Bivariate Point Process Research Articles

Nonparametric estimation and forecasting of interval-valued time series regression models with constraints

Nowadays information technology advances allow the collecting and storage of large complex datasets in many areas. Modeling and forecasting interval-valued time series (ITS) has drawn much attention over the last two decades because interval-valued observations contain more information than point-valued observations over the same period and remove undesirable noises in high-frequency data. However, most work mainly focuses on modeling a linear univariate ITS or bivariate point process. This paper proposes nonparametric regression models for interval-valued time series with imposing constraints, e.g., monotonicity. This setting with a monotonic constraint is consistent with the existing literature, which focuses on incorporating valuable empirical information in modeling and forecasts. Two constraint estimators are developed and asymptotic properties are established. Monte Carlo simulation is conducted to show the finite sample performance. An empirical application to equity premium documents that the proposed model yields a better forecast performance than some popular models in the literature.

A mechanistic bivariate point process model for crime pattern analysis

The statistical analysis of crime data has gained attention in the last decade. In particular, the availability of spatio‐temporal crime data at the event level allows us to model the incidence of crime with high precision. Point process models are the natural tool to study crime patterns. As it is well‐known that crime events often spread as a contagion process, mechanistic self‐exciting models are usually considered in this context. In this paper, we propose a mechanistic bivariate spatio‐temporal model for the first‐order intensity function of the point processes associated with the intensity of two crime types. Specifically, the model includes separate estimates of the overall temporal and spatial intensities of crime and a spatio‐temporal interaction term for each of the crime types under analysis. Regarding the spatio‐temporal term, we model how the occurrence of previous crime events (from any of the two types) influences the intensity of each type of crime under study. We consider a dataset of crime events recorded in Valencia (Spain) during the year 2017 and focus on two crime types for the analysis: property crime and robbery. The results show that there is an association between the recent occurrence of either property crimes or robberies and the intensity of both crime types. Several spatio‐temporal monitoring tools are described and discussed as well.

Asymptotic Dependence Modelling of the BRICS Stock Markets

With the use of empirical data, this paper focuses on solving financial and investment issues involving extremal dependence of 10 pairwise combinations of the 5 BRICS (Brazil, Russia, India, China, and South Africa) stock markets. Daily closing equity indices from 5 January 2010 to 6 August 2018 are used in the study. Unlike previous literature, we use bivariate point process and conditional multivariate extreme value models to investigate the extremal dependence of the stock market returns. However, it is observed that the point process was able to model many more extreme observations or exceedances that contribute to the likelihood estimation. It gives more information than the threshold excess method of the CMEV model. This study shows varying levels of low extremal dependence structure whose outcomes are highly beneficial to investors, portfolio managers and other market participants interested in maximising investment returns and financial gains.

Recovering individual-level spatial inference from aggregated binary data

Binary regression models are commonly used in disciplines such as epidemiology and ecology to determine how spatial covariates influence individuals. In many studies, binary data are shared in a spatially aggregated form to protect privacy. For example, rather than reporting the location and result for each individual that was tested for a disease, researchers may report that a disease was detected or not detected within geopolitical units. Often, the spatial aggregation process obscures the values of response variables, spatial covariates, and locations of each individual, which makes recovering individual-level inference difficult. We show that applying a series of transformations, including a change of support, to a bivariate point process model allows researchers to recover individual-level inference for spatial covariates from spatially aggregated binary data. The series of transformations preserves the convenient interpretation of desirable binary regression models that are commonly applied to individual-level data. Using a simulation experiment, we compare the performance of our proposed method under varying types of spatial aggregation against the performance of standard approaches using the original individual-level data. We illustrate our method by modeling individual-level probability of infection using a data set that has been aggregated to protect an at-risk and endangered species of bats. Our simulation experiment and data illustration demonstrate the utility of the proposed method when access to original non-aggregated data is impractical or prohibited.

A Method of Estimating the Partial Power Spectrum of a Bivariate Point Process and an Application to a Neurophysiological Data Set

In this paper, we present a method of estimating the partial power spectrum of a bivariate point process. The method is based on the simple and cross-periodograms which are smoothed by applying a simple averaging weighting scheme in order to improve the properties of the estimate. The asymptotic distribution of the estimate is shown to be approximately normal when the amount of smoothing is large. An illustrative example is presented from the field of Neurophysiology when the complex system of the muscle spindle is affected by a static gamma motoneuron.

Saltcedar (Tamarix mascatensis) inhibits growth and spatial distribution of eshnan (Seidlitzia rosmarinus) by enrichment of soil salinity in a semi-arid desert

BACKGROUND AND AIMS: Understanding the species associations in dryland sparse and patchy vegetation requires information on species intra- and interspecific interactions and their response to environmental heterogeneity. To better understand the coexistence mechanisms of saltcedar (Tamarix mascatensis) and eshnan (Seidlitzia rosmarinus) in central deserts of Iran, we analysed the spatial distributions and soil chemical properties of their mixed stands. METHODS: We applied uni- and bivariate point process statistics together with mark correlation functions and soil information under the species canopy and in the interspaces to evaluate plant-to-plant interactions. RESULTS: Tamarix and Seidlitzia were both highly aggregated in conspecific clusters, and the species were significantly segregated at small spatial scales. Interestingly, we found that the density of conspecific neighbours in Tamarix clusters was higher than Seidlitzia (approximately 9 times more), although the mean density of Tamarix (29.4 shrubs ha⁻¹) was lower than Seidlitzia (357.3 shrubs ha⁻¹). For both species, intraspecific interactions had negative effects on their size, which was more evident in Tamarix. Our results showed that the size of Seidlitzia was strongly and negatively depended on the proximity of nearby Tamarix. Additionally, the two species improved soil chemical properties and moisture under their canopy compared to bare soil; however, soil salinity under Tamarix (24.29 dS m⁻¹) was higher than Seidlitzia (1.68 dS m⁻¹). CONCLUSIONS: According to the segregated spatial pattern of the species and negative effects of Tamarix neighbourhood on Seidlitzia growth, we suggest that high soil salinity beneath Tamarix canopy likely reduces interspecific interactions between the two species and affects Seidlitzia growth. Furthermore, our results indicate no spatial mixture of Tamarix and Seidlitzia, contrary to our expectations, although we still encourage further research on the co-occurrence of the species in the region.

Improving heart rate estimation in preterm infants with bivariate point process analysis of heart rate and respiration.

Accurate estimation of heart rate dynamics in preterm infants is important for predicting recurrent episodes of severe bradycardia. We hypothesize that estimation of heart rate can be improved by including respiration as a state variable, based on mechanisms that underlie cardio-respiratory coherence. For ten preterm infants, we demonstrate that including respiration as a covariate improves estimation accuracy by an average of 11% across bradycardia severity, and reduces the maximum error by 8%. We also find that cardio-respiratory coherence increases in low frequency content just prior to severe bradycardia. Thus, incorporating respiratory information may improve models of heart rate dynamics and narrow potential features for bradycardia prediction.

Unraveling dynamics of human physical activity patterns in chronic pain conditions

Chronic pain is a complex disabling experience that negatively affects the cognitive, affective and physical functions as well as behavior. Although the interaction between chronic pain and physical functioning is a well-accepted paradigm in clinical research, the understanding of how pain affects individuals' daily life behavior remains a challenging task. Here we develop a methodological framework allowing to objectively document disruptive pain related interferences on real-life physical activity. The results reveal that meaningful information is contained in the temporal dynamics of activity patterns and an analytical model based on the theory of bivariate point processes can be used to describe physical activity behavior. The model parameters capture the dynamic interdependence between periods and events and determine a ‘signature’ of activity pattern. The study is likely to contribute to the clinical understanding of complex pain/disease-related behaviors and establish a unified mathematical framework to quantify the complex dynamics of various human activities.

Measuring the impact of option market activity on the stock market: Bivariate point process models of stock and option transactions

I apply the bivariate Autoregressive Conditional Duration model of Engle and Lunde [2003. Trade and quotes: a bivariate point process. Journal of Financial Econometrics 1, 159–188] to stock and option market transactions. The first model uses option trades and stock trades. Shocks to option trade/option trade durations have a significant impact on option trade/stock trade durations. Higher implied volatility, larger stock and option market order imbalances, larger stock trades, larger spreads, smaller depths in the stock market and faster trading in the stock and option markets are all associated with faster trading in both markets. In the second model, option trade/option trade timing leads option trade/stock quote timing and several information-related stock and option market covariates impact the expected inter-market event durations.

Modelling security market events in continuous time: Intensity based, multivariate point process models

A continuous time econometric modelling framework for multivariate financial market event (or ‘transactions’) data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A bivariate point process model of the timing of trades and mid-quote changes is then presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes in continuous time is found to be important empirically.

Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models

A continuous time econometric modelling framework for multivariate financial market event (or 'transactions') data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Sufficient conditions for identification of linear Hawkes models and for stationarity of some new non-linear Hawkes models are derived. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A computationally efficient thinning algorithm for simulation of generalised Hawkes processes is also developed. A bivariate point process model of the timing of trades and mid-quote changes is presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes is found to be important empirically.

Nonparametric One‐way Analysis of Variance of Replicated Bivariate Spatial Point Patterns

AbstractA common problem in neuropathological studies is to assess the spatial patterning of cells on tissue sections and to compare spatial patterning between disorder groups. For a single cell type, the cell positions constitute a univariate point process and interest focuses on the degree of spatial aggregation. For two different cell types, the cell positions constitute a bivariate point process and the degree of spatial interaction between the cell types is of interest. We discuss the problem of analysing univariate and bivariate spatial point patterns in the one‐way design where cell patterns have been obtained for groups of subjects. A bootstrapping procedure to perform a nonparametric one‐way analysis of variance of the spatial aggregation of a univariate point process has been suggested by Diggle, Lange and Beneš (1991). We extend their replication‐based approach to allow the comparison of the spatial interaction of two cell types between groups, to include planned comparisons (contrasts) and to assess whole groups against complete spatial randomness and spatial independence. We also accommodate several replicate tissue sections per subject. An advantage of our approach is that it can be applied when processes are not stationary, a common problem in brain tissue sections since neurons are arranged in cortical layers. We illustrate our methods by applying them to a neuropathological study to investigate abnormalities in the functional relationship between neurons and astrocytes in HIV associated dementia. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Trades and Quotes: A Bivariate Point Process

This article formulates a bivariate point process to jointly analyze trade and quote arrivals. In microstructure models, trades may reveal private information that is then incorporated into new price quotes. This article examines the speed of this information flow and the circumstances that govern it. A joint likelihood function for trade and quote arrivals is specified in a way that recognizes that an intervening trade sometimes censors the time between a trade and the subsequent quote. Models of trades and quotes are estimated for eight stocks using Trade and Quote database (TAQ) data. The essential finding for the arrival of price quotes is that information flow variables, such as high trade arrival rates, large volume per trade, and wide bid–ask spreads, all predict more rapid price revisions. This means prices respond more quickly to trades when information is flowing so that the price impacts of trades and ultimately the volatility of prices are high in such circumstances.

MORPHOMETRIC ANALYSIS OF HUMAN CORNEAL ENDOTHELIUM BY MEANS OF SPATIAL POINT PATTERNS

This paper presents a method for detecting abnormalities in spatial arrangements of cells within any tissue that can be described by different sets of relevant points. The method has been applied to the detection of subtle abnormalities in corneal endothelia. Images of this type of tissue can be characterized by two types of points: cell centroids and triple points associated with the apical intersections as it was proposed by Díaz.7 Both types of points jointly considered are modeled using a bivariate spatial point process; then a statistical analysis based on certain distributional descriptors proposed by Doguwa4,9 is carried out to discriminate severe and subtle abnormalities from controls. According to the results, the method constitutes a valid alternative to currently used techniques, which involve statistical descriptions of cell density, cell area and hexagonality, that fail in the detection of these subtle abnormalities.

Bivariate stochastic modelling of ephemeral streamflow

AbstractStreamflow time series in arid and semi‐arid regions can be characterized as a sequence of single discrete flow episodes or clusters of hydrographs separated by periods of zero discharge. Here, two point process models are presented for the joint occurrence of flow events at neighbouring river sites. The first allows for excess clustering by adding autocorrelated errors to an empirically derived seasonally varying probability of an event and is extended to the case of the joint occurrence of flow events in two catchments. The second approach is to explicitly model the occurrences of clusters of events and the bivariate point process of event occurrences within them at both sites. For the two models, the magnitude of event peaks are assumed to be drawn from continuous distributions with seasonally varying parameters. Rises and recessions in discharge are interpolated between the peaks using regression estimates of hydrographs. The models are fitted to mean daily flows at two sites in Namibia and demonstrated to provide realistic simulations of the hydrology. Copyright © 2002 John Wiley & Sons, Ltd.

Robust inference for bivariate point processes

AbstractIn the analysis of recurrent events where the primary interest lies in studying covariate effects on the expected number of events occurring over a period of time, it is appealing to base models on the cumulative mean function (CMF) of the processes (Lawless & Nadeau 1995). In many chronic diseases, however, more than one type of event is manifested. Here we develop a robust inference procedure for joint regression models for the CMFs arising from a bivariate point process. Consistent parameter estimates with robust variance estimates are obtained via unbiased estimating functions for the CMFs. In most situations, the covariance structure of the bivariate point processes is difficult to specify correctly, but when it is known, an optimal estimating function for the CMFs can be obtained. As a convenient model for more general settings, we suggest the use of the estimating functions arising from bivariate mixed Poisson processes. Simulation studies demonstrate that the estimators based on this working model are practically unbiased with robust variance estimates. Furthermore, hypothesis tests may be based on the generalized Wald or generalized score tests. Data from a trial of patients with bronchial asthma are analyzed to illustrate the estimation and inference procedures.

Characterization Results and Markov Chain Monte Carlo Algorithms Including Exact Simulation for Some Spatial Point Processes

The area-interaction process and the continuum random-cluster model are characterized in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a two-component Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a Swendsen-Wang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.

Perfect simulation of some point processes for the impatient user

Recently Propp and Wilson [14] have proposed an algorithm, called coupling from the past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect sampling using CFTP has been successfully extended to the context of point processes by, amongst other authors, Häggström et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similarly to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to an infinite state space context to produce an exact sample of the penetrable spheres mixture process and related models. Following [5] we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects against bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process.

Perfect simulation of some point processes for the impatient user

Recently Propp and Wilson [14] have proposed an algorithm, called coupling from the past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect sampling using CFTP has been successfully extended to the context of point processes by, amongst other authors, Häggström et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similarly to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to an infinite state space context to produce an exact sample of the penetrable spheres mixture process and related models. Following [5] we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects against bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process.

Correlations in bivariate point processes: some biological applications

Let (Φ1 Φ2) be a bivariate point process. Let be the probability that (Φ1 Φ2) - (0, s) (the process without the 0 and s points) verify U when we have a point of <1>1 in the origin and a point of Φ2 in s. This is the reduced cross Palm distribution. Some correlation measures for bivariate point processes based on this reduced cross Palm distribution are proposed. Their estimators and expressions under the independence and the random labelling hypothesis are considered. The differences and improvements with respect to the cross intensity function and its integrated version, the cross function (Stoyan et al. 1987), are studied. Some Monte Carlo tests for testing the independence and the random labelling hypothesis are proposed. They are applied to real bivariate point patterns: positions of hickories and maples in the Lansing Woods (Diggle 1983) and cases and controls of childhood leukaemia and lymphoma in North Humberside (Cuzick and Edwards 1990).