SPECTRAL MEASURES OF POINT PROCESSES Pierre Br emaud January 12, - PowerPoint PPT Presentation

SPECTRAL MEASURES OF POINT PROCESSES Pierre Br´ emaud January 12, 2015 P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 1 / 47

The purpose of this talk to honor Fran¸ cois for his 60-th birthday P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 2 / 47

And to opportunistically take advantage of his great popularity and the large number of friends gathered in this occasion to advertise my recently published book: Fourier Analysis and Stochastic Processes P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 3 / 47

What is it about? Consider a point process N on R with event times { T n } n ∈ Z . The “random Dirac comb” � X ( t ) := δ ( t − T n ) , n ∈ Z is not a bona fide stochastic process. In particular, one cannot define for the random Dirac comb associated with a stationary point process a power spectral measure as in the case of wide-sense stationary stochastic processes. The natural extension of the notion of power spectral density is the so-called Bartlett spectral measure Here we concentrate on the computation of such measures. P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 4 / 47

Who needs it? 1 biology (spike trains) 2 communications (ultra-wide band) 3 perhaps nobody needs it. P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 5 / 47

Some contributors M.S. Bartlett (1963), The spectral analysis of point processes, J. R. Statist. Soc. Ser. B 29 , 264-296. J. Neveu, Processus ponctuels, in ´ e de Saint Flour , Lect. Notes Ecole d’´ et´ in Math. 598 , 249-445, Springer (1976). D.J. Daley, D. Vere–Jones, An Introduction to the Theory of Point Processes , Springer, NY (1988, 2003). P. B. and L. Massouli´ e, Power Spectra of Generalized Shot Noises and Hawkes Point Processes with a random excitation, Adv. Appl. Proba. , 205-222 (2002) P. B, L. Massouli´ e, and A. Ridolfi, “Power spectra of random spike fields and related processes”, Adv. in Appl. Probab. , 37 , 4, 1116-1146 (2005). P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 6 / 47

Second moment measure Second-order: for all compact sets C , � N ( C ) 2 � E < ∞ . M 2 ( A × B ) := E [ N ( A ) N ( B )] . M 2 is the intensity measure of N × N . By Campbell’s theorem, �� � � g ( X n , X k ) E n ∈ N k ∈ N � � = R m g ( t , s ) M 2 ( dt × ds ) . R m P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 7 / 47

L 2 N ( M 2 ) The collection of functions ϕ : R m → C such that � � R m | ϕ ( t ) ϕ ( s ) | M 2 ( dt × ds ) < ∞ , R m � N ( | ϕ | ) 2 � ⇔ E < ∞ , � � N ( | ϕ | 2 ) ⇒ E [ N ( | ϕ | )] < ∞ , E < ∞ ⇒ L 2 N ( M 2 ) ⊆ L 1 C ( ν ) ∩ L 2 C ( ν ) . (where ν ( C ) := E [ N ( C )]) P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 8 / 47

Wide-sense stationary point process Second-order, plus E [ N ( C + t )] = E [ N ( C )] , and E [ N ( A + t ) N ( B + t )] = E [ N ( A ) N ( B )] . Immediate consequence: for all non-negative ϕ, ψ , ��� � �� �� E ϕ ( t ) N ( dt ) ψ ( τ + t ) N ( dt ) R R is independent of τ ∈ R . P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 9 / 47

Covariance measure Basic lemma from measure theory ( X , X ), µ loc. fin. measure on X ⊗ k , invariant by the simultaneous shifts, that is, µ (( A 1 + h ) × · · · × ( A k + h )) = µ ( A 1 × · · · × A k ) . µ on X k − 1 such that for all Then, there exists a locally finite measure ˆ non-negative measurable functions f from X k to R , � X k f ( x 1 , . . . , x k ) µ ( dx 1 × · · · × dx k ) � �� � = X k − 1 f ( x 1 , x 1 + x 2 , . . . , x 1 + x k )ˆ µ ( dx 2 × · · · × dx k ) dx 1 . X P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 10 / 47

Application to point processes M 2 (( A + t ) × ( B + t )) = M 2 ( A × B ) Therefore, for all ϕ, ψ ∈ L 2 N ( M 2 ), � � R m ϕ ( t ) ψ ∗ ( s ) M 2 ( dt × ds ) R m � �� � R m ϕ ( t ) ψ ∗ ( s + t ) dt = σ ( ds ) R m for some locally finite measure σ . In fact, σ can be identified to the intensity measure of the Palm version of a given stationary point process. P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 11 / 47

Since for ϕ, ψ ∈ L 1 C ( R m ), E [ N ( ϕ )] E [ N ( ψ )] ∗ � � � � � � R m ψ ∗ ( s ) ds = λ R m ϕ ( t ) dt λ �� � � R m ϕ ( t ) ψ ∗ ( t + s ) dt = λ 2 ds , R m For ϕ, ψ ∈ L 2 N ( M 2 ), �� � � R m ϕ ( t ) N ( dt ) , R m ψ ( s ) N ( ds ) cov � �� � R m ϕ ( t ) ψ ∗ ( t + s ) dt = Γ N ( ds ) R m where the locally finite measure Γ N := σ − λ 2 ℓ m is called the covariance measure of the stationary second-order point process N . P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 12 / 47

Covariance of the renewal process. Let N be a stationary renewal point process with renewal function R . Γ N ( dt ) = λ ( R ( dt ) − λ dt ) . Homogeneous Poisson process on the line. By the covariance formula, � ϕ ( t ) ψ ∗ ( t ) dt . cov ( N ( ϕ ) , N ( ψ )) = λ R �� � � ϕ ( t ) ψ ∗ ( t + s ) dt = λ ε 0 ( ds ) , R R and therefore, , Γ N = λε 0 . P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 13 / 47

Bartlett spectral measure The unique locally finite measure µ N such that �� � � ϕ ( ν ) | 2 µ N ( d ν ) ϕ ( t ) N ( dt ) = | � Var for all ϕ ∈ B N , where B N ⊆ L 2 N ( M 2 ) is a vector space of functions called the test function space . By polarization, for all ϕ, ψ ∈ B N , � ϕ ( ν ) � ψ ∗ ( ν ) µ N ( d ν ) . cov ( N ( ϕ ) , N ( ψ )) = � P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 14 / 47

B N should contain a class of functions rich enough to guarantee uniqueness of the measure µ N : if the locally finite measures µ 1 and µ 2 are such that � � ϕ ( ν ) | 2 µ 1 ( d ν ) = ϕ ( ν ) | 2 µ 2 ( d ν ) | � | � for all ϕ ∈ B N , then µ 1 ≡ µ 2 . Note that B N ⊆ L 1 C ( R m ) since, as we observed earlier, L 2 N ( M 2 ) ⊆ L 1 C ( R m ). In particular the Fourier transform of any ϕ ∈ B N is well-defined. J. Neveu (1976): B N contains at least the functions that are, together � 1 / | x | 2 � with their Fourier transform, O as | x | → ∞ . P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 15 / 47

Examples Poisson impulsive white noise. The covariance function is λ times the Dirac measure at the origin, and therefore its spectral measure is λ times the Lebesgue measure, therefore it admits a power spectral density that is a constant: f N ( ν ) = λ. P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 16 / 47

Examples Regular grid. Regular T -grid on R with random origin, that is N ≡ { nT + U ; n ∈ Z } where T > 0, and U is uniform random [0 , T ]. Here, λ = 1 / T . � µ N = 1 ε n T , T 2 n � =0 and we can take B N specified by the following two conditions ϕ ∈ L 1 C ( R ) ∩ L 2 C ( R ) and � � n �� � � � � ˆ ϕ � < ∞ . T n ∈ Z Note that the latter condition implies ( ℓ 1 C ( Z ) ⊂ ℓ 2 C ( Z )) � � n �� � 2 � � � ˆ ϕ < ∞ . T u � n ∈ Z P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 17 / 47

Regular grid, proof Weak Poisson summation formula : Both sides of the following equality � n � � � ϕ ( u + nT ) = 1 e 2 i π n T u . ϕ � ( ⋆ ) T T n ∈ Z n ∈ Z are well-defined, and the equality holds for almost-all u ∈ R . By ( ⋆ ), � � n � � � ϕ ( U + nT ) = 1 e 2 i π n T U ϕ ( t ) N ( dt ) = ϕ � T T R n ∈ Z n ∈ Z and therefore �� 2 � � � � � � � E ϕ ( t ) N ( dt ) � � R �� � � k � � n � � = 1 e 2 i π ( n − k T U ) ϕ ∗ ϕ � � T 2 E T T n ∈ Z k ∈ Z � �� � n � = 1 2 � � �� ϕ � . T 2 T n ∈ Z P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 18 / 47

Also �� � � E R 2 ϕ ( t ) N ( dt ) = E [ ϕ ( U + nT )] n ∈ Z � T � = 1 ϕ ( u + nT ) du = 1 ϕ ( t ) dt = 1 T � ϕ (0) . T T 0 R Therefore �� � ϕ ( t ) N ( dt ) Var R � �� � n � = 1 2 − 1 � � ϕ (0) | 2 �� ϕ � T 2 | � T 2 T n ∈ Z � � � n �� � = 1 � � 2 ϕ ( ν ) | 2 µ N ( d ν ) . �� | � ϕ � = T 2 T R n � =0 P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 19 / 47

Examples Cox process. (on R m with stochastic intensity { λ ( t ) } t ∈ R m .) Suppose that { λ ( t ) } t ∈ R m is a wss process with mean λ and Cram´ er spectral measure µ λ . Then the Bartlett spectrum of N is µ N ( d ν ) = µ λ ( d ν ) + λ d ν , and we can take B N = L 1 C ( R m ) ∩ L 2 C ( R m ). Even more, in this case B N = L 2 N ( M 2 ) P. Br´ emaud (Inria and EPFL) Point process spectra Jan. 12, 2015 20 / 47