Set estimation

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 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 (or set estimation for short) is an estimation approach which considers that measurements are represented by a set Y (most of the time a box of R^m, 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

${\displaystyle P=P_{0}\cap f^{-1}(Y)}$,

where P₀ is the prior set for the parameters. Characterizing P corresponds to a set-inversion problem .

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

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

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