Relaxed intersection explained

The relaxed intersection of m sets corresponds to the classicalintersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection.This notion can be used to solve constraints satisfaction problemsthat are inconsistent by relaxing a small number of constraints.When a bounded-error approach is considered for parameter estimation,the relaxed intersection makes it possible to be robust with respectto some outliers.

Definition

The q-relaxed intersection of the m subsets

X _{1} , ... ,X _{m} of R ^{n} ,denoted byX ^{\{q\} }=\bigcap^X_is the set of allx \in R ^{n} which belong to allX _{i} 's, exceptq at most.This definition is illustrated by Figure 1.

Define

λ (x) = card \left\{ i | x\in X _{i} \right\}.

We have

X ^{\{q\} }=\lambda ^([m-q,m]) .

Characterizing the q-relaxed intersection is a thus a set inversion problem.^[1]

Example

Consider 8 intervals:

X _{1} = [ 1 , 4 ], X _{2} = [ 2 , 4 ], X _{3} = [ 2 , 7 ], X _{4} = [ 6 , 9 ], X _{5} = [ 3 , 4 ], X _{6} = [ 3 , 7 ].

We have

X ^{\{0\} } = \emptyset, X ^{\{1\} }=[3,4],X ^{\{2\} }=[3,4], X ^{\{3\} }=[2,4] \cup [6,7],X ^{\{4\} }=[2,7], X ^{\{5\} }=[1,9],X ^{\{6\} }=]-\infty,\infty [. </math> ==Relaxed intersection of intervals== The relaxed intersection of intervals is not necessary an interval. We thus take the interval hull of the result. If <math>X_i</math>'s are intervals, the relaxed intersection can be computed with a complexity of ''m''.log(''m'') by using the [[Marzullo's algorithm]]. It suffices tosort all lower and upper bounds of the m intervals to represent thefunction λ . Then, we easily get the setX ^{\{q\} }=\lambda^([m-q,m])

which corresponds to a union of intervals.We then return thesmallest interval which contains this union.

Figure 2 shows the function

λ (x) associated to the previous example.

Relaxed intersection of boxes

To compute the q-relaxed intersection of m boxes of

R ^{n} , we project all m boxes with respect to the n axes.For each of the n groups of m intervals, we compute the q-relaxed intersection.We return Cartesian product of the n resulting intervals.^[2] Figure 3 provides anillustration of the 4-relaxed intersection of 6 boxes. Each point of thered box belongs to 4 of the 6 boxes.

Relaxed union

The q-relaxed union of

X _{1,...} ,X _{m} is defined by\overset{\{q\}}{ c up}X _{i} = c ap ^{\{m-1-q\} }X_i

Note that when q=0, the relaxed union/intersection corresponds tothe classical union/intersection. More precisely, we have

c ap ^{\{0\} }X_ =\bigcap X_i

and

\overset{\{ 0 \} }{ c up}X _{i} = c up X _{i}

De Morgan's law

If

\overline{X} denotes the complementary set of X _{i} , we have\overline{ c ap ^{\{q\} }X_i} = \overset\overline\overline{\overset{\{q\} }{ c up }X ^{\{q\} _{i} = c ap} }\overline.

As a consequence

\overline{ c ap\limits ^{\{q\} }X_i}=\overline=\bigcap^\overline

Relaxation of contractors

Let

C _{1,...} ,C _{m} be m contractors for the sets X _{1,...} ,X _{m} ,thenC([x]) = c ap ^{\{q\} }C_i([x]).

is a contractor for

X ^{\{q\} }and\overline{C}([x]) = c ap ^{\{m-q-1\} }\overline_i([x])

is a contractor for

\overline{X} ^{\{q\} }, where\overline{C} _{1,...,\overline{C}} _{m}

are contractors for

\overline{X} _{1,...} ,\overline{X} _{m.}

Combined with a branch-and-bound algorithm such as SIVIA (Set Inversion Via Interval Analysis), the q-relaxedintersection of m subsets of

R ^{n} can be computed.

Application to bounded-error estimation

The q-relaxed intersection can be used for robust localization^{[3] [4]} or for tracking.^[5]

Robust observers can also be implemented using the relaxed intersections to be robust with respect to outliers.^[6]

We propose here a simple example^[7] to illustrate the method.Consider a model the ith model output of which is given by

f _{i(p) = (1)/(\sqrt{ 2 \pi p _{2})} }\exp (-\frac)

where

p\in R ^{2} . Assume that we havef _{i(p)} \in [y _{i]}

where

t _{i} and [y _{i]} are given by the following list\{ ( 1 ,[ 0 ; 0 . 2 ]),( 2 ,[ 0 . 3 ; 2 ]),( 3 ,[ 0 . 3 ; 2 ]),( 4 ,[ 0 . 1 ; 0 . 2 ]),( 5 ,[ 0 . 4 ; 2 ]),( 6 ,[ -1 ; 0 . 1 ]) \}

The sets

λ ^{-1} (q) for different q are depicted onFigure 4.

Notes and References

1. Book: Jaulin. L.. Walter. E.. Didrit. O.. Guaranteed robust nonlinear parameter bounding. 1996. In Proceedings of CESA'96 IMACS Multiconference (Symposium on Modelling, Analysis and Simulation).
2. Jaulin. L.. Walter. E.. Guaranteed robust nonlinear minimax estimation. IEEE Transactions on Automatic Control. 2002. 47. 11 . 1857–1864 . 10.1109/TAC.2002.804479 .
3. Book: Kieffer. M.. Walter. E.. Guaranteed characterization of exact non-asymptotic confidence regions in nonlinear parameter estimation. 2013. In Proceedings of IFAC Symposium on Nonlinear Control Systems, Toulouse : France (2013).
4. Drevelle. V.. Bonnifait. Ph.. A set-membership approach for high integrity height-aided satellite positioning. GPS Solutions. 2011. 15. 4. 357–368 . 10.1007/s10291-010-0195-3 . 121728552 .
5. Langerwisch. M.. Wagner. B.. Guaranteed Mobile Robot Tracking Using Robust Interval Constraint Propagation. Intelligent Robotics and Applications. 2012. .
6. Jaulin. L.. Robust set membership state estimation; Application to Underwater Robotics. Automatica. 2009. 45. 10.1016/j.automatica.2008.06.013. 202–206.
7. Jaulin. L.. Kieffer. M.. Walter. E.. Meizel. D.. Guaranteed robust nonlinear estimation with application to robot localization . IEEE Transactions on Systems, Man, and Cybernetics - Part C: Applications and Reviews . 2002. 32. 4 . 374–381 . dead. https://web.archive.org/web/20110428224956/http://www.ensta-bretagne.fr/jaulin/robab.pdf. 2011-04-28 . 10.1109/TSMCC.2002.806747. 17436801 .