Brownian Bridge Research Articles

Limit Distribution of the φ-Divergence Based Change Point Estimator

The assumption of stationarity is too restrictive especially for long time series. This paper studies the change point problem through a change point estimator based on the φ-divergence which provides a rich set of distance like measures between pairs of distributions. The change point problem is considered in the following sub-fields: the problem of divergence estimation, testing for the homogeneity between two samples as well as estimating the time of change. The asymptotic distribution of the change point estimator is estimated by the limiting distribution of a stochastic process within given bounds through asymptotic theory surrounding the likelihood theory. The distribution is found to converge to that of a standardized Brownian bridge process.

On the Bernstein-von Mises theorem for the Dirichlet process

We establish that Laplace transforms of the posterior Dirichlet process converge to those of the limiting Brownian bridge process in a neighbourhood about zero, uniformly over Glivenko-Cantelli function classes. For real-valued random variables and functions of bounded variation, we strengthen this result to hold for all real numbers. This last result is proved via an explicit strong approximation coupling inequality.

Time and place of the maximum for one-dimensional diffusion bridges and meanders

For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and generalized Bessel meanders. We discuss the limits of the application of this method for other diffusion processes.

Simulation and tracking of fractional particles motion. From microscopy video to statistical analysis. A Brownian bridge approach

An ongoing rapid development in single particle tracking techniques has opened new possibilities for analysis of particles dynamics inside living cells. Assuming that the motion is governed by a fractional Brownian motion, we have generated a synthetic video resembling a real one from an experimental video of G-proteins and coupled with them receptors inside living cells. Next, we applied Brownian bridge method to study two fundamental analysis tasks on trajectory data: segmentation and classification in context of experimental data. We have shown that, depending on the method of dealing with missing data, the obtained results might vary significantly, obviously influencing the final conclusions.

African elephant (Loxodonta africana) select less fragmented landscapes to connect core habitats in human‐dominated landscapes

AbstractAfrican elephants (Loxodonta africana) utilise corridors to access limited resources, that is forage and water scattered across heterogeneous habitats they roam. The existence of small elephant metapopulations depend on the intactness of these corridors to access the scarce resources. Due to the sedentarisation of the previously nomadic Maasai people, elephant corridors have been exposed to increased fragmentation from human‐induced activities across the Amboseli ecosystem in Kenya. In this study, we sought to compare the scale of fragmentation between corridors and their immediate landscapes (noncorridors) in the Amboseli ecosystem, Kenya. We used a Brownian Bridge Movement Model (BBMM) to identify corridors used by elephants from global positioning system (GPS) collar data. The scale of fragmentation between corridors and noncorridors was determined using the effective mesh size fragmentation metric (meff). Our results showed that elephant corridors were significantly less fragmented (Wilcoxon sum rank test: W = 6,121.5, p < 0.05) when compared to the noncorridors. The presence of fragmentation geometries in the corridors remains a major cause of concern for wildlife managers as they have the potential to invade and constrict the existing corridors. Our results underscore the need to extend management of elephant habitats to migration corridors outside protected areas.

Spatial Ecology of Asian Water Monitors Adjacent to a Sea Turtle Nesting Beach.

Nest predation is the main cause of hatching failure for many turtle populations. For green turtles (Chelonia mydas) nesting at Chagar Hutang in Redang Island, Malaysia, Asian water monitors (Varanus salvator) are a potential nest predator. However, no studies have documented the space use of this species in coastal habitat adjacent to a sea turtle nesting beach to assess its potential impact on turtle nests. Here, we used Global Positioning System (GPS) data loggers to quantify space use of Asian water monitors in order to establish the extent to which they use sea turtle nesting areas. Asian water monitors had a diurnal activity pattern and spent most of their time in rain forest habitat behind the sea turtle nesting beach. The home range occupied by Asian water monitors varied between 0.015 and 0.198 km2 calculated by the Kernel Brownian Bridge method. The space use patterns of individual Asian water monitors varied between individuals. Two males had relatively small home ranges, whereas one male and the female had a relatively large home range. Because tracked Asian water monitors in this study rarely visited the sea turtle nesting areas, it is probable that only a few individuals are responsible for opening nests.

The value of the high, low and close in the estimation of Brownian motion

The conditional density of Brownian motion is considered given the max, B(t|max ), as well as those with additional information: B(t|close, max ), B(t|close, max , min ) where the close is the final value: B(t=1)=c and t in [0,1]. The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the statistics: the close (final value), the high (maximum), the low (minimum). Computational results displaying both the expectation and variance in time are presented and compared with the theoretical values. We tabulate the time averaged variance of Brownian motion conditional on knowing various extremal properties of the motion. The final table shows that knowing the high is more useful than knowing the final value among other results. Knowing the open, high, low and close reduces the time averaged variance to 42% of the value of knowing only the open and close (Brownian bridge).

Daily Movements, Space Use, and Habitat Selection of GPS‐tagged Northern Bobwhite in Texas

ABSTRACTDetailed information on movements, space use, and habitat selection patterns by northern bobwhite (Colinus virginianus; hereafter, bobwhite) is important for understanding population dynamics, gene flow, disease spread, and development of habitat management activities. Advances in Global Positioning System (GPS) technology has allowed researchers to collect spatial and temporal location data with increasing accuracy, which in turn has shifted the resolution at which habitat evaluations and management focus can occur. Only recently have GPS tags been deployable for small ground‐nesting birds. We conducted a preliminary examination of the spatial ecology of 74 bobwhites equipped with GPS tags during breeding and covey season on 3 private properties in north‐central Texas, USA, during 2015–2017. Specifically, we derived daily estimates of core area and range size using dynamic Brownian bridge movement models, daily distance traveled, and distance between consecutive roost sites. Bobwhites tended to move more during breeding than covey season. Consecutive roost sites were, on average, approximately 130 m apart. Daily core area and home range size of bobwhites across the calendar year were 1.99 ± 0.13 and 6.29 ± 0.85 ha, respectively. During the breeding season, bobwhites avoided hardwood communities and showed no selection or avoidance of shrub, grass, and herbaceous communities. During covey season, bobwhites selected for herbaceous, grass, and hardwood communities, and neither selected nor avoided shrub communities. Global Positioning System data on a variety of species has become the primary information used for habitat and conservation management decisions, therefore, future research on bobwhite will include the application of GPS, which our results show can increase the temporal resolution of information and be used for habitat conservation and restoration actions. © 2020 The Wildlife Society.

Reptiles on the wrong track? Moving beyond traditional estimators with dynamic Brownian Bridge Movement Models

BackgroundAnimal movement expressed through home ranges or space-use can offer insights into spatial and habitat requirements. However, different classes of estimation methods are currently instinctively applied to answer home range, space-use or movement-based research questions regardless of their widely varying outputs, directly impacting conclusions. Recent technological advances in animal tracking (GPS and satellite tags), have enabled new methods to quantify animal space-use and movement pathways, but so far have primarily targeted mammal and avian species.MethodsMost reptile spatial ecology studies only make use of two older home range estimation methods: Minimum Convex Polygons (MCP) and Kernel Density Estimators (KDE), particularly with the Least Squares Cross Validation (LSCV) and reference (href) bandwidth selection algorithms. These methods are frequently applied to answer space-use and movement-based questions. Reptile movement patterns are unique (e.g., low movement frequency, long stop-over periods), prompting investigation into whether newer movement-based methods –such as dynamic Brownian Bridge Movement Models (dBBMMs)– apply to Very High Frequency (VHF) radio-telemetry tracking data. We simulated movement data for three archetypical reptile species: a highly mobile active hunter, an ambush predator with long-distance moves and long-term sheltering periods, and an ambush predator with short-distance moves and short-term sheltering periods. We compared traditionally used estimators, MCP and KDE, with dBBMMs, across eight feasible VHF field sampling regimes for reptiles, varying from one data point every four daylight hours, to once per month.ResultsAlthough originally designed for GPS tracking studies, dBBMMs outperformed MCPs and KDE href across all tracking regimes in accurately revealing movement pathways, with only KDE LSCV performing comparably at some higher frequency sampling regimes. However, the LSCV algorithm failed to converge with these high-frequency regimes due to high site fidelity, and was unstable across sampling regimes, making its use problematic for species exhibiting long-term sheltering behaviours. We found that dBBMMs minimized the effect of individual variation, maintained low error rates balanced between omission (false negative) and commission (false positive), and performed comparatively well even under low frequency sampling regimes (e.g., once a month).ConclusionsWe recommend dBBMMs as a valuable alternative to MCP and KDE methods for reptile VHF telemetry data, for research questions associated with space-use and movement behaviours within the study period: they work under contemporary tracking protocols and provide more stable estimates. We demonstrate for the first time that dBBMMs can be applied confidently to low-resolution tracking data, while improving comparisons across regimes, individuals, and species.

Stochastic Interpolation of Sparsely Sampled Time Series via Multipoint Fractional Brownian Bridges.

We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of intermediate and nonequidistant points. Determining the optimal value of the Hurst exponent H_{opt}, appropriate to interpolate the sparse signal, is a very important step of our method. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow and discuss the implications of the non-self-similar character of the signal. The method introduced here could be instrumental in several physical problems, including astrophysics, particle tracking, and specific tailoring of surrogate data, as well as in domains of natural and social sciences.

Valuation model for Chinese convertible bonds with soft call/put provision under the hybrid willow tree

The Chinese convertible bond market has been developing rapidly in the last 10 years. However, some special characteristics of the Chinese convertible bond, such as the soft call/put provision, cause huge difficulty in the valuation. In this paper, we establish a new valuation model for the Chinese convertible bond, based on the available Chinese market data, through a hybrid willow tree approach with consideration of the underlying stock price, stochastic interest rate, and credit risk of the issuer. We employ the Brownian bridge to handle the special characteristics. Finally, we examine our model prices for the daily market closing prices for 20 Chinese convertible bonds traded from 2007 to 2017. The empirical results show the effectiveness of our valuation model under the historical and implied volatilities of the underlying stock price.

Identifying the variation in utilization density estimators and home ranges of elephant clans in Aceh, Sumatra, Indonesia

Movement ecology is fundamental to understanding animal home ranges or utilization distribution (UD), and is an important aspect in conservation management strategies. Over the years, there have been several new developments and some contention on which methods are best in determining animal movement and UD. Using data from Global Positioning System (GPS)-tracked Sumatran elephants, minimum convex polygon values (MCP), various Kernel Density Estimator bandwidths (KDE), and dynamic Brownian Bridge Movement Models (dBBMM) were compared to identify the most suitable estimators of space-use. Models were analyzed for variability of home range, goodness of fit, isopleth contour complexity, and precision in representing habitat features. dBBMM was shown to be the most efficient in their representation of elephant home range estimations when compared to other methods in terms of trade-off between type I and type II errors and their ability to classify high- and low-use areas, along with insight into variation of movement. We further discuss the implications of variability in home range estimation regarding conservation and provide recommendations for future studies using similar data.

Individual specialization in the use of space by frugivorous bats.

Natural populations are not homogenous systems but sets of individuals that occupy subsets of the species' niche. This phenomenon is known as individual specialization. Recently, several studies found evidence of individual specialization in animal diets. Diet is a critical dimension of a species' niche that affects several other dimensions, including space use, which has been poorly studied under the light of individual specialization. In this study, which harnesses the framework of the movement ecology paradigm and uses yellow-shouldered bats Sturnira lilium as a model, we ask how food preferences lead individual bats of the same population to forage mainly in different locations and habitats. Ten individual bats were radiotracked in a heterogeneous Brazilian savanna. First, we modelled intraspecific variation in space use as a network of individual bats and the landscape elements visited by them. Second, we developed two novel metrics, the spatial individual specialization index (SpatIS) and the spatial individual complementary specialization index (SpatICS). Additionally, we tested food-plant availability as a driver of interindividual differences in space use. There was large interindividual variation in space use not explained by sex or weight. Our results point to individual specialization in space use in the studied population of S. lilium, most probably linked to food-plant distribution. Individual specialization affects not only which plant species frugivores consume, but also the way they move in space, ultimately with consequences for seed dispersal and landscape connectivity.

Implementing de-biased estimators using mixed sequences

Abstract We describe an implementation of the de-biased estimator using mixed sequences; these are sequences obtained from pseudorandom and low-discrepancy sequences. We use this implementation to numerically solve some stochastic differential equations from computational finance. The mixed sequences, when combined with Brownian bridge or principal component analysis constructions, offer convergence rates significantly better than the Monte Carlo implementation.

Multiple GPS fix intervals show variations in the manner African White-backed Vultures Gyps africanus utilise space

ABSTRACT Application of spatial data is becoming increasingly important as a tool for mapping and understanding space utilisation patterns especially for critically endangered species. In this study, we tested how variation in the GPS fix interval may influence space utilisation patterns by African White-backed Vultures in Hwange National Park and the surrounding areas. In order to test our objectives, we collected data from six African White-backed Vultures using solar powered GPS-GSM units. The data were collected for periods ranging from five months to nineteen months (28 June 2016 to 30 January 2018), resulting in 89 809 total GPS fixes. We tested whether home range and core areas estimated at different fix interval differ significantly. We also tested whether African White-backed Vultures show significant site fidelity and at what sampling frequency the estimated home ranges tend to reach an asymptote. We used total individual data and seasonal data to test these hypotheses. In order to estimate home range size at different fix intervals, we used the Brownian Bridge Movement Model, because unlike other traditional estimators, it explicitly incorporates temporal sequence in GPS fixes, accounts for autocorrelation in movement data, as well as missing GPS data. Home range estimates differed significantly at different GPS fix intervals. Home range and core areas for five vultures were concentrated within conservation areas. African White-backed Vultures also showed significant site fidelity across seasons except for vulture three, which demonstrated nomadic tendencies. Results on home range asymptote were not conclusive, because of unique individual vulture space use patterns at different fix intervals. Our study was the first to track and monitor space utilisation patterns by mature African White-backed Vultures in the Hwange ecosystem. Accordingly, our results have critical conservation implications for the wide-ranging, critically endangered species.

Pricing of Presale Contracts with Macroeconomic Factors and Stochastic Basis Risk

This study analyzes the stochastic basis risk (presale price minus spot housing price) of presale housing from the viewpoint of forward pricing. We employ the modified Brownian bridge process to deal with the time-varying volatility and price convergence of presale and existing houses. The finding reveals an asymmetric effect of macroeconomic factors on the presale price in different market conditions. Such results might reflect the investors’ behaviors of overreaction and/or over-pessimism. In addition, the influence of macroeconomic variables is greater on the basis price than on the prices of existing houses, especially in a depressed housing market. Our results emphasize the importance of considering basis risk and the influence of macroeconomic factors when pricing presale houses to manage housing risk.

Assessing the exposure of three diving bird species to offshore wind areas on the U.S. Atlantic Outer Continental Shelf using satellite telemetry

AbstractAimThe United States Atlantic Outer Continental Shelf (OCS) has considerable offshore wind energy potential. Capturing that resource is part of a broader effort to reduce CO2 emissions. While few turbines have been constructed in U.S. waters, over a dozen currently planned offshore wind projects have the potential to displace marine birds, potentially leading to effective habitat loss. We focused on three diving birds identified in Europe to be vulnerable to displacement. Our research aimed to determine their potential exposure to areas designated or proposed for offshore wind development along the Atlantic OCS.MethodsSatellite tracking technology was used to determine the spatial and temporal use and movement patterns of Surf Scoters (Melanitta perspicillata), Red‐throated Loons (Gavia stellata) and Northern Gannets (Morus bassanus), and calculate their exposure to each offshore wind area. We tagged 236 adults in 2012–2015 on the Atlantic OCS from New Jersey to North Carolina; an additional 147 birds tagged in previous tracking studies were integrated into our analyses. Tracking data were analysed in two‐week intervals using dynamic Brownian bridge movement models to develop composite spatial utilization distributions. For each species, these distributions were then used to calculate the spatio‐temporal exposure to each offshore wind area.ResultsSurf Scoters and Red‐throated Loons were exposed to offshore wind areas almost exclusively during migration because these species were distributed among coastal and inshore waters during winter months. In contrast, Northern Gannets ranged over a much larger area, reaching farther offshore and south in winter, thus exhibited the greatest exposure to extant offshore wind areas.ConclusionsResults of this study provide better understanding of how diving birds use current and potential future offshore wind areas on the Atlantic OCS, and can inform permitting, risk assessment and pre‐ and post‐construction impact assessments of offshore energy infrastructure.

How linear reinforcement affects Donsker\u2019s theorem for empirical processes

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability pin (0,1), {hat{U}}_{n+1} is sampled uniformly from {hat{U}}_1, ldots , {hat{U}}_n, and with complementary probability 1-p, {hat{U}}_{n+1} is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when p<1/2, and that a further rescaling is needed when p>1/2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

Global sensitivity analysis for stochastic processes with independent increments

This paper is a first attempt to develop a numerical technique to analyze the sensitivity and the propagation of uncertainty through a system with stochastic processes having independent increments as input. Similar to Sobol’ indices for random variables, a meta-model based on Chaos expansions is used and it is shown to be well suited to address such problems. New global sensitivity indices are also introduced to tackle the specificity of stochastic processes. The accuracy and the efficiency of the proposed method is demonstrated on an analytical example with three different input stochastic processes: a Wiener process; an Ornstein–Uhlenbeck process and a Brownian bridge process. The considered output, which is function of these three processes, is a non-Gaussian process. Then, we apply the same ideas on an example without known analytical solution.

Optimal Berry-Esséen bound for maximum likelihood estimation of the drift parameter in $$\alpha $$-Brownian bridge

Let $$T>0,\alpha >\frac{1}{2}$$ . In the present paper we consider the $$\alpha $$ -Brownian bridge defined as $$dX_t=-\alpha \frac{X_t}{T-t}dt+dW_t,\, 0\le t< T$$ , where W is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $$ \alpha $$ based on the continuous observation $$\{X_s,0\le s\le t\}$$ as $$t\uparrow T$$ . We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $$\frac{1}{\sqrt{\vert \log (T-t)\vert }}$$ , as $$t\uparrow T$$ . First we compute an upper bound and then find a lower bound with the same speed using Corollary 1 and Corollary 2 of Kim et al. (J Multivar Anal 155:284–304, 2017b) respectively.