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.

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.

Is DARTEL-based voxel-based morphometry affected by width of smoothing kernel and group size? A study using simulated atrophy

Purpose: To quantify to what extent the new registration method, DARTEL (Diffeomorphic Anatomical Registration Through Exponentiated Lie Algebra), may reduce the smoothing kernel width required and investigate the minimum group size necessary for voxel-based morphometry (VBM) studies. Materials and Methods: A simulated atrophy approach was employed to explore the role of smoothing kernel, group size, and their interactions on VBM detection accuracy. Group sizes of 10, 15, 25, and 50 were compared for kernels between 0–12 mm. Results: A smoothing kernel of 6 mm achieved the highest atrophy detection accuracy for groups with 50 participants and 8–10 mm for the groups of 25 at P < 0.05 with familywise correction. The results further demonstrated that a group size of 25 was the lower limit when two different groups of participants were compared, whereas a group size of 15 was the minimum for longitudinal comparisons but at P < 0.05 with false discovery rate correction. Conclusion: Our data confirmed DARTEL-based VBM generally benefits from smaller kernels and different kernels perform best for different group sizes with a tendency of smaller kernels for larger groups. Importantly, the kernel selection was also affected by the threshold applied. This highlighted that the choice of kernel in relation to group size should be considered with care.

The Diffusion Kernel Filter

A particle filter method is presented for the discrete-time filtering problem with nonlinear ItA ` stochastic ordinary differential equations (SODE) with additive noise supposed to be analytically integrable as a function of the underlying vector-Wiener process and time. The Diffusion Kernel Filter is arrived at by a parametrization of small noise-driven state fluctuations within branches of prediction and a local use of this parametrization in the Bootstrap Filter. The method applies for small noise and short prediction steps. With explicit numerical integrators, the operations count in the Diffusion Kernel Filter is shown to be smaller than in the Bootstrap Filter whenever the initial state for the prediction step has sufficiently few moments. The established parametrization is a dual-formula for the analysis of sensitivity to gaussian-initial perturbations and the analysis of sensitivity to noise-perturbations, in deterministic models, showing in particular how the stability of a deterministic dynamics is modeled by noise on short times and how the diffusion matrix of an SODE should be modeled (i.e. defined) for a gaussian-initial deterministic problem to be cast into an SODE problem. From it, a novel definition of prediction may be proposed that coincides with the deterministic path within the branch of prediction whose information entropy at the end of the prediction step is closest to the average information entropy over all branches. Tests are made with the Lorenz-63 equations, showing good results both for the filter and the definition of prediction.

The univalent polynomial of Suffridge as a summability kernel

A positive summability trigonometric kernel {K(n)(theta)}(infinity)(n=1) is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K(n) * f} approximates every continuous 2 pi-periodic function f with the rate omega(f, 1/n), where omega(f, delta) denotes the modulus of continuity, and this provides a new proof of the classical Jackson`s theorem. Despite that it turns out that K(n)(theta) coincide with positive cosine polynomials generated by Fejer, our proof differs from others known in the literature.

Sensory adaptability of foods with added Lupin (Lupinus angustifolius) kernel fibre using pre-set criteria

Fiber-enriched white bread, muffin. pasta, orange juice, and breakfast bar were prepared with lupin (Lupin us angusti/olius) kernel fiber. Consumer panelists (n = 44) determined that all these fiber-enriched foods, except orange juice, fulfilled pre-set acceptability criteria. Fiber enrichment did not change overall acceptability (p> 0.05) of the bread and pasta, but reduced overall acceptability (p < 0.05) of the muffin, orange juice, and breakfast bar. In all fiber-enriched products, flavor was the attribute most highly correlated with overall acceptability (p < 0.05). The lupin kernel fiber used in this study therefore appears to have potential as a 'nonintrusive' ingredient in some processed cereal-based foods_ For other applications, fiber modification appears worthy of investigation to accomplish 'nonintrusive' fiber enrichment.

Lupin kernel fibre-enriched foods beneficially modify serum lipids in men

Objective: To examine the effect of a diet containing a novel legume food ingredient, Australian sweet lupin (Lupinus angustifolius) kernel fibre (LKFibre), compared to a control diet without the addition of LKFibre, on serum lipids in men.

Design: Randomized crossover dietary intervention study.

Setting: Melbourne, Australia — Free-living men.

Subjects: A total of 38 healthy males between the ages of 24 and 64 y completed the intervention.

Intervention: Subjects consumed an LKFibre and a control diet for 1 month each. Both diets had the same background menus with seven additional experimental foods that either contained LKFibre or did not. Depending on energy intake, the LKFibre diet was designed to contain an additional 17 to 30 g/day fibre beyond that of the control diet.

Results: Compared to the control diet, the LKFibre diet reduced total cholesterol (TC) (meanplusminuss.e.m.; 4.5plusminus1.7%; P=0.001), low-density lipoprotein cholesterol (LDL-C) (5.4plusminus2.2%; P=0.001), TC: high-density lipoprotein cholesterol (HDL-C) (3.0plusminus2.0%; P=0.006) and LDL-C:HDL-C (3.8plusminus2.6%; P=0.003). No effects on HDL-C, triacylglycerols, glucose or insulin were observed.

Conclusions: Addition of LKFibre to the diet provided favourable changes to some serum lipid measures in men, which, combined with its high palatability, suggest this novel ingredient may be useful in the dietary reduction of coronary heart disease risk.

Water-binding capacity and viscosity of Australian sweet lupin kernel fibre under in vitro conditions simulating the human upper gastrointestinal tract

There is currently little understanding of the physicochemical properties in the human gastrointestinal tract of Australian sweet lupin (Lupinus angustifolius) kernel fibre (LKF), a novel food ingredient with potential for the fibre enrichment of foods such as baked goods. Since physicochemical properties of dietary fibres have been related to beneficial physiological effects in vitro, this study compared water-binding capacity and viscosity of LKF with that of other fibres currently used for fibre-enrichment of baked goods, under in vitro conditions simulating the human upper gastrointestinal tract. At between 8.47 and 11.07g water/g dry solids, LKF exhibited water-binding capacities that were significantly higher (P<0.05) than soy fibre, pea hull fibre, cellulose and wheat fibre at all of the simulated gastrointestinal stages examined. Similarly, viscosity of LKF was significantly higher (P<0.05) than that of the other fibres at all simulated gastrointestinal stages. The relatively high water-binding capacity and viscosity of LKF identified in this study suggests that this novel fibre ingredient may elicit different and possibly more beneficial physiological effects in the upper human gastrointestinal tract than the conventional fibre ingredients currently used in fibre-enriched baked goods manufacture. We are now performing human studies to investigate the effect of LKF in the diet on health-related gastrointestinal events.