Uncertainty quantification
Objective. Propagate uncertainty through mathematical models.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
Uncertainty quantification propagates uncertainty in a model's inputs through to uncertainty in its outputs, using methods ranging from analytic variance propagation to Monte Carlo sampling of input distributions. For a linear combination Y = aX₁ + bX₂ with independent inputs, Var(Y) = a²Var(X₁) + b²Var(X₂), so uncertainties do not simply add. For example, if X₁ has variance 4 and X₂ has variance 9, and Y = X₁ + X₂, then Var(Y) = 4 + 9 = 13, giving a standard deviation of √13 rather than the sum of the individual standard deviations. When inputs are correlated, a covariance term must also be included.
Worked example
In an uncertainty quantification problem, output Y = X₁ + X₂ depends on two independent inputs with variances Var(X₁) = 5 and Var(X₂) = 11. Find the standard deviation of Y.
- 1Since X₁ and X₂ are independent, use Var(Y) = Var(X₁) + Var(X₂).
- 2Substitute the values: Var(Y) = 5 + 11 = 16.
- 3Take the square root of the variance to get the standard deviation: √16.
- 4Simplify √16 = 4, the standard deviation of Y.
Answer: 4
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Start practisingCommon mistake. A common mistake is adding standard deviations directly when combining independent uncertain inputs, instead of adding variances (squared standard deviations) and then taking the square root of the total.
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