A goodness-of-fit test for normality based on polynomial regression

Statistical models are often based on normal distributions and procedures for testing such distributional assumption are needed. Many goodness-of-fit tests are available. However, most of them are quite insensitive in detecting non-normality when the alternative distribution is symmetric. On the other hand all the procedures are quite powerful against skewed alternatives. A new test for normality based on a polynomial regression is presented. It is very effective in detecting non-normality when the alternative distribution is symmetric. A comparison between well known tests and this new procedure is performed by simulation study. Other properties are also investigated.     

Functional linear regression after spline transformation

Functional linear regression has been widely used to model the relationship between a scalar response and functional predictors. If the original data do not satisfy the linear assumption, an intuitive solution is to perform some transformation such that transformed data will be linearly related. The problem of finding such transformations has been rather neglected in the development of functional data analysis tools. In this paper, we consider transformation on the response variable in functional linear regression and propose a nonparametric transformation model in which we use spline functions to construct the transformation function. The functional regression coefficients are then estimated by an innovative procedure called mixed data canonical correlation analysis (MDCCA). MDCCA is analogous to the canonical correlation analysis between two multivariate samples, but is between a multivariate sample and a set of functional data. Here, we apply the MDCCA to the projection of the transformation function on the B-spline space and the functional predictors. We then show that our estimates agree with the regularized functional least squares estimate for the transformation model subject to a scale multiplication. The dimension of the space of spline transformations can be determined by a model selection principle. Typically, a very small number of B-spline knots is needed. Real and simulation data examples are further presented to demonstrate the value of this approach.     

Incorporating historical control information into quantal bioassay with Bayesian approach

A Bayesian approach with an iterative reweighted least squares is used to incorporate historical control information into quantal bioassays to estimate the dose-response relationship, where the logit of the historical control responses are assumed to have a normal distribution. The parameters from this normal distribution are estimated from both empirical and full Bayesian approaches with a marginal likelihood function being approximated by Laplace’s Method. A comparison is made using real data between estimates that include the historical control information and those that do not. It was found that the inclusion of the historical control information improves the efficiency of the estimators. In addition, this logit-normal formulation is compared with the traditional beta-binomial for its improvement in parameter estimates. Consequently the estimated dose-response relationship is used to formulate the point estimator and confidence bands for ED(100p) for various values of risk rate p and the potency for any dose level.     

Fitting finite mixtures of generalized linear regressions in R

R package flexmix provides flexible modelling of finite mixtures of regression models using the EM algorithm. Several new features of the software such as fixed and nested varying effects for mixtures of generalized linear models and multinomial regression for a priori probabilities given concomitant variables are introduced. The use of the software in addition to model selection is demonstrated on a logistic regression example.     

A simple and efficient method for variable ranking according to their usefulness for learning

The selection of a subset of input variables is often based on the previous construction of a ranking to order the variables according to a given criterion of relevancy. The objective is then to linearize the search, estimating the quality of subsets containing the topmost ranked variables. An algorithm devised to rank input variables according to their usefulness in the context of a learning task is presented. This algorithm is the result of a combination of simple and classical techniques, like correlation and orthogonalization, which allow the construction of a fast algorithm that also deals explicitly with redundancy. Additionally, the proposed ranker is endowed with a simple polynomial expansion of the input variables to cope with nonlinear problems. The comparison with some state-of-the-art rankers showed that this combination of simple components is able to yield high-quality rankings of input variables. The experimental validation is made on a wide range of artificial data sets and the quality of the rankings is assessed using a ROC-inspired setting, to avoid biased estimations due to any particular learning algorithm.     

Robust model selection with flexible trimming

The forward search provides data-driven flexible trimming of a C_p statistic for the choice of regression models that reveals the effect of outliers on model selection. An informed robust model choice follows. Even in small samples, the statistic has a null distribution indistinguishable from an F distribution. Limits on acceptable values of the C_p statistic follow. Two examples of widely differing size are discussed. A powerful graphical tool is the generalized candlestick plot, which summarizes the information on all forward searches and on the choice of models. A comparison is made with the use of M-estimation in robust model choice.     

Approximate dynamic programming-based approaches for input-output data-driven control of nonlinear processes

We propose two approximate dynamic programming (ADP)-based strategies for control of nonlinear processes using input-output data. In the first strategy, which we term ‘J-learning,’ one builds an empirical nonlinear model using closed-loop test data and performs dynamic programming with it to derive an improved control policy. In the second strategy, called ‘Q-learning,’ one tries to learn an improved control policy in a model-less manner. Compared to the conventional model predictive control approach, the new approach offers some practical advantages in using nonlinear empirical models for process control. Besides the potential reduction in the on-line computational burden, it offers a convenient way to control the degree of model extrapolation in the calculation of optimal control moves. One major difficulty associated with using an empirical model within the multi-step predictive control setting is that the model can be excessively extrapolated into regions of the state space where identification data were scarce or nonexistent, leading to performances far worse than predicted by the model. Within the proposed ADP-based strategies, this problem is handled by imposing a penalty term designed on the basis of local data distribution. A CSTR example is provided to illustrate the proposed approaches.     

Hierarchical trajectory refinement for a class of nonlinear systems

Trajectory generation for nonlinear control systems is an important and difficult problem. In this paper, we provide a constructive method for hierarchical trajectory refinement. The approach is based on the recent notion of φ-related control systems. Given a control affine system satisfying certain assumptions, we construct a φ-related control system of smaller dimension. Trajectories designed for the smaller, abstracted system are guaranteed, by construction, to be feasible for the original system. Constructive procedures are provided for refining trajectories from the coarser to the more detailed system.     

Improved deterministic algorithms for weighted matching and packing problems

Based on the method of (n,k)-universal sets, we present a deterministic parameterized algorithm for the weighted rd-matching problem with time complexity O^∗(4^{(r−1)k+o(k)}), improving the previous best upper bound O^∗(4^rk+o(k)). In particular, the algorithm applied to the unweighted 3d-matching problem results in a deterministic algorithm with time O^∗(16^k+o(k)), improving the previous best result O^∗(21.26^k). For the weighted r-set packing problem, we present a deterministic parameterized algorithm with time complexity O^∗(2^{(2r−1)k+o(k)}), improving the previous best result O^∗(2^2rk+o(k)). The algorithm, when applied to the unweighted 3-set packing problem, has running time O^∗(32^k+o(k)), improving the previous best result O^∗(43.62^k+o(k)). Moreover, for the weighted r-set packing and weighted rd-matching problems, we give a kernel of size O(k^r), which is the first kernelization algorithm for the problems on weighted versions.     

Relationship between the eccentric connectivity index and Zagreb indices

For a (molecular) graph, the first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. If G is a connected graph with vertex set V(G), then the eccentric connectivity index of G, ξ^C(G), is defined as, ∑_vi∈V(G)d_ie_i, where d_i is the degree of a vertex v_i and e_i is its eccentricity. In this report we compare the eccentric connectivity index (ξ^C) and the Zagreb indices (M₁ and M₂) for chemical trees. Moreover, we compare the eccentric connectivity index (ξ^C) and the first Zagreb index (M₁) for molecular graphs.     

Experimental investigation on adaptive robust controller designs applied to a free-floating space manipulator

This paper aims to formulate and investigate the application of various nonlinear H_∞ control methods to a free-floating space manipulator subject to parametric uncertainties and external disturbances. From a tutorial perspective, a model-based approach and adaptive procedures based on linear parametrization, neural networks and fuzzy systems are covered by this work. A comparative study is conducted based on experimental implementations performed with an actual underactuated fixed-base planar manipulator which is, following the DEM concept, dynamically equivalent to a free-floating space manipulator.     

Partially varying coefficient single index proportional hazards regression models

In this paper, the partially varying coefficient single index proportional hazards regression models are discussed. All unknown functions are fitted by polynomial B splines. The index parameters and B-spline coefficients are estimated by the partial likelihood method and a two-step Newton-Raphson algorithm. Consistency and asymptotic normality of the estimators of all the parameters are derived. Through a simulation study and the VA data example, we illustrate that the proposed estimation procedure is accurate, rapid and stable.     

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.     

On generalised asymmetric stochastic volatility models

Stochastic volatility (SV) models have been considered as a real alternative to time-varying volatility of the ARCH family. Existing asymmetric SV (ASV) models treat volatility asymmetry via the leverage effect hypothesis. Generalised ASV models that take account of both volatility asymmetry and normality violation expressed simultaneously by skewness and excess kurtosis are introduced. The new generalised ASV models are estimated using the Bayesian Markov Chain Monte Carlo approach for parametric and log-volatility estimation. By using simulated and real financial data series, the new models are compared to existing SV models for their statistical properties, and for their estimation performance in within and out-of-sample periods. Results show that there is much to gain from the introduction of the generalised ASV models.     

Distinct squares in run-length encoded strings

Squares are strings of the form ww where w is any nonempty string. Two squares ww and w^′w^′ are of different types if and only if w≠w^′. Fraenkel and Simpson [Avieri S. Fraenkel, Jamie Simpson, How many squares can a string contain? Journal of Combinatorial Theory, Series A 82 (1998) 112-120] proved that the number of square types contained in a string of length n is bounded by O(n). The set of all different square types contained in a string is called the vocabulary of the string. If a square can be obtained by a series of successive right-rotations from another square, then we say the latter covers the former. A square is called a c-square if no square with a smaller index can cover it and it is not a trivial square. The set containing all c-squares is called the covering set. Note that every string has a unique covering set. Furthermore, the vocabulary of the covering set are called c-vocabulary. In this paper, we prove that the cardinality of c-vocabulary in a string is less than , where N is the number of runs in this string.     

Guaranteed robustness bounds for matched-disturbance nonlinear systems

This paper considers input affine nonlinear systems with matched disturbances and shows how to compute an a priori upper bound of the H_∞ attenuation level achieved by the optimal L₂ controller and the suboptimal H_∞ central controller. The case where the disturbance contains a constant term is also discussed. These bounds are shown to depend only on the function mapping the control input to the performance variable. This result is used to derive a robust control design for a special, but practically important, class of non-input affine nonlinear systems consisting of the series connection of a nonlinear state and input dependent map and of a nonlinear input affine dynamical system. Approximate inversion of the nonlinear static map leads to a robust control problem which fits into the framework. The effectiveness of the theoretical results is shown by its use for the robust control design of a diesel engine test bench.     

Fuzzy and possibilistic clustering for fuzzy data

The Fuzzy k-Means clustering model (FkM) is a powerful tool for classifying objects into a set of k homogeneous clusters by means of the membership degrees of an object in a cluster. In FkM, for each object, the sum of the membership degrees in the clusters must be equal to one. Such a constraint may cause meaningless results, especially when noise is present. To avoid this drawback, it is possible to relax the constraint, leading to the so-called Possibilistic k-Means clustering model (PkM). In particular, attention is paid to the case in which the empirical information is affected by imprecision or vagueness. This is handled by means of LR fuzzy numbers. An FkM model for LR fuzzy data is firstly developed and a PkM model for the same type of data is then proposed. The results of a simulation experiment and of two applications to real world fuzzy data confirm the validity of both models, while providing indications as to some advantages connected with the use of the possibilistic approach.     

A comparison of design and model selection methods for supersaturated experiments

Various design and model selection methods are available for supersaturated designs having more factors than runs but little research is available on their comparison and evaluation. Simulated experiments are used to evaluate the use of E(s²)-optimal and Bayesian D-optimal designs and to compare three analysis strategies representing regression, shrinkage and a novel model-averaging procedure. Suggestions are made for choosing the values of the tuning constants for each approach. Findings include that (i) the preferred analysis is via shrinkage; (ii) designs with similar numbers of runs and factors can be effective for a considerable number of active effects of only moderate size; and (iii) unbalanced designs can perform well. Some comments are made on the performance of the design and analysis methods when effect sparsity does not hold.     

Construction of marginally and conditionally restricted designs using multiplicative algorithms

The problem of constructing optimal designs when some of the factors are not under the control of the experimenters is considered. Their values can be known or unknown before the experiment is carried out. Several criteria are taken into consideration to find optimal conditional designs given some prior information on the factors. In order to determine these optimal conditional designs a class of multiplicative algorithms is provided. Optimal designs are computed for illustrative, but simplistic, examples. Two real life problems in production models and a physical test for predicting morbidity in lung cancer surgery motivate the procedures provided.