![]() Note, the following explanation is an expansion of the extremely detailed article on QR Decomposition using Householder Reflections over at Wikipedia.Ī Householder Reflection is a linear transformation that enables a vector to be reflected through a plane or hyperplane. I've outlined the Householder Reflections method below. We will outline the method of Householder Reflections, which is known to be more numerically stable the the alternative Gramm-Schmidt method. There are a few different algorithms for calculating the matrices $Q$ and $R$. $Q^TQ=I$) and an upper triangular matrix $R$. Hence if you are performing a large quantity of regressions as part of a trading backtest, for instance, you will need to consider very extensively whether QR Decomposition is the best fit (excuse the pun).įor a square matrix $A$ the QR Decomposition converts $A$ into the product of an orthogonal matrix $Q$ (i.e. One of the key benefits of using QR Decomposition over other methods for solving linear least squares is that it is more numerically stable, albeit at the expense of being slower to execute. QR Decomposition is widely used in quantitative finance as the basis for the solution of the linear least squares problem, which itself is used for statistical regression analysis. In previous articles we have looked at LU Decomposition in Python and Cholesky Decomposition in Python as two alternative matrix decomposition methods. Try some of the exercises in matrix algebra in this course on intro to statistics with R.This article will discuss QR Decomposition in Python. (dense matrix), TAUCS (sparse matrix) and UMFPACK (sparse matrix) It provides efficient access to BLAS (Basic Linear Algebra Subroutines), Lapack The Matrix package contains functions that extend R to support highly dense or sparse matrices. This can help porting MATLAB applications and code to R. The matlab package contains wrapper functions and variables used to replicate MATLAB function calls as best possible. Returns a matrix.Ĭombine matrices(vectors) vertically. Y$pivot contains information on the pivoting strategy used.Ĭombine matrices(vectors) horizontally. Y$qraux a vector which contains additional information on Q. Y$qr has an upper triangle that contains the decomposition and a lower triangle that contains information on the Q decomposition. Returns the upper triangular factor, such that R'R = A. Y$v = matrix with columns contain the right singular vectors of AĬholeski factorization of A. Y$u = matrix with columns contain the left singular vectors of A Y$d = vector containing the singular values of A Ginv(A) requires loading the MASS package. Returns vector x in the equation b = Ax (i.e., A -1b) If k is a scalar, this creates a k x k identity matrix. ![]() Returns a vector containing the elements of the principal diagonal In the following examples, A and B are matrices and x and b are a vectors.Ĭreates diagonal matrix with elements of x in the principal diagonal Before proceeding you many want to review the sections on Data Types and Operators. This section will simply cover operators and functions specifically suited to linear algebra. Most of the methods on this website actually describe the programming of matrices.
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