umontreal.iro.lecuyer.functionfit
Class SmoothingCubicSpline

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

All Implemented Interfaces:

MathFunction, MathFunctionWithDerivative, MathFunctionWithFirstDerivative, MathFunctionWithIntegral

public class SmoothingCubicSpline
extends Object
implements MathFunction, MathFunctionWithFirstDerivative, MathFunctionWithDerivative, MathFunctionWithIntegral

Represents a cubic spline with nodes at (x_i, y_i) computed with the smoothing cubic spline algorithm of Schoenberg. A smoothing cubic spline is made of n + 1 cubic polynomials. The ith polynomial of such a spline, for i = 1,…, n - 1, is defined as S_i(x) while the complete spline is defined as

S(x) = S_i(x), for x∈[x_i-1, x_i].

For x < x₀ and x > x_n-1, the spline is not precisely defined, but this class performs extrapolation by using S₀ and S_n linear polynomials. The algorithm which calculates the smoothing spline is a generalization of the algorithm for an interpolating spline. S_i is linked to S_i+1 at x_i+1 and keeps continuity properties for first and second derivatives at this point, therefore S_i(x_i+1) = S_i+1(x_i+1), S'_i(x_i+1) = S'_i+1(x_i+1) and S''_i(x_i+1) = S''_i+1(x_i+1).

The spline is computed with a smoothing parameter ρ∈[0, 1] which represents its accuracy with respect to the initial (x_i, y_i) nodes. The smoothing spline minimizes

L = ρ∑_i=0^n-1w_i(y_i-S_i(x_i))² + (1 - ρ)∫_x0^x_n-1(S''(x))²dx

In fact, by setting ρ = 1, we obtain the interpolating spline; and we obtain a linear function by setting ρ = 0. The weights w_i > 0, which default to 1, can be used to change the contribution of each point in the error term. A large value w_i will give a large weight to the ith point, so the spline will pass closer to it. Here is a small example that uses smoothing splines:

int n;
double[] X = new double[n];
double[] Y = new double[n];
// here, fill arrays X and Y with n data points (x_i, y_i)
// The points must be sorted with respect to x_i.

double rho = 0.1;
SmoothingCubicSpline fit = new SmoothingCubicSpline(X, Y, rho);

int m = 40;
double[] Xp = new double[m+1]; // Xp, Yp are spline points
double[] Yp = new double[m+1];
double h = (X[n-1] - X[0]) / m; // step

for (int i = 0; i <= m; i++) {
double z = X[0] + i * h;
Xp[i] = z;
Yp[i] = fit.evaluate(z); // evaluate spline at z
}

Constructor Summary

SmoothingCubicSpline(double[] x, double[] y, double rho)
Constructs a spline with nodes at (x_i, y_i), with weights = 1 and smoothing factor ρ = rho.

SmoothingCubicSpline(double[] x, double[] y, double[] w, double rho)
Constructs a spline with nodes at (x_i, y_i), with weights w_i and smoothing factor ρ = rho.

Method Summary

double	derivative(double z) Evaluates and returns the value of the first derivative of the spline at z.
double	derivative(double z, int n) Evaluates and returns the value of the n-th derivative of the spline at z.
double	evaluate(double z) Evaluates and returns the value of the spline at z.
int	getFitPolynomialIndex(double x) Returns the index of P, the Polynomial instance used to evaluate x, in an ArrayList table instance returned by getSplinePolynomials().
double	getRho() Returns the smoothing factor used to construct the spline.
Polynomial[]	getSplinePolynomials() Returns a table containing all fitting polynomials.
double[]	getWeights() Returns the weights of the points.
double[]	getX() Returns the xi coordinates for this spline.
double[]	getY() Returns the yi coordinates for this spline.
double	integral(double a, double b) Evaluates and returns the value of the integral of the spline from a to b.

Methods inherited from class java.lang.Object

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

Constructor Detail

SmoothingCubicSpline

public SmoothingCubicSpline(double[] x,
                            double[] y,
                            double[] w,
                            double rho)

Constructs a spline with nodes at (x_i, y_i), with weights w_i and smoothing factor ρ = rho. The x_i must be sorted in increasing order.

Parameters:

x - the x_i coordinates.

y - the y_i coordinates.

w - the weight for each point, must be > 0.

rho - the smoothing parameter

Throws:

IllegalArgumentException - if x, y and z do not have the same length, if rho has wrong value, or if the spline cannot be calculated.

SmoothingCubicSpline

public SmoothingCubicSpline(double[] x,
                            double[] y,
                            double rho)

Constructs a spline with nodes at (x_i, y_i), with weights = 1 and smoothing factor ρ = rho. The x_i must be sorted in increasing order.

Parameters:

x - the x_i coordinates.

y - the y_i coordinates.

rho - the smoothing parameter

Throws:

IllegalArgumentException - if x and y do not have the same length, if rho has wrong value, or if the spline cannot be calculated.

Method Detail

evaluate

public double evaluate(double z)

Evaluates and returns the value of the spline at z.

Specified by:

evaluate in interface MathFunction

Parameters:

z - argument of the spline.

Returns:

value of spline.

integral

public double integral(double a,
                       double b)

Evaluates and returns the value of the integral of the spline from a to b.

Specified by:

integral in interface MathFunctionWithIntegral

Parameters:

a - lower limit of integral.

b - upper limit of integral.

Returns:

value of integral.

derivative

public double derivative(double z)

Evaluates and returns the value of the first derivative of the spline at z.

Specified by:

derivative in interface MathFunctionWithFirstDerivative

Parameters:

z - argument of the spline.

Returns:

value of first derivative.

derivative

public double derivative(double z,
                         int n)

Evaluates and returns the value of the n-th derivative of the spline at z.

Specified by:

derivative in interface MathFunctionWithDerivative

Parameters:

z - argument of the spline.

n - order of the derivative.

Returns:

value of n-th derivative.

getX

public double[] getX()

Returns the x_i coordinates for this spline.

Returns:

the x_i coordinates.

getY

public double[] getY()

Returns the y_i coordinates for this spline.

Returns:

the y_i coordinates.

getWeights

public double[] getWeights()

Returns the weights of the points.

Returns:

the weights of the points.

getRho

public double getRho()

Returns the smoothing factor used to construct the spline.

Returns:

the smoothing factor.

getSplinePolynomials

public Polynomial[] getSplinePolynomials()

Returns a table containing all fitting polynomials.

Returns:

Table containing the fitting polynomials.

getFitPolynomialIndex

public int getFitPolynomialIndex(double x)

Returns the index of P, the Polynomial instance used to evaluate x, in an ArrayList table instance returned by getSplinePolynomials(). This index k gives also the interval in table X which contains the value x (i.e. such that x_k < x <= x_k+1).

Returns:

Index of the polynomial check with x in the Polynomial list returned by getSplinePolynomials