com.numericalmethod.suanshu.matrix.doubles.linearsystem

Class OLSSolverByQR

java.lang.Object

com.numericalmethod.suanshu.matrix.doubles.linearsystem.OLSSolverByQR

Direct Known Subclasses:

OLSSolver

public class OLSSolverByQR
extends java.lang.Object

This class solves an over-determined system of linear equations in the ordinary least square sense. An over-determined system, represented by

Ax = y

has more rows than columns. That is, there are more equations than unknowns. One important application is linear regression, where A is the independent factors, y the dependent observations. The solution x^ minimizes:

|Ax - y|₂

That is, x^ is the best approximation that minimizes the sum of squared differences between the data values and their corresponding modeled values. The approach is called "linear" least squares since the solution depends linearly on the data.

x^ = (A^tA)^-1A^ty,

This implementation does not use the above formula to solve for x^ because of the numerical stability problem in computing A^tA. Instead, we use QR decomposition, an orthogonal decomposition method that is numerically stable.

See Also:

• Wikipedia: Linear least squares
• Wikipedia: Least squares
• Wikipedia: Orthogonal decomposition methods

Constructor Summary

Constructors

Constructor and Description
OLSSolverByQR(double epsilon) Construct an OLS solver for an over-determined system of linear equations.

Method Summary

Modifier and Type	Method and Description
Vector	solve(LSProblem problem) In the ordinary least square sense, solve Ax = y

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

OLSSolverByQR

public OLSSolverByQR(double epsilon)

Construct an OLS solver for an over-determined system of linear equations.

Parameters:

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

Method Detail

solve

public Vector solve(LSProblem problem)

In the ordinary least square sense, solve

Ax = y

Parameters:

problem - a system of linear equations

Returns:

a solution x^ that minimizes

|Ax - y|₂