vorob_t {moocore} | R Documentation |
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 |
|
sets |
|
reference |
|
maximise |
|
ve |
|
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).
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
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.
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