Relaxed intersection

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The relaxed intersection of m sets corresponds to the classical intersection 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 Problems that 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 respect to some outliers.

Definition[edit]

The q-relaxed intersection of the m subsets  X_{1},\dots ,X_{m} of R^{n}, denoted by 
X^{\{q\}}=\bigcap^{\{q\}}X_{i}
is the set of all 
x \in R^{n}
which belong to all 
X_{i}
's, except q at most. This definition is illustrated by Figure 1.

Figure 1. q-intersection of 6 sets for q=2 (red), q=3 (green), q= 4 (blue) , q= 5 (yellow).

Define 
\lambda (x) =\text{card} \left\{ i\ |\ x\in X_{i}\right\}.

We have 
X^{\{q\}}=\lambda ^{-1}([m-q,m]) .

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

Example[edit]

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 [.

Relaxed intersection of intervals[edit]

The relaxed intersection of intervals is not necessary an interval. We thus take the interval hull of the result. If X_i's are intervals, the relaxed intersection can be computed with a complexity of m.log(m) by using the Marzullo's algorithm. It suffices to sort all lower and upper bounds of the m intervals to represent the function \lambda . Then, we easily get the set


X^{\{q\}}=\lambda^{-1}([m-q,m])

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

Figure 2 shows the function  \lambda(x) associated to the previous example.

Figure 2. Set-membership function associated to the 6 intervals.

Relaxed intersection of boxes[edit]

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 an illustration of the 4-relaxed intersection of 6 boxes. Each point of the red box belongs to 4 of the 6 boxes.

Figure 3. The red box corresponds to the 4-relaxed intersection of the 6 boxes

Relaxed union[edit]

The q-relaxed union of X_1,\dots ,X_m is defined by


\overset{\{q\}}{\bigcup}X_{i}=\bigcap^{\{m-1-q\}}X_i

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


\bigcap^{\{0\}}X_{i} =\bigcap X_i

and


\overset{\{0\} }{\bigcup}X_{i} =\bigcup X_i

De Morgan's law[edit]

If \overline{X} denotes the complementary set of X_i, we have


\overline{\bigcap^{\{q\}}X_i}  = \overset{\{q\}}{\bigcup}\overline{X_i}


\overline{\overset{\{q\} }{\bigcup }X_i}=\bigcap^{\{q\}}\overline{X_i}.

As a consequence


\overline{\bigcap\limits^{\{q\}}X_i}=\overline{\overset{\{m-q-1\} }{\bigcup }X_i}=\bigcap^{\{m-q-1\}}\overline{X_i}

Relaxation of contractors[edit]

Let C_1,\dots ,C_m be m contractors for the sets X_1,\dots ,X_m , then


C([x]) =\bigcap^{\{q\}}C_i([x]).

is a contractor for X^{\{q\}} and


\overline{C}([x]) =\bigcap^{\{m-q-1\}}\overline{C}_i([x])

is a contractor for \overline{X}^{\{q\}}, where


\overline{C}_1,\dots,\overline{C}_{m}

are contractors for


\overline{X}_1,\dots ,\overline{X}_m.

Combined with a branch-and-bound algorithm such as SIVIA (Set Inversion Via Interval Analysis), the q-relaxed intersection of m subsets of R^{n} can be computed.

Application to bounded-error estimation[edit]

The q-relaxed intersection can be used for robust localization ,[3] for robust localization [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)=\frac{1}{\sqrt{2\pi p_2}}\exp (-\frac{(t_i-p_1)^{2}}{2p_2})

where p\in R^{2}. Assume that we have


f_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 \lambda^{-1}(q) for different q are depicted on Figure 4.

Figure 4. Set of all parameter vectors consistent with exactly 6-q data bars (painted red), for q=1,2,3,4,5.

References[edit]

  1. ^ Jaulin, L.; Walter, E.; Didrit, O. (1996). Guaranteed robust nonlinear parameter bounding. In Proceedings of CESA'96 IMACS Multiconference (Symposium on Modelling, Analysis and Simulation). 
  2. ^ Jaulin, L.; Walter, E. (2002). "Guaranteed robust nonlinear minimax estimation". IEEE Transaction on Automatic Control 47. 
  3. ^ Kieffer, M.; Walter, E. (2013). Guaranteed characterization of exact non-asymptotic confidence regions in nonlinear parameter estimation. In Proceedings of IFAC Symposium on Nonlinear Control Systems, Toulouse : France (2013). 
  4. ^ Drevelle, V.; Bonnifait, Ph. (2011). "A set-membership approach for high integrity height-aided satellite positioning". GPS Solutions 15 (4). 
  5. ^ Langerwisch, B.; Wagner (2012). "Guaranteed Mobile Robot Tracking Using Robust Interval Constraint Propagation". Intelligent Robotics and Applications. .
  6. ^ Jaulin, L. (2009). "Robust set membership state estimation ; Application to Underwater Robotics". Automatica 45. 
  7. ^ Jaulin, L.; Kieffer, M.; Walter, E.; Meizel, D. (2002). "Guaranteed robust nonlinear estimation, with application to robot localization". IEEE Transactions on systems, man and cybernetics; Part C Applications and Reviews 32.