True Parameters Research Articles

Mean Estimation of Truncated Mixtures of Two Gaussians: A Gradient Based Approach

Even though data is abundant, it is often subjected to some form of censoring or truncation which inherently creates biases. Removing such biases and performing parameter estimation is a classical challenge in Statistics. In this paper, we focus on the problem of estimating the means of a mixture of two balanced d-dimensional Gaussians when the samples are prone to truncation. A recent theoretical study on the performance of the Expectation-Maximization (EM) algorithm for the aforementioned problem showed EM almost surely converges for d=1 and exhibits local convergence for d>1 to the true means. Nevertheless, the EM algorithm for the case of truncated mixture of two Gaussians is not easy to implement as it requires solving a set of nonlinear equations at every iteration which makes the algorithm impractical. In this work, we propose a gradient based variant of the EM algorithm that has global convergence guarantees when d=1 and local convergence for d>1 to the true means. Moreover, the update rule at every iteration is easy to compute which makes the proposed method practical. We also provide numerous experiments to obtain more insights into the effect of truncation on the convergence to the true parameters in high dimensions.

Adaptive discounted control for piecewise deterministic Markov processes

The main goal of this paper is to study the adaptive infinite-horizon discounted continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. It is assumed that jump parameters (λ and Q), as well the continuous and boundary costs (Cg and Ci respectively), depend on an unknown parameter β⁎. It is shown that the principle of estimation and control holds, that is, the strategy consisting of choosing, at each stage n, an action according to an optimal stationary policy, where the true but unknown parameter β⁎ is replaced by its estimated value βˆn, is asymptotically discount optimal, provided that the sequence of estimators {βˆn} of β⁎ is strongly consistent, that is, βˆn converge to β⁎ almost surely. In the framework of PDMPs, the so-called discrepancy function depends on the derivative along the flow of the value function as well as on some boundary conditions, which brings new challenges in the analysis of this problem.

Optimal Sampling Plan for Freight Demand Synthesis with Mode Choice: A Case study of Bangladesh

This paper uses a comprehensive experimental design to investigate the influence of various traffic count sampling plans for Bangladesh on the performance of the Freight Origin-Destination Synthesis model with Mode Choice (FODS-MC) developed by Kalahasthi et al. FODS-MC estimates a national-level freight demand model including trip distribution, mode choice, empty truck trips, and empty rail trips, where one of the key inputs is the freight truck and rail, traffic counts. The traffic count sample comprises three types of road links (national, regional, and zilla) and one category for the rail link across the country. A Box–Behnken Design (BBD) with a response surface for each of four FODS-MC parameters (distribution, mode choice, truck empty trips, and rail empty trips) is constructed. The results showed that the response surfaces are nonlinear planes for all parameters. There is no single optimal sampling plan for FODS-MC as each model parameter demands different distribution across the truck and rail links. The random and stratified samples perform almost similarly if less than 20% of the sample is collected. Minimizing the loss functions between the estimated and true parameters shows that a random sample between 20% and 25% of the truck and rail links estimates the best freight demand model. Overall, this research develops a framework to assist public practitioners in the optimum usage of the limited time and resources in collecting the traffic count data that could estimate the freight demand and mode choice models effectively.

Effects of the Galactic Magnetic Field on the UHECR Correlation Studies with Starburst Galaxies

We estimate the biases caused by the coherent deflection of cosmic rays due to the Galactic magnetic field (GMF) in maximum likelihood analyses for searches of ultrahigh-energy cosmic-ray sources in the literature. We simulate mock event data sets with a set of assumptions for the starburst galaxy source model, coherent deflection by a GMF model, and mixed-mass composition. We then conduct a maximum likelihood analysis without accounting for the GMF in the same manner as previous studies. We find that the anisotropic fraction f ani is estimated to be systematically lower than the true value. We estimate the true parameters that are compatible with the best-fit parameters that were reported, and find that except for a narrow region with a large anisotropic fraction and a small separation angular scale, a wide parameter space is still compatible with the experimental results. We also develop a maximum likelihood method that takes the GMF model into account and confirm in Monte Carlo simulations that we can estimate the true parameters within a 1σ contour under the ideal condition that we know the event-by-event mass and the GMF.

Variational Bayes inference for hidden Markov diagnostic classification models.

Diagnostic classification models (DCMs) can be used to track the cognitive learning states of students across multiple time points or over repeated measurements. This study developed an effective variational Bayes (VB) inference method for hidden Markov longitudinal general DCMs. The simulations performed in this study verified the validity of the proposed algorithm for satisfactorily recovering true parameters. Simulation and applied data analyses were conducted to compare the proposed VB method to Markov chain Monte Carlo (MCMC) sampling. The results revealed that the parameter estimates provided by the VB method were consistent with the MCMC method with the additional benefit of a faster estimation time. The comparative simulation also indicated differences between the two methods in terms of posterior standard deviation and coverage of 95% credible intervals. Thus, with limited computational resources and time, the proposed VB method can output estimations comparable to that of MCMC.

Bayesian Analysis Methods for Two-Level Diagnosis Classification Models

Understanding whether or not different types of students master various attributes can aid future learning remediation. In this study, two-level diagnostic classification models (DCMs) were developed to represent the probabilistic relationship between external latent classes and attribute mastery patterns. Furthermore, variational Bayesian (VB) inference and Gibbs sampling Markov chain Monte Carlo methods were developed for parameter estimation of the two-level DCMs. The results of a parameter recovery simulation study show that both techniques appropriately recovered the true parameters; Gibbs sampling in particular was slightly more accurate than VB, whereas VB performed estimation much faster than Gibbs sampling. The two-level DCMs with the proposed Bayesian estimation methods were further applied to fourth-grade data obtained from the Trends in International Mathematics and Science Study 2007 and indicated that mathematical activities in the classroom could be organized into four latent classes, with each latent class connected to different attribute mastery patterns. This information can be employed in educational intervention to focus on specific latent classes and elucidate attribute patterns.

Use of Clinical Trial Simulations to Compare the Performance of Different Approaches for Population Analyses of Pediatric Pharmacokinetic Data.

The adequate characterization of the pharmacokinetics of a drug used in pediatrics is a mainstay of pediatric development programs and is critical for accurate dose selection in pediatrics. Analysis approaches can impact the estimation and characterization of pediatric pharmacokinetic parameters. Simulations were conducted to compare the performance of different approaches for analyzing pediatric pharmacokinetic data in the presence of extensive data from adult studies. Simulated clinical trial datasets were generated encompassing different scenarios that might be encountered in pediatric drug development. For each scenario, 250 clinical trials were simulated and analyzed using each of the following approaches: (1) estimating pediatric parameters using only pediatric data; (2) fixing specific parameters to adult estimates and estimating the remaining pediatric parameters using only pediatric data; (3) estimating pediatric parameters using adult parameters as informative Bayesian priors; (4) estimating pediatric parameters using combined adult and pediatric datasets with exponents for body weight effects estimated using adult and pediatric data; and (5) estimating pediatric parameters using combined adult and pediatric datasets with exponents for body weight effects estimated using pediatric data only. Each analysis approach was evaluated for its success in the estimation of true pediatric pharmacokinetic parameter values. Results demonstrated that analyzing pediatric data using a Bayesian approach generally performed best and had the lowest probability of significant bias in the estimated pediatric pharmacokinetic parameters among different scenarios evaluated. This clinical trial simulation framework can be used to inform the optimal approach for analyses of pediatric data for other pediatric drug development program scenarios beyond the cases evaluated in these analyses.

Behavioral event detection and rate estimation for autonomous vehicle evaluation

AbstractAutonomous vehicles are continually increasing their presence on public roads. However, before any new autonomous driving software can be approved, it must first undergo a rigorous assessment of driving quality. These quality evaluations typically focus on estimating the frequency of (undesirable) behavioral events. While rate estimation would be straight‐forward with complete data, in the autonomous driving setting this estimation is greatly complicated by the fact that detecting these events within large driving logs is a non‐trivial task that often involves human reviewers. In this article, we outline a streaming partial tiered event review configuration that ensures both high recall and high precision on the events of interest. In addition, the framework allows for valid streaming estimates at any phase of the data collection process, even when labels are incomplete, for which we develop the maximum likelihood estimate and show it is unbiased. Constructing honest and effective confidence intervals (CI) for these rate estimates, particularly for rare safety‐critical events, is a novel and challenging statistical problem due to the complexity of the data likelihood. We develop and compare several CI approximations, including a novel gamma CI method that approximates the exact but intractable distribution with a weighted sum of independent Poisson random variables. There is a clear trade‐off between statistical coverage and interval width across the different CI methods, and the extent of this trade‐off varies depending on the specific application settings (e.g., rare vs. common events). In particular, we argue that our proposed CI method is the best‐suited when estimating the rate of safety‐critical events where guaranteed coverage of the true parameter value is a prerequisite to safely launching a new ADS on public roads.

Structural geologic modeling and restoration using ensemble Kalman inversion

We demonstrate the use of Ensemble Kalman Inversion (EKI) for building three-dimensional, multi-fault, kinematically restorable structural geologic models, by means of a workflow in which fault geometry, the distribution of slip on a fault, and the geometry of folded horizons are all modeled. The models are constrained by observations of faults and horizons in the present deformed state together with the expectation that horizons should restore flat. Two modeling approaches are tested: restoration from the deformed state, and forward modeling from the restored state. We first test these methods on a synthetic model involving a single fault, and then apply them to a real-field example involving five faults. Models are prone to ensemble collapse, which results in underestimation of uncertainty at small ensemble sizes, but localization and covariance inflation can mitigate this issue. With these methods, EKI can recover the true parameter values in the synthetic case and produce a solution consistent with the data in the real case, as well as quantify uncertainty in both cases. EKI, therefore, shows promise as a tool for building complex, restorable structural geologic models, and it holds the potential for integration of fault kinematics with other ensemble-based subsurface modelling workflows.

A proof-of-concept pipeline to guide evaluation of tumor tissue perfusion by dynamic contrast-agent imaging: Direct simulation and inverse tracer-kinetic procedures.

Dynamic contrast-enhanced (DCE) perfusion imaging has shown great potential to non-invasively assess cancer development and its treatment by their characteristic tissue signatures. Different tracer kinetics models are being applied to estimate tissue and tumor perfusion parameters from DCE perfusion imaging. The goal of this work is to provide an in silico model-based pipeline to evaluate how these DCE imaging parameters may relate to the true tissue parameters. As histology data provides detailed microstructural but not functional parameters, this work can also help to better interpret such data. To this aim in silico vasculatures are constructed and the spread of contrast agent in the tissue is simulated. As a proof of principle we show the evaluation procedure of two tracer kinetic models from in silico contrast-agent perfusion data after a bolus injection. Representative microvascular arterial and venous trees are constructed in silico. Blood flow is computed in the different vessels. Contrast-agent input in the feeding artery, intra-vascular transport, intra-extravascular exchange and diffusion within the interstitial space are modeled. From this spatiotemporal model, intensity maps are computed leading to in silico dynamic perfusion images. Various tumor vascularizations (architecture and function) are studied and show spatiotemporal contrast imaging dynamics characteristic of in vivo tumor morphotypes. The Brix II also called 2CXM, and extended Tofts tracer-kinetics models common in DCE imaging are then applied to recover perfusion parameters that are compared with the ground truth parameters of the in silico spatiotemporal models. The results show that tumor features can be well identified for a certain permeability range. The simulation results in this work indicate that taking into account space explicitly to estimate perfusion parameters may lead to significant improvements in the perfusion interpretation of the current tracer-kinetics models.

Clustered Coefficient Regression Models for Poisson Process with an Application to Seasonal Warranty Claim Data

Motivated by a product warranty claims dataset, we propose clustered coefficient regression models in a nonhomogeneous Poisson process for recurrent event data. The proposed method, referred as CLUPP, can estimate the group structure and parameters simultaneously. In our proposed method, a penalized regression approach is used to identify the group structure. Numerical studies show that the proposed approach can identify the group structure well, and outperforms traditional methods such as hierarchical clustering and K-means. We also establish theoretical properties, which show that the proposed estimators can converge to true parameters in high probability. In the end, we apply our proposed methods to the product warranty claims dataset, which achieve better prediction than the state-of-the-art methods.

A Bayesian approach for analyzing crop yield response data with limited treatments

AbstractThis paper proposes a Bayesian multilevel modeling approach to incorporate response parameters from published studies into crop yield response estimation procedures when nonlimiting or limiting treatment levels are omitted or limited in agronomic experiments. Such circumstances may be encountered when data are from farmer‐led research, which may use nonstandardized experimental designs. The paper's focus is on maize yield response to nitrogen fertilizer, but the procedure is flexible enough to accommodate other factors that could affect crop yield response. A proof‐of‐concept Monte Carlo (MC) exercise supplements an empirical application. The MC simulation investigates the small sample properties of the proposed procedure. The empirical example uses field trial data for a maize planter experiment under different nitrogen (N) fertilizer rates. The planter trial compared mechanical planting methods to methods used in developing countries with limited access to mechanized planter technology. Some experiments had no check plots and all experiments lacked nonlimiting fertilizer rates. Linear and quadratic response functions with plateaus are used in the MC simulation and empirical application. MC results suggest that estimates were closest to true parameter values when priors for optimal N rates from published sources were used.

A unifying framework for rank and pseudo-rank based inference using nonparametric confidence distributions

Nonparametric confidence distributions estimate statistical functionals by a distribution function on the parameter space, instead of the classical point or interval estimators. The concept bears analogy to the Bayesian posterior, but is nevertheless a completely frequentist concept. In order to ensure the desired statistical properties, we require that the cumulative distribution function on the parameter space is, evaluated at the true parameter, uniformly distributed over the unit interval. Our main focus lies on developing confidence distributions for the nonparametric relative effect and some natural extensions thereof. We develop asymptotic, range preserving and—especially important in the case of small sample sizes—approximate confidence distributions based on rank and pseudo-rank procedures. Due to the close relationship between point estimators, confidence intervals and p-values, these can all be approached in a unified manner within the framework of confidence distributions. The main goal of our contribution is to make the powerful theory of confidence distributions available in a nonparametric context, that is, for situations where methods relying on parametric assumptions are not justifiable. Application of the proposed methods and interpretation of the results is demonstrated using real data sets, including ordinal, non-metric data.

Finite mixtures of matrix variate Poisson-log normal distributions for three-way count data.

Three-way data structures, characterized by three entities, the units, the variables and the occasions, are frequent in biological studies. In RNA sequencing, three-way data structures are obtained when high-throughput transcriptome sequencing data are collected for n genes across p conditions at r occasions. Matrix variate distributions offer a natural way to model three-way data and mixtures of matrix variate distributions can be used to cluster three-way data. Clustering of gene expression data is carried out as means of discovering gene co-expression networks. In this work, a mixture of matrix variate Poisson-log normal distributions is proposed for clustering read counts from RNA sequencing. By considering the matrix variate structure, full information on the conditions and occasions of the RNA sequencing dataset is simultaneously considered, and the number of covariance parameters to be estimated is reduced. We propose three different frameworks for parameter estimation: a Markov chain Monte Carlo-based approach, a variational Gaussian approximation-based approach, and a hybrid approach. Various information criteria are used for model selection. The models are applied to both real and simulated data, and we demonstrate that the proposed approaches can recover the underlying cluster structure in both cases. In simulation studies where the true model parameters are known, our proposed approach shows good parameter recovery. The GitHub R package for this work is available at https://github.com/anjalisilva/mixMVPLN and is released under the open source MIT license.

Modeling of heat-hydrodynamic processes in evaporators of low-temperature systems with intrachannel boiling of refrigerants

The introduction of new types of heat exchangers with phase transitions and the solution of problems of optimizing the design and operational characteristics are a priority within the framework of the energy saving program. Known methods for calculating the heat-hydrodynamic parameters of the flow of refrigerants often do not take into account the specifics of boiling processes at low temperatures as well as in channels with a small flow area. This paper presents the results of modeling heat transfer during the boiling of refrigerants in the channels of evaporators of heat and cold energy complexes, taking into account the true flow parameters. The proposed mathematical model of the boiling of the working substance in channels of various shapes is based on the true flow parameters which imply knowledge of the channel cross-sectional areas occupied by each of the phases. The value of the true volumetric steam content provides the most correct modeling of two-phase flows in a wide range of regime and geometric parameters. The paper uses the equations of material and heat balance in combination with the equation of heat transfer from the environment to the boiling refrigerant. The map of flow regimes is used as an empirical component. A program has been developed for calculating the proposed system of equations which is solved iteratively at each time step using the finite volume method. Comparison of calculation results with experimental data on models of round and rectangular channels with intracanal boiling of refrigerants at positive and negative saturation temperatures is performed. It is shown that the calculation error does not exceed 10 % for a round and 20 % for a rectangular flow section. The verification results showed the possibility of using the model in the framework of engineering calculations. The proposed mathematical model can be used as the basis for the calculation programs for existing evaporators and for the creation of new types of heat exchangers with in-tube boiling of the working substance. The proposed method allows optimizing both geometric and thermal-hydrodynamic parameters.

Adaptive genetic algorithm-based particle herding scheme for mitigating particle impoverishment

Particle filters (PFs) estimate the true parameter state from samples of states, called particles, that are drawn to construct an approximated posterior probability density function (PDF). Each drawn particle has its own probability of being the true state called importance weight. Both particle degeneracy and impoverishment are important problems found in generic PFs that cause particles to become trapped at local maxima. Genetic algorithms (GAs) were applied to herd the drawn particles to discover new high-likelihood state values in order to reshape the approximated posterior PDF and to enhance the accuracy of state estimation, while the particle diversity is maintained. We calculated the threshold weight for classifying particles as low-weight particles and high-weight particles. Each parent pair consists of a high-weight particle and a low-weight particle. High-weight particles are always kept unchanged. After each new offspring particle is found using GA operations, we must check its weight to ensure that it is actually superior to its low-weight parent. Otherwise, this offspring must not be allowed to replace its low-weight parent and must be destroyed. To ensure that the number of high-weight particles is adequate, some offspring particles evolve to be the new high-weight parent particles to produce high-weight offspring with further promoted diversity. Results showed that our approach significantly improves posterior PDF approximation and reduces errors in state estimation.

Robust inference for high‐dimensional single index models

AbstractWe propose a robust inference method for high‐dimensional single index models with an unknown link function and elliptically symmetrically distributed covariates, focusing on signal recovery and inference. The proposed method is built on the Huber loss and the estimation of the unknown link function is avoided. The and consistency of a Lasso estimator up to a multiplicative scalar is established. When the covariance matrix of the predictors satisfies the irrepresentable condition, our method is shown to recover the signed support of the true parameter under mild conditions. Based on a debiased Lasso estimator, we study component‐wise and group inference for the high‐dimensional index parameter. The finite‐sample performance of our method is evaluated through extensive simulation studies. An application to a riboflavin production dataset is provided to illustrate the proposed method.

GAT–GMM: Generative Adversarial Training for Gaussian Mixture Models

Generative adversarial networks (GANs) learn the distribution of observed samples through a zero-sum game between two machine players, a generator and a discriminator. While GANs achieve great success in learning the complex distribution of image, sound, and text data, they perform suboptimally in learning multimodal distribution-learning benchmarks such as Gaussian mixture models (GMMs). In this paper, we propose Generative Adversarial Training for Gaussian Mixture Models (GAT-GMM), a minimax GAN framework for learning GMMs. Motivated by optimal transport theory, we design the zero-sum game in GAT-GMM using a random linear generator and a softmax-based quadratic discriminator architecture, which leads to a nonconvex concave minimax optimization problem. We show that a gradient descent ascent (GDA) method converges to an approximate stationary minimax point of the GAT-GMM optimization problem. In the benchmark case of a mixture of two symmetric, well-separated Gaussians, we further show that this stationary point recovers the true parameters of the underlying GMM. We discuss the application of the proposed GAT-GMM framework for learning GMMs in the distributed federated learning setting, where the widely used expectation-maximization (EM) algorithm can incur great computational and communication costs. On the other hand, we show that GAT-GMM provides a scalable learning approach and a distributed GDA algorithm can still solve the GAT-GMM minimax problem without incurring extra computation costs. We numerically support our theoretical results by performing experiments which show that our minimax framework is successful in centralized learning tasks and can outperform standard EM-type algorithms in the federated setting.

Consistency of mixture models with a prior on the number of components

Abstract This article establishes general conditions for posterior consistency of Bayesian finite mixture models with a prior on the number of components. That is, we provide sufficient conditions under which the posterior concentrates on neighborhoods of the true parameter values when the data are generated from a finite mixture over the assumed family of component distributions. Specifically, we establish almost sure consistency for the number of components, the mixture weights, and the component parameters, up to a permutation of the component labels. The approach taken here is based on Doob’s theorem, which has the advantage of holding under extraordinarily general conditions, and the disadvantage of only guaranteeing consistency at a set of parameter values that has probability one under the prior. However, we show that in fact, for commonly used choices of prior, this yields consistency at Lebesgue-almost all parameter values, which is satisfactory for most practical purposes. We aim to formulate the results in a way that maximizes clarity, generality, and ease of use.

The Naive Estimator of a Poisson Regression Model with a Measurement Error

We generalize the naive estimator of a Poisson regression model with a measurement error as discussed in Kukush et al. in 2004. The explanatory variable is not always normally distributed as they assume. In this study, we assume that the explanatory variable and measurement error are not limited to a normal distribution. We clarify the requirements for the existence of the naive estimator and derive its asymptotic bias and asymptotic mean squared error (MSE). The requirements for the existence of the naive estimator can be expressed using an implicit function, which the requirements can be deduced by the characteristic of the Poisson regression models. In addition, using the implicit function obtained from the system of equations of the Poisson regression models, we propose a consistent estimator of the true parameter by correcting the bias of the naive estimator. As illustrative examples, we present simulation studies that compare the performance of the naive estimator and new estimator for a Gamma explanatory variable with a normal error or a Gamma error.