A multi-scale area-interaction model for spatio-temporal point patterns Marie-Colette van Lieshout joint work with Adina Iftimi and Francisco Montes Suay CWI and University of Twente The Netherlands ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 1/16
The area-interaction model in space Density p ( x ) = αβ n ( x ) γ − A r ( x ) with respect to unit rate Poisson process on bounded W ⊂ R 2 . Here β , is positive, n ( x ) denotes the cardinality of x and A r ( x ) the area of ( x ⊕ B (0 , r )) ∩ W. • γ = 1 : Poisson process; • γ > 1 : attraction, cf. Widom–Rowlinson (1970); • γ < 1 , inhibition, cf. Baddeley and Van Lieshout (1995). ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 2/16
Multi-scale area-interaction Consider the influence function defined for u � = v by κ j if r j − 1 < || u − v || ≤ r j κ ( u, v ) = 0 if || u − v || > r m for r 0 = 0 < r 1 < r 2 < · · · < r m and 1 = κ 1 > κ 2 > · · · > κ m > 0 and set � � � αβ n ( x ) exp p ( x ) = − log γ max x ∈ x κ ( w, x ) dw W � m � αβ n ( x ) exp � = − log γ κ j |{ w ∈ W : d ( w, x ) ∈ ( r j − 1 , r j ] | j =1 in full analogy to pairwise interaction models based on κ . Gregori, Van Lieshout and Mateu (2003). ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 3/16
Product interpretation Note that � m � αβ n ( x ) exp � p ( x ) = − log γ κ j | (( x ⊕ B (0 , r j )) \ ( x ⊕ B (0 , r j − 1 ))) ∩ W | j =1 � m � αβ n ( x ) exp � = − log γ ( κ j − κ j +1 ) | ( x ⊕ B (0 , r j )) ∩ W | j =1 m ( γ κ j − κ j +1 ) − A rj ( x ) αβ n ( x ) � = j =1 under the convention κ m +1 = 0 . Relaxing the constraint that κ j > κ j +1 , inhibition and attraction may be combined, cf. Picard et al. (2009) and Ambler and Silverman (2010). ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 4/16
Space-time area-interaction: Definition Density p ( x ) = αβ n ( x ) γ − ℓ ( x ⊕ G ) with respect to unit rate Poisson process on bounded subset W S × W T ⊂ R 2 × R . Here α > 0 is a normalizing constant, • ℓ is Lebesgue measure restricted to W S × W T , • γ > 0 is the interaction parameter, and • G is a cylinder C t r (0 , 0) = { ( y, s ) ∈ W S × W T : || y || ≤ r, | s | ≤ t } . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 5/16
Multi-scale space-time area-interaction Consider the influence function defined for u � = v by κ j if ( u S − v S , u T − v T ) ∈ G j \ G j − 1 κ (( u S , u T ) , ( v S , v T )) = 0 if ( u S − v S , u T − v T ) �∈ G m where, for r 0 < r 1 < r 2 < · · · < r m , t 0 < t 1 < t 2 < · · · < t m , G 1 ⊂ · · · ⊂ G m t j are nested cylinders C r j (0 , 0) , 1 = κ 1 > κ 2 > · · · > κ m and set � � � αβ n ( x ) exp p ( x ) = − log γ ( x,t ) ∈ x κ (( w S , w T ) , ( x, t )) dw S dw T max W S × W T � m � αβ n ( x ) exp � = − log γ κ j ℓ (( x ⊕ G j ) \ ( x ⊕ G j − 1 )) . j =1 ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 6/16
Product interpretation and Markov property Since m − ℓ ( x ⊕ G j ) p ( x ) = αβ n ( x ) � γ j j =1 for log γ j = ( κ j − κ j +1 ) log γ under the convention κ m +1 = 0 , the log conditional intensity is � p ( x ∪ { ( y, s ) } ) � log λ (( y, s ); x ) = log p ( x ) m � = log β − log γ j ℓ ((( y, s ) ⊕ G j ) \ ( x ⊕ G j )) j =1 and hence p is Markov at range 2 max { r m , t m } . Note: generalise to any γ j by relaxing the constraint κ j > κ j +1 . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 7/16
Varicella • transmitted by direct contact with the rash or by inhalation of aerosolised droplets from respiratory tract secretions of patients; • mostly mild in childhood, severe in adults; • may be fatal in neonates and immuno-compromised people; • 10 to 21 days incubation period; • itchy, vesicular rash, fever and malaise; • it takes 7 to 10 days for vesicles to dry out. ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 8/16
Data 921 cases in 16 districts in Valencia, Spain, during 2013. • Bounding region W S = [0 , 9] × [0 , 9] km 2 . • Bounding time region W T = [0 , 52] weeks. ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 9/16
Exploratory analysis – Spatial component Left: projection in space. To get rid of duplicate locations, add jitter in (0 , 0 . 01) to each coordinate. Right: estimated pair correlation function assuming stationarity and isotropy, Epanechnikov kernel 1 − t 2 � � κ ǫ ( t ) = 3 , − ǫ ≤ t ≤ ǫ, ǫ 2 4 ǫ with bandwidth according to Stoyan’s rule of thumb 0 . 15(5ˆ λ ) − 1 / 2 . Conclusion: r m ≈ 2 . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 10/16
Exploratory analysis – Temporal component Projection in time and estimated auto-correlation function assuming stationarity. Conclusion: t m ≈ 7 . 5 . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 11/16
Modelling – Covariate information Idea: add a first order interaction function based on explanatory variables. Visual inspection suggests a separable function λ ( x, t ) = βλ ( x ) Z ( t ) , x ∈ [0 , 9] 2 , t ∈ { 0 , . . . , 51 } is reasonable, where β > 0 , λ ( x ) is a non-parametric estimate of the population density, and Z ( t ) a fitted harmonic regression 3 � Z ( t ) = c 0 + ( c j cos(2 πjt/ 52) + d j sin(2 πjt/ 52)) + c ( a + bt ) j =1 rescaled by 100 . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 12/16
Results and details λ ( x ) : Known: number N i of inhabitants in 559 sections in the 16 districts. Take N i uniformly distributed points in each section, apply a Gaussian kernel smoother with σ = 0 . 15 and global edge correction, and scale by 1 , 000 . Z ( t ) : Estimate c 0 , . . . , c 3 , d 1 , . . . d 3 , c , a and b by maximum likelihood in the Gaussian family. ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 13/16
Inference Starting from r m = 2 and t m = 7 . 5 , we zoom in by discarding ranges where the estimated parameters κ oscillate around zero to arrive at the following model: � 6 � p ( x ) ∝ β n ( x ) � � λ ( x, t ) exp − κ j ℓ (( x ⊕ G j ) \ ( x ⊕ G j − 1 )) j =1 ( x,t ) ∈ x for r = (0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6) and t = (0 . 75 , 1 . 0 , 1 . 25 , 1 . 5 , 1 . 75 , 2 . 0) . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 14/16
Estimating the parameters Nguyen–Zessin–Georgii equation �� � � � = E h ( x, X \ { x } ) h ( x, X ) λ θ ( x ; X ) dx . E W S W T x ∈ X ∩ W S × W t The Takacs–Fiksel idea is to choose convenient functions h , estimate both sides, equate and solve for θ . Pseudo-likelihood: (Besag, 1977; Jensen and Møller, 1991) h ( x, X ) = ∂ ∂θ log λ θ ( x ; X ) Logistic regression: (Baddeley et al. 2014) � � h ( x, X ) = ∂ λ θ ( x ; X ) ∂θ log λ θ ( x ; X ) + ρ ( x ) with ρ ( x ) = λ ( x ) Z ( t ) / 25 . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 15/16
Validation • 99 realisations from the fitted model in MPPLIB using Metropolis–Hastings. • Monte Carlo envelopes for estimates of L inhom ( r, t = 3 r ) (Gabriel and Diggle, 2009) using stpp . ◭ � ◮ A multi-scale area-interaction model for spatio-temporal point patterns – p. 16/16
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