Math 2.0

Numerical analysis Definition

Propagation of uncertainty

Builds on Covariance matrixJacobian matrixExpected valueVariance

Definition

Let X=(X1,…,Xn)X = (X_1, \ldots, X_n) be a random vector with mean μ∈ℝn\mu \in \R^n and covariance matrix ΣX\Sigma_X. Let f ⁣:ℝn→ℝmf \colon \R^n \to \R^m be differentiable at μ\mu, with Jacobian matrix J=Df(μ)J = Df(\mu). The linear propagation of uncertainty assigns to Y=f(X)Y = f(X) the approximate mean f(μ)f(\mu) and the approximate covariance matrix ΣY=J ΣX JT.\Sigma_Y = J \, \Sigma_X \, J^{T}. For m=1m = 1 and uncorrelated XiX_i with variances σi2\sigma_i^2, this reads σY2=∑i=1n(∂f∂xi(μ))2σi2.\sigma_Y^2 = \sum_{i=1}^{n} \left( \frac{\partial f}{\partial x_i}(\mu) \right)^2 \sigma_i^2. It is the covariance of the first-order Taylor approximation f(μ)+J(X−μ)f(\mu) + J(X - \mu).

Remarks

Example: for f(x1,x2)=x1x2f(x_1, x_2) = x_1 x_2 and uncorrelated inputs, ∂f/∂x1=μ2\partial f/\partial x_1 = \mu_2 and ∂f/∂x2=μ1\partial f/\partial x_2 = \mu_1, so σY2=μ22σ12+μ12σ22\sigma_Y^2 = \mu_2^2 \sigma_1^2 + \mu_1^2 \sigma_2^2, that is (σY/μ1μ2)2=(σ1/μ1)2+(σ2/μ2)2(\sigma_Y/\mu_1\mu_2)^2 = (\sigma_1/\mu_1)^2 + (\sigma_2/\mu_2)^2 when μ1μ2≠0\mu_1 \mu_2 \ne 0. If f(x)=Ax+bf(x) = Ax + b is affine, the formula is exact: Cov(AX+b)=AΣXAT\mathrm{Cov}(AX + b) = A \Sigma_X A^T. Non-example: for f(x)=x2f(x) = x^2 and XX normal with mean 00 and variance σ2\sigma^2, the formula gives σY2=0\sigma_Y^2 = 0, while the true variance of X2X^2 is 2σ42\sigma^4; the linearization fails because f′(μ)=0f'(\mu) = 0.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Covariance matrix, concept, no evidence levelCovariance matrixconceptEuclidean space, object, no evidence levelEuclidean spaceobjectExpected value, concept, no evidence levelExpected valueconceptFunction, concept, no evidence levelFunctionconceptVariance, concept, no evidence levelVarianceconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptJacobian matrix, concept, no evidence levelJacobian matrixconceptPropagation of uncertainty, definition, no evidence levelPropagation of uncert…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Covariance matrix concept
  2. Euclidean space object
  3. Expected value concept
  4. Function concept
  5. Variance concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Partial derivative concept
  9. Jacobian matrix concept
  10. Propagation of uncertainty definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Propagation_of_uncertainty
About this page
TagZQKA90X325
URIm2://ZQKA90X325
Kinddefinition
WritingW0 stub
MSC65G99, 62P99
DefinesPropagation of uncertainty
UsesCovariance matrix, Jacobian matrix, Expected value, Variance

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