Advanced Linear Models for Data Science 1: Least Squares

Start Date: 09/15/2019

Course Type: Common Course

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Course Syllabus

We cover some basic matrix algebra results that we will need throughout the class. This includes some basic vector derivatives. In addition, we cover some some basic uses of matrices to create summary statistics from data. This includes calculating and subtracting means from observations (centering) as well as calculating the variance.

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Course Introduction

Welcome to the Advanced Linear Models for Data Science Class 1: Least Squares. This class is an intr

Course Tag

Statistics Linear Regression R Programming Linear Algebra

Related Wiki Topic

Article Example
Linear least squares (mathematics) The "numerical methods for linear least squares" are important because linear regression models are among the most important types of model, both as formal statistical models and for exploration of data-sets. The majority of statistical computer packages contain facilities for regression analysis that make use of linear least squares computations. Hence it is appropriate that considerable effort has been devoted to the task of ensuring that these computations are undertaken efficiently and with due regard to round-off error.
Linear least squares (mathematics) Fitting of linear models by least squares often, but not always, arise in the context of statistical analysis. It can therefore be important that considerations of computation efficiency for such problems extend to all of the auxiliary quantities required for such analyses, and are not restricted to the formal solution of the linear least squares problem.
Linear least squares (mathematics) Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations, where the best approximation is defined as that which minimizes the sum of squared differences between the data values and their corresponding modeled values. The approach is called "linear" least squares since the assumed function is linear in the parameters to be estimated. Linear least squares problems are convex and have a closed-form solution that is unique, provided that the number of data points used for fitting equals or exceeds the number of unknown parameters, except in special degenerate situations. In contrast, non-linear least squares problems generally must be solved by an iterative procedure, and the problems can be non-convex with multiple optima for the objective function. If prior distributions are available, then even an underdetermined system can be solved using the Bayesian MMSE estimator.
Linear least squares (mathematics) Often it is of interest to solve a linear least squares problem with an additional constraint on the solution. With constrained linear least squares, the original equation
Linear least squares (mathematics) In statistics, linear least squares problems correspond to a particularly important type of statistical model called linear regression which arises as a particular form of regression analysis. One basic form of such a model is an ordinary least squares model. The present article concentrates on the mathematical aspects of linear least squares problems, with discussion of the formulation and interpretation of statistical regression models and statistical inferences related to these being dealt with in the articles just mentioned. See outline of regression analysis for an outline of the topic.
Linear least squares (mathematics) The primary application of linear least squares is in data fitting. Given a set of "m" data points formula_205 consisting of experimentally measured values taken at "m" values formula_206 of an independent variable (formula_207 may be scalar or vector quantities), and given a model function formula_208 with formula_209 it is desired to find the parameters formula_158 such that the model function "best" fits the data. In linear least squares, linearity is meant to be with respect to parameters formula_211 so
Non-linear least squares These equations form the basis for the Gauss–Newton algorithm for a non-linear least squares problem.
Non-linear least squares may be solved for formula_59 by Cholesky decomposition, as described in linear least squares. The parameters are updated iteratively
Non-linear least squares Non-linear least squares is the form of least squares analysis used to fit a set of "m" observations with a model that is non-linear in "n" unknown parameters ("m" > "n"). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences.
Total least squares In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models.
Least squares Least squares problems fall into two categories: linear or ordinary least squares and non-linear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The non-linear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases.
Linear least squares (mathematics) The equation and solution of linear least squares are thus described as follows:
Non-linear least squares In linear least squares the objective function, "S", is a quadratic function of the parameters.
Non-linear least squares Some information is given in the corresponding section on the linear least squares page.
Polynomial least squares In mathematical statistics, polynomial least squares refers to a broad range of statistical methods for estimating an underlying polynomial that describes observations. These methods include polynomial regression, curve fitting, linear regression, least squares, ordinary least squares, simple linear regression, linear least squares, approximation theory and method of moments. Polynomial least squares has applications in radar trackers, estimation theory, signal processing, statistics, and econometrics.
Least squares For a derivation of this estimate see Linear least squares (mathematics).
Linear least squares (mathematics) Importantly, in "linear least squares", we are not restricted to using a line as the model as in the above example. For instance, we could have chosen the restricted quadratic model formula_29. This model is still linear in the formula_7 parameter, so we can still perform the same analysis, constructing a system of equations from the data points:
Linear least squares Linear least squares is a method of solving mathematics/statistical problems. It uses least squares algorithmic technique to increase accuracy of solution approximations, corresponding with a particular problem's complexity:
Least squares The most important application is in data fitting. The best fit in the least-squares sense minimizes "the sum of squared residuals" (a residual being: the difference between an observed value, and the fitted value provided by a model). When the problem has substantial uncertainties in the independent variable (the "x" variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares.
Least squares For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.