Sensitivity analysis¶
Sensitivity analysis is used in computational engineering to investigate how the uncertainty in model input parameters affects the output. It helps identify the critical input parameters that have the most significant impact on the output, and the ones that can be neglected, therefore considered as model constants and not variables. It is handy in cases where no prior knowledge or experience with the input parameters is given. Sensitivity analysis allows a better understanding of the model's behavior and aids model calibration and validation. Resolving uncertainty through sensitivity analysis is a powerful tool that can improve the reliability of predictions and reduce the risk of errors in decision-making.
Figure 1: Uncertainty and sensitivity analysis frameworks employed in computational engineering [1].
There are several methods of sensitivity analysis, including qualitative and quantitative. Qualitative methods involve expert judgment and help identify the most critical parameters. An example of qualitative analysis is plotting the input variables' scatterplots against the output. Quantitative methods, such as variance-based and moment-independent methods, involve mathematical calculations to quantify the sensitivity of the output to the input parameters. Variance-based methods are commonly used in sensitivity analysis. These methods measure each input parameter's contribution to the output's overall variance. They are particularly useful when the input parameters are correlated or when interactions between parameters need to be considered.
Figure 2: Example of qualitative sensitivity analysis of 8 model variables on two outputs [2].
Polynomial Chaos Expansion (PCE) is a technique used in variance-based sensitivity analysis to approximate complex functions using polynomial functions. PCE is handy when the input parameters have a probability distribution, allowing for the computation of the Sobol' indices, which quantify the contribution of each input parameter and interaction to the variance of the output. The use of PCE for the computation of Sobol' indices is especially powerful when dealing with complex models, as it reduces the computational cost of the analysis while still providing accurate results.
Sobol' indices quantify the contribution of individual input parameters and interactions to the output variance in a computational model. The first-order Sobol' index measures a particular parameter's contribution to the output's variance, while the total-order Sobol' index measures the contribution of the individual parameter and its interactions with other parameters to the variance of the output.
The first-order Sobol' index helps identify a model's most important input parameters, as it measures their individual contribution to the output variance. This information can be used to reduce the number of input parameters in the model, simplify the model, or improve the accuracy of the model predictions. This is referred to as the Factor Ranking setting.
The total-order Sobol' index helps identify the joint contribution of the input parameters and their interactions with the output variance. This information can be used to improve the model predictions by including the effects of parameter interactions or to identify areas where further research or data collection is needed to reduce the uncertainty in the model. Low total-order sensitivity indices suggest no influence of a variable on the variation of the output. Therefore, the variable is considered a model constant. This is referred to as the Factor Fixing setting.
Figure 3: Example of quantitative sensitivity analysis of 9 model variables on one output [3]. In this example, notice how little the influence of parameters such as kc, ct, or TRt is. The model can be further reduced to only two random variables: kBP and cBPt.
Sensitivity analysis tools¶
Do you want to test your sensitivity analysis and you do not want to struggle with code implementation? Are you curious to know how sensitivity analysis works? Visit our sensitivity analysis online tool at uqtab.com to learn more about the topic and compute the sensitivity indices of your data.
The evalutation of the sensitivity indices will be performed on your results by using pygpc python tool. This library offers an efficient computation of the polynomial chaos expansion and relative sensitivity indices of your model's input and output data.
References¶
[1] Melito, Gian Marco (2022) Sensitivity analysis for model optimization and calibration in type B aortic dissection, Ph.D. thesis, TU Graz
[2] Melito, G. M., Müller, T. S., Badeli, V., Ellermann, K., Brenn, G., & Reinbacher-Köstinger, A. (2021). Sensitivity analysis study on the effect of the fluid mechanics assumptions for the computation of electrical conductivity of flowing human blood. Reliability Engineering & System Safety, 213, 107663.
[3] Melito, G. M., Jafarinia, A., Hochrainer, T., & Ellermann, K. (2020). Sensitivity analysis of a phenomenological thrombosis model and growth rate characterisation. Journal of Biomedical Engineering and Biosciences (JBEB), 7(1), 31-40.