Nonparametric Functional Data Analysis by Frédéric Ferraty

By Frédéric Ferraty

Glossy apparatuses let us gather samples of useful facts, usually curves but in addition photos. nevertheless, nonparametric information produces priceless instruments for normal facts exploration. This ebook hyperlinks those fields of contemporary facts via explaining how sensible facts might be studied via parameter-free statistical rules. even as it indicates how sensible information could be studied via parameter-free statistical principles, and gives an unique presentation of latest nonparametric statistical equipment for useful information research.

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Before going on, let us just recall some basic definitions. 1. is a semi-norm on some space F as soon as: 1) ∀(λ, x) ∈ R × F , λ x = |λ| x 2) ∀(x, y) ∈ F × F , x + y ≤ x + y . Note that in fact, a semi-norm . is a norm except that x = 0 ⇒ x = 0. Similarly, a semi-metric d can be defined to be a metric but such that d(x, y) = 0 ⇒ x = y. 2. d is a semi-metric on some space F as soon as: 1) ∀x ∈ F, d(x, x) = 0, 2) ∀(x, y, z) ∈ F × F × F , d(x, y) ≤ d(x, z) + d(z, y). 2 Semi-Metrics as Explanatory Tool A large part of explanatory tools consists in displaying data in low-dimensional spaces.

4. d. variables! Now, if we take into account the functional feature of the data, namely if we replace xi = (χi (t1 ), . . 1 maxi d(χi ,0) , where d denotes a functional measure of closeness. 4), we get Nd = 16. 5, it appears that such a functional approach (pending of course to a right choice of the d) may override the curse of dimensionality that was observed for phoneme data. Finally, it appears that the curse of dimensionality does not affect functional data with high correlation like the spectrometric data but is dramatic for the uncorrelated ones like the phoneme data.

Fr/staph/npfda 34 3 What is a Well-Adapted Space for Functional Data? for i = 1, . . , n. , one design per unit). 2 of [RS97], the fda handwriting data in [KLMR00], or [AOV99b]). Missing data can also lead to irregularly spaced design. An extreme case of unbalanced functional data is the so-called functional sparse data context: the individuals are observed at a sparse set of time points (see the Spinal bone mineral density or Globular filtration rate examples in [JH01] and [JS03]). According to the unbalanced data setting, specific preliminary processing before any statistical studies have been developed (see [S95] for estimating a shift parameter acting on the design points, [KG92], [GK95] and [KLMR00] for time-warping method in a curve registration context, [JHS00] and [YMW05] for sparse functional data and [B03] for automatic landmark registration).

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