0.5 1economia.uc3m.es/wp-content/uploads/pdfs/papers/talk...that sup 0 x 1 Z 1 0 (S N (t) N (t)) 2...
56
University Carlos III de Madrid June 2018 Functional data analysis Lajos Horv ´ ath University of Utah, Salt Lake City
Transcript of 0.5 1economia.uc3m.es/wp-content/uploads/pdfs/papers/talk...that sup 0 x 1 Z 1 0 (S N (t) N (t)) 2...
Univ
ersi
ty
Carlos
III
de
Madrid
June
2018
Functio
nal
data
analy
sis
Lajo
sH
orvath
Univ
ersi
ty
of
Utah,
Salt
Lake
Cit
y
● ● ●●● ● ●●● ●● ●●● ●●●●● ● ● ● ●●●●● ● ● ●●●●● ●●● ●●● ●● ● ●● ●● ●●●●●●●●● ●●● ●●●●●●● ●●● ● ●●●●●●●● ●● ● ● ●●●● ● ●●●● ● ●●●● ●●● ● ● ●● ●● ● ● ● ●● ●● ● ●●●●●●●●● ● ●● ● ●●● ● ● ● ● ●● ● ●●●●● ● ●● ●●●●●●●●●●●●●● ●● ● ● ● ●●● ●●●●●● ●●● ● ●● ● ●●●● ●●●● ● ●● ●●●● ● ● ● ● ●●●● ● ● ●● ●● ●●●●●● ● ● ● ●● ●● ● ●●● ●● ●●●● ●●●●●●●● ● ●● ● ● ● ●● ●● ● ● ●● ●● ●
192.0192.5193.0193.5
Closing Price($)
1−1−
2013
1−3−
2013
●● ● ●●● ●● ●●● ●● ● ●●●●● ● ● ●● ●●● ● ● ● ● ● ●●●● ● ●● ● ●● ●●●● ● ●●● ● ● ● ● ●● ●●● ● ● ● ● ●●●●● ● ● ●● ● ●●●●● ●●● ●● ●● ●● ●● ●● ●●● ● ●● ● ●●●●●● ●● ●●●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ●● ●●●● ●●●● ● ● ●●●●●●● ● ●● ● ●● ● ●●● ●● ●● ●● ● ●● ●● ●● ●● ●●●●●● ● ● ●● ●● ● ● ●●●● ● ●● ●●●● ●●●●●●●● ●●● ● ●●● ● ●●● ●● ● ● ●●● ●●●● ● ●●●● ● ●●●● ●●● ●● ●●●● ● ●●● ● ●● ●●●● ●● ● ● ●●● ●● ● ●● ●●●●●●●● ●● ●●●●● ●●●● ● ● ● ●●●● ● ●● ● ●●●● ●●●● ●● ●● ●● ● ● ●● ●●● ● ● ● ●●●●●●● ● ●● ● ●●●●●● ● ●● ● ● ●●● ● ●● ●● ● ● ●●●●● ● ●●●● ●● ●●● ●● ●● ● ● ● ●● ●●● ● ● ● ●●●● ●● ● ●● ● ●●●● ● ●●●●●● ● ● ●●●●● ●● ●●● ●● ●● ● ●● ●●● ● ●● ●● ● ● ●●● ●● ● ● ●●● ●●● ● ● ● ●● ●●●● ● ●●● ● ●●● ● ● ● ● ●● ●● ●● ●● ●●● ● ●●●●● ● ● ● ●●● ● ●●●●●●●●● ●●● ●●● ●●●●● ●●● ●● ●●●●●● ●●● ●● ●●●● ●●●● ●●●● ●● ●● ● ●● ●● ●●● ●● ●●● ● ● ●● ●●● ● ● ●●● ●● ●● ● ● ● ●●●●●● ●● ● ●● ●● ●●●● ●● ● ●●●●●● ●●●●● ● ●●● ●●● ●● ● ●●● ● ●● ● ● ● ●● ●●● ●● ● ●●● ●●●● ● ● ● ● ● ●● ●● ●●●●●● ● ● ● ●● ● ●● ●● ●● ●● ●● ●● ● ● ● ●●● ●
192.0192.5193.0193.5
Closing Price($)
1−1−
2013
1−3−
2013
IBM
stoc
kp
rice
curv
es
IBM
/Wal
mar
tst
ock
pri
cecu
rves
−1.0−0.50.00.51.0
IBM
WM
T
Xn
(t)
mag
net
omet
erre
adin
gson
dayn
atti
met
Dag
lis,
I.A
.,K
ozyra
,J.
U.,
Kam
ide,
Y.,
Vas
sili
adis
,D
.,S
har
ma,
A.
S.,
Lie
moh
n,
M.W
.,G
onza
lez,
W.
D.,
Tsu
ruta
ni,
B.
T.
and
Lu,
G.
(200
3).
Inte
nse
spac
est
orm
s:C
riti
cal
issu
esan
dop
end
isp
ute
s.J
ourn
alof
Geo
-
phys
ical
Res
earc
h,10
8.
Tim
e in
min
utes
014
4028
8043
2057
6072
0086
4010
080
Xn
(t)
isre
adin
gfr
omth
eT
ecat
orIn
frat
ecfo
odan
dfe
edan
alyz
er
Yao
,F
.an
dM
ull
er,
H-G
.(2
010)
.F
un
ctio
nal
qu
adra
tic
regr
essi
on.
Bio
met
rika
97,
49–6
4.
850
900
950
1000
1050
2.02.53.03.54.04.5
wav
elen
gth
(nm
)
absorbance
Xn
(t)
isth
ep
ollu
tion
leve
lon
dayn
atti
met
Fer
nan
dez
de
Cas
tro,
B.,
Gu
illa
s,S
.an
dG
onza
les
Man
teig
a,W
.(2
005)
.F
un
ctio
nal
sam
ple
san
db
oot
stra
pfo
r
pre
dic
tin
gsu
lfu
rdio
xid
ele
vels
.T
echn
omet
rics
,47
,21
2–22
2.
05
1015
20
0100200300400
NO
x le
vels
Hou
rs
mg/m³
Th
esp
ace
ofsq
uar
ein
tegr
able
fun
ctio
ns
L2
isth
esp
ace
ofal
lfu
nct
ion
s{f}
such
that∫ f(t
)dt<∞
(∫ mea
n∫ 1 0
)
inn
erp
rod
uct〈f,g〉=
∫ f(t)g
(t)dt
the
nor
mis
ind
uce
dby
the
inn
erp
rod
uct‖f‖
=√ 〈f
,f〉
We
assu
me
thatE∫ X2
(t)dt<∞
and
ther
efor
eP{ω
:X
(t;ω
)∈L2}
=1.
{ϕi,i≥
1}is
anor
thon
orm
alb
asis
ofL2
Kar
hu
nen
–Loe
veex
pan
sion
:X
(t)
=∞ ∑ i=1
〈X,ϕ
i〉ϕi(t)
We
can
app
roxim
ateX
(t)
wit
hth
efi
nal
dim
ensi
onal
pro
cess∑ d i=
1〈X,ϕ
i〉ϕi(t)
and
E
∥ ∥ ∥ ∥ ∥X−d ∑ i=1
〈X,ϕ
i〉ϕi∥ ∥ ∥ ∥ ∥2
→0
asd→∞
Dim
ensi
onre
du
ctio
n–r
epla
ceX
(in
fin
ite
dim
ensi
onal
)w
ith〈X,ϕ
1〉,〈X,ϕ
2〉,...,〈X,ϕ
d〉.
How
toch
oos
eth
eb
ases
?T
he
bes
tm
ean
squ
ared
erro
r:
inf f∈L
2E‖X−ξ 1
(f)f‖2
=E‖X−ξ 1
(f1)f
1‖2,
inf f∈L
2,〈f,f
1〉=
0E‖X−
(ξ1(f
1)f
1+ξ(f
)f)‖
2=E‖X−
(ξ1(f
1)f
1+ξ 2
(f2)f
2‖2
Cov
aria
nce
oper
ator
Sol
uti
onto
the
min
imiz
atio
np
rob
lem
Cov
aria
nce
kern
el:EX
(t)
=0
andE∫ X2
(t)dt<∞
C(t,s
)=EX
(t)X
(s)
C(t,s
)is
asy
mm
etri
c,p
osit
ive
defi
nit
efu
nct
ion
–gen
eral
Hil
ber
tsp
ace
theo
ry(s
pec
tral
theo
rem
)gi
ves
ther
ear
eλ1≥λ2≥...≥
0an
dor
thon
orm
alfu
nct
ion
s{ϕ
i,i≥
1}su
chth
at
λiϕ
i(t)
=
∫ C(t,s
)ϕi(s)ds
1≤i<∞
C(t,s
)=
∞ ∑ i=1
λiϕ
i(t)ϕ
)i(s
)
(eig
enfu
nct
ion
san
dei
genva
lues
)
Tec
hn
ical
com
men
t:
Ifλd>
0an
dλd+1
=0
then
Ch
ason
lyd
eige
nfu
nct
ion
s.T
hes
eca
nb
eex
ten
ded
into
anor
thog
onal
bas
esan
d∫ C(
t,s)ϕi(s)ds
=0
for
alli>d
Par
tial
sum
sin
Hilb
ert
spac
es
Par
tial
sum
sp
roce
ss:
SN
(x,t
)=N−1/
2
bNxc ∑ i=1
Xi(t).
Th
eore
m2:
IfX
1,X
2,...,X
Nar
eiid
wit
hEX
1(t
)=
0an
dE∫ X2 1
(t)dt<∞
,th
enw
eca
nd
efin
eG
auss
ian
pro
cess
esΓN
(x,t
)su
chth
atsu
chth
at
sup
0≤x≤1
∫ 1 0
(SN
(x,t
)−
ΓN
(x,t
))2dt
P −→0,
andE
ΓN
(x,t
)=
0an
dE
ΓN
(x,t
)ΓN
(y,s
)=
min
(x,y
)C(t,s
).
Not
e
ΓN
(x,t
)D =
∞ ∑ i=1
λ1/2
iW
i(x
)ϕi(t),
wh
ereW
i(x
),1≤i<∞
are
ind
epen
den
tW
ien
erp
roce
sses
(sta
nd
ard
Bro
wn
ian
mot
ion
),i.
e.G
auss
ian
pro
cess
es
wit
hEW
i(x
)=
0an
dEW
i(x
)Wi(y)
=m
in(x,y
).
Est
imat
ion
infu
nct
ion
alm
odel
s
We
nee
dto
esti
mat
eth
eei
genva
lues
and
eige
nfu
nct
ion
s–w
en
eed
toes
tim
ateC
CN
(t,s
)=
1 N
N ∑ i=1
(Xi(t)−XN
(t))
(Xi(s)−XN
(s))
wit
hXN
(t)
=1 N
N ∑ i=1
Xi(t).
By
the
law
ofla
rge
nu
mb
ers
inH
ilb
ert
spac
esw
eh
ave
‖CN−C‖
P −→0.
Em
pir
ical
eige
nva
lues
and
orth
onor
mal
eige
nfu
nct
ion
s:
λi,Nϕi,N
(t)
=
∫ CN
(t,s
)ϕi,N
(s)ds,
1≤i<∞.
Th
eore
m3:λ1>λ2>...>λd>λd+1>
0
max
1≤i≤d|λi,N−λi|
P −→0
and
max
1≤i≤d‖ϕ
i,N−c i,Nϕi‖
P −→0,
wh
erec i,N,1≤i≤d
are
ran
dom
sign
s.
Th
era
teof
conve
rgen
ceis
exac
tlyN−1/2.
Ch
ange
poi
ntd
etec
tion
inth
em
ean
curv
e
Mod
el:Yi(t)
=µi(t)
+Xi(t),
1≤i≤N
H0
:µ1(t
)=µ2(t
)=...
=µN
(t)
inth
eL2
sen
se
agai
nst
the
alte
rnat
ive
that
ther
eis
anin
tege
rk∗
such
that
µ1(t
)=...
=µk∗(t
)6=µk∗ +
1(t
)=...
=µN
(t)
inth
eL2
sen
se.
Th
eC
US
UM
(CU
mu
lati
veS
UM
)p
roce
ssis
S◦ N
(x,t
)=N−1/2
bNxc ∑ i=1
Yi−bN
xc
N
N ∑ i=1
Yi ,
0≤x,t≤
1.
By
the
pre
vio
us
resu
ltS◦ N
(x,t
)≈
Γ◦ (x,t
)=
Γ(x,t
)−x
Γ(1,t
).C
lear
ly,
Γ◦ (x,t
)is
Gau
ssia
nw
ith
zero
mea
nan
dE
Γ◦ (x,t
)Γ◦ (y,s
)=
(min
(x,y
)−xy)C
(t,s
)
We
hav
eu
nd
erth
en
och
angeH
0:
sup
0≤x≤1
∫ (S◦ N
(x,t
))2dtD −→
sup
0≤x≤1
∫ (Γ◦ (x,t
))2dt
and
∫∫(S◦ N
(x,t
))2dtdxD −→∫∫
(Γ◦ (x,t
))2dtdx.
Rep
rese
nta
tion
:
Γ◦ (x,t
)=
∞ ∑ i=1
λ1/2
iB i
(x)ϕ
i(t),B 1,B
2,...
are
iid
Bro
wn
ian
bri
dge
s
Ap
pro
xim
atio
ns
for
the
limit
dis
trib
uti
ons
sup
0≤x≤1
∫ (Γ◦ (x,t
))2dt
=∞ ∑ i=1
λiB
2 i(x
)an
d
∫∫(Γ◦ (x,t
))2dtdx
=∞ ∑ i=1
∫ λiB
2 i(x
)dx
Ap
pro
xim
ati
on
:
sup
0≤x≤1
∫ (Γ◦ (x,t
))2dt≈
d ∑ i=1
λi,NB2 i(x
)an
d
∫∫(Γ◦ (x,t
))2dtdx≈
d ∑ i=1
λi,N
∫ B2 i,N
(x)dx
How
toch
oose
d?
smal
lλi
ises
tim
ated
wit
hla
rge
erro
rscr
ow’s
feet
λ1,N
+λ2,N
+...+
λd,N
λ1,N
+λ2,N
+λ3,N...≈
0.9
(or.9
5,0.
99)
Wh
at
hap
pen
sto
these
ap
pro
xim
ati
on
su
nd
er
the
alt
ern
ati
ve
?
Un
der
the
alte
rnat
ive
ther
eis
asy
mm
etri
cp
osit
ive
fun
ctio
nC∗
such
that‖C
N−C∗ ‖
P −→0.
Hen
ce
d ∑ i=1
λi,NB2 i(x
)≈
d ∑ i=1
λ∗ iB
2 i(x
)an
dd ∑ i=1
λi,N
∫ B2 i,N
(x)dx≈
d ∑ i=1
λ∗ i
∫ B2 i(x
)dx,
wh
ereλ∗ 1≥λ∗ 2≥...≥λ∗ d
are
the
eige
nva
lues
ofC∗ .
Fin
ite
crit
ical
valu
esu
nd
erH
0as
wel
las
un
derH
0u
sin
gth
em
eth
od
abov
e.
Th
efu
nct
ion
als
ofSN
(x,t
)co
nve
rge
to∞
inp
rob
abil
ity
un
derHA
.
Pro
ject
ion
met
hod
We
use
emp
iric
alp
roje
ctio
ns
usi
ngϕ1,N,ϕ
2,N,...,ϕ
d,N
,th
eei
gen
funct
ion
sas
soci
ated
wit
hth
ed
larg
est
eige
n-
valu
esλ1,N≥λ2,N≥...≥λd,N
ofCN
Pro
ject
ed
CU
SU
M
PN
(x)
=1 N
d ∑ `=1
1
λ`,N
bNxc ∑ i=1
〈Yi−YN,ϕ
`,N〉2
=d ∑ `=1
1
λ`,N
〈S◦ (x,·
),ϕ`,N
(·)〉2.
Th
eore
m4:
IfX
1,X
2,...,X
Nar
eii
d(H
0h
old
s)w
ithEX
1(t
)=
0an
dE∫ X2 1
(t)dt<∞
,th
enw
eh
ave
PN
(x)
D[0,1]
−→d ∑ i=1
B2 i(x
),w
her
eB 1,B
2,...
are
iid
Bro
wn
ian
bri
dge
s.
Con
dit
ion
for
con
sist
en
cy:
Un
derHA
we
pro
ject
the
diff
eren
cesYi−YN
toth
esp
ace
span
ned
byϕ∗ 1,ϕ∗ 2,...,ϕ∗ d,
the
eige
nfu
nct
ion
sas
soci
ated
wit
hth
ed
larg
est
eige
nva
lues
ofC∗ .
So
the
pro
cedu
reis
con
sist
ent
if〈E
(Yk∗−Yk∗ +
1),ϕ∗ `〉6=
0fo
rat
leas
ton
e
`∈{1,2,...,d}.
Tem
per
atu
res
inC
entr
alE
ngl
and
1780
–200
7(2
28ye
ars)
Tim
e
Degrees Celsius
0.0
0.2
0.4
0.6
0.8
1.0
−505101520
Act
ual d
aily
tem
pera
ture
sF
unct
iona
l obs
erva
tion
Mon
thly
ave
rage
s
Tab
le1:
Su
mm
ary
and
com
par
ison
ofse
gmen
tati
on.
Beg
inn
ing
and
end
ofd
ata
per
iod
inb
old
.
Appro
ach
Changepoints
FDA
1780
1808
1850
1926
1992
2007
MDA
1780
1815
1926
2007
Res
ult
s
Th
etr
adit
ion
alM
DA
use
sd
=12
dim
ensi
onal
dat
a,F
DA
use
sd
=8
pro
ject
ion
s.
Ave
rage
curv
es
Tim
e
Degrees Celsius
0.0
0.2
0.4
0.6
0.8
1.0
05101520
1808
− 1
849
1850
− 1
925
1926
− 1
991
1992
− 2
007
Dep
end
ent
pro
cess
inL
2
Au
tore
gres
sive
(1)
pro
cess
inL2
Xk(t
)=
∫ 1 0
Ψ(s,t
)Xk−1(s
)ds
+ε k
(t),−∞
<k<∞
Con
dit
ion
for
stat
ion
arit
y:
ε k(t
)ar
ein
dep
end
ent
and
iden
tica
lly
dis
trib
ute
d
Eε 0
(t)
=0
and
∫ 1 0
Eε2 0
(t)dt<∞,
∫ 1 0
∫ 1 0
|Ψ(s,t
)|mdsdt<
1w
ith
som
em≥
2.
Exp
lici
tex
pre
ssio
nfo
rth
est
atio
nar
ity
solu
tion
Xk(t
)u
sin
git
erat
ions
ofΨ
(s,t
)an
dth
eer
rorsε j,j≤k.
Mor
ed
epen
den
tp
roce
ssinL
2
Fu
nct
ion
alA
RM
A(p,q
)
Xk(t
)=
p ∑ `=1
∫ 1 0
Ψ`(s,t)Xk−`(s)ds
+ε k
(t)
+
q ∑ `=1
∫ 1 0
Θ`(s,t)ε k−`(s)ds,
t∈
[0,1
],−∞
<k<∞.
Th
ere
isn
on
eces
sary
and
suffi
cien
tco
nd
itio
nfo
rth
eex
iste
nce
ofth
est
atio
nar
yso
luti
on.
Fu
nct
ion
alli
nea
rp
roce
sses
Xk(t
)=
∞ ∑ `=0
∫ Ψ`(s,t)ε k−`(s)ds.
Con
dit
ion
for
stat
ion
arit
y:
ε k(t
)ar
ein
dep
end
ent
and
iden
tica
lly
dis
trib
ute
d
Eε 0
(t)
=0
and
∫ 1 0
Eε2 0
(t)dt<∞,
∞ ∑ `=1
‖Ψ`‖<∞.
Vol
atili
typ
roce
ssinL
2
Fu
nct
ion
alA
RC
H(1
)
Xk(t
)=ε k
(t)σ
k(t
),an
dσ2 k(t
)=δ(t)
+
∫ 1 0
Ψ(s,t
)X2 k−1(s
)ds
δ≥
0an
dΨ≥
0
Con
dit
ion
for
con
sist
ency
:
E
{ ∫ 1 0
∫ 1 0
Ψ2(t,s
)ε4 0(s
)dtds} τ <
1w
ith
som
eτ>
0
con
stan
tco
nd
itio
nal
corr
elat
ion
Fu
nct
ion
alG
AR
CH
(1,1
)
Xk(t
)=ε k
(t)σ
k(t
),
σ2 k(t
)=δ(t)
+
∫ 1 0
Ψ1(s,t
)X2 k−1(s
)ds
+
∫ 1 0
Ψ2(s,t
)Yσ2 k−1(s
)
δ≥
0an
dΨ
1≥
0Ψ
2≥
0
Ber
nou
llip
roce
sses
inH
ilber
tsp
aces
ther
eis
afu
nct
ion
ala
:S∞→
L2
such
thatXk
=a(εk,εk−1,...
)
ais
am
easu
rab
lefu
nct
ion
al
(εk:k∈Z)
are
iid
sequ
ence
sof
ran
dom
elem
ents
wit
hva
lues
inso
me
mea
sura
ble
spac
esS
ω0(t
;ω)
isjo
intl
ym
easu
rab
lein
(t,ω
)
Wea
kD
epen
den
ce
E‖X
k‖κ<∞,
and
∞ ∑ `=1
(E‖X
k−X
(`)
k‖κ
)1/κ<∞
wit
hso
me
κ>
2,
wh
ere
X(`)
k=a(εk,εk−1,...,εk−`+
1,ε
(`)
k−`,ε(`) k−`−
1,...
),
(ε(`)
k:k,`∈Z)
are
iid
cop
ies
ofε 0
dep
end
ence
onp
revio
us
inn
ovat
ion
sis
“dyin
gou
r”fa
sten
ough
Su
ms
ofw
eakl
yd
epen
den
tfu
nct
ion
alti
me
seri
es
SN
(x,t
)=
1
N1/
2
bNxc ∑ i=1
Xi(t)
Th
eore
m5:
Un
der
the
wea
kd
epen
den
ceas
sum
pti
ons,
ther
eis
ase
qu
ence
ofG
auss
ian
pro
cess
esΓN
(x,t
)w
ith
EΓN
(x,t
)=
0an
dE
ΓN
(x,t
)ΓN
(y,s
)=
min
(x,y
)D(t,s
)
such
that
sup
0≤x≤1‖α
N(x,t
)−
ΓN
(x,t
)‖P −→
0,
asN→∞
,w
her
e
D(t,s
)=
∞ ∑ `=−∞
EX
0(t
)EX`(s).
Wh
atis
the
bes
tb
asis
?th
eei
gen
fun
ctio
ns
ofC
(t,s
)=EX
1(t
)X1(s
)or
the
eige
nfu
nct
ion
sof
the
lon
gru
nco
vari
ance
fun
ctio
nD
(t,s
)?
We
pro
jectSN
(x,t
)so
we
nee
dto
pre
serv
eth
eva
riab
ilit
yinSN
(x,t
)–
use
the
eige
nfu
nct
ion
sof
the
lon
gru
n
cova
rian
cefu
nct
ionD
(t,s
).
Est
imat
ion
ofth
elo
ng
run
cova
rian
cefu
nct
ion
We
esti
mat
eD
(t,s
)w
ith
DN
(t,s
)=
∞ ∑ i=−∞K
( i h
) γi(t,s)
wit
hth
eem
pir
ical
cova
rian
ces
γi(t,s)
=γi,N
(t,s
)=
1 N
N−i ∑ j=1
(Xj(t
)−XN
(t))
(Xj+i(s)−XN
(s)),
i≥
0
1 N
N ∑ j=1−i(X
j(t
)−XN
(t))
(Xj+i(s)−XN
(s)),i<
0
and
the
sam
ple
mea
n
XN
(t)
=1 N
N ∑ `=1
X`(t).
Ass
um
pti
on
on
the
win
dow
size
:
h=h
(N)→∞
and
h(N
)
N→
0,as
N→∞
Ass
um
pti
ons
onth
eke
rnel
K(0
)=
1
Kis
sym
met
ric
arou
nd
0an
dK
(u)
=0
ifu>c
wit
hso
mec>
0
Kis
Lip
sch
itz
conti
nu
ous
on[−c,c]
Exam
ple
s:B
artl
ett
kern
el,
Par
zen
kern
el,
flat
–top
kern
elof
Pol
itis
and
soon
Th
eore
m6:
We
assu
me
that
the
wea
kB
ern
oull
ias
sum
pti
on,
and
the
con
dit
ions
onh
andK
hol
d,
then
we
hav
e‖D
N−D‖
P −→0.
How
toch
oos
eh
?M
inim
ize
the
mea
n–s
qu
ared
erro
r,i.
e.m
inim
izeE‖D
N−D‖2
.U
sin
gst
and
ard
argu
men
tsw
en
eed
that
inca
seof
the
”op
tim
al”h
for
larg
eN
(so
larg
eh
)w
eh
aveE‖D
N−EDN‖2
=‖E
DN−D‖2
.T
he
”op
tim
al”h
dep
end
son
the
un
kn
ownD
–pra
ctic
al(d
ata
dri
ven
way
s)to
choos
eh
.
Dat
ad
rive
nch
oice
ofth
ew
ind
ow
Fla
t–to
pke
rnel
:
Kf(t
;x)
=
1,0≤|t|<x
(x−
1)−1(|t|−
1),x≤|t|<
1
0,|t|≥
1,
ρi
=‖γ
i‖/∫ γ
0(t,t
)dt
Pro
ced
ure
for
choosi
ngh:
Fin
dth
efi
rst
non
–neg
ativ
ein
tege
rm
such
that
ρm+r<T√ lo
gN/N
forr
=
1,...,H
,w
her
eT>
0,an
dH
isa
pos
itiv
ein
tege
r.T
akeh
=h
wh
ereh
=dm
/xe.
Sim
ula
tion
s:FA
R∗ 1/2(1
)
Xi(t)
=1 2Xi−
1(t
)+W
i(t),
wh
ereW
i,−∞
<i<∞
are
iid
Wie
ner
pro
cess
es
0.0
0.5
1.0
0.0
0.5
1.0024
n=10
0
0.0
0.5
1.0
0.0
0.5
1.0024
n=30
0
Sim
ula
tion
s
0.0
0.5
1.0
0.0
0.5
1.0024
n=50
0
0.0
0.5
1.0
0.0
0.5
1.0
24
Th
elim
it
−4−3−2−10123
4/14
/199
74/
18/1
997
0.0
0.5
1.0
0.0
0.5
1.0
0
2
Cu
mu
lati
vein
trad
ayre
turn
s
Fu
nct
ion
alan
alys
isof
vari
ance
Mod
el
Xi,j(t
)=µi(t)
+ε i,j
(t),
t∈
[0,1
],1≤i≤k
and
1≤j≤Ni,
Mod
elas
sum
pti
ons
Eη i,j
(t)
=0,t∈
[0,1
],1≤i≤k,
and
1≤j≤Ni
for
each
i,1≤i≤k,{η
i,j,0≤j<∞}
isa
wea
kly
dep
end
ent
Ber
nou
lli
sequ
ence
the
erro
rse
qu
ence
s{η
i,j,1≤j≤Ni}
are
ind
epen
den
t
lim
N→∞
Ni
N=ai>
0,w
her
eN
=N
1+N
2+···+
Nk
FA
NO
VA
nu
llhyp
oth
esis
H0
:µ1(·)
=µ2(·)
=...
=µk(·)
FA
NO
VA
alte
rnat
ive
HA
:H
0d
oes
not
hol
d
Figure
1:Threepopulations
0.0
0.2
0.4
0.6
0.8
1.0
−1.5−1.0−0.50.00.51.01.5
Th
ree
pop
ula
tion
s
Figure
2:Threepopulationswiththesample
means
0.0
0.2
0.4
0.6
0.8
1.0
−1.5−1.0−0.50.00.51.01.5
Th
ree
pop
ula
tion
sw
ith
the
sam
ple
mea
ns
Pro
ject
ion
met
hod
Wh
atb
asis
tou
se?–
We
are
com
par
ing
the
mea
ns,
hen
ceav
erag
esm
ust
be
com
par
ed.
We
wil
lu
sea
bas
esre
late
dto
the
lon
gru
nco
vari
ance
sof
the
ind
ivid
ual
sam
ple
s.
Di
isth
elo
ng
run
cova
rian
cefu
nct
ion
ofth
eit
hp
opu
lati
on,
Di(t,s)
=∞ ∑ `=−∞
cov(X
i,0(t
),Xi,`(s)
).
We
use
the
eige
nfu
nct
ion
sof
D(t,s
)=
k ∑ i=1
Ni
NDi(t,s),
N=N
1+N
2+···+
Nk.
Est
imat
ion
ofD
DN
(t,s
)=
k ∑ i=1
Ni
NDi,N
(t,s
)
Do
we
nee
dth
eco
nsi
sten
cyofDN
(t,s
)?U
nd
erth
enu
ll–d
efin
itel
y.U
nd
erth
eal
tern
ativ
e?
Est
imat
ion
ofD
–firs
tm
eth
od
Di,N
(t,s
)=
∞ ∑ `=−∞
K
( ` h
) γi,`(t,s)
wit
hth
eem
pir
ical
cova
rian
ces
γi,`(t,s)
=γi,N
(t,s
)=
1 N
N− ∑ j=1
(Xi,j(t
)−XN
(t))
(Xi,j+i(s)−XN
(s)),
i≥
0
1 N
N ∑ j=1−`(X
i,j(t
)−XN
(t))
(Xi,j+i(s)−XN
(s)),i<
0
and
the
sam
ple
mea
nof
the
tota
lp
opu
lati
on XN
(t)
=1 N
k ∑ i=1
Ni ∑ `=1
Xi,`(t).
Con
sist
ent
only
un
der
the
nu
llhyp
oth
esis
!D
iver
ges
to∞
un
der
the
alte
rnat
ive.
Est
imat
ion
ofD
–sec
ond
met
hod
Di,N
(t,s
)=
∞ ∑ `=−∞
K
( ` h
) γi,`(t,s)
wit
hth
eem
pir
ical
cova
rian
ces
γi,`(t,s)
=γi,N
(t,s
)=
1 N
N− ∑ j=1
(Xi,j(t
)−Xi,N
i(t
))(X
i,j+i(s)−Xi,N
i(s
)),
i≥
0
1 N
N ∑ j=1−`(X
i,j(t
)−Xi,N
i(t
))(X
i,j+i(s)−Xi,N
i(s
)),i<
0
and
the
sam
ple
mea
nof
theit
hp
opu
lati
on
Xi,N
i(t
)=
1 Ni
Ni ∑ `=1
Xi,`(t)
Con
sist
ent
un
der
the
nu
llas
wel
las
un
der
the
alte
rnat
ive.
Pro
ject
ion
sb
ased
onth
efir
stm
eth
od
Em
pir
ical
eige
nva
lues
/eig
enfu
nct
ion
s:
λiϕ
i(t)
=
∫ DN
(t,s
)ϕi(s)ds
Em
pir
ical
pro
ject
ion
sof
theit
hp
opu
lati
on:
ξi,j
=(〈Xi,j,ϕ
1〉,〈X
i,j,ϕ
2〉,...,〈X
i,j,ϕ
d〉)T,
1≤j≤d
Th
eav
erag
esof
the
emp
iric
alsc
ore
vect
ors
wit
hin
each
pop
ula
tion
are
defi
ned
as
ξi·
=1 Ni
Ni ∑ j=1
ξi,j,
1≤i≤k
Est
imat
orfo
rth
eco
mm
onm
ean
assu
min
gth
atH
0h
old
s:
ξ··
=
( k ∑ i=1
NiΣ−1
i
) −1k ∑ i=1
NiΣ−1
iξi·,
wh
ere
Σi
=
{ ∫∫Di,N
i(t,s
)ϕ`(t)ϕj(s
)dtds,
1≤j,`≤d
} ,
Tes
tst
atis
tic:
TN
=k ∑ i=1
Ni
( ξ i·−ξ··) T Σ
−1
i
( ξ i·−ξ··) .
Dis
trib
uti
onofTN
un
derH
0an
dHA
Th
eore
m7:
We
assu
me
that
the
pop
ula
tion
sar
ein
dep
end
ent
wea
kly
dep
end
ent
Ber
nou
lli
shif
ts.
(H0)
Ifth
em
ean
sar
eth
esa
me,
than
we
hav
e TN
D →χ2(d
(k−
1)),
wh
ereχ2(d
(k−
1))
stan
ds
for
aχ2
ran
dom
vari
able
wit
hd(k−
1)d
egre
esof
free
dom
.
(HA
)If
atle
ast
two
mea
ns
are
diff
eren
t,th
enw
eh
ave
TN
P →∞.
Ele
ctri
city
dem
and
inA
del
aid
e,A
ust
rali
a
Dai
lyel
ectr
icit
yd
eman
dcu
rves
con
stru
cted
from
hal
f–h
ourl
ym
easu
rem
ents
ofth
eel
ectr
icit
yd
eman
din
Ad
e-la
ide
Au
stra
lia
from
7/6/
1997
to3/
31/2
007.
Eac
hd
ayis
com
pri
sed
of48
obse
rvat
ion
s
Th
eco
stof
un
serv
eden
ergy
can
be
valu
edat
thou
san
ds
ofd
olla
rsp
erM
Wh
,an
dh
ence
ther
eis
anin
centi
veto
dev
elop
accu
rate
mod
els
ofth
ed
aily
dem
and
inor
der
tore
du
ceex
cess
elec
tric
ity
gen
erat
ion
.
Diff
erin
gtr
end
sin
the
dem
and
acco
rdin
gto
the
seas
onan
dth
ed
ayof
the
wea
k
Su
pp
oseUn(t
)is
the
elec
tric
ity
dem
and
atti
met
ond
ayn
for
t∈
[0,1
],n
=1,...,N
.T
he
fun
ctio
ns
Rn(t
)=
lnDn(t
)−
lnDn(0
),t∈
[0,1
],n
=1,...,N,
are
call
edth
elo
gdi
ffer
ence
dde
man
dcu
rves
(LD
DC
’s).
Figure
3:Fivefunctional
dataob
jectsconstructed
from
half-hourlymeasurements
oftheelectricitydem
andin
AdelaideAustralia.Theverticallines
separate
thedays.
10001200140016001800
MW
1997
−07
−07
1997
−07
−11
Dem
and
curv
es
Figure
4:FiveLDDC’s
constructed
from
thecurves
inFigure
3.
−0.5−0.4−0.3−0.2−0.10.00.10.2
1997
−07
−07
1997
−07
−11
Diff
eren
ced
dem
and
curv
es
Sea
son
aleff
ect
Fir
stw
eco
nsi
der
the
pro
ble
mof
test
ing
ifth
em
ean
ofth
eL
DD
C’s
ish
omog
eneo
us
acro
ssth
efo
ur
pre
dom
inan
tse
ason
sin
Ad
elai
de:
Su
mm
er(D
ecem
ber
,Jan
uar
y,F
ebru
ary),
Fal
l(M
arch
,A
pri
l,M
ay),
Win
ter
(Ju
ne
Ju
lyA
u-
gust
),an
dS
pri
ng
(Sep
tem
ber
,N
ovem
ber
,D
ecem
ber
).W
ed
ivid
edth
ed
ata
set
con
sist
ing
of35
56d
aily
curv
esin
toth
ese
fou
rse
ason
algr
oup
sdep
end
ing
onth
em
onth
inw
hic
hth
eob
serv
atio
nw
asta
ken
.F
rom
this
sam
ple
the
obse
rvat
ion
sco
rres
pon
din
gto
the
wee
ken
ds
wer
ere
mov
edsi
nce
the
dem
and
beh
avio
ris
vast
lyd
iffer
ent
onth
ese
day
s.A
fter
rem
ovin
gth
ew
eeke
nd
sin
tota
lth
ere
are
642
obse
rvat
ion
sfr
omth
eS
pri
ng
mon
ths,
628
from
Fal
l,63
0fr
omS
um
mer
,an
d64
0fr
omW
inte
r.T
he
mea
nfu
nct
ion
sfr
omth
ese
sam
ple
sar
esh
own
inF
igu
re5
onth
en
ext
slid
e.
Wh
enth
eFA
NO
VA
test
isap
pli
edto
thes
efo
ur
pop
ula
tion
sth
ete
stre
ject
sth
enu
llhyp
oth
esis
wit
hap–
valu
ew
hic
his
less
than
10−6.
By
exam
inin
gF
igu
re5
onth
en
ext
slid
eit
app
ears
that
Spri
ng
and
Fal
lh
ave
sim
ilar
mea
nL
DD
C’s
.T
he
app
roxim
atep–
valu
eof
the
test
wh
enap
pli
edto
just
the
Sp
rin
gan
dF
all
sam
ple
sis
app
roxim
atel
y.2
1,in
dic
atin
gth
atth
ere
isn
otsu
ffici
ent
evid
ence
pre
sent
inth
ed
ata
tore
ject
the
not
ion
that
Sp
rin
gan
dF
all
hav
eth
esa
me
dem
and
pat
tern
s.
Figure
5:MeanCurves
from
each
season
constructed
from
theLDDC’stakenfrom
7/6/1997to
3/31/2007.Thep-valueoftheFANOVA
test
applied
tothis
sample
was
zero.
−0.4−0.3−0.2−0.10.00.1
Spr
ing
Fall
Sum
mer
Win
ter
Sea
son
alm
ean
curv
es
Daysof
theweekforwhichthetest
isapplied
Season
All
Weekdays
Weekends
TW
Th
MTW
Th
TW
ThF
Summer
.000
.035
.006
.750
.647
.056
Fall
.000
.003
.001
.886
.380
.023
Winter
.000
.000
.000
.257
.001
.000
Spring
.000
.002
.000
.582
.083
.001
Tab
le2:p–values
oftheFANOVA
test
when
applied
tosamplesofdailyLDDC’s
organized
accordingto
theday
oftheweekandseason.Across
thetopof
thetable
thedaysincluded
inthesample
aredisplayed.“All”denotesthatallsevendayswereincluded
(k=
7).
Dai
lyva
riat
ion
To
stu
dy
wh
eth
erth
ed
aily
pat
tern
inel
ectr
icit
yd
eman
dis
hom
ogen
eou
sac
ross
each
day
ofth
ew
eek
we
div
ided
the
dat
ase
tin
tose
ven
grou
ps
each
ofsi
ze50
8co
rres
pon
din
gto
the
day
sof
the
wee
k,
Su
nd
ayth
rou
ghS
atu
rday
,
and
then
com
pu
ted
thei
rL
DD
C’s
.D
ue
toth
ep
rior
anal
ysi
sof
the
seas
onal
tren
dab
ove
we
furt
her
grou
ped
the
dat
ain
toth
efo
ur
seas
onal
grou
ps
ofS
um
mer
,F
all,
Win
ter,
and
Sp
rin
g;ea
chsu
bsa
mp
lefo
rea
chd
ayco
nta
ined
atle
ast
120
curv
es.
Figure
6:Meancurves
foreach
day
computedfrom
theSummer
monthsbetween7/7/1997to
7/5/1998(52curves
foreach
day
).Thep-valueoftheFANOVA
test
was
less
than
10−4.
−0.3−0.2−0.10.00.10.2
M T W Th
F S Su
Dai
lyav
erag
es
Som
efu
rth
erre
sult
son
fun
ctio
nal
dat
a
Tes
tsfo
rst
atio
nar
ity
Isit
true
that
the
obse
rvat
ion
sfo
rma
stat
ion
ary
sequ
ence
?If
not
wh
atis
the
cau
seof
the
non
stat
ion
arit
y?
Ch
angi
ng
mea
n?
Ch
angi
ng
cova
rian
ce?
Ran
dom
wal
ker
rors
(un
itro
otp
rob
lem
)?
Th
eas
ym
pto
tic
dis
trib
uti
onof
the
esti
mat
orfo
rth
elo
ng
run
cova
rian
cefu
nct
ion
Itis
nor
mal
lyd
istr
ibu
ted
,an
dth
eref
ore
the
corr
esp
ond
ing
eige
nva
lues
/eig
enfu
nct
ion
are
nor
mal
too.
Op
tim
alch
oice
ofth
ew
ind
ow.
Sam
ple
bas
edse
lect
ion
ofth
ew
ind
ow.
Pos
itiv
ed
efin
ite
fun
ctio
n.
Het
eros
ced
asti
cer
rors
Th
eco
vari
ance
fun
ctio
n(l
ong
run
cova
rian
cefu
nct
ion
)m
ight
dep
end
onti
me.
Ch
ange
poi
ntw
ith
het
eros
ced
asti
cer
rors
Mod
el:Xi(t)
=∑ K k
=1βi,kf k
(t)
+ε i
(t),
Eε i
(t)
=0,
0≤t≤
1,1≤i≤N
Mot
ivat
ion
:N
elso
n–S
iege
lm
od
elfo
ryie
ldcu
rves
f k(t
)ar
egi
ven
fun
ctio
ns
βi,k
=µi,k
+b i,kEb i,k
=0
H0
:µ
1=µ
2=...
=µN
µi
=(µ
i,1,µ
i,2,...,µ
i,K
)>.
HA
:R
pos
sib
lech
ange
sin
the
mea
ns
ofth
eµ′ is
Ran
dth
eti
mes
ofth
ech
ange
sar
eu
nkn
own
(ch
ange
poi
nt
pro
ble
min
the
mea
n)
Het
eros
ced
asti
city
:th
eco
vari
ance
stru
ctu
reb
ecom
esd
iffer
ent
atti
mesi 1<i 2<...<i M
(th
ese
are
kn
own
poi
nts
)T
he
erro
rte
rmis∑ K i=
1b i,kf k
(t)
+ε i
(t),
1≤i≤N
Err
orst
ruct
ure
:B
ern
oull
ish
ifts
(bi,1,...,bi,K,εi(t)
)>=g m
(δi,δ i−1....
)i m<i m
+1,m
=0,
1,...,M
(i0
=0,i M
+1
=N
)
US
yiel
ds
curv
esfo
r5
day
s
012345
Cris
is o
f 200
8
Day
Yield Curves
●●●●●●
●
●
●
● ●●●●●●
●
●
●
● ●●●●●●
●
●
●
● ●●●●●●
●
●
●
● ●●●●●●
●
●
●
●
Sep
−10
Sep
−11
Sep
−12
Sep
−15
Sep
−16
12
34
5
US
yiel
ds
curv
esb
etw
een
Jun
e–4–
2012
and
Oct
ober
–24–
2012
US
yiel
ds
curv
esb
etw
een
Oct
ober
–18–
2005
and
Mar
ch–1
4–20
06
Pro
ject
ion
met
hod
Pro
ject
ion
sin
tosp
an(f
1,...,f
K)
(f1,f
2,...,f
Kar
eli
nea
rly
ind
epen
den
tinL2
zi
=(〈Xi,f 1〉,〈X
i,f 2〉,...,〈X
K,f
K〉)>,
1≤i≤N
Let
C={〈f i,f
j〉,
1≤i,j≤K},
ε i=
(〈ε i,f
1〉,〈εi,f 2〉,...〈ε i,f
K〉)>,bi
=(bi,1,bi,2,...,bi,K
)>.
Pro
ject
edM
od
el:
zi
=Cµi+γi,
wit
hγi
=Cbi+ε i
CU
SU
Mp
roce
ss
αN
(x)
=N−1/2
bNxc ∑ i=1
zi−
N ∑ i=1
zi
Th
eore
m1
IfH
0an
dth
eB
ern
oulli
assu
mp
tion
hol
ds,
then
αN
(x)
D →G
0(x
)inDK
([0,
1]),
wh
ereG
0is
aG
auss
ian
pro
cess
wit
hEG
0(x
)=
0an
dEG
0(x
)(G
0(y
))>
=R
(x,y
)=
exp
lici
tly
com
pu
ted
(diffi
cult
look
ing)
.
Tw
ote
stst
atis
tics
C N=
∫ 1 0
‖αN
(x)‖
2dx
andKN
=su
p0≤x≤1‖α
N(x
)‖2.
Cra
mer
–von
Mis
esst
atis
tic
un
derH
0
C ND →
∫ 1 0
‖G0(x
)‖2dx
Kar
hu
nen
–Loe
veex
pan
sion
∫ 1 0
‖G0(x
)‖2dx
=∞ ∑ `=1
λ`Z
2 `,
wh
ereZ1,Z
2,...
are
ind
epen
den
tst
and
ard
nor
mal
s,λ1≥λ2≥...
sati
sfyin
g
λ`φ
`(x
)=
∫ 1 0
R(x,y
)φ`(y)dy,
1≤`<∞,
wh
ere
theφ`’
sar
eor
thon
orm
alfu
nct
ion
s(d
efin
edon
[0,1
],w
ith
valu
esinRK
).
Ap
pro
xim
atio
n:
∞ ∑ `=1
λ`Z
2 `≈
d ∑ `=1
λ`Z
2 `w
her
ed
issu
ffici
entl
yla
rge
Issu
e:R
and
ther
efor
eλ1,λ
2,...
are
un
kn
own
Cri
tica
lva
lues
for
the
Cra
mer
–von
Mis
esst
atis
tic
LetRN
(x,y
)b
ea
con
sist
ent
esti
mat
orfo
rR
un
derH
0,
i.e.
∫ 1 0
∫ 1 0
‖RN
(x,y
)−R
(x,y
)‖2dxdy
P →0.
Ifλ1,N≥λ2,N...
sati
sfy
λ`,Nφ`,N
(x)
=
∫ 1 0
RN
(x,y
)φ`,N
(y)dy,
1≤`<∞,
then
un
derH
0w
eh
ave
λ`,N
P →λ`
for
all`.
Hen
ceu
nd
erH
0d ∑ `=1
λ`Z
2 `≈
d ∑ `=1
λ`,NZ
2 `
Con
stru
ctio
nofRN
(x,y
):ifi m−1<bN
xc≤i m
,th
en
bNxc ∑ i=1
zi
=m−1 ∑ j=1
i j ∑`=i j
−1+1
zi+
bNxc ∑
i=i m
−1+1
zi
so
E
bNxc ∑ i=1
zi bN
xc ∑ i=1
zi >
≈m−1 ∑ j=1
(ij−i j−1)D
j+
(bNxc−
i m−1)D
m,
wh
ereD
1,D
2,...,D
Mar
elo
gru
nco
vari
ance
mat
rice
s.S
imil
arfo
rmu
lafo
rth
eco
vari
ance
s
kern
elty
pe
esti
mat
ors
forD
1,D
2,...,D
Mca
nb
eu
sed
toge
tRN
(x,y
)
Th
eb
ehav
ior
ofcr
itic
alva
lues
for
the
Cra
mer
–von
Mis
esst
atis
tic
un
derHA
Ob
serv
atio
n:
ther
eisR∗ (x,y
)su
chth
at ∫ 1 0
∫ 1 0
‖RN
(x,y
)−R∗ (x,y
)‖2dxdy
P →0
and
ther
efor
eλi,N
P →λ∗ i,
wh
ereλ∗ 1≥λ∗ 2≥...
are
the
eige
nva
lues
ofR∗ (x,y
).H
ence
un
derH
0
d ∑ `=1
λ`Z
2 `≈
d ∑ `=1
λ∗ `,NZ
2 `.
Th
eore
m2
Ifth
ere
isa
chan
gein
theµi’
s(m
ean
s),
then
1 NC N
P →c 0>
0.
Th
ete
stis
con
sist
ent
Fu
nct
ion
alap
pro
ach
Mod
el
Xi(t)
=µi(t)
+η i
(t),
wit
hµi(t)
=K ∑ k=1
µi,kf k
(t)
andη i
(t)
=K ∑ k=1
b i,kf k
(t)
+ε i
(t)
H0
µ1(t
)=µ2(t
)=...
=µN
(t)
(inL2
sen
se)
Fu
nct
ion
alC
US
UM
αN
(x,t
)=N−1/2
bNxc ∑ i=1
Xi(t)−bN
xc
N
N ∑ i=1
Xi(t)
.T
heo
rem
3IfH
0an
dth
eB
ernou
lli
assu
mp
tion
hol
d,
then
we
can
defi
ne
ase
qu
ence
ofG
auss
ian
pro
cess
esΓ0 N
(x,t
)su
chth
at∫ 1 0
∫ 1 0
(αN
(x,t
)−
Γ0 N
(x,t
))2dxdt
P →0
EΓ0 N
(x,t
)=
0,E
Γ0 N
(x,t
)Γ0 N
(y,s
)=U
(x,y
;t,s
)=
exp
lici
tfo
rmu
la.
Con
sequ
ence
:
V N=
∫ 1 0
∫ 1 0
α2 N
(x,t
)dxdt
D →∫ 1 0
∫ 1 0
(Γ0(x,t
))2dxdt,
wh
ere
Γ0(x,t
)is
Gau
ssia
nw
ithE
Γ0(x,t
)=
0,E
Γ0(x,t
)Γ0(y,s
)=U
(x,y
;t,s
).
Kar
hun
en–L
oeve
exp
ansi
on ∫ 1 0
∫ 1 0
(Γ0(x,t
))2dxdt
=∞ ∑ `=1
λjZ
2 j,
wh
ereZ1,Z
2,...
are
stan
dar
dn
orm
als,λ1≥λ2≥...
λiφi(x,t
)=
∫ 1 0
∫ 1 0
U(x,y,t,s
)φi(y,s
)dyds,
1≤i<∞.
Est
imat
eU
(x,y,t,s
)w
ithUN
(x,y,t,s
)(e
stim
atio
nofM
lon
gru
nco
vari
ance
fun
ctio
ns)
sati
sfyin
g‖U
N−U‖
P →0.
Ifλ1,N≥λ2,N≥...
are
the
eige
nva
lues
ofUN
(x,y,t,s
),th
en
∫ 1 0
∫ 1 0
(Γ0(x,t
))2dxdt≈
d ∑ `=1
λ`,NZ
2 `
Dyn
amic
Nel
son
–Sie
gel
mod
elfo
ryi
eld
curv
es
K=
3,
f 1(t
)=
1,f 2
(t,λ
)=
1−e−
λt
λt
andf 2
(t,λ
)=
1−e−
λt
λt−e−
λt
λ=
3.59
(Die
bol
dan
dL
i(2
003)
and
Die
bol
dan
dR
ud
ebu
sch
(201
3))
Ou
tcom
eTab
le3:
Application
ofthetest
proceduresto
yield
curves
over
thesixsamplingperiods.
Weexpectsm
allP–Values
inperiods(1)–(4),
largein
periods
(5)–(6).
Sam
plingPeriod
Sample
Size
Method
BreakPoint
P–value
ProjSim
yes
1.8%
(1)
ProjEigen
yes
1.7%
07/08/2008
–11/28/2008
N=
100
NFEigen
yes
0.0%
ProjSim
no
99.7%
ProjEigen
no
99.9%
NFEigen
no
85.3%
ProjSim
yes
0.2%
(2)
ProjEigen
yes
0.1%
03/20/2008
–03/19/2009
N=
250
NFEigen
yes
0.0%
ProjSim
no
92.9%
ProjEigen
no
92.5%
NFEigen
no
5.1%
ProjSim
yes
0.6%
(3)
ProjEigen
yes
0.1%
10/18/2005
-03/14/2006
N=
100
NFEigen
yes
0.0%
ProjSim
no
57.6%
ProjEigen
no
52.9%
NFEigen
no
21.7%
ProjSim
yes
0.2%
(4)
ProjEigen
yes
0.0%
06/30/2005
-06/29/2006
N=
250
NFEigen
yes
0.1%
ProjSim
no
53.3%
ProjEigen
no
47.7%
NFEigen
no
73.0
%
ProjSim
yes
14.6%
(5)
ProjEigen
yes
10.8%
06/05/2012
–10/24/2012
N=
100
NFEigen
yes
3.0%
ProjSim
no
42.7%
ProjEigen
no
37.6%
NFEigen
no
29.1%
ProjSim
yes
78.0%
(6)
ProjEigen
yes
74.6%
02/16/2012
–02/14/2014
N=
250
NFEigen
yes
50.4%
ProjSim
no
70.3%
ProjEigen
no
65.6%
NFEigen
no
62.6%
Ack
now
led
gem
ents
Th
eta
lkw
asb
ased
onjo
int
rese
arch
wit
h
Pat
rick
Bar
dsl
ey(U
niv
ersi
tyof
Uta
h)
Istv
anB
erke
s(R
enyi
Inst
itu
te)
Pio
trK
okos
zka
(Col
orad
oS
tate
)G
rego
ryR
ice
(Un
iver
sity
ofW
ater
loo)
Gab
riel
You
ng
(Col
orad
oS
tate
)
