Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

5.3 Modelling Dependence

Section 5.2 discussed the case of independent input components. In the more general case of (possibly) non-independent uncertain inputs, one has to estimate the joint distribution, including θ_X parameters modelling the dependence structure:

(5.44)

In general, f_X (x¹, . . . x^p| θ_X) cannot be factorised merely into marginal components as is the case – per definition – for independent inputs:

(5.45)

The methods of increasing complexity may then be contemplated:

• linear (Pearson) correlations between inputs: the correlation (symmetric and diagonal one) matrix then provides the p(p − 1)/2 additional parameters needed within θ_X;
• rank (Spearman) correlations between inputs: an extension of the previous possibility, again parametrised by a matrix of dependence coefficients;
• copula model: this the most powerful and general approach. An additional function is inferred in order to describe the dependence structure, as well as a set of corresponding dependence parameters.

5.3.1 Linear Correlations

Definition and Estimation

The linear correlation coefficient (or Pearson coefficient) is an elementary probabilistic concept defined pairwise between uncertain inputs as follows:

(5.46)

Thus, the correlation matrix R = (ρ^ij)_i,j is symmetrical with diagonal 1, is semi-positive-definite ...