Spectral analysis of stationary logGaussian Cox in functions spaces - PowerPoint PPT Presentation

Spectral analysis of stationary log–Gaussian Cox processes Spectral analysis of stationary log–Gaussian Cox in functions spaces processes in functions spaces M. D. RUIZ- MEDINA Motivation Preliminary M. D. RUIZ-MEDINA Results The Gaussian University of Granada case III International Workshop on Advances in Functional Data Analysis 23–24, Mayo 2019

Spectral analysis of high–dimensional multivariate stationary log–Gaussian Cox processes Spectral analysis of stationary log–Gaussian Cox processes in functions spaces Stationary log–Gaussian Cox processes in ℓ 2 –spaces M. D. RUIZ- The distribution is completely characterized by the functional intensity and MEDINA the pair correlation operator (suitable class for parameter estimation) In this stationary case, under suitable conditions, estimators, based on the Motivation periodogram operator, can be constructed Preliminary Results The formulation in the infinite–dimensional setting can be achieved in a natural way (Bosq & Ruiz–Medina, 2014) The Gaussian case 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

Functional Data Analysis (FDA) techniques applied to point process estimation Spectral analysis of stationary log–Gaussian Cox processes in functions spaces An underdeveloped research area M. D. RUIZ- FDA techniques are well suited to estimate summary statistics in point MEDINA pattern analysis, which are functional in nature (see Illian et al. , 2008, pp. 271–271) Motivation Particularly, we refer to point pattern classification, based on second–order Preliminary statistics, from FDA methodologies (see, e.g., Illian et al. , pp. 135–150 in Results Baddeley et al. , 2006) The Gaussian case 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

Statistical spectral analysis on the second-order structure of stationary sequences of functional data Spectral analysis of stationary log–Gaussian Spectral analysis of stationary processes in functions spaces Cox processes in functions Panaretos and Tavakoli (2013) introduce the elements, and derive several results spaces to construct the basic building blocks , in the spectral analysis of stationary M. D. RUIZ- temporal correlated functional data sequences. The main ingredients were: MEDINA The spectral density operator Motivation The functional Discrete Fourier Transform (fDFT) Preliminary The derivation of the asymptotic law of fDFT Results The periodogram operator The Gaussian case 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

Asymptotic analysis in the frequency domain Spectral analysis of stationary log–Gaussian Cox processes in functions The points addressed in the present work spaces We consider the stationary Gaussian process case in Hilbert spaces: M. D. RUIZ- MEDINA Infinite–dimensional weighted chi-squared distributions, related to Fredholm determinant, arise in the study of Tightness Motivation A non-central limit result for the periodogram operator Preliminary Results Strong consistency of the periodogram operator in the Hilbert-Schmidt norm The Gaussian case 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

Asymptotic normality of fDFT (Panaretos and Tavakoli, 2013) Spectral analysis of stationary log–Gaussian Cox processes in functions Basic Definitions spaces Let { X t } t ∈ Z be a strictly stationary functional time series with values in M. D. RUIZ- H = L 2 ([0 , 1] , R ) . MEDINA E [ X t ] = µ ∈ L 2 ([0 , 1] , R ) , t ∈ Z Motivation E [ � X 0 � 2 L 2 ([0 , 1] , R ) ] < ∞ implies r t = L 2 E [( X s − µ ) ⊗ ( X s + t − µ )] , t , s ∈ Z Preliminary Results Pointwise definition of r t under continuity in the mean-square sense of X t The Gaussian Autocovariance operator at lag t case � 1 � � R t ( h )( τ ) = r t ( τ, σ ) h ( σ ) d σ = Cov � X 0 , h � L 2 ([0 , 1] , R ) , X t ( τ ) , τ ∈ [0 , 1] 0

Asymptotic normality of fDFT Spectral Spectral density and periodogram operators analysis of Assumption I(p) � stationary t ∈ Z � r t � p < ∞ , p = 2 , or ∞ ; then, for any ω ∈ R , the log–Gaussian following series converges in � · � p Cox processes in functions spaces � f ω ( · , · ) = 1 r t ( · , · ) exp ( − i ω t ) M. D. RUIZ- 2 π t ∈ Z MEDINA Motivation f ω ( · , · ) is called the spectral density kernel at frequency ω f ω ( · , · ) is uniformly bounded and continuous in ω with respect to � · � p Preliminary Results � 1 The spectral density operator F ω ( h )( τ ) = 0 f ω ( τ, σ ) h ( σ ) d σ, ∀ τ ∈ [0 , 1] , The Gaussian h ∈ L 2 ([0 , 1] , R ) , is self-adjoint and nonnegative definite for all ω case Assumption II � t ∈ Z �R t � 1 < ∞ , with convergence in the nuclear norm � · � 1 , the following identity holds � F ω = 1 exp ( − i ω t ) R t , 2 π t ∈ Z � �F ω � 1 ≤ 1 �R t � 1 < ∞ 2 π t ∈ Z

Asymptotic distribution of the fDFT Spectral analysis of stationary Functional Discrete Fourier Transform (fDFT) log–Gaussian Cox processes in functions spaces T − 1 � 1 X ( T ) � ( · ) = √ X t ( · ) exp ( − i ω t ) , M. D. RUIZ- ω MEDINA 2 π T t =0 Motivation � � ( · ) ∈ L 2 ([0 , 1] , C ) X ( T ) � E [ � X t � 2 Preliminary 2 ] < ∞ ⇒ P = 1 ω Results The Gaussian It is 2 π –periodic and Hermitian with respect to the case argument ω � � � � X ( T ) � � � X t � l � l < ∞ ⇒ E < ∞ E ω 2 2 Its asymptotic covariance operator is the spectral density operator (see Theorem 2.2, Panaretos and Tavakoli, 2013)

Asymptotic distribution of the fDFT Spectral analysis of stationary Theorem 2.2 (Panaretos and Tavakoli, 2013) log–Gaussian Cox processes be a strictly stationary sequence of random elements of L 2 ([0 , 1] , R ) { X t } T − 1 in functions t =0 such that spaces 2 ] < ∞ � ∞ (i) E [ � X 0 � k t 1 ,..., t k − 1 = −∞ � cum( X t 1 , . . . , X t k − 1 , X 0 ) � 2 < ∞ , k ≥ 2 M. D. RUIZ- MEDINA (ii) � t ∈ Z �R t � 1 < ∞ � � Motivation T , . . . , 2 π [( T − 1) / 2] − 2 π for ω j = lim T →∞ ω j , T ; ω j , T ∈ , ω j , T → ω j , T → ∞ , T Preliminary j = 3 , . . . , J , ω 1 := ω 1 , T = 0 , ω 2 , T := ω 2 = π Results The Gaussian � case T X ( T ) � 2 π µ → D � ω 1 − X ω 1 , T → ∞ X ( T ) � ω j , T → D � X ω j , T → ∞ , j = 2 , . . . , J , X ω j , j = 1 , 2 , independent Gaussian elements in L 2 ([0 , 1] , R ) , and in � L 2 ([0 , 1] , C ) , for j = 3 , . . . , J , with covariance operator F ω j , for every j

Main Results for the Gaussian case Spectral analysis of Convergence of the covariance operators of the fDFT, under increasing domain stationary asymptotics log–Gaussian Cox processes � � � in functions X ω j ∼ N 0 H , F ω j , j = 1 , . . . , J , spaces � � X ( T ) � ω j , T ∼ N 0 H , Q j , T , j = 1 , . . . , J , T ≥ 2 M. D. RUIZ- MEDINA F ω j and Q j , T are self–adjoint, trace, positive integral operators with kernels Motivation ∞ ∞ � � λ k ( Q j , T ) φ ( T ) j , k ⊗ φ ( T ) F ω j = λ k ( F ω j ) φ j , k ⊗ φ j , k , Q j , T = Preliminary j , k L 2 L 2 Results k =1 k =1 The Gaussian case Theorem 2.2 (Panaretos and Tavakoli, 2013) implies, as T → ∞ , � � �� � � �� X ( T ) � � M � ( h ) = E exp ω j , T ( h ) → M � X ω j ( h ) = E exp X ω j ( h ) , ∀ h X ( T ) ω j , T Theorem 2.2 (Panaretos and Tavakoli, 2013) then also implies Q j , T ( h )( h ) → F ω j ( h )( h ) , T → ∞ , h ∈ L 2 ([0 , 1] , R )

Main Results for the Gaussian case Spectral analysis of Assumption A1 . Assume, for j = 1 , . . . , J , stationary log–Gaussian Cox processes ∞ � � � in functions � Q j , T ( φ j , p )( φ j , p ) − F ω j ( φ j , p )( φ j , p ) � < ∞ spaces sup T ≥ 2 M. D. RUIZ- p =1 MEDINA Motivation Lema 1 Preliminary Results Under Assumption A1 , The Gaussian case � Q j , T − F ω j � 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