View source for Darken's equations

'''Set-membership approach'''

In [[statistics]], a random vector ''x'' is classically represented by a [[probability density function]]. 
In a set-membership approach, ''x''
is represented by a set ''X'' to which ''x'' is assumed to belong.
It means that the support of the probability distribution function of ''x'' is included inside ''X''.
On the one hand, representing random vectors by sets makes it possible
to provide less assumptions on the random variables (such as independence)
and dealing with nonlinearities is easier. On the other hand, a 
probability distribution function provides
a more accurate information than a set enclosing its support.

==Set-membership estimation==

Set membership estimation (or ''set estimation'' for short) is 
an [[Estimation theory|estimation approach]] which considers that measurements 
are represented by a set ''Y'' (most of the time a box of '''R'''<sup>''m''</sup>, where ''m''
is the number of measurements) of the 
measurement space. If ''p'' is the parameter vector and ''f'' is the model function, then
the set of all feasible parameter vectors is

<math>P =  P_0 \cap f^{-1}(Y) </math>,

where ''P''<sub>0</sub> is the prior set for the parameters. Characterizing ''P'' corresponds to a [[set inversion|set-inversion problem]]
.<ref>
{{cite journal|last1=Jaulin|first1=L.|last2=Walter|first2=E.|
title=Guaranteed nonlinear parameter estimation via interval computations|journal=Interval Computation|year=1993|url=http://www.ensta-bretagne.fr/jaulin/paper_IntervalComp94.pdf}}
</ref>

==Resolution==

When ''f'' is linear the feasible set ''P'' can be described by linear inequalities and can be approximated
using [[linear programming]] techniques
.<ref>
{{cite journal|last1=Walter|first1=E.|last2=Piet-Lahanier|first2=H.|
title=Exact Recursive Polyhedral Description of the Feasible Parameter Set for Bounded-Error Models|journal=IEEE Transactions on Automatic Control|year=1989|volume=34|issue=8}}
</ref>

When ''f'' is nonlinear,  the resolution can be performed using [[Interval arithmetic|interval analysis]].
The feasible set ''P'' is then approximated by an inner and an outer
[[subpaving]]s. The main limitation of the method is its exponential complexity with respect 
to the number of parameters
.<ref>
{{cite journal|last1=Kreinovich|first1=V.|last2=Lakeyev|first2=A.V.|last3=Rohn|first3=J.|last4=Kahl|first4=P.T.|
title=Computational Complexity and Feasibility of Data Processing and Interval Computations|journal=Reliable Computing|year=1997|volume=4|issue=4}}
</ref>

==Example==

Consider the following model

<math>\phi(p_1,p_2,t) = (t.p_1)^2+t.p_2^2+sin(p_1 + t.p_2), </math>

where ''p''<sub>1</sub> and ''p''<sub>2</sub> are the two parameters
to be estimated.
[[File:Wiki estim data.png|thumb|Figure 1. Bounded-error data]]
Assume that at times ''t''<sub>1</sub>=-1, ''t''<sub>2</sub>=1, ''t''<sub>3</sub>=2,
the following interval measurements have been collected:
[''y''<sub>1</sub>]=[-4,-2],  
[''y''<sub>2</sub>]=[4,9],  
[''y''<sub>3</sub>]=[7,11],
as illustrated by Figure 1. The corresponding measurement set (here a box) is

<math> Y=[y_1] \times [y_2] \times [y_3] </math>.

The model function is defined by

<math>f(p_1,p_2) = \begin{bmatrix} p_1^2-p_2^2+sin(p_1-p_2) \\ 
    p_1^2+p_2^2+sin(p_1+p_2)  \\ (2p_1)^2+2 p_2^2+sin(p_1+2 p_2) \end{bmatrix}</math>

The components of ''f'' are obtained using the model for each time measurement.
After solving the set inversion problem, we get the approximation depicted on Figure 2.
Red boxes are inside the feasible set ''P'' and blue boxes are outside ''P''.
[[File:Wiki estim param.png|thumb|Figure 2 Feasible set for the parameters]]

==Recursive case==

Set estimation can be used to estimate the state of a system described by state equations using a recursive implementation.
When the system is linear, the corresponding feasible set for the state vector can be described by polytopes or by ellipsoids
<ref>
{{cite journal|last1=Fogel|first1=E.|last2=Huang|first2=Y.F.|
title=On the Value of Information in System Identification - Bounded Noise Case|journal=Automatica|year=1982|volume=18|issue=2}}
</ref>
.<ref>
{{cite journal|last1=Schweppe|first1=F.C.|
title=Recursive State Estimation: unknown but Bounded Errors and System Inputs|journal=IEEE Transactions on Automatic Control|year=1968|volume=13|issue=1}}
</ref> 
When the system is nonlinear, the set can be enclosed by subpavings.
<ref>
{{cite journal|last1=Kieffer|first1=M.|last2=Jaulin|first2=L.|last3=Walter|first3=E.|
title=Guaranteed recursive nonlinear state estimation using interval Analysis|
journal=Proceedings of the 37th IEEE Conference on Decision and Control|
year=1998|volume=4|
url=http://www.ensta-bretagne.fr/jaulin/IEEEKalman.pdf}}
</ref>

==Robust case==
When outliers occur, the set estimation method generally returns an empty set. This is
due to the fact that the intersection between of sets of parameter vectors that are consistent
with the ''i''th data bar is empty. To be robust with respect to outliers,
we generally characterize the set of parameter vectors that are consistent with
all data bars except ''q'' of them. This is possible using the notion of ''q''-[[relaxed intersection]].

==References==
{{Reflist|2}}

[[Category:Estimation theory| ]]
[[Category:Statistical inference]]