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Optimal Transport, Independance Versus Indetermination Duality, Impact on a New Copula Design

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Geometric Science of Information (GSI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

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Abstract

This article leans on some previous results already presented in [10], based on the Fréchet’s works, Wilson’s entropy and Minimal Trade models in connection with the MKP transportation problem (MKP, stands for Monge-Kantorovich Problem). Using the duality between “independance” and “indetermination” structures, shown in this former paper, we are in a position to derive a novel approach to design a copula, suitable and efficient for anomaly detection in IT systems analysis.

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Notes

  1. 1.

    This explains the term: “Minimal Trade Model”.

  2. 2.

    There exist some constraints to satisfy for garanteeing the positivity of the optimal values \(\pi _{uv}\) (see [9]).

  3. 3.

    \(C(u,v)- C(u,v')- C(u',v)+ C(u',v')\ge 0\) \(\forall 0\le u \le u' \le 1\) \(\forall 0\le v \le v' \le 1\) is known as the 2-increasing property. It is nothing but the so called Monge’s condition which was coined by Alan Hoffmann in 1963 (see [6]), this is an additional link between optimal transport and copula theory.

  4. 4.

    In the last paragraph we suppose X to be at least bivariate.

  5. 5.

    Where \(\{\hat{Y}=0\} = \{P(X\le x)\le s\}\).

References

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Author information

Authors and Affiliations

  1. Thales Communications and Security, CENTAI Lab, Gennevilliers, France

    Benoit Huyot & Yves Mabiala

  2. Thales R&T Directorate and LSTA Lab, UPMC Paris VI University, Paris, France

    J. F. Marcotorchino

Authors
  1. Benoit Huyot
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  2. Yves Mabiala
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  3. J. F. Marcotorchino
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Corresponding author

Correspondence to J. F. Marcotorchino .

Editor information

Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

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Huyot, B., Mabiala, Y., Marcotorchino, J.F. (2015). Optimal Transport, Independance Versus Indetermination Duality, Impact on a New Copula Design. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_8

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  • DOI: https://doi.org/10.1007/978-3-319-25040-3_8

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