian Research Group FQM-147, and the Project PGC2018-099549-B-I00 (co-funded by Feder funds), Ministry of Science, Innovation and Universities, Spain