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Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression. [1]
Consider a set of data points, (,), (,), …, (,), and a curve (model function) ^ = (,), that in addition to the variable also depends on parameters, = (,, …,), with . It is desired to find the vector of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares = = is minimized, where the residuals (in-sample prediction errors) r i are ...
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.
Local regression or local polynomial regression, [1] also known as moving regression, [2] is a generalization of the moving average and polynomial regression. [3] Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing), both pronounced / ˈ l oʊ ɛ s / LOH-ess.
Optimal instruments regression is an extension of classical IV regression to the situation where E[ε i | z i] = 0. Total least squares (TLS) [6] is an approach to least squares estimation of the linear regression model that treats the covariates and response variable in a more geometrically symmetric manner than OLS. It is one approach to ...
The following Python code with the SymPy library will allow for calculation of the values of and to 20 digits of precision: from sympy import * def lag_weights_roots ( n ): x = Symbol ( "x" ) roots = Poly ( laguerre ( n , x )) . all_roots () x_i = [ rt . evalf ( 20 ) for rt in roots ] w_i = [( rt / (( n + 1 ) * laguerre ( n + 1 , rt )) ** 2 ...
Partial least squares (PLS) regression is a statistical method that bears some relation to principal components regression and is a reduced rank regression [1]; instead of finding hyperplanes of maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the observable variables to a new space of maximum ...
Weighted least squares (WLS), also known as weighted linear regression, [1] [2] is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (heteroscedasticity) is incorporated into the regression.