Partial least squares discriminant analysis (PLS-DA) has been available for nearly 20 years yet is poorly understood by most users. By simple examples, it is shown graphically and algebraically that for two equal class sizes, PLS-DA using one partial least squares (PLS) component provides equivalent classification results to Euclidean distance to centroids, and by using all nonzero components to linear discriminant analysis. Extensions where there are unequal class sizes and more than two classes are discussed including common pitfalls and dilemmas. Finally, the problems of overfitting and PLS scores plots are discussed. It is concluded that for classification purposes, PLS-DA has no significant advantages over traditional procedures and is an algorithm full of dangers. It should not be viewed as a single integrated method but as step in a full classification procedure. However, despite these limitations, PLS-DA can provide good insight into the causes of discrimination via weights and loadings, which gives it a unique role in exploratory data analysis, for example in metabolomics via visualisation of significant variables such as metabolites or spectroscopic peaks. Copyright © 2014 John Wiley & Sons, Ltd.

Partial least squares discriminant analysis: taking the magic away

In this paper, we intend to implement a class of fractional differential masks with high-precision. Thanks to two commonly used definitions of fractional differential for what are known as Grumwald-Letnikov and Riemann-Liouville, we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x-coordinate, positive x-coordinate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal. Moreover, by theoretical and experimental analyzing, we demonstrate the second is the best performance fractional differential mask of the proposed six ones. Finally, we discuss further the capability of multiscale fractional differential masks for texture enhancement. Experiments show that, for rich-grained digital image, the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential-based approach appears obvious better than by traditional integral-based algorithms.

Fractional Differential Mask: A Fractional Differential-Based Approach for Multiscale Texture Enhancement

Thc paper deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. A survey of the methods and results in the theory of such ordinary fractional differential equations is given. In particular, the method based on the reduction of the Cauchy-type problem for the fractional differential equations to the Volterra integral equations is discussed, and the Laplace transform, operational calculas compositional methods for the solution of linear differential equations of fractional order are presented. Problems and new trends of research are discussed.

Differential equations of fractional order:methods results and problem —I

The article gives the historical background and a brief introduction to fractional calculus. An overview of fractional calculus was also given as well as its potential applications. Examples of complex system modeled by means of fractional calculus was given.

Fractional Calculus: A Mathematical Tool from the Past for Present Engineers

Fractional calculus: A mathematical tool from the past for present engineers [Past and present]

(1981). On the Inverse of the Sum of Matrices. Mathematics Magazine: Vol. 54, No. 2, pp. 67-72.

On the Inverse of the Sum of Matrices

(1995). Derivatives of Noninteger Order. Mathematics Magazine: Vol. 68, No. 3, pp. 183-192.

Kenneth S. Miller

Papers

On the Inverse of the Sum of Matrices

Derivatives of Noninteger Order