Mixed Distribution Research Articles

Density estimation for spherical data using nonparametric mixtures

Nonparametric density estimation is studied for spherical data that may arise in many scientific and practical fields. In particular, nonparametric mixture models based on likelihood maximization are used. A nonparametric mixture has component distributions mixed together with a mixing distribution that is completely unspecified and needs to be determined from data. For mixture components, a two-parameter distribution family can be used, with one parameter as the mixing variable and the other to control the smoothness of the density estimator. For example, the popular von Mises-Fisher distributions can be readily used for this purpose. Numerical studies with various spherical data sets show that the resultant mixture-based density estimators are strong competitors with the best of the other density estimators.

Reliability analysis of cold-standby systems using a self-stratifying subset simulation method

Cold-standby systems are often featured with small failure probability and widely used as critical safety systems in current engineering fields. The existing analysis methods for cold standby systems have the problems with low-computional efficiency either for small failure probability events or for the components with mixed distribution types. To solve these problems, we propose a self-stratifying subset simulation (4S) method based on MCMC simulation and the performance function of cold-standby system. Numerical case studies are performed to demonstrate the reasonability and feasibility of the proposed 4S method. The case studies indicate that the results obtained by the proposed 4S method agree well with those obtained by the existing methods and the computation efficiency is greatly improved especially for small probability events. Moreover, the proposed 4S method was applicable for the cold-standby system cold-standby system with not only the exponentially or non-exponentially distributed components but components with more mixed distribution types.

A mixed distribution approach for low-flow frequency analysis – Part 1: Concept, performance, and effect of seasonality

Abstract. In seasonal climates with a warm and a cold season, low flows are generated by different processes so that the annual extreme series will be a mixture of summer and winter low-flow events. This leads to a violation of the homogeneity assumption for all statistics derived from the annual series and gives rise to inaccurate conclusions. In this first part of a two-paper series, a mixed distribution approach to perform frequency analysis in catchments with mixed low-flow regimes is proposed. We formulate the theoretical basis of the mixed distribution approach for the lower extremes based on annual minima series. The main strength of the model is that it allows the user to estimate return periods of summer low flows, winter low flows, and annual return periods in a theoretically sound and consistent way. Using archetypal examples, we show how the model behaves for a range of low-flow regimes, from distinct winter and summer regimes to mixed regimes where seasonal occurrence in summer and winter is equally likely. The examples show in a qualitative way the loss in accuracy one has to expect with conventional extreme value statistics performed with the annual extremes series. The model is then applied to a comprehensive Austrian data set to quantify the expected gain of using the mixed distribution approach compared to conventional frequency analysis. Results indicate that the gain of using a mixed distribution approach is indeed large. On average, the relative deviation is 21 %, 39 %, and 63 % when estimating the low flow with a 20-, 50-, and 100-year return period. For the 100-year event, 75 % of stations show a performance gain of >10 %, 41 % of stations > 50 %, and 25 % of stations > 80.6 %. This points to a broad relevance of the approach that goes beyond highly mixed seasonal regimes to include the strongly seasonal ones. We finally correlate the performance gain with seasonality indices in order to show the expected gain conditional to the strength of seasonality expressed by the ratio of average summer and winter low flow seasonality ratio (SR). For the 100-year event, the expected gain is about 70 % for SR=1.0, 20 % for SR=1.5, and 10 % for SR=2.0. The performance gain is further allocated to the spatial patterns of SR in the study area. The results suggest that the mixed estimator is relevant not only for mountain forelands but to a much wider range of catchment typologies. The mixed distribution approach provides one consistent approach for summer, winter, and annual probabilities and should be used by default in seasonal climates with a cold winter season where summer and winter low flows can occur.

Step-feed sequence batch reactor filled with surface-modified carriers can enhance nitrogen removal from low C/N ratio wastewater

It is difficult to achieve efficient nitrogen removal from low C/N ratio wastewater if there is no extra carbon source addition. In this study, integrated floating-film activated sludge (IFFAS) process filled with surface-modified carriers (rough, hydrophilic and positive-charged surface) was employed to enhance nitrogen removal performance. By applying step-feed sequence batch reactor strategy, high total nitrogen (TN) removal loading of 0.062 ± 0.010 kg/(m3∙d) could be achieved under the inflow flow ratio of 2:1:1 without external carbon source when treating low C/N ratio wastewater (C/N ratio: 3.5). In comparison, TN removal loading was only 0.031 ± 0.010 kg/(m3∙d) in activated sludge, and 0.057 ± 0.014 kg/(m3∙d) in IFFAS filled with conventional carrier system. The step-feed sequence batch reactor strategy resulted in complete utilization of influent carbon source for denitrification. The low C/N ratio also promoted microbial by-products production that enhanced nitrogen removal. Meanwhile, surface-modified carriers induced the mixed distribution of nitrifiers and denitrifiers. This study proposed a low-cost and efficient approach for nitrogen removal from low C/N ratio wastewater, showing potential in actual application.

Effect of hybrid reinforcement and number of passes on microstructure, mechanical and corrosion behavior of WE43 Mg alloy based metal matrix composite

In this work, a hybrid metal matrix composite with WE43 magnesium alloy as matrix and a hybrid mixture of NiTi, iron, and tin as reinforcement was fabricated using friction stir processing (FSP). The effect of the number of FSP passes on the grain morphology, distribution of reinforced particles, and mechanical properties were evaluated. Scanning electron microscope with energy dispersive spectroscopy was used to study the distribution of reinforced particles, a more uniform distribution was found in the double pass friction stir processed (DP-FSPed) specimen. The grain size was marginally smaller in the DP-FSPed sample compared to the single pass friction stir processed (SP-FSPed) sample. The processed samples showed an increment in hardness values, however, the distribution of hardness values was found uniform in the case of the DP-FSPed specimen, due to better mixing and homogenous distribution of reinforcement particles. The ductility of the DP-FSPed specimen increased by 126 % as compared to the SP-FSPed specimen. The corrosion rate was evaluated using the potentiodynamic polarization test and was found to be higher in the case of the DP-FSPed specimen.

Optimal estimation of high-dimensional Gaussian location mixtures

This paper studies the optimal rate of estimation in a finite Gaussian location mixture model in high dimensions without separation conditions. We assume that the number of components k is bounded and that the centers lie in a ball of bounded radius, while allowing the dimension d to be as large as the sample size n. Extending the one-dimensional result of Heinrich and Kahn (Ann. Statist. 46 (2018) 2844–2870), we show that the minimax rate of estimating the mixing distribution in Wasserstein distance is Θ((d/n)1/4+n−1/(4k−2)), achieved by an estimator computable in time O(nd2+n5/4). Furthermore, we show that the mixture density can be estimated at the optimal parametric rate Θ(d/n) in Hellinger distance and provide a computationally efficient algorithm to achieve this rate in the special case of k=2. Both the theoretical and methodological development rely on a careful application of the method of moments. Central to our results is the observation that the information geometry of finite Gaussian mixtures is characterized by the moment tensors of the mixing distribution, whose low-rank structure can be exploited to obtain a sharp local entropy bound.

Extracting information on reaction intermediates from single-molecule data resulting from parallel reaction mechanisms.

Enzymes can increase the rate of a chemical reaction by stabilizing one or more reaction intermediates. Therefore, a full understanding of how a given enzyme works requires awareness of the various reaction intermediates involved in the reaction mechanism. Single-molecule studies of enzyme activity have become an increasingly common tool used to gain insight into enzyme reaction mechanisms. At the single-molecule level, product formation by any enzyme is a stochastic process. Accordingly, there is a great deal of variation in the amount of time it takes for each molecule of product to be generated, but the shape of the probability distribution for event waiting times contains information about the underlying mechanism. With adequate statistics, it is possible to determine the number of reaction intermediates for a process and their average lifetime from the experimental dwell-time distribution, even though the intermediates cannot be directly observed. However, when product can be formed via more than one pathway, the dwell-time distribution will be a mixture of the distributions for the different pathways. We use simulated data and blinded fitting to demonstrate that it is possible to extract useful information about parallel reaction pathways from mixed data distributions. By considering the fractional error of each parameter recovered, we assess what fitting methods are most successful, and determine what conditions adversely impact accuracy. We further explore the impact of heterogeneity among the lifetimes of intermediates, naturally occurring variation among enzymes, and measurement error on our ability to accurately characterize two parallel reaction mechanisms producing the same product.

Constraining Bedrock Groundwater Residence Times in a Mountain System With Environmental Tracer Observations and Bayesian Uncertainty Quantification

AbstractGroundwater residence time distributions provide fundamental insights on the hydrological processes within watersheds. Yet, observations that can constrain groundwater residence times over broad timescales remain scarce in mountain catchment studies. We use environmental tracers (CFC‐12, SF6, 3H, and 4He) to investigate groundwater residence times along a hillslope in the East River Watershed, Colorado, USA. We develop a Bayesian inference framework that applies a Markov‐chain Monte Carlo (MCMC) approach to estimate noble gas recharge temperature, elevation, and excess‐air parameters and the resulting environmental tracer concentrations. MCMC is then used to propagate the environmental tracer uncertainties to estimates of groundwater mean residence times inferred with lumped parameter models. All samples contain 3H, CFC‐12, and SF6 in addition to terrigenic 4He, suggesting a mixture of water characterized by modern and premodern residence times. 4He exponential mean residence times range from hundreds of years at the upslope well to thousands of years at the toe‐slope well assuming average crustal production rates. We find that binary mixing residence time distributions with separate young and old mixing fractions are needed to predict the 4He, CFC‐12, SF6, and 3H observations, supporting the importance of flow path mixing in this bedrock system. Our findings that the fractured bedrock hosts groundwater with a mixture of residence times ranging from decades to millennia suggest variable recharge dynamics and flow path mixing along the hillslope and highlight the importance of characterizing groundwater systems with observations that are sensitive to transport over a broad range of residence times.

On the hazard rate of α-mixture of survival functions

The α-mixture model, as a flexible family of distributions, is an effective tool for modeling heterogeneity inS population. This article investigates the hazard rate of α-mixture in terms of hazard rates of mixed baseline distributions. In particular, when the baseline hazard rate follows either additive or multiplicative models an inverse problem to obtain the baseline hazard is solved. We, also, study the α-mixture hazard rate ordering for the ordered mixing distributions in the sense of likelihood ratio order. Sufficient conditions to order two finite α-mixtures in the sense of dispersive ordering are provided. Finally, it is shown that the hazard rate of the finite α-mixture in the multiplicative model tends to the hazard rate of the strongest (weakest) population as α → + ∞ ( − ∞ ). Several examples are presented to illustrate theoretical findings.

Thermal and Postural Effects on Fluid Mixing and Irrigation Patterns for Intraventricular Hemorrhage Treatment

Intraventricular hemorrhage is characterized by blood leaking into the cerebral ventricles and mixing with cerebrospinal fluid. A standard treatment method involves inserting a passive drainage catheter, known as an external ventricular drain (EVD), into the ventricle. EVDs have common adverse complications, including the occlusion of the catheter, that may lead to permanent neural damage or even mortality. In order to prevent such complications, a novel dual-lumen catheter (IRRAflow®) utilizing an active fluid exchange mechanism has been recently developed. However, the fluid dynamics of the exchange system have not been investigated. In this study, convective flow in a three-dimensional cerebral lateral ventricle with an inserted catheter is evaluated using an in-house lattice-Boltzmann-based fluid–solid interaction solver. Different treatment conditions are simulated, including injection temperature and patient position. Thermal and gravitational effects on medication distribution are studied using a dye simulator based on a recently-introduced (pseudo)spectral convection–diffusion equation solver. The effects of injection temperature and patient position on catheter performance are presented and discussed in terms of hematoma irrigation, vortical structures, mixing, and medication volume distribution. Results suggest that cold-temperature injections can increase catheter efficacy in terms of dye distribution and irrigation potential, both of which can be further guided by patient positioning.

Departure time choices in the morning commute with a mixed distribution of capacity

Incidents and other random factors may create variations to the transportation system and thus result in stochastic road capacity during the travel period. The realized capacity on a given day (i.e., an average value over the travel period) changes from day to day. For instance, existing empirical studies indicate that incident capacity reduction can be approximated as a continuous random variable. This study examines the morning commute problem with a road bottleneck whose capacity is constant within a day but changes stochastically from day to day. This study extends existing stochastic bottleneck model studies by considering a more general distribution of the bottleneck capacity. In particular, the capacity of bottleneck is at the designed value under good external conditions and degrades into a smaller value within an interval (i.e., the capacity degradation range) under adverse external conditions (a “mixed” distribution). Commuters’ departure time choices follow the Wardrop’s first principle in terms of their mean individual travel cost. Given the considered distribution of capacity, additional equilibrium departure/arrival patterns not identified by the literature have been identified and examined. How the mean travel cost and the mean of total travel time may vary with the capacity degradation probability and the level of capacity degradation have been analyzed. The impacts of the width of degraded capacity range have also been investigated to quantify how the results are affected if the degraded capacity is assumed as the mean value (rather than a degraded capacity range). Our results indicate that with less severe capacity degradation, the mean travel cost always decreases and improving the capacity under the “worst condition” can be more effective than improving the capacity under the “best adverse condition”. However, it is not always the case for reducing the total travel time, and the mentioned measures may exacerbate the system’s congestion, especially when the capacity degradation rarely occurs. Under a given mean capacity, the mean travel cost would be underestimated if the capacity degradation range is ignored. We also compare system performance considering different capacity distributions. It is found that the “mixed” capacity distribution defined in this study outperforms the binary capacity distribution in terms of evaluating the departure/arrival pattern and the mean travel cost. This study enhances our understanding on the morning commute problem under capacity uncertainty.

Revisiting consistency of a recursive estimator of mixing distributions

Estimation of the mixing distribution under a general mixture model is a very difficult problem, especially when the mixing distribution is assumed to have a density. Predictive recursion (PR) is a fast, recursive algorithm for nonparametric estimation of a mixing distribution/density in general mixture models. However, the existing PR consistency results make rather strong assumptions, some of which fail for a class of mixture models relevant for monotone density estimation, namely, scale mixtures of uniform kernels. In this paper, we develop new consistency results for PR under weaker conditions. Armed with this new theory, we prove that PR is consistent for the scale mixture of uniforms problem, and we show that the corresponding PR mixture density estimator has very good practical performance compared to several existing methods for monotone density estimation.

Use of a commercial indoor positioning system for monitoring resting time and moving distance in group-housed dairy calves.

To assess the usefulness of a commercially available indoor positioning system for monitoring the resting time and moving distance in group-housed dairy calves as indicators of their health status, five dairy calves were housed in a free barn, and their coordinate was recorded. The mean displacement (cm/s) within a minute showed a double-mixture distribution. Actual observations revealed that the minutes in the first distribution with shorter displacement were mostly the time that the calves spent lying. To predict the daily lying time and moving distance, a mixed distribution was divided at a threshold value. The mean sensitivity (the proportion of total minutes predicted correctly as lying, in total minutes observed lying) was more than 92%. The daily fluctuation in lying time correlated well with the actual lying time (r = 0.758, p < 0.01). The range of fluctuations was 740-1308 min/day and 724-1269 m/day for daily lying time and moving distance, respectively. The rectal temperature was correlated with daily lying time (r = 0.441, p < 0.001) and distance moved (r = 0.483, p < 0.001). The indoor positioning system can be a useful tool for early illness detection in calves before the onset of symptoms in group-housing systems.

Asymptotic tail properties of Poisson mixture distributions

SummaryCount data are omnipresent in many applied fields, often with overdispersion. With mixtures of Poisson distributions representing an elegant and appealing modelling strategy, we focus here on how the tail behaviour of the mixing distribution is related to the tail of the resulting Poisson mixture. We define five sets of mixing distributions, and we identify for each case whenever the Poisson mixture is in, close to or far from a domain of attraction of maxima. We also characterize how the Poisson mixture behaves similarly to a standard Poisson distribution when the mixing distribution has a finite support. Finally, we study, both analytically and numerically, how goodness‐of‐fit can be assessed with the inspection of tail behaviour.

OPTIMAL ALLOCATION OF FLEET OF HELICOPTERS AND AIRPORTS TO TRANSPORT PASSENGERS IN OFFSHORE MARITIME UNITS

The choice and distribution of the optimal fleet mix of helicopters to service the offshore air operation is an important logistical problem that has a potential to reduce costs considered in the oil and gas industry. The variability of helicopter models and sizes, as well as the possible operational restrictions of airport bases and airspace, requires that adequate numerical techniques of Operational Research are applied. In this context, this work develops a model to determine the trajectories and distances traveled between airport bases and offshore maritime units and vice-versa, through an airspace modeled by a directed graph. Also develop a helicopter performance model to obtain the estimated payload and flight times for each specified mission and last but not least an integer linear programming model to allocate an optimal fleet and airport mix for passenger transport through the use of different helicopter sizes. The model was applied to offshore units operating in the Santos basin due to their strategic importance in the oil and gas industry. The results obtained included a map of preferred regions for two different helicopter sizes (large and medium), the weekly flight tables showing the allocation of helicopters and airport for different demand scenarios, additionally the impact of fuel prices on different airport bases and the concentration of movements on air routes are also evaluated in order to complement the analysis.

Effects of corporate marketing strategy, brand awareness, and perceived quality on brand equity for accounting service

Brand equity and trust are consistently the most important antecedents to both behavioral and attitudinal forms of customer loyalty (Taylor et al., 2004). Accounting service refers to a service that offers bookkeeping, accounting, and reporting in alignment with both local and international accounting regulations. These services possess unique qualities, necessitating the service providers to establish themselves as reputable entities. It is crucial for them to maintain their brand equity to attract and retain customers. Thus, the study aims to identify and measure the influence of marketing mix, brand awareness, and perceived quality on brand equity for accounting services in Thanh Hoa province, Vietnam. Quantitative data were collected, including 385 businesses using accounting services at all levels across the country. The research results show that all elements of the marketing mix (product quality, price, distribution, and advertising), brand awareness, and perceived quality have a positive impact on brand equity for accounting services in Thanh Hoa province. The study also contributes some policy implications to help accounting service providers enhance their brand equity and then maintain customer satisfaction and customer loyalty.

Small universal unit for preparing, transporting and distributing liquid feed in small pig farms

The article focuses on the advantage of liquid mixed feeds in the use of mixed feeds for small farms and farms. The analysis of the scientific research works devoted to the problems of transportation and distribution of liquid mixed feed using pipelines showed that the physical and mechanical and rheological properties of such feeds have not been fully studied, and special feed pumps designed for the transfer of liquid mixed feeds with high viscosity have not been created. The article describes the structure and principle of operation of the small-sized liquid mixed feed preparation, transfer and distribution unit created by the authors. All equipment included in the unit has been modernized considering the peculiarities of small farms. Based on the results of the study, the necessary conclusions were drawn.

Toward a Globally-Applicable Uncertainty Quantification Framework for Satellite Multisensor Precipitation Products Based on GPM DPR

The usefulness of satellite multisensor precipitation products such as NASA’s 30-min, 0.1° Integrated Multi-satellitE Retrievals for the Global Precipitation Measurement (GPM) mission (IMERG) is hindered by their associated errors. Reliable estimates of uncertainty would mitigate this limitation, especially in near-real time when gauge observations are not available. However, creating such estimates is challenging, due to both the complicated nature of satellite precipitation errors and the lack of “ground-truth” data precisely in the places—including oceans, complex terrain, and developing countries—that could benefit most from satellite precipitation estimates. In this work, we use the GPM dual-frequency precipitation radar (DPR)-derived swath-based precipitation products as an alternative to ground-based observations to facilitate IMERG uncertainty estimation. We compare the suitability of two DPR-derived precipitation products, 2ADPR and 2BCMB, against higher fidelity ground validation multiradar multisensor (GV-MRMS) ground reference data over the contiguous United States. The 2BCMB is selected to train error models based on censored shifted gamma distribution (CSGD; a mixed discrete-continuous probability distribution). Uncertainty estimates from these models are compared against alternative error models trained on GV-MRMS. Using information from NASA’s Modern-Era Retrospective analysis for Research and Applications, version 2 (MERRA-2) reanalysis, we also demonstrate how IMERG uncertainty estimates can be further constrained using additional precipitation-related predictors. Though several critical issues remain unresolved, the proposed method shows promise for yielding robust uncertainty estimates in near-real time for IMERG and other similar precipitation products at their native resolution across the entire globe.

Analysis of College Students’ Test Scores Based on Two-Component Mixed Generalized Normal Distribution

In order to improve the fitting accuracy of college students’ test scores, this paper proposes two-component mixed generalized normal distribution, uses maximum likelihood estimation method and Expectation Conditional Maxinnization (ECM) algorithm to estimate parameters and conduct numerical simulation, and performs fitting analysis on the test scores of Linear Algebra and Advanced Mathematics of F University. The empirical results show that the two-component mixed generalized normal distribution is better than the commonly used two-component mixed normal distribution in fitting college students’ test data, and has good application value.

Nonparametric Maximum Likelihood Estimation Under a Likelihood Ratio Order

Comparison of two univariate distributions based on independent samples from them is a fundamental problem in statistics, with applications in a wide variety of scientific disciplines. In many situations, we might hypothesize that the two distributions are stochastically ordered, meaning intuitively that samples from one distribution tend to be larger than those from the other. One type of stochastic order that arises in economics, biomedicine, and elsewhere is the likelihood ratio order, also known as the density ratio order, in which the ratio of the density functions of the two distributions is monotone non-decreasing. In this article, we derive and study the nonparametric maximum likelihood estimator of the individual distributions and the ratio of their densities under the likelihood ratio order. Our work applies to discrete distributions, continuous distributions, and mixed continuous-discrete distributions. We demonstrate convergence in distribution of the estimator in certain cases, and we illustrate our results using numerical experiments and an analysis of a biomarker for predicting bacterial infection in children with systemic inflammatory response syndrome.