Fits a multiple linear regression model with or without an intercept.
For a list of all members of this type, see LinearRegression Members.
System.Object
Imsl.Stat.LinearRegression
Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.
Fits a multiple linear regression model with or without an intercept. If the constructor argument hasIntercept
is true, the multiple linear regression model is
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
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 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 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.
From a general linear model fitted using the 's as the weights, inner class LinearRegression.CaseStatistics can also compute predicted values, confidence intervals, and diagnostics for detecting outliers and cases that greatly influence the fitted regression. Let be a column vector containing elements of the -th row of . Let . The leverage is defined as
(In the case of linear equality restrictions on , the leverage is defined in terms of the reduced model.) Put with if the -th diagonal element of is positive and 0 otherwise. The leverage is computed as where is a solution to . The estimated variance of is given by , where . The computation of the remainder of the case statistics follows easily from their definitions.Let denote the residual
for the th case. The estimated variance of is where is the residual mean square from the fitted regression. The th standardized residual (also called the internally studentized residual) is by definition and follows an approximate standard normal distribution in large samples.The th jackknife residual or deleted residual involves the difference between and its predicted value based on the data set in which the th case is deleted. This difference equals . The jackknife residual is obtained by standardizing this difference. The residual mean square for the regression in which the th case is deleted is
The jackknife residual is defined to be and follows a distribution with degrees of freedom.Cook's distance for the th case is a measure of how much an individual case affects the estimated regression coefficients. It is given by
Weisberg (1985) states that if exceeds the 50-th percentile of the distribution, it should be considered large. (This value is about 1. This statistic does not have an distribution.)DFFITS, like Cook's distance, is also a measure of influence. For the th case, DFFITS is computed by the formula
Hoaglin and Welsch (1978) suggest that greater than is large.Often predicted values and confidence intervals are desired for combinations of settings of the effect variables not used in computing the regression fit. This can be accomplished using a single data matrix by including these settings of the variables as part of the data matrix and by setting the response equal to Double.NaN
. LinearRegression
will omit the case when performing the fit and a predicted value and confidence interval for the missing response will be computed from the given settings of the effect variables.
Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll)
LinearRegression Members | Imsl.Stat Namespace | Example1 | Example2