Error margins

[carey]

(10-28-2024 04:59 PM)born2laser Wrote:  If you calculate a value based on measured parameters that are known with uncertainty (error), the canonical way of estimating the uncertainty of the calculated value is to multiply each error by the partial derivative respect to each parameter and then add them all in quadrature (square root of the sum of the squares). This of course assumes small errors as it is a Taylor expansion and statistical independence of the parameters.

Yes, the canonical way is the way to go, however...

(10-28-2024 04:59 PM)born2laser Wrote:  Back in college I wrote a program for my 41 that calculated the derivatives numerically and used it to estimate the propagated error...

The issue with calculating derivatives numerically is that the process of numerical differentiation (taking a difference divided by a difference) dramatically increases noise (i.e., error). For example, if starting with position data, after two numerical differentiations to find acceleration, noise resulting from the numerical differentiation operations can make the signal unusable. Differentiation acts as a high-pass filter.

Automatic differentiation (as suggested by Thomas) is a better alternative.

However, if requirements are simple, the following three rules for addition & subtraction, multiplication & division, and exponentiation (special cases of the canonical way) may suffice:
Rule 1: + - : quadrature sum the errors to get the error of the result.
Rule 2: x / : quadrature sum the fractional errors to get the fractional error of the result.
Rule 3: z^n : multiply the fractional error by the exponent n to get the fractional error of the result.