Fuzzy logic modifications of the Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) is a measurement methodology based on pair-wise comparisons that relies on judgment to derive priority scales. During its implementation, one constructs hierarchies, then makes judgments or performs measurements on pairs of elements with respect to a criterion to derive preference scales, which are then synthesized throughout the structure to select the preferred alternative.One of the areas where the AHP finds application is in the subjective phases of risk assessment (RA), where it is used to structure and prioritize diverse risk factors, including the judgments of experts. Since fuzzy logic (FL) has been shown to be an effective tool for accommodating human experts and their communication of linguistic variables, there has been research aimed at modeling the fuzziness in the AHP (FAHP), and recently the focus of some of that modeling has been with respect to RA.The literature discusses more than one FAHP model, which raises the question as to which are the prominent models and what are their characteristics. In response to this question, we examine three of the most influential FAHP models. The article proceeds as follows. It begins with a brief overview of the AHP and its limitations when confronted with a fuzzy environment. This is followed by a discussion of FL modifications of the AHP. A RA-based likelihood score example is used throughout. The article ends with a commentary on the findings.     

Maximum likelihood estimation for second level fMRI data analysis with expectation trust region algorithm

The trust region method which originated from the Levenberg–Marquardt (LM) algorithm for mixed effect model estimation are considered in the context of second level functional magnetic resonance imaging (fMRI) data analysis. We first present the mathematical and optimization details of the method for the mixed effect model analysis, then we compare the proposed methods with the conventional expectation-maximization (EM) algorithm based on a series of datasets (synthetic and real human fMRI datasets). From simulation studies, we found a higher damping factor for the LM algorithm is better than lower damping factor for the fMRI data analysis. More importantly, in most cases, the expectation trust region algorithm is superior to the EM algorithm in terms of accuracy if the random effect variance is large. We also compare these algorithms on real human datasets which comprise repeated measures of fMRI in phased-encoded and random block experiment designs. We observed that the proposed method is faster in computation and robust to Gaussian noise for the fMRI analysis. The advantages and limitations of the suggested methods are discussed.     

Likelihood stabilization for ill-conditioned vector GARCH models

The likelihood of vector GARCH models is ill-conditioned because of two facts. First, when the series display high correlations, as often happens with financial data, some eigenvalues of the conditional covariance matrix are close to zero. Second, the likelihood function is very flat in the neighborhood of the optimum due to the functional form of the GARCH process. These facts explain the instability of multivariate GARCH estimation procedures. Building on this analysis, we suggest a data transformation which moves the critical eigenvalues far from zero and, therefore, improves the stability of iterative optimization methods. The transformed values are re-scaled principal components, so their interpretation is straightforward. The application of this technique is illustrated by modeling the short-run conditional correlations of four nominal exchange rates.      

分层关联表中伞形序对无约束问题的检验

概率分布间的随机序是应用概率论与统计推断中的一个重要概念. 基于交叉分类数据的趋势检验问题已被广泛地研究, 并且分层关联表广泛存在于实践中. 似然比检验方法常用于涉及随机序约束问题的检验. 对带序约束的分层关联表, 该文介绍了一种不基于模型假定的似然比检验方法, 并且给出了检验统计量的极限分布.     

Affective bases of likelihood judgments

Analyses of likelihood judgments for 515 events support the idea that the perceived likelihood of an event is partly a function of affective processes. Production of affective disturbances makes events seem unlikely. Event likelihoods also are influenced by specific kinds of people and behaviors which are involved in the events. Affective determination of likelihood is stronger when actors have institutionally‐clear identities rather than institutionally‐vague identities; indeed, affect accounts for more than half of the variance in likelihoods of events within clear institutional contexts.     

Asymptotic robustness of the normal theory likelihood ratio statistic for two-level covariance structure models

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures have been developed to analyze such data while taking into account the resulting dependence of observations. Most of these developments require a multivariate normality distribution assumption. It is important to know whether normal theory-based inference can still be valid when applied to nonnormal hierarchical data sets. Using an analytical approach for balanced data and numerical illustrations for unbalanced data, this paper shows that the likelihood ratio statistic based on the normality assumption is asymptotically robust for many nonnormal distributions. The result extends the scope of asymptotic robustness theory that has been established in different contexts.     

A bartlett-type adjustment for the likelihood ratio statistic with an ordered alternative

For testing the equality of normal variances with an increasing alternative, under the null hypothesis the likelihood ratio test statistic is asymptotically distributed as a mixture of chi-squared distributions. In this paper a Bartlett-type adjustment is proposed to improve the approximation of the null distribution of the likelihood ratio test statistic with an ordered alternative.     

Tests of ordered hypotheses in linkage in heredity

The asymptotic null distribution of the likelihood ratio test for two cases of ordered hypotheses in a particular genetic model is considered. A simple iterative process is proposed in order to get the restricted estimates. It is shown that both tests have asymptotically a chi-bar squared distribution and the same size. A simulation study is also conducted in order to compare the usual unrestricted test with the corresponding one of ordered hypotheses. Finally, the results are extended to some special cases.