Set estimation

In statistics, a random vector Template:Mvar is classically represented by a probability density function. In a set-membership approach or set estimation, Template:Mvar is represented by a set Template:Mvar to which Template:Mvar is assumed to belong. This means that the support of the probability distribution function of Template:Mvar is included inside Template:Mvar. 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 Template:Mvar (most of the time a box of Template:Tmath where Template:Mvar is the number of measurements) of the measurement space. If Template:Mvar is the parameter vector and Template:Mvar is the model function, then the set of all feasible parameter vectors is

$$P=P_0\cap f^{-1}(Y),$$

where Template:Math is the prior set for the parameters. Characterizing Template:Mvar corresponds to a set-inversion problem.

Resolution

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

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

$$\varphi (p_1,p_2,t)=(t{p}_1{)}^2+t{p}_2^2+\sin (p_1+t{p}_2),$$

where Template:Math and Template:Math are the two parameters to be estimated.

Figure 1. Bounded-error data

Assume that at times Template:Math, Template:Math, Template:Math, the following interval measurements have been collected: $$\begin{array}{cccc}[y_1]& =& [-4,& -2],\\ [y_2]& =& [4,& 9],\\ [y_3]& =& [7,& 11],\end{array}$$ as illustrated by Figure 1. The corresponding measurement set (here a box) is

$$Y=[y_1]\times [y_2]\times [y_3].$$

The model function is defined by

$$f(p_1,p_2)=\left[\begin{array}{c}{p}_1^2-p_2^2+\sin (p_1-p_2)\\ p_1^2+p_2^2+\sin (p_1+p_2)\\ (2p_1{)}^2+2p_2^2+\sin (p_1+2p_2)\end{array}\right]$$

The components of Template:Mvar 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 Template:Mvar and blue boxes are outside Template:Mvar.

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 . 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 Template:Mvarth 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 Template:Mvar of them. This is possible using the notion of Template:Mvar-relaxed intersection.

See also

• Set identification