LinearRegression (JMSL Numerical Library)

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JMSL^TM Numerical Library 3.0

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com.imsl.stat
Class LinearRegression

java.lang.Object
  com.imsl.stat.LinearRegression

All Implemented Interfaces:: Cloneable, Serializable

public class LinearRegression
extends Object
implements Serializable, Cloneable

Fits a multiple linear regression model with or without an intercept. If the constructor argument hasIntercept is true, the multiple linear regression model is

$y_i = beta _0 + beta _1 x_{i1} + beta _2 x_{i2} + , ldots + beta _k x_{ik}+ varepsilon _i ,,,,, i = 1,,2,, ldots ,,n$

where the observed values of the

's constitute the responses or values of the dependent variable, the $x_{i1}$ 's, $x_{i2}$ 's, $ldots, x_{ik}$ 's are the settings of the independent variables,

are the regression coefficients, and the

's are independently distributed normal errors each with mean zero and variance

. If hasIntercept is false,

is not included in the model.

LinearRegression computes estimates of the regression coefficients by minimizing the sum of squares of the deviations of the observed response from the fitted response

for the observations. This minimum sum of squares (the error sum of squares) is in the ANOVA output and denoted by

${rm SSE} = sumlimits_{i=1}^n w_i (y_i-hat y_i)^2$

In addition, the total sum of squares is output in the ANOVA table. For the case, hasIntercept is true; the total sum of squares is the sum of squares of the deviations of from its mean

--the so-called corrected total sum of squares; it is denoted by

${rm SST} = sumlimits_{i=1}^n w_i (y_i - bar y)^2$

For the case hasIntercept is false, the total sum of squares is the sum of squares of --the so-called uncorrected total sum of squares; it is denoted by

${rm SST} = sumlimits_{i=1}^n y_i^2$

See Draper and Smith (1981) for a good general treatment of the multiple linear regression model, its analysis, and many examples.

In order to compute a least-squares solution, LinearRegression performs an orthogonal reduction of the matrix of regressors to upper triangular form. Givens rotations are used to reduce the matrix. This method has the advantage that the loss of accuracy resulting from forming the crossproduct matrix used in the normal equations is avoided, while not requiring the storage of the full matrix of regressors. The method is described by Lawson and Hanson, pages 207-212.

See Also:: Example, Serialized Form

Nested Class Summary
`class`	`LinearRegression.CoefficientTTests` CoefficientTTests contains statistics related to the regression coefficients.

Constructor Summary
`LinearRegression(int nVariables, boolean hasIntercept)` Constructs a new linear regression object.

Method Summary
`ANOVA`	`getANOVA()` Get an analysis of variance table and related statistics.
`double[]`	`getCoefficients()` Returns the regression coefficients.
`LinearRegression.CoefficientTTests`	`getCoefficientTTests()` Returns statistics relating to the regression coefficients.
`double[][]`	`getR()` Returns a copy of the R matrix.
`int`	`getRank()` Returns the rank of the matrix.
`void`	`update(double[][] x, double[] y)` Updates the regression object with a new set of observations.
`void`	`update(double[][] x, double[] y, double[] w)` Updates the regression object with a new set of observations and weights.
`void`	`update(double[] x, double y)` Updates the regression object with a new observation.
`void`	`update(double[] x, double y, double w)` Updates the regression object with a new observation and weight.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail