An iterative algorithm for optimal variable weighting in K-means clustering

The K-means clustering method is a widely adopted clustering algorithm in data mining and pattern recognition, where the partitions are made by minimizing the total within group sum of squares based on a given set of variables. Weighted K-means clustering is an extension of the K-means method by assigning nonnegative weights to the set of variables. In this paper, we aim to obtain more meaningful and interpretable clusters by deriving the optimal variable weights for weighted K-means clustering. Specifically, we improve the weighted k-means clustering method by introducing a new algorithm to obtain the globally optimal variable weights based on the Karush-Kuhn-Tucker conditions. We present the mathematical formulation for the clustering problem, derive the structural properties of the optimal weights, and implement an recursive algorithm to calculate the optimal weights. Numerical examples on simulated and real data indicate that our method is superior in both clustering accuracy and computational efficiency.     

Sums of powers of binomial coefficients

An asymptotic series for sums of powers of binomial coefficients is derived, the general term being defined and usable with a computer symbolic language. Sums of squares of coefficients in the symmetric case are shown to have a link with classical moment problems, but this property breaks down for cubes and higher powers. Problems of remainders for the asymptotic series are mentioned. Using the reflection formula for I'(.), a continuous form for a binomial function is set up, and this becomes oscillatory outstde the usual range. A new contmued fraction emerges for the logarithm of an adjusted sum of binomial squares. The note is a contribution to the problem of the interpretation of asymptotic series and processes for their convergence acceleration.     

The asymptotic behaviour of the residual sum of squares in models with multiple break points

ABSTRACT

Models with multiple discrete breaks in parameters are usually estimated via least squares. This paper, first, derives the asymptotic expectation of the residual sum of squares and shows that the number of estimated break points and the number of regression parameters affect the expectation differently. Second, we propose a statistic for testing the joint hypothesis that the breaks occur at specified points in the sample. Our analytical results cover models estimated by the ordinary, nonlinear, and two-stage least squares. An application to U.S. monetary policy rejects the assumption that breaks are associated with changes in the chair of the Fed.     

Minimax estimators of regression functions under normalized quadratic loss functions and inequality restrictions

This paper deals with the estimation of a regression function f of unknown form by shrinked least squares estimates calculated on the basis of possibly replicated observa-tions, The estimation loss is chosen in a somewhat more realistic manner then the usual quadratic losses and is given by an adequately weighted sum of squared errors in estimat-ing the values of f at the design points, normalized by the squared norm of the regression function, Shrinked least squares (as special ridge estimators) have been proved by the suthors in special cases to be minimax under all estimatiors.

We investigate the shrinking of least squares estimators with the objective of minimiz-ing the least favourable risk. Here we assume a known lower bound for the magnitude of f and a known upper bound for the difference between f and some simple function approxi-mating f, e.g. we know that f is the sum of a quadratic polynomial and of some     

Asymptotic optimality of estimators of a linear functional eeiation if the ratio of the error variances is known 2

For the problem of estimating a linear functional relation when the ratio of the error variances is known a general class of estimators is introduced. They include as special cases the instrumental variable and replication cases and some others. Conditions are given for consistency, asymptotic normality and asymptotic optimality within this class based on the variance of the limit distribution. Fisheb's lower bound for asymptotic variances is established, and under normality the asymptotically optimal estimators are shown to be best asymptotically normal. For an inhomogeneous linear relation only estimators which are invariant with respect to a translation of the origin are considered, and asymptotically optimal invariant and, under normality, best asymptotically normal invariant estimators are obtained. Several special cases are discussed.     

Generalized Estimating Equations for Symmetric Distributions Observed with Admixture

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. Generalized estimating equations are constructed for the estimation of the mixture proportion and the location parameter of the primary component. Asymptotic normality of the estimates is demonstrated and the lower bound for the asymptotic covariance matrix is obtained. An adaptive estimation technique is proposed to obtain the estimates with nearly optimal asymptotic variances.     

Kruskal–Wallis, Multiple Comparisons and Efron Dice

The Kruskal–Wallis test is a rank–based one way ANOVA. Its test statistic is shown here to be a quadratic form among the Mann–Whitney or Kendall tau concordance measures between pairs of treatments. But the full set of such concordance measures has more degrees of freedom than the Kruskal–Wallis test uses, and the independent surplus is attributable to circularity, or non–transitive effects. The meaning of circularity is well illustrated by Efron dice. The cases of k = 3, 4 treatments are analysed thoroughly in this paper, which also shows how the full sum of squares among all concordance measures can be decomposed into uncorrelated transitive and non–transitive circularity effects. A multiple comparisons procedure based on patterns of transitive orderings among treatments is implemented. The testing of circularities involves non–standard asymptotic distributions. The asymptotic theory is deferred, but Monte Carlo permutation tests are easy to implement.     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

Selecting Regressors for Prediction Using PRESS and White t Statistics

It is shown that selecting regressors based on the prediction error sum of squares is asymptotically related to hypothesis testing with White's (1980) heteroscedasticity-consistent covariance matrix. A simulation experiment suggests that this asymptotic relation may be useful. Illustrative examples are also given.     

Some comments on two-stage selection procedures

In this paper we study the procedures of Dudewicz and Dalal ( 1975 ), and the modifications suggested by Rinott ( 1978 ), for selecting the largest mean from k normal populations with unknown variances. We look at the case k = 2 in detail, because there is an optimal allocation scheme here. We do not really allocate the total number of samples into two groups, but we estimate this optimal sample size, as well, so as to guarantee the probability of correct selection (written as P(CS)) at least P^?, 1/2 < P^? < 1 . We prove that the procedure of Rinott is “asymptotically in-efficient” (to be defined below) in the sense of Chow and Robbins ( 1965 ) for any k  2. Next, we propose two-stage procedures having all the properties of Rinott's procedure, together with the property of “asymptotic efficiency” - which is highly desirable.     

Weighted Support Vector Machine Using k-Means Clustering

The support vector machine (SVM) has been successfully applied to various classification areas with great flexibility and a high level of classification accuracy. However, the SVM is not suitable for the classification of large or imbalanced datasets because of significant computational problems and a classification bias toward the dominant class. The SVM combined with the k-means clustering (KM-SVM) is a fast algorithm developed to accelerate both the training and the prediction of SVM classifiers by using the cluster centers obtained from the k-means clustering. In the KM-SVM algorithm, however, the penalty of misclassification is treated equally for each cluster center even though the contributions of different cluster centers to the classification can be different. In order to improve classification accuracy, we propose the WKM–SVM algorithm which imposes different penalties for the misclassification of cluster centers by using the number of data points within each cluster as a weight. As an extension of the WKM–SVM, the recovery process based on WKM–SVM is suggested to incorporate the information near the optimal boundary. Furthermore, the proposed WKM–SVM can be successfully applied to imbalanced datasets with an appropriate weighting strategy. Experiments show the effectiveness of our proposed methods.     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

Generalized estimating equations for mixtures with varying concentrations

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. A parametric model is assumed for one mixture component distribution, while the others are nonparametric nuisance parameters. Generalized estimating equations (GEE) are proposed for the semi‐parametric estimation. Asymptotic normality of the GEE estimates is demonstrated and the lower bound for their dispersion (asymptotic covariance) matrix is derived. An adaptive technique is developed to derive estimates with nearly optimal small dispersion. An application to the sociological analysis of voting results is discussed. The Canadian Journal of Statistics 41: 217–236; 2013 © 2013 Statistical Society of Canada     

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X² statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h² distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G² (the power-divergence statistic of index u = 0).     

Central composite designs for estimating the optimum conditions for a second-order model

Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

An extended sweep operator for the cross validation of variable selection in linear regression

In its application to variable selection in the linear model, cross-validation is traditionally applied to an individual model contained in a set of potential models. Each model in the set is cross-validated independently of the rest and the model with the smallest cross-validated sum of squares is selected. In such settings, an efficient algorithm for cross-validation must be able to add and to delete single points quickly from a mixed model. Recent work in variable selection has applied cross-validation to an entire process of variable selection, such as Backward Elimination or Stepwise regression (Thall, Simon and Grier, 1992). The cross-validated version of Backward Elimination, for example, divides the data into an estimation and validation set and performs a complete Backward Elimination on the estimation set, while computing the cross-validated sum of squares at each step with the validation set. After doing this process once, a different validation set is selected and the process is repeated. The final model selection is based on the cross-validated sum of squares for all Backward Eliminations. An optimal algorithm for this application of cross-validation need not be efficient in adding and deleting observations from a single model but must be efficient in computing the cross-validation sum of squares from a series of models using a common validation set. This paper explores such an algorithm based on the sweep operator.     

On the Optimality of Prediction-based Selection Criteria and the Convergence Rates of Estimators

Several estimators of squared prediction error have been suggested for use in model and bandwidth selection problems. Among these are cross-validation, generalized cross-validation and a number of related techniques based on the residual sum of squares. For many situations with squared error loss, e.g. nonparametric smoothing, these estimators have been shown to be asymptotically optimal in the sense that in large samples the estimator minimizing the selection criterion also minimizes squared error loss. However, cross-validation is known not to be asymptotically optimal for some `easy' location problems. We consider selection criteria based on estimators of squared prediction risk for choosing between location estimators. We show that criteria based on adjusted residual sum of squares are not asymptotically optimal for choosing between asymptotically normal location estimators that converge at rate n ^1/2but are when the rate of convergence is slower. We also show that leave-one-out cross-validation is not asymptotically optimal for choosing between √ n -differentiable statistics but leave- d -out cross-validation is optimal when d ∞ at the appropriate rate.     

The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.