GRASPA CONFERENCE 2008 Intermediate Meeting of GRASPA-PRIN 2006-2008 on Environmental Statistics 27 and 28 March 2008 Diagnostics tools for space-time point processes Giada Adelfio and Marcello Chiodi ∗ ∗ Dipartimento di Scienze Statistiche e Matematiche “Silvio Vianelli” (DSSM), Universit` a di Palermo • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Introduction • A new diagnostic approach is introduced; it is based on a transformed version of some second-order statistics for general point processes. • Assume the existence of the conditional intensity function for each point in space and time. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Introduction • A new diagnostic approach is introduced; it is based on a transformed version of some second-order statistics for general point processes. • Assume the existence of the conditional intensity function for each point in space and time. • For a more realistic interpretation of real phenomena: Self-similarity, long-range dependence and fractal behavior are introduced, analyzing second-order moments of the underlying process that can give infor- mation about interdependence among events. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Conditional intensity function The conditional intensity function of a space-time point process can be defined as: E [ N ([ t, t + ∆ t ) × [ x , x + ∆ x ) |H t )] λ ( t, x |H t ) = lim | ∆ t ∆ x | ∆ t, ∆ x → 0 where: • H t is the space-time occurrence history of the process up to time t ; • ∆ t, ∆ x are time and space infinitesimal increments; • E [ N ([ t, t + ∆ t ) × [ x , x + ∆ x ) |H t )] is the history-dependent expected value of the number of events that occur in the volume { [ t, t + ∆ t ) × [ x , x + ∆ x ) } . • If the conditional intensity is independent of the history but dependent only on the current time and the spatial locations this supplies a non-stationary Poisson process. • A constant conditional intensity provides a stationary Poisson process. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Second-order statistics • R/S Statistic (Mandelbrot, 1965) for long-range dependence in time: log-log plot of R/S versus intervals of time has a constant slope (0 < H < 1, Hurst constant) (Clegg (2006)). • Spectrum: a point process is said to be long-range dependent if its spectral density obeys: f N ( ω ) ∼ C | ω | θ as ω → 0, for some C > 0 and some real θ ∈ ( − 1 , 0), with θ = 1 − 2 H . A process with a high spectral density at small frequencies has the greater part of the variance concentrated at low frequency cycles. • Correlation Integral: when fractal objects are self-scaling in a statistical sense (parts of the whole fit the whole in distribution) ⇒ D 2 (correlation dimension): log C 2 ( δ ) D 2 = lim log( δ ) δ → 0 with C 2 ( δ ) the correlation integral: numbers of pairs of points with Euclidean distance less than δ . Given X 1 , X 2 , . . . , X n on R 2 its estimator is (Grassberger and Procaccia, 1983): n − 1 n 2 ˆ � � C 2 ( δ ) = I ( | X i − X j | ≤ δ ) (1) n ( n − 1) i =1 j = i +1 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Second-order residuals For point processes, many methods designed to test whether the second-order properties of an observed point pattern are consistent with the stationary Poisson process have been proposed (Diggle (1983), Ripley (1976), Baddeley and Silverman (1984)). A better diagnostic tool: The intensities used in the thinning method to retain points (Schoenberg (2003)) are here used as weights in order to offset the inhomogeneity of the process. Residual process is defined just weighting second-order statistics by the inverse of the conditional intensity function (autocorrelation, K-function, spectrum, fractal dimension and R/S statistic). Main result: second-order statistics of N w ( · ) behave as the ones of a homogeneous Poisson process. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Second-order residuals For point processes, many methods designed to test whether the second-order properties of an observed point pattern are consistent with the stationary Poisson process have been proposed (Diggle (1983), Ripley (1976), Baddeley and Silverman (1984)). A better diagnostic tool: The intensities used in the thinning method to retain points (Schoenberg (2003)) are here used as weights in order to offset the inhomogeneity of the process. Residual process is defined just weighting second-order statistics by the inverse of the conditional intensity function (autocorrelation, K-function, spectrum, fractal dimension and R/S statistic). Main result: second-order statistics of N w ( · ) behave as the ones of a homogeneous Poisson process. • The approach includes all the observed points rather than the only ones retained after the application of a random thinning. • This method can be applied to processes of any dimension, provided that those statistics exist. • It gives to second-order statistics a primary role in the diagnostic aim, to interpret dependence and at- tractive/inhibitive features of observed point patterns. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
The weighted process Let N a point process defined on S ∈ R d , d ≥ 1, identified by λ ( s |F ) with respect to the filtration F on S , assumed bounded away from zero. N w is a real valued random measure: for each S then 1 � N w ( S ) = λ ∗ ( s ) dN S λ ∗ ( s ) = λ min 1 where λ ( s ) and such that there exists λ min ≤ min { λ ( s ); ( s ) ∈ S } . Assumption: N is a simple space-time point process with H i the past history up to i th event. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
The weighted process Let N a point process defined on S ∈ R d , d ≥ 1, identified by λ ( s |F ) with respect to the filtration F on S , assumed bounded away from zero. N w is a real valued random measure: for each S then � 1 N w ( S ) = λ ∗ ( s ) dN S λ ∗ ( s ) = λ min 1 where λ ( s ) and such that there exists λ min ≤ min { λ ( s ); ( s ) ∈ S } . Assumption: N is a simple space-time point process with H i the past history up to i th event. By theoretical results (Adelfio and Schoenberg (2007)) the covariance is cov[ N w ( s i , s i + δ ) , N w ( s j , s j + δ )] = 0 (2) Therefore the spectral density of the weighted process is: f N w ( ω ) = λ min 2 π ⇒ that is the power spectrum of a Poisson process with constant rate λ min Moreover N w is a martingale (see Adelfio and Schoenberg (2007)). • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Weighted R/S statistic Weighted version: 0 ≤ u ≤ d [ N w ( u + t ) − N w ( t )] − u R w ( t ; d ) = max d [ N w ( t + d ) − N w ( t )] − 0 ≤ u ≤ d [ N w ( u + t ) − N w ( t )] − u min d [ N w ( t + d ) − N w ( t )] and � 2 w ( t ; d ) = N w ( t + d ) − N w ( t ) � N w ( t + d ) − N w ( t ) S 2 − d d then R/S w = R w ( t ; d ) (3) S w ( t ; d ) ⇒ generalization of the Donsker’s functional theorem or following the martingale approach: as for homogenous Poisson processes (Mandelbrot, 1975) and for short-term dependence (3) converges in distribution to the range of a Brownian bridge. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Weighted correlation integral � Weighted correlation integral for a point process defined on A ∈ R 2 with measure | A | is: 1 ˆ � � C W ( δ ) = ω i ω j I ( | s i − s j | ≤ δ ) (4) ( λ min | A | ) 2 i j � = i with λ ( s ) the conditional intensity function defined with respect the filtration F on A , ω k = λ min λ ( s k ) , ∀ k . � Weighted correlation integral for a time process N with realizations t 1 , t 2 , . . . , t n on [0 , T ] ∈ R : n n 1 ˆ � � C W ( δ ) = ω i ω j I ( | t i − t j | ≤ δ ) (5) ( λ min T ) 2 i j � = i with ω k = λ min λ ( t k ) , ∀ k and λ ( t ) the conditional intensity function with respect the filtration H t on [0 , T ]. ⇒ as for a homogeneous Poisson process it is asymptotically normally distributed (Adelfio and Schoenberg, 2007). • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
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