Kok Seng Chua

A comparison of PCA, KPCA and ICA for dimensionality reduction in support vector machine

Abstract Recently, support vector machine (SVM) has become a popular tool in time series forecasting. In developing a successful SVM forecastor, the first step is feature extraction. This paper proposes the applications of principal component analysis (PCA), kernel principal component analysis (KPCA) and independent component analysis (ICA) to SVM for feature extraction. PCA linearly transforms the original inputs into new uncorrelated features. KPCA is a nonlinear PCA developed by using the kernel method. In ICA, the original inputs are linearly transformed into features which are mutually statistically independent. By examining the sunspot data, Santa Fe data set A and five real futures contracts, the experiment shows that SVM by feature extraction using PCA, KPCA or ICA can perform better than that without feature extraction. Furthermore, among the three methods, there is the best performance in KPCA feature extraction, followed by ICA feature extraction.

Representations of integers as sums of 32 squares

In this paper, we derive a new explicit formula for r32(n), where rk(n) is the number of representations of n as a sum of k squares. For a fixed integer k, our method can be used to derive explicit formulas for r8k(n). We conclude the paper with various conjectures that lead to explicit formulas for r2k(n), for any fixed positive integer k > 4.

Circular Summation of the 13th Powers of Ramanujan's Theta Function

An explicit formula is derived for the circular summation of the 13th power of Ramanujans theta function in terms of Dedekind eta function.

Seven-modular lattices and a septic base Jacobi identity

A quadratic Jacobi identity to the septic base is introduced and proved by means of modular lattices and codes over rings. As an application the theta series of all the 6-dimensional 7modular lattices with an Hermitian structure over Qð ffiffiffiffiffiffiffi 7 p Þ are derived.

Quadratic iterations to associated with elliptic functions to the cubic and septic base

In this paper, properties of the functions A d (q), B d (q) and C d (q) are derived. Specializing at d = 1 and 2, we construct two new quadratic iterations to π. These are analogues of previous iterations discovered by the Borweins (1987), J. M. Borwein and F. G. Garvan (1997), and H. H. Chan (2002). Two new transformations of the hypergeometric series 2 F 1 (1/3, 1/6; 1; z) are also derived.

Congruence Subgroups Associated to the Monster

Let Δ = {G : g(G) = 0,Γ0(m) ≤ G ≤ N(Γ0(m)) for some m}, where N(Γ0(m)) is the normaliser of Γ0(m) in PSL 2(R) and g(G) is the genus of H*/G. In this article, we determine all the m. Further, for each m, we list all the intermediate groups G of Γ0(m) ≤ N(Γ0(m)) such that g(G) = 0. All the intermediate groups of width 1 at ∞ are also listed in a separatetable (see www.rnath.nus.edu.sg/e-matlml/).

Explicit congruences for the partition function modulo every prime

Abstract.Let p(n) denote the number of unrestricted partitions of the positive integer n, and let m be a prime

Eisenstein lattices, Galois rings, and theta series

\geq 13

Jacobi identities, modular lattices, and modular towers

. We prove, for k = 1, explicit congruences of the form