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 θX parameters modelling the dependence structure:
(5.44)
In general, fX (x1, . . . xp| θX) 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 θX;
- 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 = (ρij)i,j is symmetrical with diagonal 1, is semi-positive-definite ...
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