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Файл:Теория информации. Лабораторный практикум в MATLAB. Учебное пособие
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Ʉɨɥɢɱɟɫɬɜɨ ɢɧɮɨɪɦɚɰɢɢ ɢ ɷɧɬɪɨɩɢɹ
ɇɚ ɤɚɤɭɸ ɜɟɥɢɱɢɧɭ ɭɦɟɧɶɲɚɟɬɫɹ ɷɧɬɪɨɩɢɹ ɜɨ ɜɬɨɪɨɦ ɫɥɭ
ɱɚɟ"
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶScript_1_3
Ɂɚɞɚɧɢɟɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ
ORJ
pa pa
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ɥɟɧɢɹɨɬɞɟɥɶɧɨɝɨɫɢɦɜɨɥɚ
ɰɟɥɨɦɂɫɫɥɟɞɨɜɚɬɶɮɭɧɤɰɢɸ
ɢɢɫɫɥɟɞɨɜɚɬɶɜɥɢɹɧɢɟɜɟɪɨɹɬɧɨɫɬɢ
ɧɚɜɟɥɢɱɢɧɭɷɧɬɪɨɩɢɢɢɫɬɨɱɧɢɤɚɜ
a
i
ɧɚɷɤɫɬɪɟɦɭɦ
fpa
>@
i
fpa
>@
i
ɩɨɹɜ
pa
i
%Script_1_6
%ɉɨɫɬɪɨɟɧɢɟ ɝɪɚɮɢɤɚ ɨɞɧɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɜ ɡɚɜɢɫɢɦɨɫɬɢ
ɨɬ ɜɟɪɨɹɬɧɨɫɬɢ ɫɢɦɜɨɥɚ
clear all; close all; clc
%Ɂɚɞɚɧɢɟ ɚɧɨɧɢɦɧɨɣ ɮɭɧɤɰɢɢ
%--------------------------fun=@(x) -x.*log2(x+(x==0));
%--------------------------%Ɂɚɞɚɧɢɟ ɡɧɚɱɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɢ
p=0:.001:1;
%Ɋɚɫɱɟɬ ɡɧɚɱɟɧɢɣ ɷɧɬɪɨɩɢɢ
y=fun(p);
%ɉɨɫɬɪɨɟɧɢɟ ɝɪɚɮɢɤɚ
plot(p,y)
grid
xlabel('ȼɟɪɨɹɬɧɨɫɬɶ ɫɢɦɜɨɥɚ p')
ylabel('Ɏɭɧɤɰɢɹ f(p), ɛɢɬ')
title('Ƚɪɚɮɢɤ ɮɭɧɤɰɢɢ f(p)')
%Ɉɩɪɟɞɟɥɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ ɩɨ ɝɪɚɮɢɤɭ
[fm,im]=max(y);
disp('Ɉɩɪɟɞɟɥɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ ɩɨ ɝɪɚɮɢɤɭ')
disp('Ɇɚɤɫɢɦɭɦ ɮɭɧɤɰɢɢ, ɛɢɬ:')
disp(fm)
disp('Ɂɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ:')
disp(p(im))
ɑɢɫɥɟɧɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ
%
[pm,fm]=fminbnd(@(x) x.*log2(x+(x==0)),.2,.7);
disp('ɑɢɫɥɟɧɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ')
disp('Ɇɚɤɫɢɦɭɦ ɮɭɧɤɰɢɢ, ɛɢɬ:')
disp(-fm)
disp('Ɂɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ:')
disp(pm)
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 1
^`
A
%Ɍɨɱɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ
pm=1/exp(1);
fm=-pm*log2(pm);
disp('Ɍɨɱɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ')
disp('Ɇɚɤɫɢɦɭɦ ɮɭɧɤɰɢɢ:')
disp(fm)
disp('Ɂɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ:')
disp(pm)
%Ɉɛɨɡɧɚɱɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɧɚ ɝɪɚɮɢɤɟ
line([0 pm],[fm fm],'LineStyle','--')
line([pm pm],[fm 0],'LineStyle','--')
hold on
plot(pm,fm,'r.')
Ɉɬɱɟɬɢɤɨɧɬɪɨɥɶɧɵɟɜɨɩɪɨɫɵ
Ɉɬɱɟɬɩɨɥɚɛɨɪɚɬɨɪɧɨɣɪɚɛɨɬɟɨɮɨɪɦɥɹɟɬɫɹɜɜɢɞɟɞɨɤɭɦɟɧ
ɬɚ06 :RUG ɢɞɨɥɠɟɧɫɨɞɟɪɠɚɬɶ
ɇɚɡɜɚɧɢɟ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɵ ɲɪɢɮɬ 7LPHV 1HZ
5RPDQ
Ɋɟɡɭɥɶɬɚɬɵɜɵɩɨɥɧɟɧɢɹɜɫɟɯɪɚɫɱɟɬɧɵɯɡɚɞɚɧɢɣɜɤɥɸɱɚɹ
ɥɢɫɬɢɧɝɢ Pɮɚɣɥɨɜ ɤɨɩɢɪɭɟɦɵɟ ɢɡ ɨɤɧɚ (GLWRU ɪɟɡɭɥɶɬɚɬɵ ɢɯ
ɜɵɩɨɥɧɟɧɢɹ ɤɨɩɢɪɭɟɦɵɟ ɢɡ ɨɤɧɚ &RPPDQG :LQGRZ ɲɪɢɮɬ
&RXULHU 1HZɚɬɚɤɠɟɩɨɥɭɱɟɧɧɵɟɝɪɚɮɢɤɢ ɤɨɩɢɪɭɟɦɵɟ ɢɡ ɨɤɧɚ
)LJXUH ɩɨɤɨɦɚɧɞɟ(GLW _ &RS\ )LJXUH
Ɂɚɳɢɬɚ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɵ ɩɪɨɜɨɞɢɬɫɹ ɧɚ ɨɫɧɨɜɟ ɩɪɟɞ
ɫɬɚɜɥɟɧɧɨɝɨɨɬɱɟɬɚɢɨɬɜɟɬɨɜɧɚɤɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ ɢɡ ɫɥɟɞɭ
ɸɳɟɝɨɫɩɢɫɤɚ
ɑɬɨɬɚɤɨɟɞɢɫɤɪɟɬɧɵɣɢɫɬɨɱɧɢɤɢɧɮɨɪɦɚɰɢɢ"
Ⱦɚɣɬɟ ɨɩɪɟɞɟɥɟɧɢɟ ɥɨɝɚɪɢɮɦɢɱɟɫɤɨɣ ɦɟɪɵ ɤɨɥɢɱɟɫɬɜɚ
ɢɧɮɨɪɦɚɰɢɢ
Ia
i
ɉɟɪɟɱɢɫɥɢɬɟ ɫɜɨɣɫɬɜɚɥɨɝɚɪɢɮɦɢɱɟɫɤɨɣɦɟɪɵɤɨɥɢɱɟɫɬɜɚ
ɢɧɮɨɪɦɚɰɢɢ
ɉɪɢ ɨɞɢɧɚɤɨɜɨɦ ɡɧɚɱɟɧɢɢ ɜɟɪɨɹɬɧɨɫɬɢ ɫɢɦɜɨɥɚ ɤɚɤɨɟ
ɡɧɚɱɟɧɢɟɤɨɥɢɱɟɫɬɜɚɢɧɮɨɪɦɚɰɢɢɛɭɞɟɬɧɚɢɛɨɥɶɲɢɦɚɤɚɤɨɟɛɭ
ɞɟɬɧɚɢɦɟɧɶɲɢɦ± ɜɵɪɚɠɟɧɧɨɟɜɞɢɬɚɯɜɧɚɬɚɯɢɥɢɜɛɢɬɚɯ"
Ⱦɚɣɬɟ ɨɩɪɟɞɟɥɟɧɢɟ ɷɧɬɪɨɩɢɢ ɞɢɫɤɪɟɬɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɫ
ɚɥɮɚɜɢɬɨɦ
aa a
ɞɢɫɤɪɟɬɧɨɣɫɥɭɱɚɣɧɨɣɜɟɥɢɱɢɧɵ
ɤɚɤ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ
m
Ia
i
Ʉɨɥɢɱɟɫɬɜɨ ɢɧɮɨɪɦɚɰɢɢ ɢ ɷɧɬɪɨɩɢɹ
Ɂɚɩɢɲɢɬɟɜɵɪɚɠɟɧɢɟɞɥɹɷɧɬɪɨɩɢɢɞɢɫɤɪɟɬɧɨɝɨɢɫɬɨɱɧɢ
ɤɚɫɬɚɬɢɫɬɢɱɟɫɤɢɧɟɡɚɜɢɫɢɦɵɯɫɢɦɜɨɥɨɜ
ɉɟɪɟɱɢɫɥɢɬɟ ɫɜɨɣɫɬɜɚ ɷɧɬɪɨɩɢɢ ɞɢɫɤɪɟɬɧɨɝɨ ɢɫɬɨɱɧɢɤɚ
ɫɬɚɬɢɫɬɢɱɟɫɤɢɧɟɡɚɜɢɫɢɦɵɯɫɢɦɜɨɥɨɜ
Ɂɚɩɢɲɢɬɟɜɵɪɚɠɟɧɢɟɞɥɹɷɧɬɪɨɩɢɢɞɢɫɤɪɟɬɧɨɝɨɢɫɬɨɱɧɢ
ɤɚɫɨɛɴɟɦɨɦɚɥɮɚɜɢɬɚmɢɧɟɡɚɜɢɫɢɦɵɦɢɪɚɜɧɨɜɟɪɨɹɬɧɵɦɢɫɢɦ
ɜɨɥɚɦɢ
Ɂɚɩɢɲɢɬɟɜɵɪɚɠɟɧɢɟ ɞɥɹ ɷɧɬɪɨɩɢɢɞɜɨɢɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ
ɧɟɡɚɜɢɫɢɦɵɯɫɢɦɜɨɥɨɜ
ɉɪɢɤɚɤɢɯ ɡɧɚɱɟɧɢɹɯɜɟɪɨɹɬɧɨɫɬɢɩɨɹɜɥɟɧɢɹiɝɨɫɢɦɜɨ
ɥɚɞɨɫɬɢɝɚɸɬɫɹɧɚɢɛɨɥɟɟɫɭɳɟɫɬɜɟɧɧɵɟɡɧɚɱɟɧɢɹɫɥɚɝɚɟɦɵɯɜɢɞɚ
ORJ
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Ʌɚɛɨɪɚɬɨɪɧɚɹɪɚɛɨɬɚʋ
¸
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¸
m
ɋɨɜɦɟɫɬɧɚɹɢɭɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹ
ɐɟɥɶɪɚɛɨɬɵɂɡɭɱɢɬɶɩɨɧɹɬɢɹɫɨɜɦɟɫɬɧɨɣɢ ɭɫɥɨɜɧɨɣ ɷɧ
ɬɪɨɩɢɢ ɨɜɥɚɞɟɬɶ ɩɪɨɝɪɚɦɦɧɵɦɢ ɫɪɟɞɫɬɜɚɦɢ ɢɯ ɪɚɫɱɟɬɚ ɜ
0$7/$%
Ɍɟɨɪɟɬɢɱɟɫɤɢɟɫɜɟɞɟɧɢɹ
ɋɨɜɦɟɫɬɧɚɹɢɭɫɥɨɜɧɚɹɷɧɬɪɨɩɢɢɩɨɡɜɨɥɹɸɬ ɭɱɢɬɵɜɚɬɶ ɫɬɚ
ɬɢɫɬɢɱɟɫɤɢɟ ɫɜɹɡɢ ɦɟɠɞɭ ɚɥɮɚɜɢɬɚɦɢ ɞɜɭɯ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɢɫ
ɬɨɱɧɢɤɨɜ ɦɟɠɞɭ ɚɥɮɚɜɢɬɚɦɢ ɧɚ ɜɯɨɞɟ ɢ ɜɵɯɨɞɟ ɤɚɧɚɥɚ ɫɜɹɡɢ
ɦɟɠɞɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦɢ ɫɢɦɜɨɥɚɦɢ ɧɚ ɜɵɯɨɞɟ ɨɞɧɨɝɨ ɢɫɬɨɱ
ɧɢɤɚ
Ɉɛɴɟɞɢɧɟɧɢɟ ɢɫɬɨɱɧɢɤɨɜ ɫ ɚɥɮɚɜɢɬɚɦɢ
Aa
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ɢɦɟɸɳɢɯɨɛɴɟɦɵmɢnɫɨɨɬɜɟɬɫɬɜɟɧɧɨɩɨɥɧɨɫɬɶɸɯɚɪɚɤɬɟɪɢɡɭ
ɟɬɫɹ ɦɚɬɪɢɰɟɣ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜ
ɢ
ɤɨɬɨɪɚɹɢɦɟɟɬɫɥɟɞɭɸɳɢɣɜɢɞ
b
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ij
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ɩɨɮɨɪɦɭɥɟɫɭɦɦɚɪɧɨɣɜɟɪɨɹɬɧɨɫɬɢɩɨɥɧɨɣɝɪɭɩɩɵɫɨɛɵɬɢɣ
Ȼɟɡɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɨɜ ɞɥɹɤɚɠɞɨɝɨ
ɚɥɮɚɜɢɬɚɨɩɪɟɞɟɥɹɸɬɫɹɫɨɨɬɧɨɲɟɧɢɹɦɢ
±ɞɥɹ ɚɥɮɚɜɢɬɚ Aɫɭɦɦɢɪɨɜɚɧɢɟ ɷɥɟɦɟɧɬɨɜɦɚɬɪɢɰɵ
ɩɨɫɬɪɨɤɚɦ
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ɢ ɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ
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ɬɨɝɨ ɤɚɤ ɩɨɹɜɢɥɫɹ ɫɢɦɜɨɥ
ɩɨɹɜɥɟɧɢɹɫɢɦɜɨɥɚ
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ij
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ɫɬɢɱɟɫɤɢɫɜɹɡɚɧɧɵɯɚɧɫɚɦɛɥɟɣAɢBɧɚɯɨɞɢɬɫɹɩɨɮɨɪɦɭɥɟ
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ɢɡɷɬɢɯɚɥɮɚɜɢɬɨɜ
ɋ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦɫɨɨɬɧɨɲɟɧɢɣ ± ɫɨɜɦɟɫɬɧɭɸ ɷɧ
ɬɪɨɩɢɸɦɨɠɧɨɩɪɟɞɫɬɚɜɢɬɶɜɞɜɭɯɷɤɜɢɜɚɥɟɧɬɧɵɯɮɨɪɦɚɯ
ɉɟɪɜɚɹɮɨɪɦɚɡɚɩɢɫɢɢɦɟɟɬɜɢɞ
_ HAB HA HBA
ȼ ɫɭɦɦɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɥɚɝɚɟɦɵɟ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ
ɫɥɟɞɭɸɳɢɟɷɧɬɪɨɩɢɢ
±ɷɧɬɪɨɩɢɹɢɫɬɨɱɧɢɤɚɫɚɥɮɚɜɢɬɨɦA
m
ORJ
H A pa pa
¦
ii
i
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 2
n
n
¸
¸
¸¸¸
¸
n
±ɭɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹɚɥɮɚɜɢɬɚBɨɬɧɨɫɢɬɟɥɶɧɨɚɥɮɚɜɢɬɚA
mn
_ ORJ _
HB A pa b pb a
¦¦
ij
mn
_ORJ_
pa pb a pb a
¦¦
iji ji
ij
ij j i
ɤɨɬɨɪɚɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣɫɪɟɞɧɸɸ ɞɨɛɚɜɨɱɧɭɸɢɧɮɨɪɦɚɰɢɸ
ɫɨɞɟɪɠɚɳɭɸɫɹ ɜ ɫɢɦɜɨɥɟ ɚɥɮɚɜɢɬɚ B ɩɨɫɥɟ ɬɨɝɨ ɤɚɤ ɫɬɚɥɚ ɢɡ
ɜɟɫɬɧɚɢɧɮɨɪɦɚɰɢɹɜɫɪɟɞɧɟɦɫɨɞɟɪɠɚɳɚɹɫɹɜɫɢɦɜɨɥɟɚɥɮɚɜɢɬɚ
Aɢɬɚɤɠɟɦɨɠɟɬɛɵɬɶɡɚɩɢɫɚɧɚɜɜɢɞɟ
m
_ _
HB A pa HBa
¦
ii
i
ɝɞɟ
n
_ _ORJ_
HBa pb a pb a
¦
ijiji
j
±ɱɚɫɬɧɚɹɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹɚɥɮɚɜɢɬɚBɩɨɨɬɧɨɲɟɧɢɸɤɫɢɦɜɨ
ɥɭ
ɤɨɬɨɪɚɹɩɪɟɞɫɬɚɜɥɹɟɬɫɨɛɨɣɫɥɭɱɚɣɧɭɸɜɟɥɢɱɢɧɭɯɚɪɚɤɬɟ
a
i
ɪɢɡɭɸɳɭɸɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶɩɪɢɯɨɞɹɳɭɸɫɹɧɚɨɞɢɧɫɢɦɜɨɥɚɧ
ɫɚɦɛɥɹ B ɩɪɢ ɭɫɥɨɜɢɢɱɬɨ ɪɟɚɥɢɡɨɜɚɥɨɫɶɤɨɧɤɪɟɬɧɨɟ ɫɨɫɬɨɹɧɢɟ
ɚɧɫɚɦɛɥɹA
a
i
Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ ɜɟɥɢɱɢɧɚ ɭɫɥɨɜɧɨɣ ɷɧ
ɬɪɨɩɢɢ
pa
i
ɭɫɥɨɜɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣ
_HB A
ɨɩɪɟɞɟɥɹɟɬɫɹɜɢɞɨɦɪɚɫɩɪɟɞɟɥɟɧɢɹɜɟɪɨɹɬɧɨɫɬɟɣ
ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ A ɢ ɜɟɥɢɱɢɧɚɦɢ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ
ji
_
pb a
_ _ _
pb a pb a pb a
§·
¨¸
_ _ _
pb a pb a pb a
3
_
¨¸
BA
¨¸
¨¸
_ _ _
pb a pb a pb a
©¹
mm nm
n
n
ȼɬɨɪɚɹɮɨɪɦɚɡɚɩɢɫɢɨɩɪɟɞɟɥɹɟɬɫɹɜɵɪɚɠɟɧɢɟɦ
_ HAB HB HAB
ɢɜɤɥɸɱɚɟɬɜɫɟɛɹɫɥɟɞɭɸɳɢɟɷɧɬɪɨɩɢɢ
ɋɨɜɦɟɫɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
¸
¸
¸¸¸
¸
m
±ɷɧɬɪɨɩɢɸɚɥɮɚɜɢɬɚB
n
ORJ
H B pb pb
¦
jj
j
±ɭɫɥɨɜɧɭɸɷɧɬɪɨɩɢɸɚɥɮɚɜɢɬɚAɨɬɧɨɫɢɬɟɥɶɧɨɚɥɮɚɜɢɬɚB
mn
_ ORJ _
HAB pab pa b
¦¦
ij
nm
_ORJ_
pb pa b pa b
¦¦
jij ij
ji
ij i j
ɗɧɬɪɨɩɢɹ
_HAB
ɩɪɟɞɫɬɚɜɥɹɟɬɫɨɛɨɣɫɪɟɞɧɸɸɞɨɛɚɜɨɱɧɭɸɢɧ
ɮɨɪɦɚɰɢɸɤɨɬɨɪɭɸɞɚɟɬ ɫɢɦɜɨɥɚɥɮɚɜɢɬɚAɩɨɫɥɟɬɨɝɨɤɚɤɫɬɚ
ɥɚɢɡɜɟɫɬɧɚɢɧɮɨɪɦɚɰɢɹɜ ɫɪɟɞɧɟɦ ɫɨɞɟɪɠɚɳɚɹɫɹ ɜ ɫɢɦɜɨɥɟ ɚɥ
ɮɚɜɢɬɚBɢɦɨɠɟɬɛɵɬɶɡɚɩɢɫɚɧɚɜɷɤɜɢɜɚɥɟɧɬɧɨɦɜɢɞɟ
n
_ _
HAB pb HAb
¦
j
jj
ɝɞɟ
m
_ _ORJ_
HAb pa b pa b
¦
jijij
i
±ɱɚɫɬɧɚɹɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹɚɥɮɚɜɢɬɚAɩɨɨɬɧɨɲɟɧɢɸɤɫɢɦɜɨ
ɥɭ
ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɩɪɢɯɨɞɹɳɭɸɫɹ ɧɚ
b
j
ɨɞɢɧ ɫɢɦɜɨɥ ɚɧɫɚɦɛɥɹ A ɩɪɢ ɭɫɥɨɜɢɢ ɱɬɨ ɪɟɚɥɢɡɨɜɚɥɨɫɶ ɤɨɧ
ɤɪɟɬɧɨɟɫɨɫɬɨɹɧɢɟ
ɚɧɫɚɦɛɥɹB
b
j
ɋɨɨɬɧɨɲɟɧɢɟ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
_HAB
ɡɚɜɢɫɢɬɨɬ ɪɚɫɩɪɟɞɟɥɟɧɢɹɜɟɪɨɹɬɧɨɫɬɟɣ
pb
j
ɫɢɦɜɨɥɨɜ
ɚɥɮɚɜɢɬɚBɢɜɟɥɢɱɢɧɷɥɟɦɟɧɬɨɜɦɚɬɪɢɰɵɭɫɥɨɜɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣ
ij
3
_
AB
_ _ _
pa b pa b pa b
§·
¨¸
_ _ _
pa b pa b pa b
¨¸
n
n
¨¸
¨¸
_ _ _
pa b pa b pa b
©¹
mm mn
_
pa b
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 2
Ɋɢɫɋɨɨɬɧɨɲɟɧɢɹɦɟɠɞɭɷɧɬɪɨɩɢɹɦɢ
HA
HB
_HAB
_HB A
HAB
ɍɹɫɧɟɧɢɸ ɫɨɨɬɧɨɲɟɧɢɣ ɦɟɠɞɭ ɪɚɫɫɦɨɬɪɟɧɧɵɦɢ ɷɧɬɪɨɩɢɹ
ɦɢɞɢɫɤɪɟɬɧɵɯɢɫɬɨɱɧɢɤɨɜɫɡɚɜɢɫɢɦɵɦɢɚɥɮɚɜɢɬɚɦɢ AɢBɫɩɨ
ɫɨɛɫɬɜɭɟɬ ɢɯ ɝɪɚɮɢɱɟɫɤɨɟ ɨɬɨɛɪɚɠɟɧɢɟ ɜ ɜɢɞɟ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ
ɞɢɚɝɪɚɦɦȼɟɧɧɚ±ɗɣɥɟɪɚɩɨɤɚɡɚɧɧɵɯɧɚɪɢɫ
ɇɚ ɷɬɢɯ ɞɢɚɝɪɚɦɦɚɯ ɡɚɲɬɪɢɯɨɜɚɧɧɵɦɢ ɨɛɥɚɫɬɹɦɢ ɨɛɨɡɧɚ
ɱɟɧɵ ɷɧɬɪɨɩɢɢ
ɬɪɨɩɢɹ
HAB
HA
ɚɬɚɤɠɟɭɫɥɨɜɧɵɟɷɧɬɪɨɩɢɢ
ɢ
HB
ɚɥɮɚɜɢɬɨɜ A ɢ B ɫɨɜɦɟɫɬɧɚɹ ɷɧ
_HAB
ɢ
_HB A
ɍɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹɢɦɟɟɬɫɥɟɞɭɸɳɢɟɫɜɨɣɫɬɜɚ
ɍɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹɟɫɬɶɜɟɥɢɱɢɧɚɨɝɪɚɧɢɱɟɧɧɚɹɧɟɪɚɜɟɧ
ɫɬɜɚɦɢ
_HAB HAdd
_HB A HBdd
ȿɫɥɢɦɟɠɞɭɚɥɮɚɜɢɬɚɦɢAɢBɢɦɟɟɬɫɹɨɞɧɨɡɧɚɱɧɚɹɫɜɹɡɶ
ɬɨɭɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹɨɛɪɚɳɚɟɬɫɹɜɧɭɥɶ
_ _ HAB HBA
ȿɫɥɢ ɚɥɮɚɜɢɬɵ A ɢ B ɫɬɚɬɢɫɬɢɱɟɫɤɢ ɧɟɡɚɜɢɫɢɦɵ ɬɨ
ɭɫɥɨɜɧɚɹɷɧɬɪɨɩɢɹɪɚɜɧɚɛɟɡɭɫɥɨɜɧɨɣ
_ HAB HA
_ HB A HB
ɋɜɨɣɫɬɜɚɫɨɜɦɟɫɬɧɨɣɷɧɬɪɨɩɢɢ
ɋɨɜɦɟɫɬɧɚɹɷɧɬɪɨɩɢɹɨɛɥɚɞɚɟɬɫɜɨɣɫɬɜɨɦɫɢɦɦɟɬɪɢɢ
HAB HBA
ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ ɟɫɬɶ ɧɟɨɬɪɢɰɚɬɟɥɶɧɚɹ ɜɟɥɢɱɢɧɚ
ɫɜɟɪɯɭɨɝɪɚɧɢɱɟɧɧɚɹɧɟɪɚɜɟɧɫɬɜɨɦ
ɋɨɜɦɟɫɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ
HAB HA HBd
ȿɫɥɢɦɟɠɞɭɚɥɮɚɜɢɬɚɦɢAɢBɢɦɟɟɬɫɹɨɞɧɨɡɧɚɱɧɚɹɫɜɹɡɶ
ɩɪɢɷɬɨɦ
HA HB
ɬɨɫɨɜɦɟɫɬɧɚɹɷɧɬɪɨɩɢɹɫɨɫɬɚɜɥɹɟɬ
HAB HA HB
ȿɫɥɢɚɥɮɚɜɢɬɵ A ɢB ɫɬɚɬɢɫɬɢɱɟɫɤɢ ɧɟɡɚɜɢɫɢɦɵɬɨɫɨɜ
ɦɟɫɬɧɚɹɷɧɬɪɨɩɢɹɪɚɜɧɚɫɭɦɦɟɢɯɷɧɬɪɨɩɢɣ
HAB HA HB
Ɋɚɫɱɟɬɧɵɟɡɚɞɚɧɢɹ
ɁɚɞɚɧɢɟɆɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɛɴɟɞɢɧɟ
ɧɢɹɚɥɮɚɜɢɬɨɜAɢBɢɦɟɟɬɜɢɞ
pa b pa b
§·
3
¨¸
pa b pa b
AB
¨¸
¨¸
pa b pa b
©¹
Ɋɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ
_
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
_HB A
ɢ
_HAB
HBa
ɫɨɜɦɟɫɬɧɭɸɷɧɬɪɨɩɢɸ
ɢ
i
HAb
§·
¨¸
¨¸
¨¸
©¹
HA
ɢ
_
ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ
j
HAB
HB
ɱɚɫɬɧɵɟ
ɩɨɮɨɪɦɭɥɚɦ
ɢ
%Script_2_1
%Ɋɚɫɱɟɬ ɭɫɥɨɜɧɨɣ ɢ ɫɨɜɦɟɫɬɧɨɣ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɨɜ Ⱥ ɢ
ȼ
%ɩɪɢ ɡɚɞɚɧɧɨɣ ɦɚɬɪɢɰɟ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ Ɋ(Ⱥ,ȼ)
clear all; close all; clc
%Ɂɚɞɚɧɢɟ ɚɧɨɧɢɦɧɨɣ ɮɭɧɤɰɢɢ
%--------------------------fun=@(x) -x.*log2(x+(x==0));
%--------------------------%Ɂɧɚɱɟɧɢɹ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
P_AB=[.1 .25;.2 0;.3 .15];
disp('Ɇɚɬɪɢɰɚ ɫɨɜɦɟɫɬɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(A,B):')
disp(P_AB)
%Ɉɩɪɟɞɟɥɟɧɢɟ ɨɛɴɟɦɨɜ ɚɥɮɚɜɢɬɨɜ
[m, n]=size(P_AB);
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 2
disp('Ɉɛɴɟɦ ɚɥɮɚɜɢɬɚ Ⱥ:')
disp(m)
disp('Ɉɛɴɟɦ ɚɥɮɚɜɢɬɚ B:')
disp(n)
%Ɋɚɫɱɟɬ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ A
p_a=sum(P_AB,2);
disp('ȼɟɪɨɹɬɧɨɫɬɢ ɫɢɦɜɨɥɨɜ ɚɥɮɚɜɢɬɚ Ⱥ:')
disp(p_a)
%Ɋɚɱɟɬ ɷɧɬɪɨɩɢɢ ɚɥɮɚɜɢɬɚ Ⱥ
H_A=sum(fun(p_a));
disp('ɗɧɬɪɨɩɢɹ ɚɥɮɚɜɢɬɚ Ⱥ, ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_A)
%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)
%ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B) - ɩɟɪɜɚɹ ɮɨɪɦɚ
%Ɋɚɫɱɟɬ
P(B|A)
P_B_A=bsxfun(@rdivide,P_AB,p_a(:));
disp('Ɇɚɬɪɢɰɚ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(B|A):')
disp(P_B_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)
%Ɋɚɫɱɟɬ ɭɫɥɨɜɧɨɣ ɷɧɬɪɨɩɢɢ H(B|A)
H_B_A=sum(p_a.*H_B_a);
disp('ɍɫɥɨɜɧɚɹ ɷɧɬɪɨɩɢɹ H(B|A), ɛɢɬ/ɫɢɦɜɨɥ:')
disp(H_B_A)
%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)
%ɋɨɜɦɟɫɬɧɚɹ ɷɧɬɪɨɩɢɹ H(A,B) - ɜɬɨɪɚɹ ɮɨɪɦɚ
%Ɋɚɫɱɟɬ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
P(A|B)
P_A_B=bsxfun(@rdivide,P_AB,p_b(:)');
disp('Ɇɚɬɪɢɰɚ
ɷɥɟɦɟɧɬɨɜ
ɦɚɬɪɢɰɵ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ P(A|B):')
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