In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a
random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. ...
''x'' is classically represented by a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
.
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
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
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
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
:
,
where ''P''
0 is the prior set for the parameters. Characterizing ''P'' corresponds to a
set-inversion problem.
Resolution
When ''f'' is linear the feasible set ''P'' can be described by linear inequalities and can be approximated using
linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
techniques.
When ''f'' is nonlinear, the resolution can be performed using
interval analysis
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to Floating point error mitigation, put bounds on rounding errors and measurement errors in numerical analysis ...
. The
feasible set
In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potent ...
''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.
Example
Consider the following model
:
where ''p''
1 and ''p''
2 are the two parameters
to be estimated.
Assume that at times ''t''
1=−1, ''t''
2=1, ''t''
3=2,
the following interval measurements have been collected:
:
1">'y''1 minus;4,−2
:
2">'y''2 ,9
:
3">'y''3 ,11
as illustrated by Figure 1. The corresponding measurement set (here a box) is
:
.
The model function is defined by
:
The components of ''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 ''P'' and blue boxes are outside ''P''.
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 of sets of parameter vectors that are consistent
with the ''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 ''q'' of them. This is possible using the notion of ''q''-
relaxed intersection.
See also
*
Set identification In statistics and econometrics, set identification (or partial identification) extends the concept of identifiability (or "point identification") in statistical models to situations where the distribution of observable variables is not informative ...
References
{{Reflist, 2
Estimation theory