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m
d = dtrue + η
d = G (mtrue ) + η.
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d (s) =
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g (s, x, m(x)) = g (s, x) · m(x)
g (s, x)
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a
m (x) dx
g (s, x) = 1 a = 0
Z
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d (s) =
g (s, x, m(x)) dx
a
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g (s, x)
g (s, x) = g(s − x)
d (s) =
H (w)
Rs
m (x) dx
0
Z s
Z ∞
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H (s − x) m (x) dx.
1
w
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y (t)
1
y (t) = m1 + m2 t − at2 .
2
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tk
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dk
1
dk = y (tk ) = m1 + t m2 − t2 m3
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ã xW˜ˆp/e.gA•nWp„wixWwund ,œˆ0pÈÓSfokgcfj ¢Hc•wix •nW£ fhmgÒwmhi˜ˆÜC˜ž•œ § ¢Scfwmx ª s
e.ã xWc¤g£¤nÃg:œçwix,pÈÓSgfhigokg?˜žqp„EÓHghmjlg:o7wokc?hmhig:pmÓSc?nW£¤g:nWokgu¢Hgkwř„gg:nÃÜfgAo7wicfhypטˆn±£,wypmÓWfog•nS£
ã xW˜ˆp/˜žp„wmxWgU˜ˆ£¤gA•œ s Ÿ„ ¤w•hig]™„gcf ¤wucfjçwix,gU™ºcdc¤£,p^]
Ôo˜³g:cWnCz¤wu™„e.gҝ•pÈwmwm˜ˆhiœ ˜¶HÝ xS§<hiÜfggue.wigc¿e*£,¢Sg:g:•h œ4®™/˜ˆ˜³œ³wm¢Hxæghm£,w^]<wyª s g:hmhicfh„•nW£±˜³œˆ³›ÎocfnW£¤˜³wm˜ˆcfnW˜³n,š¿c•jçwmx,gUokcdg ¿›
rank (G) = p
span
{V1 , V†2 , . . . , Vp }
m† = G d
G=
σj Uj VjT
ê
N (G) = span {Vp+1 , Vp+2 , . . . , Vn } R (G) =
j=1
N GT = span {Up+1 , Up+2 , . . . , Um } R (G) = span {U1 , U2 , . . . , Up }
Gm = d
2
p
X
G
m×n
Rn
p
Rm
p=m=n
•
G
GT
•
Gm = d, m = G−1 d = G† d.
•
•
«•Á
{0}
ò û pmî gþ wmx,ü/gUî LnWòc•wi•wmð˜ˆcfn ?íLï”î:ð\ò ðlî s ò
î Qí û ò þ ï”íLî:ð þ ñ ãx,… ˜žpe.‘g:| •nSp„¼¿wm~ xWº•“w Š”~k|4“kÐf· sÆt cfe.e.gnCwip:q
ã xW˜ˆp/e.gA•nWp„wixWwund ,œˆ0pÈÓSfokgcfj ˜ˆp„wihm˜ˆÜd˜ˆfœ\z¤¢, ,w/wmxW•wuc•j ˜ˆpnWc•w s
® ghig ˜žpwmx,gU Wn,˜ˆÉC ,gœˆg:?pÅwpiÉC W•hig:ppmcfœˆ ¤wm˜ˆcfn s
gkÝ,fokwmœˆÀ*˜³j ˜žpº˜³n
wix,gÒhifn,šfgucfj s
¨ nW£ ˜žp×wix,gÒgkÝ,fokwºpÈc?œ³ ,wm˜ˆcfn
wic
Ÿ„ ¤w ˜ˆp/˜ˆnWpmgnWpm˜³wm˜ˆÜfgÒwmc
•ndÀ
wihifnWpmœˆ•wm˜ˆcfn ™/˜³wmx
m† = G † d
d=n<m
•
GT
G
•
m†
•
m†
•
m
Gm = d
G
d
d + d0
d0 ∈ N G
†
ò û î þ ü/î Lò ð ?íLï”î:ð\ò ðlî ò
î Qí û ò þ ï”íLî:ð þ ñ ã ß×x,…‡˜ž»¤p‰¿e.| g:•]nS‘p | E¼¿~ „“kŠ”~|H“sºÐft · c?e.e¿g:nCwip:q
ã xW˜ˆp/e.gA•nWp„wixWwund ,œˆ0pÈÓSfokgcfj ˜žp„wmhi˜³Üd˜ž•œYz¤¢, ¤wuwixWw/cfj ˆ˜ pnWc•w s
® ghig ˜žpu.pÈc?œ³ ¤wi˜³c?ncfj e.˜ˆn,˜ˆe ,e ›În,cfhie s
˜žp/•œžpÈc.
pÈc?œ³ ,wm˜ˆcfnwmc
jlcfhufnCÀ
¨ nW£
s
b¤c ˜žp/˜ˆnWpmgnWpm˜¶wi˜³Ü?gÒwmc
•ndÀ
wihifnWpȜžwi˜³c?n
™/˜³wmx
s
p=m<n
GT
•
G
•
m†
2
•
m† + m0
Gm = d
•
m† + m0
d
m0 ∈ N (G)
m0 ∈ N (G)
ò û î þ ü/î Lò ð ?íLï”î:ð\ò ðlî ò
î Qí û ò þ ï”íLî:ð þ ñ ãx,… (˜žpe.‘|Hg:̒•nSß×p …‡ »¤‰¿| ]‘| *¼¿~ „sº“kt Š”~cf|He.“Ðfe.· gnCwip:q
ã xW˜ˆp/e.gA•nWp„wixWwund ,œˆ0pÈÓSfokgcfj ¢Hc•wix •nW£ f hmgn,c?n?wihm˜ˆÜd˜ˆfœ s
® ghig ˜žpu¿œ³gAfpÈw/piÉC W•hig:ppmcfœˆ ¤wm˜ˆcfn s
¡gUxW<Ü?g]wihmc? ,¢,œˆg˜³næ¢Hc•wmx £¤˜ˆhig:o7wi˜³c?nWp s
p < min {m, n}
•
•
G
m†
•
«¤r
GT
H *M & JM S9M 9S#U) Q ! # ! - S 9S S+&[MIQ S '
* S &/US
Q !
*þ [ û ð,ñ fò ,ñ ÿ ò :þ ï”íLî:ð þ ñ
˜žp
ò %ñ ðlî:ð þ ñ s ù ã xWg þ0ÿ òCï û ò Aþ ï\íçîAð þ ñ ¤î û ðY jlcfhwix,g
Ó,hicf¢,œˆge
ßׅ|Wk~ »W~k|4“y~7·
z¤™/x,˜žoyx˜žp Å WpÈw ˜¶j xWfp„jl ,œˆçocfœˆ ,e.nhy•nW¯ s
áÎj
zdwmx,g:n
ã xd Wp:zwmxWg!¢,˜žfp؈nÙwmxWg!šfg:hmg:hifœ³˜ˆ²gA™/£ ˜³wm˜ˆndx Üfghypmg!pÈc?œ³ ¤wi˜³c?s n٘ˆp
ob¤c˜³e.ܝ•˜ˆhiœ f˜ˆfhmnWœˆÀfokz¤gҘˆn±e
wix,wmgEhi˜³Ý±o?˜ˆpÈp gÒc•j ˜ˆ£,gnCwm˜žofœ³ˆÀ±£¤˜žpÅwihm˜ˆ¢, ¤wig:££,wy™/˜³wmxæܝ•hi˜ž•nWog z¤wmx,g
fhmhmgÒc?e okœˆc?gpmÝdguÓHwig:c okwmgAzd£ƒ™„Ü<g]foœ³ Wœˆfg:˜³pUe֙„š?g
cCc¤cf£
¢¤wyhi•g:˜ˆpmn‹cfœˆ ¤wiû ˜³c?ò n
Aþ c•ï\j0íçwiîAxWð þ ñw/okî:cdò cf hyî £¤qE˜ˆnW˜³j/wigfz?£,c•˜ˆwmfxWšfgcfhinS™/•˜ˆœ#pmggnCn,wicfhmw À s
Gm = d
Rm = G † G
• Rm = Vp VpT
•
•
Gmtrue = d
G
E [m† ] = Rm mtrue
(Rm − I) mtrue = −V0 V0T mtrue
•
E [m† ] − mtrue =
V = [Vp V0 ]
2
Cov (m† ) = σ G
•
ë
In
†
G
† T
=σ
2
σ2
Pp
Vi ViT
i=1 σi2
.
1
H '
ZU
!
M J Q ! #
S S &[MIQ S S /US Q ! ZU
') * +&
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Ï4,~ f~ÐÀm~k7»¤‡ \k·
cfh pmÉC WfhmgÒe
wihm˜³Ý
s
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•wmhi˜¶Ý
Ôc•wigfq#˜³j zdwix,gEokc?nW£¤˜³wm˜ˆcfnænd ,e¢Hgh/˜žp˜ˆn¤ÛWn,˜³wÅÀ s áÅpwix,˜žp/n,c•wi˜³c?n SpÈgjl ,œ ]
áÎj £,wy Üfg:okwmc?hƘžpÆÓHghmwm ,hi¢Hg:£
wic zChig:pm ,œ³wm˜ˆn,š˜ˆnÓSg:hÈwi ,hi¢Wwi˜³c?nÃc•jQwix,g]š?gn¤›
g:hifœ³˜ˆ²gA£±˜ˆndÜfg:hipmgÒpmcfœˆ ¤wi˜³c?n wmc z¤wmx,g:n
•
•
•
•
n×n
σ1 /σq
G cond2 (G) = kGk2 G−1 2 = σ1 /σn
m×n
q = min {m, n}
G
σq = 0
d0
d
m†
m0†
«f«
km0† −m† k2
km† k2
≤ cond (G)
kd0 −dk2
kdk2
.
ë
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1
3×3
11
12
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t4
21
22
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t5
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32
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t2
t1
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G
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Gm = 
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

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0
0
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1
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1 0 0
0 1 0
0 0 1
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1 1 1
0 √0 0
0
2 0
0 0 0
1
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G
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1
0
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0
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1
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1
√
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2















s11
s12
s13
s21
s22
s23
s31
s32
s33



 
 
 
 
 
=
 
 
 
 
 

t1
t2
t3
t4
t5
t6
t7
t8



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
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