I believe the best we can use is bilinearity of coefficients, i.e.,
Cov(aX, bY)=abCov(X,Y).
But you're right, beyond that, I think there are no rules for f with f=f(X,Y).
Edit: I suspect the answer here may fall under propagation of errors/uncertainty
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Propagation of uncertainty
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For the propagation of uncertainty through time, see
Chaos theory § Sensitivity to initial conditions.
In
statistics,
propagation of uncertainty (or
propagation of error) is the effect of
variables'
uncertainties (or
errors, more specifically
random errors) on the uncertainty of a
function based on them. When the variables are the values of experimental measurements they have
uncertainties due to measurement limitations (e.g., instrument
precision) which propagate due to the combination of variables in the function.
The uncertainty
u can be expressed in a number of ways. It may be defined by the
absolute error Δ
x. Uncertainties can also be defined by the
relative error (Δ
x)/
x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the
standard deviation, σ, which is the positive square root of the
variance. The value of a quantity and its error are then expressed as an interval
x ±
u. However, the most general way of characterizing uncertainty is by specifying its
probability distribution. If the
probability distribution of the variable is known or can be assumed, in theory it is possible to get any of its statistics. In particular, it is possible to derive
confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a
normal distribution are approximately ± one standard deviation
σ from the central value
x, which means that the region
x ±
σ will cover the true value in roughly 68% of cases.
If the uncertainties are
correlated then
covariance must be taken into account. Correlation can arise from two different sources. First, the
measurement errors may be correlated. Second, when the underlying values are correlated across a population, the
uncertainties in the group averages will be correlated.
[1]
In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the
Monte Carlo method family.
[2] For very expansive data or complex functions, the calculation of the error propagation may be very expansive so that a
surrogate model[3] or a
parallel computing strategy
[4][5][6] may be necessary.
In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.
Linear combinations
Non-linear combinations
Example formulae
Example calculations
See also
References
Further reading
External links
Last edited 1 month ago by Hellacioussatyr
