Statistics Package

The NeqSim statistics package provides tools for parameter fitting, uncertainty quantification, and data analysis for thermodynamic model development and validation.

Overview
Package Structure
Sub-Documentation
Core Concepts
Quick Start
Integration with Thermodynamic Models

Overview

The statistics package supports:

Parameter Fitting - Nonlinear regression using Levenberg-Marquardt algorithm
Monte Carlo Simulation - Uncertainty propagation and confidence intervals
Data Analysis - Smoothing, filtering, and statistical analysis
Experimental Data Management - Sample sets and experimental equipment modeling

Location: neqsim.statistics

Key Applications:

Fitting binary interaction parameters ($k_{ij}$) to experimental VLE data
Tuning EoS parameters for custom components
Uncertainty analysis for thermodynamic predictions
Regression of transport property correlations

Package Structure

statistics/
├── parameterfitting/                    # Core parameter fitting framework
│   ├── StatisticsBaseClass.java        # Abstract base for all fitting
│   ├── StatisticsInterface.java        # Interface definition
│   ├── SampleSet.java                  # Collection of experimental points
│   ├── SampleValue.java                # Single experimental data point
│   ├── BaseFunction.java               # Abstract objective function
│   ├── FunctionInterface.java          # Function interface
│   ├── NumericalDerivative.java        # Numerical differentiation
│   └── nonlinearparameterfitting/      # Nonlinear optimization
│       ├── LevenbergMarquardt.java     # L-M optimizer (least squares)
│       ├── LevenbergMarquardtAbsDev.java   # Absolute deviation
│       ├── LevenbergMarquardtBiasDev.java  # Bias deviation
│       └── LevenbergMarquardtFunction.java # Example function
│
├── montecarlosimulation/               # Uncertainty quantification
│   └── MonteCarloSimulation.java       # MC simulation runner
│
├── dataanalysis/                       # Data processing
│   └── datasmoothing/
│       └── DataSmoother.java           # Savitzky-Golay smoothing
│
├── experimentalsamplecreation/         # Sample generation
│   ├── readdatafromfile/               # File I/O for experimental data
│   └── samplecreator/
│       ├── SampleCreator.java          # Base sample creator
│       └── wettedwallcolumnsamplecreator/  # Specialized creator
│
└── experimentalequipmentdata/          # Equipment modeling
    ├── ExperimentalEquipmentData.java
    └── wettedwallcolumndata/           # Wetted wall column

Sub-Documentation

Detailed guides for each major subsystem:

Guide	Description
Parameter Fitting	Levenberg-Marquardt optimization, creating objective functions, bounds
Monte Carlo Simulation	Uncertainty propagation, confidence intervals, distribution sampling
Data Analysis	Data smoothing, filtering, statistical measures

Core Concepts

Sample Values

A SampleValue represents one experimental data point:

// Experimental value with uncertainty
double experimentalValue = 0.5;    // Measured value (e.g., pressure)
double standardDeviation = 0.05;  // Experimental uncertainty
double[] independentVariables = {300.0, 0.1};  // e.g., temperature, composition

SampleValue sample = new SampleValue(
    experimentalValue, 
    standardDeviation, 
    independentVariables
);

Sample Sets

A SampleSet is a collection of experimental points:

SampleSet sampleSet = new SampleSet();
sampleSet.add(sample1);
sampleSet.add(sample2);
sampleSet.add(sample3);

// Or from array
SampleValue[] samples = {sample1, sample2, sample3};
SampleSet sampleSet = new SampleSet(samples);

Objective Functions

Functions extend BaseFunction or LevenbergMarquardtFunction:

public class MyObjectiveFunction extends LevenbergMarquardtFunction {
    
    @Override
    public double calcValue(double[] dependentValues) {
        // params[0], params[1], ... are the fitting parameters
        // dependentValues are the independent variables (T, P, x, ...)
        
        double T = dependentValues[0];
        double x = dependentValues[1];
        
        // Calculate model prediction
        double predicted = params[0] * Math.exp(-params[1] / T) * x;
        
        return predicted;
    }
    
    @Override
    public void setFittingParams(int i, double value) {
        params[i] = value;
    }
}

Quick Start

Basic Parameter Fitting

import neqsim.statistics.parameterfitting.*;
import neqsim.statistics.parameterfitting.nonlinearparameterfitting.*;
import java.util.ArrayList;

// 1. Create objective function
MyObjectiveFunction function = new MyObjectiveFunction();

// 2. Set initial parameter guess
double[] initialGuess = {1.0, 500.0};  // Two parameters to fit
function.setInitialGuess(initialGuess);

// 3. Create experimental samples
ArrayList<SampleValue> samples = new ArrayList<>();

double[] x1 = {300.0, 0.1};  // T=300K, x=0.1
SampleValue s1 = new SampleValue(0.05, 0.005, x1);  // exp=0.05 ± 0.005
s1.setFunction(function);
samples.add(s1);

double[] x2 = {350.0, 0.2};
SampleValue s2 = new SampleValue(0.12, 0.01, x2);
s2.setFunction(function);
samples.add(s2);

// Add more samples...

// 4. Create sample set and optimizer
SampleSet sampleSet = new SampleSet(samples);
LevenbergMarquardt optimizer = new LevenbergMarquardt();
optimizer.setSampleSet(sampleSet);

// 5. Solve
optimizer.solve();

// 6. Get results
double[] fittedParams = sampleSet.getSample(0).getFunction().getFittingParams();
System.out.println("Fitted parameters: " + Arrays.toString(fittedParams));

// 7. Display results
optimizer.displayCurveFit();
optimizer.displayResult();

With Monte Carlo Uncertainty

// After solving...
optimizer.runMonteCarloSimulation(100);  // 100 Monte Carlo runs

// This generates samples with normally distributed perturbations
// around experimental values and re-fits, providing parameter distributions

Integration with Thermodynamic Models

Fitting Binary Interaction Parameters

public class KijFittingFunction extends LevenbergMarquardtFunction {
    
    @Override
    public double calcValue(double[] dependentValues) {
        double temperature = dependentValues[0];
        double pressure = dependentValues[1];
        double x_exp = dependentValues[2];  // Experimental composition
        
        // Set up thermodynamic system
        system.setTemperature(temperature);
        system.setPressure(pressure);
        
        // Set kij from fitting parameters
        ((PhaseEos) system.getPhase(0)).getMixingRule()
            .setBinaryInteractionParameter(0, 1, params[0]);
        ((PhaseEos) system.getPhase(1)).getMixingRule()
            .setBinaryInteractionParameter(0, 1, params[0]);
        
        // Flash calculation
        thermoOps.TPflash();
        
        // Return calculated liquid composition
        return system.getPhase(1).getComponent(0).getx();
    }
    
    @Override
    public void setFittingParams(int i, double value) {
        params[i] = value;
    }
}

Fitting CPA Association Parameters

public class CPAFittingFunction extends LevenbergMarquardtFunction {
    
    @Override
    public double calcValue(double[] dependentValues) {
        double T = dependentValues[0];
        double P = dependentValues[1];
        
        // params[0] = epsilon (association energy)
        // params[1] = beta (association volume)
        
        system.getComponent("water").setAssociationEnergy(params[0]);
        system.getComponent("water").setAssociationVolume(params[1]);
        
        thermoOps.TPflash();
        
        return system.getPhase(1).getDensity("kg/m3");
    }
    
    @Override
    public void setFittingParams(int i, double value) {
        params[i] = value;
    }
}

Statistical Measures

After fitting, several statistics are available:

// Solve first
optimizer.solve();

// Chi-square statistic
double chiSquare = optimizer.calcChiSquare();

// Absolute deviation statistics
optimizer.calcAbsDev();

// Covariance matrix
optimizer.calcCoVarianceMatrix();

// Parameter standard deviations
optimizer.calcParameterStandardDeviation();

// Parameter correlation matrix
optimizer.calcCorrelationMatrix();

// 95% confidence intervals
optimizer.calcParameterUncertainty();

Best Practices

Initial Guess Selection

Good initial guesses are critical for convergence:

Use physical reasoning to estimate parameter magnitude
Start with literature values when available
Try multiple starting points if convergence fails

Experimental Uncertainties

Proper uncertainty specification affects:

Weighting of data points in regression
Confidence intervals on fitted parameters
Chi-square goodness-of-fit

Parameter Bounds

Set physical bounds to prevent unphysical solutions:

double[][] bounds = {
    {0.0, 1.0},     // Parameter 0: between 0 and 1
    {100.0, 1000.0} // Parameter 1: between 100 and 1000
};
function.setBounds(bounds);

Convergence Criteria

optimizer.setMaxNumberOfIterations(100);  // Increase if needed

References

Press, W.H., et al. (2007). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.
Marquardt, D.W. (1963). An Algorithm for Least-Squares Estimation of Nonlinear Parameters. SIAM J. Appl. Math.
Savitzky, A., Golay, M.J.E. (1964). Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Anal. Chem.

Statistics Package

Comprehensive Java library for thermodynamic, physical property, and process simulation. NeqSim provides tools for PVT analysis, phase equilibria, transport properties, and full process simulation.

Statistics Package

Table of Contents

Overview

Package Structure

Sub-Documentation

Core Concepts

Sample Values

Sample Sets

Objective Functions

Quick Start

Basic Parameter Fitting

With Monte Carlo Uncertainty

Integration with Thermodynamic Models

Fitting Binary Interaction Parameters

Fitting CPA Association Parameters

Statistical Measures

Best Practices

Initial Guess Selection

Experimental Uncertainties

Parameter Bounds

Convergence Criteria

References