Michael Thomas Flanagan's Java Scientific Library

ProbabilityPlot Class:     Probablity Plots

      

Last update: 9 April 2015                                                                                                                              PERMISSION TO USE
Main Page of Michael Thomas Flanagan's Java Scientific Library

This class contains methods for plotting probability plots. It contains methods for
The details of the resultant probability plot are also printed to a text file.

Presently pobability plot methods are available for the:
               

The values of the estimated distribution parameters, e.g. μ, σ and γ for a Fréchet distribution, are obtained by minimising the sum of squares of the differences between the calculated order statistic medians and the ordered observed responses using a constrained Nelder and Mead Simplex optimisation procedure. Details of this library's java implementation of Nelder and Mead's simplex optimisation may be found in the Regression and Minimisation classes of this library.
Options are available allowing for the performance of either an unweighted minimisation or weighted minimisations.

Outlier removal methods may be found in the Outliers Class and further tests for non-normality, e.g. Shapiro-Wilk W test, may be found in the Normality Class.

Warning 1:
The Nelder and Mead Simplex procedure relies on good initial estimates of the unknown distribution parameters. For all methods these initial estimates are automatically calculated, on calling the relevant method, by an in-program inspection of the observed response data. The success of this estimation depends on the quality of the data and the complexity of the distribution. Consequently a failure to obtain a good fit in the chosen probability plot may, in some cases, represent not a choice of an inappropriate distribution but a failure in the minimisation process. For users who have prior knowledge of the range within which these parameters may lay may choose the option of entering their own initial estimates. If this option is used their supplied initial estimates are used rather than those calculated within the program. The method enabling the use of user supplied initial estimates must be called before the relevant probability plot method, e.g. if the user wishes to use their own initial estimates in the construction of a Fréchet probability plot frechetUserSuppliedInitialEstimates(mu, sigma, gamma) should be called before frechetProbabiltyPlot().

The results of a probability plot calculation should aways be examined graphically as a displayed plot as such graphical examination is one of the best methods of assessing goodness of fit. All calls to methods associated with a specific distribution will display a plot but only on the first call to such methods with the exception of the F-distribution in which case fDistributionProbabiltyPlot(nu1, nu2) must always be called first. Suppression of the plot is possible but should be used with caution.

Warning 2:
The errors of the estimated distribution parameters are estimated using linear statistical procedures assuming that these non-linear functions behave in a linear manner about the optimization minimum of the sum of squares of the data minus the corresponding order statistic median. The relevant Hessian matrix is calculated numerically. These 'pseudo-linear' error values should be treated with great caution. They are, at best, an indication of the minimum value that the error may take and in most cases probably significantly underestimate the true error. It is advised that you examine the information that is written to the output text file where the parameter-parameter correlations are given. Even though these are also 'pseudo-linear' they do give a reasonable estimate of parameter- parameter correlation and hence uncertainty. Several of the distributions do have highly correlated parameters [see text file example].

The errors reported for the probabilty plot gradient and intercept are obtained for a linear regression and are reliable.

General References:
Probability Plots: NIST/SEMATECH Engineering Statistics Handbook: 1.3.3.22. Probability Plot, http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm
Nelder and Mead Simplex Optimisation: Nelder, J.A. and Mead, R. (1965) Computer Journal, 7, 308-313.

See Regression class for alternative regression procedures.
See Stat class for the general statistical methods and functions.

import directive: import flanagan.analysis.ProbabilityPlot;

SUMMARY OF METHODS
Constructors Unweighted Regression public ProbabilityPlot(double[] dataArray)
public ProbabilityPlot(Double[] dataArray)
public ProbabilityPlot(float[] dataArray)
public ProbabilityPlot(Float[] dataArray)
public ProbabilityPlot(int[] dataArray)
public ProbabilityPlot(Integer[] dataArray)
public ProbabilityPlot(long[] dataArray)
public ProbabilityPlot(Long[] dataArray)
public ProbabilityPlot(short[] dataArray)
public ProbabilityPlot(Short[] dataArray)
public ProbabilityPlot(byte[] dataArray)
public ProbabilityPlot(Byte[] dataArray)
public ProbabilityPlot(BigDecimal[] dataArray)
public ProbabilityPlot(BigInteger[] dataArray)
public ProbabilityPlot(ArrayMaths dataArray)
public ProbabilityPlot(ArrayList<Object> dataArray)
public ProbabilityPlot(Vector<Object> dataArray)
public ProbabilityPlot(Stat dataArray)
Weighted Regression public ProbabilityPlot(double[] dataArray, double[] weights)
public ProbabilityPlot(Double[] dataArray, Double[] weights)
public ProbabilityPlot(float[] dataArray, float[] weights)
public ProbabilityPlot(Float[] dataArray, Float[] weights)
public ProbabilityPlot(int[] dataArray, int[] weights)
public ProbabilityPlot(Integer[] dataArray, Integer[] weights)
public ProbabilityPlot(long[] dataArray, long[] weights)
public ProbabilityPlot(Long[] dataArray, Long[] weights)
public ProbabilityPlot(short[] dataArray, short[] weights)
public ProbabilityPlot(Short[] dataArray, Short[] short)
public ProbabilityPlot(byte[] dataArray, byte[] weights)
public ProbabilityPlot(Byte[] dataArray, Byte[] weights)
public ProbabilityPlot(BigDecimal[] dataArray, BigDecimal[] weights)
public ProbabilityPlot(BigInteger[] dataArray, BigInteger[] weights)
public ProbabilityPlot(ArrayMaths dataArray, ArrayMaths weights)
public ProbabilityPlot(ArrayList<Object> dataArray, ArrayList<Object> weights)
public ProbabilityPlot(Vector<Object> dataArray, Vector<Object> weights)
public ProbabilityPlot(Stat dataArray, Stat weights)
Alternative Weighting Options Set weighting option public void setWeightingOption(int option)
Get weighting option public String getWeightingOption()
Suppress display of the plot Suppress display public void suppressDisplay()
Restore display public void restoreDisplay()
Gaussian Probability Plot
(Normal Probability Plot)

See below for the
Standard Gaussian Probability Plot 		Calculate and display plot public void gaussianProbabiltyPlot()
User supplied initial estimates public void gaussianUserSuppliedInitialEstimates(double mu, double sigma)
Remove user supplied initial estimates public void removeGaussianUserSuppliedInitialEstimates()
Correlation coefficient public double gaussianCorrelationCoefficient()
Gradient value public double gaussianGradient()
error public double gaussianGradientError()
Intercept value public double gaussianIntercept()
error public double gaussianInterceptError()
μ value public double gaussianMu()
error public double gaussianMuError()
σ value public double gaussianSigma()
error public double gaussianSigmaError()
Sum of squares Unweighted public double gaussianSumOfSquares()
Weighted public double gaussianWeightedSumOfSquares()
Order Statistic Medians public double[] gaussianOrderStatisticMedians()
Standard Gaussian Probability Plot
(Normal Probability Plot)

See above for the
Two Parameter Gaussian Probability Plot 		Calculate and display plot public void gaussianStandardProbabiltyPlot()
Correlation coefficient public double gaussianStandardCorrelationCoefficient()
Gradient value public double gaussianStandardGradient()
error public double gaussianStandardGradientError()
Intercept value public double gaussianStandardIntercept()
error public double gaussianStandardInterceptError()
Sum of squares Unweighted public double gaussianStandardSumOfSquares()
Weighted public double gaussianStandardWeightedSumOfSquares()
Order Statistic Medians public double[] gaussianStandardOrderStatisticMedians()
Exponential Probability Plot Calculate and display plot public void exponentialProbabiltyPlot()
User supplied initial estimates public void exponentialUserSuppliedInitialEstimates(double mu, double sigma)
Remove user supplied initial estimates public void removeExponentialUserSuppliedInitialEstimates()
Correlation coefficient public double exponentialCorrelationCoefficient()
Gradient value public double exponentialGradient()
error public double exponentialGradientError()
Intercept value public double exponentialIntercept()
error public double exponentialInterceptError()
μ value public double exponentialMu()
error public double exponentialMuError()
σ value public double exponentialSigma()
error public double exponentialSigmaError()
Sum of squares Unweighted public double exponentialSumOfSquares()
Weighted public double exponentialWeightedSumOfSquares()
Order Statistic Medians public double[] exponentialOrderStatisticMedians()
F-Distribution Probability Plot Calculate and display plot public void fDistributionProbabiltyPlot(int nu1, int nu2)
Correlation coefficient public double fDistributionCorrelationCoefficient()
Gradient value public double fDistributionGradient()
error public double fDistributionGradientError()
Intercept value public double fDistributionIntercept()
error public double fDistributionInterceptError()
Sum of squares Unweighted public double fDistributionSumOfSquares()
Weighted public double fDistributionWeightedSumOfSquares()
Order Statistic Medians public double[] fDistributionOrderStatisticMedians()
Fréchet Probability Plot
(Three Parameters)

See below for the Two Parameter Fréchet and
the Standard Fréchet Probability Plots 		Calculate and display plot public void frechetProbabiltyPlot()
User supplied initial estimates public void frechetUserSuppliedInitialEstimates(double mu, double sigma, double gamma)
Remove user supplied initial estimates public void removeFrechetUserSuppliedInitialEstimates()
Correlation coefficient public double frechetCorrelationCoefficient()
Gradient value public double frechetGradient()
error public double frechetGradientError()
Intercept value public double frechetIntercept()
error public double frechetInterceptError()
μ value public double frechetMu()
error public double frechetMuError()
σ value public double frechetSigma()
error public double frechetSigmaError()
γ value public double frechetGamma()
error public double frechetGammaError()
Sum of squares Unweighted public double frechetSumOfSquares()
Weighted public double frechetWeightedSumOfSquares()
Order Statistic Medians public double[] frechetOrderStatisticMedians()
Two parameter Fréchet Probability Plot

See above for Three Parameter Fréchet Plot
See below for Standard Fréchet Plot 		Calculate and display plot public void frechetTwoParProbabiltyPlot()
User supplied initial estimates public void frechetTwoParUserSuppliedInitialEstimates(double sigma, double gamma)
Remove user supplied initial estimates public void removeFrechetTwoParUserSuppliedInitialEstimates()
Correlation coefficient public double frechetTwoParCorrelationCoefficient()
Gradient value public double frechetTwoParGradient()
error public double frechetTwoParGradientError()
Intercept value public double frechetTwoParIntercept()
error public double frechetTwoParInterceptError()
σ value public double frechetTwoParSigma()
error public double frechetTwoParSigmaError()
γ value public double frechetTwoParGamma()
error public double frechetTwoParGammaError()
Sum of squares Unweighted public double frechetTwoParSumOfSquares()
Weighted public double frechetTwoParWeightedSumOfSquares()
Order Statistic Medians public double[] frechetTwoParOrderStatisticMedians()
Standard Fréchet Probability Plot

See above for Three Parameter Fréchet and
Two Parameter Fréchet Probability Plots 		Calculate and display plot public void frechetStandardProbabiltyPlot()
User supplied initial estimates public void frechetStandardUserSuppliedInitialEstimates(double gamma)
Remove user supplied initial estimates public void removeFrechetStandardUserSuppliedInitialEstimates()
Correlation coefficient public double frechetStandardCorrelationCoefficient()
Gradient value public double frechetStandardGradient()
error public double frechetStandardGradientError()
Intercept value public double frechetStandardIntercept()
error public double frechetStandardInterceptError()
γ value public double frechetStandardGamma()
error public double frechetStandardGammaError()
Sum of squares Unweighted public double frechetStandardSumOfSquares()
Weighted public double frechetStandardWeightedSumOfSquares()
Order Statistic Medians public double[] frechetStandardOrderStatisticMedians()
Gumbel (minimum order statistic) Probability Plot Calculate and display plot public void gumbelMinProbabiltyPlot()
User supplied initial estimates public void gumbelMinUserSuppliedInitialEstimates(double mu, double sigma)
Remove user supplied initial estimates public void removeGumbelMinUserSuppliedInitialEstimates()
Correlation coefficient public double gumbelMinCorrelationCoefficient()
Gradient value public double gumbelMinGradient()
error public double gumbelMinGradientError()
Intercept value public double gumbelMinIntercept()
error public double gumbelMinInterceptError()
μ value public double gumbelMinMu()
error public double gumbelMinMuError()
σ value public double gumbelMinSigma()
error public double gumbelMinSigmaError()
Sum of squares Unweighted public double gumbelMinSumOfSquares()
Weighted public double gumbelMinWeightedSumOfSquares()
Order Statistic Medians public double[] gumbelMinOrderStatisticMedians()
Gumbel (maximum order statistic) Probability Plot Calculate and display plot public void gumbelMaxProbabiltyPlot()
User supplied initial estimates public void gumbelMaxUserSuppliedInitialEstimates(double mu, double sigma)
Remove user supplied initial estimates public void removeGumbelMaxUserSuppliedInitialEstimates()
Correlation coefficient public double gumbelMaxCorrelationCoefficient()
Gradient value public double gumbelMaxGradient()
error public double gumbelMaxGradientError()
Intercept value public double gumbelMaxIntercept()
error public double gumbelMaxInterceptError()
μ value public double gumbelMaxMu()
error public double gumbelMaxMuError()
σ value public double gumbelMaxSigma()
error public double gumbelMaxSigmaError()
Sum of squares Unweighted public double gumbelMaxSumOfSquares()
Weighted public double gumbelMaxWeightedSumOfSquares()
Order Statistic Medians public double[] gumbelMaxOrderStatisticMedians()
Logistic Probability Plot Calculate and display plot public void logisticProbabiltyPlot()
User supplied initial estimates public void logisticUserSuppliedInitialEstimates(double mu, double beta)
Remove user supplied initial estimates public void removeLogisticUserSuppliedInitialEstimates()
Correlation coefficient public double logisticCorrelationCoefficient()
Gradient value public double logisticGradient()
error public double logisticGradientError()
Intercept value public double logisticIntercept()
error public double logisticInterceptError()
μ value public double logisticMu()
error public double logisticMuError()
β value public double logisticBeta()
error public double logisticBetaError()
Sum of squares Unweighted public double logisticSumOfSquares()
Weighted public double logisticWeightedSumOfSquares()
Order Statistic Medians public double[] logisticOrderStatisticMedians()
Pareto Probability Plot Calculate and display plot public void paretoProbabiltyPlot()
User supplied initial estimates public void paretoUserSuppliedInitialEstimates(double alpha, double beta)
Remove user supplied initial estimates public void removeParetoUserSuppliedInitialEstimates()
Correlation coefficient public double paretoCorrelationCoefficient()
Gradient value public double paretoGradient()
error public double paretoGradientError()
Intercept value public double paretoIntercept()
error public double paretoInterceptError()
α value public double paretoAlpha()
error public double paretoAlphaError()
β value public double paretoBeta()
error public double paretoBetaError()
Sum of squares Unweighted public double paretoSumOfSquares()
Weighted public double paretoWeightedSumOfSquares()
Order Statistic Medians public double[] paretoOrderStatisticMedians()
Rayleigh Probability Plot Calculate and display plot public void rayleighProbabiltyPlot()
User supplied initial estimates public void rayleighUserSuppliedInitialEstimates(double beta)
Remove user supplied initial estimates public void removeRayleighUserSuppliedInitialEstimates()
Correlation coefficient public double rayleighCorrelationCoefficient()
Gradient value public double rayleighGradient()
error public double rayleighGradientError()
Intercept value public double rayleighIntercept()
error public double rayleighInterceptError()
β value public double rayleighBeta()
error public double rayleighBetaError()
Sum of squares Unweighted public double rayleighSumOfSquares()
Weighted public double rayleighWeightedSumOfSquares()
Order Statistic Medians public double[] rayleighOrderStatisticMedians()
Weibull Probability Plot
(Three Parameters)

See below for the Two Parameter Weibull and
the Standard Weibull Probability Plots 		Calculate and display plot public void weibullProbabiltyPlot()
User supplied initial estimates public void weibullUserSuppliedInitialEstimates(double mu, double sigma, double gamma)
Remove user supplied initial estimates public void removeWeibullUserSuppliedInitialEstimates()
Correlation coefficient public double weibullCorrelationCoefficient()
Gradient value public double weibullGradient()
error public double weibullGradientError()
Intercept value public double weibullIntercept()
error public double weibullInterceptError()
μ value public double weibullMu()
error public double weibullMuError()
σ value public double weibullSigma()
error public double weibullSigmaError()
γ value public double weibullGamma()
error public double weibullGammaError()
Sum of squares Unweighted public double weibullSumOfSquares()
Weighted public double weibullWeightedSumOfSquares()
Order Statistic Medians public double[] weibullOrderStatisticMedians()
Two parameter Weibull Probability Plot

See above for Three Parameter Weibull Plot
See below for Standard Weibull Plot 		Calculate and display plot public void weibullTwoParProbabiltyPlot()
User supplied initial estimates public void weibullTwoParUserSuppliedInitialEstimates(double sigma, double gamma)
Remove user supplied initial estimates public void removeWeibullTwoParUserSuppliedInitialEstimates()
Correlation coefficient public double weibullTwoParCorrelationCoefficient()
Gradient value public double weibullTwoParGradient()
error public double weibullTwoParGradientError()
Intercept value public double weibullTwoParIntercept()
error public double weibullTwoParInterceptError()
σ value public double weibullTwoParSigma()
error public double weibullTwoParSigmaError()
γ value public double weibullTwoParGamma()
error public double weibullTwoParGammaError()
Sum of squares Unweighted public double weibullTwoParSumOfSquares()
Weighted public double weibullTwoParWeightedSumOfSquares()
Order Statistic Medians public double[] weibullTwoParOrderStatisticMedians()
Standard Weibull Probability Plot

See above for Three Parameter Weibull and
Two Parameter Weibull Probability Plots 		Calculate and display plot public void weibullStandardProbabiltyPlot()
User supplied initial estimates public void weibullStandardUserSuppliedInitialEstimates(double gamma)
Remove user supplied initial estimates public void removeWeibullStandardUserSuppliedInitialEstimates()
Correlation coefficient public double weibullStandardCorrelationCoefficient()
Gradient value public double weibullStandardGradient()
error public double weibullStandardGradientError()
Intercept value public double weibullStandardIntercept()
error public double weibullStandardInterceptError()
γ value public double weibullStandardGamma()
error public double weibullStandardGammaError()
Sum of squares Unweighted public double weibullStandardSumOfSquares()
Weighted public double weibullStandardWeightedSumOfSquares()
Order Statistic Medians public double[] weibullStandardOrderStatisticMedians()
Common methods Critical value for the correlation coefficients public double correlationCoefficientCriticalValue()
public double correlationCoefficientCriticalValue(double significance)
Significance level public void resetSignificance(double significance)
public double getSignificance()
Ordered data array public double[] orderedData()
Number of data points public int numberOfDataPoints()
Data mean public double mean()
Data standard deviation public double standardDeviation()
Data minimum public double minimum()
Data maximum public double maximum()
Return initial estimates public double[] getInitialEstimates()
Simplex step size factor Current value public double stepFactor()
Reset value public void resetStepFactor(double stepFactor)
Numerical differentiation
step, δ		 Current value public double delta()
Reset value public void resetDelta(double delta)
Standard deviation
denominator		 Set to n public void setDenominatorToN()
Set to n-1 public void setDenominatorToNminusOne()
Suppress the display of the plot public void suppressDisplay()
Restore the display of the plot public void restoreDisplay()
Suppress the output to a text file public void suppressFileOutput()
Restore the the output to a text file public void restoreFileOutput()
Suppress some error messages public void suppressErrorMessages()
Restore suppressed error messages public void restoreErrorMessages()



CONSTRUCTORS

Unweighted Regression
public ProbabilityPlot(double[] dataArray)
public ProbabilityPlot(Double[] dataArray)
public ProbabilityPlot(float[] dataArray)
public ProbabilityPlot(Float[] dataArray)
public ProbabilityPlot(int[] dataArray)
public ProbabilityPlot(Integer[] dataArray)
public ProbabilityPlot(long[] dataArray)
public ProbabilityPlot(Long[] dataArray)
public ProbabilityPlot(short[] dataArray)
public ProbabilityPlot(Short[] dataArray)
public ProbabilityPlot(byte[] dataArray)
public ProbabilityPlot(Byte[] dataArray)
public ProbabilityPlot(BigDecimal[] dataArray)
public ProbabilityPlot(BigInteger[] dataArray)
public ProbabilityPlot(ArrayMaths dataArray)
public ProbabilityPlot(ArrayList<Object> dataArray)
public ProbabilityPlot(Vector<Object> dataArray)
public ProbabilityPlot(Stat dataArray)
Usage:                      ProbabilityPlot pp = new ProbabilityPlot(dataArray);
Creates an instance of ProbabilityPlot. The data array is stored as a one dimensional array of double but may be entered as:
 The regressions used in probability plot constructions are usually unweighted. On using one of the above constructors, and if the method allowing alternative weighted regression options, setWeightingOption(option), has not been called, the Nelder and Mead simplex procedure, performed on calling a plot method, will minimise the unweighted sum of squares, SS, where

n is the number of data points entered, ‘data responsei’ is the ith entered data point [dataArray] and ‘ordered statistic mediani’ is the ith internally calculated ordered statistic median for the chosen distribution.
A weighted regression, with user supplied weights, requires the use of one of the constructors listed below.

Weighted Regression
public ProbabilityPlot(double[] dataArray, double[] weights)
public ProbabilityPlot(Double[] dataArray, Double[] weights)
public ProbabilityPlot(float[] dataArray, float public ProbabilityPlot(Float[] dataArray, Float[] weights)
public ProbabilityPlot(int[] dataArray, int[] weights)
public ProbabilityPlot(Integer[] dataArray, Integer[] weights)
public ProbabilityPlot(long[] dataArray, long[] weights)
public ProbabilityPlot(Long[] dataArray, Long[] weights)
public ProbabilityPlot(short[] dataArray, short[] weights)
public ProbabilityPlot(Short[] dataArray, Short[] short)
public ProbabilityPlot(byte[] dataArray, byte[] weights)
public ProbabilityPlot(Byte[] dataArray, Byte[] weights)
public ProbabilityPlot(BigDecimal[] dataArray, BigDecimal[] weights)
public ProbabilityPlot(BigInteger[] dataArray, BigInteger[] weights)
public ProbabilityPlot(ArrayMaths dataArray, ArrayMaths weights)
public ProbabilityPlot(ArrayList<Object> dataArray, ArrayList<Object> weights)
public ProbabilityPlot(Vector<Object> dataArray, Vector<Object> weights)
public ProbabilityPlot(Stat dataArray, Stat weights)
Usage:                      ProbabilityPlot pp = new ProbabilityPlot(dataArray, weights);
Creates an instance of ProbabilityPlot. The data array and the array of weights are stored as one dimensional arrays of double but may be entered as:
 The regressions used in probability plot constructions are usually unweighted. In such a case the data should be entered via the appropriate constructor listed previously. If a weighted regression is felt to be appropriate the weights should ideally reflect the errors in the data responses and may be entered via these constructors. The data will be analysed using a weighted nonlinear (simplex) regression. The Nelder and Mead simplex procedure, performed on calling a plot method, will minimise the weighted sum of squares, χ2, where

n is the number of data points entered, ‘data responsei’ is the ith entered data point [dataArray], ‘weighti’ is the ith entered weight [weights] and ‘ordered statistic mediani’ is the ith internally calculated ordered statistic median for the chosen distribution.
A choice of less rigorous weighting options is available via the setWeightingOption(option) method.


ALTERNATIVE WEIGHTING OPTIONS
public void setWeightingOption(int option)
Usage:                      pp.setWeightingOption(option);
The regressions used in probability plot constructions are usually unweighted. In such a case the data should be entered via the appropriate constructor. If a weighted regression is felt to be appropriate the weights should ideally reflect the errors in the data responses and should be entered via the appropriate constructor. This method allows the choice of less rigorous weighting options: If one of these options is chosen the data entered via an unweighted regression constructor will be analysed using a weighted nonlinear (simplex) regression with the weights calculated as indicated in the above option table. The Nelder and Mead simplex procedure, performed on calling a plot method, will minimise the weighted sum of squares, χ2, where

n is the number of data points entered, ‘data responsei’ is the ith entered data point [dataArray], ‘weighti’ is the ith entered weight [weights] and ‘ordered statistic mediani’ is the ith internally calculated ordered statistic median for the chosen distribution.
If the data had been previously entered via a weighted regression constructor an exception will be thrown if this method is called.

public String getWeightingOption()
Usage:                      option = pp.getWeightingOption();
This method returns the weighting option used, as a String, in the last call to a plotting method.


SUPPRESS THE DISPLAY OF THE PLOT
public void suppressDisplay()
Usage:                      pp.suppressDisplay();
This method suppresses the display of the probability plot which will be displayed on the first call to any of the methods of a specific distribution, e.g. frechetProbabiltyPlot(), frechetMu(), frechetGradient(). The plot is not displayed by calls to further methods related a specific distribution once the first call has been made. Suppression of the plot on the first call is not recommended as a graphical display of the plot is one of the best methods of checking the goodness of fit. If such suppression is needed this method should be called.

RESTORE THE DISPLAY OF THE PLOT
public void restoreDisplay()
Usage:                      pp.restoreDisplay();
This method restores the display of the probability plot which will be displayed on the first call to any of the methods of a specific distribution, e.g. frechetProbabiltyPlot(), frechetMu(), frechetGradient(). This method need only be called if the above method suppressing the display, suppressDisplay(), has been called and restoration of the display is required.



GAUSSIAN PROBABILITY PLOTS (NORMAL PROBABILITY PLOTS)

TWO PARAMETER GAUSSIAN PROBABILITY PLOT
See below for the Standard Gaussian Probability Plot

This section describes methods for plotting and analysising Gaussian Probability Plots where the Gaussian probabilty density function is defined as

Naming note: gaussian may be replaced by normal in the names of all these methods, e.g. normalProbabilityPlot() performs the same function as gaussianProbabilityPlot()

Calculate and Display a Gaussian Probabilty Plot
public void gaussianProbabilityPlot()
Usage:                      pp.gaussianProbabilityPlot();
This method calculates and displays a Gaussian probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gaussian order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called GaussianProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void gaussianUserSuppliedInitialEstimates(double mu, double sigma)
Usage:                      pp.gaussianUserSuppliedInitialEstimates(mu, sigma);
This method allows the user to override the above method's [gaussianProbabilityPlot()] automatic calculation of the initial estimates of μ and σ and enter their own initial estimates of μ [mu] and σ [sigma]. If this method is required it must be called before gaussianProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeGaussianUserSuppliedInitialEstimates()
Usage:                      pp.removeGaussianUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method gaussianUserSuppliedInitialEstimates(mu, sigma) [see immediately above]. The automatic calculation of the initial estimates of μ and σ by gaussianProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double gaussianCorrelationCoefficient()
Usage:                      rho = pp.gaussianCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Gaussian order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double gaussianGradient()
Usage:                      gradient = pp.gaussianGradient();
This method returns the gradient of the best straight line fit to the Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double gaussianGradientError()
Usage:                      gradientError = pp.gaussianGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double gaussianIntercept()
Usage:                      intercept = pp.gaussianIntercept();
This method returns the intercept of the best straight line fit to the Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double gaussianInterceptError()
Usage:                      interceptError = pp.gaussianInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return gaussian parameter μ
public double gaussianMu()
Usage:                      mu = pp.gaussianMu();
This method returns the Gaussian parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Gaussian order statistic median value. The data is first sorted into an ascending order. See also Warning.


Return the Estimated Error of the gaussian parameter μ
public double gaussianMuError()
Usage:                      muError = pp.gaussianMuError();
This method returns the estimated error of the Gaussian parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return gaussian parameter σ
public double gaussianSigma()
Usage:                      sigma = pp.gaussianSigma();
This method returns the Gaussian parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Gaussian order statistic median value. The data is first sorted into an ascending order. See also Warning.


Return the Estimated Error of the gaussian parameter σ
public double gaussianSigmaError()
Usage:                      sigmaError = pp.gaussianSigmaError();
This method returns the estimated error of the Gaussian parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double gaussianSumOfSquares()
Usage:                      ss = pp.gaussianSumOfSquares();
This method returns the unweighted sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Gaussian order statistic median value. The data is first sorted into an ascending order.

public double gaussianWeightedSumOfSquares()
Usage:                      ss = pp.gaussianWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Gaussian order statistic median value. The data is first sorted into an ascending order.

Return the Gaussian Order Statistic Medians
public double[] gaussianOrderStatisticMedians()
Usage:                      gsom = pp.gaussianOrderStatisticMedians();
This method returns the Gaussian order statistic median used in the Probability Plot.


STANDARD GAUSSIAN PROBABILITY PLOT
See above for the Two Parameter gaussianStandard Probability Plot

This section describes methods for plotting and analysising Standard Gaussian Probability Plots where the Standard Gaussian probabilty density function is defined as

Naming note: gaussianStandard may be replaced by normalStandard in the names of all these methods, e.g. normalStandardProbabilityPlot() performs the same function as gaussianStandardProbabilityPlot().

Calculate and Display a Standard Gaussian Probabilty Plot
public void gaussianStandardProbabilityPlot()
Usage:                      pp.gaussianStandardProbabilityPlot();
This method calculates and displays a Standard Gaussian probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Gaussian order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called StandardGaussianProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

Return the Correlation Coefficient
public double gaussianStandardCorrelationCoefficient()
Usage:                      rho = pp.gaussianStandardCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Standard Gaussian order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double gaussianStandardGradient()
Usage:                      gradient = pp.gaussianStandardGradient();
This method returns the gradient of the best straight line fit to the Standard Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double gaussianStandardGradientError()
Usage:                      gradientError = pp.gaussianStandardGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Standard Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double gaussianStandardIntercept()
Usage:                      intercept = pp.gaussianStandardIntercept();
This method returns the intercept of the best straight line fit to the Standard Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double gaussianStandardInterceptError()
Usage:                      interceptError = pp.gaussianStandardInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Standard Gaussian probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Gaussian order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Sum of Squares
public double gaussianStandardSumOfSquares()
Usage:                      ss = pp.gaussianStandardSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Standard Gaussian order statistic median value. The data is first sorted into an ascending order.

public double gaussianStandardWeightedSumOfSquares()
Usage:                      ss = pp.gaussianStandardWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Standard Gaussian order statistic median value. The data is first sorted into an ascending order.

Return the Standard gaussianStandard Order Statistic Medians
public double[] gaussianStandardOrderStatisticMedians()
Usage:                      gsom = pp.gaussianStandardOrderStatisticMedians();
This method returns the Standard Gaussian order statistic median used in the Probability Plot.


EXPONENTIAL PROBABILITY PLOT

This section describes methods for plotting and analysising Exponential Probability Plots where the Exponential probabilty density function is defined as


Calculate and Display a Exponential Probabilty Plot
public void exponentialProbabilityPlot()
Usage:                      pp.exponentialProbabilityPlot();
This method calculates and displays a Exponential probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Exponential order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called ExponentialProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void exponentialUserSuppliedInitialEstimates(double mu, double sigma)
Usage:                      pp.exponentialUserSuppliedInitialEstimates(mu, sigma);
This method allows the user to override the above method's [exponentialProbabilityPlot()] automatic calculation of the initial estimates of μ and σ and enter their own initial estimates of μ [mu] and σ [sigma]. If this method is required it must be called before exponentialProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeExponentialUserSuppliedInitialEstimates()
Usage:                      pp.removeExponentialUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method exponentialUserSuppliedInitialEstimates(mu, sigma) [see immediately above]. The automatic calculation of the initial estimates of μ and σ by exponentialProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double exponentialCorrelationCoefficient()
Usage:                      rho = pp.exponentialCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Exponential order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double exponentialGradient()
Usage:                      gradient = pp.exponentialGradient();
This method returns the gradient of the best straight line fit to the Exponential probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Exponential order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double exponentialGradientError()
Usage:                      gradientError = pp.exponentialGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Exponential probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Exponential order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double exponentialIntercept()
Usage:                      intercept = pp.exponentialIntercept();
This method returns the intercept of the best straight line fit to the Exponential probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Exponential order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double exponentialInterceptError()
Usage:                      interceptError = pp.exponentialInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Exponential probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Exponential order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Exponential parameter μ
public double exponentialMu()
Usage:                      mu = pp.exponentialMu();
This method returns the Exponential parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Exponential order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Exponential parameter μ
public double exponentialMuError()
Usage:                      muError = pp.exponentialMuError();
This method returns the estimated error of the Exponential parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ, σ and γ. See also Warning.

Return Exponential parameter σ
public double exponentialSigma()
Usage:                      sigma = pp.exponentialSigma();
This method returns the Exponential parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Exponential order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Exponential parameter σ
public double exponentialSigmaError()
Usage:                      sigmaError = pp.exponentialSigmaError();
This method returns the estimated error of the Exponential parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ σ and γ. See also Warning.

Return the Sum of Squares
public double exponentialSumOfSquares()
Usage:                      ss = pp.exponentialSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Exponential order statistic median value. The data is first sorted into an ascending order.

public double exponentialWeightedSumOfSquares()
Usage:                      ss = pp.exponentialWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Exponential order statistic median value. The data is first sorted into an ascending order.

Return the Exponential Order Statistic Medians
public double[] exponentialOrderStatisticMedians()
Usage:                      gsom = pp.exponentialOrderStatisticMedians();
This method returns the Exponential order statistic median used in the Probability Plot.


F-DISTRIBUTION PROBABILITY PLOT

This section describes methods for plotting and analysising F-Distribution Probability Plots where the F-Distribution probabilty density function is defined as


Calculate and Display a F-Distribution Probabilty Plot
public void fDistributionProbabilityPlot(int nu1, int nu2)
Usage:                      pp.fDistributionProbabilityPlot(nu1, nu2);
This method calculates and displays a F-Distribution probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding F-Distribution order statistic medians (abscissa), and the best fit straight line. The arguments, nu1 and nu2, are the degrees of freeedom, ν1 and ν2. The data is first sorted into an ascending order.
This method also outputs a text file, called FdistributionProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

Return the Correlation Coefficient
public double fDistributionCorrelationCoefficient()
Usage:                      rho = pp.fDistributionCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding F-Distribution order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double fDistributionGradient()
Usage:                      gradient = pp.fDistributionGradient();
This method returns the gradient of the best straight line fit to the F-Distribution probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding F-Distribution order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double fDistributionGradientError()
Usage:                      gradientError = pp.fDistributionGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the F-Distribution probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding F-Distribution order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double fDistributionIntercept()
Usage:                      intercept = pp.fDistributionIntercept();
This method returns the intercept of the best straight line fit to the F-Distribution probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding F-Distribution order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double fDistributionInterceptError()
Usage:                      interceptError = pp.fDistributionInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the F-Distribution probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding F-Distribution order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Sum of Squares
public double fDistributionSumOfSquares()
Usage:                      ss = pp.fDistributionSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding F-Distribution order statistic median value. The data is first sorted into an ascending order.

public double fDistributionWeightedSumOfSquares()
Usage:                      ss = pp.fDistributionWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding F-distribution order statistic median value. The data is first sorted into an ascending order.

Return the F-Distribution Order Statistic Medians
public double[] fDistributionOrderStatisticMedians()
Usage:                      gsom = pp.fDistributionOrderStatisticMedians();
This method returns the F-Distribution order statistic median used in the Probability Plot.


FRÉCHET PROBABILITY PLOTS

THREE PARAMETER FRÉCHET PROBABILITY PLOT
See below for the Two Parameter Fréchet and the Standard Fréchet Probability Plots

This section describes methods for plotting and analysising Fréchet Probability Plots where the Fréchet probabilty density function is defined as


Calculate and Display a Fréchet Probabilty Plot
public void frechetProbabilityPlot()
Usage:                      pp.frechetProbabilityPlot();
This method calculates and displays a Fréchet probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Fréchet order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called ThreeParameterFrechetProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void frechetUserSuppliedInitialEstimates(double mu, double sigma, double gamma)
Usage:                      pp.frechetUserSuppliedInitialEstimates(mu, sigma, gamma);
This method allows the user to override the above method's [frechetProbabilityPlot()] automatic calculation of the initial estimates of μ, σ and γ and enter their own initial estimates of μ [mu], σ [sigma] and γ [gamma]. If this method is required it must be called before frechetProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeFrechetUserSuppliedInitialEstimates()
Usage:                      pp.removeFrechetUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method frechetUserSuppliedInitialEstimates(mu, sigma, gamma) [see immediately above]. The automatic calculation of the initial estimates of μ, σ and γ by frechetProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double frechetCorrelationCoefficient()
Usage:                      rho = pp.frechetCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Fréchet order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double frechetGradient()
Usage:                      gradient = pp.frechetGradient();
This method returns the gradient of the best straight line fit to the Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double frechetGradientError()
Usage:                      gradientError = pp.frechetGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double frechetIntercept()
Usage:                      intercept = pp.frechetIntercept();
This method returns the intercept of the best straight line fit to the Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double frechetInterceptError()
Usage:                      interceptError = pp.frechetInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Fréchet parameter μ
public double frechetMu()
Usage:                      mu = pp.frechetMu();
This method returns the Fréchet parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Fréchet order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Fréchet parameter μ
public double frechetMuError()
Usage:                      muError = pp.frechetMuError();
This method returns the estimated error of the Fréchet parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Fréchet parameter σ
public double frechetSigma()
Usage:                      sigma = pp.frechetSigma();
This method returns the Fréchet parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Fréchet order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Fréchet parameter σ
public double frechetSigmaError()
Usage:                      sigmaError = pp.frechetSigmaError();
This method returns the estimated error of the Fréchet parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Fréchet parameter γ
public double frechetGamma()
Usage:                      gamma = pp.frechetGamma();
This method returns the Fréchet parameter, γ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Fréchet order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Fréchet parameter γ
public double frechetGammaError()
Usage:                      gammaError = pp.frechetGammaError();
This method returns the estimated error of the Fréchet parameter, γ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double frechetSumOfSquares()
Usage:                      ss = pp.frechetSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Fréchet order statistic median value. The data is first sorted into an ascending order.

public double frechetWeightedSumOfSquares()
Usage:                      ss = pp.frechetWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Fréchet order statistic median value. The data is first sorted into an ascending order.

Return the Fréchet Order Statistic Medians
public double[] frechetOrderStatisticMedians()
Usage:                      gsom = pp.frechetOrderStatisticMedians();
This method returns the Fréchet order statistic median used in the Probability Plot.


TWO PARAMETER FRÉCHET PROBABILITY PLOT
See above for the Three Parameter Fréchet and below for the Standard Fréchet Probability Plots

This section describes methods for plotting and analysising Two Parameter Fréchet Probability Plots where the Fréchet probabilty density function is defined as


Calculate and Display a Fréchet Probabilty Plot
public void frechetTwoParProbabilityPlot()
Usage:                      pp.frechetTwoParProbabilityPlot();
This method calculates and displays a Two Parameter Fréchet probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Fréchet order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called TwoParameterFrechetProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void frechetTwoParUserSuppliedInitialEstimates(double sigma, double gamma)
Usage:                      pp.frechetTwoParUserSuppliedInitialEstimates(sigma, gamma);
This method allows the user to override the above method's [frechetTwoParProbabilityPlot()] automatic calculation of the initial estimates of σ and γ and enter their own initial estimates of σ [sigma] and γ [gamma]. If this method is required it must be called before frechetTwoParProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeFrechetTwoParUserSuppliedInitialEstimates()
Usage:                      pp.removeFrechetTwoParUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method frechetTwoParUserSuppliedInitialEstimates(sigma, gamma) [see immediately above]. The automatic calculation of the initial estimates of σ and γ by frechetTwoParProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double frechetTwoParCorrelationCoefficient()
Usage:                      rho = pp.frechetTwoParCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Two Parameter Fréchet order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double frechetTwoParGradient()
Usage:                      gradient = pp.frechetTwoParGradient();
This method returns the gradient of the best straight line fit to the Two Parameter Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double frechetTwoParGradientError()
Usage:                      gradientError = pp.frechetTwoParGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Two Parameter Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double frechetTwoParIntercept()
Usage:                      intercept = pp.frechetTwoParIntercept();
This method returns the intercept of the best straight line fit to the Two Parameter Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double frechetTwoParInterceptError()
Usage:                      interceptError = pp.frechetTwoParInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Two Parameter Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Two Parameter Fréchet parameter σ
public double frechetTwoParSigma()
Usage:                      sigma = pp.frechetTwoParSigma();
This method returns the Two Parameter Fréchet parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Two Parameter Fréchet order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Two Parameter Fréchet parameter σ
public double frechetTwoParSigmaError()
Usage:                      sigmaError = pp.frechetTwoParSigmaError();
This method returns the estimated error of the Two Parameter Fréchet parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Two Parameter Fréchet parameter γ
public double frechetTwoParGamma()
Usage:                      gamma = pp.frechetTwoParGamma();
This method returns the Two Parameter Fréchet parameter, γ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Two Parameter Fréchet order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Two Parameter Fréchet parameter γ
public double frechetTwoParGammaError()
Usage:                      gammaError = pp.frechetTwoParGammaError();
This method returns the estimated error of the Two Parameter Fréchet parameter, γ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double frechetTwoParSumOfSquares()
Usage:                      ss = pp.frechetTwoParSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Two Parameter Fréchet order statistic median value. The data is first sorted into an ascending order.

public double frechetTwoParWeightedSumOfSquares()
Usage:                      ss = pp.frechetTwoParWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Two parameter Fréchet order statistic median value. The data is first sorted into an ascending order.

Return the Two Parameter Fréchet Order Statistic Medians
public double[] frechetTwoParOrderStatisticMedians()
Usage:                      gsom = pp.frechetTwoParOrderStatisticMedians();
This method returns the Two Parameter Fréchet order statistic median used in the Probability Plot.


STANDARD FRÉCHET PROBABILITY PLOT
See above for the Three Parameter Fréchet and the Two Parameter Fréchet Probability Plots

This section describes methods for plotting and analysising Standard Fréchet Probability Plots where the Fréchet probabilty density function is defined as


Calculate and Display a Fréchet Probabilty Plot
public void frechetStandardProbabilityPlot()
Usage:                      pp.frechetStandardProbabilityPlot();
This method calculates and displays a Standard Fréchet probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Fréchet order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called OneParameterStandardFrechetProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void frechetStandardUserSuppliedInitialEstimates(double gamma)
Usage:                      pp.frechetStandardUserSuppliedInitialEstimates(gamma);
This method allows the user to override the above method's [frechetStandardProbabilityPlot()] automatic calculation of the initial estimate of γ and enter their own initial estimate of γ [gamma]. If this method is required it must be called before frechetStandardProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeFrechetStandardUserSuppliedInitialEstimates()
Usage:                      pp.removeFrechetStandardUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method frechetStandardUserSuppliedInitialEstimates(gamma) [see immediately above]. The automatic calculation of the initial estimate of γ by frechetStandardProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double frechetStandardCorrelationCoefficient()
Usage:                      rho = pp.frechetStandardCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Standard Fréchet order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double frechetStandardGradient()
Usage:                      gradient = pp.frechetStandardGradient();
This method returns the gradient of the best straight line fit to the Standard Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double frechetStandardGradientError()
Usage:                      gradientError = pp.frechetStandardGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Standard Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double frechetStandardIntercept()
Usage:                      intercept = pp.frechetStandardIntercept();
This method returns the intercept of the best straight line fit to the Standard Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double frechetStandardInterceptError()
Usage:                      interceptError = pp.frechetStandardInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Standard Fréchet probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Fréchet order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Standard Fréchet parameter γ
public double frechetStandardGamma()
Usage:                      gamma = pp.frechetStandardGamma();
This method returns the Standard Fréchet parameter, γ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Standard Fréchet order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Standard Fréchet parameter γ
public double frechetStandardGammaError()
Usage:                      gammaError = pp.frechetStandardGammaError();
This method returns the estimated error of the Standard Fréchet parameter, γ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double frechetStandardSumOfSquares()
Usage:                      ss = pp.frechetStandardSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Standard Fréchet order statistic median value. The data is first sorted into an ascending order.

public double frechetStandardWeightedSumOfSquares()
Usage:                      ss = pp.frechetStandardWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding StandardFréchet order statistic median value. The data is first sorted into an ascending order.

Return the Standard Fréchet Order Statistic Medians
public double[] frechetStandardOrderStatisticMedians()
Usage:                      gsom = pp.frechetStandardOrderStatisticMedians();
This method returns the Standard Fréchet order statistic median used in the Probability Plot.


GUMBEL (MINIMUM ORDER STATISTIC) PROBABILITY PLOT

This section describes methods for plotting and analysising Gumbel (minimim order statistic) Probability Plots where the Gumbel (minimim order statistic) probabilty density function is defined as


Calculate and Display a Gumbel (minimim order statistic) Probabilty Plot
public void gumbelMinProbabilityPlot()
Usage:                      pp.gumbelMinProbabilityPlot();
This method calculates and displays a Gumbel (minimim order statistic) probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (minimim order statistic) order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called GumbelminimimorderstatisticProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void gumbelMinUserSuppliedInitialEstimates(double mu, double sigma)
Usage:                      pp.gumbelMinUserSuppliedInitialEstimates(mu, sigma);
This method allows the user to override the above method's [gumbelMinProbabilityPlot()] automatic calculation of the initial estimates of μ and σ and enter their own initial estimates of μ [mu] and σ [sigma]. If this method is required it must be called before gumbelMinProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeGumbelMinUserSuppliedInitialEstimates()
Usage:                      pp.removeGumbelMinUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method gumbelMinUserSuppliedInitialEstimates(mu, sigma) [see immediately above]. The automatic calculation of the initial estimates of μ and σ by gumbelMinProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double gumbelMinCorrelationCoefficient()
Usage:                      rho = pp.gumbelMinCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Gumbel (minimim order statistic) order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double gumbelMinGradient()
Usage:                      gradient = pp.gumbelMinGradient();
This method returns the gradient of the best straight line fit to the Gumbel (minimim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (minimim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double gumbelMinGradientError()
Usage:                      gradientError = pp.gumbelMinGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Gumbel (minimim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (minimim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double gumbelMinIntercept()
Usage:                      intercept = pp.gumbelMinIntercept();
This method returns the intercept of the best straight line fit to the Gumbel (minimim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (minimim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double gumbelMinInterceptError()
Usage:                      interceptError = pp.gumbelMinInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Gumbel (minimim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (minimim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Gumbel (minimim order statistic) parameter μ
public double gumbelMinMu()
Usage:                      mu = pp.gumbelMinMu();
This method returns the Gumbel (minimim order statistic) parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Gumbel (minimim order statistic) order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Gumbel (minimim order statistic) parameter μ
public double gumbelMinMuError()
Usage:                      muError = pp.gumbelMinMuError();
This method returns the estimated error of the Gumbel (minimim order statistic) parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Gumbel (minimim order statistic) parameter σ
public double gumbelMinSigma()
Usage:                      sigma = pp.gumbelMinSigma();
This method returns the Gumbel (minimim order statistic) parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Gumbel (minimim order statistic) order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Gumbel (minimim order statistic) parameter σ
public double gumbelMinSigmaError()
Usage:                      sigmaError = pp.gumbelMinSigmaError();
This method returns the estimated error of the Gumbel (minimim order statistic) parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double gumbelMinSumOfSquares()
Usage:                      ss = pp.gumbelMinSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Gumbel (minimim order statistic) order statistic median value. The data is first sorted into an ascending order.

public double gumbelMinWeightedSumOfSquares()
Usage:                      ss = pp.gumbelMinWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Gumbel (minimim order statistic) order statistic median value. The data is first sorted into an ascending order.

Return the Gumbel (minimim order statistic) Order Statistic Medians
public double[] gumbelMinOrderStatisticMedians()
Usage:                      gsom = pp.gumbelMinOrderStatisticMedians();
This method returns the Gumbel (minimim order statistic) order statistic median used in the Probability Plot.


GUMBEL (MAXIMUM ORDER STATISTIC) PROBABILITY PLOT

This section describes methods for plotting and analysising Gumbel (maximim order statistic) Probability Plots where the Gumbel (maximim order statistic) probabilty density function is defined as


Calculate and Display a Gumbel (maximim order statistic) Probabilty Plot
public void gumbelMaxProbabilityPlot()
Usage:                      pp.gumbelMaxProbabilityPlot();
This method calculates and displays a Gumbel (maximim order statistic) probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (maximim order statistic) order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called GumbelmaximimorderstatisticProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void gumbelMaxUserSuppliedInitialEstimates(double mu, double sigma)
Usage:                      pp.gumbelMaxUserSuppliedInitialEstimates(mu, sigma);
This method allows the user to override the above method's [gumbelMaxProbabilityPlot()] automatic calculation of the initial estimates of μ and σ and enter their own initial estimates of μ [mu] and σ [sigma]. If this method is required it must be called before gumbelMaxProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeGumbelMaxUserSuppliedInitialEstimates()
Usage:                      pp.removeGumbelMaxUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method gumbelMaxUserSuppliedInitialEstimates(mu, sigma) [see immediately above]. The automatic calculation of the initial estimates of μ and σ by gumbelMaxProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double gumbelMaxCorrelationCoefficient()
Usage:                      rho = pp.gumbelMaxCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Gumbel (maximim order statistic) order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double gumbelMaxGradient()
Usage:                      gradient = pp.gumbelMaxGradient();
This method returns the gradient of the best straight line fit to the Gumbel (maximim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (maximim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double gumbelMaxGradientError()
Usage:                      gradientError = pp.gumbelMaxGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Gumbel (maximim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (maximim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double gumbelMaxIntercept()
Usage:                      intercept = pp.gumbelMaxIntercept();
This method returns the intercept of the best straight line fit to the Gumbel (maximim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (maximim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double gumbelMaxInterceptError()
Usage:                      interceptError = pp.gumbelMaxInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Gumbel (maximim order statistic) probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Gumbel (maximim order statistic) order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Gumbel (maximim order statistic) parameter μ
public double gumbelMaxMu()
Usage:                      mu = pp.gumbelMaxMu();
This method returns the Gumbel (maximim order statistic) parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Gumbel (maximim order statistic) order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Gumbel (maximim order statistic) parameter μ
public double gumbelMaxMuError()
Usage:                      muError = pp.gumbelMaxMuError();
This method returns the estimated error of the Gumbel (maximim order statistic) parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Gumbel (maximim order statistic) parameter σ
public double gumbelMaxSigma()
Usage:                      sigma = pp.gumbelMaxSigma();
This method returns the Gumbel (maximim order statistic) parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Gumbel (maximim order statistic) order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Gumbel (maximim order statistic) parameter σ
public double gumbelMaxSigmaError()
Usage:                      sigmaError = pp.gumbelMaxSigmaError();
This method returns the estimated error of the Gumbel (maximim order statistic) parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double gumbelMaxSumOfSquares()
Usage:                      ss = pp.gumbelMaxSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Gumbel (maximim order statistic) order statistic median value. The data is first sorted into an ascending order.

public double gumbelMaxWeightedSumOfSquares()
Usage:                      ss = pp.gumbelMaxWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Gumbel (maximim order statistic) order statistic median value. The data is first sorted into an ascending order.

Return the Gumbel (maximim order statistic) Order Statistic Medians
public double[] gumbelMaxOrderStatisticMedians()
Usage:                      gsom = pp.gumbelMaxOrderStatisticMedians();
This method returns the Gumbel (maximim order statistic) order statistic median used in the Probability Plot.


LOGISIC PROBABILITY PLOT

This section describes methods for plotting and analysising Logistic Probability Plots where the Logistic probabilty density function is defined as


Calculate and Display a Logistic Probabilty Plot
public void logisticProbabilityPlot()
Usage:                      pp.logisticProbabilityPlot();
This method calculates and displays a Logistic probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Logistic order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called LogisticProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void logisticUserSuppliedInitialEstimates(double mu, double beta)
Usage:                      pp.logisticUserSuppliedInitialEstimates(mu, beta);
This method allows the user to override the above method's [logisticProbabilityPlot()] automatic calculation of the initial estimates of μ and β and enter their own initial estimates of μ [mu] and β [beta]. If this method is required it must be called before logisticProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeLogisticUserSuppliedInitialEstimates()
Usage:                      pp.removeLogisticUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method logisticUserSuppliedInitialEstimates(mu, beta) [see immediately above]. The automatic calculation of the initial estimates of μ and β by logisticProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double logisticCorrelationCoefficient()
Usage:                      rho = pp.logisticCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Logistic order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double logisticGradient()
Usage:                      gradient = pp.logisticGradient();
This method returns the gradient of the best straight line fit to the Logistic probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Logistic order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double logisticGradientError()
Usage:                      gradientError = pp.logisticGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Logistic probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Logistic order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double logisticIntercept()
Usage:                      intercept = pp.logisticIntercept();
This method returns the intercept of the best straight line fit to the Logistic probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Logistic order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double logisticInterceptError()
Usage:                      interceptError = pp.logisticInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Logistic probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Logistic order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Logistic parameter μ
public double logisticMu()
Usage:                      mu = pp.logisticMu();
This method returns the Logistic parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Logistic order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Logistic parameter μ
public double logisticMuError()
Usage:                      muError = pp.logisticMuError();
This method returns the estimated error of the Logistic parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Logistic parameter σ
public double logisticSigma()
Usage:                      sigma = pp.logisticSigma();
This method returns the Logistic parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Logistic order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Logistic parameter σ
public double logisticSigmaError()
Usage:                      sigmaError = pp.logisticSigmaError();
This method returns the estimated error of the Logistic parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double logisticSumOfSquares()
Usage:                      ss = pp.logisticSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Logistic order statistic median value. The data is first sorted into an ascending order.

public double logisticWeightedSumOfSquares()
Usage:                      ss = pp.logisticWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Logistic order statistic median value. The data is first sorted into an ascending order.

Return the Logistic Order Statistic Medians
public double[] logisticOrderStatisticMedians()
Usage:                      gsom = pp.logisticOrderStatisticMedians();
This method returns the Logistic order statistic median used in the Probability Plot.


PARETO PROBABILITY PLOT

This section describes methods for plotting and analysising Pareto Probability Plots where the Pareto probabilty density function is defined as


Calculate and Display a Pareto Probabilty Plot
public void paretoProbabilityPlot()
Usage:                      pp.paretoProbabilityPlot();
This method calculates and displays a Pareto probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Pareto order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called ParetoProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void paretoUserSuppliedInitialEstimates(double alpha, double beta)
Usage:                      pp.paretoUserSuppliedInitialEstimates(alpha, beta);
This method allows the user to override the above method's [paretoProbabilityPlot()] automatic calculation of the initial estimates of α and β and enter their own initial estimate of α [alpha] and β [beta]. If this method is required it must be called before paretoProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeParetoUserSuppliedInitialEstimates()
Usage:                      pp.removeParetoUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method paretoUserSuppliedInitialEstimates(beta) [see immediately above]. The automatic calculation of the initial estimates of α and β by paretoProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double paretoCorrelationCoefficient()
Usage:                      rho = pp.paretoCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Pareto order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double paretoGradient()
Usage:                      gradient = pp.paretoGradient();
This method returns the gradient of the best straight line fit to the Pareto probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Pareto order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double paretoGradientError()
Usage:                      gradientError = pp.paretoGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Pareto probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Pareto order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double paretoIntercept()
Usage:                      intercept = pp.paretoIntercept();
This method returns the intercept of the best straight line fit to the Pareto probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Pareto order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double paretoInterceptError()
Usage:                      interceptError = pp.paretoInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Pareto probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Pareto order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Pareto parameter &alpha
public double paretoAlpha()
Usage:                      alpha = pp.paretoAlpha();
This method returns the Pareto parameter, &alpha, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Pareto order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Pareto parameter &alpha
public double paretoAlphaError()
Usage:                      alphaError = pp.paretoAlphaError();
This method returns the estimated error of the Pareto parameter, &alpha. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to &alpha and &beta. See also Warning.

Return Pareto parameter &beta
public double paretoBeta()
Usage:                      beta = pp.paretoBeta();
This method returns the Pareto parameter, &beta, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Pareto order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Pareto parameter &beta
public double paretoBetaError()
Usage:                      betaError = pp.paretoBetaError();
This method returns the estimated error of the Pareto parameter, &beta. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to &alpha and &beta. See also Warning.

Return the Sum of Squares
public double paretoSumOfSquares()
Usage:                      ss = pp.paretoSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Pareto order statistic median value. The data is first sorted into an ascending order.

public double paretoWeightedSumOfSquares()
Usage:                      ss = pp.paretoWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Pareto order statistic median value. The data is first sorted into an ascending order.

Return the Pareto Order Statistic Medians
public double[] paretoOrderStatisticMedians()
Usage:                      gsom = pp.paretoOrderStatisticMedians();
This method returns the Pareto order statistic median used in the Probability Plot.


RAYLEIGH PROBABILITY PLOT

This section describes methods for plotting and analysising Rayleigh Probability Plots where the Rayleigh probabilty density function is defined as


Calculate and Display a Rayleigh Probabilty Plot
public void rayleighProbabilityPlot()
Usage:                      pp.rayleighProbabilityPlot();
This method calculates and displays a Rayleigh probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Rayleigh order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called RayleighProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void rayleighUserSuppliedInitialEstimates(double beta)
Usage:                      pp.rayleighUserSuppliedInitialEstimates(beta);
This method allows the user to override the above method's [rayleighProbabilityPlot()] automatic calculation of the initial estimate of β and enter their own initial estimate of β [beta]. If this method is required it must be called before rayleighProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeRayleighUserSuppliedInitialEstimates()
Usage:                      pp.removeRayleighUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method rayleighUserSuppliedInitialEstimates(beta) [see immediately above]. The automatic calculation of the initial estimate of β by rayleighProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double rayleighCorrelationCoefficient()
Usage:                      rho = pp.rayleighCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Rayleigh order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double rayleighGradient()
Usage:                      gradient = pp.rayleighGradient();
This method returns the gradient of the best straight line fit to the Rayleigh probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Rayleigh order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double rayleighGradientError()
Usage:                      gradientError = pp.rayleighGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Rayleigh probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Rayleigh order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double rayleighIntercept()
Usage:                      intercept = pp.rayleighIntercept();
This method returns the intercept of the best straight line fit to the Rayleigh probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Rayleigh order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double rayleighInterceptError()
Usage:                      interceptError = pp.rayleighInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Rayleigh probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Rayleigh order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Rayleigh parameter &beta
public double rayleighBeta()
Usage:                      beta = pp.rayleighBeta();
This method returns the Rayleigh parameter, &beta, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Rayleigh order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Rayleigh parameter &beta
public double rayleighBetaError()
Usage:                      betaError = pp.rayleighBetaError();
This method returns the estimated error of the Rayleigh parameter, &beta. The error is obtained as the square root of the inverse of the second derivative of the sum of squares with respect to &beta. See also Warning.

Return the Sum of Squares
public double rayleighSumOfSquares()
Usage:                      ss = pp.rayleighSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression ConstructorConstructor and the corresponding Rayleigh order statistic median value. The data is first sorted into an ascending order.

public double rayleighWeightedSumOfSquares()
Usage:                      ss = pp.rayleighWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Rayleigh order statistic median value. The data is first sorted into an ascending order.

Return the Rayleigh Order Statistic Medians
public double[] rayleighOrderStatisticMedians()
Usage:                      gsom = pp.rayleighOrderStatisticMedians();
This method returns the Rayleigh order statistic median used in the Probability Plot.


WEIBULL PROBABILITY PLOTS

THREE PARAMETER WEIBULL PROBABILITY PLOT
See below for the Two Parameter Weibull and the Standard Weibull Probability Plots

This section describes methods for plotting and analysising Three Parameter Weibull Probability Plots where the Three Parameter Weibull probabilty density function is defined as


Calculate and Display a Three Parameter Weibull Probabilty Plot
public void weibullProbabilityPlot()
Usage:                      pp.weibullProbabilityPlot();
This method calculates and displays a Three Parameter Weibull probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Three Parameter Weibull order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called ThreeParameterWeibullProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void weibullUserSuppliedInitialEstimates(double mu, double sigma, double gamma)
Usage:                      pp.weibullUserSuppliedInitialEstimates(mu, sigma, gamma);
This method allows the user to override the above method's [weibullProbabilityPlot()] automatic calculation of the initial estimates of μ, σ and γ and enter their own initial estimates of μ [mu], σ [sigma] and γ [gamma]. If this method is required it must be called before weibullProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeWeibullUserSuppliedInitialEstimates()
Usage:                      pp.removeWeibullUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method weibullUserSuppliedInitialEstimates(mu, sigma, gamma) [see immediately above]. The automatic calculation of the initial estimates of μ, σ and γ by weibullProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double weibullCorrelationCoefficient()
Usage:                      rho = pp.weibullCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Three Parameter Weibull order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double weibullGradient()
Usage:                      gradient = pp.weibullGradient();
This method returns the gradient of the best straight line fit to the Three Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Three Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double weibullGradientError()
Usage:                      gradientError = pp.weibullGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Three Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Three Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double weibullIntercept()
Usage:                      intercept = pp.weibullIntercept();
This method returns the intercept of the best straight line fit to the Three Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Three Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double weibullInterceptError()
Usage:                      interceptError = pp.weibullInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Three Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Three Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Three Parameter Weibull parameter μ
public double weibullMu()
Usage:                      mu = pp.weibullMu();
This method returns the Three Parameter Weibull parameter, μ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Three Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Three Parameter Weibull parameter μ
public double weibullMuError()
Usage:                      muError = pp.weibullMuError();
This method returns the estimated error of the Three Parameter Weibull parameter, μ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Three Parameter Weibull parameter σ
public double weibullSigma()
Usage:                      sigma = pp.weibullSigma();
This method returns the Three Parameter Weibull parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Three Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Three Parameter Weibull parameter σ
public double weibullSigmaError()
Usage:                      sigmaError = pp.weibullSigmaError();
This method returns the estimated error of the Three Parameter Weibull parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Three Parameter Weibull parameter γ
public double weibullGamma()
Usage:                      gamma = pp.weibullGamma();
This method returns the Three Parameter Weibull parameter, γ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Three Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

Return the Estimated Error of the Three Parameter Weibull parameter γ
public double weibullGammaError()
Usage:                      gammaError = pp.weibullGammaError();
This method returns the estimated error of the Three Parameter Weibull parameter, γ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double weibullSumOfSquares()
Usage:                      ss = pp.weibullSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Three Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

public double weibullWeightedSumOfSquares()
Usage:                      ss = pp.weibullWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Three Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

Return the Three Parameter Weibull Order Statistic Medians
public double[] weibullOrderStatisticMedians()
Usage:                      gsom = pp.weibullOrderStatisticMedians();
This method returns the Three Parameter Weibull order statistic median used in the Probability Plot.


TWO PARAMETER WEIBULL PROBABILITY PLOT
See above for the Three Parameter Weibull and below for the Standard Weibull Probability Plots

This section describes methods for plotting and analysising Two Parameter Weibull Probability Plots where the Weibull probabilty density function is defined as


Calculate and Display a Weibull Probabilty Plot
public void weibullTwoParProbabilityPlot()
Usage:                      pp.weibullTwoParProbabilityPlot();
This method calculates and displays a Two Parameter Weibull probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Weibull order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called TwoParameterWeibullProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void weibullTwoParUserSuppliedInitialEstimates(double sigma, double gamma)
Usage:                      pp.weibullTwoParUserSuppliedInitialEstimates(sigma, gamma);
This method allows the user to override the above method's [weibullTwoParProbabilityPlot()] automatic calculation of the initial estimates of σ and γ and enter their own initial estimates of σ [sigma] and γ [gamma]. If this method is required it must be called before weibullTwoParProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeWeibullTwoParUserSuppliedInitialEstimates()
Usage:                      pp.removeWeibullTwoParUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method weibullTwoParUserSuppliedInitialEstimates(sigma, gamma) [see immediately above]. The automatic calculation of the initial estimates of σ and γ by weibullTwoParProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double weibullTwoParCorrelationCoefficient()
Usage:                      rho = pp.weibullTwoParCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Two Parameter Weibull order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double weibullTwoParGradient()
Usage:                      gradient = pp.weibullTwoParGradient();
This method returns the gradient of the best straight line fit to the Two Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double weibullTwoParGradientError()
Usage:                      gradientError = pp.weibullTwoParGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Two Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double weibullTwoParIntercept()
Usage:                      intercept = pp.weibullTwoParIntercept();
This method returns the intercept of the best straight line fit to the Two Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double weibullTwoParInterceptError()
Usage:                      interceptError = pp.weibullTwoParInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Two Parameter Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Two Parameter Weibull order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Two Parameter Weibull parameter σ
public double weibullTwoParSigma()
Usage:                      sigma = pp.weibullTwoParSigma();
This method returns the Two Parameter Weibull parameter, σ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Two Parameter Weibull order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Two Parameter Weibull parameter σ
public double weibullTwoParSigmaError()
Usage:                      sigmaError = pp.weibullTwoParSigmaError();
This method returns the estimated error of the Two Parameter Weibull parameter, σ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return Two Parameter Weibull parameter γ
public double weibullTwoParGamma()
Usage:                      gamma = pp.weibullTwoParGamma();
This method returns the Two Parameter Weibull parameter, γ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Two Parameter Weibull order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Two Parameter Weibull parameter γ
public double weibullTwoParGammaError()
Usage:                      gammaError = pp.weibullTwoParGammaError();
This method returns the estimated error of the Two Parameter Weibull parameter, γ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double weibullTwoParSumOfSquares()
Usage:                      ss = pp.weibullTwoParSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Two Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

public double weibullTwoParWeightedSumOfSquares()
Usage:                      ss = pp.weibullTwoParWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding Two Parameter Weibull order statistic median value. The data is first sorted into an ascending order.

Return the Two Parameter Weibull Order Statistic Medians
public double[] weibullTwoParOrderStatisticMedians()
Usage:                      gsom = pp.weibullTwoParOrderStatisticMedians();
This method returns the Two Parameter Weibull order statistic median used in the Probability Plot.


STANDARD WEIBULL PROBABILITY PLOT
See above for the Three Parameter Weibull and the Two Parameter Weibull Probability Plots

This section describes methods for plotting and analysising Standard Weibull Probability Plots where the Weibull probabilty density function is defined as


Calculate and Display a One Parameter (Standard) Weibull Probabilty Plot
public void weibullStandardProbabilityPlot()
Usage:                      pp.weibullStandardProbabilityPlot();
This method calculates and displays a Standard Weibull probablity plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Weibull order statistic medians (abscissa), and the best fit straight line. The data is first sorted into an ascending order.
This method also outputs a text file, called OneParameterStandardWeibullProbabilityPlotOutput.txt, containing details of the probability plot and the fitted parameters as described below.

User supplied initial estimates
public void weibullStandardUserSuppliedInitialEstimates(double gamma)
Usage:                      pp.weibullStandardUserSuppliedInitialEstimates(gamma);
This method allows the user to override the above method's [weibullStandardProbabilityPlot()] automatic calculation of the initial estimate of γ and enter their own initial estimate of γ [gamma]. If this method is required it must be called before weibullStandardProbabilityPlot() is called. See the warning note 1 for an explanation of the possible need for this method.

Remove user supplied initial estimates
public void removeWeibullStandardUserSuppliedInitialEstimates()
Usage:                      pp.removeWeibullStandardUserSuppliedInitialEstimates();
This method removes initial estimates supplied by the user via the method weibullStandardUserSuppliedInitialEstimates(gamma) [see immediately above]. The automatic calculation of the initial estimate of γ by weibullStandardProbabilityPlot() is restored on calling this method.

Return the Correlation Coefficient
public double weibullStandardCorrelationCoefficient()
Usage:                      rho = pp.weibullStandardCorrelationCoefficient();
This method returns the correlation coefficient of the data entered via the Constructor and the corresponding Standard Weibull order statistic medians. The data is first sorted into an ascending order.

Return the Gradient
public double weibullStandardGradient()
Usage:                      gradient = pp.weibullStandardGradient();
This method returns the gradient of the best straight line fit to the Standard Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Weibull order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Gradient
public double weibullStandardGradientError()
Usage:                      gradientError = pp.weibullStandardGradientError();
This method returns the estimated error of the gradient of the best straight line fit to the Standard Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Weibull order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return the Intercept
public double weibullStandardIntercept()
Usage:                      intercept = pp.weibullStandardIntercept();
This method returns the intercept of the best straight line fit to the Standard Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Weibull order statistic medians (abscissa). The data is first sorted into an ascending order.

Return the Estimated Error of the Intercept
public double weibullStandardInterceptError()
Usage:                      interceptError = pp.weibullStandardInterceptError();
This method returns the estimated error of the intercept of the best straight line fit to the Standard Weibull probability plot, i.e. a plot of the data entered via the Constructor (ordinate) against the corresponding Standard Weibull order statistic medians (abscissa). The data is first sorted into an ascending order. The error is obtained from the linear regression.

Return Standard Weibull parameter γ
public double weibullStandardGamma()
Usage:                      gamma = pp.weibullStandardGamma();
This method returns the Standard Weibull parameter, γ, obtained from the best fit Probabilty Plot calculated as the one with the minimum sum of squares of the diferences in the data values entered via the Constructor and the corresponding Standard Weibull order statistic median value. The data is first sorted into an ascending order. See also Warning.

Return the Estimated Error of the Standard Weibull parameter γ
public double weibullStandardGammaError()
Usage:                      gammaError = pp.weibullStandardGammaError();
This method returns the estimated error of the Standard Weibull parameter, γ. The error is obtained as the square root of the appropriate diagonal of a covariance matrix obtained as the inverse of the numerically calculated matrix of second derivatives of the sum of squares with respect to μ and σ. See also Warning.

Return the Sum of Squares
public double weibullStandardSumOfSquares()
Usage:                      ss = pp.weibullStandardSumOfSquares();
This method returns the sum of squares of the diferences in the data values entered via the unweighted regression Constructor and the corresponding Standard Weibull order statistic median value. The data is first sorted into an ascending order.

public double weibullStandardWeightedSumOfSquares()
Usage:                      ss = pp.weibullStandardWeightedSumOfSquares();
This method returns the weighted sum of squares of the diferences in the data values entered via the weighted regression Constructor and the corresponding One Parameter (Standard) Weibull order statistic median value. The data is first sorted into an ascending order.

Return the Standard Weibull Order Statistic Medians
public double[] weibullStandardOrderStatisticMedians()
Usage:                      gsom = pp.weibullStandardOrderStatisticMedians();
This method returns the Standard Weibull order statistic median used in the Probability Plot.


COMMON METHODS

Critical value for the correlation coefficients
public double correlationCoefficientCriticalValue()
Usage:                      critVal = pp.correlationCoefficientCriticalValue();
This method returns the critical value for the correlation coefficient calculated for the number of data points entered and at the default significance level. This is initially set at 0.05 [5%]. See below for a method allowing this level to be reset.

public double correlationCoefficientCriticalValue(double significance)
Usage:                      critVal = pp.correlationCoefficientCriticalValue(significance);
This method returns the critical value for the correlation coefficient calculated for the number of data points entered and at the significance level entered as the argument significance.

Significance level
public void resetSignificance(double significance)
Usage:                      pp.resetSignificance(significance);
This method allows the default significance level to be reset to the value entered as the argument significance.

public double getSignificance()
Usage:                      signif = pp.getSignificance();
This method returns the default significance level.

Return the Ordered Data
public double[] orderedData()
Usage:                      orderedData = pp.orderedData();
This method returns the data entered via the Constructor sorted into an ascending order.

Mean of the Data
public double mean()
Usage:                      mean = pp.mean();
This method returns the mean of the data entered via the Constructor.

Standard Deviation of the Data
public double mean()
Usage:                      sd = pp.standardDeviation();
This method returns the standard deviation of the data entered via the Constructor. See below (setting denominator) for standard deviation denominator options.

Minimum of the Data
public double minimum()
Usage:                      min = pp.minimum();
This method returns the minimum of the data entered via the Constructor.

Maximum of the Data
public double maximum()
Usage:                      max = pp.maximum();
This method returns the maximum of the data entered via the Constructor.

Return the Initial Estimates
public double[] getInitialEstimates()
Usage:                      initialEstimates = pp.getInitialEstimates();
This method returns the initial estimates used in last call to a probability plot method.

SIMPLEX STEP SIZE
The initial step size needed by the Nelder and Mead Simplex is calculated, in general, by multiplying the absolute value of the relevant initial estimate by the step factor, stepFactor. The default value of stepFactor is 0.2. If a calculated step size equals zero it is replaced by a positive value whose value varies with the probability plot being called.

Value of the Step Size Factor, stepFactor
public double stepFactor()
Usage:                      delta = pp.stepFactor();
This method returns the value of the step size factor, stepFactor. Its default value is 0.2.

Reset the Step Size Factor, stepFactor
public void resetStepFactor()
Usage:                      pp.resetStepFactor(stepFactor);
This method allows the value of the step size factor to be alterted from the default value of 0.2 to a user supplied value, stepFactor.

NUMERICAL DIFFRENTIATION
The second derivatives used in calculating the covariance matrix in estimating the errors of the best estimates of the distribution parameters are calculated numerically

Value of the Numerical Differentiation Step, δ
public double delta()
Usage:                      delta = pp.delta();
This method returns the value of the numerical step, δ. Its default value is 1.0 x 10-2.

Reset the Numerical Differentiation Step, δ
public void resetDelta()
Usage:                      pp.resetDelta(delta);
This method allows the value of the numerical step, δ to be alterted from the default value of 1.0 x 10-2 to the user supplied value, delta.

STANDARD DEVIATION DENOMINATOR
The denominator of variances, covariances, standard deviations, skewnesses and kurtoses may be written as n or as n−1 depending on the statistical context, e.g

The default option for this class is n−1 but methods are provided for resetting this option to n

Set the denominator to n − 1
public void setDenominatorToNminusOne()
Usage:                      pp.setDenominatorToNminusOne();
This method sets the denominator to n − 1 for the instance pp. This is this class's default value and this method need only be called if the denominator needs resetting after calling the method below.

Set the denominator to n
public void setDenominatorToN()
Usage:                      pp.setDenominatorToN();
This method sets the denominator to n for the instance pp.

SUPPRESS THE DISPLAY OF THE PLOT
public void suppressDisplay()
Usage:                      pp.suppressDisplay();
This method suppresses the display of the probability plot which will be displayed on the first call to any of the methods of a specific distribution, e.g. frechetProbabiltyPlot(), frechetMu(), frechetGradient(). The plot is not displayed by calls to further methods related a specific distribution once the first call has been made. Suppression of the plot on the first call is not recommended as a graphical display of the plot is one of the best methods of checking the goodness of fit. If such suppression is needed this method should be called.

RESTORE THE DISPLAY OF THE PLOT
public void restoreDisplay()
Usage:                      pp.restoreDisplay();
This method restores the display of the probability plot which will be displayed on the first call to any of the methods of a specific distribution, e.g. frechetProbabiltyPlot(), frechetMu(), frechetGradient(). This method need only be called if the above method suppressing the display, suppressDisplay(), has been called and restoration of the display is required.


OUTPUT TO A TEXT FILE

All the above methods calling a probability plot also output a text file containing details of the plot. The title of the text file will be XxxxxxProbabilityPlotOutput.txt where Xxxxxx is the name of the distribution, e.g. LogisticProbabilityPlotOutput.txt

The file contents are

       Xxxxxxx Probability Plot [where Xxxxxx is the name of the distribution]

       Date and time

       Method used in the regression procedure

       Gradient of the plot and its error
       Intercept of the plot and its error

       Correlation Coefficient of the observed responses and the order statistic medians
       Sum of squares of differences between the observed responses and the order statistic medians

       Best estimates of the distribution parameters and the estimated errors

       Matrix of parameter-parameter correlation coefficients

Example output file

This is the output file for a three parameter Weibull distribution with μ [mu] = 2.0, β [beta] = 3.0 and γ [gamma] = 4.0

       Three Parameter Weibull Probability Plot

       Tuesday, 24 February 2015, 9.37

       Unweighted constrained simplex regression

       Gradient:   1.0               s.e. 0.13
       Intercept: 3.0382E-9     s.e. 0.6129

       Correlation Coefficient, r:    0.9943
       Critical Value for r:              0.1966
       Significance level used:        5.0%
       Sum of squares:                    0.3391

       Best Estimates of the Parameters
       mu:         2.4096      ‘pseudo-linear'’ s.e. 1.1441
       beta:        2.5075      ‘pseudo-linear'’ s.e. 1.2083
       gamma:   3.1465      ‘pseudo-linear'’ s.e. 1.7180

       ‘Pseudo-linear'’ Parameter-Parameter Correlation Coefficients
                      mu            beta          gamma
       mu          1.0000     -0.9961    -0.9630
       beta        -0.9961     1.0000     0.9629
       gamma   -0.9630     0.9629     1.0000

Note the high degree of correlation between parameters and the consequent inaccurate estimates of these parameters.
The plot is shown below:



SUPPRESS THE OUTPUT TO A TEXT FILE
public void suppressFileOutput()
Usage:                      pp.suppressFileOutput();
This method suppresses the output to a text file as described immediately above.

RESTORE THE OUTPUT TO A TEXT FILE
public void restoreFileOutput()
Usage:                      pp.restoreFileOutput();
This method restores output to a text file as described above. This method need only be called if the above method suppressing the output, suppressFileOutput(), has been called and restoration of the output is required.


SUPPRESS THE REGRESSION CLASS CONVERGENCE AND CHISQUARE ERROR MESSAGES
public void suppressErrorMessages()
Usage:                      pp.suppressErrorMessages();
This method suppresses the printing to screen of the error messages relating to lack of convergence and failure to invert the Hessian matrix in the regression procedures in the called Regression class.

RESTORE THE REGRESSION CLASS CONVERGENCE AND CHISQUARE ERROR MESSAGES
public void restoreErrorMessages()
Usage:                      pp.restoreErrorMessages();
This method restores the printing to screen of the error messages relating to lack of convergence and failure to invert the Hessian matrix in the regression procedures in the called Regression class.



EXAMPLE PROGRAM

 The example program, GaussianFit, illustrates the use of the ProbabilityPlot class in fitting examination marks to a Gaussian distribution.

Detailed description: GaussianFit
Source file: GaussianFit.java


OTHER CLASSES USED BY THIS CLASS

 This class uses the following classes in this library:

PERMISSION TO USE

Permission to use, copy and modify this software and its documentation for NON-COMMERCIAL purposes is granted, without fee, provided that an acknowledgement to the author, Dr Michael Thomas Flanagan at www.ee.ucl.ac.uk/~mflanaga, appears in all copies and associated documentation or publications.

Public listing of the source codes on the internet is not permitted.

Redistribution of the source codes or of the flanagan.jar file is not permitted.

Redistribution in binary form of all or parts of these classes is not permitted.

Dr Michael Thomas Flanagan makes no representations about the suitability or fitness of the software for any or for a particular purpose. Dr Michael Thomas Flanagan shall not be liable for any damages suffered as a result of using, modifying or distributing this software or its derivatives.



This page was prepared by Dr Michael Thomas Flanagan