Обработка данных / ElemTreat
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§5.1. $
*(6(4! 72' !3 (+,& 0 8!*!2! $. 6!# & # +,2 !*3 &! - '# "&&-. $ +!4!&, & ! !''+ 6 $(&! *($!'!3 '# 0 3 961 (*+!4&-3! 8!*!4 '2!3! .( (2# !'#!2(3!.
+,%!&'#$ *(2 & $ 8!*!2!, $ # 3 4!'+ 81&6(3 (+,&-., 8 - 31+! 1 #'" $ $!6 1 ($& &!0, '$"*-$(:;!. 3 961 ' ) 0 (*+!4&- 8!*!4 '2! $ +!4!&-.
+" !''+ 6 $(&!" $*(!3 '$"*! 8!*!4 '2!. $ +!4!& !' +,*1:#'" 2(2 72' !3 (+,&- , #(2 ! # #!4 '2! 3 # 6-. 6 $ 6 &!-3 +:)-. 72' !3 $ )"*(# +,& 6 +9 & $ 6!#,'" # #!4 - '2!0 (&(+!* '#($+ && 0 )+ 3-.
6(&& 0 /+($ (''3(# !$(:#'" '!#1(5!!, 2 /6(, ' /+('& # !!, 6$ 8!*!4 '2! $ +!4!&- X ! Y '$"*(&- & 2 # 0 81&25! &(+,& 0 *($!'!3 '#,:. +" 6 + && '#! + 9!3, 4# $ +!4!&( Y "$+" #'" & 2 # 0 81&25! 0 F 6 1/ 0 $ +!4!&- X:
Y = F(X) . |
(5.1) |
!*!4 '2(" # !" *$ +" # '6 +(#, 6 + &&- 1#$ 96 &!"
.( (2# *($!'!3 '#! $ +!4!& X ! Y. 6&(2 6+" # / , 4# )- +14!#, 2 &2 #&-0 $!6 81&25!! F(X), 2(2 ($!+ , & ). 6!3 $ '#! !*- 3 &!" *&(4 &!0 *($!'!3 0 3 && 0 Y ! (*+!4&-. *&(4 &!".
( /13 ( X.
( 3&!3, 4# *1+,#(#- -#( $' /6( ' 6 9(# '+14(0&- - / %& '#! ('3. /+($1 2). +" 2(96 / 6 + && / *&(4 &!" ( /13 &-
#( xk 72' !3 6( # $ 2(4 '#$ *&(4 &!" 81&25!! $ +!4!&1 yk =
= F(xk) + k , /6 k — / %& '#, !*3 &!" yk .
2(4 '#$ !3 ( &( !'. 1 2(*(& $ *3 9& #2+ & &! - *1+,#(#( 72' !3 ( # # #!4 '2 / *&(4 &!" 6+" # / 9 ( /13 &-
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#(. (3 #!3, 4# $ +!4!&- k 3 /1# )-#,, $ ); / $ ", 2(2 + 9!-
# +,&-3!, #(2 ! # !5(# +,&-3!.
* /+($- 2 '+ 61 #, 4# 3& / 2 (#& $# &! !*3 &!" *&(4 - &!" 81&25!! ! 8!2'! $(&& 3 *&(4 &!! ( /13 ( xk ' '+ 61:- ;!3 1' 6& &! 3 *1+,#(# $ & *$ +" # 13 &,%!#, / %& '#, $ +!4!&- F(xk) 6 &1+". 6& 2$(6 (#!4&(" / %& '#,, ' /+('& 8 31+ (2.4), '#(& # &1+ 3 # +,2 ! ) '2 & 4& 3 4!'+ $# &!0-#( $ & !*3 &&-. 1'+ $!"..
Y
yk |
Y = F(X) |
k
y = F(xk)
X
xk
!'. 1. !3 #2+ & &!" k 72' !3 (+,& / *1+,#(#( yk# *&(4 &!", (''4!#(&& / # #!4 '2 0 *($!'!3 '#! F(xk)
+ 61 # 3&!#,, 4# '+! +& '#,: #'1#'#$1 # # #!4 '2(" !&8 3(5!" 81&25! &(+,& 0 *($!'!3 '#! (5.1), # 72' !3 (+,- &-. !''+ 6 $(&!0 & 6 '#(# 4& 6+" +14 &!" 16 $+ #$ !# +,& /
!)+!9 &!" $!6( 81&25!! F(X). 7# 31 6 $ 6 &! 3 72' -
!3 ( !''+ 61 3(" *($!'!3 '#, $ +!4!& X ! Y &( '& $ # #!4 - '2!. ' ) (9 &!0 !'-$( #'" & 2 # 0 6 + && 0 81&25! 0
Y = F(X, A, B, C,...). |
(5.2) |
!6 7# 0 81&25!! *(6(&, & &( ' 6 9!# 6!& !+! & '2 +,2 |
'# - |
"&&-. ( (3 # $ A, B, C,..., *&(4 &!" 2 # -. & !*$ '#&- 6 |
$ - |
6 &!" 72' !3 $. |
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&(4(+( (''3 # !3 '+14(!, 2 /6( $' !*3 &!" *&(4 &!0 81&25!! Y $ 6"#'" ! (*+!4&-. *&(4 &!". ( /13 ( X. '+! !*3 &!"
$ 6!+!', N (*, # *1+,#(# 3 72' !3 ( "$+":#'" N ( 4!- ' + — *&(4 &!0 !*14( 3-. $ +!4!&:
xk , yk k = 1, ... , N. |
(5.3) |
( / (8!2 *($!'!3 '#! $ +!4!&- Y # X 2 6!&(#- # 4 2 (5.3)(' +(/(:#'" $ ',3( & /1+" & !*-*( $+!"&!" '+14(0&-. / %& - '# 0, $ *&!2%!. $ . 6 !*3 &!0. (2#!4 '2! &!2 /6( & 16( #'" -
6 ) (#, 4!'+ &&- *&(4 &!" ( (3 # $ A, B, C, … #(2, 4# )- 81&2- 5!" *(6(&& / $!6( (5.2) 16 $+ #$ "+( $' 3 72' !3 (+,&-3 - *1+,#(#(3 (5.3). &(4 / $ ", &!2(2!3 &() 3 ( (3 # $ A, B,
C, … & +,*" 6 )!#,'" # / , 4# )- / (8!2 81&25!! (5.2) . 6!+ 4 -* $' 72' !3 (+,&- # 42! (5.3). 7# 31 & !*) 9&-3 7#( 3() #- 8!*!2(-!''+ 6 $(# +" "$+" #'" !'2 & 2!. 6 + &&-. *&(- 4 &!0 ( (3 # $ A, B, C, ... , ! 2 # -. 6 '#!/( #'" &(!+14% ' /+('! 3 961 72' !3 (+,&-3! *1+,#(#(3! (5.3) ! # #!4 - '2 0 *($!'!3 '#,: (5.2). #( 3(# 3(#!4 '2(" 5 61 ( &(*-$( #'" -
($ *1+,#(# $ 72' !3 ( 81&25! 0 6 + && / #!- (. (06 &&(" 81&25!" &(!+14% / !)+!9 &!" !* *(6(&& / 2+(''(
(5.2) &(*-$( #'" ( %"0.
!&5! !(+,&(" # 16& '#, *(2+:4( #'" $ $-) 2 !# !" 2(4 '#$(+14 && / !)+!9 &!" 81&25!! (5.2). *3 9&- (*+!4&- 3& &!"$ 61 # / , 2(2 !)+!9 &! +(/(#, «&(!+14%!3». 6&!3 !* 7+ 3 ( &-. ' ' ) $ ('4 #( & !*$ '#&-. ( (3 # $ 81&25!! &('& $ *1+,#(# $ -# $ "$+" #'" ! * (.
# 6 *(2+:4( #'" $ # 3, 4# $ 2(4 '#$ *&(4 &!0 !'2 3-. ( (3 #-$ , , C,... $-)! (:#'" $ +!4!&-, 3!&!3!*! 1:;! '1331 2$(6 (-
# $ (*& '# 0 !*3 &&-. &( -# yk ! ' #$ #'#$1:;!. $-4!'+ &&-.
*&(4 &!0 F(xk , A, B, C,...):
N |
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Q = ∑[F (xk , A, B, C,...) − yk ]2 . |
(5.4) |
k =1
+!4!&( Q 6'#($+" # ' ) 0 6 + &&1: 3 1 #2+ & &!" 72' !3 (+,&-. *1+,#(# $ # (''4!#(&&-. # #!4 '2!. (-
43
$ 0 4('#! (5.4) '133! $(&! !'. 6!# $' 3 k = 1, . . . , N. (2
2(2 $ (5.4) 6'#($+":#'" 6 + &&- 4!'+( xk , yk , +14 &&- $$ 6 && 0 ' !! !*3 &!0, $ +!4!&1 Q 3 9& (''3(# !$(#, 2(2 81&25!: ( /13 $ , , C ! #.6.
(+ *(6(4( '$ 6!#'" 2 &(. 96 &!: 4!'+ &&-. *&(4 &!0 ( (3 #-$ , , C,..., ! 2 # -. 81&25!" Q(A, B, C, …) 6 '#!/( # '$ / 3!&!313(. * 3(# 3(#!4 '2 / (&(+!*( !*$ '#& , 4# & ). 6!3-3 1'+ $! 3 72'# 313( 81&25!! $ 6 + && 0 # 42 "$+" #'" ($ &'#$ &1+: $ 7# 0 # 42 $-. 4('#&-. !*$ 6&-. $' 3 ( /13 (3.! ($&!$(" +14 &&- $- (9 &!" 4('#&-. !*$ 6&-. &1+::
∂Q / ∂A = 0, ∂Q / ∂B = 0, ∂Q / ∂C = 0,… (5.5)
3- +14!3 '!'# 31 1 ($& &!0 6+" & !*$ '#&-. *&(4 &!0 ( (3 # $, , C ! #.6. +!4 '#$ 1 ($& &!0 ($& 2 +!4 '#$1 & !*$ '#&-..6!&'#$ && '#, ! '# #( % &!" '!'# 3- '1; '#$ && *($!'"# # $!6( 81&25!! (5.2). 6&(2 6+" $' . !3 &" 3-. &( (2#!2 ( 2- '!3! 1:;!. 81&25!0 '1; '#$1 # % &! , ' #$ #'#$1:; !3 && 3!&!3131 '133- (5.4). , 4# *&(4 &!" ( (3 # $ , , C,... , &(0- 6 &&- % &! 3 '!'# 3- 1 ($& &!0 (5.5), ) ' 4!$(:# !3 && 3!- &!313 $ +!4!&- Q, )16 # 2(*(& &!9 6+" 2 &2 #& / $!6( ( 2- '!3! 1:;!. 81&25!0.
% &! 3 '!'# 3- (5.5) "$+":#'" 6 + &&- 4!'+ $- *&(4 &!"
~ ~ ~
( (3 # $, 2 # - 3- ) *&(4!3 A, B,C ! #.6. 6'#(& $2( 7#!. 4!-
' + $ 8 31+1 (5.2) 6( # &(3 6 + &&-0 $!6 ( 2'!3! 1:; 0 81&25!!
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(5.6) |
Y |
= F (X , A, B,C |
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,, 2 /6( & !*$ '#&- ( (3 # - A, B, C,... 81&25!! (5.2) *(- |
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~ ~ ~ |
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3 & &- &( !*$ '#&- 4!'+( A, B,C …, 3 9& 6+" +:) / *&(4 &!" ( -
/13 ( X $-4!'+!#, *&(4 &! 81&25!! F 8 31+ (5.6).
# / / $ ", ( 2'!3! 1:;(" 81&25!" (5.6) "$+" #'" & 2 # --3 !)+!9 &! 3 # #!4 '2 0 *($!'!3 '#! (5.2). &(4 &!" 81&25!!
~ ~ ~
(5.6) *($!'"# # ( (3 # $ A, B,C , ... , 2 # - , $ '$ : 4 6,, $-- 4!'+"+!', ' 3 ;,: 4!' + xk , yk , ' 6 9(;!. '+14(0&- / %& -
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'#!, ! 7# 31 #(29 & !*) 9& ' 6 9(# & 2 # 1: / %& '#,.+ 6 $(# +,& , 8!*!2-72' !3 (# 6 +9 & (''4!#-$(#, / %- & '#, *1+,#(# $, +14 &&-. ' 3 ;,: ( 2'!3! 1:; 0 81&2-
5!! (5.6).
'+ 61:;!. ( (/ (8(. 6(&& 0 /+($- (''3(# !$( #'" !' +,* - $(&! 3 # 6( &(!3 &,%!. 2$(6 (# $ &( !3 (. &(!) + (' '# (- & &&-. $!6 $ ( 2'!3! 1:;!. 81&25!0.
§ 5.2. # ($#
(; $' / 3 # 6 &(!3 &,%!. 2$(6 (# $ !3 &" #'" 6+" !)+!- 9 &!" !''+ 61 3 0 *($!'!3 '#! +!& 0& 0 81&25! 0 !+! 81&25! 0, '$ 6"; 0'" 2 +!& 0& 0 ' 3 ;,: 7+ 3 ( &-. ) (* $(&!0. (- 3 &( !'#!&& 0 81&25!! +!& 0&-3 !)+!9 &! 3 &(*-$( #'" -
($ . =! 2 !3 & &! +!& 0& 0 ( 2'!3(5!! $ 2(4 '#$ $ / !)+!9 &!" !''+ 61 3 0 *($!'!3 '#! '& $(& &(
# 3, 4# +:)1: /+(621: 81&25!: ! & ). 6!3 '#! 3 9& (*+ 9!#, $ "6 0+ ( 6 +!& 0& / 4+ &(, !4 3 7# (*+ 9 &! !3 # . - %1: # 4& '#, $ & 2 # 3 ! $(+ *&(4 &!0 ( /13 (.
1'#, !* # #!4 '2!. ' ) (9 &!0 '+ 61 #, 4# *($!'!3 '#, 3 9- 61 $ +!4!&(3! X ! Y 3 9& +(/(#, +!& 0& 0 (. #" )- &( 6 + &-
& 3 ! $(+ *&(4 &!0 $ +!4!&- X). /6( 81&25!" Y(X) *( !'-$( #- '" $ '+ 61:; 3 $!6
Y = β X + γ, |
(5.7) |
/6 β ! γ — & 2 # - '# "&&- 2 788!5! -.
(' ! '1 # '+14(0, 2 /6( # #!4 '2! & $ *3 9& $-4!'+!#, *&(4 &!" β ! γ ' ! 3+ 3 0 / %& '#,:. /6( '+ 61 # &(. 6!#, β
! γ, +,*1"', 72' !3 (+,&-3! *1+,#(#(3! #! ( (5.3), !3 &"" $-% 1 3"&1#-0 3 # 6 &(!3 &,%!. 2$(6 (# $.
+" 7# / '&(4(+( $ 81&25!! (5.7) *(3 &!3 & !*$ '#&- '# "&&-
β ! γ 3 &&-3! b ! g ' #$ #'#$ && . |
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133( 2$(6 (# $ (5.4) 6+" +!& 0& 0 81&25!! (5.7) |
! ) # # $!6: |
N |
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Q = ∑(bxk + g − yk )2 . |
(5.8) |
k =1 |
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6(&& 3 '+14( '133( (5.8) '#, 81&25!" 6$1. ( /13 $ b ! g,
#.2. $ $ 6 && 0 ' !! !* N !*3 &!0 $' xk ! yk 6'#($+":# ' ) 0
6 + &&- (!*$ '#&- 72' !3 (# 1) 4!'+(. , '+ 61 # $( ,! $(#, 3 &&- b ! g 6 # . , 2( $ +!4!&( (5.8) & 6 '- #!/& # '$ / 3!&!313(. /6( $ 2(4 '#$ '# "&&-. 2 788!5! $ β ! γ +!& 0& 0 *($!'!3 '#! (5.7) 3- $ *,3 3 &(06 &&- *&(4 &!" -3 &&-. b ! g, 2 # - 3!&!3!*! 1:# '1331 (5.8) ! *(6(&&-. 4!'- +(. xk ! yk .
#-'2(&! 3!&!313( 81&25!! 6$1. 3 &&-. Q(b,g) "$+" #'" #! !4& 0 *(6(4 0 3(# 3(#!4 '2 / (&(+!*(. # 42 72'# 313( $- 4('#&- !*$ 6&- # Q, $*"#- b ! g, ) (;(:#'" $ &1+,. # 1'+ $! 6( # &(3 '!'# 31 6$1. 1 ($& &!0 6+" 6 + &!" 6$1. & !*- $ '#&-. b ! g:
∂Q = 0 ; |
∂Q = 0 . |
(5.9) |
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∂b |
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∂g |
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!88 &5! $(&! ($ 0 4('#! (5.8) ! (+/ ) (!4 '2! |
) (* - |
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$(&!" !$ 6"# '!'# 31 (5.9) 2 '#(&6( #& 0 8 3 : |
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b∑xk2 + g∑xk = ∑xk yk , |
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k =1 |
k =1 |
k =1 |
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N |
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b∑xk |
+ gN = ∑ yk . |
(5.10) |
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k =1 |
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k =1 |
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# )-4&(" '!'# 3( 6$1. +!& 0&-. (+/ ) (!4 '2!. 1 ($& &!0 6+" 6$1. & !*$ '#&-. b ! g. 6!&'#$ && % &! 3 9& *( !'(#, $ '+ 61:; 3 $!6 :
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N ∑xk yk − |
∑xk ∑ yk |
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b = |
k =1 |
k =1 |
k =1 |
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N ∑xk2 − ∑xk |
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∑xk2 |
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∑ yk |
− ∑xk |
∑xk yk |
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g = |
k =1 |
k =1 |
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k =1 |
k =1 |
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(5.11) |
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N ∑xk2 − |
∑xk |
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k =1 |
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* !'. 1 4 $!6& , 4# |
! 16(+ &!! ( |
2'!3! 1:; 0 81&25!! |
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# 72' !3 (+,&-. # 4 2 '133( (5.4) !+! (5.8) )16 # & / (&!4 && $ * ('#(#,. + 6 $(# +,& , &(06 &&-0 72'# 313 81&25!! Q(b,g) "$- +" #'" !3 && 3!&!313 3. +" '# / / 3(# 3(#!4 '2 / 6 2(*(# +,- '#$( 7# / 1#$ 96 &!" # )1 #'" $-4!'+!#, *&(2 2$(6 (#!4& 0 8 3-
2 |
Q |
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∂ |
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∂ |
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∂ |
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∂b |
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− |
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# 16& 1) 6!#,'", 4# 6+" 81&25!! Q(b,g) $!6( (5.8) 6(&&(" 2$(6 (#!4&(" 8 3( $' /6( + 9!# +,&(, 4# "$+" #'" 6 '#(# 4&-3 1'+ $! 3 3!&!313( 7# 0 81&25!! ! *&(4 &!". ( (3 # $, 6 - +" 3-. 8 31+(3! (5.11).
(2!3 ) (* 3, 6+" +14 &!" &(!+14% 0 ($ '3-'+ 3 # 6( &(!- 3 &,%!. 2$(6 (# $) !)+!9 && 0 +!& 0& 0 81&25!! (5.7) '+ 61 #
)(*! $(#,'" &( *1+,#(#(. 72' !3 $ xk , yk $ $ 6 && 0 ' !! !* N !*3 &!0. 8 31+(3 (5.11) & ). 6!3 $-4!'+!#, *&(4 &!" $ +!4!& b ! g, *(# 3 6'#($!#, !. $ 81&25!: (5.7) $3 '# '# "&&-.
& !*$ '#&-. ( (3 # $ β ! γ ' #$ #'#$ && . 788!5! -, $-4!'- +" 3- 8 31+(3 (5.11), +(/(:#'" &(!+14%!3! !)+!9 &&-3!
*&(4 &!"3! ( 5 &2(3!) ( (3 # $ β ! γ +!& 0& 0 81&25!! (5.7).+14 &&(" !)+!9 &&(" 81&25!"
~ |
(X ) = bX |
+ g |
(5.12) |
Y |
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3 9 # !' +,* $(#,'" 6+" $-4!'+ &!0 *&(4 &!0 Y(X) |
! !*$ +,- |
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& 3 *&(4 &!! ( /13 ( X.
& 2 # -. 8!*!4 '2!. *(6(4(. # #!4 '2! 6 2(*-$( #'", 4# '$ ) 6&-0 4+ & +!& 0& 0 81&25!! (5.7) '$ 31 '3-'+1 6 +9 & ($-
47
&"#,'" &1+: γ = 0. # *&(4!#, 4# 3- !3 3 4('#&-0 '+14(0 +!& 0& 0 *($!'!3 '#!, 2 # 1: !& /6( &(*-$(:# "3 5! &(+,& 0:
Y = β0 X . |
(5.13) |
1&25!" (5.13) ' 6 9!# +!%, 6!& & !*$ '#&-0 |
( (3 # β0 — |
1/+ $ 0 2 788!5! . !'+ && *&(4 &! ( (3 # ( β0 # !" & 6( #, ! & 6 +9 & )-#, $-4!'+ & ' 3 ;,: 72' !3 (+,&-. *1+,#(-
# $.
'+! $ #(2 3 '+14( *1+,#(#- !*3 &!0 xk, yk 8 3(+,& 6'#(-
$!#, $ 8 31+- (5.11), # $ +!4!&( g (2#!4 '2! $' /6( )16 # #+!- 4(#,'" # &1+" !*-*( !'1#'#$!" & 1'# (&!3-. '+14(0&-. / %& -
'# 0. 6&(2 +14 && *&(4 &! g ≠ 0 & )16 # !3 #, 8!*!4 '2 / '3-'+(.
+" $-4!'+ &!" 1/+ $ / 2 788!5! ( β0 81&25!! (5.13) '+ 61 # $& $, !3 &!#, 3 # 6 &(!3 &,%!. 2$(6 (# $. !*$ '#&(" '# "&&("
β0 $ 81&25!! (5.13) *(3 &" #'" 3 && 0 b0. +" +!& 0& 0 *($!'!3 - '#! $!6( (5.13) '133( Q "$+" #'" 81&25! 0 6!&'#$ && / ( /13 (
b0:
N |
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Q = ∑(b0 xk − yk )2 . |
(5.14) |
k =1 |
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!*$ 6&1: 81&25!! Q(b0) ( (3 # 1 b0 |
! ($&!$( 3 &1+:. |
N |
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2 ∑[b0 xk − yk ]xk = 0 . |
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k =1 |
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* +14 && / 6!&'#$ && / 1 ($& &!" $- (*!3 $ +!4!&1 b0 4 -* *1+,#(#- 72' !3 ( xk , yk :
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∑xk yk |
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b0 = |
k =1 |
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(5.15) |
N |
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∑xk2 |
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k =1
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/2 $!6 #,, 4# $# (" !*$ 6&(" 81&25!! Q(b0) b0 $' /6(+ 9!# +,&(, 7# 31 &(06 &&(" $ +!4!&( (5.15) 6 +" # 3!&!313
81&25!! Q(b0).
31+( (5.15) 6( # ' ' ) $-4!'+ &!" &(!+14% / !)+!9 &!"
1/+ $ / 2 788!5! ( β0 "3 5! &(+,& 0 *($!'!3 '#! (5.13) &( '& $ $ 6 &&-. !*3 &!0 xk , yk (k = 1,..., N).
§ 5.3. ! & & #
($
$ 0 *(6(4 0 3 # 6( &(!3 &,%!. 2$(6 (# $ 6+" 81&25!0 $!6( (5.7) "$+" #'" $-4!'+ &! ( (3 # $ 81&25!! — 1/+ $ / 2 788!5!-( ! '$ ) 6& / 4+ &(. ! !' +,* $(&!! 8 31+ (5.11) '+ 61 # - &!3(#,, 4# 6+" 2(96 0 2 &2 #& 0 ' !! $ 6 &&-. !*3 &!0 $' $ +!4!&- xk ! yk (k = 1,…, N) — 6 + &&- 4!'+(. + 6 $(# +,& , $ +!4!&- b ! g, $-4!'+ &&- 8 31+(3 (5.11), #(29 "$+":#'" -6 + &&-3! 4!'+(3!. , $ ); / $ ", *1+,#(#- !*3 &!0 6- '#($+":# ' ) 0 '+14(0&- $ +!4!&-, 4# 19 )'196(+ ', $ /+($ 2.
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