Error Budgets
An itemized accounting of every error source in a prediction, added up so the total uncertainty is defensible rather than guessed.
Adding Up the Errors
A prediction is only as trustworthy as the accounting behind its uncertainty. An error budget is an itemized list of every significant error source in a computed result, with a magnitude for each and a rule for combining them into a total. It converts a vague sense that a result is roughly right into a defensible statement of how wrong it could be and why.
Typical Line Items
- Discretization error, from the convergence and order-of-accuracy study.
- Iterative and round-off error, usually small but not always negligible.
- Input parameter uncertainty, from measurement and estimation.
- Model-form uncertainty, from approximations and omitted physics.
- Statistical error, for methods that sample.
Combining Terms
How terms combine depends on their nature. Independent random errors add in quadrature, the square root of the sum of squares, because they partly cancel. Systematic biases and bounds add linearly, because they can align in the worst case. A budget must state which rule it uses for each term; mixing them silently understates or overstates the total.
import math
random_terms = [0.012, 0.008, 0.015] # combine in quadrature
systematic_terms = [0.010, 0.005] # combine linearly
rand = math.sqrt(sum(t*t for t in random_terms))
syst = sum(systematic_terms)
total = math.sqrt(rand*rand + syst*syst)
print(f'random {rand:.3f}, systematic {syst:.3f}, total {total:.3f}')The Discipline
The value of an error budget is as much organizational as numerical. Building it forces every error source to be named, measured or bounded, and defended. A source that cannot be quantified must at least be listed as unquantified, so the reader knows the total is a lower bound. In design work the budget feeds directly into margin decisions: the design must tolerate the full budgeted uncertainty, not just the central estimate. A prediction reported without an error budget invites the reader to supply their own, usually pessimistic, guess.