ODR Fit with Uncertainties

This Python script performs Orthogonal Distance Regression (ODR) on data with uncertainties in both the x and y variables. It reads data from a CSV file, performs the ODR analysis, and creates plots for the data, fit, residuals, and parameter correlation ellipses.

Features

• Reads data from a CSV file with columns for x, y, and their uncertainties (dx and dy)
• Performs ODR analysis using the scipy.odr module
• Calculates chi-square, reduced chi-square, and p-value for the fit
• Creates a figure with subplots for:
  • Data points with error bars and the best-fit line
  • Residuals plot with error bars
  • Parameter correlation ellipses (2D confidence regions) for the slope and intercept
• Prints the regression results, including the best-fit parameters, their uncertainties, correlation coefficient, chi-square, reduced chi-square, and p-value

Dependencies

• Python 3.x
• NumPy
• Pandas
• SciPy
• Matplotlib

Installation

Clone this repository, and install the dependencies. Detailed installation guide for beginners here: https://github.com/hezy/ODR-example/blob/main/Installation_Guide.md

Usage

1. Prepare a CSV file with the following columns: x, dx, y, dy

   • x: x-values of the data points
   • dx: uncertainties in the x-values
   • y: y-values of the data points
   • dy: uncertainties in the y-values

2. Run the script:

python odr-fit.py data.csv

(change to your file name)

3. The script will read the data from the CSV file, perform the ODR analysis, and save the regression results in your working directory:

fit_results.txt

• Contains complete regression analysis results
• Includes parameter correlations and statistical measures
• Shows full covariance matrix in scientific notation

Plot files:

• fit_plot.png: Data and fit line, with automatic style adjustment
• residuals_plot.png: Includes propagated uncertainties from both x and y
• correlation_ellipses.png: Parameter correlation ellipses at 1σ, 2σ, and 3σ confidence levels

Visualization Features

• Automatic smart plotting: The tool dynamically chooses between showing point markers or error bars only, based on the relative size of uncertainties
  • Shows error bars only when uncertainties are significant (>1% of data range)
  • Shows points with error bars when uncertainties are smaller

Example

An example CSV file (data.csv) is provided with the script that produced it (generate-fake-data.py). The data represents measurements with uncertainties in both x and y variables.

Running the script with the provided example data will produce the following output:

Regression Results:
-----------------
Slope: 2.147160 ± 0.016329
Intercept: -0.359677 ± 0.137575

Covariance matrix:
 2.369639e-04 -1.720251e-03
-1.720251e-03  1.682074e-02
Pearson's Correlation coefficient: -0.765758

Chi-square: 14.627831
Degrees of freedom: 13
Reduced chi-square: 1.125218
P-value: 0.331159

Best Practices

• Verify your uncertainties before analysis
• Review the residuals plot to assess fit quality
• Use the p-value and reduced chi-square to evaluate goodness-of-fit
• Check the correlation ellipses to understand parameter interdependencies

Troubleshooting

Common Issues

• If your CSV file isn't being read correctly, ensure it has exactly these column names: 'x', 'dx', 'y', 'dy'
• The script requires uncertainties (dx, dy) to be positive values
• For best visualization results, ensure your uncertainties are reasonable compared to your data range

Error Messages

The script will provide clear error messages if:

• The input file cannot be found
• The CSV is missing required columns
• The data contains invalid values

Similar Tools and Alternatives

Here are some free alternatives for fitting data with uncertainties:

Graphical Tools

• EddingtonGUI: User-friendly GUI for curve fitting with uncertainties
• Fityk: Versatile curve fitting tool that supports:
  • Both x and y error bars in fitting
  • Various weighting schemes
  • Linear and non-linear fitting
  • Both GUI and command-line interface
• Veusz: Python-based plotting and fitting tool
  • Supports fitting with y-errors
  • Good plotting capabilities
  • Can be scripted
• Grace and its fork QtiGrace: Traditional scientific plotting tools
  • Support fitting with y-errors
  • Extensive plotting features
  • Note: do not support x-errors in fitting

Python Libraries & Tools

• lmfit: Flexible curve fitting with parameter bounds
• emcee: Bayesian approach using MCMC
• kmpfit: Part of Kapteyn Package, supports ODR

Julia Alternatives

• LsqFit.jl: Non-linear least squares with weights
• CurveFit.jl: Basic curve fitting capabilities

All tools listed above are free and open source. Most support at least y-error weighted fitting, but full ODR (with both x and y errors) is less common. This script provides ODR capabilities similar to commercial tools, using the robust scipy.odr implementation.

If you prefer a graphical interface for ODR analysis, consider using EddingtonGUI, which provides:

• A user-friendly graphical interface
• Similar ODR fitting capabilities
• Interactive plot manipulation
• Export options for results and figures