Set Estimation

In statistics, a random vector is classically represented by a probability density function.
In a set-membership approach or set estimation, is represented by a set to which is assumed to belong. This means that the support of the probability distribution function of is included inside . On the one hand, representing random vectors by sets makes it possible to provide fewer 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 approach which considers that measurements are represented by a set (most of the time a box of where is the number of measurements) of the measurement space. If is the parameter vector and is the model function, then the set of all feasible parameter vectors is

$P = P_0 \cap f^(Y) ,$

where is the prior set for the parameters. Characterizing corresponds to a set-inversion problem.

Resolution

When is linear the feasible set can be described by linear inequalities and can be approximated using linear programming techniques.

When is nonlinear, the resolution can be performed using interval analysis. The feasible set 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.

Example

Consider the following model

$\phi(p_1,p_2,t) = (t p_1)^2+t p_2^2+\sin(p_1 + t p_2),$

where and are the two parameters
to be estimated.

Assume that at times , , ,
the following interval measurements have been collected:
\begin
_1&=& 4, & -2 \\
_2&=& quad\! 4, & 9 \\
_3&=& quad\! 7, & 11
\end
as illustrated by Figure 1. The corresponding measurement set (here a box) is

Y= _1\times _2\times _3

The model function is defined by

$f(p_1,p_2) = \begin p_1^2-p_2^2+\sin(p_1-p_2) \\ p_1^2+p_2^2+\sin(p_1+p_2) \\ (2p_1)^2+2 p_2^2+\sin(p_1+2 p_2) \end$

The components of 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 and blue boxes are outside .

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

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

Robust case

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 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 of them. This is possible using the notion of -relaxed intersection.

See also

*Set identification

References

{{Reflist, 2

Estimation theory