*â€¦People who wish to analyze nature without using mathematics must settle for a reduced understandingâ€¦*

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.

*â€¦Because I had worked in the closest possible way with physicists and engineers, I knew that our data can never be preciseâ€¦*

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.

*â€¦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 beginningâ€¦*

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.Â providing informative data which facilitate the process of parameter identification and model selection.

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

2. Levinson N: **Wienerâ€™s life**. *Bull Amer Math Soc* 1966, **72**:1â€“32.

3. Oâ€™Hagan A, Oakley JE: **Probability is perfect, but we canâ€™t elicit it perfectly**. *Reliability Engineering & System Safety* 2004, **85**:239â€“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**:1543â€“1556.

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

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].

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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