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Файл:Теория информации. Лабораторный практикум в MATLAB. Учебное пособие
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ɋɨɜɦɟɫɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
disp(P_A_B)
%Ɋɚɫɱɟɬ ɱɚɫɬɧɵɯ ɭɫɥɨɜɧɵɯ ɷɧɬɪɨɩɢɣ H(A|b)
H_A_b=sum(fun(P_A_B),1);
disp('ɑɚɫɬɧɵɟ ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ H(A|b), ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_A_b)
%Ɋɚɫɱɟɬ ɭɫɥɨɜɧɨɣ ɷɧɬɪɨɩɢɢ H(A|B)
H_A_B=sum(p_b.*H_A_b);
disp('ɍɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ H(A|B), ɛɢɬ/ɫɢɦɜɨɥ:');
disp(H_A_B)
%Ɋɚɫɱɟɬ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ H(A,B)
H_AB=H_B+H_A_B;
disp('ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B), ɛɢɬ/ɫɢɦɜɨɥ - ɜɬɨɪɚɹ
ɮɨɪɦɚ:')
disp(H_AB)
%Ɋɚɫɱɟɬ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ H(A,B)
H_AB=sum(sum(fun(P_AB),1));
disp('ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B), ɛɢɬ/ɫɢɦɜɨɥ - ɨɛɳɚɹ
ɮɨɪɦɭɥɚ:')
disp(H_AB)
ɁɚɞɚɧɢɟɆɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɛɴɟɞɢɧɟ
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Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
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HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
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HAb
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
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HB
HAB
ɩɨɮɨɪɦɭɥɚɦ
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ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_2_1
ɁɚɞɚɧɢɟɆɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɛɴɟɞɢɧɟ
ɧɢɹɚɥɮɚɜɢɬɨɜAɢBɢɦɟɟɬɜɢɞ
§·
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AB
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ɱɚɫɬɧɵɟ
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 2
Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
_
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
i
ɢ
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HAb
HA
ɢ
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
HB
HAB
ɩɨɮɨɪɦɭɥɚɦ
ɱɚɫɬɧɵɟ
ɢ Ɉɬɥɢɱɚɸɬɫɹ ɥɢ ɜɟɥɢɱɢɧɵ ɷɧɬɪɨɩɢɣ
HB_HBA_HAB
ɢ
HAB
ɨɬ ɡɧɚɱɟɧɢɣ ɫɨɨɬɜɟɬɫɬɜɭɸ
ɳɢɯɷɧɬɪɨɩɢɣɫɢɫɬɟɦɵɪɚɫɫɦɨɬɪɟɧɧɨɣɜɩɪɟɞɵɞɭɳɟɣɡɚɞɚɱɟ"
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_2_1
ɁɚɞɚɧɢɟɆɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɛɴɟɞɢɧɟ
ɧɢɹɚɥɮɚɜɢɬɨɜAɢBɢɦɟɟɬɜɢɞ
§·
3
AB
Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
_
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
¨¸
¨¸
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ɢ
i
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HA
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ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
HB
HAB
ɩɨɮɨɪɦɭɥɚɦ
ɱɚɫɬɧɵɟ
ɢ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_2_1
ɁɚɞɚɧɢɟɆɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɛɴɟɞɢɧɟ
ɧɢɹɚɥɮɚɜɢɬɨɜAɢBɢɦɟɟɬɜɢɞ
HA
§·
3
Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
¨¸
AB
¨¸
¨¸
©¹
HA
_
i
ɢ
_
HAb
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
ɢ
HAB
HB
ɩɨɮɨɪɦɭɥɚɦ
ɢ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_2_1
ɁɚɞɚɧɢɟɆɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɛɴɟɞɢɧɟ
ɧɢɹɚɥɮɚɜɢɬɨɜAɢBɢɦɟɟɬɜɢɞ
ɱɚɫɬɧɵɟ
ɋɨɜɦɟɫɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
§·
3
¨¸
AB
¨¸
¨¸
©¹
Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
_
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
HA
ɢ
ɢ
i
_
HAb
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
HB
HAB
ɩɨɮɨɪɦɭɥɚɦ
ɱɚɫɬɧɵɟ
ɢ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_2_1
Ɂɚɞɚɧɢɟȼɟɪɨɹɬɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹ ɫɢɦɜɨɥɨɜ ɢɫɬɨɱɧɢɤɚ ɫ
ɚɥɮɚɜɢɬɨɦ A ɫɨɫɬɚɜɥɹɸɬ
pa
pa
pa
Ɇɚɬɪɢɰɚ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜɢɫɬɨɱɧɢɤɚ
BɩɨɫɥɟɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜɢɡɚɥɮɚɜɢɬɚAɢɦɟɟɬɜɢɞ
_ _ _
pb a pb a pb a
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pb a pb a pb a
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Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
_
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
HA
ɢ
ɢ
i
_
HAb
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
HAB
HB
ɱɚɫɬɧɵɟ
ɩɨɮɨɪɦɭɥɚɦ
ɢ
%Script_2_7
%Ɋɚɫɱɟɬ ɭɫɥɨɜɧɨɣ ɢ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ Ⱥ ɢ
ȼ
%ɩɪɢ ɡɚɞɚɧɧɨɦ ɜɟɤɬɨɪɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ
Ⱥ
%ɢ ɦɚɬɪɢɰɟ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(B|A)
clear all; close all; clc
%Ɂɚɞɚɧɢɟ ɚɧɨɧɢɦɧɨɣ ɮɭɧɤɰɢɢ
%--------------------------fun=@(x) -x.*log2(x+(x==0));
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 2
%--------------------------%Ɂɧɚɱɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ Ⱥ
p_a=[.2 .3 .5];
disp('ȼɟɪɨɹɬɧɨɫɬɢ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ Ⱥ:')
disp(p_a)
%Ɂɧɚɱɟɧɢɹ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
P_B_A=[.5 .5 0; 0 .666 .334; 0 .4 .6];
disp('Ɇɚɬɪɢɰɚ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(B|A):')
disp(P_B_A)
%Ɉɩɪɟɞɟɥɟɧɢɟ ɨɛɴɟɦɨɜ ɚɥɮɚɜɢɬɨɜ
[m, n]=size(P_B_A);
disp('Ɉɛɴɟɦ ɚɥɮɚɜɢɬɚ Ⱥ:')
disp(m)
disp('Ɉɛɴɟɦ ɚɥɮɚɜɢɬɚ B:')
disp(n)
%Ɋɚɱɟɬ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɚ Ⱥ
H_A=sum(fun(p_a));
disp('ɗɧɬɪɨɩɢɹ ɚɥɮɚɜɢɬɚ Ⱥ, ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_A)
%Ɋɚɫɱɟɬ ɱɚɫɬɧɵɯ ɭɫɥɨɜɧɵɯ ɷɧɬɪɨɩɢɣ H(B|a)
H_B_a=sum(fun(P_B_A),2);
disp('ɑɚɫɬɧɵɟ ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ H(B|a), ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_B_a)
ɭɫɥɨɜɧɨɣ ɷɧɬɪɨɩɢɢɢ H(B|A)
%Ɋɚɫɱɟɬ
H_B_A=sum(p_a'.*H_B_a);
disp('ɍɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ H(B|A), ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_B_A)
%Ɋɚɫɱɟɬ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
P(Ⱥ,ȼ)
P_AB=bsxfun(@times,P_B_A,p_a(:));
disp('Ɇɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(A,B):')
disp(P_AB)
%Ɋɚɫɱɟɬ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ B
p_b=sum(P_AB,1);
disp('ȼɟɪɨɹɬɧɨɫɬɢ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ B:')
disp(p_b)
%Ɋɚɱɟɬ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɚ B
H_B=sum(fun(p_b));
disp('ɗɧɬɪɨɩɢɹ ɚɥɮɚɜɢɬɚ B, ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_B)
%Ɋɚɫɱɟɬ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
P(A|B)
P_A_B=bsxfun(@rdivide,P_AB,p_b(:)');
disp('Ɇɚɬɪɢɰɚ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(A|B):')
ɋɨɜɦɟɫɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
disp(P_A_B)
%Ɋɚɫɱɟɬ ɱɚɫɬɧɵɯ ɭɫɥɨɜɧɵɯ ɷɧɬɪɨɩɢɣ H(A|b)
H_A_b=sum(fun(P_A_B),1);
disp('ɑɚɫɬɧɵɟ ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ H(A|b), ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_A_b)
%Ɋɚɫɱɟɬ ɭɫɥɨɜɧɨɣ ɷɧɬɪɨɩɢɢɢ H(A|B)
H_A_B=sum(p_b.*H_A_b);
disp('ɍɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ H(A|B), ɛɢɬ/ɫɢɦɜɨɥ:');
disp(H_A_B)
%Ɋɚɫɱɟɬ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ H(A,B)
H_AB=H_A+H_B_A;
disp('ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B), ɛɢɬ/ɫɢɦɜɨɥ - ɩɟɪɜɚɹ
ɮɨɪɦɚ:')
disp(H_AB)
%Ɋɚɫɱɟɬ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ H(A,B)
H_AB=H_B+H_A_B;
disp('ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B), ɛɢɬ/ɫɢɦɜɨɥ - ɜɬɨɪɚɹ
ɮɨɪɦɚ:')
disp(H_AB)
%Ɋɚɫɱɟɬ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ H(A,B)
H_AB=sum(sum(fun(P_AB),1));
disp('ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B), ɛɢɬ/ɫɢɦɜɨɥ - ɨɛɳɚɹ
ɮɨɪɦɭɥɚ
disp(H_AB)
:')
Ɂɚɞɚɧɢɟȼɟɪɨɹɬɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹ ɫɢɦɜɨɥɨɜ ɢɫɬɨɱɧɢɤɚ ɫ
ɚɥɮɚɜɢɬɨɦ A ɫɨɫɬɚɜɥɹɸɬ
pa
pa
pa
Ɇɚɬɪɢɰɚ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜɢɫɬɨɱɧɢɤɚ
BɩɨɫɥɟɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜɢɡɚɥɮɚɜɢɬɚAɢɦɟɟɬɜɢɞ
§·
3
¨¸
_
BA
¨¸
¨¸
©¹
Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
ɢ
i
_
ɢ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_2_7
_
HAb
HA
ɢ
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
HB
HAB
ɩɨɮɨɪɦɭɥɚɦ
ɱɚɫɬɧɵɟ
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 2
Ɉɬɱɟɬɢɤɨɧɬɪɨɥɶɧɵɟɜɨɩɪɨɫɵ
Ɉɬɱɟɬɩɨɥɚɛɨɪɚɬɨɪɧɨɣɪɚɛɨɬɟɨɮɨɪɦɥɹɟɬɫɹɜɜɢɞɟɞɨɤɭɦɟɧ
ɬɚ06:RUGɢɞɨɥɠɟɧɫɨɞɟɪɠɚɬɶ
ɇɚɡɜɚɧɢɟ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɵ ɲɪɢɮɬ 7LPHV 1HZ
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ɥɢɫɬɢɧɝɢ Pɮɚɣɥɨɜ ɤɨɩɢɪɭɟɦɵɟ ɢɡ ɨɤɧɚ (GLWRU ɪɟɡɭɥɶɬɚɬɵ ɢɯ
ɜɵɩɨɥɧɟɧɢɹ ɤɨɩɢɪɭɟɦɵɟ ɢɡ ɨɤɧɚ &RPPDQG :LQGRZ ɲɪɢɮɬ
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Ɂɚɳɢɬɚ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɵ ɩɪɨɜɨɞɢɬɫɹ ɧɚ ɨɫɧɨɜɟ ɩɪɟɞ
ɫɬɚɜɥɟɧɧɨɝɨɨɬɱɟɬɚɢɨɬɜɟɬɨɜɧɚɤɨɧɬɪɨɥɶɧɵɟɜɨɩɪɨɫɵ ɢɡ ɫɥɟɞɭ
ɸɳɟɝɨɫɩɢɫɤɚ
Ⱦɚɣɬɟɨɩɪɟɞɟɥɟɧɢɟɦɚɬɪɢɰɵ
ɫɨɜɦɟɫɬɧɵɯɜɟɪɨɹɬɧɨ
3
AB
ɫɬɟɣɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜ ɞɜɭɯ ɫɬɚɬɢɫɬɢɱɟɫɤɢ ɫɜɹɡɚɧɧɵɯ ɚɥɮɚɜɢ
ɬɨɜ
Aa
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ɑɟɦɭɪɚɜɧɚɫɭɦɦɚɜɫɟɯɷɥɟɦɟɧɬɨɜɦɚɬɪɢɰɵ
ɑɟɦɭɪɚɜɧɵɫɭɦɦɵɜɫɟɯɷɥɟɦɟɧɬɨɜɫɬɪɨɤɦɚɬɪɢɰɵ
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ɑɟɦɭ ɪɚɜɧɵ ɫɭɦɦɵ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɫɬɨɥɛɰɨɜ ɦɚɬɪɢɰɵ
"
3
AB
Ɂɚɩɢɲɢɬɟ ɜɵɪɚɠɟɧɢɟ ɫɜɹɡɵɜɚɸɳɟɟ ɫɨɜɦɟɫɬɧɵɟ ɜɟɪɨɹɬ
ɧɨɫɬɢ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɨɜ A ɢ B ɫ ɛɟɡɭɫɥɨɜɧɵɦɢ ɢ ɭɫɥɨɜɧɵɦɢ
ɜɟɪɨɹɬɧɨɫɬɹɦɢ
Ɂɚɩɢɲɢɬɟ ɨɛɳɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ
ɞɜɭɯɫɬɚɬɢɫɬɢɱɟɫɤɢɫɜɹɡɚɧɧɵɯɚɥɮɚɜɢɬɨɜ
ɉɪɢɜɟɞɢɬɟ ɩɟɪɜɭɸ ɮɨɪɦɭ ɡɚɩɢɫɢ ɞɥɹ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨ
ɩɢɢ
ɉɪɢɜɟɞɢɬɟ ɜɬɨɪɭɸ ɮɨɪɦɭ ɡɚɩɢɫɢ ɞɥɹ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨ
ɩɢɢ
Ⱦɚɣɬɟɨɩɪɟɞɟɥɟɧɢɟɱɚɫɬɧɨɣɭɫɥɨɜɧɨɣɷɧɬɪɨɩɢɢ
Ⱦɚɣɬɟ ɨɩɪɟɞɟɥɟɧɢɟ ɦɚɬɪɢɰɵ
ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨ
3
_BA
ɫɬɟɣɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜɚɥɮɚɜɢɬɨɜAɢB
ɂɡɨɛɪɚɡɢɬɟɞɢɚɝɪɚɦɦɵȼɟɧɧɚ±ɗɣɥɟɪɚɩɨɹɫɧɹɸɳɢɟɫɨ
ɨɬɧɨɲɟɧɢɹɦɟɠɞɭɷɧɬɪɨɩɢɹɦɢ
ɉɟɪɟɱɢɫɥɢɬɟɫɜɨɣɫɬɜɚɭɫɥɨɜɧɨɣɷɧɬɪɨɩɢɢ
ɋɨɜɦɟɫɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
ȼɤɚɤɢɯɫɥɭɱɚɹɯɭɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹɩɪɢɧɢɦɚɟɬɧɚɢɦɟɧɶ
ɲɟɟɢɧɚɢɛɨɥɶɲɟɟɡɧɚɱɟɧɢɟ"
ɉɟɪɟɱɢɫɥɢɬɟɫɜɨɣɫɬɜɚɫɨɜɦɟɫɬɧɨɣɷɧɬɪɨɩɢɢ
ȼɤɚɤɢɯɫɥɭɱɚɹɯɫɨɜɦɟɫɬɧɚɹɷɧɬɪɨɩɢɹɩɪɢɧɢɦɚɟɬɧɚɢɦɟɧɶ
ɲɟɟɢɧɚɢɛɨɥɶɲɟɟɡɧɚɱɟɧɢɟ"
Ʌɚɛɨɪɚɬɨɪɧɚɹɪɚɛɨɬɚʋ
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Ⱦɢɫɤɪɟɬɧɵɟɢɫɬɨɱɧɢɤɢɢɧɮɨɪɦɚɰɢɢ
ɐɟɥɶ ɪɚɛɨɬɵ ɂɡɭɱɢɬɶ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɢɫɤɪɟɬɧɵɯ ɢɫɬɨɱ
ɧɢɤɨɜ ɢɧɮɨɪɦɚɰɢɢ ɨɜɥɚɞɟɬɶɩɪɨɝɪɚɦɦɧɵɦɢ ɫɪɟɞɫɬɜɚɦɢ ɢɯɪɚɫ
ɱɟɬɚɜ0$7/$%
Ɍɟɨɪɟɬɢɱɟɫɤɢɟɫɜɟɞɟɧɢɹ
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹɦɨɞɟɥɶɞɢɫɤɪɟɬɧɨɝɨɢɫɬɨɱɧɢɤɚɨɩɪɟɞɟɥɹɟɬ
ɫɹ ɫɬɚɬɢɫɬɢɱɟɫɤɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɜɵɪɚɛɚɬɵɜɚɟɦɵɯ ɢɦ ɢɧ
ɮɨɪɦɚɰɢɨɧɧɵɯɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ
ɉɭɫɬɶ ɢɫɬɨɱɧɢɤ ɫ ɚɥɮɚɜɢɬɨɦ
Aaa a
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ɝɟɧɟɪɢɪɭɟɬ
m
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kɦ ɲɚɝɟ ɧɚ ɜɵɯɨɞɟ ɢɫɬɨɱɧɢɤɚ ɧɚɛɥɸɞɚɟɬɫɹ ɧɟɤɨɬɨɪɵɣ ɫɢɦɜɨɥ
ɬɚɤ ɱɬɨ ©ɮɪɚɝɦɟɧɬªɞɥɢɧɨɣ
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ɫɢɦɜɨɥɨɜ ɢɡ ɜɫɟɣɩɨɫɥɟɞɨ
ɜɚɬɟɥɶɧɨɫɬɢɢɦɟɟɬɜɢɞ
aa a a
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L
ɢɨɩɢɫɵɜɚɟɬɫɹɫɨɜɦɟɫɬɧɵɦɪɚɫɩɪɟɞɟɥɟɧɢɟɦɜɟɪɨɹɬɧɨɫɬɟɣ
pa a a a
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iii iii i
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pa
i
ɭɫɥɨɜɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɫɢɦɜɨɥɚ
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L
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ɩɨɫɥɟ ɫɢɦɜɨɥɚ
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pa a
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Ⱦɢɫɤɪɟɬɧɵɣ ɢɫɬɨɱɧɢɤ ɹɜɥɹɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ ɟɫɥɢ ɫɨɜ
ɦɟɫɬɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣɜɜɵɪɚɠɟɧɢɢ ɩɪɢ ɥɸ
ɛɵɯɡɧɚɱɟɧɢɹɯLɧɟɡɚɜɢɫɢɬɨɬɜɵɛɨɪɚɧɚɱɚɥɶɧɨɣɬɨɱɤɢɲɚɝɚ k
ɨɬɫɱɟɬɚɜɪɟɦɟɧɢ
ɋɬɚɰɢɨɧɚɪɧɵɣ ɞɢɫɤɪɟɬɧɵɣ ɢɫɬɨɱɧɢɤ ɹɜɥɹɟɬɫɹ ɷɪɝɨɞɢɱɟ
ɫɤɢɦ ɟɫɥɢ ɟɝɨ ɜɟɪɨɹɬɧɨɫɬɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɨɥɭɱɟɧɧɵɟ ɩɪɢ
ɢɫɫɥɟɞɨɜɚɧɢɢ ɨɞɧɨɣ ɞɨɫɬɚɬɨɱɧɨ ɞɥɢɧɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
±
a
i
Ⱦɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ ɢɧɮɨɪɦɚɰɢɢ
m
m
p
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L
ɬɟɨɪɟɬɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨɣ ɞɥɢɧɵ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɛɥɢɡɤɨɣ ɤ
ɟɞɢɧɢɰɟ ɬɟɨɪɟɬɢɱɟɫɤɢ ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ ɫɨɜɩɚɞɚɸɬ ɫ ɟɝɨ ɫɬɚɬɢ
ɫɬɢɱɟɫɤɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɩɨɥɭɱɚɟɦɵɦɢ ɩɨ ɚɧɫɚɦɛɥɸ ɜɫɟɯ
ɪɟɚɥɢɡɚɰɢɣɜɵɪɚɛɚɬɵɜɚɟɦɵɯɢɫɬɨɱɧɢɤɨɦ
Ⱦɢɫɤɪɟɬɧɵɣ ɢɫɬɨɱɧɢɤ ɫ ɩɚɦɹɬɶɸ Lɝɨ ɩɨɪɹɞɤɚ ± ɷɬɨ ɞɢɫ
ɤɪɟɬɧɵɣɢɫɬɨɱɧɢɤɭɤɨɬɨɪɨɝɨɫɬɚɬɢɫɬɢɱɟɫɤɚɹɫɜɹɡɶɦɟɠɞɭɩɨɫɥɟ
ɞɨɜɚɬɟɥɶɧɵɦɢ ɫɢɦɜɨɥɚɦɢ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɧɚ
L
ɫɢɦɜɨɥɨɜ ɢ
ɞɥɹ ɥɸɛɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɜɦɟɫɬɧɵɦ
ɪɚɫɩɪɟɞɟɥɟɧɢɟɦɜɟɪɨɹɬɧɨɫɬɟɣ
ɗɧɬɪɨɩɢɹɫɬɚɰɢɨɧɚɪɧɨɝɨɷɪɝɨɞɢɱɟɫɤɨɝɨɞɢɫɤɪɟɬɧɨɝɨ ɢɫɬɨɱ
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pa a a a
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Ⱦɢɫɤɪɟɬɧɵɣ ɢɫɬɨɱɧɢɤ ɛɟɡ ɩɚɦɹɬɢɫɩɚɦɹɬɶɸɝɨɩɨɪɹɞ
ɤɚ± ɷɬɨɞɢɫɤɪɟɬɧɵɣɢɫɬɨɱɧɢɤɭɤɨɬɨɪɨɝɨɜɫɟɫɢɦɜɨɥɵɞɥɹɥɸ
ɛɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢ ɧɟɡɚɜɢɫɢɦɵ ɢ ɫɨɜ
ɦɟɫɬɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɜɧɨ ɩɪɨɢɡɜɟɞɟɧɢɸ
ɜɟɪɨɹɬɧɨɫɬɟɣɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜ
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɷɬɨɬ ɢɫɬɨɱɧɢɤɩɨɥɧɨɫɬɶɸ ɡɚɞɚɧ ɟɫɥɢɡɚɞɚɧɜɟɤ
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S
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m
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L
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ɧɭɥɥɢ
Ⱦɨɤɚɡɚɧɨ ɱɬɨ ɜɫɹɤɢɣ ɫɬɚɰɢɨɧɚɪɧɵɣ ɢɫɬɨɱɧɢɤ ɛɟɡ ɩɚɦɹɬɢ
ɬɚɤɠɟɹɜɥɹɟɬɫɹɷɪɝɨɞɢɱɟɫɤɢɦ
ɋɬɚɰɢɨɧɚɪɧɵɣɢɷɪɝɨɞɢɱɟɫɤɢɣɢɫɬɨɱɧɢɤɛɟɡɩɚɦɹɬɢ ɢ ɫ
ɪɚɜɧɨɜɟɪɨɹɬɧɵɦɢ ɫɢɦɜɨɥɚɦɢ Ɍɚɤɨɣ ɢɫɬɨɱɧɢɤ ɩɨɥɧɨɫɬɶɸ ɡɚɞɚɧ
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ɂɧɨɝɞɚɷɬɨɬɢɫɬɨɱɧɢɤɧɚɡɵɜɚɸɬɢɫɬɨɱɧɢɤɨɦɏɚɪɬɥɢ
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ɧɵɣɢɫɬɨɱɧɢɤɩɨɥɧɨɫɬɶɸɡɚɞɚɧɟɫɥɢɢɡɜɟɫɬɧɵɥɢɛɨɜɟɤɬɨɪɜɟɪɨ
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