Fuzzy & Possibility Interest Group

Biweekly meetings

Everyone is welcome to attend.  F&PIG now meets via Mattermost (hosted by Université de technologie de Compiègne) on alternating Wednesdays at 5pm CET.

You can join the collaboration with the link: https://team.picasoft.net/signup_user_complete/?id=86wwg9ccdtyubdqg3bdks1j89c&md=link&sbr=su where the initial interface is in French, but once you've joined via email + username, you can change the language to English.   Thereafter, the Mattermost collaboration can be accessed via a browser, by navigating to https://team.picasoft.net/fnpig/channels/town-square.  There is also a handy desktop app which can be downloaded for free from https://mattermost.com/download/?utm_source=mattermost&utm_medium=in-product&utm_content=menu_item_external_link&uid=19potgpg6fngtrwgn441tigoyr&sid=qqztsmoh43b78bfron8kb31c1o&redirect_source=about-mm-com#desktop. To get started on the desktop app, specify "https://team.picasoft.net" (no quotes) when it asks for the Server URL (the URL of your Mattermost server) and "F&PIG" when it asks for the  Server display name (the name that will be displayed in your server list).  Once loaded, Mattermost will continue to run in the system tray, so it should always be accessible.

Previously meeting were held via Zoom https://liverpool-ac-uk.zoom.us/j/92016191644?pwd=MzlCcTU3RDVENHlrUXFUdlNmVXhKQT09  hosted by Ander Gray (Ander.Gray@liverpool.ac.uk).

Please ignore any Teams meeting links.

Alternate Wednesdays

    18:00 EEST/EET (Helskinki, Sophia, Athens)
    17:00 CEST /CET (Stockholm, Berlin, Paris)
    16:00 BST/GMT (London)
    12:00 BRT (Rio de Janeiro)
    11:00 ET (New York)
    09:00 MT (Albuquerque, El Paso)
    08:00 PT (Los Angeles)
    23:00 SST (Singapore)

N.B. Some times vary with the asynchronous changes into or out of daylight savings.  The base time is 17:00 Paris time.

Find out more

PUBLIC WEBPAGE: https://sites.google.com/view/fnpig/home

THIS GOOGLE SITE: https://sites.google.com/view/fnpig/

OVERLEAF PAPER: https://www.overleaf.com/project/5fc4ef23867d1b2a646b8bfb

Members

This interest group is open to everyone. To be added to the list, please ask Ander Gray (ander.gray@ukaea.uk). You can remove yourself anytime.  Current and honorary members include 

Michael Balch, msb1@alexvalcon-llc.com
Ander Gray, ander.gray@ukaea.uk
Nick Gray, ngg@liverpool.ac.uk
Marco De Angeles, marco.deangelis09@gmail.com, marco.de-angelis@strath.ac.uk
Peter Hristov,  P.Hristov2@liverpool.ac.uk
Scott Ferson, sandp8@gmail.com, ferson@liverpool.ac.uk
Thomas Koenecke, tom.koenecke@itm.uni-stuttgart.de
Adolphus Lye, snrltsa@nus.edu.sg
Krasymyr Tretiak, krasymyr.t@gmail.com
Alex Wimbush, a.wimbush.1@bham.ac.uk
Robin Langley, rsl21@cam.ac.uk
Dominik Hose, dominik.hose@outlook.de
Kari Sentz, ksentz@lanl@gov, kari.sentz@gmail.com

There is currently no group mailing list, but to address all active members, copy this string to your email's send field:   ander.gray@ukaea.uk;a.wimbush.1@bham.ac.uk;N.G.Gray@liverpool.ac.uk;tom.koenecke@itm.uni-stuttgart.de;marco.de-angelis@strath.ac.uk;snrltsa@nus.edu.sg;P.Hristov2@liverpool.ac.uk;sandp8@gmail.com;ferson@liverpool.ac.uk

Software

See the Code page.

Ander's Julia package FuzzyArithmetic.jl is under development. Access to the Risk Institute GitHub account is required. Access can be granted on request.  Julia version pba.jl is live and under development.

Scott's R source code union.r is under development.  His source code library for pba.r is changing slowly.  Other libraries such as the more complete pbox.r, the Matlab implementation riskCalc4Matlab.m, and the commercial software RAMAS Risk Calc are also accessible through Scott upon request.

Nick's Python module pba.py for probability bounds analysis is under development.

 

Documents in development

Notes : https://www.overleaf.com/read/jbdnvfhtctpm
https://www.overleaf.com/project/5fc4ef23867d1b2a646b8bfb OVERLEAF PAPER

Links

This webpage is https://sites.google.com/view/fnpig.
Joint paper: GIT Repository

Other resources

FnPIG's ISIPTA paper: Dependent Possibilistic Arithmetic using Copulas

Noémie Le Carrer
thesis:  A possibilistic framework for interpreting ensemble predictions in weather forecasting and aggregate imperfect sources of information
talk: Making sense of ensemble predictions in weather forecasting: Can possibility theory overcome the limitations of standard probabilistic interpretations?

Dominik Hose (Universität Stuttgart)

thesis:  Possibilistic Reasoning with Imprecise Probabilities: Statistical Inference and Dynamic Filtering
paper: Possibilistic calculus as a conservative counterpart to probabilistic calculus
talk: The embarrassingly simple calculus of possibility theory
slides: A fuzzy approach to uncertainty quantification with imprecise probabilities
code: Matlab Script for Imprecise-Probability-to-Possibility Transformation

Sebastien Destercke (Université de Technologie de Compiègne)
paper: A Consonant Approximation of the Product of Independent Consonant Random Sets

Thierry Denoeux (Université de Technologie de Compiègne)
paper: Marie-Hélène Masson and Thierry Denœux. 2006. Inferring a possibility distribution from empirical data. Fuzzy Sets and Systems 157(3):319–340. ISSN 0165-0114, https://doi.org/10.1016/j.fss.2005.07.007

Didier Dubois (Université de Toulouse III — Paul Sabatier)
talk: Representing uncertainty: randomness vs. incomplete information 20pp
talk: Possibility and probability in risk analysis 46pp
talk: Possibility theory: the missing link between logic and probability? 66pp
paper: Risk-informed decision-making in the presence of epistemic uncertainty 2pp
talk: Uncertainty theories: a unified view: from imprecise probability to possibility theory 65pp
paper: Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities

Cédric Baudrit (Université de Toulouse III — Paul Sabatier)
paper link: Practical representations of incomplete probabilistic knowledge
uploaded paper: Practical representations of incomplete probabilistic knowledge
thesis: Représentation et propagation de connaissances imprécises et incertaines

Scott Ferson
slides: Fuzzy arithmetic and the lazy risk analyst conjecture

Troffaes, Matthias CM, Enrique Miranda, and Sebastien Destercke:
paper: On the connection between probability boxes and possibility measures.

Williamson's argument

Bob Williamson wrote extensively about fuzzy arithmetic in his 1989 PhD thesis Probabilistic Arithmetic (University of Queensland, 163 ff).  He concluded 

The theory of fuzzy numbers, as developed to date, does essentially nothing new compared to the theory of random variables (when proper account is taken of missing independence information), and would appear to be of little value in engineering applications. (Williamson 1990, page 166, italics in the original)

Integrating fuzzy and p-box methods

Baudrit and Dubois (2006; Baudrit 2005; Dubois <<>>), inter alia, have suggested that possibilistic calculations retain some knowledge that is lost by a probability bounds analysis.  This implies that, in the same way that moment projection enriched the basic boundary cdf calculations suggested by Williamson and Downs (1990) to form probability bounds analysis, it may be the case that possibilistic calculations could be incorporated to tighten or otherwise improve calculation outputs. The loosest version of this idea is that the calculations are parallel and both the inputs and the outputs are mutually informed by the separate calculations, as depicted below.

Focus and purpose

In the figure below, inputs are transformed into outputs according to some specified mathematical operation. The mathematical operation might be a convolution or a non-convolutive operation like mixture, composition, or intersection, one or a sequence of unary transformations such as log or square root, summarisation functions such as mean or range, or any arbitrary combination of these various operations. (Note that we  use the word 'convolution' to refer to the distributional analog of a binary arithmetic operation, including addition but also multiplication, subtraction, division, powers, minimum, etc., or something more complex that is a sequence or composition of such operations.) The mathematical operation includes specification of what is known about intervariable dependencies.

The left-to-right arrows in the figure denote the mathematical operations within each of three propagation theories. The bottom arrow denotes the propagation of moment information, by which, for instance, one may be able to compute the mean and variance of a mathematical function of uncertain quantities if one know the means and variances—or at least bounds on the means and variances—of those inputs. The middle left-to-right arrow denotes the theory developed by Makarov (1981), Frank et al. (1987) Williamson and Downs (1990) and extended in probability bounds analysis. The top arrow denotes the propagation of fuzzy numbers or more generally possibility distributions through a specified mathematical operation.  

The arrows between the three levels correspond to the implications, if any, that structures in one theory have for structures in the other theory. For instance, knowing the mean and variance (bottom level) of an uncertain number implies constraints on its cdfs (middle level) via the Chebyshev inequality. Conversely, knowing bounds on the cdfs likewise implies bounds on the moments.

The central question at hand is whether integrating these three propagation theories can often or substantially improve the results. To assess this idea, we want to be able to

• 1. identify the credal set implied by a given PoD (possibility distribution), or at least be able to say whether a given distribution is in its credal set),

  2. identify the credal set matching a given p-box,

  3. identify the credal set matching a given random set/DSS (Dempster-Shafer structure),

  4. characterize an uncertain number with a PoD given empirical knowledge about it,

  5. characterize an uncertain number with a p-box given the same empirical knowledge,

  6. characterize an uncertain number with a DSS given the same empirical knowledge,

  7. find a credal set imperfectly modelled by a p-box and a possibility distribution with different deficiencies,

  8. construct a data structure that integrates a p-box and a possibility distribution so as to combine the strengths of each, 

  9. extract modal, or any other information, from a convolution A + B of uncertain quantities A and B using the integration of the two theories than can be done using either theory alone, and

  10. answer simple existence questions such as (i) Does any given credal set imply a unique p-box, (ii) Does a given credal set likewise correspond to a single possibility distribution? and (iii) If not, is there a canonical choice that can be made so that we can have uniqueness in practice?

Further issues about the interplay of possibility distributions and probability boxes 

***Information that can be extracted from possibility distributions
bounds on expectation
core (singleton point masses)
support
lower bound (1-alpha) on an alpha-cut
upper bound [0,alpha] on double tail risks of the sublevel set at alpha {X: pi(X) <= alpha} = complement of alpha cut
conservative bounds on p-box, the bounding distributions of which are in the credal set

***How to specify possibility distributions
possibility density
mode
range

***Information from p-boxes
bounds on quantiles
bounds on tail risks
core
support
bounds on moments
family
crappy bounds on internal risks, Pr(a<X<b)

***How to specify p-box
range
moments
constraints on quantiles or (exceedance) probabilities
shape (family, unimodality, symmetry, positivity, etc.)
mode
bounds on probability densities
parametric distribution families with interval parameters
data (fitting, predictive distributions from c-boxes)

***Extra
inference with c-boxes

***What p-boxes cannot do
reveal tight bounds on the probability of internal risks Pr(a < X < b)
retain mode and modality information