Backward elimination model construction for regression and classification using leave-one-out criteria

A fundamental principle in practical nonlinear data modeling is the parsimonious principle of constructing the minimal model that explains the training data well. Leave-one-out (LOO) cross validation is often used to estimate generalization errors by choosing amongst different network architectures (M. Stone, "Cross validatory choice and assessment of statistical predictions", J. R. Stast. Soc., Ser. B, 36, pp. 117-147, 1974). Based upon the minimization of LOO criteria of either the mean squares of LOO errors or the LOO misclassification rate respectively, we present two backward elimination algorithms as model post-processing procedures for regression and classification problems. The proposed backward elimination procedures exploit an orthogonalization procedure to enable the orthogonality between the subspace as spanned by the pruned model and the deleted regressor. Subsequently, it is shown that the LOO criteria used in both algorithms can be calculated via some analytic recursive formula, as derived in this contribution, without actually splitting the estimation data set so as to reduce computational expense. Compared to most other model construction methods, the proposed algorithms are advantageous in several aspects; (i) There are no tuning parameters to be optimized through an extra validation data set; (ii) The procedure is fully automatic without an additional stopping criteria; and (iii) The model structure selection is directly based on model generalization performance. The illustrative examples on regression and classification are used to demonstrate that the proposed algorithms are viable post-processing methods to prune a model to gain extra sparsity and improved generalization.

A robust nonlinear identification algorithm using PRESS statistic and forward regression

This letter introduces a new robust nonlinear identification algorithm using the Predicted REsidual Sums of Squares (PRESS) statistic and for-ward regression. The major contribution is to compute the PRESS statistic within a framework of a forward orthogonalization process and hence construct a model with a good generalization property. Based on the properties of the PRESS statistic the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation.

Automatic nonlinear predictive model-construction algorithm using forward regression and the PRESS statistic

An automatic nonlinear predictive model-construction algorithm is introduced based on forward regression and the predicted-residual-sums-of-squares (PRESS) statistic. The proposed algorithm is based on the fundamental concept of evaluating a model's generalisation capability through crossvalidation. This is achieved by using the PRESS statistic as a cost function to optimise model structure. In particular, the proposed algorithm is developed with the aim of achieving computational efficiency, such that the computational effort, which would usually be extensive in the computation of the PRESS statistic, is reduced or minimised. The computation of PRESS is simplified by avoiding a matrix inversion through the use of the orthogonalisation procedure inherent in forward regression, and is further reduced significantly by the introduction of a forward-recursive formula. Based on the properties of the PRESS statistic, the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation. Numerical examples are used to demonstrate the efficacy of the algorithm.

Particle swarm optimization aided orthogonal forward regression for unified data modeling

We propose a unified data modeling approach that is equally applicable to supervised regression and classification applications, as well as to unsupervised probability density function estimation. A particle swarm optimization (PSO) aided orthogonal forward regression (OFR) algorithm based on leave-one-out (LOO) criteria is developed to construct parsimonious radial basis function (RBF) networks with tunable nodes. Each stage of the construction process determines the center vector and diagonal covariance matrix of one RBF node by minimizing the LOO statistics. For regression applications, the LOO criterion is chosen to be the LOO mean square error, while the LOO misclassification rate is adopted in two-class classification applications. By adopting the Parzen window estimate as the desired response, the unsupervised density estimation problem is transformed into a constrained regression problem. This PSO aided OFR algorithm for tunable-node RBF networks is capable of constructing very parsimonious RBF models that generalize well, and our analysis and experimental results demonstrate that the algorithm is computationally even simpler than the efficient regularization assisted orthogonal least square algorithm based on LOO criteria for selecting fixed-node RBF models. Another significant advantage of the proposed learning procedure is that it does not have learning hyperparameters that have to be tuned using costly cross validation. The effectiveness of the proposed PSO aided OFR construction procedure is illustrated using several examples taken from regression and classification, as well as density estimation applications.

Probability density estimation with tunable kernels using orthogonal forward regression

A generalized or tunable-kernel model is proposed for probability density function estimation based on an orthogonal forward regression procedure. Each stage of the density estimation process determines a tunable kernel, namely, its center vector and diagonal covariance matrix, by minimizing a leave-one-out test criterion. The kernel mixing weights of the constructed sparse density estimate are finally updated using the multiplicative nonnegative quadratic programming algorithm to ensure the nonnegative and unity constraints, and this weight-updating process additionally has the desired ability to further reduce the model size. The proposed tunable-kernel model has advantages, in terms of model generalization capability and model sparsity, over the standard fixed-kernel model that restricts kernel centers to the training data points and employs a single common kernel variance for every kernel. On the other hand, it does not optimize all the model parameters together and thus avoids the problems of high-dimensional ill-conditioned nonlinear optimization associated with the conventional finite mixture model. Several examples are included to demonstrate the ability of the proposed novel tunable-kernel model to effectively construct a very compact density estimate accurately.

Regression based D-optimality experimental design for sparse kernel density estimation

This paper derives an efficient algorithm for constructing sparse kernel density (SKD) estimates. The algorithm first selects a very small subset of significant kernels using an orthogonal forward regression (OFR) procedure based on the D-optimality experimental design criterion. The weights of the resulting sparse kernel model are then calculated using a modified multiplicative nonnegative quadratic programming algorithm. Unlike most of the SKD estimators, the proposed D-optimality regression approach is an unsupervised construction algorithm and it does not require an empirical desired response for the kernel selection task. The strength of the D-optimality OFR is owing to the fact that the algorithm automatically selects a small subset of the most significant kernels related to the largest eigenvalues of the kernel design matrix, which counts for the most energy of the kernel training data, and this also guarantees the most accurate kernel weight estimate. The proposed method is also computationally attractive, in comparison with many existing SKD construction algorithms. Extensive numerical investigation demonstrates the ability of this regression-based approach to efficiently construct a very sparse kernel density estimate with excellent test accuracy, and our results show that the proposed method compares favourably with other existing sparse methods, in terms of test accuracy, model sparsity and complexity, for constructing kernel density estimates.

Radial basis function classifier construction using particle swarm optimisation aided orthogonal forward regression

We develop a particle swarm optimisation (PSO) aided orthogonal forward regression (OFR) approach for constructing radial basis function (RBF) classifiers with tunable nodes. At each stage of the OFR construction process, the centre vector and diagonal covariance matrix of one RBF node is determined efficiently by minimising the leave-one-out (LOO) misclassification rate (MR) using a PSO algorithm. Compared with the state-of-the-art regularisation assisted orthogonal least square algorithm based on the LOO MR for selecting fixednode RBF classifiers, the proposed PSO aided OFR algorithm for constructing tunable-node RBF classifiers offers significant advantages in terms of better generalisation performance and smaller model size as well as imposes lower computational complexity in classifier construction process. Moreover, the proposed algorithm does not have any hyperparameter that requires costly tuning based on cross validation.

Power properties if invariant tests for spatial autocorrelation in linear regression

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included

Neural networks for identification of nonlinear systems under random piecewise polynomial disturbances

The problem of identification of a nonlinear dynamic system is considered. A two-layer neural network is used for the solution of the problem. Systems disturbed with unmeasurable noise are considered, although it is known that the disturbance is a random piecewise polynomial process. Absorption polynomials and nonquadratic loss functions are used to reduce the effect of this disturbance on the estimates of the optimal memory of the neural-network model.

Parameter estimation based on stacked regression and evolutionary algorithms

A new parameter-estimation algorithm, which minimises the cross-validated prediction error for linear-in-the-parameter models, is proposed, based on stacked regression and an evolutionary algorithm. It is initially shown that cross-validation is very important for prediction in linear-in-the-parameter models using a criterion called the mean dispersion error (MDE). Stacked regression, which can be regarded as a sophisticated type of cross-validation, is then introduced based on an evolutionary algorithm, to produce a new parameter-estimation algorithm, which preserves the parsimony of a concise model structure that is determined using the forward orthogonal least-squares (OLS) algorithm. The PRESS prediction errors are used for cross-validation, and the sunspot and Canadian lynx time series are used to demonstrate the new algorithms.

The effect of test meal monounsaturated fatty acid: saturated fatty acid ratio on postprandial lipid metabolism

Epidemiological evidence shows that a diet high in monounsaturated fatty acids (MUFA) but low in saturated fatty acids (SFA) is associated with reduced risk of CHD. The hypocholesterolaemic effect of MUFA is known but there has been little research on the effect of test meal MUFA and SFA composition on postprandial lipid metabolism. The present study investigated the effect of meals containing different proportions of MUFA and SFA on postprandial triacylglycerol and non-esterified fatty acid (NEFA) metabolism. Thirty healthy male volunteers consumed three meals containing equal amounts of fat (40 g), but different proportions of MUFA (12, 17 and 24% energy) in random order. Postprandial plasma triacylglycerol, apolipoprotein B-48, cholesterol, HDL-cholesterol, glucose and insulin concentrations and lipoprotein lipase (EC 3.1.1.34) activity were not significantly different following the three meals which varied in their levels of SFA and MUFA. There was a significant difference in the postprandial NEFA response between meals. The incremental area under the curve of postprandial plasma NEFA concentrations was significantly (P = 0.03) lower following the high-MUFA meal. Regression analysis showed that the non-significant difference in fasting NEFA concentrations was the most important factor determining difference between meals, and that the test meal MUFA content had only a minor effect. In conclusion, varying the levels of MUFA and SFA in test meals has little or no effect on postprandial lipid metabolism.

Induced random fields in the LiHoxY1-xF4 quantum ising magnet in a transverse magnetic field

The LiHoxY1-xF4 magnetic material in a transverse magnetic field Bxx̂ perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x̂ direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising ẑ direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.