Least Squares Support Vector Machines

Abstract

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.

Nonlinear system identification via direct weight optimization

“…Very many approaches have been suggested, attempted and tested. See among many references, e.g., (Chen and Billings, 1992;Harris et al, 2002;Roll et al, 2002;Sjöberg et al, 1995;Suykens et al, 2002;Vidyasagar, 1997).…”

Support Vector Machine Identification of Subspace Hammerstein Models

“…However, to apply centering constraints (14)  new parameter y  should be added to (12) to get [13] …”

“…However, the necessity to state the neural network topology in terms of the number of nodes and layers, and the necessity to compute non-convex optimization make its implementation difficult. Lately, support vector machines (SVMs) and least squares support vector machines (LS-SVMs) have revealed excellent abilities in estimating linear and nonlinear functions ( [11], [12]). Goethals et al [13] considered the extension of the N4SID family of subspace model identification schemes to the Hammerstein-type of nonlinear system.…”

Forecasting Model Based on LSSVM and ABC for Natural Resource Commodity

“…Least Squares Support Vector Machines (LSSVM) reformulates the original SVM algorithm. It has been proposed by Suykens and Vandewalle [15] for the purpose to solve short term load prediction problems. LSSVM is reported to consume less computational effort in the huge-scale problem compared to standard SVM's.…”

“…They perform structural risk minimization, which was introduced to machine learning by Vapnik [14] the original SVM algorithm and is reported to consume less computational effort in a huge-scale problem compared to standard SVM's [15]. For LSSVM, the regularization and kernel(s) parameters are known as hyper-parameters.…”