Some Geometric Methods for Constructing Decision Criteria Based On Two-Dimensional Parameters

This paper reviews two types of geometric methods proposed in recent years for defining statistical decision rules based on 2-dimensional parameters that characterize treatment effect in a medical setting. A common example is that of making decisions, such as comparing treatments or selecting a best dose, based on both the probability of efficacy and the probability toxicity. In most applications, the 2-dimensional parameter is defined in terms of a model parameter of higher dimension including effects of treatment and possibly covariates. Each method uses a geometric construct in the 2-dimensional parameter space based on a set of elicited parameter pairs as a basis for defining decision rules. The first construct is a family of contours that partitions the parameter space, with the contours constructed so that all parameter pairs on a given contour are equally desirable. The partition is used to define statistical decision rules that discriminate between parameter pairs in term of their desirabilities. The second construct is a convex 2-dimensional set of desirable parameter pairs, with decisions based on posterior probabilities of this set for given combinations of treatments and covariates under a Bayesian formulation. A general framework for all of these methods is provided, and each method is illustrated by one or more applications.     

Particle filtering for partially observed Gaussian state space models

Summary. Solving Bayesian estimation problems where the posterior distribution evolves over time through the accumulation of data has many applications for dynamic models. A large number of algorithms based on particle filtering methods, also known as sequential Monte Carlo algorithms, have recently been proposed to solve these problems. We propose a special particle filtering method which uses random mixtures of normal distributions to represent the posterior distributions of partially observed Gaussian state space models. This algorithm is based on a marginalization idea for improving efficiency and can lead to substantial gains over standard algorithms. It differs from previous algorithms which were only applicable to conditionally linear Gaussian state space models. Computer simulations are carried out to evaluate the performance of the proposed algorithm for dynamic tobit and probit models.     

Modeling Interval Time Series with Space–Time Processes

We consider interval-valued time series, that is, series resulting from collecting real intervals as an ordered sequence through time. Since the lower and upper bounds of the observed intervals at each time point are in fact values of the same variable, they are naturally related. We propose modeling interval time series with space–time autoregressive models and, based on the process appropriate for the interval bounds, we derive the model for the intervals’ center and radius. A simulation study and an application with data of daily wind speed at different meteorological stations in Ireland illustrate that the proposed approach is appropriate and useful.     

Sequential tuning of complex computer models

We propose a method that uses a sequential design instead of a space filling design for estimating tuning parameters of a complex computer model. The goal is to bring the computer model output closer to the real system output. The method fits separate Gaussian process (GP) models to the available data from the physical experiment and the computer experiment and minimizes the discrepancy between the predictions from the GP models to obtain estimates of the tuning parameters. A criterion based on the discrepancy between the predictions from the two GP models and the standard error of prediction for the computer experiment output is then used to obtain a design point for the next run of the computer experiment. The tuning parameters are re-estimated using the augmented data set. The steps are repeated until the budget for the computer experiment data is exhausted. Simulation studies show that the proposed method performs better in bringing a computer model closer to the real system than methods that use a space filling design.     

Transformation of non positive semidefinite correlation matrices

In multivariate statistics, estimation of the covariance or correlation matrix is of crucial importance. Computational and other arguments often lead to the use of coordinate-dependent estimators, yielding matrices that are symmetric but not positive semidefinite. We briefly discuss existing methods, based on shrinking, for transforming such matrices into positive semidefinite matrices, A simple method based on eigenvalues is also considered. Taking into account the geometric structure of correlation matrices, a new method is proposed which uses techniques similar to those of multidimensional scaling.     

Statistical inference for oscillation processes

ABSTRACT

A new model for time series with a specific oscillation pattern is proposed. The model consists of a hidden phase process controlling the speed of polling and a nonparametric curve characterizing the pattern, leading together to a generalized state space model. Identifiability of the model is proved and a method for statistical inference based on a particle smoother and a nonparametric EM algorithm is developed. In particular, the oscillation pattern and the unobserved phase process are estimated. The proposed algorithms are computationally efficient and their performance is assessed through simulations and an application to human electrocardiogram recordings.     

Modular-transform based clustering

Spectral clustering uses eigenvectors of the Laplacian of the similarity matrix. It is convenient to solve binary clustering problems. When applied to multi-way clustering, either the binary spectral clustering is recursively applied or an embedding to spectral space is done and some other methods, such as K-means clustering, are used to cluster the points. Here we propose and study a K-way clustering algorithm – spectral modular transformation, based on the fact that the graph Laplacian has an equivalent representation, which has a diagonal modular structure. The method first transforms the original similarity matrix into a new one, which is nearly disconnected and reveals a cluster structure clearly, then we apply linearized cluster assignment algorithm to split the clusters. In this way, we can find some samples for each cluster recursively using the divide and conquer method. To get the overall clustering results, we apply the cluster assignment obtained in the previous step as the initialization of multiplicative update method for spectral clustering. Examples show that our method outperforms spectral clustering using other initializations.     

Classification using semiparametric mixtures

A new density-based classification method that uses semiparametric mixtures is proposed. Like other density-based classifiers, it first estimates the probability density function for the observations in each class, with a semiparametric mixture, and then classifies a new observation by the highest posterior probability. By making a proper use of a multivariate nonparametric density estimator that has been developed recently, it is able to produce adaptively smooth and complicated decision boundaries in a high-dimensional space and can thus work well in such cases. Issues specific to classification are studied and discussed. Numerical studies using simulated and real-world data show that the new classifier performs very well as compared with other commonly used classification methods.     

Estimating multiple-membership logit models with mixed effects: indirect inference versus data cloning

Multiple-membership logit models with random effects are models for clustered binary data, where each statistical unit can belong to more than one group. The likelihood function of these models is analytically intractable. We propose two different approaches for parameter estimation: indirect inference and data cloning (DC). The former is a non-likelihood-based method which uses an auxiliary model to select reasonable estimates. We propose an auxiliary model with the same dimension of parameter space as the target model, which is particularly convenient to reach good estimates very fast. The latter method computes maximum likelihood estimates through the posterior distribution of an adequate Bayesian model, fitted to cloned data. We implement a DC algorithm specifically for multiple-membership models. A Monte Carlo experiment compares the two methods on simulated data. For further comparison, we also report Bayesian posterior mean and Integrated Nested Laplace Approximation hybrid DC estimates. Simulations show a negligible loss of efficiency for the indirect inference estimator, compensated by a relevant computational gain. The approaches are then illustrated with two real examples on matched paired data.     

Confidence intervals based on the deviance statistic for the hyperparameters in state space models

The main objective of this work is to evaluate the performance of confidence intervals, built using the deviance statistic, for the hyperparameters of state space models. The first procedure is a marginal approximation to confidence regions, based on the likelihood test, and the second one is based on the signed root deviance profile. Those methods are computationally efficient and are not affected by problems such as intervals with limits outside the parameter space, which can be the case when the focus is on the variances of the errors. The procedures are compared to the usual approaches existing in the literature, which includes the method based on the asymptotic distribution of the maximum likelihood estimator, as well as bootstrap confidence intervals. The comparison is performed via a Monte Carlo study, in order to establish empirically the advantages and disadvantages of each method. The results show that the methods based on the deviance statistic possess a better coverage rate than the asymptotic and bootstrap procedures.     

A mathematical comparison of the members of an infinite family of agglomerative clustering algorithms

This article presents an algebraic analysis of agglomerative clustering method algorithms, which results in a graphic portrayal of these algorithms and a classification scheme for these algorithms based on the degree of distortion perpetrated on the object space by the algorithms in each group.     

Balanced Asymmetrical Nearly Orthogonal Designs for first and second order effect estimation

A method for constructing asymmetrical (mixed-level) designs, satisfying the balancing and interaction estimability requirements with a number of runs as small as possible, is proposed in this paper. The method, based on a heuristic procedure, uses a new optimality criterion formulated here. The proposed method demonstrates efficiency in terms of searching time and optimality of the attained designs. A complete collection of such asymmetrical designs with two- and three-level factors is available. A technological application is also presented.     

MAXIMUM LIKELIHOOD ESTIMATION OF GROSS FLOWS

A method of maximum likelihood estimation of gross flows from overlapping stratified sample data is developed. The approach taken is model-based and the EM algorithm is used to solve the estimation problem. Inference is thus based on information from the total sample at each time period. This can be contrasted with the conventional approach to gross flows estimation which only uses information from the overlapping sub-sample. An application to estimation of flows of Australian cropping and livestock industries farms into and out of an “at risk” situation over the period 1979–84 is presented, as well as a discussion of extensions to more complex sampling situations.     

Bayesian scale space analysis of temporal changes in satellite images

We consider the detection of land cover changes using pairs of Landsat ETM+ satellite images. The images consist of eight spectral bands and to simplify the multidimensional change detection task, the image pair is first transformed to a one-dimensional image. When the transformation is non-linear, the true change in the images may be masked by complex noise. For example, when changes in the Normalized Difference Vegetation Index is considered, the variance of noise may not be constant over the image and methods based on image thresholding can be ineffective. To facilitate detection of change in such situations, we propose an approach that uses Bayesian statistical modeling and simulation-based inference. In order to detect both large and small scale changes, our method uses a scale space approach that employs multi-level smoothing. We demonstrate the technique using artificial test images and two pairs of real Landsat ETM+satellite images.     

Perfect sampling algorithm for small <Emphasis Type="Italic">m</Emphasis>×<Emphasis Type="Italic">n</Emphasis> contingency tables

A Markov chain is proposed that uses coupling from the past sampling algorithm for sampling m×n contingency tables. This method is an extension of the one proposed by Kijima and Matsui (Rand. Struct. Alg., 29:243–256, 2006). It is not polynomial, as it is based upon a recursion, and includes a rejection phase but can be used for practical purposes on small contingency tables as illustrated in a classical 4×4 example.     

A reflected feature space for CART

We present an algorithm for learning oblique decision trees, called HHCART(G). Our decision tree combines learning concepts from two classification trees, HHCART and Geometric Decision Tree (GDT). HHCART(G) is a simplified HHCART algorithm that uses linear structure in the training examples, captured by a modified GDT angle bisector, to define splitting directions. At each node, we reflect the training examples with respect to the modified angle bisector to align this linear structure with the coordinate axes. Searching axis parallel splits in this reflected feature space provides an efficient and effective way of finding oblique splits in the original feature space. Our method is much simpler than HHCART because it only considers one reflected feature space for node splitting. HHCART considers multiple reflected feature spaces for node splitting making it more computationally intensive to build. Experimental results show that HHCART(G) is an effective classifier, producing compact trees with similar or better results than several other decision trees, including GDT and HHCART trees.     

A continuous repeated measurements model:the experimental design problem with fixed time constraints

A repeated measurements model where the within-subject response is modeled as a continuous time regression is analyzed using reproducing kernel Hilbert space methods, Parzen (1961). The basic goal is to study the statistical design problem when experimental units and measurements are "free", but the total measurement time is fixed. This kind of situation arises when the experiment uses scarce resources such as space or oceanographic platforms or a magnetic resonance imaging (MRI) device. In this situation the only constraint is the total time allocated to the experiment.     

An alternative method for global and partial comparison of two diagnostic systems based on ROC curves

In this paper, an alternative method for the comparison of two diagnostic systems based on receiver operating characteristic (ROC) curves is presented. ROC curve analysis is often used as a statistical tool for the evaluation of diagnostic systems. However, in general, the comparison of ROC curves is not straightforward, in particular, when they cross each other. A similar difficulty is also observed in the multi-objective optimization field where sets of solutions defining fronts must be compared with a multi-dimensional space. Thus, the proposed methodology is based on a procedure used to compare the performance of distinct multi-objective optimization algorithms. In general, methods based on the area under the ROC curves are not sensitive to the existence of crossing points between the curves. The new approach can deal with this situation and also allows the comparison of partial portions of ROC curves according to particular values of sensitivity and specificity of practical interest. Simulations results are presented. For illustration purposes, considering real data from newborns with very low birthweight, the new method was applied in order to discriminate the better index for evaluating the risk of death.     

中国城市民生发展的区域差异测度、来源分解与形成机理

本文基于熵值法构建城市民生发展指数,利用Dagum基尼系数和方差分解方法首次从空间和结构两个视角考察我国城市民生发展的差异大小及来源,并运用关系数据分析范式研究其形成机理。研究发现:我国城市民生发展水平持续向好,但2010年之后民生发展不均衡现象显著;民生发展总体差异主要来源于区域间差异,其中东部和中部区域间差异贡献最大,东部区域内差异贡献显著高于中部、西部地区;社会发展差异和生态建设差异是民生发展差异的主要结构来源;人民生活差异、科技创新差异对民生发展差异的影响均从东向西依次减弱,社会发展差异的作用则依次增强,生态建设差异的影响在中部地区最为突出,经济发展差异对东部地区的影响较弱;人民生活差异、科技创新差异上升是导致2010年之后民生发展差异扩大的主要原因。本研究丰富了城市民生评价体系的构建方案,为揭示我国城市民生发展差异状况、探索协同提升对策提供了重要借鉴经验。     

Minimal–maximal correlation-type goodness-of-fit tests

ABSTRACT

In this paper we consider correlation-type tests based on plotting points which are modifications to the simultaneous closeness probability plotting points as recently introduced in the literature. In particular, we consider a maximal correlation test and a minimal correlation test. Furthermore, we provide two methods to carry out each test, where one method uses plotting points which are data dependent and the other test uses plotting points which are not. Some numerical properties on the associated correlation statistics are provided for various distributions, as well as a comprehensive power study to assess their performance in comparison to correlation-type tests based on more traditional plotting points. Two illustrative examples are also provided to demonstrate the tests. Finally, we make some observations and provide ideas for future work.