Variogram Analysis of Response Surfaces (VARS): A Powerful Global Sensitivity Analysis Method

A diagram showing the input importance of different machine-learning models
Figure 6 (Panigrahi et al., 2025)

Variogram Analysis of Response Surfaces (VARS) is a state-of-the-art framework for conducting Global Sensitivity Analysis (GSA), developed by Dr. S. Razavi and Dr. H.V. Gupta in 2016, offers a unified and rigorous approach grounded in the theory of variogram to quantify the influence of input variables on model outputs. Unlike conventional GSA techniques like Sobol and Morris, VARS captures both the magnitude and scale of sensitivity across a model’s response surface. Its computational efficiency and robustness make it ideal for handling complex, high-dimensional problems.

Over the years, several specialized packages of VARS have been developed to extend its capabilities, including:

  • G-VARS: While VARS assumes a uniform distribution for its parameters and samples accordingly, this is a package for GSA of models with correlated, nonuniformly distributed factors. This package applied the inverse Nataf transformation to sample the joint and conditional distributions of factors.
  • D-VARS: This package also called data-driven VARS characterizes the relationship strength between inputs and outputs by investigating their covariograms that works on any sample of input‐output data or pre‐computed model evaluations.
  • X-VARS: This package was developed to enable explainable artificial intelligence (XAI) through the lens of sensitivity analysis. Its key strength lies in its ability to characterize the entire response surface of a machine learning model, rather than focusing only on the local behavior within the available input–output data. This makes it applicable to a wide range of model types, including connectionist, kernel‑based, and tree‑based ML approaches.

These tools have been implemented in both Python and MATLAB, providing accessible, high-performance packages for researchers and engineers. Its robustness has led to widespread adoption in sensitivity and uncertainty analysis across various fields.

For more details on the methodology, application guidance, and access to educational resources, including tutorial videos and published articles, please visit Dr. Razavi’s YouTube channel and the referenced publications.

VARS implementations in Python and MATLAB are openly available through GitHub repositories. VARS‐TOOL is licensed under the GNU General Public License, Version 3.0 or later, Copyright (C) 2015‐21 Saman Razavi, University of Saskatchewan.

  • Razavi, S., & Gupta, H. V. (2016a). A new framework for comprehensive, robust, and efficient global sensitivity analysis: 1. Theory. Water Resources Research, 52(1), Article 1. https://doi.org/10.1002/2015WR017558
  • Razavi, S., & Gupta, H. V. (2016b). A new framework for comprehensive, robust, and efficient global sensitivity analysis: 2. Application. Water Resources Research, 52(1), Article 1. https://doi.org/10.1002/2015WR017559