Explaining Support Vector Machines: A Color Based Nomogram.

PROBLEM SETTING: Support vector machines (SVMs) are very popular tools for classification, regression and other problems. Due to the large choice of kernels they can be applied with, a large variety of data can be analysed using these tools. Machine learning thanks its popularity to the good performance of the resulting models. However, interpreting the models is far from obvious, especially when non-linear kernels are used. Hence, the methods are used as black boxes. As a consequence, the use of SVMs is less supported in areas where interpretability is important and where people are held responsible for the decisions made by models. OBJECTIVE: In this work, we investigate whether SVMs using linear, polynomial and RBF kernels can be explained such that interpretations for model-based decisions can be provided. We further indicate when SVMs can be explained and in which situations interpretation of SVMs is (hitherto) not possible. Here, explainability is defined as the ability to produce the final decision based on a sum of contributions which depend on one single or at most two input variables. RESULTS: Our experiments on simulated and real-life data show that explainability of an SVM depends on the chosen parameter values (degree of polynomial kernel, width of RBF kernel and regularization constant). When several combinations of parameter values yield the same cross-validation performance, combinations with a lower polynomial degree or a larger kernel width have a higher chance of being explainable. CONCLUSIONS: This work summarizes SVM classifiers obtained with linear, polynomial and RBF kernels in a single plot. Linear and polynomial kernels up to the second degree are represented exactly. For other kernels an indication of the reliability of the approximation is presented. The complete methodology is available as an R package and two apps and a movie are provided to illustrate the possibilities offered by the method

Density Estimation using Support Vector Machines

this report we describe how the Support Vector (SV) technique of solving linear operator equations can be applied to the problem of density estimation [4]. We present a new optimization procedure and set of kernels closely related to current SV techniques that guarantee the monotonicity of the approximation. This technique estimates densities with a mixture of bumps (Gaussian-like shapes), with the usual SV property that only some coefficients are non-zero. Both the width and the height of each bump is chosen adaptively, by considering a dictionary of several kernel functions. There is empirical evidence that the regularization parameter gives good control of the approximation, and that the choice of this parameter is universal with respect to the number of sample points, and can even be fixed with good results. 2 The density estimation proble

Support Vector Density Estimation

> (x); (1.1) 1. `(x) = ( 1; x ? 0 0; otherwise Generic author design sample pages 1999/07/12 15:50 2 Support Vector Density Estimation where instead of knowing the distribution function F (x) we are given the iid (independently and identically distributed) data x 1 ; : : : ; x ` (1.2) generated by F . The problem of density estimation is known to be ill-posed. "Ill-posed" means that when finding p that satisfies the equality Ap = F , where A is a linear operator, we can have large deviations in solution p corresponding to small deviations in F . In our terms, a small change in the distribution function of the continuous random variable X can cause large change

Support Vector Machine - Reference Manual

this document will describe these programs. To find out more about SVMs, see the bibliography. We will not describe how SVMs work here

Specification and prediction of nickel mobilization using artificial intelligence methods

Groundwater and soil pollution from pyrite oxidation, acid mine drainage generation, and release and transport of toxic metals are common environmental problems associated with the mining industry. Nickel is one toxic metal considered to be a key pollutant in some mining setting; to date, its formation mechanism has not yet been fully evaluated. The goals of this study are 1) to describe the process of nickel mobilization in waste dumps by introducing a novel conceptual model, and 2) to predict nickel concentration using two algorithms, namely the support vector machine (SVM) and the general regression neural network (GRNN). The results obtained from this study have shown that considerable amount of nickel concentration can be arrived into the water flow system during the oxidation of pyrite and subsequent Acid Drainage (AMD) generation. It was concluded that pyrite, water, and oxygen are the most important factors for nickel pollution generation while pH condition, SO 4, HCO3, TDS, EC, Mg, Fe, Zn, and Cu are measured quantities playing significant role in nickel mobilization. SVM and GRNN have predicted nickel concentration with a high degree of accuracy. Hence, SVM and GRNN can be considered as appropriate tools for environmental risk assessment. © Versita Sp. z o.o