umontreal.iro.lecuyer.functionfit
Class LeastSquares

java.lang.Object
  umontreal.iro.lecuyer.functionfit.LeastSquares

public class LeastSquares
extends Object

This class implements different linear regression models, using the least squares method to estimate the regression coefficients. Given input data x_ij and response y_i, one want to find the coefficients β_j that minimize the residuals of the form (using matrix notation)

r = min_β| Y - Xβ|₂,

where the L₂ norm is used. Particular cases are

r = min_β∑_i(y_i - β₀ - ∑_j=1^kβ_jx_ij)².

for k regressor variables x_j. The well-known case of the single variable x is

r = min_{α, β}∑_i(y_i - α - βx_i)².

Sometimes, one wants to use a basis of general functions ψ_j(t) with a minimization of the form

r = min_β∑_i(y_i - ∑_j=1^kβ_jψ_j(t_i))².

For example, we could have ψ_j(t) = e^-λ_jt or some other functions. In that case, one has to choose the points t_i at which to compute the basis functions, and use a method below with x_ij = ψ_j(t_i).

Constructor Summary

LeastSquares()

Method Summary

static double[]	calcCoefficients(double[][] X, double[] Y) Computes the regression coefficients using the least squares method.
static double[]	calcCoefficients(double[] X, double[] Y) Computes the regression coefficients using the least squares method.
static double[]	calcCoefficients(double[] X, double[] Y, int deg) Computes the regression coefficients using the least squares method.
static double[]	calcCoefficients0(double[][] X, double[] Y) Computes the regression coefficients using the least squares method.

Methods inherited from class java.lang.Object

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

Constructor Detail

LeastSquares

public LeastSquares()

Method Detail

calcCoefficients

public static double[] calcCoefficients(double[] X,
                                        double[] Y)

Computes the regression coefficients using the least squares method. This is a simple linear regression with 2 regression coefficients, α and β. The model is

y = α + βx.

Given the n data points (X_i, Y_i), i = 0, 1,…,(n - 1), the method computes and returns the array [α, β].

Parameters:

X - the regressor variables

Y - the response

Returns:

the regression coefficients

calcCoefficients

public static double[] calcCoefficients(double[] X,
                                        double[] Y,
                                        int deg)

Computes the regression coefficients using the least squares method. This is a linear regression with a polynomial of degree deg = k and k + 1 regression coefficients β_j. The model is

y = β₀ + ∑_j=1^kβ_jx^j.

Given the n data points (X_i, Y_i), i = 0, 1,…,(n - 1), the method computes and returns the array [β₀, β₁,…, β_k]. Restriction: n > k.

Parameters:

X - the regressor variables

Y - the response

Returns:

the regression coefficients

calcCoefficients0

public static double[] calcCoefficients0(double[][] X,
                                         double[] Y)

Computes the regression coefficients using the least squares method. This is a model for multiple linear regression. There are k + 1 regression coefficients β_j, and k regressors variables x_j. The model is

y = β₀ + ∑_j=1^kβ_jx_j.

There are n data points Y_i, X_ij, i = 0, 1,…,(n - 1), and each X_i is a k-dimensional point. Given the response Y[i] and the regressor variables X[i][j], i = 0, 1,…,(n - 1), j = 0, 1,…,(k - 1), the method computes and returns the array [β₀, β₁,…, β_k]. Restriction: n > k + 1.

Parameters:

X - the regressor variables

Y - the response

Returns:

the regression coefficients

calcCoefficients

public static double[] calcCoefficients(double[][] X,
                                        double[] Y)

Computes the regression coefficients using the least squares method. This is a model for multiple linear regression. There are k regression coefficients β_j, j = 0, 1,…,(k - 1) and k regressors variables x_j. The model is

y = ∑_j=0^k-1β_jx_j.

There are n data points Y_i, X_ij, i = 0, 1,…,(n - 1), and each X_i is a k-dimensional point. Given the response Y[i] and the regressor variables X[i][j], i = 0, 1,…,(n - 1), j = 0, 1,…,(k - 1), the method computes and returns the array [β₀, β₁,…, β_k-1]. Restriction: n > k.

Parameters:

X - the regressor variables

Y - the response

Returns:

the regression coefficients