# Probabilistic Set-membership Approach for Robust Regression

@article{Jaulin2010ProbabilisticSA, title={Probabilistic Set-membership Approach for Robust Regression}, author={Luc Jaulin}, journal={Journal of Statistical Theory and Practice}, year={2010}, volume={4}, pages={155-167} }

Interval constraint propagation methods have been shown to be efficient and reliable to solve difficult nonlinear bounded-error estimation problems. However they are considered as unsuitable in a probabilistic context, where the approximation of a probability density function by a set cannot be accepted as reliable. This paper shows how probabilistic estimation problems can be transformed into a set estimation problem by assuming that some rare events will never happen. Since the probability of… Expand

#### 17 Citations

Probabilistic Set-Membership State Estimator

- Computer Science
- 2011

A new probabilistic approach is proposed which makes it possible to use classical set-membership observers which are robust with respect to outliers and is illustrated on a localization of robots in situations where there exist a large number of outliers. Expand

Inner and Outer Approximations of Probabilistic Sets

- Mathematics
- 2014

This paper proposes a set-membership method to characterize a probabilistic set , i.e., a set enclosing the true value for the parameter vector of a parametric system with a given probability. The… Expand

Nonlinear set-membership identification using a Bayesian approach

- Computer Science, Mathematics
- 2014 IEEE Conference on Control Applications (CCA)
- 2014

The paper illustrates how the Bayesian approach can be used to approximate the feasible parameter set (FPS) by assuming uniform distributed estimation error and flat model prior probability distributions. Expand

Set-membership nonlinear regression approach to parameter estimation

- Mathematics
- Journal of Process Control
- 2018

Abstract This paper introduces set-membership nonlinear regression (SMR), a new approach to nonlinear regression under uncertainty. The problem is to determine the subregion in parameter space… Expand

Set-membership identification and fault detection using a bayesian framework

- Mathematics, Computer Science
- 2013 Conference on Control and Fault-Tolerant Systems (SysTol)
- 2013

The paper shows that, assuming uniform distributed measurement noise and flat model prior probability distribution, the Bayesian approach leads to the same feasible parameter set than the set-membership strips technique and, additionally, can deal with models nonlinear in the parameters. Expand

Set-membership identification and fault detection using a bayesian framework

- Mathematics
- 2013

This paper deals with the problem of set-membership identification and fault detection using a Bayesian framework. The paper presents how the set-membership model estimation problem can be… Expand

Nonlinear set-membership identification and fault detection using a Bayesian framework: Application to the wind turbine benchmark

- Computer Science, Engineering
- 52nd IEEE Conference on Decision and Control
- 2013

The paper shows that the Bayesian approach, assuming uniform distributed measurement noise and flat model prior probability distribution, leads to the same feasible parameter set as the set-membership technique. Expand

Guaranteed Nonlinear Parameter Estimation with Additive Gaussian Noise

- Mathematics
- 2020

In this paper we propose a new approach for nonlinear parameter estimation under additive Gaussian noise. We provide an algorithm based on interval analysis and set inversion which computes an inner… Expand

How to Detect Possible Additional Outliers: Case of Interval Uncertainty∗

- 2021

In many practical situations, measurements are characterized by interval uncertainty – namely, based on each measurement result, the only information that we have about the actual value of the… Expand

Set-membership identification: Bayesian approach vs subpavings approach

- Computer Science, Mathematics
- 21st Mediterranean Conference on Control and Automation
- 2013

It is shown that the Bayesian approach, assuming uniform distributed estimation error and flat model prior probability distributions, leads to the same feasible parameter set than the subpavings technique. Expand

#### References

SHOWING 1-10 OF 28 REFERENCES

Computing minimal-volume credible sets using interval analysis; application to bayesian estimation

- Mathematics, Computer Science
- IEEE Transactions on Signal Processing
- 2006

This paper provides an algorithm able to compute accurate inner and outer approximations of minimal-volume credible sets, in a guaranteed way, based on interval analysis and an application to nonlinear parameter estimation, inA Bayesian context, is treated. Expand

Box particle filtering for nonlinear state estimation using interval analysis

- Mathematics, Computer Science
- Autom.
- 2008

An extension of the particle filter algorithm able to handle interval data and using interval analysis and constraint satisfaction techniques is proposed and experiments using actual data for global localization of a vehicle show the usefulness and the efficiency of this approach. Expand

Probabilities, intervals, what next? extension of interval computations to situations with partial information about probabilities

- Mathematics
- 2004

In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x1,… Expand

Outlier Detection under Interval Uncertainty: Algorithmic Solvability and Computational Complexity

- Mathematics, Computer Science
- Reliab. Comput.
- 2005

The computational complexity of these outlier detection problems is analyzed, efficient algorithms that solve some of these problems (under reasonable conditions) are provided, and related open problems are list. Expand

Constructing belief functions from sample data using multinomial confidence regions

- Mathematics
- 2006

Abstract The transferable belief model is a subjectivist model of uncertainty in which an agent’s beliefs at a given time are modeled using the formalism of belief functions. Belief functions that… Expand

Clouds, Fuzzy Sets, and Probability Intervals

- Computer Science, Mathematics
- Reliab. Comput.
- 2004

The basic theoretical and numerical properties of clouds are discussed, and they are related to histograms, cumulative distribution functions, and likelihood ratios, and to consistent possibility and necessity measures of Jamison and Lodwick. Expand

Constructing belief functions from sample data using multinomial confidence regions

- Computer Science
- Int. J. Approx. Reason.
- 2006

The proposed solution verifies two “reasonable” properties with respect to PX : it is less committed than PX with some user-defined probability, and it converges towards PX in probability as the size of the sample tends to infinity. Expand

Upper and Lower Probabilities Induced by a Multivalued Mapping

- Mathematics, Computer Science
- Classic Works of the Dempster-Shafer Theory of Belief Functions
- 2008

A distinctive feature of the present approach is a rule for conditioning, or more generally, arule for combining sources of information, as discussed in Sects. Expand

Algorithmic Power from Declarative Use of Redundant Constraints

- Mathematics, Computer Science
- Constraints
- 2004

This paper shows that computation of an unstable recurrence relation can be improved and shows that, by adding as redundant constraints instances of Taylor's theorem, one can obtain convergence that appears to be quadratic. Expand

Random sets and fuzzy interval analysis

- Mathematics
- 1991

Abstract A general expression of function with random set-valued arguments is stated, which encompasses Zadeh's extension principle as well as functions of random variables, and interval analysis. A… Expand