public class OLSMultipleLinearRegression extends AbstractMultipleLinearRegression
Implements ordinary least squares (OLS) to estimate the parameters of a multiple linear regression model.
The regression coefficients, b, satisfy the normal equations:
XT X b = XT y To solve the normal equations, this implementation uses QR decomposition
of the X matrix. (See QRDecomposition for details on the
decomposition algorithm.) The X matrix, also known as the design matrix,
has rows corresponding to sample observations and columns corresponding to independent
variables. When the model is estimated using an intercept term (i.e. when
isNoIntercept is false as it is by default), the X
matrix includes an initial column identically equal to 1. We solve the normal equations
as follows:
XTX b = XT y
(QR)T (QR) b = (QR)Ty
RT (QTQ) R b = RT QT y
RT R b = RT QT y
(RT)-1 RT R b = (RT)-1 RT QT y
R b = QT y Given Q and R, the last equation is solved by back-substitution.
| Constructor and Description |
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OLSMultipleLinearRegression()
Create an empty OLSMultipleLinearRegression instance.
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OLSMultipleLinearRegression(double threshold)
Create an empty OLSMultipleLinearRegression instance, using the given
singularity threshold for the QR decomposition.
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| Modifier and Type | Method and Description |
|---|---|
double |
calculateAdjustedRSquared()
Returns the adjusted R-squared statistic, defined by the formula
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protected RealVector |
calculateBeta()
Calculates the regression coefficients using OLS.
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protected RealMatrix |
calculateBetaVariance()
Calculates the variance-covariance matrix of the regression parameters.
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RealMatrix |
calculateHat()
Compute the "hat" matrix.
|
double |
calculateResidualSumOfSquares()
Returns the sum of squared residuals.
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double |
calculateRSquared()
Returns the R-Squared statistic, defined by the formula
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double |
calculateTotalSumOfSquares()
Returns the sum of squared deviations of Y from its mean.
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void |
newSampleData(double[] y,
double[][] x)
Loads model x and y sample data, overriding any previous sample.
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void |
newSampleData(double[] data,
int nobs,
int nvars)
Loads model x and y sample data from a flat input array, overriding any previous sample.
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protected void |
newXSampleData(double[][] x)
Loads new x sample data, overriding any previous data.
|
calculateErrorVariance, calculateResiduals, calculateYVariance, estimateErrorVariance, estimateRegressandVariance, estimateRegressionParameters, estimateRegressionParametersStandardErrors, estimateRegressionParametersVariance, estimateRegressionStandardError, estimateResiduals, getX, getY, isNoIntercept, newYSampleData, setNoIntercept, validateCovarianceData, validateSampleDatapublic OLSMultipleLinearRegression()
public OLSMultipleLinearRegression(double threshold)
threshold - the singularity thresholdpublic void newSampleData(double[] y,
double[][] x)
throws MathIllegalArgumentException
y - the [n,1] array representing the y samplex - the [n,k] array representing the x sampleMathIllegalArgumentException - if the x and y array data are not
compatible for the regressionpublic void newSampleData(double[] data,
int nobs,
int nvars)
Loads model x and y sample data from a flat input array, overriding any previous sample.
Assumes that rows are concatenated with y values first in each row. For example, an input
data array containing the sequence of values (1, 2, 3, 4, 5, 6, 7, 8, 9) with
nobs = 3 and nvars = 2 creates a regression dataset with two
independent variables, as below:
y x[0] x[1] -------------- 1 2 3 4 5 6 7 8 9
Note that there is no need to add an initial unitary column (column of 1's) when
specifying a model including an intercept term. If AbstractMultipleLinearRegression.isNoIntercept() is true,
the X matrix will be created without an initial column of "1"s; otherwise this column will
be added.
Throws IllegalArgumentException if any of the following preconditions fail:
data cannot be nulldata.length = nobs * (nvars + 1)nobs > nvarsThis implementation computes and caches the QR decomposition of the X matrix.
newSampleData in class AbstractMultipleLinearRegressiondata - input data arraynobs - number of observations (rows)nvars - number of independent variables (columns, not counting y)public RealMatrix calculateHat()
Compute the "hat" matrix.
The hat matrix is defined in terms of the design matrix X by X(XTX)-1XT
The implementation here uses the QR decomposition to compute the hat matrix as Q IpQT where Ip is the p-dimensional identity matrix augmented by 0's. This computational formula is from "The Hat Matrix in Regression and ANOVA", David C. Hoaglin and Roy E. Welsch, The American Statistician, Vol. 32, No. 1 (Feb., 1978), pp. 17-22.
Data for the model must have been successfully loaded using one of
the newSampleData methods before invoking this method; otherwise
a NullPointerException will be thrown.
NullPointerException - unless method newSampleData has been
called beforehand.public double calculateTotalSumOfSquares()
Returns the sum of squared deviations of Y from its mean.
If the model has no intercept term, 0 is used for the
mean of Y - i.e., what is returned is the sum of the squared Y values.
The value returned by this method is the SSTO value used in
the R-squared computation.
NullPointerException - if the sample has not been setAbstractMultipleLinearRegression.isNoIntercept()public double calculateResidualSumOfSquares()
SingularMatrixException - if the design matrix is singularNullPointerException - if the data for the model have not been loadedpublic double calculateRSquared()
R2 = 1 - SSR / SSTOwhere SSR is the
sum of squared residuals
and SSTO is the total sum of squares
If there is no variance in y, i.e., SSTO = 0, NaN is returned.
NullPointerException - if the sample has not been setSingularMatrixException - if the design matrix is singularpublic double calculateAdjustedRSquared()
Returns the adjusted R-squared statistic, defined by the formula
R2adj = 1 - [SSR (n - 1)] / [SSTO (n - p)]where SSR is the
sum of squared residuals,
SSTO is the total sum of squares, n is the number
of observations and p is the number of parameters estimated (including the intercept).
If the regression is estimated without an intercept term, what is returned is
1 - (1 - calculateRSquared()) * (n / (n - p))
If there is no variance in y, i.e., SSTO = 0, NaN is returned.
NullPointerException - if the sample has not been setSingularMatrixException - if the design matrix is singularAbstractMultipleLinearRegression.isNoIntercept()protected void newXSampleData(double[][] x)
Loads new x sample data, overriding any previous data.
The inputx array should have one row for each sample
observation, with columns corresponding to independent variables.
For example, if
x = new double[][] {{1, 2}, {3, 4}, {5, 6}}
then setXSampleData(x) results in a model with two independent
variables and 3 observations:
x[0] x[1]
----------
1 2
3 4
5 6
Note that there is no need to add an initial unitary column (column of 1's) when specifying a model including an intercept term.
This implementation computes and caches the QR decomposition of the X matrix once it is successfully loaded.
newXSampleData in class AbstractMultipleLinearRegressionx - the rectangular array representing the x sampleprotected RealVector calculateBeta()
Data for the model must have been successfully loaded using one of
the newSampleData methods before invoking this method; otherwise
a NullPointerException will be thrown.
calculateBeta in class AbstractMultipleLinearRegressionSingularMatrixException - if the design matrix is singularNullPointerException - if the data for the model have not been loadedprotected RealMatrix calculateBetaVariance()
Calculates the variance-covariance matrix of the regression parameters.
Var(b) = (XTX)-1
Uses QR decomposition to reduce (XTX)-1 to (RTR)-1, with only the top p rows of R included, where p = the length of the beta vector.
Data for the model must have been successfully loaded using one of
the newSampleData methods before invoking this method; otherwise
a NullPointerException will be thrown.
calculateBetaVariance in class AbstractMultipleLinearRegressionSingularMatrixException - if the design matrix is singularNullPointerException - if the data for the model have not been loadedCopyright © 2003–2017. All rights reserved.