In statistical theory, a statistic that is function of sample observations is used to estimate distribution parameter. This statistic is called unbiased estimate if its expectation is equal to theoretical parameter. P...In statistical theory, a statistic that is function of sample observations is used to estimate distribution parameter. This statistic is called unbiased estimate if its expectation is equal to theoretical parameter. Proving whether or not a statistic is unbiased estimate is very important but this proof may require a lot of efforts when statistic is complicated function. Therefore, this research facilitates this proof by proposing a theorem which states that the expectation of variable x 〉 0 is u if and only if the limit of logarithm expectation of x approaches logarithm of u. In order to make clear of this theorem, the research gives an example of proving correlation coefficient as unbiased estimate by taking advantages of this theorem.展开更多
Efficient iterative unsupervised machine learning involving probabilistic clustering analysis with the expectation-maximization(EM)clustering algorithm is applied to categorize reservoir facies by exploiting latent an...Efficient iterative unsupervised machine learning involving probabilistic clustering analysis with the expectation-maximization(EM)clustering algorithm is applied to categorize reservoir facies by exploiting latent and observable well-log variables from a clastic reservoir in the Majnoon oilfield,southern Iraq.The observable well-log variables consist of conventional open-hole,well-log data and the computer-processed interpretation of gamma rays,bulk density,neutron porosity,compressional sonic,deep resistivity,shale volume,total porosity,and water saturation,from three wells located in the Nahr Umr reservoir.The latent variables include shale volume and water saturation.The EM algorithm efficiently characterizes electrofacies through iterative machine learning to identify the local maximum likelihood estimates(MLE)of the observable and latent variables in the studied dataset.The optimized EM model developed successfully predicts the core-derived facies classification in two of the studied wells.The EM model clusters the data into three distinctive reservoir electrofacies(F1,F2,and F3).F1 represents a gas-bearing electrofacies with low shale volume(Vsh)and water saturation(Sw)and high porosity and permeability values identifying it as an attractive reservoir target.The results of the EM model are validated using nuclear magnetic resonance(NMR)data from the third studied well for which no cores were recovered.The NMR results confirm the effectiveness and accuracy of the EM model in predicting electrofacies.The utilization of the EM algorithm for electrofacies classification/cluster analysis is innovative.Specifically,the clusters it establishes are less rigidly constrained than those derived from the more commonly used K-means clustering method.The EM methodology developed generates dependable electrofacies estimates in the studied reservoir intervals where core samples are not available.Therefore,once calibrated with core data in some wells,the model is suitable for application to other wells that lack core data.展开更多
This paper develops a new method of parametric estimate, which is named as 'synthesized expected Bayesian method'. When samples of products are tested and no failure events occur, the definition of expected Ba...This paper develops a new method of parametric estimate, which is named as 'synthesized expected Bayesian method'. When samples of products are tested and no failure events occur, the definition of expected Bayesian estimate is introduced and the estimates of failure probability and failure rate are provided. After some failure information is introduced by making an extra-test, a synthesized expected Bayesian method is defined and used to estimate failure probability, failure rate and some other parameters in exponential distribution and Weibull distribution of populations. Finally, calculations are performed according to practical problems, which show that the synthesized expected Bayesian method is feasible and easy to operate.展开更多
In the study, a method of solving ANOVA problems based on an unbalanced three-way mixed effects model with interaction for data when factors A and B are fixed, and factor C is random was presented, and the required EM...In the study, a method of solving ANOVA problems based on an unbalanced three-way mixed effects model with interaction for data when factors A and B are fixed, and factor C is random was presented, and the required EMS was derived. Under each of the appropriate null hypotheses, it was observed that none of the derived EMS was unbiased for the other. Unbiased estimators of the mean squares were determined to test hypotheses. With the unbiased estimators,?appropriate F-statistics as well as their corresponding pseudo-degrees of freedom were obtained. The theoretical results presented in the paper were?illustrated using a numerical example.展开更多
Fisher [1] proposed a simple method to combine p-values from independent investigations without using detailed information of the original data. In recent years, likelihood-based asymptotic methods have been developed...Fisher [1] proposed a simple method to combine p-values from independent investigations without using detailed information of the original data. In recent years, likelihood-based asymptotic methods have been developed to produce highly accurate p-values. These likelihood-based methods generally required the likelihood function and the standardized maximum likelihood estimates departure calculated in the canonical parameter scale. In this paper, a method is proposed to obtain a p-value by combining the likelihood functions and the standardized maximum likelihood estimates departure of independent investigations for testing a scalar parameter of interest. Examples are presented to illustrate the application of the proposed method and simulation studies are performed to compare the accuracy of the proposed method with Fisher’s method.展开更多
文摘In statistical theory, a statistic that is function of sample observations is used to estimate distribution parameter. This statistic is called unbiased estimate if its expectation is equal to theoretical parameter. Proving whether or not a statistic is unbiased estimate is very important but this proof may require a lot of efforts when statistic is complicated function. Therefore, this research facilitates this proof by proposing a theorem which states that the expectation of variable x 〉 0 is u if and only if the limit of logarithm expectation of x approaches logarithm of u. In order to make clear of this theorem, the research gives an example of proving correlation coefficient as unbiased estimate by taking advantages of this theorem.
文摘Efficient iterative unsupervised machine learning involving probabilistic clustering analysis with the expectation-maximization(EM)clustering algorithm is applied to categorize reservoir facies by exploiting latent and observable well-log variables from a clastic reservoir in the Majnoon oilfield,southern Iraq.The observable well-log variables consist of conventional open-hole,well-log data and the computer-processed interpretation of gamma rays,bulk density,neutron porosity,compressional sonic,deep resistivity,shale volume,total porosity,and water saturation,from three wells located in the Nahr Umr reservoir.The latent variables include shale volume and water saturation.The EM algorithm efficiently characterizes electrofacies through iterative machine learning to identify the local maximum likelihood estimates(MLE)of the observable and latent variables in the studied dataset.The optimized EM model developed successfully predicts the core-derived facies classification in two of the studied wells.The EM model clusters the data into three distinctive reservoir electrofacies(F1,F2,and F3).F1 represents a gas-bearing electrofacies with low shale volume(Vsh)and water saturation(Sw)and high porosity and permeability values identifying it as an attractive reservoir target.The results of the EM model are validated using nuclear magnetic resonance(NMR)data from the third studied well for which no cores were recovered.The NMR results confirm the effectiveness and accuracy of the EM model in predicting electrofacies.The utilization of the EM algorithm for electrofacies classification/cluster analysis is innovative.Specifically,the clusters it establishes are less rigidly constrained than those derived from the more commonly used K-means clustering method.The EM methodology developed generates dependable electrofacies estimates in the studied reservoir intervals where core samples are not available.Therefore,once calibrated with core data in some wells,the model is suitable for application to other wells that lack core data.
基金This work was supported partly by the Zhejiang Province National Natural Science Foundation of China under Grant 100026And this work was supported in part by the Zhejiang Province Education Committee Foundation of China under Grant 20031024.
文摘This paper develops a new method of parametric estimate, which is named as 'synthesized expected Bayesian method'. When samples of products are tested and no failure events occur, the definition of expected Bayesian estimate is introduced and the estimates of failure probability and failure rate are provided. After some failure information is introduced by making an extra-test, a synthesized expected Bayesian method is defined and used to estimate failure probability, failure rate and some other parameters in exponential distribution and Weibull distribution of populations. Finally, calculations are performed according to practical problems, which show that the synthesized expected Bayesian method is feasible and easy to operate.
文摘In the study, a method of solving ANOVA problems based on an unbalanced three-way mixed effects model with interaction for data when factors A and B are fixed, and factor C is random was presented, and the required EMS was derived. Under each of the appropriate null hypotheses, it was observed that none of the derived EMS was unbiased for the other. Unbiased estimators of the mean squares were determined to test hypotheses. With the unbiased estimators,?appropriate F-statistics as well as their corresponding pseudo-degrees of freedom were obtained. The theoretical results presented in the paper were?illustrated using a numerical example.
文摘Fisher [1] proposed a simple method to combine p-values from independent investigations without using detailed information of the original data. In recent years, likelihood-based asymptotic methods have been developed to produce highly accurate p-values. These likelihood-based methods generally required the likelihood function and the standardized maximum likelihood estimates departure calculated in the canonical parameter scale. In this paper, a method is proposed to obtain a p-value by combining the likelihood functions and the standardized maximum likelihood estimates departure of independent investigations for testing a scalar parameter of interest. Examples are presented to illustrate the application of the proposed method and simulation studies are performed to compare the accuracy of the proposed method with Fisher’s method.
