١
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ
ﺱﺭﺪﻣ : ﺩﻮﻨﺷﻮﺧ ﺪﻴﺠﻤﻟﺍﺪﺒﻋ ﻥﺍﻮﻨﻋ ﺯﺍﻭﺮﭘ ﯼﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﻦﻴﻤﺨﺗ ﻭ ﻢﺘﺴﻴﺳ ﻲﻳﺎﺳﺎﻨﺷ
ﻪﺘﺧﺎﻨﺷﺎﻧ ﻢﺘﺴﻴﺳ ﻢﺘﻳﺭﻮﮕﻟﺍ
ﻲﻳﺎﺳﺎﻨﺷ
ﻢﺘﺴﻴﺳ ﻱﺎﻫﺮﺘﻣﺍﺭﺎﭘ
ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﺕﺎﻌﺑﺮﻣ
ﯼﺎﻫ ﻝﺪﻣ ARX
ﻢﻴﻫﺍﻮﺧ ﻲﻣ ﻭ ﺎﻫ ﺕﺭﻮﺻ ﻪﺑ ﻢﻳﺭﺍﺩ ﺍﺭ ﺎﻫ ﻩﺩﺍﺩ ﺯﺍ ﻱﺍ ﻪﺘﺳﺩ ﺪﻴﻨﻛ ﺽﺮﻓ ﻢﻴﻫﺩ ﺭﻮﺒﻋ ﻥﺁ ﺯﺍ ﺍﺭ ﺮﻳﺯ ﻲﻨﺤﻨﻣ :
ﺪﻨﻳﺁ ﺖﺳﺪﺑ ﺪﻳﺎﺑ ﻪﻛ ﺪﻨﺘﺴﻫ ﻲﻟﻮﻬﺠﻣ ﻱﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﺎﻫ .Regression model ﻝﺪﻣ ﻦﺘﺷﻮﻧ ﻉﻮﻧ ﻦﻳﺍ ﻪﺑ ﻭ ﺩﻮﺷ ﻲﻣ ﻪﺘﻔﮔ
ﺩﻮﺷ ﻲﻣ ﻩﺪﻴﻣﺎﻧ ﺭﻮﺳﺮﮔﺭ .
ﻡﻮﻠﻌﻣ ﻭ ﺖﺳﺍ ﻢﺘﺴﻴﺳ ﻲﻌﻗﺍﻭ ﻲﺟﻭﺮﺧ ﻪﻛ ﺲﭘ
ﺪﻨﺘﺴﻫ ﻪﻟﺎﺴﻣ .
i
) ( ), (i i y
(i)
T
) y
T(i
٢
ﻦﺷﻮﻟﻮﻧﺎﮐ ﯼﻭﺭ ﺯﺍ ﻥﻮﻴﺳﺮﮔﺭ ﻝﺪﻣ•
= ∗ ℎ = ℎ .
= ℎ . + ℎ . + ⋯ + ℎ . +
= … . ℎ ℎ
⋮ ℎ
+
= . +
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﻪﻠﭘ ﺦﺳﺎﭘ ﯼﻭﺭ ﺯﺍ ﻥﻮﻴﺳﺮﮔﺭ ﻝﺪﻣ•
ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﻪﻠﭘ ﺦﺳﺎﭘ ﯽﻠﮐ ﻡﺮﻓ LTI
= + + + ⋯ +
ﻪﻄﺑﺍﺭ ﻦﻳﺍ ﺭﺩ ﻭ ﻢﺘﺴﻴﺳ ﯼﺎﻫ ﺐﻄﻗ ﺎﻫ s
ﺪﻨﺘﺴﻫ ﻝﻮﻬﺠﻣ ﺎﻫk .
ﻪﻟﺎﺴﻣ ﮏﻳ ﻪﻟﺎﺴﻣ
ﺖﺳﺍ ﯽﻄﺧﺮﻴﻏ ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ .
ﻢﻴﻨﮐ ﺽﺮﻓ ﺎﻣﺍ ﻢﻳﺭﺍﺩ ﺍﺭs
:
= = 1 … . ⋮ +
= . +
٣
ﻥﺎﻣﺯ ﺎﺑ ﺮﻴﻐﺘﻣ ﻝﺎﻨﮕﻴﺳ ﮏﻳ ﯼﺍﺮﺑ ﻥﻮﻴﺳﺮﮔﺭ ﻝﺪﻣ• ﻅﺎﺤﻟ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﯽﻄﺧ ﺏﺎﺘﺷ ﺽﺮﻓ ﺎﺑ ﻭ ﺭﺍﺩﺍﺭ ﺎﺑ ﺍﺭ ﺎﻤﻴﭘﺍﻮﻫ ﮏﻳ ﺖﻴﻌﻗﻮﻣ ﺮﮔﺍ
ﺖﻓﺮﮔ ﻩﺮﻬﺑ ﺭﻮﺳﺮﮔﺭ ﺭﺍﺩﺮﺑ ﺯﺍ ﻥﺍﻮﺗ ﯽﻣ ﻩﺪﻨﻳﺁ ﺕﺎﻈﺤﻟ ﻦﻴﻤﺨﺗ ﯼﺍﺮﺑ ،ﻢﻴﻳﺎﻤﻧ :
] [
2 ];
1 1 [
; .
] 2 ].[
1 1 [
2 1
0 0 2
0 0 2 2 0
0
a v x θ t t u
e θ u x y
e a v x t t x
y
e at t v x x
T t t T t t t
t t
t
t t
=
= +
=
=
+
=
=
+ +
= +
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ
ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﻱﺍﺮﺑ ﻥﻮﻴﺳﺮﮔﺭ ﻝﺪﻣ• ﻲﻠﻳﺪﺒﺗ ﻊﺑﺎﺗ ﺪﻴﻨﻛ ﺽﺮﻓ ﻢﻳﺭﺍﺩ ﻞﺑﺎﻘﻣ ﺕﺭﻮﺻ ﻪﺑ :
ﺩﺮﻴﮔ ﻡﺎﺠﻧﺍ ﻱﺯﺎﺳ ﻪﺘﺴﺴﮔ ﺍﺪﺘﺑﺍ ﺪﻳﺎﺑ ﻝﺪﻣ ﻞﻴﻜﺸﺗ ﻱﺍﺮﺑ
ﺪﻨﺘﺴﻴﻧ ﺮﺑﺍﺮﺑ ﺎﻣﻭﺰﻟ ﻪﺘﺴﺴﮔ ﻭ ﻪﺘﺳﻮﻴﭘ ﺐﻳﺍﺮﺿ ) .
ﺭﻮﺗﺍﺮﭘﺍ ﻭ ﻱﺯﺎﺳ ﻪﺘﺴﺴﮔ ﺲﻳﺪﻧﺍ ﺢﻴﺿﻮﺗ
ﻪﺘﺴﺴﮔ ﺱﻮﻜﻌﻣ (
0 1
1
0 1
1
...
) ...
) ( (
) (
b s
b s b
a s
a s s a
s G u
s y
m m m m
n n n n
) ( ) ( ) ( )
(z y t B z u t
A
۴
2
1
] )
( 2 [ 1
) ( ) ˆ( ) ( ) (
) ˆ(
t
i
T
T T
t y J
t y t y t y t e
t y
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﻛ ﻪﻴﻀﻗ • ﻲﺴﻳﺮﺗﺎﻣ ﻥﺎﻳﺩﺍﺮﮔ ﻦﻴﻧﺍﻮﻗ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺎﺑ ﻩﺪﺷ ﻲﻓﺮﻌﻣ ﻪﻨﻳﺰﻫ ﻊﺑﺎﺗ ﻥﺩﺮﻛ ﻪﻨﻴﻤﻛ ﻱﺍﺮﺑ
ﻢﻳﺭﺍﺩ :
y y
J grad for
t y t
y t
y J
T T
T T
T t T
i
T t
i
T
1 1
2
1
) ˆ (
ˆ
0 ) (
] )
( [ ] )
( 2 [
] 1 )
( 2 [
1
۵
ﻲﻠﺻﺍ ﻪﻄﺑﺍﺭ • ﻂﺧ ﺝﺭﺎﺧ ﻲﻳﺎﺳﺎﻨﺷ ﻭ ﻲﻨﺤﻨﻣ ﺵﺯﺍﺮﺑ ﺭﺩ ﻩﺩﺎﻔﺘﺳﺍ ﺩﺭﻮﻣ ﻭ LS
ﺩﻮﺷ ﻲﻣ ﻩﺩﺎﻔﺘﺳﺍ ﺰﻴﻧ ﻲﻫﺩ ﻥﺯﻭ ﺯﺍ ﻲﻨﺤﻨﻣ ﺵﺯﺍﺮﺑ ﺭﺩ• .
(W)
ﻱﺎﻨﻏ ﺎﻳ ﻚﻳﺮﺤﺗ ﻁﺮﺷ ﻥﺍﻮﻨﻋ ﺎﺑ ﻭ ﺖﺳﺍ ﻲﻤﻬﻣ ﻪﻟﺎﺴﻣ ﻱﺮﻳﺬﭘ ﺱﻮﻜﻌﻣ• ﺩﻮﺷ ﻲﻣ ﺩﺎﻳ ﻥﺁ ﺯﺍ ﻲﻳﺎﺳﺎﻨﺷ ﻝﺎﻨﮕﻴﺳ .
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﻪﻄﺑﺍﺭ ﺶﻳﺎﻤﻧ ﺮﮕﻳﺩ ﻉﻮﻧ• :
T
) . .(
) .
ˆ (
∑ ∑
1 1 1
N
t
t t N
t
T t
tu u y
u θ
=
¬
=
=
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﯽﺒﻧﺎﺟ ﻪﺘﮑﻧ ﺪﻨﭼ ﺲﯾﺮﺗﺎﻣ صاﻮﺧ يرادﺮﺑ ﻪﻧﻮﻤﻧ داﺪﻌﺗ زا ﻞﻘﺘﺴﻣ تﺎﻌﺑﺮﻣ ﻦﯾﺮﺘﻤﮐ ﻪﻟدﺎﻌﻣ ﺪﻨﯾاﺮﻓ • ﺖﺳا N
: :ﺪﺷﺎﺑ ﯽﻣP×P ﺲﯾﺮﺗﺎﻣ ﮏﯾ قﻮﻓ ﺲﯾﺮﺗﺎﻣ
≡ × . × ≡ ×
≡ × . × 1 ≡ × 1
ﻪﺠﯿﺘﻧ رد :
Θ = ( ) . ≡ × . × 1 ≡ × 1
۶
ﯽﺒﻧﺎﺟ ﻪﺘﮑﻧ ﺪﻨﭼ ﺎﻄﺧ ﻊﺑﺎﺗ نﺪﺷ ﻢﻣﺮﺘﺴﮐا ﯽﺳرﺮﺑ
ﺖﺒﺜﻣ ﺲﯾﺮﺗﺎﻣ ﻪﮐ ﺎﺠﻧآ زا )
ﻪﻤﯿﻧ ( ﻦﯾا و هﺪﺷ ﺖﺒﺜﻣ مود ﻖﺘﺸﻣ اﺬﻟ ﺖﺳا ﻦﯿﻌﻣ
ﺪﺷﺎﺑ ﯽﻣ ﺎﻄﺧ نﺪﺷ ﻞﻗاﺪﺣ ﯽﻨﻌﻣ ﻪﺑ مﻮﻬﻔﻣ .
ﺮﻌﻘﺗ و بﺪﺤﺗ مﻮﻬﻔﻣ و مود ﻖﺘﺸﻣ • U
U ) U U y U (
)) e e ( ( )
e e (
T T
T
T T
2
= θ 2 + 2
θ ¬
∂
∂
θ =
∂
∂
∂θ
= ∂
∂θ
∂
2 2
U UT
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﺕﺎﻌﺑﺮﻣ ﻞﻗﺍﺪﺣ ﻲﻳﺎﺳﺎﻨﺷ ﺖﻴﻠﺑﺎﻗ ﺲﯾﺮﺗﺎﻣ يﺮﯾﺬﭘ سﻮﮑﻌﻣ تﺎﻌﺑﺮﻣ ﻦﯾﺮﺘﻤﮐ ﻪﻟﺎﺴﻣ ﯽﺗﺎﯿﺿﺎﯾر هﺮﮔ ﻦﯾﺮﺘﻤﻬﻣ •
لﺎﺣﺪﺑ ﺲﯾﺮﺗﺎﻣ ،ﺲﯾﺮﺗﺎﻣ نﺎﻨﯿﻣﺮﺗد نﺪﺷ ﺮﻔﺻ ﮏﯾدﺰﻧ (Ill condition)
ﯽﯾﺎﺳﺎﻨﺷ و
ﻻﺎﺑ يﺎﻄﺧ و ﻦﯿﯾﺎﭘ ﺖﻗد ﺎﺑ دﺎﯾز ﺎﻫﺮﺘﻣارﺎﭘ يﺎﻄﺧ ﺎﻣا بﻮﻠﻄﻣ ﯽﺟوﺮﺧ ﺎﻄﺧ زا ﺪﻨﺗرﺎﺒﻋ ﯽﯾﺎﺳﺎﻨﺷ ﺖﯿﻠﺑﺎﻗ ﻪﺑ طﻮﺑﺮﻣ ﻊﺑﺎﻨﻣ :
يدورو ندﻮﺒﻧ ﯽﻨﻏ Persistency Excitation (PE)
ﻪﺑ ﺮﺠﻨﻣ ﻪﮐ لﺎﺣﺪﺑ
ﺪﺷ ﺪﻫاﻮﺧ ﺲﯾﺮﺗﺎﻣ نﺪﺷ .
ﯽﻌﻗاو ﻢﺘﺴﯿﺳ ﻪﺒﺗﺮﻣ زا لﺪﻣ ﻪﺒﺗﺮﻣ ندﻮﺑ ﺮﺘﺸﯿﺑ )
ﺪﺘﻓا ﯽﻣ قﺎﻔﺗا ترﺪﻧ ﻪﺑ ﻪﮐ
( .ﺪﺷ ﺪﻫاﻮﺧ قﻮﻓ ﺲﯾﺮﺗﺎﻣ نﺪﺷ ﺮﻔﺻ ﻪﺑ ﺮﺠﻨﻣ
ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﺖﻟﺎﺣ رد ﻢﺘﺴﯿﺳ ﯽﯾﺎﺳﺎﻨﺷ )
هﺪﻨﯾآ رد ﺮﺘﺸﯿﺑ ﺢﯿﺿﻮﺗ (
U UT
U UT
٧
ﺕﺎﻌﺑﺮﻣ ﻞﻗﺍﺪﺣ ﺵﻭﺭ ﯽﺳﺪﻨﻫ ﺮﻴﺒﻌﺗ ﺎﻄﺧ ﺭﺍﺩﺮﺑ
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﺕﺎﻌﺑﺮﻣ ﻞﻗﺍﺪﺣ ﺵﻭﺭ ﯽﺳﺪﻨﻫ ﺮﻴﺒﻌﺗ
ﯽﻌﻗﺍﻭ ﯽﺟﻭﺮﺧ ﺭﺍﺩﺮﺑ Y ﺎﻫﺭﻮﺳﺮﮔﺭ ﺯﺍ ﻪﺘﻓﺎﻳ ﻞﮑﺷ ﺭﺍﺩﺮﺑ
ﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﻭ ﺭﺍﺪﻘﻣ ﻦﻳﺮﺘﻤﮐ ﻪﮐ ﺖﺳﺍ ﯽﺘﻗﻭE
ﯽﻨﻌﻳ ﺪﺷﺎﺑ ﺩﻮﻤﻋ ﺎﻫﺭﻮﺳﺮﮔﺭ ﺮﺑ :
ﺖﺳﺍ ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﻪﻟﺩﺎﻌﻣ ﻥﺎﻤﻫ ﺕﺭﺎﺒﻋ ﻦﻳﺍ ﻪﮐ .
٨
ﺵﻭﺭ ﯽﺣﻼﺻﺍ Singular Value Decomposition (SVD)
SVD ﻪﻳﺰﺠﺗ ﻒﻳﺮﻌﺗ
ﻪﻳﺰﺠﺗ ﺲﻳﺮﺗﺎﻣ ﺮﻫ ﻊﻗﺍﻭ ﺭﺩSVD
ﯽﻣ ﻪﻳﺰﺠﺗ ﺮﻳﺯ ﻞﮑﺷ ﻪﺑ ﺲﻳﺮﺗﺎﻣ ﻪﺳ ﻪﺑ ﺍﺭ M
ﺪﻨﮐ :
= Σ
∗ﯼﺎﻫ ﺲﻳﺮﺗﺎﻣ ﺖﺳﺍ ﺱﺎﻴﻘﻣ ﺲﻳﺮﺗﺎﻣ ﻭ ﺪﻨﺘﺴﻫ ﺪﻣﺎﻌﺘﻣ ﻭ ﻥﺍﺭﻭﺩ ﺲﻳﺮﺗﺎﻣU ,V
. ﻪﻳﺰﺠﺗ ﻦﻳﺍ .ﺖﺳﺍ ﯽﻌﻗﺍﻭ ﻩﺯﺍﺪﻧﺍ ﮏﻳ ﻭ ﻥﺍﺭﻭﺩ ﻭﺩ ﯼﺍﺭﺍﺩ ﺲﻳﺮﺗﺎﻣ ﺮﻫ ،ﺮﮕﻳﺩ ﺕﺭﺎﺒﻋ ﻪﺑ ﺩﺭﺍﺩ ﺩﺮﺑﺭﺎﮐ ﺭﺎﻴﺴﺑ ﺰﻴﻧ ﻡﻭﺎﻘﻣ ﻝﺮﺘﻨﮐ ﺭﺩ ﻩﺪﻨﻨﮐ .
U’U=I ﻭ
V’V=I Σ
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﺵﻭﺭ ﯽﺣﻼﺻﺍ Singular Value Decomposition (SVD)
٩
ﻝﺎﺜﻣ :
M=[1 2 1;3 4 5;4 6 1]
M =
1 2 1 3 4 5 4 6 1
[A,B,C]=svd(M)
C =
-0.5096 0.1872 0.8398 -0.7429 0.3966 -0.5393 -0.4340 -0.8987 -0.0631 B =
9.9209 0 0 0 3.2372 0 0 0 0.3114 A =
-0.2449 0.0252 -0.9692 -0.6724 -0.7246 0.1510 -0.6985 0.6887 0.1944
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺭﺩ ﺩﺮﺑﺭﺎﮐ ﺕﺭﺎﺒﻋ ﺭﺩ y=Ux+e
ﺲﻳﺮﺗﺎﻣ ﺵﻭﺭ ﺎﺑ ﺍﺭ U
ﻢﻴﻨﮐ ﯽﻣ ﻪﻳﺰﺠﺗSVD :
U=PRQ
ﺭﺩ ﻦﻴﻓﺮﻃ ﺏﺮﺿ :P’
ﻞﺣ ﻭ ﺭﻮﺳﺮﮔﺭ ﺪﻳﺪﺟ ﻪﻟﺩﺎﻌﻣ ﺲﻴﺗﺎﻣ ﯼﺎﺠﺑ ﻪﮐ ﺖﺳﺍ ﻦﻳﺍ ﺵﻭﺭ ﯽﻠﺻﺍ ﺖﻳﺰﻣ • ﺲﻳﺮﺗﺎﻣU
ﯼﺎﻫ ﻪﻳﺍﺭﺩ ﯼﻭﺭ ﺯﺍ ﻪﮐ ﻪﺘﻓﺮﮔﺭﺍﺮﻗR
ﺍﺭ ﺩﻮﺷ ﯽﻣ ﯽﺗﺭﻻﻮﮕﻨﻴﺳ ﺎﻳ ﻥﺪﺷ ﻝﺎﺣﺪﺑ ﺚﻋﺎﺑ ﻪﮐ ﻲﻳﺎﻫ ﻪﻳﺍﺭﺩ ﻥﺍﻮﺗ ﯽﻣ ﻥﺁ ﺮﻔﺻ ﻪﺑ ﮏﻳﺩﺰﻧ ﺩﺮﮐ ﻑﺬﺣ .
Qx x
, e P e , y P y
e Rx y
e P RQx y
P
e P PRQx P
y P
* T
* T
*
*
*
* T
T
T T
T
=
=
=
+
⇒ = +
=
+
=
x*
١٠
ﺕﺎﻌﺑﺮﻣ ﻞﻗﺍﺪﺣ ﯼﺭﺎﻣﺁ ﺮﻴﺒﻌﺗ زا ﺪﻨﺗرﺎﺒﻋ ﺎﻫﺮﮔ ﻦﯿﻤﺨﺗ يرﺎﻣآ ﯽﺳرﺮﺑ رد ﯽﻠﺻا رﺎﯿﻌﻣ ود :
ﻦﯿﻤﺨﺗ سﺎﯾﺎﺑ ﻦﺘﺷاﺪﻧ • Bias
ﻦﯿﻤﺨﺗ ﺲﻧﺎﯾراو ﻦﯾﺮﺘﻤﮐ • Variance (Cov)
] ˆ )) ˆ (
ˆ ))(
ˆ ( [(
ˆ ) (
ˆ ) (
E
TE E
Cov E b
σx1
σx2
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﻦﯿﻤﺨﺗ ندﻮﺑ سﺎﯾﺎﺑ نوﺪﺑ ﻂﯾاﺮﺷ
طﺮﺷ ﻪﺳ ﻖﻘﺤﺗ ﺎﺑ ﺮﻔﺻ سﺎﯾﺎﺑ
: AU=I (ﻒﻟا
ب ( E(Ae)=E(A)E(e)
ج ( E(e)=0 ترﻮﺻ رد تﺎﻌﺑﺮﻣ ﻦﯾﺮﺘﻤﮐ ﻦﯿﻤﺨﺗ ياﺮﺑ قﻮﻓ طﺮﺷ ﻪﺳ ﺮﻫ ﺪﻫد ﯽﻣ نﺎﺸﻧ ﺎﻫ ﯽﺳرﺮﺑ
ندﻮﺑ ﺪﯿﻔﺳ ﺰﯾﻮﻧ ضﺮﻓ ﺪﻧراﺮﻗﺮﺑ e
. e
U y
ˆ ) ( E b
U ) U U ( A ˆ Ay
T T
+ θ
=
θ
¬ θ
=
=
= θ
1
¬
) Ae ( E ] ) I AU [(
E b
)]
e U ( A [ E )
Ay ( E b
+ θ
¬
⇒ =
θ
¬ + θ
= θ
¬
⇒ =
١١
ﻦﯿﻤﺨﺗ ﺲﻧﺎﯾراﻮﮐ ﻦﯿﻤﺨﺗ ندﻮﺑ سﺎﯾﺎﺑ نوﺪﺑ ضﺮﻓ
ندﻮﺑ ﻦﯿﻌﻣ ضﺮﻓﺎﺑ A
] ˆ ) ˆ )(
[(
ˆ) (
0 ˆ)
( E T
Cov E b
) ( ˆ) (
) ( ) ( ) ( ˆ)
(
) ) )
)((
) ((
) ) )) (
)(
) (
((
ˆ) (
) ) )(
((
ˆ) ( ˆ
2 T
T T
T
T T T
AA Cov
A E e Cov A E A Aee E Cov
Ae I
AU Ae I
AU E
e U A e
U A E Cov
Ay Ay
E Cov
e U y
Ay
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ تﺎﻌﺑﺮﻣ ﻦﯾﺮﺘﻤﮐ رد ﺲﻧﺎﯾراﻮﮐ
ﻪﯿﻀﻗ BLUE Best of Linear Unbiased Estimator ﺖﺳا ﯽﻄﺧ سﺎﯾﺎﺑ نوﺪﺑ ﺮﮕﻨﯿﻤﺨﺗ ﻦﯾﺮﺘﻬﺑ تﺎﻌﺑﺮﻣ ﻞﻗاﺪﺣ شور .
يﺎﻫﺮﮔ ﻦﯿﻤﺨﺗ ﺮﯾﺎﺳ ﻪﺑ ﺖﺒﺴﻧ ار ﺲﻧﺎﯾراﻮﮐ ﺲﯾﺮﺗﺎﻣ ﻦﯾﺮﺘﻤﮐ تﺎﻌﺑﺮﻣ ﻞﻗاﺪﺣ شور دراد سﺎﯾﺎﺑ نوﺪﺑ ﯽﻄﺧ .
1 2
2( ) ( )
ˆ)
( AA U U
Cov T T
١٢
ﻝﺎﺜﻣ ﻚﻳ ) :
ﻲﻳﺎﺳﺎﻨﺷ ﺮﻳﺯ ﻭ ﻲﻳﺎﺳﺎﻨﺷ ﻕﻮﻓ ﻉﻮﺿﻮﻣ (
ﺕﺎﻌﺑﺮﻣ ﻦﻳﺮﺘﻤﮐ ﺵﻭﺭ ﻢﻴﻧﺰﺑ ﻦﻴﻤﺨﺗ ﺍﺭ ﻲﺟﻭﺮﺧ ﺭﻮﺳﺮﮔﺭ ﻒﻠﺘﺨﻣ ﻱﺎﻫ ﻝﺪﻣ ﺎﺑ ﻢﻴﻫﺍﻮﺧ ﻲﻣ ﻝﺎﺣ• :
١٣
ﻉﻮﺿﻮﻣ Over-parameterization ﻭ
Under-parameterization
