## Uncertainty Analysis in Modelling by the Point Estimate Method (PEM)

Uncertain Inputs

Click the button to update the calculations displayed in the main panel after changes in the slider positions and numeric values, respectively.

### Why uncertainty analysis in modelling matters

…People who wish to analyze nature without using mathematics must settle for a reduced understanding…
R. P. Feynman [1]

Mathematical models have become an everyday tool in science and industry. Here, to name but a few applications, mathematical models help to gain a deeper understanding of complex systems or to improve manufacturing processes.

…Because I had worked in the closest possible way with physicists and engineers, I knew that our data can never be precise…
N. Wiener [2]

To a certain extent, variability exists in almost all physical systems. The quantification of this uncertainty, as well as its proper representation might be challenging by itself [3]. When focusing on measurement data, $$y^{data}(t_k)$$, by way of example, exact values of measured quantities cannot be derived in practice, because of limitations in the measurement equipment or because of the inherent variability (process noise) [4] of the system under study. Consequently, the uncertainty in measurements has to be addressed carefully and its effect onto the estimated model parameters, $$\hat{\theta}$$, and/or simulation results, $$y^{sim}(t)$$, has to be investigated adequately.

…uncertainty assessment is not just something to be added after the completion of the modelling work. Instead uncertainty should be seen as a red thread throughout the modelling study starting from the very beginning…
J. C. Refsgaard [5]

For this purpose the so-called Point Estimate Method might be an interesting approach - worth to be considered for various disciplines/applications. For instance, in Schenkendorf2014 [6] PEM has been successfully applied to Optimal Experimental Design, i.e. providing informative data which facilitate the process of parameter identification and model selection.

### References

1. Www.feynman.com [http://www.feynman.com]

2. Levinson N: Wiener’s life. Bull Amer Math Soc 1966, 72:1–32.

3. O’Hagan A, Oakley JE: Probability is perfect, but we can’t elicit it perfectly. Reliability Engineering & System Safety 2004, 85:239–248.

4. Stengel RF: Optimal Control and Estimation. Dover Publications; 1994.

5. Refsgaard JC, Sluijs JP van der, Hojberg AL, Vanrolleghem PA: Uncertainty in the environmental modelling process - a framework and guidance. Environmental Modelling & Software 2007, 22:1543–1556.

6. Schenkendorf R: Optimal experimental design for parameter identification and model selection. PhD thesis. Otto-von-Guericke University, Magdeburg, Germany; 2014.

### Gompertz Function

The intention of this example is to demonstrate the performance of PEM in case of uncertainty propagation. In detail, the induced uncertainty about the time-dependent outcome of the so-called Gompertz function is analysed

$y(a,b,c,t) = a\cdot e^{-b\cdot e^{-c\cdot t}}$

The detailed specifications of the applied distributions are given by

$a \sim \mathcal{N}(5,0.1)$ $b \sim \mathcal{N}(2,0.1)$ $c \sim \mathcal{N}(3,0.1)$

(By changing the sliders on the left side the standard deviation of the parameters can be modified.)

In a subsequent step, PEM is applied to determine the mean, $$E[y(t)]$$, and the variance, $$\sigma_{y}^2(t)$$. The numerical results are illustrated in the figure above. In addition to the expectation value $$E[y(t)]$$ the 95% confidence intervel (CI) $$CI=\{E[y(t)] \pm 3\cdot \sqrt{\sigma_{y}^2(t)}\}$$ is presented. In comparison to Monte Carlo simulations, the proposed concept provides working results, i.e, the uncertainty about $$y(t)$$ is approximated properly by a minimum of computational load [1].

#### References

1. Schenkendorf R: Optimal experimental design for parameter identification and model selection. PhD thesis. Otto-von-Guericke University, Magdeburg, Germany; 2014.

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