Error Loss Function Research Articles

Statistical inference for Gompertz distribution under adaptive type-II progressive hybrid censoring

Gompertz distribution is a significant and commonly used lifetime distribution, which plays an important role in reliability engineering. In this paper, we study the statistical inference of Gompertz distribution based on adaptive Type-II hybrid progressive censored schemes. From the perspective of frequentist, we derive the point estimations through the method of maximum likelihood estimation (MLE) and the existence of MLE is proved. Besides MLE, we propose the stochastic EM algorithm to reduce complexity and simplify computing. We also apply the method of Bootstraps (Bootstrap-p and Bootstrap-t) to construct confidence intervals. From Bayesian aspect, the Bayes estimates of the unknown parameters are evaluated by applying the MCMC method, the average length and coverage rate of credible intervals are also carried out. The Bayes inference is based on the squared error loss function and LINEX loss function. Furthermore, a numerical simulation is conducted to assess the performance of the proposed methods. Finally, a real-life example is considered to illustrate the application and development of the inference methods. In summary, the Bayesian method seems to perform the best among all approaches, while other approaches also present different advantages.

Tunable Neural Networks for Multi-Material Image Formation from Spectral CT Measurements.

Quantitative estimation of multi-material density images is an important goal for Spectral CT imaging. However, material decomposition is a poorly-conditioned nonlinear inverse problem. Maximum-likelihood model-based material decomposition results in very noisy material density image estimates. One increasingly popular strategy for noise reduction is to apply deep neural networks for multi-material image formation. The most common loss function is mean squared error with respect to supervised target images such as ground truth or higher-dose cases. However, we believe that the mean-squared error loss function has several issues for multi-material image formation. In this work, we present a new loss function which includes multiple noise realizations with separate weights on covariance and bias for joint denoising of all material bases. By modulating these weights, it is possible to tune the image quality of neural network output images. To demonstrate our proposed approach, we conducted a simulation of a water/calcium/gadolinium spectral CT imaging scenario using a deep neural network for multi-material image denoising. Our results show that by changing the weights of our proposed loss function, it is possible to control the tradeoff between variance and bias for individual materials as well as the control over the bias coupling between materials.

A new extension of the beta generator of distributions

Abstract In this paper, we introduced a new extension of the beta generator of distributions. Some important properties of the model are discussed, such as the series representation, quantile function, moments, Rényi entropy, and order statistics. Three special members are discussed, namely, the new extended beta exponential (NEBE), new extended beta uniform (NEBU), and new extended beta half logistic (NEBH). The model parameters estimation was achieved by the method of maximum likelihood and the Bayes estimation under square error loss function (SEL); the estimators were examined by simulation studies and it was a very good result as discussed by the bias and mean square error (MSE) of the estimators. Two real data applications are provided to illustrate the importance of the new model in practice.

Asymmetric and robust loss function driven least squares support vector machine

Least squares support vector machine (LSSVM) considerably simplifies problem solving, however, there are restrictions. The first is that it treats samples on both sides of the proximal hyperplane equally and does not differentiate them based on locations; the second is that it is sensitive to noise and outliers. To address the aforementioned issues, this paper proposes an asymmetric robust least squares support vector machine termed QTLS by combining Quadratic Type Squared Error Loss Function (QTSELF) with LSSVM, in which QTSELF is introduced into machine learning for the first time. On the one hand, QTLS can impose various penalties on samples according to their locations and place greater emphasis on samples that are susceptible to misclassification. On the other hand, it improves the model robustness by imposing a tiny penalty on noise or outliers located far from the proximal hyperplane. Using Rademacher complexity theory, we investigate the generalization capacity of QTLS. The RMSProp technique is utilized to solve both linear and nonlinear QTLS. Extensive experiments indicate the effectiveness of QTLS in addressing binary classification problems.

Raindrop Size Distribution Prediction by an Improved Long Short-Term Memory Network

The observation of and research on raindrop size distribution (DSD) is important for mastering and understanding the mutual restriction relationship between cloud dynamics and cloud microphysics in a process of precipitation; it also plays an irreplaceable role in many fields, such as radar meteorology, weather modification, boundary layer land surface processes, aerosols, etc. Using more than 1.7 million minutes of raindrop data observed with 17 laser disdrometers at 17 stations in Anhui Province, China, from 7 August 2009 to 30 April 2020, a DSD training dataset was constructed. Furthermore, the data are fitted to a normalized Gamma function and used to obtain its three parameters, i.e., the normalized intercept Nw, the mass weighted average diameter Dm, and the shape factor μ. Based on the long short-term memory network (LSTM), a DSD Gamma distribution prediction network (DSDnet) was designed. In the process of modeling based on DSDnet, a self-defined loss function (SLF) was proposed in order to improve the DSD prediction by increasing the weight values in the poor fitting regions according to the common mean square error loss function (MLF). By means of the training dataset, a DSDnet-based model was trained to realize the prediction of Nw, Dm, and μ minute-to-minute over the course of 30 min, and then was evaluated by the test dataset according to three indicators, namely, mean relative error (MRE), mean absolute error (MAE), and correlation coefficient (CC). The CC of lgNw, Dm, and μ can reach 0.93403, 0.90934, and 0.89741 for 12-min predictions, and 0.87559, 0.85261, and 0.84564 for 30-min predictions, respectively, which means that the DSD prediction accuracy within 30 min can basically reach the application level. Furthermore, the 12- and 30-min predictions of 3 precipitation processes were taken as examples to fully demonstrate the application effect of model. The prediction effects of Nw and Dm are better than that of μ, and the stratiform precipitation is better than the convective and convective-stratiform mixed cloud precipitation.

Adera2.0: A Drug Repurposing Workflow for Neuroimmunological Investigations Using Neural Networks.

Drug repurposing in the context of neuroimmunological (NI) investigations is still in its primary stages. Drug repurposing is an important method that bypasses lengthy drug discovery procedures and focuses on discovering new usages for known medications. Neuroimmunological diseases, such as Alzheimer’s, Parkinson’s, multiple sclerosis, and depression, include various pathologies that result from the interaction between the central nervous system and the immune system. However, the repurposing of NI medications is hindered by the vast amount of information that needs mining. We previously presented Adera1.0, which was capable of text mining PubMed for answering query-based questions. However, Adera1.0 was not able to automatically identify chemical compounds within relevant sentences. To challenge the need for repurposing known medications for neuroimmunological diseases, we built a deep neural network named Adera2.0 to perform drug repurposing. The workflow uses three deep learning networks. The first network is an encoder and its main task is to embed text into matrices. The second network uses a mean squared error (MSE) loss function to predict answers in the form of embedded matrices. The third network, which constitutes the main novelty in our updated workflow, also uses a MSE loss function. Its main usage is to extract compound names from relevant sentences resulting from the previous network. To optimize the network function, we compared eight different designs. We found that a deep neural network consisting of an RNN neural network and a leaky ReLU could achieve 0.0001 loss and 67% sensitivity. Additionally, we validated Adera2.0’s ability to predict NI drug usage against the DRUG Repurposing Hub database. These results establish the ability of Adera2.0 to repurpose drug candidates that can shorten the development of the drug cycle. The workflow could be download online.

Statistical Inference of Weighted Exponential Distribution under Joint Progressive Type-II Censoring

The weighted exponential distribution is a promising skewed distribution in the life-testing experiment. The joint progressive type-II censoring (JPC) scheme is an effective approach to reducing costs. In this paper, we consider the estimates of parameters of the weighted exponential distribution with the JPC data. Two populations, whose scale parameters are the same but the shape parameters of which are different, are chosen to be studied. We first evaluate the parameters with the maximum likelihood method. Because the maximum likelihood estimates of parameters cannot be obtained in closed form, we apply the Newton–Raphson method in this part. Bayesian estimates and the corresponding credible intervals under the squared error loss function are computed by using the Markov Chain Monte Carlo method. After that, we use the bootstrap method to calculate the associated confidence intervals of the unknown parameters. A simulation has been performed to test the feasibility of the above methods and real data analysis is also provided for illustrative purposes.

A New 3-Parameter Bounded Beta Distribution: Properties, Estimation, and Applications

This study presents a new three-parameter beta distribution defined on the unit interval, which can have increasing, decreasing, left-skewed, right-skewed, approximately symmetric, bathtub, and upside-down bathtub shaped densities, and increasing, U, and bathtub shaped hazard rates. This model can define well-known distributions with various parameters and supports, such as Kumaraswamy, beta exponential, exponential, exponentiated exponential, uniform, the generalized beta of the first kind, and beta power distributions. We present a comprehensive account of the mathematical features of the new model. Maximum likelihood methods and a Bayesian method under squared error and linear exponential loss functions are presented; also, approximate confidence intervals are obtained. We present a simulation study to compare all the results. Two real-world data sets are analyzed to demonstrate the utility and adaptability of the proposed model.

Sparse phase retrieval using a physics-informed neural network for Fourier ptychographic microscopy.

In this paper, we report a sparse phase retrieval framework for Fourier ptychographic microscopy using the recently proposed principle of physics-informed neural networks. The phase retrieval problem is cast as training bidirectional mappings from the measured image space with random noise and the object space to be reconstructed, in which the image formation physics and convolutional neural network are integrated. Meanwhile, we slightly modify the mean absolute error loss function considering the signal characteristics. Two datasets are used to validate this framework. The results indicate that the proposed framework is able to reconstruct sparsely sampled data using a small aperture overlapping rate without additional data driving whereas conventional methods cannot.

A Discrete Exponential Generalized-G Family of Distributions: Properties with Bayesian and Non-Bayesian Estimators to Model Medical, Engineering and Agriculture Data

This paper introduces a new flexible probability tool for modeling extreme and zero-inflated count data under different shapes of hazard rates. Many relevant mathematical and statistical properties are derived and analyzed. The new tool can be used to discuss several kinds of data, such as “asymmetric and left skewed”, “asymmetric and right skewed”, “symmetric”, “symmetric and bimodal”, “uniformed”, and “right skewed with a heavy tail”, among other useful shapes. The failure rate of the new class can vary and can take the forms of “increasing-constant”, “constant”, “monotonically dropping”, “bathtub”, “monotonically increasing”, or “J-shaped”. Eight classical estimation techniques—including Cramér–von Mises, ordinary least squares, L-moments, maximum likelihood, Kolmogorov, bootstrapping, and weighted least squares—are considered, described, and applied. Additionally, Bayesian estimation under the squared error loss function is also derived and discussed. Comprehensive comparison between approaches is performed for both simulated and real-life data. Finally, four real datasets are analyzed to prove the flexibility, applicability, and notability of the new class.

Bayesian Life Analysis of Generalized Chen's Population Under Progressive Censoring

Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.

Insulator and Burst Fault Detection Using an Improved Yolov3 Algorithm

Insulators play an important role in the operation of outdoor high-voltage transmission lines. However, insulators are installed in outdoor environments for long periods and thus failures are inevitable. It is necessary to conduct timely insulator inspection and maintenance. In this paper, an improved Yolov3 target detection network (Yolov3-CK) is proposed in order to achieve higher detection accuracy and speed. First, Yolov3-CK uses the CIOU loss function instead of the mean square error loss function from Yolov3. Second, the Yolov3-CK model uses cluster analysis of the priori box via the k -means++ algorithm to obtain a priori box size that is more suitable for the detection of insulators and their burst faults. Finally, we use a dataset obtained by performing data enhancement on the China power line insulator dataset to train and test the data-enhanced Yolov3-CK model. The mean precision of Yolov3-CK reaches 91.67% with 47.9 frames processed per second. Yolov3-CK provides better detection accuracy and a higher processing rate than Faster RCNN, SSD, and Yolov3. Therefore, the Yolov3-CK model is more suitable for the detection of insulators and their burst faults.

Metode Bayesian untuk Estimasi Parameter Distribusi Eksponensial pada Data Tersensor

Parameter is a value that describe the characteristics of a population. But the parameterof a real data, the value is unknown. To estimate the value of the parameter,there are several methods, which are maximum likelihood estimation method (MLE)and Bayesian parameter estimation method. In Bayesian method, the prior informationis applied to update the current data. The prior is determined based on the informationin the data. This mini thesis is using censored data with exponential distribution, andusing the conjugate prior. Followed by squared error loss function (SELF), the estimatedvalue function ot the λ parameter is ˆλ =α+Σ_{i=1}^{n}δ_{i}β+Σ_{i=1}^{n}t_{i}with α and β are hyperparameters,Σ_{i=1}^{n}δ_{i} is the number of objects that experienced the event and Σ_{i=1}^{n}t_{i} is the numberof the survival time. When the function was applied on Stanford heart transplant data,the value of ˆλ = 0.00089, which means the patient’s failure (death) probability is lowand the patient’s probability to survive is high.

Estimation and prediction for Burr type III distribution based on unified progressive hybrid censoring scheme

The present communication develops the tools for estimation and prediction of the Burr-III distribution under unified progressive hybrid censoring scheme. The maximum likelihood estimates of model parameters are obtained. It is shown that the maximum likelihood estimates exist uniquely. Expectation maximization and stochastic expectation maximization methods are employed to compute the point estimates of unknown parameters. Based on the asymptotic distribution of the maximum likelihood estimators, approximate confidence intervals are proposed. In addition, the bootstrap confidence intervals are constructed. Furthermore, the Bayes estimates are derived with respect to squared error and LINEX loss functions. To compute the approximate Bayes estimates, Metropolis–Hastings algorithm is adopted. The highest posterior density credible intervals are obtained. Further, maximum a posteriori estimates of the model parameters are computed. The Bayesian predictive point, as well as interval estimates, are proposed. A Monte Carlo simulation study is employed in order to evaluate the performance of the proposed statistical procedures. Finally, two real data sets are considered and analysed to illustrate the methodologies established in this paper.

Analysis of unit-Weibull based on progressive type-II censored with optimal scheme

Using progressive Type-II censoring data, this study deals with the estimation of parameters of the Unit-Weibull distribution using two classical methods and the Bayesian method. In the classical methods, maximum likelihood and the maximum product of spacing (MPS) methods are used to obtain the parameters of the model by utilizing the Newton–Raphson method. On the basis of observed Fisher information matrix, approximate confidence intervals for the unknown parameters are obtained. In addition, two bootstrap methods are used to obtain confidence intervals for the unknown parameters of the model. In the Bayesian estimation, we have considered both likelihood function as well as product of spacing function to estimate the model parameters. Bayes estimators are obtained under squared error loss function using independent gamma density priors for the unknown model parameters. Since closed-form of the Bayes estimators is not available, the Metropolis–Hastings algorithm is proposed to approximate the Bayes estimates. In addition, highest posterior density credible intervals are obtained. Further, using different optimally criteria, an optimal scheme has been proposed. A simulation study is conducted to assess the statistical performance of all the estimators. To demonstrate the proposed methodology a real data analysis is provided to illustrate all the statistical inferential procedures developed in the paper.

Arctan-Based Family of Distributions: Properties, Survival Regression, Bayesian Analysis and Applications

In this paper, a new class of the continuous distributions is established via compounding the arctangent function with a generalized log-logistic class of distributions. Some structural properties of the suggested model such as distribution function, hazard function, quantile function, asymptotics and a useful expansion for the new class are given in a general setting. Two special cases of this new class are considered by employing Weibull and normal distributions as the parent distribution. Further, we derive a survival regression model based on a sub-model with Weibull parent distribution and then estimate the parameters of the proposed regression model making use of Bayesian and frequentist approaches. We consider seven loss functions, namely the squared error, modified squared error, weighted squared error, K-loss, linear exponential, general entropy, and precautionary loss functions for Bayesian discussion. Bayesian numerical results include a Bayes estimator, associated posterior risk, credible and highest posterior density intervals are provided. In order to explore the consistency property of the maximum likelihood estimators, a simulation study is presented via Monte Carlo procedure. The parameters of two sub-models are estimated with maximum likelihood and the usefulness of these sub-models and a proposed survival regression model is examined by means of three real datasets.

Inferences and Engineering Applications of Alpha Power Weibull Distribution Using Progressive Type-II Censoring

As an extension of the standard Weibull distribution, a new crucial distribution termed alpha power Weibull distribution has been presented. It can model decreasing, increasing, bathtub, and upside-down bathtub failure rates. This research investigates the estimation of model parameters and some of its reliability characteristics using progressively Type-II censored data. To get estimates of unknown parameters, reliability, and hazard rate functions, the maximum likelihood, and Bayesian estimation approaches are studied. To acquire estimated confidence intervals for unknown parameters and reliability characteristics, the maximum likelihood asymptotic properties are used. The Markov chain Monte Carlo approach is used in Bayesian estimation to provide Bayesian estimates under squared error and LINEX loss functions. Furthermore, the highest posterior density credible intervals of the parameters and reliability characteristics are determined. A Monte Carlo simulation study is used to investigate the accuracy of various point and interval estimators. In addition, various optimality criteria are used to choose the best progressive censoring schemes. Two real data from the engineering field are analyzed to demonstrate the applicability and significance of the proposed approaches. Based on numerical results, the Bayesian procedure for estimating the parameters and reliability characteristics of alpha power Weibull distribution is recommended. The analysis of two real data sets showed that the alpha power Weibull distribution is a good model to investigate engineering data in the presence of progressive Type-II censoring.

Bayesian Inference for Geometric Process with Lindley Distribution and its Applications

The geometric process (GP) plays an important role in the reliability theory and life span models. It has been used extensively as a stochastic model in many areas of application. Therefore, the parameter estimation problem is very crucial in a GP. In this study, the parameter estimation problem for GP is discussed under the assumption that [Formula: see text] has a Lindley distribution with parameter [Formula: see text]. The maximum likelihood (ML) estimators of [Formula: see text] and [Formula: see text] of the GP and their asymptotic distributions are derived. A test statistic is developed based on ML estimators for testing whether [Formula: see text] or not. The same problem is also studied by using Bayesian methods. Bayes estimators of the unknown model parameters are obtained under squared error loss function (SELF) using uniform and gamma priors on the ratio [Formula: see text] and [Formula: see text] parameters. It is not possible to obtain Bayes estimators in explicit forms. Therefore, Markov Chain Monte Carlo (MCMC), Lindley (LD), and Tierney–Kadane (T–K) methods are used to estimate the parameters [Formula: see text] and [Formula: see text] in GP. The efficiencies of the ML estimators are compared with Bayes estimators via an extensive Monte Carlo simulation study. It is seen that the Bayes estimators perform better than the ML estimators. Two real-life examples are also presented for application purposes. The first data set concerns the coal mining disaster. The second is the number of COVID-19 patients in Turkey.

Statistical inference for a constant-stress partially accelerated life tests based on progressively hybrid censored samples from inverted Kumaraswamy distribution.

In this article, we investigate the problem of point and interval estimations under constant-stress partially accelerated life tests. The lifetime of items under use condition is assumed to follow the two-parameter inverted Kumaraswamy distribution. Based on Type-I progressively hybrid censored samples, the maximum likelihood and Bayesian methods are applied to estimate the model parameters as well as the acceleration factor. Under linear exponential, general entropy and squared error loss functions, Bayesian method outcomes are obtained. In addition, interval estimation is achieved by finding approximately confidence intervals for the parameters, as well as credible intervals. To investigate the accuracy of the obtained estimates and to compare the performance of confidence intervals, a Monte Carlo simulation is developed. Finally, a set of real data is analyzed to demonstrate the estimation procedures.

Probabilistic Prediction of Dst Storms One‐Day‐Ahead Using Full‐Disk SoHO Images

AbstractWe present a new model for the probability that the disturbance storm time (Dst) index exceeds −100 nT, with a lead time between 1 and 3 days. Dst provides essential information about the strength of the ring current around the Earth caused by the protons and electrons from the solar wind, and it is routinely used as a proxy for geomagnetic storms. The model is developed using an ensemble of Convolutional Neural Networks that are trained using Solar and Heliospheric Observatory (SoHO) images (Michelson Doppler Imager, Extreme ultraviolet Imaging Telescope, and Large Angle and Spectrometric Coronagraph). The relationship between the SoHO images and the solar wind has been investigated by many researchers, but these studies have not explicitly considered using SoHO images to predict the Dst index. This work presents a novel methodology to train the individual models and to learn the optimal ensemble weights iteratively, by using a customized class‐balanced mean square error (CB‐MSE) loss function tied to a least‐squares based ensemble. The proposed model can predict the probability that Dst < −100 nT 24 hr ahead with a True Skill Statistic (TSS) of 0.62 and Matthews Correlation Coefficient (MCC) of 0.37. The weighted TSS and MCC is 0.68 and 0.47, respectively. An additional validation during non‐Earth‐directed CME periods is also conducted which yields a good TSS and MCC score.