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LIBSVM and LIBLINEAR are two popular open source machine learning libraries, both developed at the National Taiwan University and both written in C++ though with a C API. LIBSVM implements the sequential minimal optimization (SMO) algorithm for kernelized support vector machines (SVMs), supporting classification and regression. [1]
Hierarchical linear models (or multilevel regression) organizes the data into a hierarchy of regressions, for example where A is regressed on B, and B is regressed on C. It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms ...
where Q = A T A and c = −A T y. This problem is convex , as Q is positive semidefinite and the non-negativity constraints form a convex feasible set. [ 7 ]
A simple example is fitting a line in two dimensions to a set of observations. Assuming that this set contains both inliers, i.e., points which approximately can be fitted to a line, and outliers, points which cannot be fitted to this line, a simple least squares method for line fitting will generally produce a line with a bad fit to the data including inliers and outliers.
IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set, for example, by minimizing the least absolute errors rather than the least square errors.
If the assumptions of OLS regression hold, the solution = (), with =, is an unbiased estimator, and is the minimum-variance linear unbiased estimator, according to the Gauss–Markov theorem. The term λ n I {\displaystyle \lambda nI} therefore leads to a biased solution; however, it also tends to reduce variance.
Deming regression (total least squares) also finds a line that fits a set of two-dimensional sample points, but (unlike ordinary least squares, least absolute deviations, and median slope regression) it is not really an instance of simple linear regression, because it does not separate the coordinates into one dependent and one independent ...
In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L 1 and L 2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. [1]