Sparse kernel regression modeling using combined locally regularized orthogonal least squares and D-optimality experimental design

The note proposes an efficient nonlinear identification algorithm by combining a locally regularized orthogonal least squares (LROLS) model selection with a D-optimality experimental design. The proposed algorithm aims to achieve maximized model robustness and sparsity via two effective and complementary approaches. The LROLS method alone is capable of producing a very parsimonious model with excellent generalization performance. The D-optimality design criterion further enhances the model efficiency and robustness. An added advantage is that the user only needs to specify a weighting for the D-optimality cost in the combined model selecting criterion and the entire model construction procedure becomes automatic. The value of this weighting does not influence the model selection procedure critically and it can be chosen with ease from a wide range of values.

Kernel density construction using orthogonal forward regression

An automatic algorithm is derived for constructing kernel density estimates based on a regression approach that directly optimizes generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. Local regularization is incorporated into the density construction process to further enforce sparsity. Examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample Parzen window density estimate.

Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization

This paper presents an efficient construction algorithm for obtaining sparse kernel density estimates based on a regression approach that directly optimizes model generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. A local regularization method is incorporated naturally into the density construction process to further enforce sparsity. An additional advantage of the proposed algorithm is that it is fully automatic and the user is not required to specify any criterion to terminate the density construction procedure. This is in contrast to an existing state-of-art kernel density estimation method using the support vector machine (SVM), where the user is required to specify some critical algorithm parameter. Several examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample optimized Parzen window density estimate. Our experimental results also demonstrate that the proposed algorithm compares favorably with the SVM method, in terms of both test accuracy and sparsity, for constructing kernel density estimates.

An orthogonal forward regression technique for sparse kernel density estimation

Using the classical Parzen window (PW) estimate as the desired response, the kernel density estimation is formulated as a regression problem and the orthogonal forward regression technique is adopted to construct sparse kernel density (SKD) estimates. The proposed algorithm incrementally minimises a leave-one-out test score to select a sparse kernel model, and a local regularisation method is incorporated into the density construction process to further enforce sparsity. The kernel weights of the selected sparse model are finally updated using the multiplicative nonnegative quadratic programming algorithm, which ensures the nonnegative and unity constraints for the kernel weights and has the desired ability to reduce the model size further. Except for the kernel width, the proposed method has no other parameters that need tuning, and the user is not required to specify any additional criterion to terminate the density construction procedure. Several examples demonstrate the ability of this simple regression-based approach to effectively construct a SKID estimate with comparable accuracy to that of the full-sample optimised PW density estimate. (c) 2007 Elsevier B.V. All rights reserved.

Identification of nonlinear systems using generalized kernel models

Nonlinear system identification is considered using a generalized kernel regression model. Unlike the standard kernel model, which employs a fixed common variance for all the kernel regressors, each kernel regressor in the generalized kernel model has an individually tuned diagonal covariance matrix that is determined by maximizing the correlation between the training data and the regressor using a repeated guided random search based on boosting optimization. An efficient construction algorithm based on orthogonal forward regression with leave-one-out (LOO) test statistic and local regularization (LR) is then used to select a parsimonious generalized kernel regression model from the resulting full regression matrix. The proposed modeling algorithm is fully automatic and the user is not required to specify any criterion to terminate the construction procedure. Experimental results involving two real data sets demonstrate the effectiveness of the proposed nonlinear system identification approach.

Kernel classifier construction using orthogonal forward selection and boosting with Fisher ratio class separability measure

A greedy technique is proposed to construct parsimonious kernel classifiers using the orthogonal forward selection method and boosting based on Fisher ratio for class separability measure. Unlike most kernel classification methods, which restrict kernel means to the training input data and use a fixed common variance for all the kernel terms, the proposed technique can tune both the mean vector and diagonal covariance matrix of individual kernel by incrementally maximizing Fisher ratio for class separability measure. An efficient weighted optimization method is developed based on boosting to append kernels one by one in an orthogonal forward selection procedure. Experimental results obtained using this construction technique demonstrate that it offers a viable alternative to the existing state-of-the-art kernel modeling methods for constructing sparse Gaussian radial basis function network classifiers. that generalize well.

Fast kernel classifier construction using orthogonal forward selection to minimise leave-one-out misclassification rate

We propose a simple yet computationally efficient construction algorithm for two-class kernel classifiers. In order to optimise classifier's generalisation capability, an orthogonal forward selection procedure is used to select kernels one by one by minimising the leave-one-out (LOO) misclassification rate directly. It is shown that the computation of the LOO misclassification rate is very efficient owing to orthogonalisation. Examples are used to demonstrate that the proposed algorithm is a viable alternative to construct sparse two-class kernel classifiers in terms of performance and computational efficiency.

A kernel-based two-class classifier for imbalanced data sets

Many kernel classifier construction algorithms adopt classification accuracy as performance metrics in model evaluation. Moreover, equal weighting is often applied to each data sample in parameter estimation. These modeling practices often become problematic if the data sets are imbalanced. We present a kernel classifier construction algorithm using orthogonal forward selection (OFS) in order to optimize the model generalization for imbalanced two-class data sets. This kernel classifier identification algorithm is based on a new regularized orthogonal weighted least squares (ROWLS) estimator and the model selection criterion of maximal leave-one-out area under curve (LOO-AUC) of the receiver operating characteristics (ROCs). It is shown that, owing to the orthogonalization procedure, the LOO-AUC can be calculated via an analytic formula based on the new regularized orthogonal weighted least squares parameter estimator, without actually splitting the estimation data set. The proposed algorithm can achieve minimal computational expense via a set of forward recursive updating formula in searching model terms with maximal incremental LOO-AUC value. Numerical examples are used to demonstrate the efficacy of the algorithm.

A forward-constrained regression algorithm for sparse kernel density estimation

Using the classical Parzen window (PW) estimate as the target function, the sparse kernel density estimator is constructed in a forward-constrained regression (FCR) manner. The proposed algorithm selects significant kernels one at a time, while the leave-one-out (LOO) test score is minimized subject to a simple positivity constraint in each forward stage. The model parameter estimation in each forward stage is simply the solution of jackknife parameter estimator for a single parameter, subject to the same positivity constraint check. For each selected kernels, the associated kernel width is updated via the Gauss-Newton method with the model parameter estimate fixed. The proposed approach is simple to implement and the associated computational cost is very low. Numerical examples are employed to demonstrate the efficacy of the proposed approach.

Sparse kernel density estimation technique based on zero-norm constraint

A sparse kernel density estimator is derived based on the zero-norm constraint, in which the zero-norm of the kernel weights is incorporated to enhance model sparsity. The classical Parzen window estimate is adopted as the desired response for density estimation, and an approximate function of the zero-norm is used for achieving mathemtical tractability and algorithmic efficiency. Under the mild condition of the positive definite design matrix, the kernel weights of the proposed density estimator based on the zero-norm approximation can be obtained using the multiplicative nonnegative quadratic programming algorithm. Using the -optimality based selection algorithm as the preprocessing to select a small significant subset design matrix, the proposed zero-norm based approach offers an effective means for constructing very sparse kernel density estimates with excellent generalisation performance.

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.

In vitro assessment of the bioaccessibility of brominated flame retardants in indoor dust using a colon extended model of the human gastrointestinal tract

An in vitro colon extended physiologically based extraction test (CEPBET) which incorporates human gastrointestinal tract (GIT) parameters (including pH and chemistry, solid-to-fluid ratio, mixing and emptying rates) was applied for the first time to study the bioaccessibility of brominated flame retardants (BFRs) from the 3 main GIT compartments (stomach, small intestine and colon) following ingestion of indoor dust. Results revealed the bioaccessibility of γ-HBCD (72%) was less than that for α- and β-isomers (92% and 80% respectively) which may be attributed to the lower aqueous solubility of the γ-isomer (2 μg L−1) compared to the α- and β-isomers (45 and 15 μg L−1 respectively). No significant change in the enantiomeric fractions of HBCDs was observed in any of the studied samples. However, this does not completely exclude the possibility of in vivo enantioselective absorption of HBCDs, as the GIT cell lining and bacterial flora – which may act enantioselectively – are not included in the current CE-PBET model. While TBBP-A was almost completely (94%) bioaccessible, BDE-209 was the least (14%) bioaccessible of the studied BFRs. Bioaccessibility of tri-hepta BDEs ranged from 32–58%. No decrease in the bioaccessibility with increasing level of bromination was observed in the studied PBDEs.

Heat kernel smoothing via Laplace-Beltrami eigenfunctions and its application to subcortical structure modeling

We present a new subcortical structure shape modeling framework using heat kernel smoothing constructed with the Laplace-Beltrami eigenfunctions. The cotan discretization is used to numerically obtain the eigenfunctions of the Laplace-Beltrami operator along the surface of subcortical structures of the brain. The eigenfunctions are then used to construct the heat kernel and used in smoothing out measurements noise along the surface. The proposed framework is applied in investigating the influence of age (38-79 years) and gender on amygdala and hippocampus shape. We detected a significant age effect on hippocampus in accordance with the previous studies. In addition, we also detected a significant gender effect on amygdala. Since we did not find any such differences in the traditional volumetric methods, our results demonstrate the benefit of the current framework over traditional volumetric methods.