Functional Nonparametric Prediction Methodologies

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Functional Nonparametric Prediction Methodologies

• Markus Kuusisto

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Prediction problem

• Chemometric data, 215 samples
• functional sample curves ?i (spectra)
• scalar response yi (fat content)
• How to predict fat content from spectrum
• using whole spectrometric curve, continuity
• combining nonparametric consepts and functional
  variable modeling
• Following slides presents three different
  approaches

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Different approaches

• Regression, conditional expectation
• r(?) E(Y ? ?)
• y r(?)
• conditional c.d.f, median
• (?,y) P(Y ? y ? ?)
• m(?) (?,y) ? 1/2
• y m(?)
• Can also be used to calculate confidence band
• conditional p.d.f, mode
• (?,y) (?,y)
• ?(?) arg sup (?,y) , y ? S
• y ?(?)

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Properties of models

• r, , are required to be continuous
• Continuity-type models
• convergence results can be obtained
• If also Lipschits-type
• function will never have a slope steeper than
  Lipschitz constant
• precise rate of convergence can be found
• Difficulties
• infinite dimensional space of constrains
  (nonparametric model)
• infinite dimensional space of the functional
  feature of the explanatory variable

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Kernel Estimators regression

• Estimating the regression
• K is an asymmetrical kernel
• h is bandwidth of kernel
• r(?new) is weighted average of Y

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Kernel Estimators conditional c.d.f

• Estimating the conditional c.d.f
• g is bandwith
• H is c.d.f of Kernel type 0
• m(?) (?,y) ? 1/2

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Integrated symmetrical Kernels
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Kernel Estimators conditional p.d.f

• Estimating the conditional p.d.f
• (?,y) (?,y)