Variance-based Sensitivity Analysis Methods

Pearson's \(\eta^2\)

Each input is sorted and binned, and the conditional means of the output are calculated for each bin.
The variances of the conditional means are normalized by the total variance to compute the first-order effects.

Explanation:
Pearson’s \(\eta^2\) determines how much of the total variance of the output can be explained by a specific input parameter \(X_i\).
It can also be used when input is discrete (e.g. categorical).
Continuous data is grouped into bins of \(X_i\).
For each bin or category of \(X_i\), the average outcome is determined.
Comparing the average outcomes of the different bins allows to estimate the influence of the parameter – e.g. if there are large differences in output, then the parameter is important for the model’s response.
This is quantified by comparing the variance in output due to \(X_i\) with the total variance of the output.

• \(\eta^2 = 1\): all of the output variance is due to this parameter
• \(\eta^2 = 0\): the parameter has no effect on the output

Pearson’s \(\eta^2\) delivers an estimate for the first-order effect; for continuous data and very small bins and large sample sizes, the error becomes negligible.

Implementation:
This method is implemented in Sensitivity Analysis → Setup → First-Order Effects → Pearson \(\eta^2\).

User-specified software options:

Sampling methods:

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

High-Dimensional Model Representation (HDMR)

Each first- and higher-order effect is evaluated separately using B-splines in order to estimate the sensitivity indices for each input parameter or parameter interaction.

Explanation:
HDMR is a functional decomposition technique used to analyse how a model’s output depends on multiple input parameters.
It expresses the model output as the sum of effects – first-order (individual parameters), second-order (parameter pairs), third-order (triplets) etc.
All parameter interactions are evaluated separately to determine the influence of each input and interaction.
The SA-Toolbox uses B-splines (piecewise polynomial functions, see Figure below) as basis functions to estimate the individual effects.
For each effect, its contribution to the variance of the output is estimated.
From this, the variances of the lower-order terms are subtracted (recursive subtraction) to isolate the contribution of the order under consideration.
For example, a third-order effect is determined by subtracting all first- and second-order effects:

From these variance terms, the sensitivity indices are derived by dividing each variance contribution by the total output variance.
The first- and higher-order indices quantify how much of the output variance is caused by each parameter or interaction.
HDMR is suitable for models with a large number of input variables.

Implementation:
This method is implemented in Sensitivity Analysis → Setup → \(2^{nd}\) Order Interactions using HDMR.

User-specified software options:

Sampling methods:
Compatible with all sampling methods.

Random Balanced Design - Fourier Amplitude Sensitivity Test (RBD-FAST)

The output is reordered according to the input of interest, and the variance contributions are detected from a Fast Fourier Transform.
This is a version of RBD-FAST that is suitable for application to existing data sets.

Explanation:
RBD-FAST is a computationally efficient method that scales well for models with a large number of input parameters.
The method estimates how much of the model output variance is explained by each input variable (first-order effect).
Higher-order interactions between input parameters are not captured by RBD-FAST.

The method is based on sorting: The input of interest is sorted, and the output is reordered accordingly.
This is then mapped into a periodic signal, such that the variance contributions can be detected from a Fast Fourier Transform.
The sum of the squared amplitudes of the first few (typically: 10) harmonics indicate how strongly that input affects the total variance of the output.

Implementation:
This method is implemented in Sensitivity Analysis → Setup → First-Order Effects → RBD-FAST.

Cosine Sensitivity Index (COSI)

The output is reordered according to the input of interest, and the variance contributions are detected from a Discrete Cosine Transform (DCT).
This is a version of RBD-FAST that is suitable for application to existing data sets.

Explanation:
The cosine sensitivity index (COSI) provides a fast estimation of variability contributions.
The method is based on sorting: The input of interest is sorted. For each parameter, the output is reordered accordingly.
A Discrete Cosine Transform is then used to ascertain how much structure is contained in that relationship.
Each output sequence is represented as a sum of cosine waves of different frequencies.
A user-specified number of the amplitudes are squared and summed up to estimate the first-order effect for each input parameter.

Implementation:
This method is implemented in Sensitivity Analysis → Setup → First-Order Effects → COSI.

Conditional Linear Method Sensitivity Index (CLMSI)

The input is sorted and binned; the method utilizes a linear spline approach to predict the output within each bin, with a penalty term applied for non-smooth transit between the bin boundaries.

Implementation:
This method is implemented in Sensitivity Analysis → Setup → First-Order Effects → CLMSI.