Stochastic smoothing of point processes for wildlife-vehicle collisions on road networks

AbstractWildlife-vehicle collisions on road networks represent a natural problem between human populations and the environment, that affects wildlife management and raise a risk to the life and safety of car drivers. We propose a statistically principled method for kernel smoothing of point pattern data on a linear network when the first-order intensity depends on covariates. In particular, we present a consistent kernel estimator for the first-order intensity function that uses a convenient relationship between the intensity and the density of events location over the network, which also exploits the theoretical relationship between the original point process on the network and its transformed process through the covariate. We derive the asymptotic bias and variance of the estimator, and adapt some data-driven bandwidth selectors to estimate the optimal bandwidth. The performance of the estimator is analysed through a simulation study under inhomogeneous scenarios. We present a real data analysis on wildlife-vehicle collisions in a region of North-East of Spain.

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Mapping the Intensity Function of a Non-stationary Point Process in Unobserved Areas

10.21203/rs.3.rs-1119582/v1 ◽

2021 ◽

Author(s):

Edith Gabriel ◽

Francisco Rodriguez-Cortes ◽

Jérôme Coville ◽

Jorge Mateu ◽

Joël Chadoeuf

Keyword(s):

Point Process ◽

Stationary Point ◽

Point Processes ◽

Pair Correlation ◽

General Setting ◽

Real Data ◽

Intensity Function ◽

Fredholm Equation ◽

Point Pattern ◽

Stationary Point Process

Abstract Seismic networks provide data that are used as basis both for public safety decisions and for scientific research. Their configuration affects the data completeness, which in turn, critically affects several seismological scientific targets (e.g., earthquake prediction, seismic hazard...). In this context, a key aspect is how to map earthquakes density in seismogenic areas from censored data or even in areas that are not covered by the network. We propose to predict the spatial distribution of earthquakes from the knowledge of presence locations and geological relationships, taking into account any interactions between records. Namely, in a more general setting, we aim to estimate the intensity function of a point process, conditional to its censored realization, as in geostatistics for continuous processes. We define a predictor as the best linear unbiased combination of the observed point pattern. We show that the weight function associated to the predictor is the solution of a Fredholm equation of second kind. Both the kernel and the source term of the Fredholm equation are related to the first- and second-order characteristics of the point process through the intensity and the pair correlation function. Results are presented and illustrated on simulated non-stationary point processes and real data for mapping Greek Hellenic seismicity in a region with unreliable and incomplete records.

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Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study

Axioms ◽

10.3390/axioms10020102 ◽

2021 ◽

Vol 10 (2) ◽

pp. 102

Author(s):

Maya Briani ◽

Emiliano Cristiani ◽

Paolo Ranut

Keyword(s):

Road Network ◽

Real Data ◽

Second Order ◽

Heavy Vehicles ◽

First Order ◽

Multi Scale ◽

Stop And Go ◽

Scale Models ◽

Road Space

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste–Venice and its branches, provided by Autovie Venete S.p.A.

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A new metric between distributions of point processes

Advances in Applied Probability ◽

10.1017/s0001867800002731 ◽

2008 ◽

Vol 40 (03) ◽

pp. 651-672 ◽

Cited By ~ 1

Author(s):

Dominic Schuhmacher ◽

Aihua Xia

Keyword(s):

Point Processes ◽

Relative Difference ◽

Continuous Functions ◽

Upper And Lower Bounds ◽

Point Pattern ◽

Multiple Point ◽

Finite Point ◽

Lipschitz Continuous Functions ◽

Lipschitz Continuous ◽

Comprehensive Collection

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅ 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅ 1 and its induced Wasserstein metric d̅ 2 for point process distributions are given, including examples of useful d̅ 1-Lipschitz continuous functions, d̅ 2 upper bounds for the Poisson process approximation, and d̅ 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅ 1 in applications.

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A stochastic process whose successive intervals between events form a first order Markov chain — I

Journal of Applied Probability ◽

10.1017/s0021900200114470 ◽

1968 ◽

Vol 5 (03) ◽

pp. 648-668

Author(s):

D. G. Lampard

Keyword(s):

Markov Chain ◽

Point Processes ◽

Electronic Device ◽

Statistical Properties ◽

Time Intervals ◽

First Order ◽

Stochastic Point Process ◽

Order Markov Chain ◽

The One ◽

Stochastic Point Processes

In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind. Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

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First-order autoregressive gamma sequences and point processes

Advances in Applied Probability ◽

10.2307/1426429 ◽

1980 ◽

Vol 12 (3) ◽

pp. 727-745 ◽

Cited By ~ 196

Author(s):

D. P. Gaver ◽

P. A. W. Lewis

Keyword(s):

Parameter Estimation ◽

Point Processes ◽

Random Variables ◽

Innovation Process ◽

Sums Of Random Variables ◽

First Order ◽

Wide Range ◽

Serial Correlations ◽

Generating Sequences ◽

Stieltjes Transforms

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

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First-order autoregressive gamma sequences and point processes

Advances in Applied Probability ◽

10.1017/s0001867800035473 ◽

1980 ◽

Vol 12 (03) ◽

pp. 727-745 ◽

Cited By ~ 35

Author(s):

D. P. Gaver ◽

P. A. W. Lewis

Keyword(s):

Parameter Estimation ◽

Point Processes ◽

Random Variables ◽

Innovation Process ◽

Sums Of Random Variables ◽

First Order ◽

Wide Range ◽

Serial Correlations ◽

Generating Sequences ◽

Stieltjes Transforms

It is shown that there is an innovation process {∊ n } such that the sequence of random variables {X n } generated by the linear, additive first-order autoregressive scheme X n = pXn-1 + ∊ n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

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Determination of the Order of a Markov Chain for Daily Rainfall Data of North East India: "Application AIC criterion"

Journal of Applied and Natural Science ◽

10.31018/jans.v10i1.1583 ◽

2018 ◽

Vol 10 (1) ◽

pp. 80-87

Author(s):

Surobhi Deka

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Akaike Information Criterion ◽

Markov Chain Model ◽

Information Criterion ◽

Chain Model ◽

Adequate Model ◽

First Order ◽

North East India ◽

North East

The paper aims at demonstrating the application of the Akaike information criterion to determine the order of two state Markov chain for studying the pattern of occurrence of wet and dry days during the rainy season (April to September) in North-East India. For each station, each day is classified as dry day if the amount of rainfall is less than 3 mm and wet day if the amount of rainfall is greater than or equal to 3 mm. We apply Markov chain of order up to three to the sequences of wet and dry days observed at seven distantly located stations in North East region of India. The Markov chain model of appropriate order for analyzing wet and dry days is determined. This is done using the Akaike Information Criterion (AIC) by checking the minimum of AIC estimate. Markov chain of order one is found to be superior to the majority of the stations in comparison to the other order Markov chains. More precisely, first order Markov chain model is an adequate model for the stations North Bank, Tocklai, Silcoorie, Mohanbari and Guwahati. Further, it is observed that second order and third order Markov chains are competing with first order in the stations Cherrapunji and Imphal, respectively. A fore-knowledge of rainfall pattern is of immense help not only to farmers, but also to the authorities concerned with planning of irrigation schemes. The outcomes are useful for taking decisions well in advance for transplanting of rice as well as for other input management and farm activities during different stages of the crop growing season.

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The Monotonic Polynomial Graded Response Model: Implementation and a Comparative Study

Applied Psychological Measurement ◽

10.1177/0146621620909897 ◽

2020 ◽

Vol 44 (6) ◽

pp. 465-481

Author(s):

Carl F. Falk

Keyword(s):

Latent Variable ◽

Kernel Smoothing ◽

Marginal Likelihood ◽

Expectation Maximization Algorithm ◽

Real Data ◽

Response Model ◽

Response Functions ◽

Graded Response Model ◽

Graded Response ◽

Maximum Marginal Likelihood

We present a monotonic polynomial graded response (GRMP) model that subsumes the unidimensional graded response model for ordered categorical responses and results in flexible category response functions. We suggest improvements in the parameterization of the polynomial underlying similar models, expand upon an underlying response variable derivation of the model, and in lieu of an overall discrimination parameter we propose an index to aid in interpreting the strength of relationship between the latent variable and underlying item responses. In applications, the GRMP is compared to two approaches: (a) a previously developed monotonic polynomial generalized partial credit (GPCMP) model; and (b) logistic and probit variants of the heteroscedastic graded response (HGR) model that we estimate using maximum marginal likelihood with the expectation–maximization algorithm. Results suggest that the GRMP can fit real data better than the GPCMP and the probit variant of the HGR, but is slightly outperformed by the logistic HGR. Two simulation studies compared the ability of the GRMP and logistic HGR to recover category response functions. While the GRMP showed some ability to recover HGR response functions and those based on kernel smoothing, the HGR was more specific in the types of response functions it could recover. In general, the GRMP and HGR make different assumptions regarding the underlying response variables, and can result in different category response function shapes.

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Selective interaction of a Poisson and renewal process: first-order stationary point results

Journal of Applied Probability ◽

10.1017/s0021900200034938 ◽

1970 ◽

Vol 7 (02) ◽

pp. 359-372 ◽

Cited By ~ 7

Author(s):

A. J. Lawrance

Keyword(s):

Stationary Point ◽

Joint Distribution ◽

Point Processes ◽

First Order ◽

Response Interval ◽

Selective Interaction ◽

Counting Distribution ◽

Stationary Point Process ◽

Equilibrium Process ◽

Stationary Point Processes

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

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