# Taylor decomposition importance factors¶

The importance factors derived from a Taylor expansion are defined to rank the sensitivity of the inputs to the output for central dispersion analysis.

Let us denote by the input random vector. Assume that the marginals of are independent. Suppose that is a real function of the input, i.e. . Assume that the order 1 Taylor expansion of the function $h$ at the point is exact, i.e.

where:

• is the mean of the input random vector,

• is the partial derivative of the model with respect to the i-th input variable, evaluated at the point .

Therefore the expectation of is:

The independence of the marginals implies:

where:

• is the variance of the output variable,

• is the variance of the i-th input variable.

Let be the importance factor of the i-th input variable, defined by:

Therefore, the importance factors sum to one:

Each coefficient is a linear estimate of the number of units change in the variable as a result of a unit change in the variable . This first term depends on the physical units of the variables and is meaningful only when the units of the model are known. In the general case, as the variables have different physical units, if , it is not possible to compare with . This is the reason why the importance factor are normalized. These factors enable to make the results comparable independently of the original units of the inputs of the model.
To summarize, the coefficients represent a linear estimate of the change in the output variable caused by a small change in the input variable . The importance factors are independent of the original units of the model, and are comparable with each other.

These are also called importance factors derived from perturbation methods.

API:

Examples: