Functional Nonparametric
Prediction Methodologies
Markus Kuusisto
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
Different approaches
•
Regression, conditional expectation
– r() = E(Y| = )
– y’ = r’()
•
conditional c.d.f, median
–
–
–
–
•
FY (,y) = P(Y  y |  = )
m() = FY (,y)  1/2
y’ = m’()
Can also be used to calculate confidence band
conditional p.d.f, mode
– f Y (,y) =  FY (,y)
y
– () = arg supf Y
– y’ = ’()
(,y) , y  S
Properties of models
• r, FY, fY 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
Kernel Estimators: regression
• Estimating the regression
1
Y
K
(
h
d (  new ,  i ))
i1 i
n
r (  new ) 
1
K
(
h
d (  new ,  i ))
i1
n
– K is an asymmetrical kernel
– h is bandwidth of kernel
• r(new) is weighted average of Y
1
K
(
h
d (  new ,  i ))
i1
n
wh (  new ) 

n
i 1
K (h d (  new ,  i ))
1
1
Kernel Estimators: conditional c.d.f
• Estimating the conditional c.d.f
1
1
K
(
h
d
(

,

))
H
(
g
( y  Yi ))
i1
new
i
n
FY (  new , y) 
• g is bandwith
• H is c.d.f of Kernel type 0
u
H (u)   K 0 (v)dv


• m() = FY (,y)  1/2
1
K
(
h
d (  new ,  i ))
i1
n
Integrated symmetrical Kernels
Kernel Estimators: conditional p.d.f
•
Estimating the conditional p.d.f


– fY (,y) = FY (,y)
y

1
K
(
h
d
(

,

))
H
(
g
( y  Yi ))
i1
new
i
y
1
n
 
FY (  new , y ) 
y
1
1
K
(
h
d
(

,

))
K
(
g
( y  Yi ))
i1
new
i
0
g
n
fY (  new , y ) 
1
K
(
h
d (  new ,  i ))
i1
n
1
1
K
(
h
d (  new ,  i ))
i1
n