Package com.imsl.math

Class BsLeastSquares

java.lang.Object
com.imsl.math.BSpline
com.imsl.math.BsLeastSquares
All Implemented Interfaces:
Serializable, Cloneable
public class BsLeastSquares extends BSpline
Extension of the BSpline class to compute a least squares spline approximation to data points.

Let's make the identifications

n = xData.length

x = xData

f = yData

m = nCoef

k = order

For convenience, we assume that the sequence x is increasing, although the class does not require this.

By default, k = 4, and the knot sequence we select equally distributes the knots through the distinct \({x_i}'s\). In particular, the m + k knots will be generated in \([x_1, x_n]\) with k knots stacked at each of the extreme values. The interior knots will be equally spaced in the interval.

Once knots \({\bf t}\) and weights w are determined, then the spline least-squares fit to the data is computed by minimizing over the linear coefficients \(a_j\)

$$\sum_{i=0}^{n-1} {w_i\biggl[f_i-\sum_{j=1}^{m}{a_jB_j(x_i)}\biggr]^2}$$

where the \(B_j, j = 1, ..., m\) are a (B-spline) basis for the spline subspace.

This algorithm is based on the routine L2APPR by deBoor (1978, p. 255).

See Also:

  • Field Details

    • nCoef

      protected int nCoef
      Number of B-spline coefficients.

    • weight

      protected double[] weight
      The weight array of length n, where n is the number of data points fit.

  • Constructor Details

    • BsLeastSquares

      public BsLeastSquares(double[] xData, double[] yData, int nCoef)
      Constructs a least squares B-spline approximation to the given data points.
      Parameters:
      xData - A double array containing the x-coordinates of the data.
      yData - A double array containing the y-coordinates of the data. The arrays xData and yData must have the same length.
      nCoef - An int denoting the linear dimension of the spline subspace. It should be smaller than the number of data points and greater than or equal to the order of the spline (whose default value is 4).

    • BsLeastSquares

      public BsLeastSquares(double[] xData, double[] yData, int nCoef, int order)
      Constructs a least squares B-spline approximation to the given data points.
      Parameters:
      xData - A double array containing the x-coordinates of the data.
      yData - A double array containing the y-coordinates of the data. The arrays xData and yData must have the same length.
      nCoef - An int denoting the linear dimension of the spline subspace. It should be smaller than the number of data points and greater than or equal to the order of the spline.
      order - An int denoting the order of the spline.

    • BsLeastSquares

      public BsLeastSquares(double[] xData, double[] yData, int nCoef, int order, double[] weight, double[] knot)
      Constructs a least squares B-spline approximation to the given data points.
      Parameters:
      xData - A double array containing the x-coordinates of the data.
      yData - A double array containing the y-coordinates of the data. The arrays xData and yData must have the same length.
      nCoef - An int denoting the linear dimension of the spline subspace. It should be smaller than the number of data points and greater than or equal to the order of the spline.
      order - An int denoting the order of the spline.
      weight - A double array containing the weights for the data. The arrays xData, yData and weights must have the same length.
      knot - A double array containing the knot sequence for the spline.