com.numericalmethod.suanshu.matrix.doubles.linearsystem
Class OLSSolver

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.linearsystem.OLSSolver

public class OLSSolver
extends java.lang.Object

This class solves a system of over-determined linear equations in the ordinary least square sense.

An over-determined system, which is represented by

Ax = y

has more rows than columns. That is, there are more equations than unknowns. One application of this is linear regression, where A is the independent factors of a data set. y is the dependent observation of the data set.

The solution x_hat is such that

|| Ax - y ||2

is minimized. That is, it 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,

x_hat = (AtA)-1Aty,

depends linearly on the data.

Our implementations do not use the above formula to solve for x_hat because of the numerical stability problem in computing AtA. Instead, we use the orthogonal decomposition methods.

See Also:

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

Nested Class Summary

static class

OLSSolver.Method the methods available to solve an Ordinary Least Square problem

Field Summary

double

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

Constructor Summary

OLSSolver(Matrix A)
Construct an OLS solver for a matrix A, where Ax = y.

OLSSolver(Matrix A, OLSSolver.Method method, double epsilon)
Construct an OLS solver for a matrix A, where Ax = y

Method Summary

Vector

solve(Vector y) 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

Field Detail

epsilon

public final double epsilon

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

Constructor Detail

OLSSolver

public OLSSolver(Matrix A,
                 OLSSolver.Method method,
                 double epsilon)

Construct an OLS solver for a matrix A, where

Ax = y

Parameters:

A - a matrix

method - one of the OLSSolver.Method to use

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

OLSSolver

public OLSSolver(Matrix A)

Construct an OLS solver for a matrix A, where

Ax = y

.

Parameters:

A - a matrix

Method Detail

solve

public Vector solve(Vector y)

In the ordinary least square sense, solve

Ax = y

.

Parameters:

y - a vector

Returns:

a solution x_hat such that

|| Ax - y ||2

is minimized.