E-malmasakazu.nada@boj.or.jp E-malkouchrou.kamada@boj.or.jp
wavelet J.J. Morlet D. D. Gabor uncertanty prncple Conway and Frame Schlecher
wave let
DWT: dscrete wavelet transformcwt: contnuous wavelet transform AUSSMATLABS-PlusMathematca CD-ROMMathematca
MODWT h L h k =. k = L h k =. k = L h kh k+n =. n k = L Percval and Walden Gençay, Selçuk and Whtcher k+ n >L,,, h,, h L,,,
orthonormalty gquadrature mrror relatonshp g =( ) h L + h =( ) g L +. L g k =. k = L g k =. k = L g kg k +n =. n k = L g kh k +n =. n k = h =,, g =,. g k =
L L DLDL L / L (k ) h k =. =,, L / k = vanshng momentsl / D4 = D4 h = 4 3 3 + 3 3 + 3 3,,, 4 4 4, g + = 4 3 3+ 3 3 3 3,,, 4 4 4. D4 D
D D LA MB
D / 4 LAL least asymmetrc wavelet MBL mnmum bandwdth wavelet
D D LA MB Gençay, Selçuk and Whtcher
S. S. Mallat x = (x,,x N )N h w L, t = k x{( t k) mod N } + k= w h. Nt N/A mod N AN N xg v v L g x, = t k k= {( t k) mod N } +. N/ N
w x x, x, x 3, x 4 g h v x + x x, 4 + x 3 w x x, x 4 x 3 g h v x 4 + x 3 + x + x w x 4 + x 3 x x g = [, ], h = [, ]
v v w v v v x v h w w,t = L h k v, {( N/ } + k= t k ) mod. v g v v, t L = g kv, {( t k) mod N/ } + k=. w v N/ w v v w 3 v 3 + w + v + v w + v + w + = (, w +,,, w +,,)v + = (, v +,,, v +,,)
+ w + h + v + g v v = L L, t hk w + g +,{( t + k )mod N / } + kv k= k= +,{( t+ k )mod N / } +.
(, 3, 3, 3,,, 3, 4) g,h ( 4,,, 7) (, 6,, ) g,h ( 4, 9) (4, 5) g,h (5) (3) g = [, ], h = [, ]
v w v N/ N/ W V h h4 g g 4 h h g g 3 3 h g h g g h 4 4 h g 3 3 h h g g 4 4 h h g g 3 3 h h g 4 4 g h3 h g3 g,. w = W v, v = V v. W V U N/ N/ w = U v v, W U. V w v
UU T = U T U = I U T UI +U T + w U = T + v v T T + + w+ + V+ v+ + W = v. U T + h g U T + ( N/ N/ )w + v + w + v + N/ U + T h = h4 h3 h h h h h 4 3 h h h h 4 3 h h h h 4 3 g g g g 4 3 g g g g 4 3 g g g g 4 3, g 4 g 3 g g w v + + w + w = +, v+ v +,, N/, N/. U T + T w U + + = v v +. U T + N / N / W T + V T + W T + w + +V T +v + = v.
v w w v w d d v v w s v s w d s d s s d x s x s = d + s. x x, x, x 3, x 4 s x + x x, + x x, 4 + x 3 x, 4 + x 3 x d + x x, x x x, 4 + x 3, 4 x 3 x 4 + x 3 + x + x s, 4 x 4 + x 3 + + x x, 4 + x x x, 4 + x 3 + x + x 4 4 x 4 + x 3 + d x 4 x 3 + x + x, x 4 x 3 + x + x, 4 4 x 4 + x 3 x x, 4 x 4 + x 3 x x 4 g = [, ], h = [, ]
r = k= d kwavelet rough x = r + s. s d s d s s 3 s d 3
(, 3, 3, 3,,, 3, 4) g,h ( 4, 4,,,,, 7, 7) g,h (,, 6, 6,,,, ) 4 ( 4, 4, 4, 4, 9, 9, 9, 9) 4 ( 4, 4, 4, 4, 5, 5, 5, 5) g,h 8 (5, 5, 5, 5, 5, 5, 5, 5) 8 ( 3, 3, 3, 3, 3, 3, 3, 3) g = [, ], h = [, ]
HT: hard thresholdng HT( w, j w, j for w, j >, ) =. soft thresholdng unversal thresholdng Gençay, Selçuk and Whtcher
w, j j =.5 HT w w HT w HT d HT s HT (= s ) x HT HT k = for for k, k +. = (w ). Walker
g,h ( 4,,, 7) (, 6,, ) g,h ( 4, 9) (4, 5) g,h (5) (3) g = [, ], h = [, ]
zz z z T z + UU T = U T U = I v = v T = T v w +, v ( T T +) U +U + w v + + = wt + + + T w v +v + = w + v + +. v x = x = w + v. w = = w = s = V T V T v. VV T = I s =( V T T T T V v ) V V T v T T T T V v = v v v = v V V V =. d = V T V T W T w. WW T = I VV T = I
T T T T T T d = ( V V W w ) V V T T T T T = w WV V V V W w = w w = w W T w. x x x
w w w 3 w 4 j MODWTmaxmal overlap dscrete wavelet transform MODWT h = h/, g = g /. MODWT v x
v h w w, t L = h k v, { t ( k ) mod N} + k=. v g v v, t = L g kv, { t ( k ) mod N} + k=. +w + h + v + g v v, = L t h kw { t + + L +, ( k ) modn } + g kv k +, { t + ( k ) mod N } + = k=. MODWT MODWT MODWT MODWT
4 = 5 7 D4 D
w w w 3 w 4 w 5 v 5
MODWT MODWT DWT
d d d 3 d 4 d 5 s 5
s 5 d 5 d 4 d 3 d
w w w 3 w 4 w 5 v 5
d d d 3 d 4 d 5 s 5
s 5 d 5 d 4 d 3 d
Ramsey and Lampart
D4 D4 = h + h + h 3 + h 4 =..h +.h +.h 3 + 3.h 4 =. 3, 5, 7, 9 9 h h + 7 h + 5 h 3+ 3 4 = ( + 9) h + h + ( + 9) h + ( + 9) h 3+ ( 3 9) 4 = ( h + h + h 3 + 3 h4) + 9 ( h + h + h3 + h4) =. D =,,5 = D4 D4 D D D4D D4 384 = 7 3
D4D D4D D4 D D4 D
D D
D D
d d d 3 d 4 d 5 d d d 3 d 4 d 5 D D d d d 3 d 4 d 5 d d d 3 d 4 d 5 D D d d d 3 d 4 d 5 d d d 3 d 4 d 5
c,t =, +,. y (p),t +, t. c, t y (p),t p p y,t p D4D D4D D4D,
t t d d d 3 d 4 d 5 s 5 D d d d 3 d 4 d 5 s 5 D d d d 3 d 4 d 5 s 5
, D4D MODWTw,t ( w,t ) w,t E ( w,t ) ( w,t )=E( w,t ). N t=w,t /N x
MODWT L ( )(L )+w,t t = L,, N w, t t = L ( w,t) =. N L + N DWT N/ t= w,t /(N/ )MODWT L (L )( )+ w, t t = L,, N/ / N w, t t = L ( w ) =,t N / L +. DWT DWTMODWT DWT ( w,t )/ MODWT DL = ( 9 ) ( ) = 7 DMODWT
DWTDWTMODWT DWTMODWT ICSS: terated cumulatve sums of squaresinclán and Tao k p = L q = N/ k w t= p, t =. ( k = p,, q ) C, k q w t= p, t (w, t )=w, t C, k p q p k C,k (k p +)/(q p +) C,k (k p +)/(q p +) Percval and Walden D
+ D = max( D, D ), + k p + = max C k q p D, k, = max k C D, k k q p p. Inclán and TaoD P( D j+ > z) ( ) exp[ j ( q p + j= ) z ]. D z p D k = k DWT MODWT p =L q = NC,k D k w t= p, t C, k = q w, t t= p. ( k p,, q ) = + D = max( D, D ), + k p + = D max C k, k q p = max k C D, k k q p p., Dk = k MODWTDWT DWT w,t MODWT w,t MODWT
a = p a = p c = q c = q w,t (t = a,, c) D kcw, t (t = a,, c )D kc c c b b a = c a = cc = q c = q m b b j b j b j w,t (t = b j,, b j+ )D b p b m+ q j + b j b j ICSS D DWT MODWT p DWTDWT MODWT
p
BBx t = x t ( B) x t = t. /<</ < < /x = x /<< x x Hoskng x
(,t w ) C ( ). CJ = + ln( ( w )),t ln( ) +. ( =,, J ) =/( +) Var ()=/4Var ( )
C. K. Chu, C. K., Introducton to Wavelets, New York: Academc Press, 99 Conway, P., and D. Frame, Spectral Analyss of New Zealand Output Gaps Usng Fourer and Wavelet Technques, Reserve Bank of New Zealand Dscusson Paper, DP/6,. Gençay, R., F. Selçuk, and B. Whtcher, An Introducton to Wavelets and Other Flterng Methods n Fnance and Economcs, San Dego: Academc Press,. Hoskng, J. R. M., Fractonal Dfferencng, Bometrka, 68 (), 98, pp. 65-76. Inclán, D., and G. C. Tao, Use of Cumulatve Sums of Squares for Retrospectve Detecton of Changes of Varance, Journal of the Amercan Statstcal Assocaton, Theory and Methods, 89, 994, pp. 93-93. Percval, D. B., and A. T. Walden, Wavelet Methods for Tme Seres Analyss, Cambrdge: Cambrdge Unversty Press,. Ramsey, J. B., and C. Lampart, The Decomposton of Economc Relatonshps by Tme Scale Usng Wavelets: Expendture and Income, Studes n Nonlnear Dynamcs and Econometrcs, 3 (), 998, pp. 3-4. Schlecher, C., An Introducton to Wavelets for Economsts, Bank of Canada Workng Paper, -3,. Walker, J. S., A Prmer on Wavelets and ther Scentfc Applcaton, Boca Raton: Chapman & Hall/CRC, 999.