On the Use of Kriging Models to Approximate Deterministic Computer Models

2004 ◽  
Author(s):  
Jay D. Martin ◽  
Timothy W. Simpson
Keyword(s):  
Likelihood Estimation ◽  
Information Criterion ◽  
Computer Models ◽  
Kriging Model ◽  
Estimation Methods ◽  
Test Problems ◽  
Computationally Efficient ◽  
Design And Optimization ◽  
Parameter Estimation Methods ◽  
Computationally Efficient Algorithms

The use of kriging models for approximation and metamodel-based design and optimization has been steadily on the rise in the past decade. The widespread usage of kriging models appears to be hampered by (1) the lack of guidance in selecting the appropriate form of the kriging model, (2) computationally efficient algorithms for estimating the model’s parameters, and (3) an effective method to assess the resulting model’s quality. In this paper, we compare (1) Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) parameter estimation methods for selecting a kriging model’s parameters given its form and (2) and an R2 of prediction and the corrected Akaike Information Criterion for assessing the quality of the created kriging model, permitting the comparison of different forms of a kriging model. These methods are demonstrated with six test problems. Finally, different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models.

scholarly journals ESTIMASI PARAMETER PADA MODEL COX MULTIVARIAT DENGAN METODE MAXIMUM PARTIAL LIKELIHOOD ESTIMATION

2012 ◽  
Vol 4 (1) ◽  
pp. 185
Author(s):  
Irfan Wahyudi ◽  
Purhadi Purhadi ◽  
Sutikno Sutikno ◽  
Irhamah Irhamah
Keyword(s):  
Parameter Estimation ◽  
Likelihood Function ◽  
Cox Model ◽  
Hessian Matrix ◽  
Likelihood Estimation ◽  
Partial Likelihood ◽  
Estimation Methods ◽  
Hazard Functions ◽  
Score Vector ◽  
Parameter Estimation Methods

Multivariate Cox proportional hazard models have ratio property, that is the ratio of  hazard functions for two individuals with covariate vectors  z1 and  z2 are constant (time independent). In this study we talk about estimation of prameters on multivariate Cox model by using Maximum Partial Likelihood Estimation (MPLE) method. To determine the appropriate estimators  that maximize the ln-partial likelihood function, after a score vector and a Hessian matrix are found, numerical iteration methods are applied. In this case, we use a Newton Raphson method. This numerical method is used since the solutions of the equation system of the score vector after setting it equal to zero vector are not closed form. Considering the studies about multivariate Cox model are limited, including the parameter estimation methods, but the methods are urgently needed by some fields of study related such as economics, engineering and medical sciences. For this reasons, the goal of this study is designed to develop parameter estimation methods from univariate to multivariate cases.

Methods for Parameter Estimation of the Negative Binomial-Generalized Exponential Distribution

2017 ◽  
Vol 866 ◽  
pp. 383-386
Author(s):  
Sirinapa Aryuyuen ◽  
Issaraporn Thiamsorn
Keyword(s):  
Parameter Estimation ◽  
Negative Binomial ◽  
Likelihood Estimation ◽  
Estimation Methods ◽  
Mean Square ◽  
Generalized Exponential Distribution ◽  
Chi Square ◽  
Average Mean Square Error ◽  
Parameter Estimation Methods ◽  
Better Than

Abstract. We proposed several estimation methods for the parameters of the negative binomial-generalized exponential (NB-GE) distribution. In the simulation study, the maximum likelihood estimation (MLE) with nlm function seems to have the most efficiency to estimate the parameters and of the NB-GE distribution when it compares with method of the moments (MM) and MLE with optim function by using the average mean square error (AMSE) for a criteria. The AMSE values of each parameter estimation methods are decreasing when the sample size increasing. Moreover, the example dataset is illustrated. Based on the chi-square values for the fitting distribution via the MLE with nlm function is better than other estimation methods.

scholarly journals Volatility Fitting Performance of QGARCH(1,1) Model with Student-t, GED, and SGED Distributions

2020 ◽  
Vol 11 (2) ◽  
pp. 97-104
Author(s):  
Didit Budi Nugroho ◽  
Bintoro Ady Pamungkas ◽  
Hanna Arini Parhusip
Keyword(s):  
Stock Exchange ◽  
Linear Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Information Criterion ◽  
Estimation Methods ◽  
Data Sets ◽  
Non Linear ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

The research had two objectives. First, it compared the performance of the Generalized Autoregressive Conditional Heteroscedasticity (1,1) (GARCH) and Quadratic GARCH (1,1) (QGARCH)) models based on the fitting to real data sets. The model assumed that return error follows four different distributions: Normal (Gaussian), Student-t, General Error Distribution (GED), and Skew GED (SGED). Maximum likelihood estimation was usually employed in estimating the GARCH model, but it might not be easily applied to more complicated ones. Second, it provided two ways to evaluate the considered models. The models were estimated using the Generalized Reduced Gradient (GRG) Non-Linear method in Excel’s Solver and the Adaptive Random Walk Metropolis (ARWM) in the Scilab program. The real data in the empirical study were Financial Times Stock Exchange Milano Italia Borsa (FTSEMIB) and Stoxx Europe 600 indices over the daily period from January 2000 to December 2017 to test the conditional variance process and see whether the estimation methods could adapt to the complicated models. The analysis shows that GRG Non-Linear in Excel’s Solver and ARWM methods have close results. It indicates a good estimation ability. Based on the Akaike Information Criterion (AIC), the QGARCH(1,1) model provides a better fitting than the GARCH(1,1) model on each distribution specification. Overall, the QGARCH(1,1) with SGED distribution best fits both data.

Computational Improvements to Estimating Kriging Metamodel Parameters

2009 ◽  
Vol 131 (8) ◽  
Author(s):  
Jay D. Martin
Keyword(s):  
Response Surface ◽  
Spatial Correlation ◽  
Likelihood Estimation ◽  
Kriging Model ◽  
Model Parameters ◽  
Computationally Efficient ◽  
Spatial Correlation Function ◽  
Computational Burden ◽  
Kriging Metamodel ◽  
Gradient Based

The details of a method to reduce the computational burden experienced while estimating the optimal model parameters for a Kriging model are presented. A Kriging model is a type of surrogate model that can be used to create a response surface based a set of observations of a computationally expensive system design analysis. This Kriging model can then be used as a computationally efficient surrogate to the original model, providing the opportunity for the rapid exploration of the resulting tradespace. The Kriging model can provide a more complex response surface than the more traditional linear regression response surface through the introduction of a few terms to quantify the spatial correlation of the observations. Implementation details and enhancements to gradient-based methods to estimate the model parameters are presented. It concludes with a comparison of these enhancements to using maximum likelihood estimation to estimate Kriging model parameters and their potential reduction in computational burden. These enhancements include the development of the analytic gradient and Hessian for the log-likelihood equation of a Kriging model that uses a Gaussian spatial correlation function. The suggested algorithm is similar to the SCORING algorithm traditionally used in statistics.

A Study on the Use of Kriging Models to Approximate Deterministic Computer Models

2003 ◽  
Author(s):  
Jay D. Martin ◽  
Timothy W. Simpson
Keyword(s):  
Cross Validation ◽  
Computer Experiments ◽  
Likelihood Estimation ◽  
Higher Dimensions ◽  
Kriging Model ◽  
One Dimension ◽  
Test Problems ◽  
The Past ◽  
Different Types ◽  
Similarities And Differences

The use of kriging models for approximation and global optimization has been steadily on the rise in the past decade. The standard approach used in the Design and Analysis of Computer Experiments (DACE) is to use an Ordinary kriging model to approximate a deterministic computer model. Universal and Detrended kriging are two alternative types of kriging models. In this paper, a description on the basics of kriging is given, highlighting the similarities and differences between these three different types of kriging models and the underlying assumptions behind each. A comparative study on the use of three different types of kriging models is then presented using six test problems. The methods of Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) for model parameter estimation are compared for the three kriging model types. A one-dimension problem is first used to visualize the differences between the different models. In order to show applications in higher dimensions, four two-dimension and a 5-dimension problem are also given.

scholarly journals NGSremix: A software tool for estimating pairwise relatedness between admixed individuals from next-generation sequencing data

2021 ◽  
Author(s):  
Anne Krogh Nøhr ◽  
Kristian Hanghøj ◽  
Genis Garcia Erill ◽  
Zilong Li ◽  
Ida Moltke ◽  
...  
Keyword(s):  
Next Generation Sequencing ◽  
Genetic Research ◽  
Likelihood Estimation ◽  
Software Tool ◽  
Estimation Methods ◽  
Next Generation Sequencing Data ◽  
Next Generation ◽  
Sequencing Data ◽  
Ngs Data ◽  
Generation Sequencing

Abstract Estimation of relatedness between pairs of individuals is important in many genetic research areas. When estimating relatedness, it is important to account for admixture if this is present. However, the methods that can account for admixture are all based on genotype data as input, which is a problem for low-depth next-generation sequencing (NGS) data from which genotypes are called with high uncertainty. Here we present a software tool, NGSremix, for maximum likelihood estimation of relatedness between pairs of admixed individuals from low-depth NGS data, which takes the uncertainty of the genotypes into account via genotype likelihoods. Using both simulated and real NGS data for admixed individuals with an average depth of 4x or below we show that our method works well and clearly outperforms all the commonly used state-of-the-art relatedness estimation methods PLINK, KING, relateAdmix, and ngsRelate that all perform quite poorly. Hence, NGSremix is a useful new tool for estimating relatedness in admixed populations from low-depth NGS data. NGSremix is implemented in C/C ++ in a multi-threaded software and is freely available on Github https://github.com/KHanghoj/NGSremix.

scholarly journals Evaluating the Observed Log-Likelihood Function in Two-Level Structural Equation Modeling with Missing Data: From Formulas to R Code

Psych ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 197-232
Author(s):  
Yves Rosseel
Keyword(s):  
Structural Equation ◽  
Structural Equation Models ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Missing At Random ◽  
Equation Modeling ◽  
Computationally Efficient ◽  
Log Likelihood ◽  
Efficient Expression ◽  
Special Case

This paper discusses maximum likelihood estimation for two-level structural equation models when data are missing at random at both levels. Building on existing literature, a computationally efficient expression is derived to evaluate the observed log-likelihood. Unlike previous work, the expression is valid for the special case where the model implied variance–covariance matrix at the between level is singular. Next, the log-likelihood function is translated to R code. A sequence of R scripts is presented, starting from a naive implementation and ending at the final implementation as found in the lavaan package. Along the way, various computational tips and tricks are given.

scholarly journals A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 25 ◽  
Author(s):  
Ehab Almetwally ◽  
Randa Alharbi ◽  
Dalia Alnagar ◽  
Eslam Hafez
Keyword(s):  
Statistical Model ◽  
Least Squares ◽  
Weighted Least Squares ◽  
Likelihood Estimation ◽  
Linear Representation ◽  
Rate Function ◽  
Estimation Methods ◽  
Unknown Parameters ◽  
Least Squares Estimators ◽  
New Distribution

This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new lifetime distribution with two-parameter is introduced by a combination of inverted Topp-Leone distribution and modified Kies family to produce the modified Kies inverted Topp-Leone (MKITL) distribution, which covers a lot of application that both the traditional inverted Topp-Leone and the modified Kies provide poor fitting for them. This new distribution has many valuable properties as simple linear representation, hazard rate function, and moment function. We made several methods of estimations as maximum likelihood estimation, least squares estimators, weighted least-squares estimators, maximum product spacing, Crame´r-von Mises estimators, and Anderson-Darling estimators methods are applied to estimate the unknown parameters of MKITL distribution. A numerical result of the Monte Carlo simulation is obtained to assess the use of estimation methods. also, we applied different data sets to the new distribution to assess its performance in modeling data.

scholarly journals A Distribution for Instantaneous Failures

Stats ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 247-258 ◽  
Author(s):  
Pedro L. Ramos ◽  
Francisco Louzada
Keyword(s):  
Residual Life ◽  
Likelihood Estimation ◽  
Estimation Methods ◽  
Mean Residual Life ◽  
Parameter Distribution ◽  
Coverage Probabilities ◽  
Instantaneous Failures ◽  
The Relationship ◽  
Mean Variance ◽  
New Distribution

A new one-parameter distribution is proposed in this paper. The new distribution allows for the occurrence of instantaneous failures (inliers) that are natural in many areas. Closed-form expressions are obtained for the moments, mean, variance, a coefficient of variation, skewness, kurtosis, and mean residual life. The relationship between the new distribution with the exponential and Lindley distributions is presented. The new distribution can be viewed as a combination of a reparametrized version of the Zakerzadeh and Dolati distribution with a particular case of the gamma model and the occurrence of zero value. The parameter estimation is discussed under the method of moments and the maximum likelihood estimation. A simulation study is performed to verify the efficiency of both estimation methods by computing the bias, mean squared errors, and coverage probabilities. The superiority of the proposed distribution and some of its concurrent distributions are tested by analyzing four real lifetime datasets.

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