Uncertainty Quantification

Mathematical Modeling · Axiom Academy

LESSON Uncertainty Quantification Unit 7: Data-Driven Modeling - Mathematical Modeling Before we can quantify uncertainty, we must understand where it comes from. Uncertainty in mathematical models arises from two fundamentally different sources: When inputs have uncertainty, how does that uncertainty flow through our model to affect the output? For a function y = f(x) where x has uncertainty, we can approximate the output uncertainty using Taylor expansion. For a single variable with small uncertainty: For multiple independent variables, the uncertainties add in quadrature: 3. Confidence Intervals for Parameters When we estimate model parameters from data, we obtain not just point estimates but also measures of their uncertainty. For least-squares fitting, the parameter covariance matrix captures this information. where J is the Jacobian matrix of the model with respect to parameters. The diagonal elements give the variance of each parameter estimate. While confidence intervals describe uncertainty in parameter estimates, prediction intervals describe the uncertainty in future observations. Prediction intervals are always wider because they include both parameter uncertainty and inherent noise. Confidence interval for the mean: Where we expect the true mean response to lie Prediction interval: Where we expect individual future observations to lie Beyond analytical formulas, several computational methods are commonly used for uncertainty quantification in complex models:

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