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Visual Methods

Cumulative Sum of Normalized Reordered Output (CUSUNORO)

This method delivers a graphical representation of how the model output deviates from its mean as a function of each input parameter \(X_i\).

Explanation:
The output is standardized by subtracting the mean and dividing by the standard deviation.
For each input parameter \(X_i\), this standardized output is reordered according to the ranked values of \(X_i\), i.e. there is one output sequence for each input parameter.
The cumulative sums of these normalized sequences are computed and plotted against the empirical cumulative distribution function (CDF) (rank) of \(X_i\):

  • A flat (horizontal) curve indicates that the conditional mean of the input parameter \(X_i\) has little or no influence on model behaviour.
  • A positive slope indicates a positive correlation between \(X_i\) and the output \(Y\).
  • A negative slope indicates a negative correlation between \(X_i\) and \(Y\).
  • The steepness of the slope shows the local sensitivity of Y to changes of \(X_i\).
  • The amplitude (overall height or depth of the curve) reflects the overall strength of the relationship between \(X_i\) and \(Y\).

Implementation:
The CUSUNORO plot is generated automatically with each sensitivity analysis run, regardless of the selected method, and even if no method has been selected from the “First-Order Effects”, “Second Order Interaction using HDMR” and “Checkerboard SA” tabs.

The method does not provide numerical values for sensitivity indices.

The CUSUNORO plot is displayed in Sensitivity Analysis → Results → Plot → CUSUNORO Plot.

User-specified software options:
N/A.

Sampling methods:

  • Compatible with all sampling methods.
  • Assumes independent parameters.

References

  • Plischke, Elmar (2019): The Cusunoro Curve: A Visual Tool for Global Sensitivity Analysis. In: Michael Beer und Enrico Zio (Hg.): Proceedings of the 29th European Safety and Reliability Conference (ESREL). 22–26 September 2019, Hannover, Germany. Europe, Singapore: European Safety and Reliability Association; Research Publishing Services, 2748–2753. DOI: 10.3850/978-981-11-2724-3_0153-cd.
  • Plischke, Elmar (2012): An adaptive correlation ratio method using the cumulative sum of the reordered output. In: Reliability Engineering & System Safety 107, 149–156. DOI: 10.1016/j.ress.2011.12.007.
  • Implementation adapted from Python code: https://github.com/tupui/batman/blob/master/batman/visualization/density.py

Checkerboard

This method allows for the visual representation of higher-order effects and provides regionalized information.

Explanation:
The checkerboard method shows how the sensitivity of the output varies locally across two user-specified input parameters.
All input parameters \(X_i\) are used to calculate the sensitivity indices, but instead of returning one SI per parameter, the method calculates separate local SI for each of the regions defined by the two chosen input parameters.

The two selected input parameters are ranked and scaled to the interval [0,1].
The data is divided into a user-specified number of bins.
Arranged along the x- and y-axes, the bins of the two chosen parameters form a square grid.
In each grid cell, the local sensitivity is calculated independently for each of the parameters, using the COSI variance-based method.
The sensitivity index of each input parameter \(X_i\) is therefore not a single value, but a matrix of local values which are displayed as a heatmap.
Consequently, there are as many heatmaps as there are input parameters.

In each heatmap, the two chosen input parameters appear on the x- and y-axes, and the colour represents the local SI for the respective input parameter \(X_i\).

Checkerboard

Implementation:
This method is implemented in Sensitivity Analysis → Setup → Checkerboard SA.
The method does not return numerical values for the local SI.
The Checkerboard plot in the ‘plot’ tab of the ‘results’ window shows a series of heatmaps where the local SI are colour coded according to a continuous colourmap.

Notes

  • The drop-down lists contain all data columns in the original dataset, including those marked as “exclude” or “output”. Ensure that only inputs are selected. Selecting an output or excluded column will not trigger a warning in the GUI, but the analysis will fail to run.
  • Unsuitable entries, e.g. values outside the permitted range, will not trigger a warning message in the GUI.

Sampling methods:
Compatible with all sampling methods.

References

User-specified software options:

Description Data Type Range Default
Partition variables - x
From the drop-down menu, select one of the dataset’s columns as the x-axis input parameter.		 N/A N/A N/A
Partition variables - y
From the drop-down menu, select one of the dataset’s columns as the y-axis input parameter.		 N/A N/A N/A
Scale by Local Variance
Controls the scale covered by the colour map.

Behaviour
  • The same colour map will be applied to all heatmaps.
  • Ticked: SI values are normalized by the largest local SI; the colour map covers the range [0, 1].
  • Unticked: SI values are not normalized; the range covered by the colour map depends on the data. In case of highly localized sensitivity or noisy data/artefacts, the interval [0, 1] may be exceeded.
 N/A N/A N/A
Grid Size
Controls the resolution of the grid by determining how many bins each of the two input parameters is divided into.

Behaviour
  • Low values reflect coarse partitioning which may miss detail and underestimate sensitivity.
  • High values reflect finer resolution, but may be subject to noise. If the value is too high, there are few samples in each bin which can cause unstable estimates. If any of the grid cells contain one or no samples, the analysis will fail to run.
  • Recommended value for the grid size \(p\) depends on sample size \(s\) to ensure a sufficient number of samples per grid cell: \(p \geq \sqrt{\frac{s}{64}}\).
 integer (positive) not specified 7
Colormap
Controls the colour scheme used for the heatmaps.		 N/A N/A N/A