In statistics, a random vector x is classically represented by a probability density function. In a set-membership approach or set estimation, x is represented by a set X to which x is assumed to belong. This 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.
HINT:
In this Dossier
Set estimation in the context of Estimation theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.In estimation theory, two approaches are generally considered:
- The probabilistic approach (described in this article) assumes that the measured data is random with a probability distribution dependent on the parameters of interest
- The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector.