Wikipedia / Linear model
Wikipedia / Linear model
In statistics, the term linear model refers to any model which assumes linearity in the system. 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 and the independent variables is formulated as
may be nonlinear are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the regression coefficients in a linear way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely
.
Given that estimation is undertaken on the basis of a least squares are determined by minimising a sum of squares function
From this, it can readily be seen that the "linear" aspect of the model means the following:
for which minimisation is a relatively simple problem;
making it easy to find the minimising values;
;
.
Time series models
An example of a linear time series model is an autoregressive moving average model} in a time series can be written in the form
are random variables representing innovations as a linear function of past values of the same time series and of current and past values of the innovations. This particular aspect of the structure means that it is relatively simple to derive relations for the mean and covariance, 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