2. Propagation of Uncertainty

Attribution

This chapter was written by Sandra Verhagen. Find out more here.

In engineering and sciences we often work with functions of random variables, since when estimating or modelling something, the output is a function of the random input variables, see Fig. 2.1

Fig. 2.1 Output \(X\) of a model is function of random input \(Y\).

Some simple examples are:

• conversion of temperature measured in degrees Celsius to temperature in degrees Fahrenheit: \(T_f = q(T_c)=\frac{9}{5}T_c+32\)

• taking the mean of \(m\) repeated measurements \(Y_i\): \(\hat{X}=q(Y_1,\ldots,Y_m)=\frac{1}{m}\sum_{i=1}^m Y_i\)

• subsurface temperature \(T_z\) as a function of depth \(Z\) and surface temperature \(T_0\) and known \(a\): \(T_z = T_0 + aZ\)

• wind loading \(F\) on a building as function of area of building face \(A\), wind pressure \(P\), drag coefficient \(C\): \(F = A\cdot P\cdot C\)

• Evaporation \(Q\) using Bowen Ratio Energy Balance as function of the net radiation \(R\), ground heat flux \(G\), bowen ratio \(B\): \(Q =q(R,G,B) =\frac{R-G}{1-B}\)

Fig. 2.2 shows an example of the distribution of the average July temperature in a city, both in degrees Celsius and degrees Fahrenheit. Due to the change of units, the PDF is transformed, the mean is shifted and the variance changed.

Fig. 2.2 Distribution of temperature in degrees Celsius and degrees Fahrenheit.

The question we are interested in is: how does the statistical uncertainty in input data propagate in the output variables?

In this part, we will first introduce random vectors, covariance, correlation and covariance matrices, as well as the multivariate normal distribution, uncertainty classifications, and then look at the propagation laws that we need.