Chapter 1
Overview

An uncertainty analysis allow propagating uncertainties in parameters, derived variable initial conditions and data variables through the model in order to obtain a model output uncertainty.

Uncertainty Analysis Module Areas:

Right navigator: Source and target component selection and configuration.
Central area: Results area.

Starting an uncertainty analysis

In order to start an uncertainty analysis follow these steps:

Make sure the simulation in the Simulation module is properly configured.
Add source components to the uncertainty analysis. These are model components (parameters, data variables and/or derived variable initial conditions) which are uncertain and for which the uncertainty will be propagated through the model.
Assign uncertainty distributions to the source components.
Add target components to the uncertainty analysis. These are model components (algebraic and derived variables) for which we want to know the uncertainty caused by the uncertainty of the source components.
Specify the output interval for the target components at which uncertainty results will be calculated.
Specify the number of model evaluations that will be done.
Press Start to start the uncertainty analysis.

Chapter 2
Uncertainty Settings

The Uncertainty Analysis module toolbar gives access to two settings:

Number of evaluations: Determines how many model evaluations will be performed in order to propagate the source component uncertainty to the target components.
Enable/Disable Graphs: Allows switching off the graphs during the analysis. Switching off the graphs greatly increases the analysis speed since no intermediate calculations need to be made.

Chapter 3
Source Components

Source components are model components (parameters, data variables and/or derived variable initial conditions) which are uncertain and for which the uncertainty will be propagated through the model.

Add source components by clicking the 'Add' button next to the Source Components item in the right navigator.
Remove a source component by selecting it and clicking the 'Rem' button.

Source Component Properties

Distribution (for parameters and derived variable initial conditions): Type of uncertainty distribution of the source component.

Uniform (continuous): A continuous distribution where each value between a minimum and maximum has the same probability of occurring.

Minimum: The minimum value for this source component distribution.
Maximum: The maximum value for this source component distribution.

Triangular (continuous): A distribution which is characterized by a triangular shape defined by a minimum, mode and maximum.

Minimum: The minimum value for this source component distribution.
Mode: The source component value for which this distribution reaches its peak.
Maximum: The maximum value for this source component distribution.

Normal (continuous): A continuous distribution with the typical bell-shaped probability density function.

Mean: Mean for this source component distribution.
Std. dev.: Standard deviation for this source component distribution.

Error type (for data variables): Type of error associated with a data variable.

Absolute:

Std. dev.: The measurement error standard deviation for this data variable. Based on this value, a normal distribution with mean 0 and the given standard deviation will be generated. From this distribution a sample will be generated that is added to each data point of the data variable.

Relative:

Percentage: Error percentage relative to the measured value. From a normal distribution with mean 0 and standard deviation 1, a sample will be generated. These values will be scaled before they are added to the measured value such that a value of 1 corresponds to the given percentage of the measured value.

Allow negative: Allow the finally generated data points to be negative otherwise negative values will be turned into zero values.

Sampling

In order to propagate the source component uncertainty through the model, samples need to be taken from the source component uncertainty distributions. The number of samples is defined in the toolbar of the module ('Number of evaluations'). The PhytoSim Uncertainty Analysis modules uses Latin Hypercube Sampling (see Helton and Davis, 2003) to make sure that all uncertainty distributions are evenly sampled.

Once an uncertainty analysis is started, the final uncertainty distribution for each source component is shown when a source component is selected. This allows checking if the generated samples follow the uncertainty distribution as defined by the source component properties. During the analysis the uncertainty distribution of the already evaluated samples is also shown. As the analysis proceeds, the current uncertainty distribution will converge to the final uncertainty distribution.

References

Helton J.C. and Davis F.J. (2003) Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliability Engineering & System Safety, 81(1), 23-69.
De Pauw D.J.W., Steppe K. and De Baets B. (2008) Unravelling the output uncertainty of a tree water flow and storage model using several global sensitivity analysis methods. Biosystems engineering, 101(1), 87-99.

Chapter 4
Target Components

Target Component Properties

Interval: The interval at which uncertainty output will be generated for this target component.

Results

For each of the samples taken from the source component uncertainty distributions, a model evaluation is performed. After all N (total number of evaluations) evaluations are performed, the output uncertainty distribution at each time instance of each target component can be assessed. The PhytoSim Uncertainty Analysis module summarizes the distribution based on the minimum, mean and maximum values and the 5th, 50th (median) and 95th percentiles. The 95th percentile represents the target component value for which 95% of the model evaluations were below and 5% above. These distribution summary values are visualised on a graph when a target component is selected from the right navigator.

For each target component a time averaged (over all time instances) variance of the distribution is also calculated. This variance is shown in the lower graph of the results area for each of the target components as a function of the evaluation number. The flattening out of these variances is an indication that a sufficient number of evaluations is performed in order to obtain a proper target component uncertainty distribution.

Chapter 5
Examples

Following example workspaces can be found in the Examples/PhytoSimUncertainty folder of your PhytoSim folder:

Irrigation/Staci/Staci.psw: Illustration of parameters, derived variable initial conditions and data variables used as source components.
Misc/Linear/Linear.psw: Basic example of an uncertainty analysis of a simple linear model using both a uniform and a triangular distribution.

Caution: Example workspaces installed in the Examples folder by the modules are overwritten each time PhytoSim is started. Any modification made to these files will be lost. When you want to use an example workspace as a starting point for your own work, first save a copy of it in the Workspaces folder.