How does SVM classify new points after being trained?
Support Vector Machines (SVMs) are supervised learning models that can be used for classification and regression tasks. In the context of classification, SVMs aim to find a hyperplane that separates different classes of data points. Once trained, SVMs can be used to classify new points by determining which side of the hyperplane they fall on.
To understand how SVMs classify new points, let's first discuss the training process. During training, an SVM learns the optimal hyperplane by finding support vectors, which are the data points closest to the decision boundary. The decision boundary is defined by a function that takes into account the support vectors and their associated weights. This function is also known as the decision function or the discriminant function.
When a new point is to be classified, the SVM applies the decision function to that point. The decision function calculates the signed distance between the new point and the decision boundary. The sign of this distance determines the class to which the new point belongs. If the distance is positive, the point is classified as belonging to one class, and if it is negative, it is classified as belonging to the other class.
Mathematically, the decision function can be represented as:
f(x) = sign(Σ_i α_i y_i K(x_i, x) + b)
where:
– f(x) is the output of the decision function for the new point x.
– Σ_i represents the sum over all support vectors.
– α_i is the weight associated with the i-th support vector.
– y_i is the class label (+1 or -1) of the i-th support vector.
– K(x_i, x) is the kernel function that measures the similarity between the i-th support vector and the new point x.
– b is the bias term.
The kernel function is a important component of SVMs as it allows them to handle non-linearly separable data. It implicitly maps the input space into a higher-dimensional feature space, where the data becomes linearly separable. Common kernel functions include the linear kernel, polynomial kernel, and radial basis function (RBF) kernel.
To illustrate the classification process, consider a simple example where we have two classes, red and blue, and the SVM has been trained on a set of data points from these classes. The decision boundary found by the SVM separates the red and blue points in the feature space. When a new point is presented, the SVM calculates its distance to the decision boundary using the decision function. If the distance is positive, the point is classified as red, and if it is negative, it is classified as blue.
It's worth noting that SVMs can also provide a measure of confidence or probability for the classification. This is achieved by using methods such as Platt scaling or by directly optimizing the SVM to output probabilities.
After being trained, SVMs classify new points by applying the decision function to these points. The decision function calculates the signed distance between the new point and the decision boundary, allowing the SVM to determine the class to which the point belongs. The kernel function plays a important role in SVMs by mapping the data into a higher-dimensional feature space, where it becomes linearly separable.
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