com.numericalmethod.suanshu.stats.regression.linear.ols
Class Residuals

java.lang.Object
  com.numericalmethod.suanshu.stats.regression.linear.Residuals
      com.numericalmethod.suanshu.stats.regression.linear.ols.Residuals

public class Residuals
extends Residuals

Residual analysis of the results of an Ordinary Least Square linear regression model.

Once a regression model has been constructed, it may be important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters. Commonly used checks of goodness of fit include the R-squared, analysis of the pattern of residuals and hypothesis testing. Statistical significance can be checked by an F-test of the overall fit, followed by t-tests of individual parameters.

See Also:

• Wikipedia: Regression diagnostics
• Wikipedia: Coefficient of determination
• Wikipedia: Residual sum of squares
• Wikipedia: Explained sum of squares
• Wikipedia: Total sum of squares

Field Summary

double	AR2 diagnostic measure: the adjusted R-squared
double	f diagnostic measure: F statistics mean of regression / mean squared error = sum((y_i_hat-y_mean)^2) / mean squared error [(TSS-RSS)/n] / [RSS/(m-n)] y_i_hat are the fitted values of the regression.
ImmutableMatrix	hHat projection matrix H-hat
ImmutableVector	leverage leverage; the bigger the leverage for an observation, the bigger influence on the prediction
double	R2 diagnostic measure: the R-squared
double	RSS diagnostic measure: the sum of squared residuals, Σ(ε^2)
double	stderr the standard error of the residuals
double	TSS diagnostic measure: the total sum of squares, Σ((y-y_mean)^2)
ImmutableVector	wFitted the weighted, fitted values
ImmutableVector	wResiduals the weighted residuals

Fields inherited from class com.numericalmethod.suanshu.stats.regression.linear.Residuals

fitted, problem, residuals

Method Summary

ImmutableVector	standardized() standard residual = residual / v1 / RSS / (m-n)
ImmutableVector	studentized() studentized residual = standardized * sqrt((m-n-1) / (n-m-standardized))

Methods inherited from class java.lang.Object

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

Field Detail

wFitted

public final ImmutableVector wFitted

the weighted, fitted values

wResiduals

public final ImmutableVector wResiduals

the weighted residuals

stderr

public final double stderr

the standard error of the residuals

RSS

public final double RSS

diagnostic measure: the sum of squared residuals, Σ(ε^2)

TSS

public final double TSS

diagnostic measure: the total sum of squares, Σ((y-y_mean)^2)

R2

public final double R2

diagnostic measure: the R-squared

AR2

public final double AR2

diagnostic measure: the adjusted R-squared

f

public final double f

diagnostic measure: F statistics

mean of regression / mean squared error = sum((y_i_hat-y_mean)^2) / mean squared error [(TSS-RSS)/n] / [RSS/(m-n)]

y_i_hat are the fitted values of the regression.

See Also:

"p.69, equation (2.60). Applied linear regression models. Kutner, Nachtsheim and Neter. 4th edition."

hHat

public final ImmutableMatrix hHat

projection matrix H-hat

See Also:

"p.168 Section 8.1, Chapter 8. Applied Linear Regression, 3rd edition, 2005. Sanford Weisberg. Wiley-Interscience."

leverage

public final ImmutableVector leverage

leverage; the bigger the leverage for an observation, the bigger influence on the prediction

Method Detail

standardized

public ImmutableVector standardized()

standard residual = residual / v1 / RSS / (m-n)

Returns:

standardized residuals

studentized

public ImmutableVector studentized()

studentized residual = standardized * sqrt((m-n-1) / (n-m-standardized))

Returns:

studentized residuals

See Also:

• "p.90 (4.15). Section 4.3. Regression Analysis by Example, 3rd edition, 2000. Chatterjee, Hadi and Price. Wiley Series in Probability and Statistics."
• @see Wikipedia: Studentized residual