Sobol Indices

Description

Sobol indices show how much of the variation in a model's output can be traced to each of its inputs. The analyst sets a range or probability distribution for every input and runs the model on a large, carefully designed sample of input combinations. The variance (spread) of the outputs is then split into shares. A first-order index is the share caused by one input varying on its own, and a total-order index also counts every interaction that input takes part in. A large gap between the two signals that an input matters mainly through interactions, and optional second-order indices show which pairs of inputs interact.

Example Use Cases

Explainability

Analysing a climate prediction model to determine which atmospheric parameters (temperature, humidity, pressure) contribute most to rainfall forecast uncertainty, helping meteorologists understand which measurements need the highest precision.

A risk team validating a portfolio valuation model asks which economic indicators (interest rates, inflation, GDP growth) drive most of the variation in predicted portfolio value. A large gap between an indicator's total-order and first-order indices shows it acts mainly through interactions, which tells the team which scenarios to stress-test together.

Limitations

The standard sampling scheme needs N(2D+2) model runs for D inputs (N(D+2) without second-order indices), with N often in the thousands. A slow model, or one with dozens of inputs, quickly becomes impractical. Grouping related inputs, or screening first with a cheaper method such as Morris, reduces the cost.
Assumes the inputs vary independently. If real inputs are correlated, for example income and postcode, the sample contains combinations the model never meets in practice, and the indices can mislead about which inputs matter.
Needs a probability distribution or range for every input, and the indices depend on that choice, because widening one input's range raises its share of the variance. If the ranges do not reflect the real deployment population, the ranking describes a model the users will never face.
Variance is the only yardstick. When the output is heavily skewed or has several peaks, variance shares can be unreliable guides to which inputs matter, and a moment-independent method such as PAWN may suit better.
Estimates carry sampling error. With too few runs the confidence intervals can be as wide as the indices themselves, so small indices cannot be told apart from zero. Report the intervals SALib returns and check that the ranking holds when N is increased.

Resources

Research Papers

Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates

I.M Sobol′•Feb 1, 2001

Importance measures in global sensitivity analysis of nonlinear models

Toshimitsu Homma and Andrea Saltelli•Apr 1, 1996

Software Packages

UQpy

Dec 1, 2017

UQpy (Uncertainty Quantification with python) is a general purpose Python toolbox for modeling uncertainty in physical and mathematical systems.

SALib

May 30, 2013

Sensitivity Analysis Library in Python. Contains Sobol, Morris, FAST, and other methods.

Tutorials

Basics — SALib's documentation

Salib Developers•Jan 1, 2025

Documentations

Sobol indices — UQpy v4.2.0 documentation

Uqpyproject Developers•Jan 1, 2017

API — SALib documentation

Related Techniques

Name	Description	Assurance Goals
SHapley Additive exPlanations	SHAP (SHapley Additive exPlanations) explains one prediction by sharing out the gap between that prediction and a baseline, usually the model's average output over a background dataset, among the input features. Each feature receives a signed score, where positive means it pushed the prediction above the baseline and negative means it pushed it below. The scores are Shapley values from game theory, meaning each is the feature's average effect over every order in which the features could be added, and together they add up exactly to the gap. Exact calculation is costly, so tools usually estimate the scores by sampling or use shortcuts for particular model types, such as a fast exact method for tree ensembles, and averaging the absolute scores over many predictions gives a global ranking of features. In a fairness use, the difference in average output between two groups is shared out feature by feature in the same way, showing which inputs account for the gap.	Explainability Fairness
Permutation Importance	Permutation Importance measures how much a trained model relies on each input feature. For each feature, the values in a dataset are shuffled at random, which breaks the link between that feature and the outcome, and the model is scored again. The fall in score, for example in accuracy, is the feature's importance: a large fall means the model depends on the feature, and little or no fall means it does not. Repeating the shuffle several times shows how much the result varies. The method works with any model, but it shows what the model uses, not what causes the outcome in the world.	Explainability
Partial Dependence Plots	Partial Dependence Plots show how changing one or two features affects a model's predictions on average. The technique works by varying the selected feature(s) across their full range whilst keeping all other features fixed at their original values, then averaging the predictions. This creates a clear visualisation of whether increasing or decreasing a feature tends to increase or decrease predictions, and reveals patterns like linear trends, plateaus, or threshold effects that help explain model behaviour.	Explainability