ZinovyevBook
.pdf
% " MX & "" ! . ' " *' ! -. 4+* ""#$ #) &:
|
1 |
|
1 |
N |
|
MX » |
X i , S » S = |
(X i - MX )(X i - MX )T . |
|||
N |
|
||||
|
i |
N -1i=1 |
|||
2.2.1. ' / "#$ & .-" "'
, & 6 -. " +, + " ""#$ – % "
6 4$! "$ "0 . . % " 6 " ! . " ' " ) ""#$ – -" & 6 ( ' %& ""#$ -. 4+* % ", % & ' #$ . "+5 . " ' -' " ' & ' # ! !*' ! 0 "& !. $"#$ & " '
ηk = Fk (ξ1 ,ξ2 ,...,ξm ) , k = 1…m’ , m’ < m.
9 "& Fk *' ' ) 6 " F $" / -' " ' Rm -' " ' Rm’. 7' ' ) 6 " 6" #) '+ ! ' & . ) ., %' )# " " ) ""#$ X . & . '+ -""#( & ' (, & &- ' ' 6 *4 ( & % ' $"! . ( - 1' . -) "
"0 . . #) ! ' ) 6 " F -"" / & ' ) 6 " ( & ' ( $" " ! "0 . J, . 6" -% '+
%"# . ' # & 4 " ! . " ' -' " ' -" &.
. ' / "#$ & .-" "' F – " & ' " (" ' / " +" " . "" ' ) 6 ", '. .
Fk (ξ1,ξ2 ,...,ξm ) = c1k (ξ1 − μ1 ) + ... + cmk (ξm − μm
" - " ) ""#$ " % " ! -" &, " & # *' ! !
|
/ μ j = |
1 |
N |
) , |
|
xij – |
|
|
|||
|
|
N i=1 |
|
" & 100 "'# cij
m |
m |
cik2 = 1, |
cik c jk = 0 , i, j = 1…m, i ¹ j. |
k=1 |
k =1 |
& ' ! J : |
|
J = Dη1 + ... + Dηm′ , Dξ1 + ... + Dξm
/ D – #% " - % (" ( % "#.
3 / " 1' . & ' *, & % ' $" "" ( "0 .
" «)8! " "" (» -. 4+* " #$ -" & η1…ηm -
$"#$ -" &.
" # *' ' & * " . ""- "' "" * " (" * & .) " * $"#$ -" &, & ' !
$ -%$ " . ""- "' ""#$ " ("#$ & .) " ( ) ' " "" . " ) ""#$ " ) +5 ( -(.
5 . % "$6 " ! -( / " ( & .-" "'#. ! 1' / " )$. 5 '+ %
D(l1 X ) → max ,
l1
/ l1 – &'-' & . " ' m, - " . & l1l1T = 1. &' l1 . 6" -' !'+ & & " %"#( &' -' " '
""#$, ' / (l1, X i )l1 – ' %& -& &' X i " &' l1 .
6 ., %' ' . &' ""#$ ! ! ' ! "' "" (, '. .
E(X ) ≡ X = 0 2 /
|
D(l X ) = E(l X )2 |
= E(l XX T lT ) = l E(XX T )lT |
= l SlT , |
||||||||
1 |
|
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
||
/ S – & "" ! . ' " ) ""#$ X. |
|
||||||||||
. 0 "& * "/ "6 ϕ(l |
,λ) = l |
SlT − λ(l lT −1) , ' / |
|||||||||
|
∂ϕ |
|
|
|
|
|
1 |
1 |
1 |
1 1 |
|
|
= 2SlT − 2λlT |
= 0, |
|
|
|
|
|
||||
|
∂lT |
|
|
|
|
|
|||||
|
|
1 |
1 |
|
|
|
|
|
|
||
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
(S − λI )lT |
= 0 , |
|
|
|
|
|
|
||||
|
|
|
|
1 |
|
|
|
|
|
|
|
' '+ |
|
lT – |
) ' ""#( |
&' & "" ( . ' #. |
|||||||
|
|
|
|
1 |
|
|
|
|
|
|
|
D(l X ) = l SlT |
= λ , " % ' ! ' /, |
%' )# |
D(l X ) ' / . & . ., |
||||||||
1 |
|
|
1 |
1 |
|
|
|
|
1 |
|
|
" 6" #) '+ . & . +" ) ' "" " % ". . ""#(
) ' ""#( &', ' % *4 ( 1' . " % " * ' "-"
-( / " ( & .-" "'# -' " '.
k-o (k = 2…m) " # ' ! ' & ! " . ""- "' "" ! " (" ! & .) " ! $"#$ -" &, & ' ! " & " k-1 -# 4 . / "#. & .-" "' . $ -%$ " . ""- "' ""#$ " ("#$ & .) " (, " & ""#$ -# 4 . k-1 / "#. & .-" "' . ) ' " "" . " ) ""#$ " ) +5 ( -(.
6" -& '+, %' k-! / " ! & .-" "' ' ! ) ' ""#. &' . & "" ( . ' # ""#$, & ' #( ' ' ' ' k-. - % " ) ' "" . " % " *.
. ' ., %' 5 " % "$6 " ! / "#$ & .-" "' " ! ! ' ! " "'"#. '" ' +" . "# . 5' ) "#$ -" &. 1' . - -. " ". .' ""# " . *' ! ' &, %' )# -" & )# . "# -' .#$ . 5' )$.
m’ / "#$ & .-" "' . 6" " '!" '+ - -' " ' . " '
m’. /& -"!'+, %' .. & ' ' !" ( ' ' % & ""#$ 1' / - -' " ' " ." 6 "" ( " N (% ' % &) ' ' %" (
- , «" )8! " "" (» -. 4+* m’ / "#$ & .-" "', ' '+
N (Dξm'+1 + ... + Dξm ) = N (λm'+1 + ... + λm ) , / λm'+1 ,…,λm – " . "+5 -
% " ) ' ""# " % " !. ' * ' " ' ! -"!'"#. 6" 1& ' . +" ( ' & "" / - -' " ':
+ 1. + -
X1 ,…, X N , m’
&
m’, * -
m’ .
& 6 . 4 1& ' . +"#$ ( ' - -' " ' / "#$ & .-" "'.
. - - $ & . "+5 . % -. ""#$ -. 4+* " (" / -) " !:
m
zij = c jk xik , j = 1…m’, i = 1…N,
k =1
Z = CX,
+ xik – k-! & " ' &' ""#$ Xi, zij – j-! & " ' i-(
' %& ""#$ " & ' . - -' " ' . "+5 ( . " ' Rm’. 6" . ' '+ 1' 0 . # & & -& * ' % & ""#$$" / -' " ' Rm’.
. ' . % "#
N N
M = (Xi − X j )2 ,
i=1 j=1
N N
M (C) = (Zi − Z j )2 .
i=1 j=1
$ .# – .. & ' ' !" ( . 6 . 6"#.
-. )8 &' $" . -' " ' Rm’. . & % ' . # & 6 " ! ..# & ' - -"#$ ' !" ( . 6 ' %& . ""#$
% " M-M(C). 6" -& '+ [4], %'
M − M (L) = min{M − M (C)} = N 2 (λm'+1 + ... + λm ),
C
/ L – . ' , *4 ! -& * ' % & ""#$ - -' " ', " '!" ' " m’ / "#$ & .-" "'. ' * '
+ 2. + m’ ,
*
, ,
m’ )
) )
.
& " , . . & 6 " ! ' !" ( " % & " ' /
. 6 -!.#., "!*4 . . 6"# -# ' % & " % . & " '. ) " % . 
H − H (C)
, /
H = {hij }, hij = (X i , X j ) ,
H (C) = {hij (C)}, hij (C) = (Zi , Z j ) ,
- 
A
– & " . . ' # A. & # ' !, %'
H − H (L) = min H − H (C) = N 2 (λ2m'+1 + ... + λ2m ) .
C
2 '+ -
+ 3. + m’ ,
*
, ,
m’ )
, ) )
, * )
.
& " , '. ' ., %' & "" " % ' ) " « "' "" '» ""#$ " ! ! ' ! -" -+"#.. ""#$ " ' -), ' / . ' % & ( "' ) & ' % & -"
" " %". ' % ! 0 . &$ ( ' 1-3 ) ' 5+ ' ., %' . ' " (" / - -' " ', " '!" ' / " / "# & .-" "'# " . ' '+ " (" ." / ), -' "" " -#$ / "#$ & .-" "'$ - $!4 % ' %& / . ' % & / "'.
2.2.2. ' ""#( / '. "$6 " ! / "#$ & .-" "'
-+ ! 1& ' . +" 3 ( ' 1 - -' " ', " '!" '#$ " / "# & .-" "'#, . 6" -6 '+ ' ""#( / '. "$6 " ! -( / " ( & .-" "'# ( . [41,53]). : . & '+ -!. * -' " ' ""#$, "" * -. ' % & . " " .
y = at + b ,
' & *, %' .. & ' ' !" ( ' ' % & ""#$ 1' ( -!. (
. " . +". 7' .. , " !
N
Q = (X i − ati − b)2
i=1
! ! ' ! & ' ., & ' #( . 6" ." . '+ -. 4+*
*4 ( -' ( - #:
. ! -+"#. &' . a b. ' ! / '.
' ' $ 5 /:
1 1. ""#$ &'$ a b -! ' ! " ) {ti}, i = 1…N:
|
∂Q |
= −2(X i − ati − b)a = −2(X i − b)a − 2a2ti = 0 , |
|||||||||
|
|
||||||||||
|
∂ti |
(X i − b)a |
|
|
|
||||||
ti |
= |
. |
|
|
|||||||
|
|
a2 |
|
|
|
||||||
|
|
|
|
|
|
|
|
|
|
||
1 2. "" . " ) {ti} -!*' ! " # & " '# |
|||||||||||
&' a b: |
|
|
|
|
|||||||
|
∂Q |
|
N |
|
|
|
|||||
|
∂a |
|
= −2 (X i − ati − b)ti = 0 |
||||||||
|
|
|
i=1 |
|
|
, |
|||||
|
∂Q |
|
N |
|
|
||||||
|
|
|
|
|
|||||||
= −2 (X i − ati − b) = 0 |
|||||||||||
|
∂b |
|
|||||||||
|
|
i=1 |
|
|
|
||||||
|
N |
|
|
+ b |
N |
|
N |
|
|||
|
a t 2 |
t |
i |
= X t |
i |
||||||
|
|
|
|
i |
|
|
i |
||||
|
i=1 |
|
|
i=1 |
|
i=1 |
, |
||||
|
|
|
|
||||||||
|
N |
|
|
+ bN = |
N |
|
|||||
|
a ti |
X i |
|
||||||||
|
i=1 |
|
|
|
|
i=1 |
|
||||
%' ' m ' . " ("#$ " " ( 2x2 ! -" ! $ & .-" "' &' a b.
. / '. ' " ' !, & / |
Q |
< ε , / |
|
Q |
|||
|
|
Q – . " " % "# Q ' *, ε – . ! % ".
. 4 ' ' & / -) "$6 " ! -( / " ( & .-" "'# ' ' ' ., %' " /& ) )4 ' ! " % (, & / " & ' # ""# 6 ' "-"# " % " !. -' -' –
' ' ' *4 ( .. ' % ' ! " '" " % ", ' ' &
/ . - -& ' !. 2 /, "-"#$ ""#$ " ', ' b '
|
b = |
1 |
N |
&' " / " % " ! $ & " ': |
|
X i , " % – " & ' #( |
|
|
|||
|
|
N i=1 |
|
«100 &' "#(» &' " /. &' a % -"#$ ""#$ '
"-" -( / " ( & .-" "'#, % "-"#$ – «100 &' " *» -* / " * & .-" "'.
! ' /, %' )# " (' ' * / " * & .-" "', -'-*'
*4 . ) .:
1. % '# ' ! ." 6 ' &' -#$ |
' '& X ′: |
X i′ = X i − ati − b . 7' ." 6 ' 6 ' -' " ', |
' / " +" . |
-( / " ( & .-" "', . " '+* " " . "+5
. " ' $" / -' " ' ""#$.
2. ! " / ." 6 ' &' % '# ' ! -! / " ! & .-" "'. " ) ' ' ( / " ( & .-" "' ( $" / " ) ""#$.
! "$6 " ! ' '+ ( / " ( & .-" "'# 4 ' ! ." 6 ' ' #$ ' '& ! " / -! ' ! -! / " ! & .-" "', '. .
2.2.3. " (" / 0 &' " / "
-." ., %' . ' / "#$ & .-" "' . 6 ' )#'+ 0 . " & & % -' . 0 "& " J & % ' « $" " !» "0 . - "" . ' ) 6 " F $" / -' " '
-' " ' . "+5 ( . " '. . ' / "#$ & .-" "' & % ' 0 "& " J # '-' ! «)8! " "" (» -. 4+* " #$ & " ' - .
. 0 &' " / |
" |
& 6 . |
&' |
""#$ Xi |
||
-' ! ' ! " ) m’ " % " ( 0 &' yi1 ,…, yim' : |
|
|||||
|
|
m' |
|
|
|
|
xij − μ j = q jk yik + u j , j = 1…m’, |
|
|
||||
|
|
k=1 |
|
|
|
|
X i − |
|
= QYi + U , |
|
|
|
|
X |
|
|
|
|
||
/ μ j – " " % " j-/ -" &, |
xij – " % " j-/ -" & ! i- |
|||||
/ )8 &', qij – «" / &» |
0 &', uj – |
' ' %" ! |
% (" ! |
|||
& .-" "'. 1' . #-"!*' ! !: |
|
|
||||
Eyk = 0, Euk = 0 , Dyk = 1,
y1i ,…, yim' , u1 ,…,um' - -" " & "#.
& % ' F #) ' ! " (" -) " & " ' ' &, %' )# #-"! + 1' !, ' / . & . . 0 "& "
J (F ) = 1− |
|
|
|
R − R |
ˆ |
|
|
|
2 |
, |
|
|
|
|
|||||||
|
|
|
|
X |
|
|
|
|
|
|
|
|
|
|
X |
|
|
|
|
|
/ R |
X |
– & ! "" ! . ' $"#$ -" &, |
R ˆ |
– |
|
|
X |
||
& ! "" ! . ' «-& (» -' " ' 0 &' |
ˆ |
|
||
X i = QYi , |
||||
A
– & " . . ' # A.
! " . #& &. ' " / & Q . . m’×m
4 ' ! ' " (" ' ) 6 " -' " ' 0 &' (. " ' m’) $" -' " ' (. " ' m). +' '
-% ' ! ." 6 ' ""#$ Xˆ i = QYi , & ' -' ." 6 ' .
$"#$ ""#$ ' %" '+* % (" ( & .-" "'# U. 7'
." 6 ' 6 ' " & ' . - -' " ' . " ' m’, " '!" ' . " m’ ' ) . ' # Q. ' Q $"#( " )
0 &' y1 |
,…, ym' , i = |
1…N #) *' ! |
' & . ) ., |
%' )# |
||
i |
|
i |
|
ˆ |
|
|
& ! "" ! |
. ' |
" ) ""#$ |
' %" |
|||
X . & . +" |
||||||
- & ! "" * . ' $" / " ) ""#$. 2 & . ) ., & ' . & % ' « $" "" (» "0 .
! ! ' ! + )8! " " " - -" &, $ ." (
& "" '.
<' & ' ! «5 . (» & .-" "'# U, ' )#%" -/ *', %' " " ' ' -" ! ""#$ -% " " m-. " . " . +" .
-" * " #. " . " % ".. 2 / & "" !
. ' -" ! U . ' / " +"#( :
V = E(UU T ) , V = diag(v11...vmm ) , vii = Dui .
$" -" ""#$ - -/ ' !
"' ""#., ' / & "" ! . '
S = QQT +V
/ & " # *' - . ' (Q,V),
' !*4 * 1' . *. % ", %' " ' & 5 " 4 ' ', ' " . "" 5 " . ! ! ' ! (QC,V), / C – -+" ' / " +" -) " (-' &'-' )
. ' # Q). 1' ( / . -% " . 5 " % 0 &' " / " ! ! ' ! " " " %"#., -1' . ! 5 " ! " )$.
#) '+ & &-) -" ' +"# - -6 " ! ( '$ . ' # Q. . ' 1'$ - -6 " ( ' ""#( . ' 5 " !
% #/ ! ' *4 . )..
" % ' ! " -) 6 " . ' # V=V(0).
=/ 1. % . " -) 6 " . ' # Ψ = QQT, '. . Ψ(0) = S
–V(0).
=/ 2. 3 -. 4+* Ψ(0) -! . " -) 6 " . ' # Q. / '. -6 ' ! -% " ! " )$. ( ' %" '.
-" ' +"# - -6 " ! ' &' . ' # Q-+ *' ! ! = / 2 / '.. . ' . !:
1) QTQ – / " +" ! . ' , -% . / " +"# 1 . "'#
%"# -! % "# -! & )# " !. 2 /
A = QT Q = diag(λ1...λm′ ) , λ1 > λ2 > .. > λm , ΨQ = QQT Q = AQ ,
' '+ m’ ' ) q1…qm’ . ' # Q ' !*' " " !.
Ψqi − λi qi = 0 " ) ' ""# " % " ! . ' # Ψ. 3 ) ' ""# &'
. ' # Ψ, ' % *4 -#. m’ - % " ) ' ""#. " % " !.
' !' % " -) 6 " . ' # " / & Q.
2) QTVQ – / " +" ! . ' , -% . / " +"# 1 . "'#
%"# -! % "# -! & )# " !. 2 /
A = QTVQ = diag(λ ...λ |
m′ |
) , |
λ > λ |
2 |
> .. > λ |
m |
, |
|
|
1 |
|
1 |
|
|
|
|
|||
ΨQ = V −1QQTVQ = AV −1Q , |
|
|
|
|
|
|
|||
' '+ m’ ' ) q1…qm’ |
. ' # Q |
' !*' |
" " !. |
||||||
Ψqi − λivii−1qi = 0 " ) )4 ""# |
) ' ""# " % " ! |
. ' # Ψ. |
|||||||
) )4 ""# ) ' ""# &' . ' # Ψ, ' % *4 -#. m’ -
% " ) )4 ""#. ) ' ""#. " % " !. ' !*' % " -) 6 " . ' # " / & Q.
. " & ' # ' /. "" . " ) ""#$, -+ ! . ' / "#$ & .-" "' . ' # 0 &' " / ", . 6" -' '+ " ("# . ""#$. 9 &' % & 1' . ' # ' !' - +"
" (" ." / ) ."+5 ( . " ', " & ' - *' !
$"# ""#. 7' - -' " ' & # ' ! " & ' . .#-' . +"#. $ /$ " ("#$ ." / ) ( ' ( 6
. " '. % . ' / "#$ & .-" "' -' . +" '+ & *% ' ! ' ., %' -& ""#$ . & . +" -!'
-* $"#$ ""#$. % . ' 0 &' " / " " % " ! -" & -& ( . & . +" - $6 " $"# " % " ! -" & .# ." ( & ! . 3 ' . ' '+, %' %, & / ' ' %"# - (.."# ' !" ! -' "" / - -' " ') " &, ) . ' *' $"# +' '# (1' ' " ' ! ) "" -"!'", . ' '+ 1 " ' &'
. ' # Q).
2.3. ! " &
&
-' " ', " '!" ' " m’ / "#$ & .-" "' ) '
( ' . «. " . . ' ' %" ( - » – " ( & '
' !" ! ' ' % & ""#$ 1' / - -' " ' . " . "
$ /$ " ("#$ - -' " ' . " ' m’.
, 6 ' ! - -&' " ( ! -' " ! «/ "#$ - $" ' (» – " " ("#$ ." / ) (, ) *4$ ' . 6 -' . +"#. ( '..
' ' . 6" ' ' '+ ) '# [58,66,67], & ' #$ ' !' ! ' & - $" ' -+ " ., "-., . ' .& . +" / - -) !.
-. ' % & ( . ' & " ' ' % &
." / ) ! " ' " (" ' -"#$ ) 6 " ( ' &' ""#$), ' % -' " ! " / " % "" / " " (" /
." / ) !, ' '+ #% " & " ' & 6 ( / ' % &, ! ! ' ! +. ' .& ( ' %& " ! #% " (.
# . / " % ) . '+ -', " & '. .*4
." / ) ) ' - -/ '+ ! / " % ""#. ' ! & " %" . % $ ' % &. / . ., .# ) . ' '+
.
-' ., %' .# . . . 4 "" * -' " ' ""#$
, " & ' * ) ' " '!/ '+ ! & . ." / ). ! 1' /
' ! -"" ! - . 6 ..
3 . ' ., %' " ! "$ ' +& -6 " (
-' " ' ""#$ " ' ' %" ! ' " " ! ." / ) !. #
6"#, & . ' /, ) '+ "0 . ( ' ., & & # ! !*' ! " " . . 2 '+ " & " %". ." 6 ' 6"# )#'+
"# & . ! / . / " 6 / '. SOM (Self-Organizing Maps – . / " *4 ! & '# ,$" ") " )$.
" '+ " ' +& & & # ! !*' ! " ., " ' &6 & & ! !*' ! !., !. '. .
. ' ., "-. . " ." / ). : . % ' '+, %'
# ) *' -!. / +" * / & / " +" * '& ( .. .17). 2 / '" 5 " ! ' -!*' ! ' ' ""#. ) . – . 6" % ' '+, %' & 6 #( (& . & ("$) " -!. / +" ( '& . ' % '# -#$, .+ ' #$ '. . (, " / & / " +" ( – 5 '+ -#$, " '+ ' #$ '. . (. ." / ) ! ! ' ! '$. "#., '& -!. / +" (, ' & 6 #( . ',
' ' ' "", 5 '+ -#$ (, '. .
. ' . / '. -' " ! '& – / '.
* , (SOM) / . 0 & , ' &6 / '. -' " ! .
2.4. 'SOM |
( |
3 . / " *4 ! & '# |
,$" " (SOM) – 1' |
. 0 ""#( / '. ,
' '+ -' " ! N ' % & ""#$ -. 4+* . "+5 / % ' % &- ) (samples). , 6 #( ) -' ! ' . "! ' ) ( & +" / 4 " ""#$. +' ' ' & ( . "# ""# -' !*' ! -"" ( 5 )& ( --& . –
" & ' %" / ' !" ! ' ' %& ""#$ ) 6 (5 / & " ( ) :
|
|
1 |
N |
|
MSE = |
|
(X i − yBMU (X i ))2 , |
||
|
||||
|
|
N i=1 |
||
/ yBMU (X i ) – ) 6 (5 ( & ' %& ""#$ Xi ) .
«*. "& (» . ' SOM & + ', %' -% . ! - / -. " " ' . () ) & # ' ! -""#. ) .-! % ". # . / ' )#'+ -' "# -!. / +" (
/ & / " +" ( '&. 3 " " 1' ( '& # +' ' ( ' ! / '. SOM & # *' ! " . -' " ' ""#$, %' '
- . 4 " ! ' % & ""#$ - ) 6 (5 . . % . " (
'$. " ( '& . 6" '+ '+ ""#.-5 . $"#( "' / '. SOM.
1.3 '& " ' ! – . 4 ' ! -' " '
""#$. ' (5 . "' . ! ! ' ! % (" -6 " ,
/ ( "' – . 4 " '& -' " ', " '!" ' . " / "# & .-" "'#. 3 4 ' *' /, ) . " 100 &' "# $.# " ( .., "-., [70]).
2.#) ' ! % ("#. ) . - -! & ' %& ""#$ Xi.
3.3 $ '& #) ' ! ) 6 (5 ( & ' %& Xi. ) " % . / &' % yBMU (BMU – Best Matching Unit).
4.# '& / *' ! - "-" * & Xi - - :
( y j )′ = y j + h(r( y j , yBMU ),t)(X i − y j ) , j = 1…p,