Modified linear discriminant analysis using block covariance matrix in high-dimensional data

Among many classification methods, linear discriminant analysis (LDA) is a favored tool due to its simplicity, robustness, and predictive accuracy but when the number of genes is larger than the number of observations, it cannot be applied directly because the within-class covariance matrix is singular. Also, diagonal LDA (DLDA) is a simpler model compared to LDA and has better performance in some cases. However, in reality, DLDA requires a strong assumption based on mutual independence. In this article, we propose the modified LDA (MLDA). MLDA is based on independence, but uses the information that has an effect on classification performance with the dependence structure. We suggest two approaches. One is the case of using gene rank. The other involves no use of gene rank. We found that MLDA has better performance than LDA, DLDA, or K-nearest neighborhood and is comparable with support vector machines in real data analysis and the simulation study.     

Decomposition of Variance for Spatial Cox Processes

Abstract. Spatial Cox point processes is a natural framework for quantifying the various sources of variation governing the spatial distribution of rain forest trees. We introduce a general criterion for variance decomposition for spatial Cox processes and apply it to specific Cox process models with additive or log linear random intensity functions. We moreover consider a new and flexible class of pair correlation function models given in terms of normal variance mixture covariance functions. The proposed methodology is applied to point pattern data sets of locations of tropical rain forest trees.     

Evaluating the Use of Spatial Covariance Structures in the Analysis of Cardiovascular Imaging Data

This article investigates the choice of working covariance structures in the analysis of spatially correlated observations motivated by cardiac imaging data. Through Monte Carlo simulations, we found that the choice of covariance structure affects the efficiency of the estimator and power of the test. Choosing the popular unstructured working covariance structure results in an over-inflated Type I error possibly due to a sample size not large enough relative to the number of parameters being estimated. With regard to model fit indices, Bayesian Information Criterion outperforms Akaike Information Criterion in choosing the correct covariance structure used to generate data.     

Multivariate process dispersion monitoring without subgrouping

The memory-type adaptive and non-adaptive control charts are among the best control charts for detecting small-to-moderate changes in the process parameter(s). In this paper, we propose the Crosier CUSUM (CCUSUM), EWMA, adaptive CCUSUM (ACCUSUM) and adaptive EWMA (AEWMA) charts for efficiently monitoring the changes in the covariance matrix of a multivariate normal process without subgrouping. Using extensive Monte Carlo simulations, the length characteristics of these control charts are computed. It turns out that the ACCUSUM and AEWMA charts perform uniformly and substantially better than the CCUSUM and EWMA charts when detecting a range of shift sizes in the covariance matrix. Moreover, the AEWMA chart outperforms the ACCUSUM chart. A real dataset is used to explain the implementation of the proposed control charts.     

USING AN UNCERTAINTY MODEL TO ASSESS SALES RESPONSE TO ADVERTISING*

Implications of a probabilistic “random walk” model of incremental sales response to advertising are developed for various timing patterns of advertising expenditures. Maximum likelihood procedures for assessing advertising effectiveness and for estimating a decay (forgetting) rate are developed and applied to artificial data of known configuration and are used to assess the impact over time of a brochure program on mail-order sales. Results are also compared to those from alternative models involving various lag patterns in advertising effects.     

Designing and integrating composite networks for monitoring multivariate gaussian pollution fields

Networks of ambient monitoring stations are used to monitor environmental pollution fields such as those for acid rain and air pollution. Such stations provide regular measurements of pollutant concentrations. The networks are established for a variety of purposes at various times so often several stations measuring different subsets of pollutant concentrations can be found in compact geographical regions. The problem of statistically combining these disparate information sources into a single 'network' then arises. Capitalizing on the efficiencies so achieved can then lead to the secondary problem of extending this network. The subject of this paper is a set of 31 air pollution monitoring stations in southern Ontario. Each of these regularly measures a particular subset of ionic sulphate, sulphite, nitrite and ozone. However, this subset varies from station to station. For example only two stations measure all four. Some measure just one. We describe a Bayesian framework for integrating the measurements of these stations to yield a spatial predictive distribution for unmonitored sites and unmeasured concentrations at existing stations. Furthermore we show how this network can be extended by using an entropy maximization criterion. The methods assume that the multivariate response field being measured has a joint Gaussian distribution conditional on its mean and covariance function. A conjugate prior is used for these parameters, some of its hyperparameters being fitted empirically.     

Stochastic Choice and Consistency in Decision Making Under Risk: An Experimental Study

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time.     

Standard errors for EM estimation

The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.     

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

In this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.