Class OLSMultipleLinearRegression
 All Implemented Interfaces:
MultipleLinearRegression
Implements ordinary least squares (OLS) to estimate the parameters of a multiple linear regression model.
The regression coefficients, b
, satisfy the normal equations:
X^{T} X b = X^{T} 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:
X^{T}X b = X^{T} y
(QR)^{T} (QR) b = (QR)^{T}y
R^{T} (Q^{T}Q) R b = R^{T} Q^{T} y
R^{T} R b = R^{T} Q^{T} y
(R^{T})^{1} R^{T} R b = (R^{T})^{1} R^{T} Q^{T} y
R b = Q^{T} y
Given Q
and R
, the last equation is solved by backsubstitution.

Constructor Summary
ConstructorDescriptionCreate an empty OLSMultipleLinearRegression instance.OLSMultipleLinearRegression
(double threshold) Create an empty OLSMultipleLinearRegression instance, using the given singularity threshold for the QR decomposition. 
Method Summary
Modifier and TypeMethodDescriptiondouble
Returns the adjusted Rsquared statistic, defined by the formula \(R_\mathrm{adj}^2 = 1  \frac{\mathrm{SSR} (n  1)}{\mathrm{SSTO} (n  p)}\) where SSR is thesum of squared residuals
, SSTO is thetotal sum of squares
, n is the number of observations and p is the number of parameters estimated (including the intercept).protected RealVector
Calculates the regression coefficients using OLS.protected RealMatrix
Calculates the variancecovariance matrix of the regression parameters.Compute the "hat" matrix.double
Returns the sum of squared residuals.double
Returns the RSquared statistic, defined by the formula \(R^2 = 1  \frac{\mathrm{SSR}}{\mathrm{SSTO}}\) where SSR is thesum of squared residuals
and SSTO is thetotal sum of squares
double
Returns the sum of squared deviations of Y from its mean.void
newSampleData
(double[] y, double[][] x) Loads model x and y sample data, overriding any previous sample.void
newSampleData
(double[] data, int nobs, int nvars) Loads model x and y sample data from a flat input array, overriding any previous sample.protected void
newXSampleData
(double[][] x) Loads new x sample data, overriding any previous data.Methods inherited from class org.hipparchus.stat.regression.AbstractMultipleLinearRegression
calculateErrorVariance, calculateResiduals, calculateYVariance, estimateErrorVariance, estimateRegressandVariance, estimateRegressionParameters, estimateRegressionParametersStandardErrors, estimateRegressionParametersVariance, estimateRegressionStandardError, estimateResiduals, getX, getY, isNoIntercept, newYSampleData, setNoIntercept, validateCovarianceData, validateSampleData

Constructor Details

OLSMultipleLinearRegression
public OLSMultipleLinearRegression()Create an empty OLSMultipleLinearRegression instance. 
OLSMultipleLinearRegression
public OLSMultipleLinearRegression(double threshold) Create an empty OLSMultipleLinearRegression instance, using the given singularity threshold for the QR decomposition. Parameters:
threshold
 the singularity threshold


Method Details

newSampleData
Loads model x and y sample data, overriding any previous sample. Computes and caches QR decomposition of the X matrix. Parameters:
y
 the [n,1] array representing the y samplex
 the [n,k] array representing the x sample Throws:
MathIllegalArgumentException
 if the x and y array data are not compatible for the regression

newSampleData
public 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) withnobs = 3
andnvars = 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()
istrue
, 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 > nvars
This implementation computes and caches the QR decomposition of the X matrix.
 Overrides:
newSampleData
in classAbstractMultipleLinearRegression
 Parameters:
data
 input data arraynobs
 number of observations (rows)nvars
 number of independent variables (columns, not counting y)

calculateHat
Compute the "hat" matrix.
The hat matrix is defined in terms of the design matrix X by X(X^{T}X)^{1}X^{T}
The implementation here uses the QR decomposition to compute the hat matrix as Q I_{p}Q^{T} where I_{p} is the pdimensional 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. 1722.
Data for the model must have been successfully loaded using one of the
newSampleData
methods before invoking this method; otherwise aNullPointerException
will be thrown. Returns:
 the hat matrix
 Throws:
NullPointerException
 unless methodnewSampleData
has been called beforehand.

calculateTotalSumOfSquares
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
Rsquared
computation. Returns:
 SSTO  the total sum of squares
 Throws:
NullPointerException
 if the sample has not been set See Also:

calculateResidualSumOfSquares
public double calculateResidualSumOfSquares()Returns the sum of squared residuals. Returns:
 residual sum of squares
 Throws:
MathIllegalArgumentException
 if the design matrix is singularNullPointerException
 if the data for the model have not been loaded

calculateRSquared
public double calculateRSquared()Returns the RSquared statistic, defined by the formula \(R^2 = 1  \frac{\mathrm{SSR}}{\mathrm{SSTO}}\) where SSR is thesum of squared residuals
and SSTO is thetotal sum of squares
If there is no variance in y, i.e., SSTO = 0, NaN is returned.
 Returns:
 Rsquare statistic
 Throws:
NullPointerException
 if the sample has not been setMathIllegalArgumentException
 if the design matrix is singular

calculateAdjustedRSquared
public double calculateAdjustedRSquared()Returns the adjusted Rsquared statistic, defined by the formula \(R_\mathrm{adj}^2 = 1  \frac{\mathrm{SSR} (n  1)}{\mathrm{SSTO} (n  p)}\) where SSR is the
sum of squared residuals
, SSTO is thetotal 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.
 Returns:
 adjusted RSquared statistic
 Throws:
NullPointerException
 if the sample has not been setMathIllegalArgumentException
 if the design matrix is singular See Also:

newXSampleData
protected void newXSampleData(double[][] x) Loads new x sample data, overriding any previous data.
The input
x
array should have one row for each sample observation, with columns corresponding to independent variables. For example, ifx = 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.
 Overrides:
newXSampleData
in classAbstractMultipleLinearRegression
 Parameters:
x
 the rectangular array representing the x sample

calculateBeta
Calculates the regression coefficients using OLS.Data for the model must have been successfully loaded using one of the
newSampleData
methods before invoking this method; otherwise aNullPointerException
will be thrown. Specified by:
calculateBeta
in classAbstractMultipleLinearRegression
 Returns:
 beta
 Throws:
MathIllegalArgumentException
 if the design matrix is singularNullPointerException
 if the data for the model have not been loaded

calculateBetaVariance
Calculates the variancecovariance matrix of the regression parameters.
Var(b) = (X^{T}X)^{1}
Uses QR decomposition to reduce (X^{T}X)^{1} to (R^{T}R)^{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 aNullPointerException
will be thrown. Specified by:
calculateBetaVariance
in classAbstractMultipleLinearRegression
 Returns:
 The beta variancecovariance matrix
 Throws:
MathIllegalArgumentException
 if the design matrix is singularNullPointerException
 if the data for the model have not been loaded
