Vorob'ev threshold, expectation and deviation

Compute Vorob'ev threshold, expectation and deviation. Also, displaying the symmetric deviation function is possible. The symmetric deviation function is the probability for a given target in the objective space to belong to the symmetric difference between the Vorob'ev expectation and a realization of the (random) attained set.

Usage

vorob_t(x, sets, reference, maximise = FALSE)

vorob_dev(x, sets, reference, ve = NULL, maximise = FALSE)

Arguments

x: matrix()|data.frame()
Matrix or data frame of numerical values that represents multiple sets of points, where each row represents a point. If sets is missing, the last column of x gives the sets.
sets: integer()
Vector that indicates the set of each point in x. If missing, the last column of x is used instead.
reference: numeric()
Reference point as a vector of numerical values.
maximise: logical()
Whether the objectives must be maximised instead of minimised. Either a single logical value that applies to all objectives or a vector of logical values, with one value per objective.
ve: matrix()
Vorob'ev expectation, e.g., as returned by vorob_t().

Value

vorob_t returns a list with elements threshold, ve, and avg_hyp (average hypervolume)

vorob_dev returns the Vorob'ev deviation.

Details

Let \(\mathcal{A} = \{A_1, \dots, A_n\}\) be a multi-set of \(n\) sets \(A_i \subset \mathbb{R}^d\) of mutually nondominated vectors, with finite (but not necessarily equal) cardinality. If bounded by a reference point \(\vec{r}\) that is strictly dominated by any point in any set, then these sets can be seen a samples from a random closed set (Molchanov 2005) .

Let the \(\beta\)-quantile be the subset of the empirical attainment function \(\mathcal{Q}_\beta = \{\vec{z}\in \mathbb{R}^d : \hat{\alpha}_{\mathcal{A}}(\vec{z}) \geq \beta\}\).

The Vorob'ev expectation is the \(\beta^{*}\)-quantile set \(\mathcal{Q}_{\beta^{*}}\) such that the mean value hypervolume of the sets is equal (or as close as possible) to the hypervolume of \(\mathcal{Q}_{\beta^{*}}\), that is, \(\text{hyp}(\mathcal{Q}_\beta) \leq \mathbb{E}[\text{hyp}(\mathcal{A})] \leq \text{hyp}(\mathcal{Q}_{\beta^{*}})\), \(\forall \beta > \beta^{*}\). Thus, the Vorob'ev expectation provides a definition of the notion of mean nondominated set.

The value \(\beta^{*} \in [0,1]\) is called the Vorob'ev threshold. Large differences from the median quantile (0.5) indicate a skewed distribution of \(\mathcal{A}\).

The Vorob'ev deviation is the mean hypervolume of the symmetric difference between the Vorob'ev expectation and any set in \(\mathcal{A}\), that is, \(\mathbb{E}[\text{hyp}(\mathcal{Q}_{\beta^{*}} \ominus \mathcal{A})]\), where the symmetric difference is defined as \(A \ominus B = (A \setminus B) \cup (B \setminus A)\). Low deviation values indicate that the sets are very similar, in terms of the location of the weakly dominated space, to the Vorob'ev expectation.

For more background, see Binois et al. (2015); Molchanov (2005); Chevalier et al. (2013) .

References

Mickaël Binois, David Ginsbourger, Olivier Roustant (2015). “Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations.” European Journal of Operational Research, 243(2), 386–394. doi:10.1016/j.ejor.2014.07.032 .

Clément Chevalier, David Ginsbourger, Julien Bect, Ilya Molchanov (2013). “Estimating and Quantifying Uncertainties on Level Sets Using the Vorob'ev Expectation and Deviation with Gaussian Process Models.” In Dariusz Ucinski, Anthony C. Atkinson, Maciej Patan (eds.), mODa 10–Advances in Model-Oriented Design and Analysis, 35–43. Springer International Publishing, Heidelberg, Germany. doi:10.1007/978-3-319-00218-7_5 .

Ilya Molchanov (2005). Theory of Random Sets. Springer.

Author

Mickael Binois

Examples

data(CPFs)
res <- vorob_t(CPFs, reference = c(2, 200))
res$threshold
#> [1] 44.14062
res$avg_hyp
#> [1] 8943.333
# Now print Vorob'ev deviation
vd <- vorob_dev(CPFs, ve = res$ve, reference = c(2, 200))
vd
#> [1] 3017.13