Transformation of Bimodal Probability Distributions Into Possibility Distributions

At the application level, it is important to be able to define the measurement result as an interval that will contain an important part of the distribution of the measured values, that is, a coverage interval. This practice acknowledged by the International Organization for Standardization (ISO) Guide is a major shift from the probabilistic representation. It can be viewed as a probability/possibility transformation by viewing possibility distributions as encoding coverage intervals. In this paper, we extend previous works on unimodal distributions by proposing a possibility representation of bimodal probability distributions. Indeed, U-shaped distributions or Gaussian mixture distribution are not very rare in the context of physical measurements. Some elements to further propagate such bimodal possibility distributions are also exposed. The proposed method is applied to the case of three independent or positively correlated C-grade resistors in series and compared with the Guide to the Expression of Uncertainty in Measurement (GUM) and Monte Carlo methods.

[1]  楊素芬.  Statistical process control for multiple stream processes , 2001 .

[2]  G. Mauris,et al.  Approximating Coverage Intervals of Well-Known Continuous Probability Distributions by Possibility Distributions , 2007, 2007 IEEE International Workshop on Advanced Methods for Uncertainty Estimation in Measurement.

[3]  George C. Runger,et al.  Statistical Process Control of Multiple Stream Processes , 1995 .

[4]  F. Pavese,et al.  A metrologist viewpoint on some statistical issues concerning the comparison of non-repeated measurement data, namely MRA Key Comparisons , 2006 .

[5]  Michał K. Urbański,et al.  Fuzzy approach to the theory of measurement inexactness , 2003 .

[6]  D. Dubois,et al.  Fuzzy real algebra: Some results , 1979 .

[7]  Laurent Foulloy,et al.  Fuzzy modeling of measurement data acquired from physical sensors , 2000, IEEE Trans. Instrum. Meas..

[8]  G. Mauris,et al.  Transformation of bimodal probability distributions into possibility distributions , 2008, 2008 IEEE International Workshop on Advanced Methods for Uncertainty Estimation in Measurement.

[9]  G. Mauris,et al.  A Comparison of Possibility and Probability Approaches for Modelling Poor Knowledge on Measurement Distribution , 2007, 2007 IEEE Instrumentation & Measurement Technology Conference IMTC 2007.

[10]  C. Perruchet.  SOME DIFFERENCES BETWEEN THE APPLIED STATISTICAL APPROACH FOR MEASUREMENT UNCERTAINTY THEORY AND THE TRADITIONAL APPROACH IN METROLOGY AND TESTING , 2004 .

[11]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[12]  D. Dubois,et al.  On Possibility/Probability Transformations , 1993 .

[13]  Alessandro Ferrero,et al.  The random-fuzzy variables: a new approach to the expression of uncertainty in measurement , 2004, IEEE Transactions on Instrumentation and Measurement.

[14]  Gilles Mauris,et al.  Representing and Approximating Symmetric and Asymmetric Probability Coverage Intervals by Possibility Distributions , 2009, IEEE Transactions on Instrumentation and Measurement.

[15]  P M Harris,et al.  Software Support for Metrology Best Practice Guide No. 6. Uncertainty evaluation. , 2006 .

[16]  Maurice G. Kendall,et al.  The advanced theory of statistics , 1945 .

[17]  Gilles MAURIS.  PROPAGATION OF MEASUREMENT UNCERTAINTY EXPRESSED BY A POSSIBILITY DISTRIBUTION WITH COVERAGE-INTERVAL-BASED SEMANTICS , 2006 .

[18]  Gilles Mauris.  Expression of Measurement Uncertainty in a Very Limited Knowledge Context: A Possibility Theory-Based Approach , 2007, IEEE Transactions on Instrumentation and Measurement.

[19]  Serafín Moral.  Construction of a Probability Distribution from a Fuzzy Information , 1986 .

[20]  Didier Dubois,et al.  Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities , 2004, Reliab. Comput..

[21]  G. Mauris,et al.  A fuzzy approach for the expression of uncertainty in measurement , 2001 .