World Library  
Flag as Inappropriate
Email this Article

Linear model

Article Id: WHEBN0000017904
Reproduction Date:

Title: Linear model  
Author: World Heritage Encyclopedia
Language: English
Subject: Multivariate adaptive regression splines, Generalized randomized block design, Hat matrix, Analysis of variance, Errors-in-variables models
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Linear model

In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term is also used in time series analysis with a different meaning. In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related statistical theory is possible.

Linear regression models

For the regression case, the statistical model is as follows. Given a (random) sample (Y_i, X_{i1}, \ldots, X_{ip}), \, i = 1, \ldots, n the relation between the observations Yi and the independent variables Xij is formulated as

Y_i = \beta_0 + \beta_1 \phi_1(X_{i1}) + \cdots + \beta_p \phi_p(X_{ip}) + \varepsilon_i \qquad i = 1, \ldots, n

where \phi_1, \ldots, \phi_p may be nonlinear functions. In the above, the quantities εi are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the regression coefficients, βj in a linear way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely

\hat{Y}_i = \beta_0 + \beta_1 \phi_1(X_{i1}) + \cdots + \beta_p \phi_p(X_{ip}) \qquad (i = 1, \ldots, n),

are linear functions of the βj.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters βj are determined by minimising a sum of squares function

S = \sum_{i = 1}^n \left(Y_i - \beta_0 - \beta_1 \phi_1(X_{i1}) - \cdots - \beta_p \phi_p(X_{ip})\right)^2 .

From this, it can readily be seen that the "linear" aspect of the model means the following:

  • the function to be minimised is a quadratic function of the βj for which minimisation is a relatively simple problem;
  • the derivatives of the function are linear functions of the βj making it easy to find the minimising values;
  • the minimising values βj are linear functions of the observations Yi;
  • the minimising values βj are linear functions of the random errors εi which makes it relatively easy to determine the statistical properties of the estimated values of βj.

Time series models

An example of a linear time series model is an autoregressive moving average model. Here the model for values {Xt} in a time series can be written in the form

X_t = c + \varepsilon_t + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,

where again the quantities εt are random variables representing innovations which are new random effects that appear at a certain time but also affect values of X at later times. In this instance the use of the term "linear model" refers to the structure of the above relationship in representing Xt as a linear function of past values of the same time series and of current and past values of the innovations.[1] This particular aspect of the structure means that it is relatively simple to derive relations for the mean and covariance properties of the time series. Note that here the "linear" part of the term "linear model" is not referring to the coefficients φi and θi, as it would be in the case of a regression model, which looks structurally similar.

Other uses in statistics

There are some other instances where "nonlinear model" is used to contrast with a linearly structured model, although the term "linear model" is not usually applied. One example of this is nonlinear dimensionality reduction.

See also

References

  1. ^ Priestley, M.B. (1988) Non-linear and Non-stationary time series analysis, Academic Press. ISBN 0-12-564911-8
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.