A unified framework on defining depth for point process using function smoothing

Xu, Zishen; Wang, Chenran; Wu, Wei

doi:10.1016/j.csda.2022.107545

Cited by 2 publications

(4 citation statements)

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“…We note that if k = 0, then f s (x, y) = 0 is a constant function in F 0 . Similarly to the result on the temporal point process in [18], we can show that the mapping from the spatial point process space S to the space of the smoothed process F is a bijection. Mathematical details are given in Appendix B.…”

Section: Mapping Between Spatial Point Process and Bivariate Functionsupporting

confidence: 58%

“…In this paper, the transformed function is called the smoothed function or the smoothed point process. Xu et al [18] first adopted a Gaussian kernel function for the temporal point process within a given domain, and then used the conventional L 2 metric on the smoothed functions. However, this distance involves numerical integration as there is no closed-form expression available in general.…”

Section: Mapping Between Spatial Point Process and Bivariate Functionmentioning

confidence: 99%

“…The generalized Mahalanobis depth [16] was the first depth method defined on the temporal point process. In Qi et al [17], Dirichlet depth was proposed to overcome the boundary issue, and in Xu et al [18] a smoothing approach was adopted to define depth using a functional depth on the smoothed process. In Zhou et al [19], ILR depth was developed via the classical Isometric Log-Ratio (ILR) transformation to address the non-Euclidean issues in the point process space.…”

Section: Introductionmentioning

confidence: 99%

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Statistical Depth in Spatial Point Process

Zhou,

2024

Mathematics

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Statistical depth is widely used as a powerful tool to measure the center-outward rank of multivariate and functional data. Recent studies have introduced the notion of depth to the temporal point process, which exhibits randomness in the cardinality as well as distribution in the observed events. The proposed methods can well capture the rank of a point process in a given time interval, where a critical step is to measure the rank by using inter-arrival events. In this paper, we propose to extend the depth concept to multivariate spatial point process. In this case, the observed process is in a multi-dimensional location and there are no conventional inter-arrival events in the temporal process. We adopt the newly developed depth in metric space by defining two different metrics, namely the penalized metric and the smoothing metric, to fully explore the depth in the spatial point process. The mathematical properties and the large sample theory, as well as depth-based hypothesis testings, are thoroughly discussed. We then use several simulations to illustrate the effectiveness of the proposed depth method. Finally, we apply the new method in a real-world dataset and obtain desirable ranking performance.

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Section: Mapping Between Spatial Point Process and Bivariate Functionsupporting

confidence: 58%

Section: Mapping Between Spatial Point Process and Bivariate Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Statistical Depth in Spatial Point Process

Zhou,

2024

Mathematics

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“…To address these issues, we propose a new, rank-based likelihood framework for modeling the spike train data. Our framework is motivated by recent developments in center-outward ranks on point process [5][6][7]. Our proposed model describes the likelihood of a spike train as the product of two terms.…”

Section: Introductionmentioning

confidence: 99%

A novel point process model for neuronal spike trains

Ma,

2024

Front. Appl. Math. Stat.

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Point process provides a mathematical framework for characterizing neuronal spiking activities. Classical point process methods often focus on the conditional intensity function, which describes the likelihood at any time point given its spiking history. However, these models do not describe the central tendency or importance of the spike train observations. Based on the recent development on the notion of center-outward rank for point process, we propose a new modeling framework on spike train data. The new likelihood of a spike train is a product of the marginal probability on the number of spikes and the probability of spike timings conditioned on the same number. In particular, the conditioned distribution is calculated by adopting the well-known Isometric Log-Ratio transformation. We systematically compare the new likelihood with the state-of-the-art point process likelihoods in terms of ranking, outlier detection, and classification using simulations and real spike train data. This new framework can effectively identify templates as well as outliers in spike train data. It also provides a reasonable model, and the parameters can be efficiently estimated with conventional maximum likelihood methods. It is found that the proposed likelihood provides an appropriate ranking on the spike train observations, effectively detects outliers, and accurately conducts classification tasks in the given data.

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A unified framework on defining depth for point process using function smoothing

Cited by 2 publications

References 21 publications

Statistical Depth in Spatial Point Process

Statistical Depth in Spatial Point Process

A novel point process model for neuronal spike trains

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