Spatio-temporal point process statistics: A review

González, Jonatan A.; Rodríguez‐Cortés, Francisco J.; Cronie, Ottmar; Mateu, Jorge

doi:10.1016/j.spasta.2016.10.002

Cited by 96 publications

(75 citation statements)

References 84 publications

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“…The interaction structure is described by second‐order characteristics and can be analysed by using the pair correlation function

g false{(u, v), (s, l) false} = \frac{λ^{(2)} false{(u, v), (s, l) false}}{λ (u, v) λ (s, l)}, false(boldu, v false), false(bolds, l false) \in W \times T,

where λ (2) (·,·) is the second‐order product density function and λ (2) {( u , v ),( s , l )} d( u , v ) d( s , l ) for ( u , v )≠( s , l ) may be interpreted as the probability that there is a point from X in each of the infinitesimal regions around ( u , v ) and ( s , l ) of spatiotemporal volumes d( u , v ) and d( s , l ) respectively (see for example González et al . ()). The process X is called second‐order intensity‐reweighted stationary (SOIRS) if

g {(u, v), (s, l)} = \overset{false¯}{g} (u - s, v - l),

for any ( u , v ),( s , l ) ∈ W × T , where

\overset{g}{true}

is some non‐negative function.…”

Section: Spatiotemporal Distributionmentioning

confidence: 94%

“…where λ .2/ .·, ·/ is the second-order product density function and λ .2/ {.u, v/, .s, l/} d.u, v/ d.s, l/ for .u, v/ = .s, l/ may be interpreted as the probability that there is a point from X in each of the infinitesimal regions around .u, v/ and .s, l/ of spatiotemporal volumes d.u, v/ and d.s, l/ respectively (see for example González et al (2016)). The process X is called second-order intensityreweighted stationary (SOIRS) if…”

Section: Spatiotemporal Distributionmentioning

confidence: 99%

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Analysis of Tornado Reports Through Replicated Spatiotemporal Point Patterns

González

Hahn

Mateu

2019

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary Understanding the spatiotemporal distribution of tornado events is increasingly imperative, not only because of the natural phenomenon itself and its tremendous complexity but also because we can potentially reduce the risks that they entail. In particular, the US regions are particularly susceptible to tornadoes and they are the focus and motivation of our statistical analysis. Tornado reports can be treated as spatiotemporal point patterns, and we develop some methods for the analysis of replicated spatiotemporal patterns to identify significant structural differences between cold and warm seasons along the years. We extend some existing spatial techniques to the spatiotemporal context to test the null hypothesis that two (or more) observed spatiotemporal point patterns with replications are realizations of point processes that have the same second‐order descriptors. In particular, we develop a non‐parametric test to approximate the null distribution of the test statistics. We present intensive simulation studies that demonstrate the validity and power of our test and apply our methods to the motivating problem of tornadoes.

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“…The interaction structure is described by second‐order characteristics and can be analysed by using the pair correlation function

g false{(u, v), (s, l) false} = \frac{λ^{(2)} false{(u, v), (s, l) false}}{λ (u, v) λ (s, l)}, false(boldu, v false), false(bolds, l false) \in W \times T,

g {(u, v), (s, l)} = \overset{false¯}{g} (u - s, v - l),

for any ( u , v ),( s , l ) ∈ W × T , where

\overset{g}{true}

is some non‐negative function.…”

Section: Spatiotemporal Distributionmentioning

confidence: 94%

Section: Spatiotemporal Distributionmentioning

confidence: 99%

Analysis of Tornado Reports Through Replicated Spatiotemporal Point Patterns

González

Hahn

Mateu

2019

Journal of the Royal Statistical Society Series C: Applied Statistics

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“…The K ‐function and the product density function provide a global measure of the covariance structure by summing over the contributions from each event observed in the process. A more complete review about spatial and spatiotemporal point processes can be found in Baddeley, Rubak, and Turner (); Diggle (); González et al (); Illian et al (); and Møller and Waagepetersen ().…”

Section: Statistical Methodologymentioning

confidence: 99%

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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“…We observe n events

{false{false({boldu}_{i}, s_{i} false) false}}_{i = 1}^{n}

of distinct points of X within a bounded spatio‐temporal region

W \times T \subset {double-struckR}^{d - 1} \times double-struckR

, with volume | W |>0, and with length | T |>0, where n ≥ 0 is not fixed in advance. In the sequel, N ( B ) denotes the number of events of the process falling in a bounded region B ⊂ W × T ; for more details, see Diggle () and González, Rodríguez‐Cortés, Cronie, and Mateu (). We assume that the basic point process X is completely stationary, that is, for all

boldu \in {double-struckR}^{d - 1}

and all real s , it holds that

X \overset{=}{true} X_{false(boldu, s false)}

, where X ( u , s ) is the translated process given by X ( u , s ) ={[ u 1 + u , s 1 + s ],[ u 2 + u , s 2 + s ],…}.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…We consider a spatio-temporal point process with no multiple points as a random countable subset X of R d−1 × R, where a point (u, s) ∈ X corresponds to an event at u ∈ R d−1 occurring at time s ∈ R. We observe n events {(u i , s i )} n i=1 of distinct points of X within a bounded spatio-temporal region W × T ⊂ R d−1 × R, with volume |W| > 0, and with length |T| > 0, where n ≥ 0 is not fixed in advance. In the sequel, N(B) denotes the number of events of the process falling in a bounded region B ⊂ W × T; for more details, see Diggle (2013) and González, Rodríguez-Cortés, Cronie, and Mateu (2016). We assume that the basic point process X is completely stationary, that is, for all u ∈ R d−1 and all real s, it holds that X  = X (u,s) , where X (u,s) is the translated process given by X (u,s)…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Spatio‐temporal classification in point patterns under the presence of clutter

et al. 2019

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We consider the problem of detection of features in the presence of clutter for spatio‐temporal point patterns. In previous studies, related to the spatial context, Kth nearest‐neighbor distances to classify points between clutter and features. In particular, a mixture of distributions whose parameters were estimated using an expectation‐maximization algorithm. This paper extends this methodology to the spatio‐temporal context by considering the properties of the spatio‐temporal Kth nearest‐neighbor distances. For this purpose, we make use of a couple of spatio‐temporal distances, which are based on the Euclidean and the maximum norms. We show close forms for the probability distributions of such Kth nearest‐neighbor distances and present an intensive simulation study together with an application to earthquakes.

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Spatio-temporal point process statistics: A review

Cited by 96 publications

References 84 publications

Analysis of Tornado Reports Through Replicated Spatiotemporal Point Patterns

Analysis of Tornado Reports Through Replicated Spatiotemporal Point Patterns

Testing for local structure in spatiotemporal point pattern data

Spatio‐temporal classification in point patterns under the presence of clutter

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