Polynomial regression

Summary

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x). 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. The explanatory (independent) variables resulting from the polynomial expansion of the "baseline" variables are known as higher-degree terms. Such variables are also used in classification settings. History Polynomial regression models are usually fit using the method of least squa

Official source

https://en.wikipedia.org/wiki/Polynomial_regression

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (30)

Related publications

Related people (1)

Related people

Related units (1)

Related units

Related concepts (18)

Related concepts

Related courses (31)

Participants of this course will master computational techniques frequently used in mathematical finance applications. Emphasis will be put on the implementation and practical aspects.

Machine learning and data analysis are becoming increasingly central in sciences including physics. In this course, fundamental principles and methods of machine learning will be introduced and practised.

The course provides an introduction to econometrics. The objective is to learn how to make valid (i.e., causal) inference from economic data. It explains the main estimators and present methods to deal with endogeneity issues.

Related courses

Related lectures (57)

Related lectures

Robust inference for generalized linear models

Sahar Hosseinian Ehrensberger

Generalized Linear Models have become a commonly used tool of data analysis. Such models are used to fit regressions for univariate responses with normal, gamma, binomial or Poisson distribution. Maximum likelihood is generally applied as fitting method. In the usual regression setting the least absolute-deviations estimator (L1-norm) is a popular alternative to least squares (L2-norm) because of its simplicity and its robustness properties. In the first part of this thesis we examine the question of how much of these robustness features carry over to the setting of generalized linear models. We study a robust procedure based on the minimum absolute deviation estimator of Morgenthaler (1992), the Lq quasi-likelihood when q = 1. In particular, we investigate the influence function of these estimates and we compare their sensitivity to that of the maximum likelihood estimate. Furthermore we particularly explore the Lq quasi-likelihood estimates in binary regression. These estimates are difficult to compute. We derive a simpler estimator, which has a similar form as the Lq quasi-likelihood estimate. The resulting estimating equation consists in a simple modification of the familiar maximum likelihood equation with the weights wq(μ). This presents an improvement compared to other robust estimates discussed in the literature that typically have weights, which depend on the couple (xi, yi) rather than on μi = h(xiT β) alone. Finally, we generalize this estimator to Poisson regression. The resulting estimating equation is a weighted maximum likelihood with weights that depend on μ only.

EPFL2009

Polynomial autoregressions have been most of the time discarded as being unrealistic. Indeed, for such processes to be stationary, strong assumptions on the parameters and on the noise are necessary. For example, the distribution of the latter has to have finite support. Nevertheless, the use of polynomials has been advocated by a few authors: Cox (1991), Cobb and Zacks (1988) and Chan and Tong (1994). From a model-free perspective, that is using parametric families of functions (e.g. Fourier series, wavelets, neural networks, MARS, ...) as approximators of the optimal predictor, there is no more concern about stationarity. Still, polynomials have not been used within this framework; the reason is the now well known "curse of dimensionality". Indeed, when high lag orders are used to forecast, a common situation in time series, polynomial autoregressions imply too many parameters to estimate. We will introduce a new family of predictors based on polynomials and on a projection scheme. A censoring procedure allows to solve the instability inherent to polynomials. The forecasting method is parsimonious (that is non-linearity is introduced with a small amount of parameters) and thus allows to forecast noisy time series of short to moderate length.

1996

Tractability of interpretability via selection of group-sparse models

Luca Baldassarre, Nirav Bhan, Volkan Cevher

Group-based sparsity models are proven instrumental in linear regression problems for recovering signals from much fewer measurements than standard compressive sensing. A promise of these models is to lead to “interpretable” signals for which we identify its constituent groups, however we show that, in general, claims of correctly identifying the groups with convex relaxations would lead to polynomial time solution algorithms for an NP-hard problem. Instead, leveraging a graph-based understanding of group models, we describe group structures which enable correct model identification in polynomial time via dynamic programming. We also show that group structures that lead to totally unimodular constraints have tractable relaxations. Finally, we highlight the non-convexity of the Pareto frontier of group-sparse approximations and what it means for tractability.

2013