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HWAHAK KONGHAK Vol. 39, No. 2, April, 2001, pp. 163-175(Journal of the Korean Institute of Chemical Engineers)
���� ��� �� �� �� � ��
���������†����*
����� ���� �����, *�����(1999� 10 20� ��, 2001� 2 20� ��)
A Study on System Identification Using Wavelet Transformation
Wook Jin Baek, Jeong Woo Han, Sung Ju Kang† and Chang Bock Chung*
Department of Chemical Engineering, *Faculty of Applied Chemical Engineering, Chonnam National University, Kwangju 500-757, Korea(Received 20 October 1999; accepted 20 February 2001)
� �
���(Fourier) ��� �� �� � ��� ����(wavelet) ��� �� ����(de-noising, ����, !�
"#$� %& ')�( )*+ ,-. /,0 1). 2 3 de-noising� ���� ��� 4567 8,�� 3� �9.(
:; /�+ ��< =>0 1). De-noising ?@A B,�A shrinkage CDE �� 6,0A threshold F GH�A "
I� JK1). L ���(A (. )M NO PA QRE ST� UV� WX �� threshold Y �Z 6,[\ ]@
^ (. _`� , 2 8,a. de-noising� �� Y �Z �,�� ST bc+ QRE 2de f� QR^ [gh $
i(system identification)� 6,[j 2 Ok (. _`�l).
Abstract − The wavelet transformation, which was developed in order to overcome the defects of traditional Fourier trans-
formation, is applied to many fields of study in various ways-for example, de-noising, data compression and mathematic appli-
cations such as solving partial differential equations, etc. De-noising is one of the main application areas of the wavelettransformation and has been studied by many researchers. The effect of de-noising depends upon the shrinkage function and
the method of choosing the threshold value for the function. The objective of this work is to analyze the results of applying var-
ious threshold algorithms according to characteristics for signals and noise level. By applying the de-noising to the system
identification, we compared the performances of signals which went through the de-noising process with those of signals with-
out de-noising.
Key words: Wavelet, De-noising, Shrinkage, Threshold, System Identification
†E-mail: [email protected]
1. � �
��� ��� �� �� � � ��, ����� ��
�� ��� �� ��� !"�� �#$ sharp spike% �� ��
% &'( )�* �+) �,� �-.(Gibb's phenomena)� �
� �,. % /0)�1 2 34 �� 56789:, de-noising,
1;<=, �>?@, ABC�, D�EF, GHI JKFL MN O P
Q IR�S ,TU VWX YV7� �,[1]. Z [ de-noising� 2
34 ��L \]�E V IRXS Donoho[2-4], Johnstone[3-6], Gao
[7-11] O� LM Y6U !_� `a7� �,. bD� �cd% ��
`e f� ABX UKg� � 2 34 ��� V)P bD -#
% )� Jh !_789: PQ i��jk 5678,.
lKmn� _)� Jh9X opq 1;% V)� Black-Box m
nr Jh� 'V)� st, \uIL v- lKL opq 1;� ,T
U wxL bD� y�)� �9zX K{U lKmn� _)� |}
,. 2 34� \U ~� ��7�S 2 34 ��� V)P �
S ��U ���E wxL bD� �(�) C�U AB% �� �,
$� lKmn� _)� �U ��� F� ~� !_� �& � � '
� 6]78,. Palavajjhala O(1996)� � 2 34 ��(discrete
wavelet transform)� V)P prefilter �K� #� AB% ��� F
�� V)�9:[12], Nikolaou O(1998)� 2 34 ��� V)
� AB <=� �M FIR Jh� �>��� !_% )�,[13]. �U
Carrier O(1998)� 2 34 ��� -|� ��% �U lK * �L
F�� �V)��[14], Flehmig O(1998)� noisy ��� �� @K A
B�L s�� F�)� !_� 2 34� V)�,[15].
� !_�S� Table 1� K�U PQ � 2 34 de-noising i�
�jk� lK mn� _)� ��� F� !_� �V)P bD -#
�K� #� AB% VU st bD -# �K� #¡ �� AB
% VU stL �¢� £¤)��, PQ de-noising i��jL �¢
W £¤)P �¥,. Z�� ABL ��� ¦§ de-noising �
163
164 �������������
% ��� F� �K� �Vg� �L �¢� i©�� �)P SX ,
§ ��� �� ª 5L �«mn� V)��, bDL ¬�� \U
�� i©�� �)P, ,TU bD-AB£(Noise-to-Signal Ratio, NSR)
% �� bD� ®®L �« mn� ¯�)�9: de-noising�K�S
VU 2 34 ��L U JhE Multi-Resolution Analysis(MRA)L °
±(level)� ¦§ �¢� £¤)�,.
2. � �
2-1. ���� ��
��kL ²³� VU ABL cd� MN� H 1800�\ ´
Joseph Fourier� KLU sine� cosine� V)P ��% ]�)� �
�� ���S �W78,. ZQ$ ��� ��L ����% _�)�
sine, cosine��� �� �© !" AB sharp spike MN�
�-% �� �� ��� �, �+U ����L µ¶� \ª7
8,[1]. �U �·� cd� K�% ¸�� ]�¹ � º� ». -
�7| �·� cd� K�% �� $¼½ � �� Short Time Fourier
Transform(STFT) 56789$ �U M>W(resolution)� �K7
� ».� �`,. % /0)�1 ����� �� � ,T):
cd�-�· ]� �¢U 2 34 �� 5678�, 1980�\ [
� � S. Mallat� I. Daubechies O� LM �"WX 6�7| PQ
IR�S ^V7� �,.
2 34 ��� m(¾) 2 34(mother wavelet)L scaling� trans-
lation�K� LM ,TU ����% ¿�)� ��� cd� �· K
�% ¸�� �+)À ]�¹ � �,� Á.� �� �,. � scale
� �· K�% y�)� translation� cd� K�% y�U,[1, 16].
2-1-1. !" 2 34 ��(Continuous Wavelet Transform, CWT)
CWT� ©Â Ã� F (1)XS KL7� Ä� MN)�1 )� A
B x(t) m 2 34 Å �� ���� ψ(t)L convolution wx%
�9:, *� lÆ0Ç�(complex conjugate) ,.
(1)
P�S s� scale È5�� : τ� translation È5�� ,(s, τÉR).
2-1-2. � 2 34 ��(Discrete Wavelet Transform, DWT)
CWT% ��)� �MS� �( �K µ¶),. F (2)� LM ®
®L È5 ��k� �(�Ê F (3)� � 2 34 ��(Discrete
Wavelet Transform, DWT)Å U,.
(2)
(3)
���9X s0 τ0L �� 2 1� 'V)� m, k� ]�((sampling)
�K� 'V7� È5�� ,.
DWT� ËÌ�% V)P CWTL ��� �¢)À )�9$ �
» CWTL ]�(Í ��� �)zX µ¶U �� Ω £¨
Ï� ,. QU ».� �Ð)�1 1989� Mallet� LM DWT% �
, ÑÒ� ¨Ï�9X ��¹ � �� �Ó� i��jE MRA Jh
-Ô78,[1, 16-18].
2-1-3. Multi-Resolution Analysis(MRA)
De-noising�� ���9X �" � 2 34(fast discrete wavelet)
i��jE MRA� 'VÍ,. MRA� 2 34 ��L �Õ9X Ö�
(filter)% V):, Ö�kL פ�� V)P ��Ø� ��Ù ÚE
Jh ,. Ö�L wx� 'V)� m 2 34� ¦Å ²K7: SX
פ)� �� Ö�(low-pass filter)�¹� )� scaling ��[F (4)] �
� Ö�(high-pass filter)�¹� )� 2 34 ��[F (5)]% 'VU,.
(4)
(5)
(6)
P�S h(m)� �� � �� � g(m)� �� � �� ,. �
��k� dyadic down sampling �K� #� ABL convolution� L
M ��7: KÛ(% �M � 'VÍ,. m 2 349X 'VU
Daubechies 3 2 34� Fig. 1� $¼Ü8,.
Fig. 2(a) (b)� F (4) (5)% ÝÞßWX W�U Ä : °±� ¦
ÅS IM(decomposition)�K� �_�(reconstruction)�K� #¡À
Í,. IM �K�S �� µ�� LM IMÍ AB% approximation �
�(cA)Å )� �� µ�� LM IMÍ AB� detail ��(cD)Å U,.
� IMÍ ® ��� 5�% +�9X ÚPc� dyadic down-sampling
�K� #¡: IMÍ approximation��� ,D °±� �a)�*
VÍ,. �_� �K� IMÍ ��k� ²³)� �K9X IM�K�
S +�9X Ú|à 5�% á >xX �á)� dyadic up-sampling
�K� #â AB% 0áU,.
°±� ¦§ ]�( �K� LM �· M>W� �+7� filtering�
K� LM cd� ·ã �+Í,. ¦ÅS °±� ¦Å �·� cd�
� \M ,TU M>W% �À 7�*, � MRAÅ �� [¶U
CWTx τ s,( ) 1
s-------- x t( )ψ∗ t τ–
s---------
dt∫=
s s0m= τ kτ0s0
m=,
DWTx m k,( ) 1
s0m
----------- x t( )ψ∗ s0m–t kτ0–( ) td∫= m k, Z∈
φ x( ) h m( )m Z∈∑ 2φ 2x m–( )=
ψ x( ) g m( ) 2φ 2x m–( )m Z∈∑=
g m( ) 1–( )mh 1 m–( )=
2
Table 1. Applied threshold method for each shrinkage
ShrinkageThreshold method
Minimax Universal SURE Hybrid GCV
Hard √ √ √ √Soft √ √ √ √ √Non-Negative Garrote √ √ √ √ √Firm √
Fig. 1. ‘Daubechies 3’ wavelets.
���� �39� �2� 2001� 4�
��� ��� ��� ��� ��� �� �� 165
ä� Í,[17, 18].
2-2. De-noising
De-noising� ���9X bD� å AB� £M cd�� ¬,� .
� æÔ)P bD� -#)� Jh ,.
2 34 ��� VU de-noising �K� ¬À 3»�X ç| `
,[2, 5, 7].
1 »� −IM�K°±� ßè)�, ßèÍ °± IM�K� �a
2 »� − Detail ��k� threshold �K� �V
®®L °±� \M threshold Jh� ²K)�, shrinkage ��� �V
3 »� −�_� �K
�KÍ detail �� �éL approximation ��% V)P � ��
ª êë »�� de-noising �K�S vì�9X [¶U »�XS threshold
Jh�S ��Í threshold�E λ� V)P shrinkage ��� LM
�cd K�% y�)� detail ��k� �KU,.
2-2-1. Threshold �L ßè
Shrinkage ��� �V��� �U � _)� Jh ,. Threshold
� íî f9� P�Ù bD� Î y�¹ Ä �, �\X threshold
� íî ¬� ABL [¶U K�� ïv7| ð ñ |� I��
R�¹ Ä ,. ¦ÅS ® °±� \M �+U threshold �� ²K)�
Jh� [¶U �-� Í,. � \U PQ Jhk ��7| ��
� !_�S� minimax, universal, SURE, hybrid, GCV Jh� 'V)
�,.
�Minimax
Minimax risk bound% �ÇX )� λ�� ��U,.
®®L shrinkage ��� \U λ�k� Gao� LM ��Í �� '
V)�,[7-9].
�Universal
F (7)� à detail ��L 5�� LM ��7zX �� �K £¤
� ·»)� bDL ¬�� f� st� �¢ ò,� ióô �,[6].
(7)
P�S nj� °± j�S detail ��L 5� ,.
�Stein’s Unbiased Risk Estimate(SURE)
SURE criterion� �ÇX )� �� ßèU,[3, 6, 10, 19].
SURE Jh� ��� 0b)� I�� ñ |�% �ó)� �,
³��E Jh ,. MRAL °±� ¦Å �� 7| : bDL ¬�
� õ�Þ �¢ ò,� ióô �,. jêë °±�SL SURE criterion
� F (8)X ]�Í,.
SURE criterion :
(8)
P�S #� �ö� �÷)� ��kL 5� � xøy� ª � [�S
f� �� ßè)� �B ,.
�Hybrid
©Â� KLÍ hybrid KL (I), (II)� LM bDL ¬�� f9�
SURE Jh ò �9zX universal Jh� ßè)�, Z ù��
SURE universal Jh [ f� λ�� ßè)� �KÍ SURE Jh
,[3]. jêë °±L st ,D� Ã� Jh� LMS λ�� ßèU,.
(I) E st: bDL ¬�� f� stX universal Jh�
ßè.
(II) E st: bDL ¬�� ð stX ��S �úU J
h� ¦Å λ�� ßè
(9)
�Generalized Cross-Validation(GCV)
!"�� �-X )zX soft shrinkage non-negative garrote shrinkage
� �V7: !"�� �� hard shrinkage�� �V¹ � º� thresh-
old Jh ,. GCV��� F (10)� à detail���L ��X ç|
ô �� ��� bD� \U '�F µ¶ º,. ® °±� \M
GCV ��% �ÇX )� δ�� ��)� % threshold �(λ)9X
VU,. ×û �Ç�� ü� ��� ��L Jh Å ¹ � �� �
��K £¤� 0b),. � ® °±�S δ� 0�Su� cDjL �\
�� ý�X )� �� 'V)�� þÿ I¹ h(golden section method)
� V)P GCV ��L �Ç�� ²K)�,[20-24].
(10)
P�S nj,0� °±j�S 0�� �� detail ��L 5� � cDj,δ� °
± j�S c|` threshold �(δ)� V)P soft, non-negative garrote
shrinkage� LM �KÍ detail �� ,. ||x||2� -� norm� $¼Ü
� �B ,.
2-2-2. Shrinkage ��
� !_�S� � � shrinkage ��% 'V)�,. Hard shrinkage
� keep-or-kill Jh9X, detail ��L +\� λ�, f� st detail
��% 09X �KU,. Soft shrinkage� shrink-or-killXS detail ��
L +\� λ�, f� st detail ��% 09X )� Z ùL detail
��k� soft shrinkage ��� LM ÚPc� Û� ,. Soft shrinkage
λ 2 nj( )log=
SURE λ cD,( ) nj 2 # i: cDi λ≤{ } cDi λ∧( )2
i 1=
nj
∑+⋅–=
Sj2 Qj nj⁄≤
Sj2 Qj nj⁄≤
Sj2 nj
1– cDi2 1–( )
i 1=
nj
∑= , Qj nj( )3 2⁄2log=
GCVj δ( )
1nj
---- cDj cDj δ,– 2
nj 0,
nj
------- 2
-----------------------------------=
Fig. 2. (a) Decomposition process of MRA, (b) Reconstruction processof MRA.
HWAHAK KONGHAK Vol. 39, No. 2, April, 2001
166 �������������
� £¤� ð �� �� ��% Ú� � hard shrinkage� £M ñ |
�� ¬À $¼�,. Hard shrinkage� !"� � ��� soft shrink-
age�, I� ¬� ÔKU G ,[3, 4, 7]. kL ».� �Ð)�1
�, f� ñ |� ÔK�� �� Non-Negative Garrote shrinkage
� -Ô78,[8]. Firm shrinkage� semi-soft shrinkageÅ�W ): hard
shrinkage soft shrinkage% �³U [· wx ,. Minimax Jh��
'V)�9: Gao� LM ��Í �� 'V)�,. Detail ��L +\
� λ1�, f� st detail ��% 09X )� λ2� �, ð st�
detail ��% �é): $� st� soft shrinkage��� LM detail
�� �� �KU,[9].
® Jh� LM detail ��% �K)� �U Fk� ©Â Ã,.
�Hard
(11)
�Soft
(12)
�Non-Negative Garrote
(13)
�Firm(Semi-soft)
(14)
P�S sign� uB% ²KM c� signum function ,.
2-2-3. Scale
Shrinkage ��k� threshold �� LM �t7� ��� �+U λ�Lßè� ©c [¶),. °± Lé�� �� MRA�S bD �t� I
y% ¦Ò� white noiseE st� °±� \M bDL �� �ó)
�©W 7$ colored noise% �� st�� °±� ¦§ detail ��L >
~~�% �óMR U,[11]. ¦ÅS °±� ¦Å bDL ¬�% �óU
threshold �L �� µ¶)À 7�* % scale� V)P �KU,.
F (15)% V)P _U �KÍ ]G% ®®L threshold�� �M
�VU,. ���9X white noise� st�W °±� �ó)P scale�
VU threshold �� ��)� Ä �� �³),� ióô �,[6].
(15)
P�S MAD� 09Xu�L median absolute deviation � 0.6745�
�t� Iy% �WÞ )� �KE1 ,.
2-3. �� �(System Identification)
��� F�� opq * �Xu� ���� \U ¸� mn� �K
)� � � mnrL U �9X �Ù black box modeling Å U,.
opq * �Xu� lK mn� ²K)� �MS� ,DL 4»�%
#¡À Í,[25, 26].
»� 1. opq 1; ��
lK� \U ò� mn� �� �MS� M� lK� \U �IU K
�% y�)� opq * �� µ¶),.
»� 2. mnL ßè
mn _�% ßè)� ßèÍ mn� \U � �·!� ²KU
,. »� opq�L ��(Í ßw mn _�� oq u(t) pq y(t)
% !~��� F (16)9X ]�� � �,[26].
(16)
F (16)�SL A(q), B(q), C(q), D(q) � F(q)� q−1� \U ,�F9
X ©Â à ]�Í,.
P�S nk� �·!, e(t)� white noise% $¼Ü� q−1� Backward
shift operatorX ?% k�, q−1y(t)=y(t−1)L ~�% �E,.
� !_�S� È5 �� B% V)� Finite Impulse Response(FIR)
mn� B F% V)� Output Error(OE) mn� 'V)�,.
»� 3. lK mnL È5 �� �K
ßèÍ mn� LM ��Í pq * �� v«� LM �|` pq
* �� �³)WÞ ��L È5 ��k� �� �MS� ��U �
»� µ¶),. ��� F��S� QU �»�� loss function
Š): ���9X F (17)L wx% �� ��-��(Mean Square
Error, MSE)L wx% ��,.
(17)
P�S N� mn �K� 'VÍ * �L 5�, y(t)� @Kpq¡,
� �· t�SL �K pq� LHU,.
»� 4. mn validation
�KÍ mn 'V��� �IÙ ò�% ��)� »�X mn�
�K)� �K(»� 3)� 'V7 �� * �% VU,. Validation
�K� mn � ²K�W ~�Í,. �% �����S ��� F
��K� aU �, �÷¹�U validation²�% �9� [»)� �L
mn _�% ßèU,. ���� �� ���� ¦§ loss function
L �Ç KW% VU,[25].
2-3-1. FIR mn
lK� \U '�F(�· !, mn � O) µ¶ º,� Á.
� �� �9$ Î� �L È5 ��k� µ¶X U,� ».� ��
�,. FIR mn� �M �� �· !� ,§ mn _�(OE, BJ O)�
V¹ � �9: F (18)X ]�Í,[25, 26].
(18)
2-3-2. OE mn
OE mn� ßw � �· mn� \M �Á ×û� � £¤� ·
»U Jh ,[25, 26].
(19)
OE mn� lK� \U '�F µ¶): �� FIR mn� V)
P �·!� ��,. OE mnL loss function� F (20)9X ]�¹ �
�9: mn È5��� \M £ßw ,. mn� \U È5 ��k
L �� loss function� �Ç( �9XS ²K7�* loss function È
5 ��k� \MS £ßw zX £ßw ��INh� VMR U,.
Thard cD λ;( ) 0 for cD λ≤cD for cD λ≤
=
Tsoft cD λ;( ) 0 for cDλ≤sign cD( ) cD λ–( ) for cD λ≥
=
TGarrot cD λ;( ) 0 for cDλ≤
cD 1 λ cD⁄( )2–{ } for cD λ≤
=
TFirm cD λ1; λ2,( )
0 for cDλ1≤
sign cD( )λ2 cD λ1–( )
λ2 λ1–----------------------------- for λ1 cD λ2≤<
cD for cDλ2>
=
σjMAD cD j( )
0.6745---------------------------=
A q( )y t( ) B q( )F q( )-----------u t nk–( ) c q( )
D q( )------------e t( )+=
A q( ) 1 a1q1–+Λ a+ naq
na–B q( ) b1q 1–
+Λ bnbqnb–
,+=,+=
F q( ) 1+f1q1– +Λ+fnfq
nf– C q( ) 1 c1q 1– +Λ+cncqnc– ,+=,=
D q( ) 1 d1q1– +Λ d+ ndq
nd–+=
V loss1N---- y t( ) y t( )–( )2
t 1=
N
∑ 1N---- ε2 t( )
t 1=
N
∑= =
y t( )
y t( ) B q( )u t nk–( )=
y t( ) B q( )F q( )-----------u t nk–( ) e t( )+=
VOE1N---- εOE
2 t( ) 1N---- y t( ) y t( )–[ ]2
t 1=
N
∑=t 1=
N
∑=
���� �39� �2� 2001� 4�
��� ��� ��� ��� ��� �� �� 167
(20)3. Simulation
2,0005L Pseudo-Random Binary Sequence(PRBS)% oq9X )�
Table 2L È5 �� F (19)�S bD �� -ùU OE mn� V
)P oq9Xu� pq AB% ¿�)P % å ABX 'V)�,.
� �«mn 1� �«mn 2L SX ,§ ��� �� ª 5L AB%
¿�)�9: ® mnL ��� ,D� Ã,[27].
�«mn 1: �· ! 2E 3�
�«mn 2: �· ! º� 3 � ^ �
¿�Í å AB� NSR 5, 10, 30, 50, 75, 100%% �WÞ �t� I
y% �� white noise� �)P @K pq AB(measured output signal)
% ¿�)�9: NSRL KL� F (21)� Ã,.
(21)
1N----= y t( ) B q( )
F q( )-----------u t nk–( )–
2
t 1=
N
∑
NSR %( ) Variance of NoiseVariance of True Signal---------------------------------------------------------------- 100 %( )×=
Fig. 3. True(solid lines) and various measured signals(dashed lines) of test model #1.
Fig. 4. True(solid lines) and various measured signals(dashed lines) of test model #2.
Table 2. OE parameters applied on each test model
B F
Model #1 [0 0 0 0.075-0.0425 0.005] [1-2.4 1.91-0.504]Model #2 [0-0.175-0.365-0.18] [1-2.46 2.0-0.5376]
HWAHAK KONGHAK Vol. 39, No. 2, April, 2001
168 �������������
bD ¯�Í @KAB� ®®L de-noising i��j� �V)��
�¢L !WX� ��-��% 'V)�,. MRA �K�SL °±�
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MRA% VU de-noising �KL °±� \U �� i© �� �
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Fig. 5. Frequency vs. amplitude plot using fast fourier transform.
Table 3. Simulation results of MRA level with minimum average MSEfor various NSR values
NSR(%) 5 10 30 50 75 100
LevelTest model #1 2 2-3 3 3 3-4 4Test model #2 1 2 2 3 3 4
Fig. 6. Effect on level of MRA for test model #1(Soft shrinkage case with various threshold methods). “circle” =Minimax, “ square” =Universal,“ triangle up” =SURE, “ diamond”=Hybrid, “ triangle down” =GCV.
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HWAHAK KONGHAK Vol. 39, No. 2, April, 2001
Table 4. MSE of de-noised signal for test model #1 (Universal threshold case with various shrinkage methods)
NSR(%)MSE
(Without de-noising)Level Hard Soft
Non-Negative Garrote
5 Mean : 0.0986 2Mean 0.0283 0.0281† 0.0283
Std 0.0014 0.0014 0.0014
10 Mean : 0.1972 2Mean 0.0528 0.0529 0.0525†
Std 0.0026 0.0026 0.0026
30 Mean : 0.5912 3Mean 0.1052 0.1041† 0.1041†
Std 0.0069 0.0065 0.0065
50 Mean : 0.9856 3Mean 0.1549 0.1532† 0.1532†
Std 0.0102 0.0097 0.0097
75 Mean : 1.4788 3Mean 0.2185 0.2156† 0.2157
Std 0.0169 0.0158 0.0158
100 Mean : 1.9708 4Mean 0.2638 0.2603† 0.2603†
Std 0.0174 0.0160 0.0160
Table 5. MSE of de-noised signal for test model #1 (Soft shrinkage case with various threshold methods)
NSR(%)MSE
(Without de-noising)Level Minimax Universal SURE Hybrid GCV
5 Mean : 0.0986 2Mean 0.0283 0.0281† 0.0286 0.0281† 0.0286
Std 0.0014 0.0014 0.0015 0.0014 0.0017
10 Mean : 0.1972 2-3Mean 0.0531(2) 0.0529(2) 0.0524(3) 0.0525(2) 0.0521(3) †
Std 0.0027 0.0026 0.0042 0.0026 0.0044
30 Mean : 0.5912 3Mean 0.1062 0.1041† 0.1067 0.1041† 0.1067
Std 0.0072 0.0065 0.0087 0.0065 0.0101
50 Mean : 0.9856 3Mean 0.1569 0.1532† 0.1598 0.1532† 0.1603
Std 0.0107 0.0097 0.0134 0.0097 0.0172
75 Mean : 1.4788 3-4Mean 0.2216(3) 0.2156(3) † 0.2270(3) 0.2156(3) † 0.2273(4)
Std 0.0177 0.0158 0.0194 0.0158 0.0257
100 Mean : 1.9708 4Mean 0.2676 0.2603 0.2661 0.2599† 0.2671
Std 0.0189 0.0160 0.0264 0.0264 0.0346
The index in parentheses indicates MRA level
Fig. 7. Results of validation based on FIR method for test model #1(Soft shrinkage case with various threshold methods).
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Fig. 8. Results of validation based on FIR method for test model #2(Soft shrinkage case with various threshold methods).
Fig. 9. Results of validation based on OE method for test model #1(Soft shrinkage case with various threshold methods).
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HWAHAK KONGHAK Vol. 39, No. 2, April, 2001
Fig. 10. Results of validation based on OE method for test model #2(Soft shrinkage case with various threshold methods).
Fig. 11. Model validation for test model #1(Soft shrinkage case withSURE threshold method). “heavy solid line”=True signal, “heavydashed line”=Measured signal, “circle”=Predicted signal(with-out de-noising), “light dashed line”=De-noised signal, “square”=Predicted signal(with de-noising).
Fig. 12. Model validation for test model #2(Soft shrinkage case withSURE threshold method). “heavy solid line”=True signal, “heavydashed line”=Measured signal, “circle”=Predicted signal(withoutde-noising), “light dashed line”=De-noised signal, “square”= Pre-dicted signal(with de-noising).
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Table 6. The optimal MRA levels in model validation of FIR method
NSR(%) 5 10 30 50 75 100
LevelModel #1 2 2 2 2 3 3Model #2 1 1 2 2 2 3
Table 7. Average MSE in model validation of FIR method for test model #1(Soft shrinkage case with various threshold methods)
NSR(%) Without de-noising Level Minimax Universal SURE Hybrid GCV
5Mean : 0.0302
Std :0.0022
1 Mean 0.0233 0.0233 0.0233 0.0233 0.0234Std 0.0011 0.0011 0.0011 0.0011 0.0011
2Mean 0.0215† 0.0215† 0.0215† 0.0215† 0.0216Std 0.0009 0.0009 0.0009 0.0009 0.0009
3Mean 0.0329 0.0335 0.0311 0.0334 0.0304Std 0.0014 0.0015 0.0020 0.0015 0.0029
4Mean 0.0931 0.0943 0.0858 0.0943 0.0832Std 0.0024 0.0023 0.0099 0.0022 0.0122
30Mean : 0.0963Std : 0.0138
1 Mean 0.0521 0.0520 0.0522 0.0520 0.0523Std 0.0065 0.0065 0.0065 0.0065 0.0068
2Mean 0.0340 0.0339† 0.0342 0.0339† 0.0343Std 0.0042 0.0042 0.0043 0.0042 0.0043
3Mean 0.0393 0.0396 0.0383 0.0396 0.0377Std 0.0038 0.0039 0.0042 0.0039 0.0043
4Mean 0.0976 0.0989 0.0865 0.0988 0.0840Std 0.0070 0.0070 0.0124 0.0070 0.0157
50Mean : 0.1444Std : 0.0213
1 Mean 0.0731 0.0728 0.0731 0.0728 0.0731Std 0.0113 0.0111 0.0112 0.0111 0.0114
2Mean 0.0449 0.0448 0.0452 0.0448 0.0451Std 0.0061 0.0062 0.0062 0.0062 0.0064
3Mean 0.0442 0.0444 0.0433† 0.0443 0.0436Std 0.0074 0.0074 0.0072 0.0074 0.0079
4Mean 0.0987 0.1006 0.0870 0.1005 0.0854Std 0.0077 0.0075 0.0154 0.0074 0.0181
100Mean : 0.2704Std : 0.0502
1 Mean 0.1316 0.1313 0.1318 0.1313 0.1317Std 0.0276 0.0277 0.0278 0.0277 0.0278
2Mean 0.0740 0.0735 0.0744 0.0735 0.0746Std 0.0167 0.0168 0.0170 0.0168 0.0175
3Mean 0.0570 0.0568 0.0572 0.0567† 0.0572Std 0.0155 0.0156 0.0155 0.0156 0.0156
4Mean 0.1061 0.1076 0.0990 0.1074 0.0966Std 0.0155 0.0152 0.0180 0.0153 0.0202
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Table 9. Average MSE in model validation of OE method for test model #1(Soft shrinkage case with various threshold methods)
NSR(%) Without de-noising Level Minimax Universal SURE Hybrid GCV
5Mean : 0.0193Std : 0.0015
1Mean 0.0183 0.0183 0.0183 0.0183 0.0183
Std 0.0003 0.0003 0.0003 0.0003 0.0003
2Mean 0.0180 0.0179† 0.0179† 0.0179† 0.0180
Std 0.0002 0.0002 0.0003 0.0002 0.0003
3Mean 0.0235 0.0241 0.0224 0.0241 0.0221
Std 0.0010 0.0009 0.0013 0.0008 0.0016
4Mean 0.0507 0.0528 0.0467 0.0528 0.0447
Std 0.0035 0.0019 0.0053 0.0019 0.0065
30Mean : 0.0267Std : 0.0106
1Mean 0.0211 0.0211 0.0211 0.0211 0.0210
Std 0.0028 0.0028 0.0028 0.0028 0.0028
2Mean 0.0195† 0.0195† 0.0195† 0.0195† 0.0196
Std 0.0012 0.0011 0.0012 0.0011 0.0013
3Mean 0.0248 0.0250 0.0238 0.0249 0.0236
Std 0.0023 0.0025 0.0026 0.0024 0.0026
4Mean 0.0524 0.0543 0.0463 0.0540 0.0445
Std 0.0062 0.0061 0.0082 0.0058 0.0094
50Mean : 0.0345Std : 0.0160
1Mean 0.0229 0.0230 0.0230 0.0230 0.0229
Std 0.0053 0.0054 0.0054 0.0054 0.0053
2Mean 0.0209 0.0207† 0.0209 0.0207† 0.0208
Std 0.0022 0.0019 0.0022 0.0019 0.0019
3Mean 0.0248 0.0252 0.0241 0.0252 0.0241
Std 0.0030 0.0030 0.0030 0.0030 0.0030
4Mean 0.0495 0.0533 0.0435 0.0532 0.0437
Std 0.0076 0.0058 0.0092 0.0058 0.0099
Table 8. Average MSE in model validation of FIR method for test model #2(Soft shrinkage case with various threshold methods)
NSR(%) Without de-noising Level Minimax Universal SURE Hybrid GCV
5Mean : 0.0186Std : 0.0015
1Mean 0.0181† 0.0182 0.0182 0.0182 0.0181†
Std 0.0012 0.0011 0.0011 0.0011 0.0012
2Mean 0.0238 0.0240 0.0237 0.0240 0.0237
Std 0.0013 0.0013 0.0014 0.0013 0.0013
3Mean 0.0431 0.0441 0.0417 0.0441 0.0412
Std 0.0014 0.0014 0.0024 0.0014 0.0033
30Mean : 0.0749Std : 0.0087
1Mean 0.0429 0.0429 0.0429 0.0429 0.0429
Std 0.0059 0.0059 0.0059 0.0059 0.0059
2Mean 0.0362 0.0364 0.0360† 0.0364 0.0361
Std 0.0048 0.0048 0.0048 0.0048 0.0048
3Mean 0.0475 0.0485 0.0454 0.0485 0.0446
Std 0.0051 0.0050 0.0057 0.0050 0.0064
50Mean : 0.1214Std : 0.0208
1Mean 0.0629 0.0629 0.0629 0.0629 0.0629
Std 0.0104 0.0104 0.0105 0.0104 0.0104
2Mean 0.0467 0.0468 0.0466† 0.0468 0.0466†
Std 0.0075 0.0075 0.0077 0.0075 0.0076
3Mean 0.0529 0.0537 0.0522 0.0537 0.0520
Std 0.0052 0.0054 0.0052 0.0053 0.0054
100Mean : 0.2373Std : 0.0328
1Mean 0.1085 0.1083 0.1087 0.1083 0.1087
Std 0.0162 0.0163 0.0162 0.0163 0.0162
2Mean 0.0708 0.0705 0.0711 0.0705 0.0713
Std 0.0124 0.0127 0.0121 0.0127 0.0121
3Mean 0.0662 0.0666 0.0651† 0.0666 0.0655
Std 0.0098 0.0098 0.0094 0.0098 0.0101
HWAHAK KONGHAK Vol. 39, No. 2, April, 2001
174 �������������
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ä'U ²�% �8,.
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~)P £¤)�� bD� -#U AB Z4 �� AB% V)P
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L ¬� Z�� MRA °±L �� i© �� �)P SX ,§ ��
� �� AB% 'V)�� ,TU ¬�L NSR� °±� 'V)�,.
2 34 de-noisingL st shrinkage Jh $ threshold Jh� ¦
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ä'�� {E)�,.
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noising �K� #� AB% VU st� Z4 �� st� £MS
Table 9. Continued.
NSR(%) Without de-noising Level Minimax Universal SURE Hybrid GCV
100Mean : 0.0526Std : 0.0382
1Mean 0.0289 0.0286 0.0287 0.0286 0.0289
Std 0.0097 0.0092 0.0095 0.0092 0.0097
2Mean 0.0239† 0.0239† 0.0240 0.0239† 0.0239†
Std 0.0039 0.0040 0.0039 0.0040 0.0039
3Mean 0.0273 0.0277 0.0269 0.0277 0.0269
Std 0.0055 0.0051 0.0052 0.0051 0.0048
4Mean 0.0523 0.0562 0.0489 0.0561 0.0485
Std 0.0122 0.0109 0.0125 0.0109 0.0137
Table 10. Average MSE in model validation of OE method for test model #2 (Soft shrinkage case with various threshold methods).
NSR (%) Without de-noising Level Minimax Universal SURE Hybrid GCV
5Mean : 0.0077†
Std : 0.0021
1Mean 0.0115 0.0116 0.0116 0.0116 0.0115
Std 0.0005 0.0005 0.0005 0.0005 0.0005
2Mean 0.0200 0.0202 0.0199 0.0202 0.0200
Std 0.0007 0.0007 0.0009 0.0007 0.0007
3Mean 0.0435 0.0461 0.0418 0.0460 0.0414
Std 0.0021 0.0013 0.0032 0.0013 0.0038
30Mean : 0.0089†
Std : 0.0013
1Mean 0.0116 0.0117 0.0116 0.0117 0.0116
Std 0.0014 0.0014 0.0014 0.0014 0.0014
2Mean 0.0201 0.0205 0.0196 0.0205 0.0197
Std 0.0014 0.0015 0.0018 0.0015 0.0018
3Mean 0.0410 0.0445 0.0382 0.0442 0.0373
Std 0.0048 0.0042 0.0062 0.0043 0.0068
50Mean : 0.0115†
Std : 0.0044
1Mean 0.0118 0.0120 0.0120 0.0120 0.0119
Std 0.0022 0.0022 0.0022 0.0022 0.0022
2Mean 0.0205 0.0208 0.0197 0.0208 0.0200
Std 0.0027 0.0025 0.0029 0.0025 0.0028
3Mean 0.0428 0.0461 0.0412 0.0460 0.0414
Std 0.0061 0.0052 0.0068 0.0052 0.0068
100Mean : 0.0158Std : 0.0088
1Mean 0.0133 0.0130† 0.0133 0.0130† 0.0133
Std 0.0031 0.0028 0.0032 0.0028 0.0031
2Mean 0.0208 0.0219 0.0208 0.0219 0.0207
Std 0.0039 0.0044 0.0045 0.0044 0.0045
3Mean 0.0412 0.0453 0.0390 0.0455 0.0391
Std 0.0072 0.0075 0.0076 0.0074 0.0079
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cDj,δ : the vector of shrunk coefficients for given threshold δ at level j
e(t) : white noise
k, m : sampling parameter
N : the number of data used model estimation
nj,0 : the number of zero elements at level j
nk : time delay
s : scale parameter
t : time
u(t) : input signal at time t
V loss : loss function
||x||2 : square norm of x
xøy : minimum of x and y
y(t) : output signal at time t
: estimated output at time t
Z : integer
# : the number of data
� �� ��
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: estimated standard deviation at level j
τ : translation parameter
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δ : threshold value minimizing GCV function
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(1995).
2. Donoho, D. L.: “Wavelet Shrinkage and W. V. D.: A Ten-minute Tour,”
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1200(1995).
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Soc., B(57), 301(1995).
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319(1997).
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HWAHAK KONGHAK Vol. 39, No. 2, April, 2001