## Class OLSMultipleLinearRegression

• All Implemented Interfaces:
MultipleLinearRegression

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 Summary

Constructors
Constructor Description
OLSMultipleLinearRegression()
Create an empty OLSMultipleLinearRegression instance.
OLSMultipleLinearRegression​(double threshold)
Create an empty OLSMultipleLinearRegression instance, using the given singularity threshold for the QR decomposition.
• ### Method Summary

All Methods
Modifier and Type Method Description
double calculateAdjustedRSquared()
Returns the adjusted R-squared 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 the total sum of squares, n is the number of observations and p is the number of parameters estimated (including the intercept).
protected RealVector calculateBeta()
Calculates the regression coefficients using OLS.
protected RealMatrix calculateBetaVariance()
Calculates the variance-covariance matrix of the regression parameters.
RealMatrix calculateHat()
Compute the "hat" matrix.
double calculateResidualSumOfSquares()
Returns the sum of squared residuals.
double calculateRSquared()
Returns the R-Squared statistic, defined by the formula $$R^2 = 1 - \frac{\mathrm{SSR}}{\mathrm{SSTO}}$$ where SSR is the sum of squared residuals and SSTO is the total sum of squares
double calculateTotalSumOfSquares()
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
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### 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 Detail

• #### newSampleData

public void newSampleData​(double[] y,
double[][] x)
throws MathIllegalArgumentException
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 sample
x - 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) with nobs = 3 and nvars = 2 creates a regression dataset with two independent variables, as below:

   y   x  x
--------------
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 null
• data.length = nobs * (nvars + 1)
• nobs > nvars

This implementation computes and caches the QR decomposition of the X matrix.

Overrides:
newSampleData in class AbstractMultipleLinearRegression
Parameters:
data - input data array
nobs - number of observations (rows)
nvars - number of independent variables (columns, not counting y)
• #### calculateHat

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.

Returns:
the hat matrix
Throws:
NullPointerException - unless method newSampleData 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 R-squared computation.

Returns:
SSTO - the total sum of squares
Throws:
NullPointerException - if the sample has not been set
AbstractMultipleLinearRegression.isNoIntercept()
• #### calculateResidualSumOfSquares

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

public double calculateRSquared()
Returns the R-Squared statistic, defined by the formula $$R^2 = 1 - \frac{\mathrm{SSR}}{\mathrm{SSTO}}$$ where 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.

Returns:
R-square statistic
Throws:
NullPointerException - if the sample has not been set
MathIllegalArgumentException - if the design matrix is singular

public double calculateAdjustedRSquared()

Returns the adjusted R-squared 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 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.

Returns:
Throws:
NullPointerException - if the sample has not been set
MathIllegalArgumentException - if the design matrix is singular
AbstractMultipleLinearRegression.isNoIntercept()
• #### 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, 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  x
----------
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 class AbstractMultipleLinearRegression
Parameters:
x - the rectangular array representing the x sample
• #### calculateBeta

protected RealVector 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 a NullPointerException will be thrown.

Specified by:
calculateBeta in class AbstractMultipleLinearRegression
Returns:
beta
Throws:
MathIllegalArgumentException - if the design matrix is singular
NullPointerException - if the data for the model have not been loaded
• #### calculateBetaVariance

protected 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.

Specified by:
calculateBetaVariance in class AbstractMultipleLinearRegression
Returns:
The beta variance-covariance matrix
Throws:
MathIllegalArgumentException - if the design matrix is singular
NullPointerException - if the data for the model have not been loaded