mathematics//statistics//uncertainty propagation
Uncertainty propagation is the computation of how the uncertainty of the inputs of a function carries into the uncertainty of its output, and engineers use it to bring every error source of a system into the same units at the same point: metres of drone position, degrees at a thermocouple's output, bar at a pressure controller. It takes each source's spread and the function that links it to the output; it returns the output's spread, which an error budget then adds up.
Uncertainty propagation is the computation of how the uncertainty of the inputs of a function carries into the uncertainty of its output, and engineers use it to bring every error source of a system into the same units at the same point: metres of drone position, degrees at a thermocouple's output, bar at a pressure controller. It takes each source's spread and the function that links it to the output; it returns the output's spread, which an error budget then adds up.
For small errors the function is replaced by its slope (linearization). A source xxx with standard deviation σx\sigma_xσx affecting an output y=f(x)y=f(x)y=f(x) contributes σy≈∣∂f/∂x∣σx\sigma_y\approx\left|\partial f/\partial x\right|\sigma_xσy≈∣∂f/∂x∣σx; for vectors of correlated sources the slopes form the Jacobian JJJ and the covariances travel together,
Σy≈J Σx JT.\Sigma_y\approx J\,\Sigma_x\,J^{\mathsf T}.Σy≈JΣxJT.
Σx\Sigma_xΣx is the covariance matrix of the inputs and Σy\Sigma_yΣy that of the outputs. This is the same computation an extended Kalman filter performs at every prediction step, with the dynamics' Jacobian in the place of JJJ (covariance propagation).
Three common sources fall out at once. A quantization step Δ\DeltaΔ gives a uniform error of standard deviation Δ/12\Delta/\sqrt{12}Δ/12 (a position sent with 1 cm resolution carries 3 mm). An uncompensated delay τ\tauτ on a drone flying at speed vvv shifts its position by vτv\tauvτ: 50 ms at 5 m/s is 25 cm. A clock offset δt\delta tδt between two sensors does the same, v δtv,\delta tvδt.
It is valid while the function is close to straight over the spread. When the curvature matters at the scale of the uncertainty (an angle of a few tens of degrees through a sine, a ratio whose denominator can approach zero), the linear answer is wrong and a Monte Carlo method that pushes samples through the true function is the safer route, as the metrology guide (GUM) and its supplement recommend.
Correlations must travel with the variances. Dropping the off-diagonal terms of Σx\Sigma_xΣx treats a latency and a clock offset that both grow with speed as independent, and underestimates their combined effect; correlated errors and biases add linearly, independent random ones in quadrature (root sum square).
The result is only as good as the inputs. A sensor σ\sigmaσ taken from an optimistic datasheet propagates into an optimistic output, which is why each source's number should come from measurement (noise characterization).