N k=1 α k y k = 0 0 ≤ α k ≤ c, k = 1, ..., N. Note: w and ϕ(x k) are not calculated. • Mercer condition: K(x k , x l) = ϕ(x k) T ϕ(x l) • Obtained classifier: y(x) = sign[ N k=1 α k y k K(x, x k) + b] with α k positive real constants, b real constant, that follow as solution to the QP problem. Non-zero α k are called support values and the corresponding data points are called support vectors. The bias term b follows from KKT conditions. • Some possible kernels K(•, •): K(x, x k) = x T k x (linear SVM) K(x, x k) = (x T k x + 1) d (polynomial SVM of degree d) K(x, x k) = exp{− x − x k 2 2 /σ 2 } (RBF SVM) K(x, x k) = tanh(κ x T k x + θ) (MLP SVM) • In the case of RBF and MLP kernel, the number of hidden units corresponds to the number of support vectors.

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Least squares support vector machines (LS-SVM) is an SVM version which involves equality instead of inequality constraints and works with a least squares cost function. In this way, the solution follows from a linear Karush-Kuhn-Tucker system instead of a quadratic programming problem. However, sparseness is lost in the LS-SVM case and the estimation of the support values is only optimal in the case of a Gaussian distribution of the error variables. In this paper, we discuss a method which can overcome these two drawbacks. We show how to obtain robust estimates for regression by applying a weighted version of LS-SVM. We also discuss a sparse approximation procedure for weighted and unweighted LS-SVM. It is basically a pruning method which is able to do pruning based upon the physical meaning of the sorted support values, while pruning procedures for classical multilayer perceptrons require the computation of a Hessian matrix or its inverse. The methods of this paper are illustrated for RBF kernels and demonstrate how to obtain robust estimates with selection of an appropriate number of hidden units, in the case of outliers or non-Gaussian error distributions with heavy tails.

In the past decades, food scientists have been searching for natural alternatives to replace synthetic antioxidants. In order to evaluate the potential of microalgae as new source of safe antioxidants, 32 microalgal biomass samples were screened for their antioxidant capacity using three antioxidant assays, and both total phenolic content and carotenoid content were measured. Microalgae were extracted using a one-step extraction with ethanol/water, and alternatively, a three-step fractionation procedure using successively hexane, ethyl acetate, and water. Antioxidant activity of the extracts varied strongly between species and further depended on growth conditions and the solvent used for extraction. It was found that industrially cultivated samples of Tetraselmis suecica, Botryococcus braunii, Neochloris oleoabundans, Isochrysis sp., Chlorella vulgaris, and Phaeodactylum tricornutum possessed the highest antioxidant capacities in this study and thus could be a potential new source of natural antioxidants. The results from the different types of extracts clearly indicated that next to the well-studied carotenoids, phenolic compounds also contribute significantly to the antioxidant capacity of microalgae.

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