Expectation Maximization Research Articles

Short-term prediction of horizontal winds in the mesosphere and lower thermosphere over coastal Peru using a hybrid model

The mesosphere and lower thermosphere (MLT) are transitional regions between the lower and upper atmosphere. The MLT dynamics can be investigated using wind measurements conducted with meteor radars. Predicting MLT winds could help forecast ionospheric parameters, which has many implications for global communications and geo-location applications. Several literature sources have developed and compared predictive models for wind speed estimation. However, in recent years, hybrid models have been developed that significantly improve the accuracy of the estimates. These integrate time series decomposition and machine learning techniques to achieve more accurate short-term predictions. This research evaluates a hybrid model that is capable of making a short-term prediction of the horizontal winds between 80 and 95 km altitudes on the coast of Peru at two locations: Lima (12°S, 77°W) and Piura (5°S, 80°W). The model takes a window of 56 data points as input (corresponding to 7 days) and predicts 16 data points as output (corresponding to 2 days). First, the missing data problem was analyzed using the Expectation Maximization algorithm (EM). Then, variational mode decomposition (VMD) separates the components that dominate the winds. Each resulting component is processed separately in a Long short-term memory (LSTM) neural network whose hyperparameters were optimized using the Optuna tool. Then, the final prediction is the sum of the predicted components. The efficiency of the hybrid model is evaluated at different altitudes using the root mean square error (RMSE) and Spearman’s correlation (r). The hybrid model performed better compared to two other models: the persistence model and the dominant harmonics model. The RMSE ranged from 10.79 to 27.04 m s−1, and the correlation ranged from 0.55 to 0.94. In addition, it is observed that the prediction quality decreases as the prediction time increases. The RMSE at the first step reached 6.04 m s−1 with a correlation of 0.99, while at the sixteenth step, the RMSE increased up to 30.84 m s−1 with a correlation of 0.5.

Unsupervised machine learning with different sampling strategies and topographic factors for distinguishing between landslide source and runout areas to improve landslide inventory production

This study derived 12 topographical and hydrological factors related to landslides from a 10-m digital elevation model. Three unsupervised machine learning algorithms were employed to distinguish between the features of landslide sources and runout areas for Typhoon Morakot. Two sampling strategies were designed in this study. The first strategy involved creating a data set by pairing landslide sources and runout areas on the basis of the size of polygonal areas recorded in the inventory and then exploring the effectiveness of feature separation with k-means++ as the primary method and EM (expectation maximization) and DBSCAN (Density-Based Spatial Clustering of Applications with Noise) as supplementary methods. Because of practical challenges associated with the inability to determine whether remote sensing results correspond to landslide sources or runout areas, the first sampling strategy was used to test the feasibility of the adopted algorithms. In the second sampling strategy, the effectiveness of feature separation was tested by dividing polygon samples into several intervals on the basis of their sizes, thereby verifying the practicability of the adopted algorithms in real-world operations. This study verified that combining unsupervised machine learning algorithms with topographic factors facilitates the effective separation of landslide sources and runout areas, thereby enhancing the quality of landslide inventories and reducing the cost of landslide inventory creation. Moreover, it indicated that meter-scale topographic data yield classification results similar to those obtained from manually created landslide inventories based on centimeter-scale stereo aerial photos. The proposed method effectively balances the cost and quality of landslide inventories.

Novel Adaptive Hidden Markov Model Utilizing Expectation–Maximization Algorithm for Advanced Pipeline Leak Detection

Novel Adaptive Hidden Markov Model Utilizing Expectation–Maximization Algorithm for Advanced Pipeline Leak Detection

Applying the Expectation-Maximization Algorithm to Multivariate Signal Extraction

Understanding time series signal extraction is vital for federal agencies, especially as data collection accelerates. One natural extension is moving from univariate to multivariate signal extraction, which offers the promise of reducing extraction error by exploiting cross-sectional relationships. However, such an extension ushers in new computational challenges, viz. larger parameter spaces and more complicated objective functions. The Expectation-Maximization (EM) algorithm provides a methodology to implicitly compute (or approximate) maximum likelihood estimates (MLEs). This paper provides methodology for applying the EM algorithm to a class of latent component multivariate time series models that allow for a nuanced specification of the unobserved signal. We derive an explicit formula for the maximization step, which facilitates computation speed while also improving the stability of the algorithm. Numerical studies demonstrate EM’s ability to efficiently compute MLEs in low-dimensional systems, while also providing feasible estimates in moderate-dimensional systems where MLEs are infeasible to compute. Applications to monthly housing starts and daily immigration are provided.

Accelerated distributed expectation-maximization algorithms for the parameter estimation in multivariate Gaussian mixture models

Rapid development for modeling big data requires effective and efficient methods for estimating the parameters involved. Although several accelerated Expectation-Maximization algorithms have been developed, there still exist two major concerns: reducing computational cost and improving model estimation accuracy. We propose three distributed-like algorithms for multivariate Gaussian mixture models, which can accelerate speed and improve estimation accuracy. The first algorithm is distributed algorithm, which is used to speed up the calculation of classic algorithms and improve its estimation accuracy by averaging the one-step estimators obtained from distributed operators. The second algorithm is distributed online algorithm, which is a distributed stochastic approximation procedure that performs online updates when reading online data. The final algorithm is called distributed monotonically over-relaxed algorithm, which uses an over-relaxation factor and a distributing strategy to improve the estimation accuracy of multivariate Gaussian mixture models. We investigate the stability, sensitivity, convergence, and robustness of these algorithms in a numerical study. We also apply these algorithms to three real data sets for validation.

Estimation and Bayesian Prediction of the Generalized Pareto Distribution in the Context of a Progressive Type-II Censoring Scheme

The generalized Pareto distribution plays a significant role in reliability research. This study concentrates on the statistical inference of the generalized Pareto distribution utilizing progressively Type-II censored data. Estimations are performed using maximum likelihood estimation through the expectation–maximization approach. Confidence intervals are derived using the asymptotic confidence intervals. Bayesian estimations are conducted using the Tierney and Kadane method alongside the Metropolis–Hastings algorithm, and the highest posterior density credible interval estimation is accomplished. Furthermore, Bayesian predictive intervals and future sample estimations are explored. To illustrate these inference techniques, a simulation and practical example are presented for analysis.

The potential of mixing model of wind speed distribution in Algerian High Plateaus

The evaluation of wind energy relies primarily on the probability density function PDF, which corresponds well with the wind speed data. Single PDFs are widely used in the assessment of wind. In contrast, homogeneous or heterogeneous mixed models are rarely used, especially in Algeria, where the bimodal wind speed distribution is expected. This research aims to investigate the potential of heterogeneous PDFs Generalized Extreme Value-Weibul and Normal-Extreme Value PDFs in assessing wind energy at three meteorological stations in the high plateau against the single widespread PDFs Weibull and GEV by analyzing five years of archived wind speed data. The estimation of mixed model parameters is obtained by applying the Expectation-Maximization algorithm, and the identification of the appropriate PDF is obtained by four goodness-of-fit (GOF) criteria and compared with the widespread single distributions. The results show that the mixed model surpasses widely the single model for all the GOF criteria used at the three selected sites. The proposed mixed model fits all the wind speed distributions related to unimodal and bimodal regimes.

Height-Resolved Analysis of Indoor Airborne Microbiome: Comparison with Floor Dust-Borne Microbiome and the Significance of Shoe Sole Dust.

Exposure to the indoor airborne microbiome is closely related to the air that individuals breathe. However, the floor dust-borne microbiome is commonly used as a proxy for indoor airborne microbiome, and the spatial distribution of indoor airborne microbiome is less well understood. This study aimed to characterize indoor airborne microorganisms at varying heights and compare them with those in floor dust. An assembly of three horizontally and three vertically positioned Petri dishes coated with mineral oil was applied for passive air sampling continuously at three heights without interruption. The airborne microbiomes at the three different heights showed slight stratification and differed significantly from those found in the floor dust. Based on the apportionment results from the fast expectation-maximization algorithm (FEAST), shoe sole dust contributed approximately 4% to indoor airborne bacteria and 14% to airborne fungi, a contribution that is comparable to that from the floor dust-borne microbiome. The results indicated that floor dust may not be a reliable proxy for indoor airborne microbiome. Moreover, the study highlights the need for height-resolved studies of indoor airborne microbiomes among humans in different activity modes and life states. Additionally, shoe sole-dust-associated microorganisms could potentially be a source to "re-wild" the indoor microbiota.

Prevalence estimates for COVID-19-related health behaviors based on the cheating detection triangular model

BackgroundSurvey studies in medical and health sciences predominantly apply a conventional direct questioning (DQ) format to gather private and highly personal information. If the topic under investigation is sensitive or even stigmatizing, such as COVID-19-related health behaviors and adherence to non-pharmaceutical interventions in general, DQ surveys can lead to nonresponse and untruthful answers due to the influence of social desirability bias (SDB). These effects seriously threaten the validity of the results obtained, potentially leading to distorted prevalence estimates for behaviors for which the prevalence in the population is unknown. While this issue cannot be completely avoided, indirect questioning techniques (IQTs) offer a means to mitigate the harmful influence of SDB by guaranteeing the confidentiality of individual responses. The present study aims at assessing the validity of a recently proposed IQT, the Cheating Detection Triangular Model (CDTRM), in estimating the prevalence of COVID-19-related health behaviors while accounting for cheaters who disregard the instructions.MethodsIn an online survey of 1,714 participants in Taiwan, we obtained CDTRM prevalence estimates via an Expectation-Maximization algorithm for three COVID-19-related health behaviors with different levels of sensitivity. The CDTRM estimates were compared to DQ estimates and to available official statistics provided by the Taiwan Centers for Disease Control. Additionally, the CDTRM allowed us to estimate the share of cheaters who disregarded the instructions and adjust the prevalence estimates for the COVID-19-related health behaviors accordingly.ResultsFor a behavior with low sensitivity, CDTRM and DQ estimates were expectedly comparable and in line with official statistics. However, for behaviors with medium and high sensitivity, CDTRM estimates were higher and thus presumably more valid than DQ estimates. Analogously, the estimated cheating rate increased with higher sensitivity of the behavior under study.ConclusionsOur findings strongly support the assumption that the CDTRM successfully controlled for the validity-threatening influence of SDB in a survey on three COVID-19-related health behaviors. Consequently, the CDTRM appears to be a promising technique to increase estimation validity compared to conventional DQ for health-related behaviors, and sensitive attributes in general, for which a strong influence of SDB is to be expected.

Dynamic mode decomposition based on expectation–maximization algorithm for simultaneous system identification and denoising

The present study proposes a novel dynamic mode decomposition (DMD) that can simultaneously estimate the reduced-order model, the original signal, and the system/observation noise model only from the noisy data. An expectation–maximization (EM)-algorithm DMD (EMDMD) combines DMD and the parameter adjustment of the linear dynamical system (LDS) based on the EM algorithm. The initial parameters based on the linearity of the reduced-order data are set by using DMD. Subsequently, the log-likelihood of the complete data is maximized by adjusting the LDS parameters while separating the noise. The proposed algorithm is applied to the benchmark data of the short-fat and tall-skinny data matrices with different noise and the time-series velocity fields of the flow around a circular cylinder and the separated flow around an airfoil. The performance of EMDMD in terms of system identification and noise separation from the noisy data is evaluated, and the EMDMD shows the highest system identification and noise separation performance in all data.

An Expectation-Maximization framework for Personalized Itinerary Recommendation with POI Categories and Must-see POIs

In this paper, we introduce a novel deterministic method based on Expectation Maximization (EM) to solve the rather complex problem of designing a tourist trip or Personalized Itinerary Recommendation (PIR). PIR objective is to recommend a personalized tour consisting of successive Points of Interest (POIs), which maximizes user satisfaction and respects user time-frame constraints. On top of that, the POIs are divided into categories, in order for travelers to be able to set limits on the maximum (and minimum) number of POIs that belong to one category and are included in the itinerary. In the proposed framework, emphasis is given on the POIs sequence selection, which exploits the customized POI recommendations offered by a recommender system. Additionally, the proposed methodology with POIs categories is able to solve the TourMustSee problem, so that the tour includes a set of POIs that must be visited. The proposed system has been successfully incorporated into a mobile app, offering a complete tourist trip design. The high performance, resilience, and computational efficiency of the proposed framework are demonstrated by experimental findings and comparisons to existing approaches on numerous synthetic and real datasets.

HCMV detection in Asian gastric cancer RNA-seq data sets and clinical validation in Indian GC patients reveals the HCMV-GC specific gene signatures.

Gastric cancer (GC) prevalence is very high in the Asian population. Oncogenic viruses play a crucial role in the progression of different types of cancers. Through reanalysis of clinical RNA-seq data sets derived from Asian GC patients, this study identified the presence of human cytomegalovirus (HCMV) in Asian GC tumors, next to the well-studied association of EBV. Clinical recruitment of the Indian GC cohort and screening for HCMV presence identified a 14.28% occurrence, similar to that observed in the bioinformatics analysis. A combinatorial approach of rank-based meta-analysis and ranking of groups based on an expectation-maximization algorithm identified that the upregulated LINC02864 and MAGEA10 correlated with poor survival of GC patients and downregulated tumor suppressor genes enriching for gastric acid secretion pathway to be associated with HCMV-positive GC patients, revealing the progressive role of HCMV infection in GC. Genes that discriminate between different stages of GC were identified through feature selection implemented in a machine-learning approach. LTF and KLK10 expressions were found to be specifically dysregulated by HCMV and can also indicate the GC stages. The results of this study will guide future studies to identify the functional role of these genes in the HCMV-associated GC.IMPORTANCENearly 75% of gastric cancer (GC) cases reported globally are from the Asian population. Most existing public databases, such as TCGA, comprise only a fractional portion of data derived from Asian ancestry. This study identified EBV and human cytomegalovirus (HCMV)'s higher detection in GC patients. The presence and role of EBV associated with GC are well-known, and the observation of HCMV prompted us to validate the findings in a small cohort of 40 Indian GC patients. We observed a 14.28% occurrence of HCMV in the Indian cohort, similar to that observed from next-generation sequencing. A combinatorial approach of rank-based meta-analysis and ranking of groups based on an expectation-maximization algorithm identified that the upregulated LINC02864 and MAGEA10 correlated with poor survival of GC patients and downregulated tumor suppressor genes enriching for gastric acid secretion pathway to be associated with HCMV-positive GC patients, revealing the progressive role of HCMV infection in GC.

Predicting demand for new products in fashion retailing using censored data

In today’s highly competitive fashion retail market, it is crucial to have accurate demand forecasting systems, namely for new products. Many experts have used machine learning techniques to forecast product sales. However, sales that do not happen due to lack of product availability are often ignored, resulting in censored demand and service levels that are lower than expected. Motivated by the relevance of this issue, we developed a two-stage approach to forecast the demand for new products in the fashion retail industry. In the first stage, we compared four methods of transforming historical sales into historical demand for products already commercialized. Three methods used sales-weighted averages to estimate demand on the days with stock-outs, while the fourth method employed an Expectation-Maximization (EM) algorithm to account for potential substitute products affected by stock-outs of preferred products. We then evaluated the performance of these methods and selected the most accurate one for calculating the primary demand for these historical products. In the second stage, we predicted the demand for the products of the following collection using Random Forest, Deep Neural Networks, and Support Vector Regression algorithms. In addition, we applied a model that consisted of weighting the demands previously calculated for the products of past collections that were most similar to the new products. We validated the proposed methodology using a European fashion retailer case study. The results revealed that the method using the Expectation-Maximization algorithm had the highest potential, followed by the Random Forest algorithm. We believe that this approach will lead to more assertive and better-aligned decisions in production management.

Film hole layout optimization for multi-row film cooling plate in continuous parameter space under non-uniform inlet temperature distribution

Film cooling is an advanced external cooling technology for protecting gas turbine blades from excessive temperatures. To counteract non-uniform heat loads caused by hot streak and swirl effects at the combustion chamber outlet, fine design of film hole positions is necessary. However, the complexity due to the numerous film holes and their positional parameters presents challenges in the independent optimization of film hole positions in a continuous parameter space. This study proposed an optimization method grounded in the divide and conquer approach to address optimization for film hole layout. The film hole layout was decomposed into multiple single rows and optimized systematically by iteratively refining the single-row film hole layout along the streamwise direction using the expectation maximization algorithm. Optimized layouts for five different inlet temperature distributions were obtained using the proposed method and adiabatic wall temperature distributions were compared and analyzed for optimized and staggered layouts. Additionally, the cooling performance of the optimized layout was evaluated against the staggered layout under conjugate heat transfer conditions and the robustness of the optimized layout under various blowing ratios and peak temperature positions was tested. The results demonstrated that the proposed method can optimize the positions of 52 film holes within 0.3h, and it was generalized to various inlet temperature distributions. By enhancing lateral interactions among downstream film jets, the lifting effect of kidney vortex in the optimized layout was weakened, bringing the film jets closer to the wall and significantly increasing coolant coverage during streamwise development. Comparative analysis reveals the superior cooling performance of the optimized layout at the blowing ratio of 1.0, evidenced by a 15.1% reduction in total input heat transfer rate and a 12.3% enhancement in overall cooling efficiency relative to the staggered layout. Under conditions with blowing ratios ranging from 0.5 to 1.5 and peak temperature position deviations, the average wall temperature of the optimized layout consistently remained lower than the staggered layout, indicating the robustness of the optimized layout to variations in blowing ratio and hot streak position.

A novel random-effect Birnbaum-Saunders distribution for reliability assessment considering accelerated mechanism equivalence

ABSTRACT Birnbaum-Saunders (B-S) distribution has been widely used in reliability assessment of products. However, the basic B-S distribution fails in characterizing the variation of damage caused by different units. In this paper, a novel random-effect B-S distribution which considers the unit heterogeneity is proposed. First, the novel distribution incorporates the inverse gamma distribution into the shape parameter of B-S distribution, and statistical inferences of the novel distribution are thoroughly investigated. Then, an extension of the proposed B-S distribution is developed to handle accelerated life test (ALT) data. In order to ensure the validity of ALTs, a coefficient of variation hypothesis test method employing the necessary condition for mechanism equivalence of the proposed distribution is executed. To estimate the parameters of the novel B-S distribution with explanatory variables, a two-step algorithm combining analogue least square and expectation maximization (ALS-EM) is proposed based on the data with constant coefficients of variation. A simulation is conducted to verify the effectiveness of the proposed ALS-EM method. Finally, two actual engineering cases are utilized to show the advantages of the random-effect B-S distribution in terms of goodness-of-fit and reliability assessment precision.

Multivariate Zero-Inflated INAR(1) Model with an Application in Automobile Insurance

The objective of this article is to propose a comprehensive solution for analyzing multidimensional non-life claim count data that exhibits time and cross-dependence, as well as zero inflation. To achieve this, we introduce a multivariate INAR(1) model, with the innovation term characterized by either a multivariate zero-inflated Poisson distribution or a multivariate zero-inflated hurdle Poisson distribution. Additionally, our modeling framework accounts for the impact of individual and coverage-specific covariates on the mean parameters of each model, thereby facilitating the computation of customized insurance premiums based on varying risk profiles. To estimate the model parameters, we employ a novel expectation-maximization (EM) algorithm. Our model demonstrates satisfactory performance in the analysis of European motor third-party liability claim count data.

The spike-and-slab quantile LASSO for robust variable selection in cancer genomics studies.

Data irregularity in cancer genomics studies has been widely observed in the form of outliers and heavy-tailed distributions in the complex traits. In the past decade, robust variable selection methods have emerged as powerful alternatives to the nonrobust ones to identify important genes associated with heterogeneous disease traits and build superior predictive models. In this study, to keep the remarkable features of the quantile LASSO and fully Bayesian regularized quantile regression while overcoming their disadvantage in the analysis of high-dimensional genomics data, we propose the spike-and-slab quantile LASSO through a fully Bayesian spike-and-slab formulation under the robust likelihood by adopting the asymmetric Laplace distribution (ALD). The proposed robust method has inherited the prominent properties of selective shrinkage and self-adaptivity to the sparsity pattern from the spike-and-slab LASSO (Roc̆ková and George, J Am Stat Associat, 2018, 113(521): 431-444). Furthermore, the spike-and-slab quantile LASSO has a computational advantage to locate the posterior modes via soft-thresholding rule guided Expectation-Maximization (EM) steps in the coordinate descent framework, a phenomenon rarely observed for robust regularization with nondifferentiable loss functions. We have conducted comprehensive simulation studies with a variety of heavy-tailed errors in both homogeneous and heterogeneous model settings to demonstrate the superiority of the spike-and-slab quantile LASSO over its competing methods. The advantage of the proposed method has been further demonstrated in case studies of the lung adenocarcinomas (LUAD) and skin cutaneous melanoma (SKCM) data from The Cancer Genome Atlas (TCGA).

Detection of diffusion anisotropy from an individual short particle trajectory

In parallel with advances in microscale imaging techniques, the fields of biology and materials science have focused on precisely extracting particle properties based on their diffusion behavior. Although the majority of real-world particles exhibit anisotropy, their behavior has been studied less than that of isotropic particles. In this study, we introduce a method for estimating the diffusion coefficients of individual anisotropic particles using short-trajectory data on the basis of a maximum likelihood framework. Traditional estimation techniques often use mean-squared displacement (MSD) values or other statistical measures that inherently remove angular information. Instead, we treated the angle as a latent variable and used belief propagation to estimate it while maximizing the likelihood using the expectation-maximization algorithm. Compared to conventional methods, this approach facilitates better estimation of shorter trajectories and faster rotations, as confirmed by numerical simulations and experimental data involving bacteria and quantum rods. Additionally, we performed an analytical investigation of the limits of detectability of anisotropy and provided guidelines for the experimental design. In addition to serving as a powerful tool for analyzing complex systems, the proposed method will pave the way for applying maximum likelihood methods to more complex diffusion phenomena. Published by the American Physical Society 2024

Image Quality of Cardiac Silicon Photomultiplier PET/CT Using an Infant Phantom of Extremely Low Birth Weight.

The lack of pediatrics-specific equipment for nuclear medicine imaging has resulted in insufficient diagnostic information for newborns, especially low-birth-weight infants. Although PET offers high spatial resolution and low radiation exposure, its use in newborns is limited. This study investigated the feasibility of cardiac PET imaging using the latest silicon photomultiplier (SiPM) PET technology in infants of extremely low birth weight (ELBW) using a phantom model. Methods: The study used a phantom model representing a 500-g ELBW infant with brain, cardiac, liver, and lung tissues. The cardiac tissue included a 3-mm-thick defect mimicking myocardial infarction. Organ tracer concentrations were calculated assuming 18F-FDG myocardial viability scans and 18F-flurpiridaz myocardial perfusion scans and were added to the phantom organs. Imaging was performed using an SiPM PET/CT scanner with a 5-min acquisition. The data acquired in list mode were reconstructed using 3-dimensional ordered-subsets expectation maximization with varying iterations. Image evaluation was based on the depiction of the myocardial defect compared with normal myocardial accumulation. Results: Increasing the number of iterations improved the contrast of the myocardial defect for both tracers, with 18F-flurpiridaz showing higher contrast than 18F-FDG. However, even at 50 iterations, both tracers overestimated the defect accumulation. A bull's-eye image can display the flow metabolism mismatch using images from both tracers. Conclusion: SiPM PET enabled cardiac PET imaging in a 500-g ELBW phantom with a 1-g heart. However, there were limitations in adequately depicting these defects. Considering the image quality and defect contrast,18F-flurpiridaz appears more desirable than 18F-FDG if only one of the two can be used.

HMM for discovering decision-making dynamics using reinforcement learning experiments.

Major depressive disorder (MDD), a leading cause of years of life lived with disability, presents challenges in diagnosis and treatment due to its complex and heterogeneous nature. Emerging evidence indicates that reward processing abnormalities may serve as a behavioral marker for MDD. To measure reward processing, patients perform computer-based behavioral tasks that involve making choices or responding to stimulants that are associated with different outcomes, such as gains or losses in the laboratory. Reinforcement learning (RL) models are fitted to extract parameters that measure various aspects of reward processing (e.g. reward sensitivity) to characterize how patients make decisions in behavioral tasks. Recent findings suggest the inadequacy of characterizing reward learning solely based on a single RL model; instead, there may be a switching of decision-making processes between multiple strategies. An important scientific question is how the dynamics of strategies in decision-making affect the reward learning ability of individuals with MDD. Motivated by the probabilistic reward task within the Establishing Moderators and Biosignatures of Antidepressant Response in Clinical Care (EMBARC) study, we propose a novel RL-HMM (hidden Markov model) framework for analyzing reward-based decision-making. Our model accommodates decision-making strategy switching between two distinct approaches under an HMM: subjects making decisions based on the RL model or opting for random choices. We account for continuous RL state space and allow time-varying transition probabilities in the HMM. We introduce a computationally efficient Expectation-maximization (EM) algorithm for parameter estimation and use a nonparametric bootstrap for inference. Extensive simulation studies validate the finite-sample performance of our method. We apply our approach to the EMBARC study to show that MDD patients are less engaged in RL compared to the healthy controls, and engagement is associated with brain activities in the negative affect circuitry during an emotional conflict task.