Linear predictor function

In statistics and in machine learning, a linear predictor function is a linear function (linear combination) of a set of coefficients and explanatory variables (independent variables), whose value is used to predict the outcome of a dependent variable. This sort of function usually comes in linear regression, where the coefficients are called regression coefficients. However, they also occur in various types of linear classifiers (e.g. logistic regression, perceptrons, support vector machines, and linear discriminant analysis), as well as in various other models, such as principal component analysis and factor analysis. In many of these models, the coefficients are referred to as "weights". The basic form of a linear predictor function for data point i (consisting of p explanatory variables), for i = 1, ..., n, is where , for k = 1, ..., p, is the value of the k-th explanatory variable for data point i, and are the coefficients (regression coefficients, weights, etc.) indicating the relative effect of a particular explanatory variable on the outcome. It is common to write the predictor function in a more compact form as follows: The coefficients β0, β1, ..., βp are grouped into a single vector β of size p + 1. For each data point i, an additional explanatory pseudo-variable xi0 is added, with a fixed value of 1, corresponding to the intercept coefficient β0. The resulting explanatory variables xi0(= 1), xi1, ..., xip are then grouped into a single vector xi of size p + 1. This makes it possible to write the linear predictor function as follows: using the notation for a dot product between two vectors. An equivalent form using matrix notation is as follows: where and are assumed to be a (p+1)-by-1 column vectors, is the matrix transpose of (so is a 1-by-(p+1) row vector), and indicates matrix multiplication between the 1-by-(p+1) row vector and the (p+1)-by-1 column vector, producing a 1-by-1 matrix that is taken to be a scalar.