Обработка данных / ElemTreat
.pdf!6& , 4# $ +!4!&- Sb ! Sg 1)-$(:# ' '# 3 2 +!4 '#$( !*3 - &!0 N.
+1%! !&( 6 $ !# +,& / ! $(+( 6+" 2(96 / ( (3 # ( β ! γ6'#($+" #'" 2(2 !*$ 6 &! ' #$ #'#$1:; / ' 6& 2$(6 (#!4- & / #2+ & &!" &( 2 788!5! #,:6 (.
(6( 3'" 6 + && 0 6 $ !# +,& 0 $ "#& '#,: α. * #()+!5-!+ 9 &!" 3 !*$+ 2( 3 2 788!5! #,:6 ( tα, N − 2 6+" 7# 0 6 $ -!# +,& 0 $ "#& '#! ! 4!'+( '# & 0 '$ ) 6-, ($& / (N − 2), /6 N — 4!'+ $ 6 &&-. !*3 &!0 ( *&(4 &!0 xk ! yk .
/6( 6 $ !# +,&-0 ! $(+ 6+" 1/+ $ / 2 788!5! ( β 3 9&
*( !'(#, $ $!6 : |
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b − tα, N − 2 Sb ≤ β ≤ b + tα, N − 2 Sb . |
(5.19) |
$ !# +,&-0 ! $(+ 6+" '$ ) 6& / 4+ &( γ 6'#($!#'" (&(- + /!4& :
g − tα, N − 2Sg ≤ γ ≤ g + tα, N − 2 Sg . |
(5.20) |
6'#(& $2( $ $- (9 &!" (5.19) ! (5.20) $ +!4!& b, g, Sb ! Sg, 2 # -- $-4!'+":#'" 8 31+(3 (5.11), (5.16) ! (5.18) ' !' +,* $(&! 3
*1+,#(# $ $ 6 && 0 ' !! !*3 &!0 xk, yk (k = 1,..., N), 6( # &(3 4!'+ $- *&(4 &!" / (&!5 6 $ !# +,&-. ! $(+ $ !'2 3-. ( (- 3 # $ +!& 0& 0 ( 2'!3! 1:; 0 81&25!!.
(2 12(*-$(+ ', $ /+($ 2, +1%! !&( 6 $ !# +,& / ! $(+(6'#($+" # ' ) 0 ()' +:#&1: / %& '#, $-4!'+" 3 0 $ +!4!&- 6+" $-) (&& 0 6 $ !# +,& 0 $ "#& '#!. 7# 31 $- (9 &!" 6+" ()' +:#&-. / %& '# 0 +14 &&-. !)+!9 &&-. *&(4 &!0 ( (-
3 # $ β ! γ 3 9& 2 (#2 *( !'(#, $ '+ 61:; 3 $!6 :
β = tα, N − 2 Sb , γ = tα, N − 2 Sg . |
(5.21) |
# 3 '+14( , 2 /6( 72' !3 (+,&- *1+,#(#- ( 2'!3! 1- :#'" +!& 0& 0 81&25! 0 ) * '$ ) 6& / 4+ &(, & ). 6!3 (''4!#(#, / (&!5- 6 $ !# +,& / ! $(+( 6+" 6!&'#$ && / & !*$ '#& /
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'!3 '#! (5.13). 7# 3 '+14( $ 2(4 '#$ 5 ( ! $(+( $ *,3 3 &(0- 6 && 8 31+ (5.15) *&(4 &! b0. (3 #!3, 4# $ +!4!&( b0, $--
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'+14(0& 0 $ +!4!&- b0 # !'#!&& / 1/+ $ / 2 788!5! ( β0:
Sb0 = |
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(5.22) |
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(N − 1)∑xk2 |
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k =1 |
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+1%! !&( 6 $ !# +,& / ! $(+( 6+" 1/+ $ / 2 788!5! ( β0 ($&( ' 6& 2$(6 (#!4& 31 #2+ & &!: Sb0, 13& 9 && 31 &( 2 78- 8!5! #,:6 ( tα, N −1 6+" 1'#(& $+ && 0 6 $ !# +,& 0 $ "#& - '#! α ! 4!'+( '# & 0 '$ ) 6-, ($& / (N − 1). $ !# +,&-0 !&-
# $(+ 6+" 1/+ $ / 2 788!5! ( β0 "3 5! &(+,& 0 *($!'!- 3 '#! (5.13) *( !% #'" $ $!6 :
b0 – tα,N−1 Sb0 ≤ β0 ≤ b0 + tα,N−1 Sb0. |
(5.23) |
)' +:#&(" / %& '#, 1/+ $ / 2 788!5! ( β0 ($&(: |
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Δβ0 = tα,N−1 Sb0 . |
(5.24) |
*(2+:4 &! 6(&& / ( (/ (8( (''3 # !3 ; 6&1 (2#!4 - '21: *(6(41. & /6( $ . 6 8!*!4 '2 / !''+ 6 $(&!" # )1 #'" $-4!'-
+"#, *&(4 &!" *($!'!3 0 3 && 0 Y 6+" *&(4 &!0 3 && 0 x, 2 -
# - & !'1#'#$1:# ' 6! !*3 &&-. xk (k = 1, ..., N). 7#!. '+1- 4(". '# '#$ && !' +,* $(#, &(06 &&1: ( 2'!3! 1:;1: 81&2- 5!:.
(''3 # !3 '+14(0, 2 /6( $ 2(4 '#$ !)+!9 && 0 *($!'!3 '#! Y = F(X) !3 &" #'" +!& 0&(" 81&25!" (5.12), ( (3 # - b ! g 2 -
# 0 6 +9&- $-4!'+"#,'" 8 31+(3 (5.11). ) *&(4!3 xp *&(4 &!
( /13 (, 6+" 2 # / # )1 #'" &(0#! *&(4 &! 81&25!! F(xp). - /6( !'2 3-3 !)+!9 &&-3 *&(4 &! 3 "$+" #'" 4!'+
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(5.25) |
yp = bxp + g . |
, & ). 6!3 $' 3&!#,, 4# )( *&(4 &!" b ! g "$+":#'"
!)+!9 &&-3!. / %& '#, Δγ 3 9 # )-#, ! #! $(&( 2(2 '6$!/ / (8!2( ( 2'!3! 1:; 0 81&25!! (5.12) $$ .-$&!*, ( - / %& '#, Δβ — 2(2 & ) +,%! $ #- #& '!# +,& & 2 # 0 ' 6& 0 # 42!. )( 8(2# ( )1'+ $+!$(:# '+14(0&-0 .( (2# $ +!- 4!&- ~yp , 6 +" 3 0 $- (9 &! 3 (5.25). 4 $!6& , 4# $ +!4!&(,
*(6(&&(" 81&25! 0 '+14(0&-. $ +!4!& b ! g, #(29 "$+" #'" '+14(0& 0.#':6( '+ 61 # $-$ 6, 4# '+ $-4!'+ &!" *&(4 &!" 81&25!! F(xp) ≈ ~yp # )1 #'" '# !#, 6+" & / 6 $ !# +,&-0 ! $(+.
'+! $ +!4!&( ~yp , $ ); / $ ", '+14(0&(", # &( 6 +9&( .( (2-
# !* $(#,'" 6 + && 0 ' 6& 2$(6 (#!4& 0 / %& '#,:. - /+('& 3(# 3(#!4 '2 0 '#(#!'#!2 7#( / %& '#, $- (9( #'" '+ - 61:; 0 8 31+ 0:
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Sy = |
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(xp − x ) |
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(5.26) |
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(N − 2) |
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∑(xk − x ) |
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k =1 |
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(6( 3'" 6 + && 0 6 $ !# +,& 0 |
$ "#& '#,: α. * #()+!5- |
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*&(4 &!" yp |
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(5.27) |
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yp = tα, N − 2 S y . |
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$-$ 6 !* '+ 6& / $- (9 &!" ' '# !# $ # 3, 4# |
/ %- |
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& '#, $-4!'+" 3 0 $ +!4!&- )-'# ('# # ' 16(+ &! 3 ( /13 ( xp
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# ' 6& / *&(4 &!" !*3 &&-. *&(4 &!0 xk , #. . # 4!'+( x , 6 - + && / 8 31+ 0 (5.17).
$ !# +,&-0 ! $(+ 6+" *&(4 &!" 81&25!! F(xp) 3 9& *( !- '(#, $ $!6 :
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(5.28) |
yp − tα, N − 2 S y ≤ F (xp )≤ yp + tα, N − 2S y . |
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!3 , !++:'# ! 1:;!0 *($!'!3 '#, %! !&- |
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6 $ !# +,& / ! $(+( # $ +!4!&- ( /13 ( x.
* (5.26) "3 '+ 61 #, 4# %! !&( 6 $ !# +,& / ! $(+( (5.28) 3!&!3(+,&( 6+" xp = x . 7# 31 !)+!9 &&1: *($!'!3 '#,
(5.12) 5 + ' ) (*& !' +,* $(#, 6+" $-4!'+ &!" *&(4 &!0 81&25!! # ( /13 $ xp, 2 # - (' +(/(:#'" $)+!*! ' 6& / *&(4 &!" x .
! 16(+ &!! $ +!4!&- xp $ +:)1: '# &1 # ' 6& / *&(4 &!" x %! !&( 6 $ !# +,& / ! $(+( )-'# $ * ('#( #. ( !'. 2 . %
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'# +, %! 2!3, 4# # " # (2#!4 '2!0 '3-'+.
+ 6 $(# +,& , !)+!9 &&1: *($!'!3 '#,, +14 &&1: 3 # 6 3 &(!3 &,%!. 2$(6 (# $, ('& !3 &"#, 6+" *(6(4 72'# ( +"5!!.
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-40 |
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-60 |
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-80 |
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'. 2. !3 *($!'!3 '#! %! !&- 6 $ !# +,& / ! $(+( 6+" *&(4 &!" 81&25!! Y = F(x) # ( /13 ( x
=# !. $(" +!&!" — / (8!2 +!& 0& 0 ( 2'!3! 1:; 0 81&25!!, ' + %&- +!&!! — / (&!5- 6 $ !# +,& / ! $(+(
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xmax
§ 5.4. # ($# &*
#*
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3 ( x *&(4 &!" !''+ 61 3 0 81&25!! Y !*3 ":#'" & '2 +,2 (*.)-4& 7# !. 6!#'" 6 +(#, $ '+14("., 2 /6( '+14(0&(" / %& '#, !*3 &!" 8!*!4 '2 0 $ +!4!&- Y 6 $ +,& $ +!2(, #. . *&(4!# +,&$-%( # !) &1: / %& '#,.
$ 6 3 '+ 61:;! ) *&(4 &!":
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yks — *&(4 &!" 81&25!!, |
+14 &&- $ |
$# &-. !*3 &!". ! |
8!2'! $(&& 3 *&(4 &!! ( /13 ( xk , (s = 1,…, mk); |
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mk — 2 +!4 '#$ $# &-. !*3 &!0 |
! x = xk . |
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/6( +& 2 +!4 '#$ |
+14 &&-. *&(4 &!0 81&25!! yks $- (- |
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*!#'" '133 0 |
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N0 = ∑mk , |
(5.29) |
k =1
/6 N — 2 +!4 '#$ (*+!4&-. *&(4 &!0 ( /13 ( xk (k = 1, …N).+" 2(96 / *&(4 &!" ( /13 ( xk $-4!'+" #'" ' 6& '#(#!'#!4 -
'2 *&(4 &! 81&25!! ' /+('& (2.1)
mk
∑ yks
yk = |
s =1 |
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(5.30) |
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mk |
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! ' 6& 2$(6 (#!4&-0 (*) ' 8 31+ (2.4)
mk
∑(yks − yk )2
S0k = |
s=1 |
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(5.31) |
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mk (mk −1) |
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-) ($ 6 + &&1: 6 $ !# +,&1: $ "#& '#, α, 3 9& -
'# !#, 6 $ !# +,&- ! $(+- 6+" 2(96 / *&(4 &!" 81&25!! Y(xk) 3 # 6 3, !'(&&-3 $ /+($ 2:
yk − tα,νk S0k ≤ Y (xk ) ≤ yk + tα,νk S0k . |
(5.32) |
788!5! - #,:6 ( tα, νk ) 1#'" 6+" 4!'+( '# & 0 '$ ) -
6- νk = mk − 1.
& $, (''3 # !3 '# 0, & (' '# (& &&-0 '+14(0, 2 /6( # -#!4 '2! !*-'2(&!" *$ +":# 1#$ 96(#,, 4# !''+ 61 3(" *($!'!-
3 '#, Y(X) "$+" #'" +!& 0& 0, #. . !'-$( #'" 81&25! 0 $!6( (5.7).
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/6( *1+,#(#- !*3 &!0 yks 3 9& 6'#($!#, $ $!6 |
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yks = β xk + γ + εks , |
(5.33) |
/6 εks — *&(4 &!" & 2 # 0 '+14(0& 0 $ +!4!&- ε, .( (2# !*1:; 0 $' '+14(0&- 8(2# -, $+!":;! &( *1+,#(# !*3 &!".
(2 ! (& , # #!4 '21: +!& 0&1: 81&25!: (5.7) *(3 &!3 !- )+!9 && 0 (5.12), 2 788!5! - 2 # 0 b ! g "$+":#'" !)+!9 &- &-3! *&(4 &!"3! ( (3 # $ β ! γ 81&25!! (5.7) ' #$ #'#$ && . +"
&(. 96 &!" &(!+14% / !)+!9 &!" 4!'+ $-. *&(4 &!0 b ! g $ 6(&- & 3 '+14( & +,*" 8 3(+,& !3 &"#, 8 31+- (5.11). , & )-
. 6!3 $- (*!#, !)+!9 &&- *&(4 &!" ( (3 # $ β ! γ 4 * - *1+,#(#- !*3 &!0 xk ! yks .
+" &(. 96 &!" &(!+14%!. !)+!9 &!0 b ≈ β ! g ≈ γ $& $, !- 3 &!3 3 # 6 &(!3 &,%!. 2$(6 (# $. '#($!3 '1331 2$(6 (# $ #2+ -
& &!0 !*3 &&-. *&(4 &!0 81&25!! yks # (''4!#-$( 3-. !- )+!9 && 0 81&25!! (5.12) 6+" ' #$ #'#$1:;!. *&(4 &!0 ( /13 &-
# $ xk:
N mk |
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2 |
N mk |
2 |
Q0 = ∑∑(yks − Y |
(xk )) |
= ∑∑(yks − bxk − g ) . (5.34) |
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k =1 s =1 |
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+!4!&( Q0 $ 6(&& 3 '+14( 6'#($+" #'" 6$ 0& 0 '133 0. |
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$&1# && 0 '133 '+ 9 &! |
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$ 6!#'" (*+!4&-3 *&(4 &!"3 81&2- |
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5!! yks ! 8!2'! $(&& 3 *&(4 &!! ( /13 ( xk , ( $ $& %& 0 —(*+!4&-3 *&(4 &!"3 ( /13 ( xk (k = 1, …N).
*$ 6 3 $ 2$(6 (# 2(96 '+(/( 3 '133- (5.34)
(y |
ks |
− bx − g )2 |
= y2 |
+ b2 x2 + g 2 − 2bx y |
ks |
− 2gy |
ks |
+ 2bgx (5.35) |
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! $ 6 3 #6 +,& $-4!'+ &!" % '#! $&1# &&!. '133. |
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+" ) + 16 )& / |
6'#($+ &!" *1+,#(# $ 6 |
+&!# +,& $$ - |
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6 3 |
$ +!4!&1 q2 |
— '1331 2$(6 (# $ #2+ & &!0 !*3 &&-. *&(4 &!0 |
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81&25!! yks # '$ !. ' 6&!. *&(4 &!0 |
yk — ! |
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mk |
mk |
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qk2 = ∑(yks − yk )2 |
= ∑ yks2 |
− mk yk2 . |
(5.36) |
s =1 |
s =1 |
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/6( $(" $&1# &&"" '133( $ (5.34) $- (*!#'" #(2:
mk
∑ yks2 = mk yk2 + qk2 .
s=1
, $ +!4!&( Q0 3 9 # )-#, 6'#($+ &( '133! $(&! 3 # +,- 2 !&6 2'(3 (*+!4&-. ( /13 $:
N |
N |
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Q0 = ∑mk yk2 + ∑qk2 + b2 |
∑mk xk2 + N0 g 2 − |
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k =1 |
k =1 |
k =1 |
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− 2b∑mk xk yk − 2g∑mk yk + 2bg∑mk xk . |
(5.37) |
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k =1 |
k =1 |
k =1 |
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/+('& 3 # 61 &(!3 &,%!. 2$(6 (# $, (''3 # !3 '1331 (5.37) 2(2 81&25!: ( (3 # $ b ! g, 2 # - )16 3 $( ,! $(#, 6 6 '#!9 - &!" 3!&!313( 81&25!! Q0(b, g). (2 ! $ § 5.2, $-4!'+!3 4('#&- -
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b ! g:
N |
N |
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b∑mk xk2 + g∑mk xk = ∑mk xk yk , |
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k =1 |
k =1 |
k =1 |
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b∑mk xk + N0 g = ∑mk yk . |
(5.38) |
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k =1 |
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% &! 7# 0 '!'# 3- 3 9& *( !'(#, $ $!6 :
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N0 ∑mk xk yk − |
∑mk xk ∑mk yk |
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b = |
k =1 |
k =1 |
k =1 |
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N0 ∑mk xk2 − ∑mk xk |
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k =1 |
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(5.39) |
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∑mk xk2 |
∑mk yk |
− ∑mk xk ∑mk xk yk |
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g = |
k =1 |
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k =1 |
k =1 |
k =1 |
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N0 ∑mk xk2 − ∑mk xk |
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k =1 |
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/2 *(3 #!#,, 4# '+! $' 4!'+( mk 6!&(2 $-, # 8 31+- (5.39)#+!4(:#'" # 8 31+ (5.11) # +,2 *(3 & 0 $ +!4!& yk &( ' 6&! *&(- 4 &!" yk . ! 7# 3 $ *&(3 &(# +". $ +!4!&( N0 *(3 &!#'" &( N.
(2!3 ) (* 3, 3- +14!+! !)+!9 &&- *&(4 &!" ( (3 # $ +!& 0& 0 ( 2'!3! 1:; 0 81&25!! 6+" '+14(" $# &-. !*3 -
&!0. +" $-4!'+ &!0 *&(4 &!0 !'2 3 0 81&25!! # +:) / ( /13 ( x $& $, '+ 61 # !' +,* $(#, !)+!9 &&1: 81&25!: (5.12), & ' ( (- 3 # (3!, 6'4!#(&&-3! 8 31+(3 (5.39).
§ 5.5. ! & & % &*
$ !# +,&- ! $(+- 6+" ( (3 # $ β ! γ $ '+14( $# &-. !*3 &!0 )16 3 '# !#,, 2(2 ! (& , +,*1"', +14 &&-3! !)+!-
9 &&-3! *&(4 &!"3! b ! g. ! ('4 # '+ 61 # 14 '#,, 4# &( %! !&1 6 $ !# +,&-. ! $(+ $ '1; '#$ && $+!" # (*) ' !*3 &&-.
*&(4 &!0 yks #& '!# +,& ' 6&!. $ +!4!& |
yk . |
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!)+!9 &!" (5.12), ! ( (3 # (. b ! g, |
6 +" 3-. 8 31+(3! |
(5.39). |
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(5.40) |
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yk = Y (xk ) = bxk + g . |
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6'#($!3 *&(4 &!" yk $ 6$ 0&1: '1331 (5.34) |
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N mk |
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Q0 |
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(5.41) |
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= ∑∑(yks − yk ) , |
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k =1 s =1 |
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( *(# 3 ) (*1 3 |
$&1# &&:: '1331: |
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mk |
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2 |
mk |
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mk |
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~2 |
= |
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∑(yks − yk ) |
= ∑ yks |
− 2 yk ∑ yks + mk yk |
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s =1 |
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s =1 |
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s =1 |
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mk |
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~2 |
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~2 |
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2 |
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= ∑ yks |
− 2mk yk yk |
+ mk yk = mk yk + qk |
2mk yk yk + mk yk |
= |
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s =1 |
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~ |
2 |
2 |
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= mk (yk − yk ) |
+ qk . |
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+ 6 $(# +,& , $ +!4!&( Q0 3 9 # )-#, |
6'#($+ &( $ $!6 6$1. |
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'133: |
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N
Q0 = ∑mk (yk
k =1
~ 2 |
N |
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2 |
(5.42) |
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− yk ) |
+ ∑qk . |
k =1
$(" '133( .( (2# !*1 # #2+ & &!" ' 6&!. 72' !3 (+,- &-. *&(4 &!0 # (''4!#(&&-. ( 2'!3! 1:; 0 81&25!! (5.12).
# (" 6 +" #'" (*) ' 3 72' !3 (+,&-. *&(4 &!0 yks #& - '!# +,& ' #$ #'#$1:;!. ' 6&!. yk .
6 + 9!3, 4# '+14(0&(" $ +!4!&( ε, 2 # (" .( (2# !*1 # ' - $ 21 & '#, $' . '+14(0&-. 8(2# $, !'2(9(:;!. *1+,#(# !*3 - &!", 64!&" #'" & 3(+,& 31 (/(1'' $ 31) *(2 &1 (' 6 + &!". - /6( 3(# 3(#!4 '2(" '#(#!'#!2( 6( # '+ 61:;1: '#1: 5 61 1
'# &!" 6 $ !# +,&-. ! $(+ $ 6+" ( (3 # $ β ! γ ( 2'!- 3! 1:; 0 +!& 0& 0 81&25!! (5.12).
+,*1"', *1+,#(#(3! !*3 &!0, $-4!'+!3 2$(6 (# ' 6& 2$(6-(#!4& / #2+ & &!"
60