Modelling, Estimation and Applications of Second-Order Spatio-Temporal Characteristics of Point Processes

Rodríguez‐Cortés, Francisco J.; Javier, Francisco

doi:10.6035/14019.2014.338537

Cited by 2 publications

(7 citation statements)

References 92 publications

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“…In this paper, the Palm expectation of the product density LISTA function (Mateu & Rodríguez‐Cortés, ; Rodríguez‐Cortés, ; Rodríguez‐Cortés, Ghorbani, & Mateu, ) and an edge‐corrected estimator are discussed in detail. With some examples, we have shown the behaviour of the LISTA surfaces for some simulated sets of points.…”

Section: Resultsmentioning

confidence: 99%

“…Values of the estimated spatiotemporal surface of the product density at a point ( r , t ) larger than ρ 2 indicate that the interevent distances around ( r , t ) are relatively more frequent than in a homogeneous Poisson process, which is typical of a cluster process. Conversely, values of the spatiotemporal surface of

{\overset{ρ^{false(2 false)}}{true}}_{ϵ, δ} false(r, t false)

smaller than ρ 2 indicate that the corresponding distances are rare, which is typical of an inhibition process; for more details, see Gabriel et al (); Rodríguez‐Cortés (); and Rodríguez‐Cortés, Ghorbani, Mateu, and Stoyan ().…”

Section: Statistical Methodologymentioning

confidence: 99%

“…A rigorous theoretical and computational study of an unbiased estimator for the K ‐function under several edge corrections is presented in Gabriel (). In both cases of stationarity and isotropy and of SOIRS and isotropy, the spatiotemporal product density function is proportional to the derivative of K ( r , t ) with respect to r and t , and for a stationary spatiotemporal point process X , the relationship between both parts is given by the following:

ρ^{false(2 false)} false(r, t false) = \frac{ρ^{2}}{4 π r} \frac{\partial^{2} K false(r, t false)}{\partial r \partial t},

where ρ 2 K ( r , t ) can be interpreted as the expected number of ordered pairs of distinct points per unit volume of the observation window with pairwise distance and time lag less than r and t , respectively; see Rodríguez‐Cortés () and Rodríguez‐Cortés, Ghorbani, Mateu, and Stoyan ().…”

Section: Statistical Methodologymentioning

confidence: 99%

“…An unbiased edge‐corrected kernel‐based estimator for the product density LISTA function (Mateu & Rodríguez‐Cortés, ; Rodríguez‐Cortés, ; Rodríguez‐Cortés, Ghorbani, & Mateu, ) is given by the following:

{\overset{ρ^{false(2 false) i}}{true}}_{ϵ, δ} false(r, t false) = \frac{n - 1}{4 π r false| B false|} \sum_{j \neq i} \frac{κ_{ϵ, δ} false(‖ u_{i} - u_{j} ‖ - r, false| s_{i} - s_{j} false| - t false)}{w_{1} false(u_{i}, ‖ u_{i} - u_{j} ‖ false) w_{0} false(s_{i}, false| s_{i} - s_{j} false| false)},

with r > ϵ >0, t > δ >0, for ( u i , s i )∈ W × T , i =1,…, n and where the edge‐effect factors are given by the following:

\frac{1}{w_{1} ({boldu}_{i}, ‖ {boldu}_{i} - {boldu}_{j} ‖)} = \frac{2 π ‖ {boldu}_{i} - {boldu}_{j} ‖}{| \partial b ({boldu}_{i}, ‖ {boldu}_{i} - {boldu}_{j} ‖) \cap W {false|}_{1}} and \frac{1}{w_{0} (s_{i}, | s_{i} - s_{j} |)} = \frac{2}{| \partial b (s_{i}, | s_{i} - s_{j} |) \cap T {false|}_{0}} .

…”

Section: Local Second‐order Characteristicsmentioning

confidence: 99%

“…[(N(B) − 1) 2 ] = |B| 2 2 + |B| , which can be obtained through Dobiński's formula for the moments (Berge, 1971); for more details, see the Appendix. An unbiased edge-corrected kernel-based estimator for the product density LISTA function Rodríguez-Cortés, 2014; is given by the following:…”

Section: Product Density Lista Functionsmentioning

confidence: 99%

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Testing for local structure in spatiotemporal point pattern data

et al. 2017

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The detection of clustering structure in a point pattern is one of the main focuses of attention in spatiotemporal data mining. Indeed, statistical tools for clustering detection and identification of individual events belonging to clusters are welcome in epidemiology and seismology. Local second-order characteristics provide information on how an event relates to nearby events. In this work, we extend local indicators of spatial association (known as LISA functions) to the spatiotemporal context (which will be then called LISTA functions). These functions are then used to build local tests of clustering to analyse differences in local spatiotemporal structures. We present a simulation study to assess the performance of the testing procedure, and we apply this methodology to earthquake data

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Section: Resultsmentioning

confidence: 99%

{\overset{ρ^{false(2 false)}}{true}}_{ϵ, δ} false(r, t false)

Section: Statistical Methodologymentioning

confidence: 99%

ρ^{false(2 false)} false(r, t false) = \frac{ρ^{2}}{4 π r} \frac{\partial^{2} K false(r, t false)}{\partial r \partial t},

Section: Statistical Methodologymentioning

confidence: 99%

{\overset{ρ^{false(2 false) i}}{true}}_{ϵ, δ} false(r, t false) = \frac{n - 1}{4 π r false| B false|} \sum_{j \neq i} \frac{κ_{ϵ, δ} false(‖ u_{i} - u_{j} ‖ - r, false| s_{i} - s_{j} false| - t false)}{w_{1} false(u_{i}, ‖ u_{i} - u_{j} ‖ false) w_{0} false(s_{i}, false| s_{i} - s_{j} false| false)},

with r > ϵ >0, t > δ >0, for ( u i , s i )∈ W × T , i =1,…, n and where the edge‐effect factors are given by the following:

\frac{1}{w_{1} ({boldu}_{i}, ‖ {boldu}_{i} - {boldu}_{j} ‖)} = \frac{2 π ‖ {boldu}_{i} - {boldu}_{j} ‖}{| \partial b ({boldu}_{i}, ‖ {boldu}_{i} - {boldu}_{j} ‖) \cap W {false|}_{1}} and \frac{1}{w_{0} (s_{i}, | s_{i} - s_{j} |)} = \frac{2}{| \partial b (s_{i}, | s_{i} - s_{j} |) \cap T {false|}_{0}} .

…”

Section: Local Second‐order Characteristicsmentioning

confidence: 99%

Section: Product Density Lista Functionsmentioning

confidence: 99%

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Testing for local structure in spatiotemporal point pattern data

et al. 2017

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Estructuras estocásticas notables en procesos puntuales espacio-temporales y medidas de riesgo bivariantes

Villa¹

2023

Preprint

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La modelización estadística constituye un conjunto de formulaciones matemáticas que proporcionan un mejor entendimiento de los fenómenos, especialmente en los que subyacen relaciones entre variables que a su vez poseen influencia aleatoria. En este trabajo de tesis se realiza la modelización estadística del fenómeno delictivo, mediante el desarrollo de estructuras estocásticas en procesos puntuales espacio-temporales, con el propósito de reproducir eventos de la forma más exacta posible a la realidad en función de su evolución en el espacio, tiempo y espacio-tiempo.

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Modelling, Estimation and Applications of Second-Order Spatio-Temporal Characteristics of Point Processes

Cited by 2 publications

References 92 publications

Testing for local structure in spatiotemporal point pattern data

Testing for local structure in spatiotemporal point pattern data

Estructuras estocásticas notables en procesos puntuales espacio-temporales y medidas de riesgo bivariantes

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