Author: psteitz
Date: Sun Jan 4 10:38:29 2009
New Revision: 731320
URL: http://svn.apache.org/viewvc?rev=731320&view=rev
Log:
Javadoc only. Cleanup formatting.
Modified:
commons/proper/math/trunk/src/java/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.java
Modified:
commons/proper/math/trunk/src/java/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.java
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/java/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.java?rev=731320&r1=731319&r2=731320&view=diff
==============================================================================
---
commons/proper/math/trunk/src/java/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.java
(original)
+++
commons/proper/math/trunk/src/java/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.java
Sun Jan 4 10:38:29 2009
@@ -27,27 +27,27 @@
* multiple linear regression model.</p>
*
* <p>OLS assumes the covariance matrix of the error to be diagonal and with
- * equal variance.
- * <pre>
- * u ~ N(0, sigma^2*I)
- * </pre></p>
+ * equal variance.</p>
+ * <p>
+ * u ~ N(0, σ<sup>2</sup>I)
+ * </p>
*
* <p>The regression coefficients, b, satisfy the normal equations:
- * <pre>
- * X^T X b = X^T y
- * </pre></p>
+ * <p>
+ * X<sup>T</sup> X b = X<sup>T</sup> y
+ * </p>
*
* <p>To solve the normal equations, this implementation uses QR decomposition
* of the X matrix. (See {...@link QRDecompositionImpl} for details on the
* decomposition algorithm.)
- * <pre>
- * 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
- * </pre>
+ * </p>
+ * <p>X<sup>T</sup>X b = X<sup>T</sup> y <br/>
+ * (QR)<sup>T</sup> (QR) b = (QR)<sup>T</sup>y <br/>
+ * R<sup>T</sup> (Q<sup>T</sup>Q) R b = R<sup>T</sup> Q<sup>T</sup> y <br/>
+ * R<sup>T</sup> R b = R<sup>T</sup> Q<sup>T</sup> y <br/>
+ * (R<sup>T</sup>)<sup>-1</sup> R<sup>T</sup> R b =
(R<sup>T</sup>)<sup>-1</sup> R<sup>T</sup> Q<sup>T</sup> y <br/>
+ * R b = Q<sup>T</sup> y
+ * </p>
* Given Q and R, the last equation is solved by back-subsitution.</p>
*
* @version $Revision$ $Date$
@@ -83,13 +83,14 @@
* <p>Compute the "hat" matrix.
* </p>
* <p>The hat matrix is defined in terms of the design matrix X
- * by X(X^TX)^-1X^T
- * <p>
+ * by X(X<sup>T</sup>X)<sup>-1</sup>X<sup>T</sup>
+ * </p>
* <p>The implementation here uses the QR decomposition to compute the
- * hat matrix as QIpQ^T 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.
+ * hat matrix as Q I<sub>p</sub>Q<sup>T</sup> where I<sub>p</sub> 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,
+ * <i>The American Statistician</i>, Vol. 32, No. 1 (Feb., 1978), pp.
17-22.
*
* @return the hat matrix
*/
@@ -134,10 +135,11 @@
}
/**
- * Calculates the variance on the beta by OLS.
- * <pre>
- * Var(b)=(X'X)^-1
- * </pre>
+ * <p>Calculates the variance on the beta by OLS.
+ * </p>
+ * <p>Var(b) = (X<sup>T</sup>X)<sup>-1</sup>
+ * </p>
+ *
* @return The beta variance
*/
protected RealMatrix calculateBetaVariance() {
@@ -147,10 +149,10 @@
/**
- * Calculates the variance on the Y by OLS.
- * <pre>
- * Var(y)=Tr(u'u)/(n-k)
- * </pre>
+ * <p>Calculates the variance on the Y by OLS.
+ * </p>
+ * <p> Var(y) = Tr(u<sup>T</sup>u)/(n - k)
+ * </p>
* @return The Y variance
*/
protected double calculateYVariance() {
