Modeling Functional Data Using Quasi-Arithmetic Means References

Modeling Functional Data Using Quasi-Arithmetic Means Cuvelier Etienne & Noirhomme-Fraiture Monique FUNDP, Facult´e d’Informatique, 21, rue Grand Gagnage, 5000 Namur Probability distributions are central tools for probabilistic modeling in data mining. In functional data analysis, as functional random variable can be considered as stochastic process, the probability distribution have been studied largely, but with rather strong hypotheses like for Markov process. In this paper we just suppose that any considered function is in L2 (D), the space of real measurable functions u(t) defined on a real interval D. Let (Ω, A, P ) a probability space, a functional random variable (frv) is any function from D × Ω → R such for any t ∈ D, X(t, .) is a real random variable on (Ω, A, P ). X(t, .) can be considered as a stochastic process. If, for f, g ∈ L2 (D), we define the pointwise order between f and g on D as follows : ∀t ∈ D, f (t) ≤ g(t) ⇐⇒ f ≤D g

(1)

then the functional cumulative distribution function (fcdf ) of a frv X on L 2 (D) computed at u ∈ L2 (D) is given by : FX,D (u)

=

P [X ≤D u]

(2)

If we define the surface of distributions of an frv as follow : G : D × R → [0, 1] : (x, y) 7→ P [X(x) ≤ y]

(3)

then we define the Quasi-Arithmetic Mean of Margins Limit (QAMML) distribution of X by :   Z 1 · φ (G [t, u(t)]) dt (4) FX,D (u) = φ−1 |D| D where φ is a generator of Archimedean Copulas. We can show, that, if we R 1 note p = kDk G [t, u(t)] dt the arithmetic mean of G [t, u(t)] over D, then D FX,D (u) ≤ p and FX,D (u) = p ⇔ G [t, u(t)] = p ∀t ∈ D. And we use this property for a Quantile based classification for functional data.

References [1] Cuvelier, E. and Noirhomme-Fraiture M., (2007), Classification de fonctions continues l’aide d’une distribution et d’une densit´e d´efinies dans un espace de dimension infinie, Extraction et gestion des connaissances EGC’2007, 679690. [2] Ramsay, J.O. and Silverman, B.W., (2005), Functional Data Analysis, Springer : New-York. [3] Nelsen, R.B.,(1999), An introduction to copulas, Springer : London

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