org.apache.commons.math.distribution.TDistributionImpl Java Examples

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * tDistribution.cumulativeProbability(-t);
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Removes the observation (x,y) from the regression data set.
 * <p>
 * Mirrors the addData method.  This method permits the use of
 * SimpleRegression instances in streaming mode where the regression
 * is applied to a sliding "window" of observations, however the caller is
 * responsible for maintaining the set of observations in the window.</p>
 *
 * The method has no effect if there are no points of data (i.e. n=0)
 *
 * @param x independent variable value
 * @param y dependent variable value
 */
public void removeData(double x, double y) {
    if (n > 0) {
        double dx = x - xbar;
        double dy = y - ybar;
        sumXX -= dx * dx * (double) n / (n - 1d);
        sumYY -= dy * dy * (double) n / (n - 1d);
        sumXY -= dx * dy * (double) n / (n - 1d);
        xbar -= dx / (n - 1.0);
        ybar -= dy / (n - 1.0);
        sumX -= x;
        sumY -= y;
        n--;

        if (n > 2) {
            distribution = new TDistributionImpl(n - 2);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = FastMath.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = FastMath.abs(r * FastMath.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * tDistribution.cumulativeProbability(-t);
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Adds the observation (x,y) to the regression data set.
 * <p>
 * Uses updating formulas for means and sums of squares defined in
 * "Algorithms for Computing the Sample Variance: Analysis and
 * Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J.
 * 1983, American Statistician, vol. 37, pp. 242-247, referenced in
 * Weisberg, S. "Applied Linear Regression". 2nd Ed. 1985.</p>
 *
 *
 * @param x independent variable value
 * @param y dependent variable value
 */
public void addData(double x, double y) {
    if (n == 0) {
        xbar = x;
        ybar = y;
    } else {
        double dx = x - xbar;
        double dy = y - ybar;
        sumXX += dx * dx * (double) n / (n + 1d);
        sumYY += dy * dy * (double) n / (n + 1d);
        sumXY += dx * dy * (double) n / (n + 1d);
        xbar += dx / (n + 1.0);
        ybar += dy / (n + 1.0);
    }
    sumX += x;
    sumY += y;
    n++;

    if (n > 2) {
        distribution = new TDistributionImpl(n - 2);
    }
}
 

/**
 * Removes the observation (x,y) from the regression data set.
 * <p>
 * Mirrors the addData method.  This method permits the use of
 * SimpleRegression instances in streaming mode where the regression
 * is applied to a sliding "window" of observations, however the caller is
 * responsible for maintaining the set of observations in the window.</p>
 *
 * The method has no effect if there are no points of data (i.e. n=0)
 *
 * @param x independent variable value
 * @param y dependent variable value
 */
public void removeData(double x, double y) {
    if (n > 0) {
        double dx = x - xbar;
        double dy = y - ybar;
        sumXX -= dx * dx * (double) n / (n - 1d);
        sumYY -= dy * dy * (double) n / (n - 1d);
        sumXY -= dx * dy * (double) n / (n - 1d);
        xbar -= dx / (n - 1.0);
        ybar -= dy / (n - 1.0);
        sumX -= x;
        sumY -= y;
        n--;

        if (n > 2) {
            distribution = new TDistributionImpl(n - 2);
        }
    }
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
@Test
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = FastMath.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            Assert.assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the 
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 * 
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix); 
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 *
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = FastMath.abs(r * FastMath.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * tDistribution.cumulativeProbability(-t);
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Adds the observation (x,y) to the regression data set.
 * <p>
 * Uses updating formulas for means and sums of squares defined in
 * "Algorithms for Computing the Sample Variance: Analysis and
 * Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J.
 * 1983, American Statistician, vol. 37, pp. 242-247, referenced in
 * Weisberg, S. "Applied Linear Regression". 2nd Ed. 1985.</p>
 *
 *
 * @param x independent variable value
 * @param y dependent variable value
 */
public void addData(double x, double y) {
    if (n == 0) {
        xbar = x;
        ybar = y;
    } else {
        double dx = x - xbar;
        double dy = y - ybar;
        sumXX += dx * dx * (double) n / (n + 1d);
        sumYY += dy * dy * (double) n / (n + 1d);
        sumXY += dx * dy * (double) n / (n + 1d);
        xbar += dx / (n + 1.0);
        ybar += dy / (n + 1.0);
    }
    sumX += x;
    sumY += y;
    n++;

    if (n > 2) {
        distribution = new TDistributionImpl(n - 2);
    }
}
 

/**
 * Removes the observation (x,y) from the regression data set.
 * <p>
 * Mirrors the addData method.  This method permits the use of
 * SimpleRegression instances in streaming mode where the regression
 * is applied to a sliding "window" of observations, however the caller is
 * responsible for maintaining the set of observations in the window.</p>
 *
 * The method has no effect if there are no points of data (i.e. n=0)
 *
 * @param x independent variable value
 * @param y dependent variable value
 */
public void removeData(double x, double y) {
    if (n > 0) {
        double dx = x - xbar;
        double dy = y - ybar;
        sumXX -= dx * dx * (double) n / (n - 1d);
        sumYY -= dy * dy * (double) n / (n - 1d);
        sumXY -= dx * dy * (double) n / (n - 1d);
        xbar -= dx / (n - 1.0);
        ybar -= dy / (n - 1.0);
        sumX -= x;
        sumY -= y;
        n--;

        if (n > 2) {
            distribution = new TDistributionImpl(n - 2);
        }
    }
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
@Test
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix);
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = FastMath.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            Assert.assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Returns a matrix of p-values associated with the (two-sided) null
 * hypothesis that the corresponding correlation coefficient is zero.
 * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
 * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
 * a value with absolute value greater than or equal to <br>
 * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
 * <p>The values in the matrix are sometimes referred to as the 
 * <i>significance</i> of the corresponding correlation coefficients.</p>
 * 
 * @return matrix of p-values
 * @throws MathException if an error occurs estimating probabilities
 */
public RealMatrix getCorrelationPValues() throws MathException {
    TDistribution tDistribution = new TDistributionImpl(nObs - 2);
    int nVars = correlationMatrix.getColumnDimension();
    double[][] out = new double[nVars][nVars];
    for (int i = 0; i < nVars; i++) {
        for (int j = 0; j < nVars; j++) {
            if (i == j) {
                out[i][j] = 0d;
            } else {
                double r = correlationMatrix.getEntry(i, j);
                double t = Math.abs(r * Math.sqrt((nObs - 2)/(1 - r * r)));
                out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
            }
        }
    }
    return new BlockRealMatrix(out);
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix); 
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}
 

/**
 * Verify that direct t-tests using standard error estimates are consistent
 * with reported p-values
 */
public void testStdErrorConsistency() throws Exception {
    TDistribution tDistribution = new TDistributionImpl(45);
    RealMatrix matrix = createRealMatrix(swissData, 47, 5);
    PearsonsCorrelation corrInstance = new PearsonsCorrelation(matrix); 
    RealMatrix rValues = corrInstance.getCorrelationMatrix();
    RealMatrix pValues = corrInstance.getCorrelationPValues();
    RealMatrix stdErrors = corrInstance.getCorrelationStandardErrors();
    for (int i = 0; i < 5; i++) {
        for (int j = 0; j < i; j++) {
            double t = Math.abs(rValues.getEntry(i, j)) / stdErrors.getEntry(i, j);
            double p = 2 * (1 - tDistribution.cumulativeProbability(t));
            assertEquals(p, pValues.getEntry(i, j), 10E-15);
        }
    }
}