Vorob: Vorob'ev threshold, expectation and deviation

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Vorob'ev threshold, expectation and deviation

Description

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().

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 \citepMolchanov2005theory.

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 \citetBinGinRou2015gaupar,Molchanov2005theory,CheGinBecMol2013moda.

Value

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

vorob_dev returns the Vorob'ev deviation.

Author(s)

Mickael Binois

References

\insertAllCited

Examples

data(CPFs)
res <- vorob_t(CPFs, reference = c(2, 200))
res$threshold
res$avg_hyp
# Now print Vorob'ev deviation
vd <- vorob_dev(CPFs, ve = res$ve, reference = c(2, 200))
vd

moocore documentation built on Aug. 8, 2025, 6:12 p.m.