Uncertainty ranking: SRC and SRRC¶
Standard Regression Coefficients (SRC) deal with analyzing the influence the random vector has on a random variable . We measure linear relationships that exist between and the different input variables .
The principle of the multiple linear regression model consists in attempting to find the function that links the variable to the variables by means of a linear model:
where are constant parameters and is a random variable with zero mean and standard deviation independent of the input variables . If the random variables are independent and with finite variance , the variance of can be estimated as follows:
The SRC coefficient is defined by (see [borgonovo2017] page 131, eq. 14.3):
for . The estimators for the regression coefficients , and the standard deviation are obtained from a sample of . The SRC coefficients are defined as the estimators of the coefficients :
where is the estimate of the regression coefficient ,
is the sample standard
deviation of the sample of the input variable
and is the sample standard
deviation of the sample of the output variable .
The absolute value of this estimated
contribution is by definition between 0 and 1. The closer it is to 1,
the greater the impact the variable has on the variance of
computeSRC method to compute the SRC
Before estimating the SRC coefficients,
we mush check the quality of the linear regression:
if the linear regression model
is a poor fit to the data, then the SRC coefficients are useless.
See e.g. the
MetaModelValidation class to validate
the linear model against a test data set.
The index, which is the contribution of to the variance of , is sometimes described in the literature as the “importance factor”, because of the similarity between this approach to linear regression and the method of cumulative variance which uses the term importance factor. This importance factor is also named “squared SRC” coefficient (see [borgonovo2017] page 131, eq. 14.5):
The squared SRC coefficients of a linear model are equal to its
If the model is linear, the first-order Sobol’ index is equal
to the total Sobol’ index since there is no interaction.
computeSquaredSRC method to compute the squared SRC
If the input random variables are dependent, then the SRC is not a valid importance measure anymore (see [daveiga2022] remark 4 page 33). In this case, the partial correlation coefficient (PCC) has been suggested, but this index is rather a measure of the linear relationship between the input and the output. Other indices such as the Lindeman-Merenda-Gold (LMG) have been suggested in the dependent case (see [daveiga2022] page 33).
Standard Rank Regression Coefficients (SRRC) are SRC coefficients
computed on the ranked input variables
and the ranked output variable .
They are useful when the relationship between
and is not linear (so SRC cannot be used),
but only monotonic.
computeSRRC method to compute the SRRC