Point-pattern analysis on the sphere

Robeson, Scott M.; Li, Ao; Huang, Chunfeng

doi:10.1016/j.spasta.2014.10.001

Cited by 24 publications

(30 citation statements)

References 29 publications

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“…The spherical analogue of the K function was described briefly by Ripley (, p. 173) and developed further by Robeson et al (). For a homogeneous point process X , following Ripley (, p. 190, Appendix), define

K (r) = \frac{1}{λ} double-struckE ([], \sum_{x \in bold-italicX ∖ {u}} bold1 ({}, d (u, x) \leq r) 2.03em | u \in bold-italicX), 0 \leq r \leq π,

where λ is theintensity of X and u is any point in

{double-struckS}^{2}

; the vertical bar indicates that the right side is an expectation with respect to the Palm distribution P u .…”

Section: Summary Functionsmentioning

confidence: 99%

“…Following standard practice for two‐dimensional point patterns, and serve as the “benchmarks” of complete randomness against which other point processes may be compared (Ripley, ; Diggle, ; Cressie, ; Baddeley et al, ). Equation appears in Ripley ) and Robeson et al (). To prove (, we apply Slivnyak's Theorem: the expectation in is simply the expected number of points falling in b( u , r ), namely, λ |b( u , r )|=2 π λ (1− cos r ), yielding .…”

Section: Summary Functionsmentioning

confidence: 99%

“…If X is homogeneous and has a second‐moment intensity function λ 2 ( u , v ), then, from ,

K (r) = \frac{1}{λ^{2} | B |} \int_{B} \int_{{double-struckS}^{2}} bold1 ({}, d (u, v) \leq r) λ_{2} (u, v) d u d v = \frac{1}{| B |} \int_{B} \int_{{double-struckS}^{2}} bold1 ({}, d (u, v) \leq r) g (d (u, v)) d u d v,

where g ( r ) is the pair correlation function. By a change of variables,

bold-italicK (r) = 2 π \int_{0}^{r} g (t) \sin t d t, 0 \leq r \leq π,

and differentiating gives

\begin{matrix} g (r) = \frac{K^{'} (r)}{2 π \sin r}, 0 < r < π; \end{matrix}

the corresponding expression for λ 2 ( u , v ) is given in Ripley () and Robeson et al (). Note however that the pair correlation function may not exist in general.…”

Section: Summary Functionsmentioning

confidence: 99%

“…Point process techniques routinely deal with data observed in a restricted subset of the spatial domain. However, statistical techniques for point processes on the sphere (Ripley, ; Bahcall & Soneira, ; Scott & Tout, ; Raskin, ; Robeson et al, ; Møller et al, ) are still surprisingly underdeveloped. In principle, this is just a matter of adapting the existing methodology for point patterns in two‐dimensional space (Diggle, ; Møller & Waagepetersen, ; Baddeley et al, ) to the sphere.…”

Section: Introductionmentioning

confidence: 99%

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Point pattern analysis on a region of a sphere

Lawrence

Baddeley

Milne

et al. 2016

Stat

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K (r) = \frac{1}{λ} double-struckE ([], \sum_{x \in bold-italicX ∖ {u}} bold1 ({}, d (u, x) \leq r) 2.03em | u \in bold-italicX), 0 \leq r \leq π,

where λ is theintensity of X and u is any point in

{double-struckS}^{2}

; the vertical bar indicates that the right side is an expectation with respect to the Palm distribution P u .…”

Section: Summary Functionsmentioning

confidence: 99%

Section: Summary Functionsmentioning

confidence: 99%

“…If X is homogeneous and has a second‐moment intensity function λ 2 ( u , v ), then, from ,

K (r) = \frac{1}{λ^{2} | B |} \int_{B} \int_{{double-struckS}^{2}} bold1 ({}, d (u, v) \leq r) λ_{2} (u, v) d u d v = \frac{1}{| B |} \int_{B} \int_{{double-struckS}^{2}} bold1 ({}, d (u, v) \leq r) g (d (u, v)) d u d v,

where g ( r ) is the pair correlation function. By a change of variables,

bold-italicK (r) = 2 π \int_{0}^{r} g (t) \sin t d t, 0 \leq r \leq π,

and differentiating gives

\begin{matrix} g (r) = \frac{K^{'} (r)}{2 π \sin r}, 0 < r < π; \end{matrix}

the corresponding expression for λ 2 ( u , v ) is given in Ripley () and Robeson et al (). Note however that the pair correlation function may not exist in general.…”

Section: Summary Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Point pattern analysis on a region of a sphere

Lawrence

Baddeley

Milne

et al. 2016

Stat

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show abstract

“…The analysis of spatial point patterns came to prominence in geography during the late 1950s and early 1960s (Gatrell, et al, 1995). Spatial point patterns can be characterized as ranging from dispersed to clustered, with random point patterns having elements of both dispersion and clustering (Robeson, et al, 2014). It has been proven that spatial pattern analysis can be used on different problems.…”

Section: Point Pattern Analysismentioning

confidence: 99%

Using spatial point pattern analysis as supplement for object-based image classification of tree clusters

Tanada¹,

Blanco²

2016

GEOBIA 2016: Solutions and Synergies

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ABSTRACT:Single tree detection is important for monitoring tree plantations in order to estimate crop yield, which can be used for assessing food and biofuels. In the field of remote sensing, there have been studies regarding single tree extraction using Light Detection and Ranging (LiDAR) point cloud data. Canopy Height Model (CHM) can be simply derived from these LiDAR data where trees can be analysed using different physical characteristics such as tree crown width, and height. However, using these physical characteristics can't be very useful for identifying which trees are within a plantation, a forest, or a single tree. This is the motivation for developing algorithms of automated differentiation of random trees from plantation trees. Object Based Image Analysis (OBIA) was used to acquire the locations of individual trees. Trees were separated from other land cover in the image using a CHM derived from LiDAR point cloud data. Using watershed algorithm in eCognition, trees with good canopy spacing was easily distinguished. A spatial point pattern analysis was done from the extracted locations of trees to differentiate regularly patterned points from randomly distributed points. Nearest neighbour statistics was done to provide measurement of the distribution of the extracted points. The ZScore parameter from nearest neighbor was then added to the layers in eCognition as supplement to the available height derivatives such as mean, standard deviation, and different textural parameters. Supervised type of classification was used in order to classify the trees on the image. Training samples of each tree from the image were entered in Support Vector Machine (SVM) to classify the different trees. Two sets of validation were done to compare the results of using the object parameters only, and the object parameters and Z-Score. Addition of the Z-Score from the object parameters increased the obtained accuracy on both validation sets. Classification accuracy based on validation set 1 and set 2 increased by about 20% and 6%, respectively. Additional spatial pattern analysis methods and validation can further prove the use of such layers in improving classification of trees.

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Point pattern analysis and classification on compact two-point homogeneous spaces evolving time

Frías

Torres

Ruiz-Medina

2023

Stoch Environ Res Risk Assess

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This paper introduces a new modeling framework for the statistical analysis of point patterns on a manifold defined by a connected and compact two-point homogeneous space, including the special case of the sphere. The presented approach is based on temporal Cox processes driven by a -valued log-intensity. Different aggregation schemes on the manifold of the spatiotemporal point-referenced data are implemented in terms of the time-varying discrete Jacobi polynomial transform of the log-risk process. The n -dimensional microscale point pattern evolution in time at different manifold spatial scales is then characterized from such a transform. The simulation study undertaken illustrates the construction of spherical point process models displaying aggregation at low Legendre polynomial transform frequencies (large scale), while regularity is observed at high frequencies (small scale). K -function analysis supports these results under temporal short, intermediate and long range dependence of the log-risk process.

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Point-pattern analysis on the sphere

Cited by 24 publications

References 29 publications

Point pattern analysis on a region of a sphere

Point pattern analysis on a region of a sphere

Using spatial point pattern analysis as supplement for object-based image classification of tree clusters

Point pattern analysis and classification on compact two-point homogeneous spaces evolving time

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