SmoothingCubicSpline (Java Libraries for Stochastic Simulation)

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

All Implemented Interfaces:

MathFunction, MathFunctionWithDerivative, MathFunctionWithFirstDerivative, MathFunctionWithIntegral
```
public class SmoothingCubicSpline
extends java.lang.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

Constructors
Constructor and Description
`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

Methods
Modifier and Type	Method and Description
`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 x_i coordinates for this spline.
`double[]`	`getY()` Returns the y_i 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:
    
    java.lang.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:
    
    java.lang.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 methodgetSplinePolynomials

Class SmoothingCubicSpline

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

SmoothingCubicSpline

SmoothingCubicSpline

Method Detail

evaluate

integral

derivative

derivative

getX

getY

getWeights

getRho

getSplinePolynomials

getFitPolynomialIndex