On the Noise Model of Support Vector Machine Regression

Support Vector Machines Regression (SVMR) is a regression technique which has been recently introduced by V. Vapnik and his collaborators (Vapnik, 1995; Vapnik, Golowich and Smola, 1996). In SVMR the goodness of fit is measured not by the usual quadratic loss function (the mean square error), but by a different loss function called Vapnik"s $epsilon$- insensitive loss function, which is similar to the "robust" loss functions introduced by Huber (Huber, 1981). The quadratic loss function is well justified under the assumption of Gaussian additive noise. However, the noise model underlying the choice of Vapnik's loss function is less clear. In this paper the use of Vapnik's loss function is shown to be equivalent to a model of additive and Gaussian noise, where the variance and mean of the Gaussian are random variables. The probability distributions for the variance and mean will be stated explicitly. While this work is presented in the framework of SVMR, it can be extended to justify non-quadratic loss functions in any Maximum Likelihood or Maximum A Posteriori approach. It applies not only to Vapnik's loss function, but to a much broader class of loss functions.

Output from the 1998 Product Development Value Stream Workshop: A Framework for Understanding Information Flow in the Product Development Process

The descriptions below and the attached diagrams are outputs of the 1998 LAI Product Development Focus Team workshop on the Value Chain in Product Development. A working group at that workshop was asked to model the product development process: in terms of the phases of product development and their interfaces, boundaries and outputs. Their work has proven to be generally useful to LAI researchers and industry members, and so is formalized here.

Model Order Reduction for Determining Bubble Parameters to Attain a Desired Fluid Surface Shape

In this paper, a new methodology for predicting fluid free surface shape using Model Order Reduction (MOR) is presented. Proper Orthogonal Decomposition combined with a linear interpolation procedure for its coefficient is applied to a problem involving bubble dynamics near to a free surface. A model is developed to accurately and efficiently capture the variation of the free surface shape with different bubble parameters. In addition, a systematic approach is developed within the MOR framework to find the best initial locations and pressures for a set of bubbles beneath the quiescent free surface such that the resultant free surface attained is close to a desired shape. Predictions of the free surface in two-dimensions and three-dimensions are presented.