OLSMultipleLinearRegression (Apache Commons Math 3.3 API)

org.apache.commons.math3.stat.regression

## Class OLSMultipleLinearRegression

- java.lang.Object
- - org.apache.commons.math3.stat.regression.AbstractMultipleLinearRegression
  - - org.apache.commons.math3.stat.regression.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.

  Since:
    2.0
  Version:
    $Id: OLSMultipleLinearRegression.java 1591624 2014-05-01 11:54:06Z tn $

- - ### Constructor Summary

    Constructors
    Constructor and 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

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `double` | `calculateAdjustedRSquared()` Returns the adjusted R-squared statistic, defined by the formula
    `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
    `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.apache.commons.math3.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
      Since:
        3.3

  - ### 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[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 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()
                                            throws MathIllegalArgumentException

      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:
        `MathIllegalArgumentException` - if the sample has not been set or does not contain at least 3 observations
      Since:
        2.2
      See Also:
        `AbstractMultipleLinearRegression.isNoIntercept()`

    - #### calculateResidualSumOfSquares

          public double calculateResidualSumOfSquares()

      Returns the sum of squared residuals.
      Returns:
        residual sum of squares
      Since:
        2.2

    - #### calculateRSquared

          public double calculateRSquared()
                                   throws MathIllegalArgumentException

      Returns the R-Squared statistic, defined by the formula

           R2 = 1 - SSR / SSTO

      where SSR is the `sum of squared residuals` and SSTO is the `total sum of squares`
      Returns:
        R-square statistic
      Throws:
        `MathIllegalArgumentException` - if the sample has not been set or does not contain at least 3 observations
      Since:
        2.2

    - #### calculateAdjustedRSquared

          public double calculateAdjustedRSquared()
                                           throws MathIllegalArgumentException

      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))

      Returns:
        adjusted R-Squared statistic
      Throws:
        `MathIllegalArgumentException` - if the sample has not been set or does not contain at least 3 observations
      Since:
        2.2
      See Also:
        `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[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 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

    - #### 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

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