Set estimation

Set membership estimation (or set estimation for short) is an estimation approach which considers that measurements are represented by a set {{mvar|Y}} (most of the time a box of {{tmath|\R^m,}} where {{mvar|m}} is the number of measurements) of the measurement space. If {{mvar|p}} is the parameter vector and {{mvar|f}} is the model function, then the set of all feasible parameter vectors is

$$P\; =\; P\_0\; \backslash cap\; f^\{-1\}(Y)\; ,$$

where {{math|P₀}} is the prior set for the parameters. Characterizing {{mvar|P}} corresponds to a set-inversion problem.

{{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}}

When {{mvar|f}} is linear the feasible set {{mvar|P}} can be described by linear inequalities and can be approximated using linear programming techniques.

{{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|pages=911–915 |doi=10.1109/9.29443}}

When {{mvar|f}} is nonlinear, the resolution can be performed using interval analysis. The feasible set {{mvar|P}} is then approximated by an inner and an outer subpavings. The main limitation of the method is its exponential complexity with respect to the number of parameters.

{{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}}

Consider the following model

$$\backslash phi(p\_1,p\_2,t)\; =\; (t\; p\_1)^2+t\; p\_2^2+\backslash sin(p\_1\; +\; t\; p\_2),$$

where {{math|p₁}} and {{math|p₂}} are the two parameters

to be estimated.

File:Wiki estim data.png

Assume that at times {{math|1=t₁ = −1}}, {{math|1=t₂ = 1}}, {{math|1=t₃ = 2}},

the following interval measurements have been collected:

$$\backslash begin\{array\}\{rclr\}$$

[y_1] &=& [-4, & -2], \\

{}[y_2] &=& [\quad\! 4, & 9], \\

{}[y_3] &=& [\quad\! 7, & 11],

\end{array}

as illustrated by Figure 1. The corresponding measurement set (here a box) is

$$Y=[y\_1]\; \backslash times\; [y\_2]\; \backslash times\; [y\_3].$$

The model function is defined by

$$f(p\_1,p\_2)\; =\; \backslash begin\{bmatrix\}$$

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}

The components of {{mvar|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 {{mvar|P}} and blue boxes are outside {{mvar|P}}.

File:Wiki estim param.png

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

{{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|pages=229–238 |doi=10.1016/0005-1098(82)90110-8}}

.

{{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|pages=22–28 |doi=10.1109/tac.1968.1098790}}

When the system is nonlinear, the set can be enclosed by subpavings.

{{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}}

When outliers occur, the set estimation method generally returns an empty set. This is

due to the fact that the intersection between sets of parameter vectors that are consistent

with the {{mvar|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 {{mvar|q}} of them. This is possible using the notion of {{mvar|q}}-relaxed intersection.

• Set identification

{{Reflist|2}}

Category:Estimation theory