jhplot.stat

Class Interpolator

java.lang.Object

jhplot.stat.Interpolator

public class Interpolator
extends java.lang.Object

Perform interpolation of a data using different algorithm. This class is useful for data smoothing and determination of background levels. Weigths for each data points can be given.

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
class	Interpolator.Cubic represents a cubic polynomial.

Constructor Summary

Constructors

Constructor and Description
Interpolator(double[] aX, double[] aY) Initialize interpolator
Interpolator(H1D h1) Initialize interpolator for a histogram.
Interpolator(P1D p1d) Initialize interpolator for P1D array.

Method Summary

Modifier and Type	Method and Description
double	getMaxValue() Get min value of data in X
double	getMinValue() Get maximum value of data in X
SmoothingCubicSpline	interpolateCubicSpline(double rho, int option) Calculate a spline with nodes at (x, y), with weights w and smoothing factor rho.
PolynomialSplineFunction	interpolateLoess(double bandwidth, int robustnessIters, double accuracy, int option) Performs a Local Regression Algorithm (also Loess, Lowess) for interpolation of data points.
Interpolator.Cubic[]	interpolateNatuarlCubicSpline() Natural cubic splines interpolation.
void	setRange(double xMin, double xMax) Redefine smooth interval for input data.
P1D	smoothAverage(boolean isWeighted, int k) Smooth data by averaging over a moving window.
P1D	smoothCubicSpline() Perform a cubic interpolatory spline.
P1D	smoothGauss(double standardDeviation) Computes a Gaussian smoothed version of data.
P1D	smoothLoess(double bandwidth, int robustnessIters, double accuracy) Performs a Local Regression Algorithm (also Loess, Lowess) for interpolation of data points.
P1D	smoothLoess(double bandwidth, int robustnessIters, double accuracy, int option) Performs a Local Regression Algorithm (also Loess, Lowess) for interpolation of data points.
P1D	smoothSpline() Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.

Methods inherited from class java.lang.Object

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

Constructor Detail

Interpolator

public Interpolator(double[] aX,
                    double[] aY)

Initialize interpolator

Parameters:

aX - is the array of x data

aY - is the array of y data

Interpolator

public Interpolator(P1D p1d)

Initialize interpolator for P1D array. Only X and Y values are taken. Errors are ignored. One can include weigths using 1/(errors*errors) as given on the Y-axis. (can be set with add(x,y,error) method of P1D.

Parameters:

p1d - p1d values x-y and y-error is used to estimate weight (1/error*error)

Interpolator

public Interpolator(H1D h1)

Initialize interpolator for a histogram. Errors on the histogram heights are considered as weights 1/(error*error)

Parameters:

h1 - input for interpolation. X - mean value in a bin, y is a height and weight is 1/(error*error)

Method Detail

setRange

public void setRange(double xMin,
                     double xMax)

Redefine smooth interval for input data.

Parameters:

xMin - minimal value for X

xMax - maximal value for Y

interpolateLoess

public PolynomialSplineFunction interpolateLoess(double bandwidth,
                                                 int robustnessIters,
                                                 double accuracy,
                                                 int option)

Performs a Local Regression Algorithm (also Loess, Lowess) for interpolation of data points. For reference, see William S. Cleveland - Robust Locally Weighted Regression and Smoothing Scatterplots

Compute an interpolating function by performing a loess fit on the data at the original abscissa and then building a cubic spline. Weights of the input data are not taken into account.

Parameters:

bandwidth - - when computing the loess fit at a particular point, this fraction of source points closest to the current point is taken into account for computing a least-squares regression. A sensible value is usually 0.25 to 0.5.

robustnessIters - How many robustness iterations are done. A sensible value is usually 0 (just the initial fit without any robustness iterations) to 4.

Returns:

polynomial functions.

smoothLoess

public P1D smoothLoess(double bandwidth,
                       int robustnessIters,
                       double accuracy,
                       int option)

Performs a Local Regression Algorithm (also Loess, Lowess) for interpolation of data points. For reference, see William S. Cleveland - Robust Locally Weighted Regression and Smoothing Scatterplots. This method computes a weighted loess fit on the data at the original abscissa.

Compute an interpolating function by performing a loess fit on the data at the original abscissa and then building a cubic spline.

Parameters:

bandwidth - - when computing the loess fit at a particular point, this fraction of source points closest to the current point is taken into account for computing a least-squares regression. A sensible value is usually 0.25 to 0.5.

robustnessIters - How many robustness iterations are done. A sensible value is usually 0 (just the initial fit without any robustness iterations) to 4.

accuracy - If the median residual at a certain robustness iteration is less than this amount, no more iterations are done. A typical value is 2

option - treatment of errors on data points:
option=0. Errors on X and Y are ignored and weights=1 for all points
option=1. Errors on Y are used to calculate weights as 1/errorY, where errorY is error on Y value. Errors on X are ignored. This option works for histograms where error on heights can be used to determine the weight of each data point, while X-values do not have errors.
option=2. Same as "1", but weights are 1/(errorY*errorY), where errorY is error on Y values
option=3. Same as "1", but weights are given as 1/(errorY*errorX). This works best for P1D which contains errors on X and Y.
option=4. same as "2", but weights on X and Y are calculated as 1/(errorX*errorX*errorY*errorY)
option=5. weights for points are given by errorY
option=6. weights for points are given by errorY*errorX

Returns:

P1D with smoothed result. Abscissa is the same as in the original input.

interpolateCubicSpline

public SmoothingCubicSpline interpolateCubicSpline(double rho,
                                                   int option)

Calculate a spline with nodes at (x, y), with weights w and smoothing factor rho. 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.

Parameters:

rho - smoothing factor rho.

option - treatment of errors on data points:
option=0. Errors on X and Y are ignored and weights=1 for all points
option=1. Errors on Y are used to calculate weights as 1/errorY, where errorY is error on Y value. Errors on X are ignored. This option works for histograms where error on heights can be used to determine the weight of each data point, while X-values do not have errors.
option=2. Same as "1", but weights are 1/(errorY*errorY), where errorY is error on Y values
option=3. Same as "1", but weights are given as 1/(errorY*errorX). This works best for P1D which contains errors on X and Y.
option=4. same as "2", but weights on X and Y are calculated as 1/(errorX*errorX*errorY*errorY)
option=5. weights for points are given by errorY
option=6. weights for points are given by errorY*errorX

Returns:

SmoothingCubicSpline

smoothLoess

public P1D smoothLoess(double bandwidth,
                       int robustnessIters,
                       double accuracy)

Performs a Local Regression Algorithm (also Loess, Lowess) for interpolation of data points. For reference, see William S. Cleveland - Robust Locally Weighted Regression and Smoothing Scatterplots. This method computes a weighted loess fit on the data at the original abscissa.

Compute an interpolating function by performing a loess fit on the data at the original abscissa and then building a cubic spline. If errors on Y are given (from a histogram or P1D), assume the weighths equal 1/(errorY*errorY).

Parameters:

bandwidth - - when computing the loess fit at a particular point, this fraction of source points closest to the current point is taken into account for computing a least-squares regression. A sensible value is usually 0.25 to 0.5.

robustnessIters - How many robustness iterations are done. A sensible value is usually 0 (just the initial fit without any robustness iterations) to 4.

accuracy - If the median residual at a certain robustness iteration is less than this amount, no more iterations are done.

option - treatment of errors on data points:
option=0. Errors on X and Y are ignored and weights=1 for all points
option=1. Errors on Y are used to calculate weights as 1/error. Errors on X are ignored. This option works for histograms where error on the heights can be used to determine the weight of each data point, while X-values do not have errors.
option=2. Same as "1", but weights are 1/(error*error), where error is on Y values
option=3. Same as "1", but errors on X are also taken into account. This works best for P1D which contains errors on X and Y. Weights are 1/(errorX*errorY)
option=4. same as "2", but errors on X and Y are calculated as 1/(errorX*errorX*errorY*errorY)

Returns:

P1D with smoothed result. Abscissa is the same as in the original input.

smoothSpline

public P1D smoothSpline()

Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set. The interpolate(double[], double[]) method returns a PolynomialSplineFunction consisting of n cubic polynomials, defined over the subintervals determined by the x values, x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest knot point and strictly less than the largest knot point is computed by finding the subinterval to which x belongs and computing the value of the corresponding polynomial at x - x[i] where i is the index of the subinterval. See PolynomialSplineFunction for more details. The interpolating polynomials satisfy:

The value of the PolynomialSplineFunction at each of the input x values equals the corresponding y value. Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials "match up" at the knot points, as do their first and second derivatives).

The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, Numerical Analysis, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.

Returns:

P1D with smoothed result. Abscissa is the same as in the original input.

smoothCubicSpline

public P1D smoothCubicSpline()

Perform a cubic interpolatory spline.

Returns:

P1D with smoothed data

smoothGauss

public P1D smoothGauss(double standardDeviation)

Computes a Gaussian smoothed version of data.

The data are smoothed by discrete convolution with a kernel approximating a Gaussian impulse response with the specified standard deviation.

Parameters:

standardDeviation - The standard deviation of the Gaussian smoothing kernel which must be non-negative or an IllegalArgumentException will be thrown. If zero, the P1D object will be returned with no smoothing applied.

Returns:

A Gaussian smoothed version of the histogram.

smoothAverage

public P1D smoothAverage(boolean isWeighted,
                         int k)

Smooth data by averaging over a moving window.

It is smoothed by averaging over a moving window of a size specified by the method parameter: if the value of the parameter is k then the width of the window is 2*k + 1. If the window runs off the end of the P1D only those values which intersect the histogram are taken into consideration. The smoothing may optionally be weighted to favor the central value using a "triangular" weighting. For example, for a value of k equal to 2 the central bin would have weight 1/3, the adjacent bins 2/9, and the next adjacent bins 1/9. Errors are kept the same as before.

Parameters:

isWeighted - Whether values Y will be weighted using a triangular weighting scheme favoring bins near the central bin.

k - The smoothing parameter which must be non-negative. If zero, the histogram object will be returned with no smoothing applied.

Returns:

A smoothed version of data

getMaxValue

public double getMaxValue()

Get min value of data in X

Returns:

getMinValue

public double getMinValue()

Get maximum value of data in X

Returns:

interpolateNatuarlCubicSpline

public Interpolator.Cubic[] interpolateNatuarlCubicSpline()

Natural cubic splines interpolation. calculates the natural cubic spline that interpolates y[0], y[1], ... y[n] The first segment is returned as C[0].a + C[0].b*u + C[0].c*u^2 + C[0].d*u^3 0<=u <1 the other segments are in C[1], C[2], ... C[n-1]

Returns:

Cubic function