One-class classification

In machine learning, one-class classification (OCC), also known as unary classification or class-modelling, is an approach to the training of binary classifiers in which only examples of one of the two classes are used. Examples include the monitoring of helicopter gearboxes, motor failure prediction, or assessing the operational status of a nuclear plant as 'normal': In such scenarios, there are few, if any, examples of the catastrophic system states – rare outliers – that comprise the second class. Alternatively, the class that is being focused on may cover a small, coherent subset of the data and the training may rely on an information bottleneck approach. In practice, counter-examples from the second class may be used in later rounds of training to further refine the algorithm.

In machine learning, one-class classification (OCC), also known as unary classification or class-modelling, is an approach to the training of binary classifiers in which only examples of one of the two classes are used. Examples include the monitoring of helicopter gearboxes, motor failure prediction, or assessing the operational status of a nuclear plant as 'normal': In such scenarios, there are few, if any, examples of the catastrophic system states – rare outliers – that comprise the second class. Alternatively, the class that is being focused on may cover a small, coherent subset of the data and the training may rely on an information bottleneck approach. In practice, counter-examples from the second class may be used in later rounds of training to further refine the algorithm. \min_{r,c} r^2 \text{ subject to, } ||\Phi(x_i) - c||^2 \le r^2 \;\; \forall i = 1, 2, ..., n However, the above formulation is highly restrictive, and is sensitive to the presence of outliers. Therefore, a flexible formulation, that allow for the presence of outliers is formulated as shown below, \min_{r,c,\zeta} r^2 + \frac{1}{\nu n}\sum_{i=1}^{n}\zeta_i \text{subject to, } ||\Phi(x_i) - c||^2 \le r^2 + \zeta_i \;\; \forall i = 1, 2, ..., n From the Karush–Kuhn–Tucker conditions for optimality, we get c = \sum_{i=1}^{n}\alpha_i\Phi(x_i), where the \alpha_i 's are the solution to the following optimization problem: \max_\alpha \sum_{i=1}^{n}\alpha_i\kappa(x_i, x_i) - \sum_{i, j = 1}^{n}\alpha_i\alpha_j\kappa(x_i, x_j) subject to, \sum_{i=1}^{n}\alpha_i = 1 \text{ and } 0 \le \alpha_i \le \frac{1}{\nu n} \text{for all } i = 1,2,...,n. The introduction of kernel function provide additional flexibility to the One-class SVM (OSVM) algorithm. p_{\mathcal{N}}(z;\mu;\Sigma) = \frac{1}{(2\pi)^{\frac{d}{2}} |\Sigma|^\frac{1}{2}}\exp\left\{-\frac{1}{2}(z-\mu)^T \Sigma^{-1} (z - \mu)\right\} Where, \mu is the mean and \Sigma is the covariance matrix. Computing the inverse of covariance matrix ( \Sigma^{-1} ) is the costliest operation, and in the cases where the data is not scaled properly, or data has singular directions pseudo-inverse \Sigma^+ is used to approximate the inverse, and is calculated as \Sigma^T(\Sigma \Sigma^T)^{-1} . \varepsilon_{k-center} = \max_i ( \min_k || x_i - \mu_k ||^2 ) The algorithm uses forward search method with random initialization, where the radius is determined by the maximum distance of the object, any given ball should capture. After the centers are determined, for any given test object z the distance can be calculated as, d_{k-centr}(z) = \min_k || z - \mu_k ||^2

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