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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(12),9081-9095
PublishedOnlineDecember2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1112958
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Huber¼ê§-èëêO§É~Šuÿ§šà¨v§1wz•{
ParameterEstimationand
OutliersDetectionBasedon
NonconvexPenalized
Regression
ZunhaoZhang,DingtaoPeng
∗
,YanyanSu
SchoolofMathematicsandStatistics,GuizhouUniversity,GuiyangGuizhou
Received:Nov.26
th
,2022;accepted:Dec.21
st
,2022;published:Dec.30
th
,2022
∗ÏÕŠö"
©ÙÚ^:܃Ê,$½7,€òò.Äušà¨v£8ëêOÚÉ~Šuÿ[J].A^êÆ?Ð,2022,11(12):
9081-9095.DOI:10.12677/aam.2022.1112958
܃Ê
Abstract
Thispaperpresentsamethodtoimplementparameterestimationandoutliersde-
tectionformultiplelinearregressionmodelsbasedonoptimizationtheory.First,a
regressionmodelbasedontheHuberlossfunctionandthe`
0
penaltytermisdevel-
oped,andthe`
0
penaltyterminthismodelisfurtherrelaxedtotheCapped-`
1
penalty
tofacilitatethesolution;andsecond,thedirectionalstabilityp ointoftherelaxation
problemischaracterized,andtheequivalenceoftheoriginalandrelaxationproblems
isestablished.Finally,weproposeasmoothmodelfortherelaxationproblemand
provetheconsistencyofthestablepointofsmoothmodelandtherelaxationproblem.
Keywords
HuberFunction,RobustParameterEstimation,OutliersDetection,Nonconvex
Penalty,SmoothingMethod
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.11129589083A^êÆ?Ð
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DOI:10.12677/aam.2022.11129589084A^êÆ?Ð
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DOI:10.12677/aam.2022.11129589085A^êÆ?Ð
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DOI:10.12677/aam.2022.11129589086A^êÆ?Ð
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)a
T
i
(x−x
∗
) =
1
n
n
X
i=1
p
X
j=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
ij
(x
j
−x
∗
j
).(14)
∀ε
1
∈(0,1),bx= (bx
1
,...,bx
p
)
>
∈R
p
÷v
bx
j
=



(1−ε
1
)x
∗
j
,j∈P
1
(x
∗
),
x
∗
j
,j∈P
2
(x
∗
).
duP
1
(x
∗
) 6= ∅,(bx−x
∗
)Ø´"•þ.du(13)é?¿x∈R
p
þ¤á,K
n
X
i=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
T
i
(bx−x
∗
)=
n
X
i=1
p
X
j=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
ij
(bx
j
−x
∗
j
)
=ε
1
n
X
i=1
X
j∈P
1
(x
∗
)
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
ij
(−x
∗
j
)
≤ε
1





n
X
i=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
i





1
·
X
j∈P
1
(bx)
|x
∗
j
|.(15)
,˜•¡,
p
X
j=1
ϕ
0
(x
∗
j
;bx
j
−x
∗
j
) =
X
j∈P
1
(bx)
sgn(x
∗
j
)(−ε
1
x
∗
j
)
v
= −ε
1
X
j∈P
1
(bx)
sgn(x
∗
j
)(x
∗
j
)
v
= −
ε
1
v
X
j∈P
1
(bx)
|x
∗
j
|.(16)
DOI:10.12677/aam.2022.11129589087A^êÆ?Ð
܃Ê
d(13)-(16),
λ
1
ε
1
v
X
j∈P
1
(bx)
|x
∗
j
|≤
ε
1
n





n
X
i=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
i





1
X
j∈P
1
(bx)
|x
∗
j
|,
du
P
j∈P
1
(x
∗
)
|x
∗
j
|6= 0,òÙž
1
v
≤
1
λ
1
n










n
X
i=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)a
i










1
≤M(x
∗
,e
∗
),
ù†M(x
∗
,e
∗
) <
1
v
gñ,lP
1
(x
∗
) = ∅.
(ii)bP
3
(e
∗
) 6= ∅.d½n2.1,x= x
∗
,K
1
n
n
X
i=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)(e
i
−e
∗
i
)+λ
2
n
X
i=1
ϕ
0
(e
∗
i
;e
i
−e
∗
i
) ≥0,∀e∈R
n
.(17)
∀ε
2
∈(0,1),be= (be
1
,...,be
n
)
>
∈R
n
÷v
be
i
=



(1−ε
2
)e
∗
i
,i∈P
3
(e
∗
),
e
∗
i
,i∈P
4
(e
∗
).
duP
3
(e
∗
)š˜,(be−e
∗
)Ø´"•þ,?
n
X
i=1
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)(be
i
−e
∗
i
)=ε
2
X
i∈P
3
(e
∗
)
h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)e
∗
i
≤ε
2
X
i∈P
3
(e
∗
)
|h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)|·|e
∗
i
|.(18)
,˜•¡,
n
X
i=1
ϕ
0
(e
∗
i
;be
i
−e
∗
i
) =
X
i∈P
3
(e
∗
)
ϕ
0
(e
∗
i
;be
i
−e
∗
i
) = −
ε
2
v
X
i∈P
3
(e
∗
)
sgn(e
∗
i
)(e
∗
i
) = −
ε
2
v
X
i∈P
3
(e
∗
)
|e
∗
i
|.(19)
d(17)-(19),
λ
2
ε
2
v
X
i∈P
3
(e
∗
)
|e
∗
i
|≤
ε
2
n
X
i∈P
3
(e
∗
)
|h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)|·|e
∗
i
|≤
ε
2
n
n
X
i=1
|h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)|
X
i∈P
3
(e
∗
)
|e
∗
i
|,
1
v
≤
1
λ
2
n
n
X
i=1
|h
0
α
(a
T
i
x
∗
+e
∗
i
−y
i
)|.
du
P
i∈P
3
(e
∗
)
|e
∗
i
|6= 0,òÙž
1
v
≤M(x
∗
,e
∗
),†bgñ,P
3
(e
∗
) = ∅.y..
DOI:10.12677/aam.2022.11129589088A^êÆ?Ð
܃Ê
3.2.¯K(7)†(8))d5
e¡/Ͻn3.1ïÄ¯K(7)†tµ¯K(8)Û•`)ƒm'X.
½n3.2(x
∗
,e
∗
)∈R
p+n
÷vM(x
∗
,e
∗
)<
1
v
,K(x
∗
,e
∗
)´tµ¯K(8)Û•`)…=
(x
∗
,e
∗
)´¯K(7)Û•`).
y².(1)(x
∗
,e
∗
)∈R
p+n
´tµ¯K(8)Û•`),dÚn2.1,(x
∗
,e
∗
)´tµ¯K(8)
••-½:.d½n3.1,Φ
1
(x
∗
)=kx
∗
k
0
,Φ
2
(e
∗
)=ke
∗
k
0
.5¿,∀(x,e)∈R
p+n
,okΦ
1
(x)≤
kxk
0
,Φ
2
(e) ≤kek
0
.l,∀(x,e) ∈R
p+n
,k
H(x
∗
,e
∗
)+λ
1
kx
∗
k
0
+λ
2
ke
∗
k
0
=H(x
∗
,e
∗
)+λ
1
Φ
1
(x
∗
)+λ
2
Φ
2
(e
∗
)
≤H(x,e)+λ
1
Φ
1
(x)+λ
2
Φ
2
(e)
≤H(x,e)+λ
1
kxk
0
+λ
2
kek
0
.
Ïd,(x
∗
,e
∗
)´¯K(7)Û•`).
(2)(x
∗
,e
∗
)∈R
p+n
´¯K(7)Û•`),b§Ø´tµ¯K(8)Û•`).
(x
0
,e
0
)´tµ¯K(8)Û•`),d½n3.1,Φ(x
0
) = kx
0
k
0
,Φ(e
0
) = ke
0
k
0
.u´,
H(x
0
,e
0
)+λ
1
kx
0
k
0
+λ
2
ke
0
k
0
=H(x
0
,e
0
)+λ
1
Φ
1
(x
0
)+λ
2
Φ
2
(e
0
)
<H(x
∗
,e
∗
)+λ
1
Φ
1
(x
∗
)+λ
2
Φ
2
(e
∗
)
≤H(x
∗
,e
∗
)+λ
1
kx
∗
k
0
+λ
2
ke
∗
k
0
.
ù†(x
∗
,e
∗
)´¯K(7)Û•`)gñ,Ïd(x
∗
,e
∗
) ∈R
p+n
´tµ¯K(8)Û•`).
4.tµ¯K(8)1wz•{
duæ^šàš1wK,tµ¯K(8)´š1w`z,¦)tµ¯K(8)š~;.˜a
•{´1wz•{.1wz•{˜‡Ø%¯K´1wzcü‡.)˜—5.
!k‰Ñtµ¯K(8)1wz.,,y²1w¯K-½:†tµ¯K(8)••-½
:äk˜—5.
4.1.¯K(8)1wz.
Capped-`
1
¼ê´˜aš~;.òU]¼ê,§Œ±•ü‡à ¼ê(DC)[21,25],Ù1
wz¼êEÄuÙàL«:
ϕ(t) = min

1,
|t|
v

= 1−max

0,1−
|t|
v

= g(t)−h(|t|),(20)
DOI:10.12677/aam.2022.11129589089A^êÆ?Ð
܃Ê
Ù¥
g(t) = 1,h(|t|) = max

0,1−
|t|
v

.
Pm(t) = |t|,N´yÙ••ê•:
m
0
(t;d) =



|d|,t= 0,
t
|t|
d,t6= 0.
dug(t)´1w,·‚•Iòh(|t|)1wz.®•1w¼êEܼêE´1w,Ïd·‚Œ±©
OE¼êh(s)†S¼êm(t)1w¼ê
e
h
µ
(s)†em
µ
(t),2ò§‚EÜ, BŒëYŒ‡1w
¼ê
e
h
µ
(t) =
e
h
µ
◦em
µ
(t).
m(t)1wz:m(t)1w¼ê•:
em
µ
(t) =
p
t
2
+µ,µ>0,(21)
Ù¥µ´1wzëê.em
µ
(t)êÚ••ê©O•:
em
0
µ
(t) =
t
p
t
2
+µ
,em
0
µ
(t;d) =em
0
µ
(t)d=
td
p
t
2
+µ
.
Œ±y²1w¼êem
µ
(t)÷vXe5Ÿ[21]:
(1)(1w¼ê˜—Âñ5):
lim
w→t,µ↓0
em
µ
(w) = m(t),∀t∈R.
(2)(••ê˜—Âñ5†f˜—Âñ5):
lim
w→t,µ↓0
em
0
µ
(w;d) =lim
w→t,µ↓0
em
0
µ
(w)d= m
0
(t)d= m
0
(t,d),∀t∈R\{0},∀d∈R,
limsup
w→0,µ↓0
em
0
µ
(w;d) =limsup
w→0,µ↓0
td
p
t
2
+µ
= |d|= m
0
(0,d),∀d∈R.
h(s)1wz:éh(s) = max{0,1−
s
v
},Œ±æ^bs(z,µ) =
1
2

z+
p
z
2
+µ
2

51wzܼ
êmax{0,z}[21,26].2òz= 1−
s
v
“\bs(z,µ),h(s)1wz¼ê•
e
h
µ
(s) = bs(1−
s
v
,µ) =
1
2

1−
s
v
+
r
(1−
s
v
)
2
+µ
2

.(22)
N´y²1w¼ê
e
h
µ
(s)äkXe5Ÿ:
(1)(1w¼ê˜—Âñ5):
lim
w→s,µ↓0
e
h
µ
(w) =
1
2

1−
s
v
+|1−
s
v
|

= h(s),∀s∈R.
DOI:10.12677/aam.2022.11129589090A^êÆ?Ð
܃Ê
(2)(••ê˜—Âñ5†f˜—Âñ5):
lim
w→s,µ↓0
e
h
0
µ
(w;d) = h
0
(s,d),∀s∈R\{0},∀d∈R,
limsup
w→0,µ↓0
e
h
0
µ
(w) = h
0
+
(0).
(Ü1w¼êem
µ
(x)Ú1w¼ê
e
h
µ
(s),·‚tµ¯K(8)1wz.Xe:
min
x,e
e
F
µ
(x,e) := H(x,e)+λ
1
p+λ
2
n−λ
1
p
X
j=1
e
h
µ
◦em
µ
(x
j
)−λ
2
n
X
i=1
e
h
µ
◦em
µ
(e
i
).(23)
4.2.1w¯K(23)†tµ¯K(8)-½:˜—5
¯K(23)´1w,Ù-½:´¦∇
e
F
µ
(x,e) = 0:.
!y²1w¯K(23)-½:†tµ¯K(8)••-½:äk˜—5.
½Â•I8:
P
1
(x) = {j∈{1,...,p}: x
j
6= 0},P
2
(x) = {j∈{1,...,p}: x
j
= 0},
Q
2
(e) = {i∈{1,...,n}: e
i
6= 0},Q
2
(e) = {i∈{1,...,n}: e
i
= 0}.
½n4.13µ=µ
k
ž,(x
∗
µ
k
,e
∗
µ
k
)´1w¯K(23)-½:,K1wëêµ
k
↓0(k→∞)ž,
{(x
∗
µ
k
,e
∗
µ
k
)}
∞
k=1
?¿à:Ñ´tµ¯K(8)••-½:.
y².(x
∗
,e
∗
)•:{(x
∗
µ
k
,e
∗
µ
k
)}
∞
k=1
˜‡à:,Ø”
lim
k→∞
x
∗
µ
k
= x
∗
,lim
k→∞
e
∗
µ
k
= e
∗
.
Ï•(x
∗
µ
k
,e
∗
µ
k
)´1w¯K(23)-½:,∇
e
F
µ
(x
∗
µ
k
,e
∗
µ
k
) = 0.K∀d
(1)
∈R
p
,∀d
(2)
∈R
n
,k
0=
e
F
0
µ
(x
∗
µ
k
,e
∗
µ
k
;d
(1)
,d
(2)
) = h∇
e
F
0
µ
(x
∗
µ
k
,e
∗
µ
k
),d
(1)
,d
(2)
i
=h∇H(x
∗
µ
k
,e
∗
µ
k
),d
(1)
,d
(2)
i−λ
1
p
X
j=1
e
h
0
µ
◦em
µ
(x
∗
µ
k
(j)
)em
0
µ
(x
∗
µ
k
(j)
)d
(1)
j
−λ
2
n
X
i=1
e
h
0
µ
◦em
µ
(e
∗
µ
k
(i)
)em
0
µ
(e
∗
µ
k
(i)
)d
(2)
i
=h∇H(x
∗
µ
k
,e
∗
µ
k
),d
(1)
,d
(2)
i−λ
1


X
j∈P
1
(x
∗
)
+
X
j∈P
2
(x
∗
)


e
h
0
µ
◦em
µ
(x
∗
µ
k
(j)
)em
0
µ
(x
∗
µ
k
(j)
)d
(1)
j
−λ
2


X
j∈Q
1
(e
∗
)
+
X
j∈Q
2
(e
∗
)


e
h
0
µ
◦em
µ
(e
∗
µ
k
(i)
)em
0
µ
(e
∗
µ
k
(i)
)d
(2)
i
.(24)
DOI:10.12677/aam.2022.11129589091A^êÆ?Ð
܃Ê
Šâ1w¼êem
µ
(x)Ú
e
h
µ
(t)5Ÿ,éª(24)ü>4•(k→∞),K∀d
(1)
∈R
p
,∀d
(2)
∈R
n
,k
0=lim
k→∞
e
F
0
µ
(x
∗
µ
k
,e
∗
µ
k
;d
(1)
,d
(2)
)
=lim
k→∞
h∇H(x
∗
µ
k
,e
∗
µ
k
),d
(1)
,d
(2)
i−λ
1
X
j∈P
1
(x
∗
)
lim
k→∞
e
h
0
µ
◦em
µ
(x
∗
µ
k
(j)
)em
0
µ
(x
∗
µ
k
(j)
)d
(1)
j
−λ
1
X
j∈P
2
(x
∗
)
lim
k→∞
e
h
0
µ
◦em
µ
(x
∗
µ
k
(j)
)em
0
µ
(x
∗
µ
k
(j)
)d
(1)
j
−λ
2
X
i∈Q
1
(e
∗
)
lim
k→∞
e
h
0
µ
◦em
µ
(e
∗
µ
k
(i)
)em
0
µ
(e
∗
µ
k
(i)
)d
(2)
i
−λ
2
X
i∈Q
2
(e
∗
)
lim
k→∞
e
h
0
µ
◦em
µ
(e
∗
µ
k
(i)
)em
0
µ
(e
∗
µ
k
(i)
)d
(2)
i
≤lim
k→∞
h∇H(x
∗
µ
k
,e
∗
µ
k
),d
(1)
,d
(2)
i−λ
1
X
j∈P
1
(x
∗
)
lim
k→∞
e
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