Linear Regression¶
Linear regression is a method for modeling the relationship between a dependent variable (which may be a vector) and one or more explanatory variables by fitting linear equations to observed data.
Details¶
Let \((x_1, \ldots, x_p)\) be a vector of input variables and \(y=(y_1, \ldots, y_k)\) be the response. For each \(j=1, \ldots ,k\), the linear regression model has the format [Hastie2009]:
Here \(x_i\), \(i=1, \ldots,p\), are referred to as independent variables, and \(y_j\) are referred to as dependent variables or responses.
Linear regression is called:
Simple Linear Regression (if there is only one explanatory variable)
Multiple Linear Regression (if the number of explanatory variables \(p > 1\))
Training Stage¶
Let \((x_{11}, \ldots, x_{1p}, y_1, \ldots, x_{n1}, \ldots, x_{np}, y_n)\) be a set of training data, \(n \gg p\). The matrix \(X\) of size \(n \times p\) contains observations \(x_{ij}\), \(i=1, \ldots, n\), \(j = 1, \ldots, p\) of independent variables.
To estimate the coefficients \((\beta_{0j}, \ldots, \beta_{pj})\) one these methods can be used:
Normal Equation system
QR matrix decomposition
Prediction Stage¶
Linear regression based prediction is done for input vector \((x_1, \ldots, x_p)\) using the equation \(y_j = \beta_{0j} + \beta_{1j}x_1 + \ldots + \beta_{pj}x_p\) for each \(j=1, \ldots, k\).
Usage of Training Alternative¶
To build a Linear Regression model using methods of the Model Builder class of Linear Regression, complete the following steps:
Create a Linear Regression model builder using a constructor with the required number of responses and features.
Use the
setBeta
method to add the set of pre-calculated coefficients to the model. Specify random access iterators to the first and the last element of the set of coefficients [ISO/IEC 14882:2011 §24.2.7]_.Note
If your set of coefficients does not contain an intercept,
interceptFlag
is automatically set toFalse
, and toTrue
, otherwise.Use the
getModel
method to get the trained Linear Regression model.Use the
getStatus
method to check the status of the model building process. IfDAAL_NOTHROW_EXCEPTIONS
macros is defined, the status report contains the list of errors that describe the problems API encountered (in case of API runtime failure).
Note
If after calling the getModel
method you use the setBeta
method to update coefficients,
the initial model will be automatically updated with the new \(\beta\) coefficients.
Examples¶
Note
There is no support for Java on GPU.