Sensitivity Analysis

1. Introduction

Some models are complex or have a large number of parameters. In these situations it is often difficult to determine a ranking of the most influential parameters and to estimate their possible interactions. Various methods exist to address this issue, we focus here on sensitivity analysis using Morris' method.

For example, let's look at the animated image below. It shows the effect of a thrust on a solid and the displacement F that results from this. In short, we observe the behaviour of the function F, the displacement, with respect to a parameter X, the form. We therefore obtain a variation \frac{\Delta F}{\Delta X}, which is the quantity by which the observed function varied between two values of the parameter.

Illustration of the elementary effect: influence of the variation in shape "X" of a solid on the displacement "F" at equal stress (from the presentation of [PFE M. Taupin 2018]).

The idea of sensitivity analysis is to perform enough simulations to estimate the average effect of the variation of a parameter on the observed output.

2. Principle of Morris' method

This method is known as a " One at a time »method, where only one parameter is changed at each simulation. The variation of the objective function F in relation to a parameter X_i is calculated by finite difference between two sets of identical parameters with the exception of a single parameter: see the figure below where the parameters are changed one by one and the so-called "elementary" effect of each is evaluated.

Thus, if there are d parameters, we need d+1 simulations to assess the effect of a variation in each of them, i.e. \frac{\Delta F}{\Delta X_1},\frac{\Delta F}{\Delta X_2},...,\frac{\Delta F}{\Delta X_d}.

In order to calculate an average elementary effect \frac{\Delta F}{\Delta X_i}, on génère à nouveau Afin de calculer un effet élémentaire moyen d+1 parameter sets and repeat the operation a number of times r These are called "repetitions".

Table showing the modification of a different parameter each time the simulation is repeated (from [PFE M. Taupin 2018]).

In total, we will therefore have achieved n=r\times(d+1) simulations. A commonly accepted value in the literature is r \sim 10or a total number of evaluations in the order of n \sim 10 \times d

The series of elementary effects is then processed \frac{\Delta F}{\Delta X_i} of each parameter in order to calculate an average elementary effect \mu_i and a standard deviation \sigma_i. For the record, more the standard deviation of a series is large, the more the values are scattered around the average. For example, the following figure represents the variation of an objective function with respect to parameter X_1.

Rappel sur les gaussiennes : moyenne et écart-type  de l’effet élémentaire du paramètre [latex]X_1[/latex] sur la fonction objectif « SET » (+/- 2 écarts-types correspondent à ~95% des valeurs sous la courbe) (tiré du PFE M. Taupin 2018]).

The classification of the parameters is then done according to the graph (\mu,\sigma)as presented below. Influential parameters have a high average elementary effect: a variation in the parameter will result in a significant variation in the objective function. If the standard deviation of the series of elementary effects is small, the effect is considered linear. If this standard deviation is large, especially if it is of the order of magnitude of the elementary effect (i. e. greater than or equal to the first bisector), the effect is non-linear or causes interactions with other parameters.

Classification of parameters according to the mean of their elementary effects and their dispersion (standard deviation of the series of elementary effects) (from the M2R thesis of [F. PEYRE, 2017]).

3. Example

An example of a result applied to the building energy simulation of large volumes is given in this paragraph.

On veut connaître les paramètres les plus influents dans le cadre de la rénovation d’une verrière. Les paramètres choisis sont l’épaisseur, la conductivité, la capacité thermique massique, la densité, l’absorptivité solaire & visible ainsi que l’émissivité de surface de la dalle de sol ;  le facteur g et le facteur U du vitrage.

The objective function is the seasonal average of the differences between the SET index at time t and the center of the so-called comfortable SET range (i.e. the average of 22.2 and 25.5°C SET). The winter and summer seasons are simulated: it can be seen that the classification of the parameters is not the same according to the season. The order of magnitude of the average elementary effect is \sim 1 K] deviation from the comfort SET.

Example of a sensitivity analysis result using the Morris method, obtained with the SAlib package.