mirror of
https://github.com/eddyem/lectures.git
synced 2025-12-06 18:45:18 +03:00
570 lines
23 KiB
TeX
570 lines
23 KiB
TeX
\documentclass[a4paper,12pt]{extarticle}
|
||
\usepackage{/home/eddy/ed, verbatim}
|
||
\title{ðÒÁËÔÉËÕÍ \No3: ÐÏÇÒÅÛÎÏÓÔÉ, ÍÅÔÏÄ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ×}
|
||
\author{}\date{}\nocolon
|
||
|
||
\long\def\task#1{\noindent\leavevmode\refstepcounter{sect}\llap{\textbf{\thesect}\;}\indent\textit{#1}}
|
||
\def\t#1{{\upshape\ttfamily #1}}
|
||
|
||
\begin{document}
|
||
\maketitle
|
||
\section{ðÏÇÒÅÛÎÏÓÔÉ}
|
||
\subsection{}
|
||
îÁÊÄÅÍ ÏÂÝÕÀ ÓÒÅÄÎÀÀ ÓÏ×ÏËÕÐÎÏÓÔÉ, ÓÏÓÔÏÑÝÅÊ ÉÚ ÓÌÅÄÕÀÝÉÈ ÔÒÅÈ ÇÒÕÐÐ:
|
||
$$\renewcommand{\arraystretch}{0}
|
||
\begin{tabular}{|r|c|c|c|c|c|c|}
|
||
\hbox to 5cm{}&\hbox to 1.3cm{}& \hbox to 1.3cm{}& \hbox to 1.3cm{}&
|
||
\hbox to 1.3cm{}& \hbox to 1.3cm{}& \hbox to 1.3cm{}\\
|
||
\hline
|
||
\strut çÒÕÐÐÁ& \multicolumn{2}{|c|}{I} & \multicolumn{2}{|c|}{II} &
|
||
\multicolumn{2}{|c|}{III} \\
|
||
\hline
|
||
\strut úÎÁÞÅÎÉÅ ÐÒÉÚÎÁËÁ&1&3&2&4
|
||
&3&6\\
|
||
\strut þÁÓÔÏÔÁ ÐÒÉÚÎÁËÁ&11&34&22&28&31&14\\\hline
|
||
\strut ïÂßÅÍ ×ÙÂÏÒËÉ&\multicolumn{2}{|c|}{$11+34=45$}&\multicolumn{2}{|c|}{$22+28=50$}&
|
||
\multicolumn{2}{|c|}{$31+14=45$}\\
|
||
\hline
|
||
\end{tabular}
|
||
$$
|
||
|
||
äÌÑ ÎÁÞÁÌÁ ÎÁÊÄÅÍ ÇÒÕÐÐÏ×ÙÅ ÓÒÅÄÎÉÅ: $\aver{x_1}$, $\aver{x_2}$ É~$\aver{x_3}$:
|
||
\begin{verbatim}
|
||
x1 = (11*1 + 34*3)/45
|
||
x2 = (22*2 + 28*4)/50
|
||
x3 = (31*3 + 14*6)/45
|
||
\end{verbatim}
|
||
|
||
ôÅÐÅÒØ ÎÁÊÄÅÍ ÉÈ ÓÒÅÄÎÅÅ:
|
||
\begin{verbatim}
|
||
X = (x1*45 + x2*50 + x3*45)/(45 + 50 + 45)
|
||
\end{verbatim}
|
||
|
||
ïÄÎÁËÏ, ÐÒÉ ÒÁÂÏÔÅ Ó ÂÏÌØÛÉÍÉ ÍÁÓÓÉ×ÁÍÉ ÄÁÎÎÙÈ ÌÕÞÛÅ ÉÓÐÏÌØÚÏ×ÁÔØ ÐÒÅÉÍÕÝÅÓÔ×Á
|
||
ÍÁÔÒÉÞÎÏÊ ÁÌÇÅÂÒÙ:
|
||
\begin{verbatim}
|
||
xi = [1 3 2 4 3 6];
|
||
ni = [11 34 22 28 31 14 ];
|
||
N = sum(ni)
|
||
X = sum(xi.*ni/N)
|
||
\end{verbatim}
|
||
|
||
îÁÊÄÅÍ\Ë ÇÅÎÅÒÁÌØÎÕÀ ÄÉÓÐÅÒÓÉÀ\Î É ÇÅÎÅÒÁÌØÎÏÅ ÓÒÅÄÎÅË×ÁÄÒÁÔÉÞÎÏÅ ÏÔËÌÏÎÅÎÉÅ
|
||
ÄÁÎÎÏÊ ×ÙÂÏÒËÉ:
|
||
\begin{verbatim}
|
||
D = sum(ni.*(xi-X).^2)/N
|
||
sigma=sqrt(D)
|
||
\end{verbatim}
|
||
ëÒÏÍÅ ÔÏÇÏ, ÏÐÒÅÄÅÌÉÔØ ÓÒÅÄÎÅË×ÁÄÒÁÔÉÞÎÏÅ ÏÔËÌÏÎÅÎÉÅ ÒÑÄÁ~$x$ ÍÏÖÎÏ ÐÒÉ ÐÏÍÏÝÉ
|
||
ËÏÍÁÎÄÙ \verb'std(x)'.
|
||
|
||
\subsection{}
|
||
òÁÓÓÍÏÔÒÉÍ ÒÑÄ ÉÚÍÅÒÅÎÉÊ ÎÅËÏÔÏÒÏÊ ÆÉÚÉÞÅÓËÏÊ ×ÅÌÉÞÉÎÙ~$x$. òÅÚÕÌØÔÁÔÙ
|
||
ÓÅÒÉÉ ÉÚÍÅÒÅÎÉÊ ÚÁÄÁÎÙ ÔÁÂÌÉÃÅÊ ($\nu_i$~-- ÞÁÓÔÏÔÁ ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÅÇÏ ÚÎÁÞÅÎÉÑ~$x_i$):
|
||
$$
|
||
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|}
|
||
\hline
|
||
$x_i$& 31 & 28 & 34 & 26 & 35 & 30 & 34 & 32 & 40 & 20 \\
|
||
$\nu_i$& 20 & 12 & 10 & 5 & 7 & 20 & 12 & 19 & 4 & 2 \\
|
||
\hline
|
||
\end{tabular}
|
||
$$
|
||
éÚ×ÅÓÔÎÏ, ÞÔÏ ÎÅËÏÔÏÒÙÅ ÒÅÚÕÌØÔÁÔÙ ÍÏÇÕÔ ÂÙÔØ ÚÁ×ÅÄÏÍÏ ÏÛÉÂÏÞÎÙÍÉ.
|
||
îÁÍ ÎÅÏÂÈÏÄÉÍÏ ÏÃÅÎÉÔØ ÓÒÅÄÎÅÅ ÚÎÁÞÅÎÉÅ ÄÁÎÎÏÊ ×ÅÌÉÞÉÎÙ, ÉÓËÌÀÞÉ× ÏÛÉÂÏÞÎÙÅ
|
||
ÒÅÚÕÌØÔÁÔÙ.
|
||
óÏÓÔÁ×ÉÍ ÍÁÓÓÉ×Ù ×ÅÌÉÞÉÎÙ~$x$ É ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÉÈ ÞÁÓÔÏÔ~$n$:
|
||
\begin{verbatim}
|
||
x = [ 31 28 34 26 35 30 34 32 40 20];
|
||
n = [ 20 12 10 5 7 20 12 19 4 2];
|
||
\end{verbatim}
|
||
ïÔÏÂÒÁÚÉ× ÄÁÎÎÙÅ ÎÁ ÇÒÁÆÉËÅ (\verb"plot(x,n,'o')") ÍÏÖÎÏ ÚÁÍÅÔÉÔØ, ÞÔÏ ÄÅÊÓÔ×ÉÔÅÌØÎÏ
|
||
ÎÅËÏÔÏÒÙÅ ÚÎÁÞÅÎÉÑ ÓÉÌØÎÏ ÏÔËÌÏÎÑÀÔÓÑ ÏÔ ÐÏÌÏÖÅÎÉÑ, ËÏÔÏÒÏÅ ÏÎÉ ÚÁÎÉÍÁÌÉ ÂÙ ÐÒÉ
|
||
ÎÏÒÍÁÌØÎÏÍ ÒÁÓÐÒÅÄÅÌÅÎÉÉ.
|
||
|
||
îÁÊÄÅÍ ÓÒÅÄÎÅÅ ÚÎÁÞÅÎÉÅ ×ÅÌÉÞÉÎÙ~$x$ É ÅÅ ÓÒÅÄÎÅË×ÁÄÒÁÔÉÞÎÏÅ ÏÔËÌÏÎÅÎÉÅ:
|
||
\begin{verbatim}
|
||
X = sum(x.*n)/sum(n)
|
||
sigma = sqrt(sum(n.*(x-X).^2)/sum(n))
|
||
\end{verbatim}
|
||
ïÐÒÅÄÅÌÉÍ ÇÒÁÎÉÃÙ ÄÏ×ÅÒÉÔÅÌØÎÏÇÏ ÉÎÔÅÒ×ÁÌÁ $[a,b]$ × ÐÒÅÄÅÌÁÈ ÔÒÅÈ~$\sigma$:
|
||
\begin{verbatim}
|
||
a = X-3*sigma
|
||
b = X+3*sigma
|
||
\end{verbatim}
|
||
ôÅÐÅÒØ ÉÓËÌÀÞÉÍ ÉÚ ×ÙÂÏÒËÉ ÚÎÁÞÅÎÉÑ, ×ÙÈÏÄÑÝÉÅ ÚÁ ÐÒÅÄÅÌÙ ÉÎÔÅÒ×ÁÌÁ. ðÒÉ ÐÏÍÏÝÉ
|
||
ÆÕÎËÃÉÉ \verb'find' ÍÏÖÎÏ ÎÁÊÔÉ ÉÎÄÅËÓÙ ÞÌÅÎÏ× ÍÁÓÓÉ×Á, ÕÄÏ×ÌÅÔ×ÏÒÑÀÝÉÈ ÚÁÄÁÎÎÏÍÕ
|
||
ÕÓÌÏ×ÉÀ. éÓËÌÀÞÉÔØ ÌÉÛÎÉÅ ÜÌÅÍÅÎÔÙ ÍÏÖÎÏ ÔÁË:
|
||
\begin{verbatim}
|
||
idx = find(x < a);
|
||
x(idx) = []; n(idx) = [];
|
||
idx = find(x > b);
|
||
x(idx) = []; n(idx) = [];
|
||
\end{verbatim}
|
||
|
||
ôÅÐÅÒØ ÐÏ×ÔÏÒÉÍ ×ÙÞÉÓÌÅÎÉÅ \verb'X' É \verb'sigma', \verb'a' É \verb'b'.\Ë
|
||
äÌÑ ÔÏÇÏ, ÞÔÏÂÙ ×ÙÚ×ÁÔØ ÉÚ ÉÓÔÏÒÉÉ ËÏÍÁÎÄ ÓÔÒÏËÕ, ÎÁÞÉÎÁÀÝÕÀÓÑ Ó ÏÐÒÅÄÅÌÅÎÎÙÈ
|
||
ÓÉÍ×ÏÌÏ×, ÎÁÂÅÒÉÔÅ ÏÄÉÎ--Ä×Á ÐÅÒ×ÙÈ ÓÉÍ×ÏÌÁ É ÎÁÖÍÉÔÅ ËÌÁ×ÉÛÕ <<××ÅÒÈ>>\Î.
|
||
ôÁËÉÍ ÏÂÒÁÚÏÍ ÍÏÖÎÏ ÂÙÓÔÒÏ ×ÙÚ×ÁÔØ ÉÚ ÉÓÔÏÒÉÉ ËÏÍÁÎÄ ÎÕÖÎÕÀ ×ÁÍ ËÏÍÁÎÄÕ, ÎÅ
|
||
ÐÅÒÅÂÉÒÁÑ ×ÓÅ ÐÒÏÍÅÖÕÔÏÞÎÙÅ.
|
||
|
||
ôÅÐÅÒØ ÐÒÏ×ÅÒÉÍ, ÎÅ ×ÌÉÑÅÔ ÌÉ <<ÐÏÄÏÚÒÉÔÅÌØÎÏÅ>> ÚÎÁÞÅÎÉÅ $x=40$ ÎÁ ÔÏÞÎÏÓÔØ
|
||
ÉÚÍÅÒÅÎÉÑ. îÁÊÄÅÍ ÍÅÄÉÁÎÕ ÎÁÛÅÇÏ ÒÑÄÁ É ÏÃÅÎÉÍ ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ ÐÏ ÍÅÄÉÁÎÅ.
|
||
äÌÑ ÜÔÏÇÏ ÎÁÍ ÎÅÏÂÈÏÄÉÍÏ ÐÏÓÔÒÏÉÔØ ÎÏ×ÙÊ ×ÅËÔÏÒ~\verb'newx', × ËÏÔÏÒÏÍ ÚÎÁÞÅÎÉÑ
|
||
×ÅÌÉÞÉÎÙ~$x$ ÂÕÄÕÔ ÓÏÄÅÒÖÁÔØÓÑ ÓÔÏÌØËÏ ÒÁÚ, ËÁËÏ×Á ÉÈ ÞÁÓÔÏÔÁ:
|
||
\begin{verbatim}
|
||
newx = []; for a = [1:length(n)]
|
||
newx = [newx ones(1,n(a)).*x(a)];
|
||
endfor
|
||
med = median(newx)
|
||
a = med-3*sigma
|
||
b = med+3*sigma
|
||
\end{verbatim}
|
||
äÅÊÓÔ×ÉÔÅÌØÎÏ, ÚÎÁÞÅÎÉÅ $x=40$ ×ÙÂÉ×ÁÅÔÓÑ ÉÚ ÄÏ×ÅÒÉÔÅÌØÎÏÇÏ ÉÎÔÅÒ×ÁÌÁ.
|
||
õÄÁÌÉÍ ÅÇÏ:
|
||
\begin{verbatim}
|
||
idx = find(x==40);
|
||
x(idx) = []; n(idx) = [];
|
||
\end{verbatim}
|
||
É ÎÁÊÄÅÍ~$\aver{x}$, ÂÌÉÚËÏÅ Ë ÉÓÔÉÎÎÏÍÕ:
|
||
\begin{verbatim}
|
||
X = sum(x.*n)/sum(n)
|
||
sigma = sqrt(sum(n.*(x-X).^2)/sum(n))
|
||
a = X-3*sigma
|
||
b = X+3*sigma
|
||
find(x>b)
|
||
find(x<a)
|
||
\end{verbatim}
|
||
éÔÁË, ×ÓÅ ÏÓÔÁ×ÛÉÅÓÑ ÚÎÁÞÅÎÉÑ~$x_i$ ÕÄÏ×ÌÅÔ×ÏÒÑÀÔ ËÒÉÔÅÒÉÀ
|
||
<<ÔÒÅÈ ÓÉÇÍ>>, ÓÌÅÄÏ×ÁÔÅÌØÎÏ, ÍÏÖÎÏ ÚÁÐÉÓÁÔØ ÏÔ×ÅÔ: $x=31.3\pm2.3$.
|
||
|
||
\subsection{}
|
||
ôÅÐÅÒØ ÏÐÒÅÄÅÌÉÍ ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ ×ÅÌÉÞÉÎÙ $\aver{x}$ Ó
|
||
ÎÁÄÅÖÎÏÓÔØÀ~95\% ÐÒÉ ÐÏÍÏÝÉ ÒÁÓÐÒÅÄÅÌÅÎÉÑ óÔØÀÄÅÎÔÁ. äÌÑ ÜÔÏÇÏ × Octave ÓÕÝÅÓÔ×ÕÅÔ
|
||
ÆÕÎËÃÉÑ~\verb'ttest'. ÷ ÐÒÏÓÔÅÊÛÅÍ ÓÌÕÞÁÅ ×ÉÄÁ \verb'h=ttest(x)' ÏÎÁ ×ÏÚ×ÒÁÝÁÅÔ
|
||
×ÅÒÏÑÔÎÏÓÔØ ÏÔËÌÏÎÅÎÉÑ ÇÉÐÏÔÅÚÙ Ï ÎÏÒÍÁÌØÎÏÍ ÒÁÓÐÒÅÄÅÌÅÎÉÉ ×ÅÌÉÞÉÎÙ~$x$ Ó
|
||
ÍÁÔÅÍÁÔÉÞÅÓËÉÍ ÏÖÉÄÁÎÉÅÍ $\mean{x}=0$. ðÒÏ×ÅÒËÁ ÄÁÓÔ ÒÅÚÕÌØÔÁÔ:~1. äÅÊÓÔ×ÉÔÅÌØÎÏ,
|
||
ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ÎÁÛÅÊ ×ÅÌÉÞÉÎÙ ÄÁÌÅËÏ ÎÅ ÒÁ×ÎÏ ÎÕÌÀ. ÷ÔÏÒÏÊ ÁÒÇÕÍÅÎÔ
|
||
ÆÕÎËÃÉÉ \verb'ttest' ÚÁÄÁÅÔ ÐÒÅÄÐÏÌÁÇÁÅÍÏÅ ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ. ðÒÏ×ÅÒÉÍ:
|
||
\verb'h=ttest(x,X)'. ðÏÌÕÞÁÅÍ: \verb'h=0'. ô.Å., ÍÏÖÎÏ ÐÒÉÎÑÔØ ÇÉÐÏÔÅÚÕ
|
||
Ï ÇÁÕÓÓÏ×ÏÊ ÆÏÒÍÅ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ×ÅÌÉÞÉÎÙ~$x$ ÏËÏÌÏ ÅÅ ÓÒÅÄÎÅÇÏ ÚÎÁÞÅÎÉÑ.
|
||
ïÃÅÎÉÔØ 95\%-Ê ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ ×ÅÌÉÞÉÎÙ~$x$ ÍÏÖÎÏ ÐÒÉ ÐÏÍÏÝÉ ÒÁÓÛÉÒÅÎÎÏÇÏ
|
||
×Ù×ÏÄÁ ÆÕÎËÃÉÉ \verb'ttest' × ÆÏÒÍÅ \verb'[h,p,ci]=ttest(x,X)'. ÷ ÜÔÏÍ ÓÌÕÞÁÅ
|
||
ÐÁÒÁÍÅÔÒ \verb'h' ÓÏÏÂÝÁÅÔ Ï ÓÔÅÐÅÎÉ ÎÅÎÁÄÅÖÎÏÓÔÉ ÇÉÐÏÔÅÚÙ, \verb'p' ÒÁ×ÅÎ
|
||
×ÅÒÏÑÔÎÏÓÔÉ ÓÏ×ÐÁÄÅÎÉÑ ×ÅÌÉÞÉÎÙ~\verb'X' Ó ÍÁÔÅÍÁÔÉÞÅÓËÉÍ ÏÖÉÄÁÎÉÅÍ ÒÑÄÁ~\verb'x',
|
||
\verb'ci' ÓÏÏÂÝÁÅÔ ÇÒÁÎÉÃÙ 95\%-ÇÏ ÄÏ×ÅÒÉÔÅÌØÎÏÇÏ ÉÎÔÅÒ×ÁÌÁ.
|
||
ïÐÒÅÄÅÌÉÍ ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ ÄÌÑ ÎÁÛÅÇÏ ÒÑÄÁ ÂÅÚ ÉÓËÌÀÞÅÎÉÑ ÚÁ×ÅÄÏÍÏ ÌÏÖÎÙÈ
|
||
ÒÅÚÕÌØÔÁÔÏ× É Ó ÉÈ ÉÓËÌÀÞÅÎÉÅÍ:
|
||
\begin{verbatim}
|
||
x = [ 31 28 34 26 35 30 34 32 40 20];
|
||
n = [ 20 12 10 5 7 20 12 19 4 2];
|
||
newx = []; for a = [1:length(n)]
|
||
newx = [newx ones(1,n(a)).*x(a)];
|
||
endfor
|
||
|
||
[h,p,ci] = ttest(newx, X)
|
||
newx = []; for a = 1:8
|
||
newx = [newx ones(1,n(a)).*x(a)];
|
||
endfor
|
||
[h,p,ci] = ttest(newx, X)
|
||
\end{verbatim}
|
||
|
||
éÔÁË, × ÏÂÏÉÈ ÓÌÕÞÁÑÈ ÇÉÐÏÔÅÚÁ Ï ÓÏÏÔ×ÅÔÓÔ×ÉÉ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ×ÅÌÉÞÉÎÙ~$x$
|
||
ÎÏÒÍÁÌØÎÏÍÕ ÒÁÓÐÒÅÄÅÌÅÎÉÀ ÐÒÉÎÉÍÁÅÔÓÑ, ÏÄÎÁËÏ, ×Ï ×ÔÏÒÏÍ ÓÌÕÞÁÅ ×ÅÒÏÑÔÎÏÓÔØ
|
||
ÏÐÒÅÄÅÌÅÎÉÑ ÍÁÔÅÍÁÔÉÞÅÓËÏÇÏ ÏÖÉÄÁÎÉÑ~$\mean{x}$ ×ÙÛÅ, É ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ
|
||
ÕÖÅ, ÞÔÏ Ñ×ÎÏ Ó×ÉÄÅÔÅÌØÓÔ×ÕÅÔ Ï ÂÏÌØÛÅÊ ÎÁÄÅÖÎÏÓÔÉ ×ÙÞÉÓÌÅÎÉÊ.
|
||
|
||
\subsection{}
|
||
Octave ÐÒÅÄÏÓÔÁ×ÌÑÅÔ ÏÇÒÏÍÎÙÊ ÎÁÂÏÒ ÉÎÓÔÒÕÍÅÎÔÁÌØÎÙÈ ÓÒÅÄÓÔ×. ïÄÎÁËÏ, ÐÒÉ ÒÁÂÏÔÅ Ó ÂÏÌØÛÉÍ
|
||
ËÏÌÉÞÅÓÔ×ÏÍ ÏÄÎÏÏÂÒÁÚÎÙÈ ÄÁÎÎÙÈ ÐÒÉÈÏÄÉÔÓÑ ÍÎÏÇÏ ÒÁÚ ÐÏ×ÔÏÒÑÔØ ÏÄÎÉ É ÔÅ ÖÅ ËÏÍÁÎÄÙ. üÔÕ ÚÁÄÁÞÕ
|
||
ÍÏÖÎÏ ÕÐÒÏÓÔÉÔØ, ÓÏÚÄÁ×\Ö ÓËÒÉÐÔ\Î (ÉÌÉ m-ÆÁÊÌ). óËÒÉÐÔ ÐÒÅÄÓÔÁ×ÌÑÅÔ ÓÏÂÏÊ ÏÐÉÓÁÎÉÅ É ÒÅÁÌÉÚÁÃÉÀ
|
||
ÐÏÌØÚÏ×ÁÔÅÌØÓËÏÊ ÆÕÎËÃÉÉ, ËÏÔÏÒÁÑ ×ÙÚÙ×ÁÅÔÓÑ ÉÚ ËÏÍÁÎÄÎÏÊ ÓÔÒÏËÉ Octave ÁÎÁÌÏÇÉÞÎÏ
|
||
ÌÀÂÏÊ ËÏÍÁÎÄÅ, ÏÄÎÁËÏ ÍÏÖÅÔ ÓÏÄÅÒÖÁÔØ ÚÎÁÞÉÔÅÌØÎÏÅ ËÏÌÉÞÅÓÔ×Ï ÉÎÓÔÒÕËÃÉÊ,
|
||
ÏÂÌÅÇÞÁÀÝÉÈ ÒÁÂÏÔÕ ÐÏÌØÚÏ×ÁÔÅÌÑ.
|
||
|
||
M-ÆÁÊÌ ÍÏÖÅÔ ÓÏÄÅÒÖÁÔØ ÌÀÂÙÅ ÉÎÓÔÒÕËÃÉÉ. åÓÌÉ ÏÎ ÎÅ ÎÁÞÉÎÁÅÔÓÑ ÓÏ ÓÌÏ×Á
|
||
\verb'function', ×ÙÐÏÌÎÑÅÔÓÑ ×ÓÅ ÅÇÏ ÓÏÄÅÒÖÉÍÏÅ. õÄÏÂÎÅÅ,
|
||
ÏÄÎÁËÏ, ÓÏÚÄÁÔØ m-ÆÁÊÌ × ×ÉÄÅ ÆÕÎËÃÉÉ, ÐÒÉÎÉÍÁÀÝÅÊ × ËÁÞÅÓÔ×Å ÁÒÇÕÍÅÎÔÏ×
|
||
ÎÅÏÂÈÏÄÉÍÙÅ ÐÅÒÅÍÅÎÎÙÅ É ×ÏÚ×ÒÁÝÁÀÝÅÊ ÏÐÒÅÄÅÌÅÎÎÙÅ ×ÅÌÉÞÉÎÙ.
|
||
|
||
úÁÇÏÌÏ×ÏË ÆÁÊÌÁ ÆÕÎËÃÉÉ ÉÍÅÅÔ ×ÉÄ
|
||
\begin{verbatim}
|
||
%
|
||
% ëÏÍÍÅÎÔÁÒÉÊ, ÏÔÏÂÒÁÖÁÀÝÉÊÓÑ ÐÒÉ ××ÅÄÅÎÉÉ ËÏÍÁÎÄÙ help ÉÍÑ_ÆÕÎËÃÉÉ
|
||
%
|
||
function [×ÏÚ×ÒÁÝÁÅÍÙÅ ×ÅÌÉÞÉÎÙ] = ÉÍÑ_ÆÕÎËÃÉÉ(×ÈÏÄÎÙÅ, ÁÒÇÕÍÅÎÔÙ)
|
||
\end{verbatim}
|
||
|
||
äÁÌÅÅ ÓÌÅÄÕÀÔ ÏÐÅÒÁÔÏÒÙ, ×ÙÐÏÌÎÑÅÍÙÅ × ÔÅÌÅ ÆÕÎËÃÉÉ. åÓÌÉ ÐÏÓÌÅ ËÏÍÁÎÄÙ ×Ù
|
||
ÐÒÏÐÕÓÔÉÔÅ ÓÉÍ×ÏÌ ÔÏÞËÉ Ó ÚÁÐÑÔÏÊ, ÅÅ ×Ù×ÏÄ ÂÕÄÅÔ ÏÔÏÂÒÁÖÅÎ ÎÁ ÜËÒÁÎÅ.
|
||
|
||
éÔÁË, ÓÏÚÄÁÄÉÍ m-ÆÁÊÌ, ÏÓÕÝÅÓÔ×ÌÑÀÝÉÊ ÐÒÏ×ÅÒËÕ ×ÙÂÏÒËÉ ÎÁ ËÏÒÒÅËÔÎÏÓÔØ
|
||
ÐÒÉ ÐÏÍÏÝÉ ËÒÉÔÅÒÉÑ <<ÔÒÅÈ ÓÉÇÍ>>:
|
||
\verb'three_s.m'.
|
||
{\small
|
||
\verbatiminput{Materials4Pract/03/three_s.m}
|
||
}
|
||
|
||
ðÒÉ ÚÁÐÕÓËÅ ÓËÒÉÐÔÁ ÐÏ ÕÍÏÌÞÁÎÉÀ Octave ÅÇÏ ÉÝÅÔ × ÔÅËÕÝÅÊ ÄÉÒÅËÔÏÒÉÉ. ïÄÎÁËÏ, ÍÏÖÎÏ ÄÏÂÁ×ÉÔØ ÌÀÂÕÀ
|
||
ÄÉÒÅËÔÏÒÉÀ ÓÏ ÓËÒÉÐÔÁÍÉ × ÓÐÉÓÏË ÐÏÉÓËÁ (\t{path}) ÐÒÉ ÐÏÍÏÝÉ ËÏÍÁÎÄÙ \t{addpath}.
|
||
|
||
úÁÐÕÓÔÉÔØ ÄÁÎÎÙÊ ÓËÒÉÐÔ ÍÏÖÎÏ ËÏÍÁÎÄÏÊ \verb'[X sigma] = three_s(x,n)'.
|
||
|
||
\subsection{}
|
||
úÁÞÁÓÔÕÀ ÆÉÚÉËÕ-ÜËÓÐÅÒÉÍÅÎÔÁÔÏÒÕ ÐÒÉÈÏÄÉÔÓÑ ÐÒÏ×ÅÒÑÔØ ÎÕÌÅ×ÕÀ ÇÉÐÏÔÅÚÕ Ï
|
||
ÒÁ×ÅÎÓÔ×Å ÓÒÅÄÎÉÈ Ä×ÕÈ ÎÅÚÁ×ÉÓÉÍÙÈ ÎÁÂÏÒÏ× ÄÁÎÎÙÈ.
|
||
ðÕÓÔØ × ÒÅÚÕÌØÔÁÔÅ ÏÄÎÏÇÏ ÉÚÍÅÒÅÎÉÑ ÎÅËÏÔÏÒÏÊ ÆÉÚÉÞÅÓËÏÊ ×ÅÌÉÞÉÎÙ~$x$ ÂÙÌ
|
||
ÐÏÌÕÞÅÎ ÒÑÄ ÄÁÎÎÙÈ:
|
||
\begin{verbatim}
|
||
x1 = [ 47.78 36.40 35.66 8.93 40.42 54.16 51.76 44.32 46.19 50.75];
|
||
\end{verbatim}
|
||
úÁÔÅÍ ÂÙÌÏ ÐÒÏÉÚ×ÅÄÅÎÏ ÎÅÚÁ×ÉÓÉÍÏÅ ÉÚÍÅÒÅÎÉÅ ÜÔÏÊ ÖÅ ÆÉÚÉÞÅÓËÏÊ ×ÅÌÉÞÉÎÙ
|
||
ÐÒÉ ÄÒÕÇÉÈ ÕÓÌÏ×ÉÑÈ ÜËÓÐÅÒÉÍÅÎÔÁ. ðÒÉ ÜÔÏÍ ÂÙÌ ÐÏÌÕÞÅÎ ÒÑÄ:
|
||
\begin{verbatim}
|
||
x2 = [ 44.09 46.75 44.20 7.99 47.74 75.07 62.48 44.43 34.73 55.26];
|
||
\end{verbatim}
|
||
ôÒÅÂÕÅÔÓÑ ÐÒÏ×ÅÒÉÔØ ÎÕÌÅ×ÕÀ ÇÉÐÏÔÅÚÕ Ï ÒÁ×ÅÎÓÔ×Å ÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ
|
||
ÄÁÎÎÙÈ ×ÅÌÉÞÉÎ.
|
||
|
||
äÌÑ ÐÒÏ×ÅÒËÉ ÄÁÎÎÏÊ ÇÉÐÏÔÅÚÙ ÓÕÝÅÓÔ×ÕÅÔ ÆÕÎËÃÉÑ Octave \verb'ttest2',
|
||
\verb'ttest2(x1,x2)' ÄÁÓÔ ÏÔ×ÅÔ: \verb'ans=0', Ô.Å.
|
||
ÇÉÐÏÔÅÚÁ Ï ÎÅÒÁ×ÅÎÓÔ×Å ÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ ÎÁÛÉÈ Ä×ÕÈ ÒÑÄÏ× ÏÔËÌÏÎÅÎÁ
|
||
ÎÁ 95\%-Í ÕÒÏ×ÎÅ. äÌÑ ÏÐÒÅÄÅÌÅÎÉÑ ÄÏ×ÅÒÉÔÅÌØÎÏÇÏ ÉÎÔÅÒ×ÁÌÁ É ×ÅÒÏÑÔÎÏÓÔÉ
|
||
ÒÁ×ÅÎÓÔ×Á ÍÁÔÅÍÁÔÉÞÅÓËÉÈ ÏÖÉÄÁÎÉÊ ×ÏÓÐÏÌØÚÕÅÍÓÑ ÒÁÓÛÉÒÅÎÎÙÍ ×Ù×ÏÄÏÍ ËÏÍÁÎÄÙ:
|
||
\begin{verbatim}
|
||
[h p ci] = ttest2(x1, x2)
|
||
\end{verbatim}
|
||
ôÁËÉÍ ÏÂÒÁÚÏÍ, ×ÅÒÏÑÔÎÏÓÔØ ÔÏÇÏ, ÞÔÏ ÍÁÔÅÍÁÔÉÞÅÓËÉÅ ÏÖÉÄÁÎÉÑ ×ÙÂÏÒÏË ÒÁ×ÎÙ,
|
||
ÓÏÓÔÁ×ÌÑÅÔ ÌÉÛØ~$p=51\%$, ÐÒÉ ÜÔÏÍ ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ ÍÁÔÅÍÁÔÉÞÅÓËÏÇÏ
|
||
ÏÖÉÄÁÎÉÑ ÒÁÚÎÏÓÔÉ $x_1-x_2$ ÄÏÓÔÁÔÏÞÎÏ ÛÉÒÏË: $c_i=[-19.2,9.9]$, Ô.Å.
|
||
ÍÁÔÅÍÁÔÉÞÅÓËÉÅ ÏÖÉÄÁÎÉÑ ÄÁÎÎÙÈ ÒÑÄÏ× ÍÏÇÕÔ ÒÁÚÎÉÔØÓÑ ÎÁ~$4.6$ ÓÏ ÓÒÅÄÎÅË×ÁÄÒÁÔÉÞÎÙÍ
|
||
ÏÔËÌÏÎÅÎÉÅÍ~$\sigma=14.6$.
|
||
|
||
âÏÌØÛÁÑ ÛÉÒÉÎÁ ÄÏ×ÅÒÉÔÅÌØÎÏÇÏ ÉÎÔÅÒ×ÁÌÁ ÇÏ×ÏÒÉÔ Ï ÔÏÍ, ÞÔÏ ÄÁÎÎÙÅ × ÒÑÄÁÈ~$x_1$
|
||
É~$x_2$ ÐÏÌÕÞÅÎÙ Ó ÎÉÚËÏÊ ÎÁÄÅÖÎÏÓÔØÀ. ïÄÎÁËÏ, ÎÁÊÄÑ ÍÅÄÉÁÎÙ ÒÑÄÏ×~$x_1$,
|
||
$x_2$ É ÓÏ×ÍÅÝÅÎÎÏÇÏ ÒÑÄÁ $(x_1;x_2)$ ÍÏÖÎÏ ÐÏÐÙÔÁÔØÓÑ Ó ÄÏÓÔÁÔÏÞÎÏ ×ÙÓÏËÏÊ
|
||
ÓÔÅÐÅÎØÀ ×ÅÒÏÑÔÎÏÓÔÉ ÏÃÅÎÉÔØ ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ ×ÅÌÉÞÉÎÙ~$x$.
|
||
|
||
|
||
\section{íÅÔÏÄ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ×}
|
||
\subsection{}
|
||
óÌÕÞÁÊÎÁÑ ÐÏÇÒÅÛÎÏÓÔØ ÆÉÚÉÞÅÓËÏÇÏ ÉÚÍÅÒÅÎÉÑ ÉÍÅÅÔ ÐÒÉÒÏÄÕ, ÁÎÁÌÏÇÉÞÎÕÀ ÂÅÌÏÍÕ ÛÕÍÕ,
|
||
ÐÏÜÔÏÍÕ ÄÌÑ ÎÁÞÁÌÁ ÒÁÓÓÍÏÔÒÉÍ ÐÒÏÓÔÅÊÛÉÅ ÍÅÔÏÄÙ ÏÞÉÓÔËÉ ÏÄÎÏÍÅÒÎÙÈ ÓÉÇÎÁÌÏ× ×ÉÄÁ
|
||
$y=y(t)$ ÏÔ ÛÕÍÏ×.
|
||
|
||
éÓÐÏÌØÚÕÅÍ ÍÁÓÓÉ× ÉÚ ÄÅÓÑÔÉ ÓÉÇÎÁÌÏ×, ÚÁÛÕÍÌÅÎÎÙÈ Ó ÏÄÉÎÁËÏ×ÙÍ ÕÒÏ×ÎÅÍ~SNR:
|
||
\begin{verbatim}
|
||
x=[0:0.05:20];
|
||
y=sin(x*10).*(0.5+sawtooth(x*pi/5)/2);
|
||
for a=[1:10]
|
||
y1(a,:)=awgn(y,1,'measured');
|
||
endfor
|
||
\end{verbatim}
|
||
|
||
÷ÉÄ ÃÉËÌÁ \verb'for' ÏÔÌÉÞÁÅÔÓÑ ÏÔ ÑÚÙËÏ× ÐÒÏÇÒÁÍÍÉÒÏ×ÁÎÉÑ ×ÒÏÄÅ C: ÃÉËÌ
|
||
ÐÏÏÞÅÒÅÄÎÏ ÐÅÒÅÂÉÒÁÅÔ ×ÓÅ ÚÎÁÞÅÎÉÑ ÐÅÒÅÍÅÎÎÏÊ \verb'a'. åÓÌÉ ÂÙ ÍÙ ÚÁÒÁÎÅÅ
|
||
ÉÎÉÃÉÁÌÉÚÉÒÏ×ÁÌÉ ÅÅ ÍÁÓÓÉ×ÏÍ, ÍÏÖÎÏ ÂÙÌÏ ÂÙ ÐÒÏÓÔÏ ÎÁÐÉÓÁÔØ \verb'for a'.
|
||
ãÉËÌ \verb'for' ÚÁËÁÎÞÉ×ÁÅÔÓÑ ËÏÍÁÎÄÏÊ \verb'endfor'. ä×ÏÅÔÏÞÉÅ × ÁÄÒÅÓÁÃÉÉ \verb'y(a,:)' ÏÚÎÁÞÁÅÔ,
|
||
ÞÔÏ ÍÙ ×ÙÂÉÒÁÅÍ\Ö ×ÓÅ\Î ÜÌÅÍÅÎÔÙ ÐÏ ×ÔÏÒÏÊ ËÏÏÒÄÉÎÁÔÅ (Ô.Å. ÐÒÉÒÁ×ÎÉ×ÁÎÉÅ ÐÒÏÉÚ×ÏÄÉÔÓÑ Ë ÃÅÌÏÊ
|
||
ÓÔÒÏËÅ). åÝÅ ÏÄÎÉÍ ÏÔÌÉÞÉÅÍ ÏÔ ÑÚÙËÏ× ÐÒÏÇÒÁÍÍÉÒÏ×ÁÎÉÑ Ñ×ÌÑÅÔÓÑ ÄÉÎÁÍÉÞÅÓËÏÅ
|
||
ÒÁÓÛÉÒÅÎÉÅ ÍÁÔÒÉÃ: ÎÅÔ ÎÅÏÂÈÏÄÉÍÏÓÔÉ × ÎÁÞÁÌÅ ÒÁÂÏÔÙ Ó ÎÅÊ ÓÏÏÂÝÁÔØ ÅÅ ÐÒÅÄÅÌØÎÙÊ
|
||
ÒÁÚÍÅÒ.
|
||
|
||
éÔÁË, ÍÙ ÐÏÌÕÞÉÌÉ ÍÁÓÓÉ× \verb'y1', × ÓÔÒÏËÁÈ ËÏÔÏÒÏÇÏ ÓÏÄÅÒÖÁÔÓÑ ÚÁÛÕÍÌÅÎÎÙÅ
|
||
×ÁÒÉÁÎÔÙ ÏÄÎÏÇÏ É ÔÏÇÏ ÖÅ ÓÉÇÎÁÌÁ. íÏÖÎÏ ÏÔÏÂÒÁÚÉÔØ ÉÈ ×ÓÅ ÇÒÁÆÉÞÅÓËÉ ËÏÍÁÎÄÏÊ
|
||
\verb'plot(x,y1)', Á ÍÏÖÎÏ É Ó ÏÒÉÇÉÎÁÌÏÍ: \verb'plot(x,y,"linewidth",2, x, y1)'.
|
||
ïÃÅÎÉÔØ ÚÁÛÕÍÌÅÎÎÏÓÔØ ÓÉÇÎÁÌÁ ÍÏÖÎÏ ËÏÍÁÎÄÏÊ \verb"plot(y,y1,'.')". åÓÌÉ ÂÙ
|
||
ÓÉÇÎÁÌÙ × \verb'y1' ÓÏ×ÐÁÄÁÌÉ Ó \verb'y', ÍÙ Õ×ÉÄÅÌÉ ÂÙ ÏÔÒÅÚÏË Ó ËÏÜÆÆÉÃÉÅÎÔÏÍ
|
||
ÎÁËÌÏÎÁ~1. þÅÍ ÄÁÌØÛÅ ÆÏÒÍÁ ÐÏÌÕÞÅÎÎÏÊ ÆÉÇÕÒÙ ÏÔ ÔÁËÏÇÏ ÏÔÒÅÚËÁ, ÔÅÍ ÂÏÌØÛÅ
|
||
ÚÁÛÕÍÌÅÎÎÏÓÔØ ÓÉÇÎÁÌÁ.
|
||
|
||
äÌÑ ×ÏÓÓÔÁÎÏ×ÌÅÎÉÑ ÓÉÇÎÁÌÁ ÉÚ ÄÅÓÑÔÉ ÉÚÍÅÒÅÎÉÊ ÐÏÐÒÏÂÕÅÍ ÕÓÒÅÄÎÉÔØ ÎÁÂÏÒÙ ÓÉÇÎÁÌÏ×
|
||
É ÎÁÊÔÉ ÉÈ ÍÅÄÉÁÎÕ:
|
||
\begin{verbatim}
|
||
y_mean = mean(y1);
|
||
y_med = median(y1);
|
||
plot(x,[y;y_mean;y_med]);
|
||
legend("original", "mean", "median");
|
||
\end{verbatim}
|
||
|
||
ïÂÁ ×ÏÓÓÔÁÎÏ×ÌÅÎÎÙÈ ÓÉÇÎÁÌÁ ÉÍÅÀÔ ÐÒÉÍÅÒÎÏ ÏÄÉÎÁËÏ×ÙÅ ×ÅÌÉÞÉÎÙ
|
||
É ÄÏ×ÏÌØÎÏ ÂÌÉÚËÉ Ë ÒÅÁÌØÎÏÊ ÆÕÎËÃÉÉ (ÏÓÏÂÅÎÎÏ ÎÁ ÕÞÁÓÔËÁÈ Ó ÂÏÌØÛÏÊ ÁÍÐÌÉÔÕÄÏÊ
|
||
ÓÉÇÎÁÌÁ). ïÄÎÁËÏ, ËÁË ÍÙ Õ×ÉÄÉÍ ×ÐÏÓÌÅÄÓÔ×ÉÉ, ÅÓÌÉ Ë ÓÉÇÎÁÌÕ ÄÏÂÁ×ÌÅÎ ÛÕÍ
|
||
ÔÉÐÁ <<ÓÏÌØ/ÐÅÒÅÃ>>, ÍÅÄÉÁÎÎÁÑ ÆÉÌØÔÒÁÃÉÑ ÂÕÄÅÔ ÒÁÂÏÔÁÔØ ÎÁÍÎÏÇÏ ÜÆÆÅËÔÉ×ÎÅÅ
|
||
ÆÉÌØÔÒÁÃÉÉ ÐÏ ÓÒÅÄÎÅÍÕ ÁÒÉÆÍÅÔÉÞÅÓËÏÍÕ.
|
||
|
||
\subsection{}
|
||
òÁÓÓÍÏÔÒÉÍ ÌÉÎÅÊÎÕÀ ÚÁ×ÉÓÉÍÏÓÔØ $y=ax+b$, ÚÁÄÁÎÎÕÀ ÔÁÂÌÉÞÎÏ × ×ÉÄÅ~$y=y(x)$.
|
||
äÌÑ ÏÐÒÅÄÅÌÅÎÉÑ ÍÅÔÏÄÏÍ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ× ËÏÜÆÆÉÃÉÅÎÔÏ× ÌÉÎÅÊÎÏÊ (Á ÔÁËÖÅ ×ÙÓÛÉÈ ÓÔÅÐÅÎÅÊ)
|
||
ÚÁ×ÉÓÉÍÏÓÔÉ ÓÌÕÖÉÔ ÆÕÎËÃÉÑ \verb'polyfit(x,y,n)'. ïÎÁ ÓÏÄÅÒÖÉÔ ÔÒÉ ÁÒÇÕÍÅÎÔÁ: $x$~-- ×ÅËÔÏÒ
|
||
ÁÒÇÕÍÅÎÔÁ, $y$~-- ×ÅËÔÏÒ ÆÕÎËÃÉÉ, $n$~-- ÓÔÅÐÅÎØ ÁÐÐÒÏËÓÉÍÉÒÕÀÝÅÇÏ ÐÏÌÉÎÏÍÁ. åÅ ÒÅÚÕÌØÔÁÔ ×
|
||
ÐÒÏÓÔÅÊÛÅÍ ÓÌÕÞÁÅ ÐÒÅÄÓÔÁ×ÌÑÅÔ ÓÏÂÏÊ ×ÅËÔÏÒ ËÏÜÆÆÉÃÉÅÎÔÏ× (ÎÁÞÉÎÁÑ ÓÏ ÓÔÁÒÛÅÊ ÓÔÅÐÅÎÉ).
|
||
åÓÌÉ ÆÕÎËÃÉÀ ×ÙÚ×ÁÔØ ËÁË \verb'[p,S] = polyfit(x,y,n)', ×ÅËÔÏÒ~$p$ ÂÕÄÅÔ ÓÏÄÅÒÖÁÔØ ËÏÜÆÆÉÃÉÅÎÔÙ,
|
||
Á × ÓÔÒÕËÔÕÒÅ~$S$ ÂÕÄÕÔ ÓÏÄÅÒÖÁÔØÓÑ ÔÁËÉÅ ÄÁÎÎÙÅ, ËÁË ÓÔÅÐÅÎÉ Ó×ÏÂÏÄÙ~(df) É ÎÏÒÍÁ ÏÔËÌÏÎÅÎÉÊ
|
||
ÄÁÎÎÙÈ ÏÔ ÁÐÐÒÏËÓÉÍÉÒÕÀÝÅÊ ËÒÉ×ÏÊ (normr). äÌÑ ×ÏÓÓÔÁÎÏ×ÌÅÎÉÑ ÐÏÌÕÞÅÎÎÏÊ ÚÁ×ÉÓÉÍÏÓÔÉ
|
||
ÉÓÐÏÌØÚÕÅÔÓÑ ÆÕÎËÃÉÑ \verb'polyval(p,x)', ÇÄÅ $p$~-- ÐÏÌÕÞÅÎÎÙÊ ÆÕÎËÃÉÅÊ \verb'polyfit'
|
||
×ÅËÔÏÒ ËÏÜÆÆÉÃÉÅÎÔÏ×, $x$~-- ×ÅËÔÏÒ ÁÒÇÕÍÅÎÔÁ. ÷ ÔÁËÏÍ ×ÉÄÅ ÆÕÎËÃÉÑ ×ÏÚ×ÒÁÝÁÅÔ ×ÅËÔÏÒ
|
||
×ÏÓÓÔÁÎÏ×ÌÅÎÎÏÊ ÆÕÎËÃÉÉ. ÷ ×ÉÄÅ \verb'[y, delta] = polyval(p,x,S)' ÆÕÎËÃÉÑ ×ÏÚ×ÒÁÝÁÅÔ ÍÁÓÓÉ×
|
||
ÐÏÇÒÅÛÎÏÓÔÅÊ (Ô.Å. × ËÁÖÄÏÊ ÔÏÞËÅ ×ÏÓÓÔÁÎÏ×ÌÅÎÎÙÅ ÚÎÁÞÅÎÉÑ ÆÕÎËÃÉÉ ÍÏÖÎÏ ÐÒÅÄÓÔÁ×ÉÔØ ×
|
||
×ÉÄÅ $y=y\pm delta$, Ô.Å. ÏÃÅÎÉÔØ ÁÂÓÏÌÀÔÎÕÀ ÐÏÇÒÅÛÎÏÓÔØ ×ÏÓÓÔÁÎÏ×ÌÅÎÉÑ ÍÏÖÎÏ ÐÒÉ ÐÏÍÏÝÉ
|
||
ËÏÍÁÎÄÙ \verb'mean(delta)'.
|
||
|
||
îÁÊÄÅÍ ËÏÜÆÆÉÃÉÅÎÔÙ ÍÏÄÅÌØÎÏÊ ÚÁ×ÉÓÉÍÏÓÔÉ. ðÕÓÔØ $y=7.15x+4.22$. ðÏÓÔÒÏÉÍ
|
||
×ÅËÔÏÒÙ, ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÉÅ ÁÒÇÕÍÅÎÔÕ É ÆÕÎËÃÉÉ:
|
||
$$
|
||
\verb'x = [0:100]; y = 7.15*x + 4.22;'
|
||
$$
|
||
úÁÛÕÍÉÍ ÓÉÇÎÁÌ ÄÌÑ ÐÏÌÕÞÅÎÉÑ ÒÁÚÂÒÏÓÁ ÔÏÞÅË~$y_i$:
|
||
$$
|
||
\verb"y1 = awgn(y,10,'measured');"
|
||
$$
|
||
ïÔÏÂÒÁÚÉÍ ÎÁ ÜËÒÁÎÅ ÏÂÁ ÒÑÄÁ: \verb"plot(x,y,x,y1,'o')" (ÚÁÐÉÓØ \verb"'o'"
|
||
ÏÚÎÁÞÁÅÔ, ÞÔÏ ÇÒÁÆÉË ÂÕÄÅÔ ÏÔÏÂÒÁÖÁÔØÓÑ ËÒÕÖË\'ÁÍÉ). òÁÚÂÒÏÓ ÄÁÎÎÙÈ
|
||
ÄÏÓÔÁÔÏÞÎÏ ×ÅÌÉË. ïÐÒÅÄÅÌÉÍ ËÏÜÆÆÉÃÉÅÎÔ ËÏÒÒÅÌÑÃÉÉ: \verb'corr(x,y1)'.
|
||
ïÎ ÄÏ×ÏÌØÎÏ ÂÌÉÚÏË Ë ÅÄÉÎÉÃÅ, ÓÌÅÄÏ×ÁÔÅÌØÎÏ, ÍÙ ÍÏÖÅÍ ÐÏÐÙÔÁÔØÓÑ ÐÏÌÕÞÉÔØ
|
||
ËÏÜÆÆÉÃÉÅÎÔÙ ÌÉÎÅÊÎÏÊ ÚÁ×ÉÓÉÍÏÓÔÉ É ×ÏÓÓÔÁÎÏ×ÉÔØ ÆÕÎËÃÉÀ:
|
||
\begin{verbatim}
|
||
[p,S] = polyfit(x,y1,1); % ËÏÜÆÆÉÃÉÅÎÔÙ a É b
|
||
[y2, delta] = polyval(p,x,S); % ×ÏÓÓÔÁÎÏ×ÌÅÎÎÙÊ ×ÅËÔÏÒ
|
||
plot(x,y1,'o',x,[y;y2]) % ×ÓÅ ÔÒÉ ÇÒÁÆÉËÁ
|
||
legend("noicy", "original", "fitted");
|
||
mean(delta) % ÁÂÓÏÌÀÔÎÁÑ ÏÛÉÂËÁ
|
||
mean(delta)/mean(y) % ÏÔÎÏÓÉÔÅÌØÎÁÑ ÏÛÉÂËÁ
|
||
\end{verbatim}
|
||
|
||
\subsection{}
|
||
íÏÖÎÏ ÎÁÊÔÉ ÐÒÉÂÌÉÖÅÎÉÅ ÍÅÔÏÄÏÍ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ× É ÄÒÕÇÉÍ ÓÐÏÓÏÂÏÍ. ðÕÓÔØ
|
||
$Y$~-- ×ÅËÔÏÒ--ÓÔÏÌÂÅà ÚÎÁÞÅÎÉÊ ÆÕÎËÃÉÉ, $A=(a,b)^{\rm Tr}$~-- ×ÅËÔÏÒ--ÓÔÏÌÂÅÃ
|
||
ËÏÜÆÆÉÃÉÅÎÔÏ×
|
||
ÒÁÚÌÏÖÅÎÉÑ. ôÏÇÄÁ ÕÓÌÏ×ÉÅ $y_i=ax_i+b$ ÍÏÖÎÏ ÐÒÅÄÓÔÁ×ÉÔØ × ×ÉÄÅ ÍÁÔÒÉÞÎÏÇÏ
|
||
ÐÒÏÉÚ×ÅÄÅÎÉÑ $Y=XA$. ÷ÔÏÒÏÊ ÓÔÏÌÂÅà ÍÁÔÒÉÃÙ~$X$ ÃÅÌÉËÏÍ ÓÏÓÔÏÉÔ ÉÚ ÅÄÉÎÉÃ, Á
|
||
× ÐÅÒ×ÏÍ ÎÁÈÏÄÉÔÓÑ ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏÓÔØ ÚÎÁÞÅÎÉÊ~$x_i$. ÷ ÜÔÏÍ ÓÌÕÞÁÅ ÎÁÈÏÖÄÅÎÉÅ
|
||
ËÏÜÆÆÉÃÉÅÎÔÏ× Ó×ÏÄÉÔÓÑ Ë ÒÅÛÅÎÉÀ ÓÉÓÔÅÍÙ ÌÉÎÅÊÎÙÈ ÕÒÁ×ÎÅÎÉÊ $y_i=ax_i+b$,
|
||
ÄÁÀÝÅÍÕ ÍÉÎÉÍÁÌØÎÕÀ ÎÅ×ÑÚËÕ. ôÁËÏÅ ÒÅÛÅÎÉÅ ÎÁÈÏÄÉÔÓÑ ÐÒÉ ÐÏÍÏÝÉ ÏÐÅÒÁÃÉÉ
|
||
ÌÅ×ÏÓÔÏÒÏÎÎÅÇÏ ÍÁÔÒÉÞÎÏÇÏ ÄÅÌÅÎÉÑ: $X\backslash Y$. òÅÛÉÍ ÐÒÅÄÙÄÕÝÉÊ ÐÒÉÍÅÒ ÔÁËÉÍ
|
||
ÓÐÏÓÏÂÏÍ.
|
||
\begin{verbatim}
|
||
X = [x' ones(size(x'))]; % ÓÏÚÄÁÅÍ ÍÁÔÒÉÃÕ ÁÒÇÕÍÅÎÔÁ
|
||
% (Ô.Ë. x É y1 - ÓÔÒÏËÉ, ÔÒÁÎÓÐÏÎÉÒÕÅÍ ÉÈ)
|
||
A = X\y1' % ÎÁÈÏÄÉÍ ËÏÜÆÆÉÃÉÅÎÔÙ
|
||
% É ÏÔÏÂÒÁÖÁÅÍ ÉÈ ÎÁ ÜËÒÁÎÅ
|
||
\end{verbatim}
|
||
|
||
ðÏÌÕÞÅÎÎÙÅ ÚÎÁÞÅÎÉÑ ÄÏÌÖÎÙ ÂÙÔØ ÐÒÉÍÅÒÎÏ ÒÁ×ÎÙ ÎÁÊÄÅÎÎÙÍ ÐÒÅÄÙÄÕÝÉÍ ÓÐÏÓÏÂÏÍ.
|
||
ëÁË ÍÙ Õ×ÉÄÉÍ ÄÁÌÅÅ, ÔÁËÏÊ ÓÐÏÓÏ ÎÁÈÏÖÄÅÎÉÑ ËÏÒÎÅÊ ÁÐÐÒÏËÓÉÍÁÃÉÉ ÐÒÉÇÏÄÅÎ ÎÅ
|
||
ÔÏÌØËÏ ÄÌÑ ÐÏÌÉÎÏÍÉÁÌØÎÙÈ, ÎÏ É ÄÌÑ ÍÎÏÇÉÈ ÄÒÕÇÉÈ ÆÕÎËÃÉÊ.
|
||
|
||
\subsection{}
|
||
ðÏÐÒÏÂÕÅÍ ÓÏÚÄÁÔØ Ë×ÁÄÒÁÔÉÞÎÕÀ ÚÁ×ÉÓÉÍÏÓÔØ É ÁÐÐÒÏËÓÉÍÉÒÏ×ÁÔØ ÅÅ ÍÅÔÏÄÏÍ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ×.
|
||
ðÕÓÔØ ÚÁ×ÉÓÉÍÏÓÔØ ÎÁ ÏÔÒÅÚËÅ $[0,100]$ ÉÍÅÅÔ ×ÉÄ $y=2.4x^2-0.87x+2.13$. óÏÚÄÁÄÉÍ ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÉÅ
|
||
ÍÁÓÓÉ×Ù ÄÁÎÎÙÈ, ÄÏÂÁ×ÉÍ ÛÕÍ Ó SNR=20\,Äâ É ÏÔÏÂÒÁÚÉÍ ÏÂÁ ÓÉÇÎÁÌÁ ÎÁ ÇÒÁÆÉËÅ:
|
||
\begin{verbatim}
|
||
x = [1:100];
|
||
y = 2.4*x.^2-0.87*x+2.13;
|
||
y1 = awgn(y,20,'measured');
|
||
plot(x,[y;y1]);
|
||
\end{verbatim}
|
||
|
||
ôÅÐÅÒØ ÓÏÚÄÁÄÉÍ ×ÅËÔÏÒ ËÏÜÆÆÉÃÉÅÎÔÏ× ÁÐÐÒÏËÓÉÍÁÃÉÉ ÐÏÌÉÎÏÍÏÍ ×ÔÏÒÏÊ ÓÔÅÐÅÎÉ
|
||
×ÏÓÓÔÁÎÏ×ÉÍ ÆÕÎËÃÉÀ É ÏÔÏÂÒÁÚÉÍ ÎÁ ÇÒÁÆÉËÅ:
|
||
\begin{verbatim}
|
||
[p, S] = polyfit(x, y1, 2);
|
||
[y2, DELTA] = polyval(p, x, S);
|
||
plot(x,[y;y2]);
|
||
legend("original", "restored")
|
||
\end{verbatim}
|
||
|
||
ïÔÏÂÒÁÚÉÍ ÎÁ ÜËÒÁÎÅ ÎÁÊÄÅÎÎÙÅ ËÏÜÆÆÉÃÉÅÎÔÙ:
|
||
\verb'p'. òÁÓÓÞÉÔÁÅÍ ÓÒÅÄÎÅÅ Ë×ÁÄÒÁÔÉÞÎÏÅ ÏÔËÌÏÎÅÎÉÅ ÁÐÐÒÏËÓÉÍÁÃÉÉ (\verb'mean(DELTA)').
|
||
ôÁËÖÅ ÒÁÓÓÞÉÔÁÅÍ ÏÔÎÏÓÉÔÅÌØÎÕÀ ÏÛÉÂËÕ ÁÐÐÒÏËÓÉÍÁÃÉÉ \verb'mean(DELTA)/mean(y1)'.
|
||
|
||
é ×ÔÏÒÏÊ ÓÐÏÓÏÂ:
|
||
\begin{verbatim}
|
||
X = [(x.^2)' x' ones(size(x'))];
|
||
A = X\y1'
|
||
\end{verbatim}
|
||
|
||
\subsection{}
|
||
ïÄÎÁËÏ, ÞÁÝÅ ×ÓÅÇÏ ÆÕÎËÃÉÏÎÁÌØÎÙÅ ÚÁ×ÉÓÉÍÏÓÔÉ ÉÍÅÀÔ ÉÎÙÅ ×ÉÄÙ ÚÁ×ÉÓÉÍÏÓÔÉ. äÏÐÕÓÔÉÍ, ÎÁÍ
|
||
ÉÚ×ÅÓÔÎÏ, ÞÔÏ ÉÚÍÅÒÑÅÍÁÑ ×ÅÌÉÞÉÎÁ ÉÚÍÅÎÑÅÔÓÑ ÐÏ ÚÁËÏÎÕ
|
||
\begin{equation}
|
||
y=a_0+a_1\e^{-t}+a_2te^{-t}.
|
||
\label{exp_y}
|
||
\end{equation}
|
||
äÌÑ ÁÐÐÒÏËÓÉÍÁÃÉÉ ÔÁËÏÊ ÆÕÎËÃÉÅÊ ÍÏÖÎÏ ÐÒÅÄÓÔÁ×ÉÔØ ÕÒÁ×ÎÅÎÉÅ~\eqref{exp_y} × ÍÁÔÒÉÞÎÏÍ
|
||
×ÉÄÅ $Y=TA$, ÇÄÅ $T$~-- ÆÕÎËÃÉÏÎÁÌØÎÁÑ ÍÁÔÒÉÃÁ, Õ ËÏÔÏÒÏÊ × ÐÅÒ×ÏÍ ÓÔÏÌÂÃÅ
|
||
ÒÁÚÍÅÝÅÎÙ ÅÄÉÎÉÃÙ (ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÕÍÎÏÖÅÎÉÀ ÎÁ~$a_0$), ×Ï ×ÔÏÒÏÍ~--- ÆÕÎËÃÉÑ
|
||
$\e^{-t}$, Á × ÔÒÅÔØÅÍ~--- $t\e^{-t}$. îÁÊÔÉ ËÏÜÆÆÉÃÉÅÎÔÙ~$A$ ÍÏÖÎÏ ÐÒÉ ÐÏÍÏÝÉ
|
||
ÏÐÅÒÁÔÏÒÁ ÌÅ×ÏÇÏ ÄÅÌÅÎÉÑ: $A=T\backslash Y$.
|
||
\begin{verbatim}
|
||
t = [0 0.3 0.8 1.1 1.6 2.3]'; % ÓÒÁÚÕ ××ÏÄÉÍ ÄÁÎÎÙÅ × ÓÔÏÌÂÃÁÈ
|
||
y = [0.6 0.67 1.01 1.35 1.47 1.25]';
|
||
T = [ones(size(t)) exp(-t) t.*exp(-t)];
|
||
A = T\y
|
||
\end{verbatim}
|
||
ôÅÐÅÒØ ÏÔÏÂÒÁÚÉÍ ÄÁÎÎÙÅ ÎÁ ÇÒÁÆÉËÅ:
|
||
\begin{verbatim}
|
||
x = [0:0.1:2.5]';
|
||
Y = [ones(size(x)) exp(-x) x.*exp(-x)]*A;
|
||
plot(x,Y, t,y,'o')
|
||
\end{verbatim}
|
||
|
||
\subsection{}
|
||
äÌÑ ËÏÒÒÅËÃÉÉ ÎÁ×ÅÄÅÎÉÑ É ÓÏÐÒÏ×ÏÖÄÅÎÉÑ ÔÅÌÅÓËÏÐÏ× ÉÓÐÏÌØÚÕÅÔÓÑ óëî~--- ÓÉÓÔÅÍÁ ËÏÒÒÅËÃÉÉ
|
||
ÎÁ×ÅÄÅÎÉÑ, ËÏÔÏÒÁÑ ÕÞÉÔÙ×ÁÅÔ ÒÁÚÌÉÞÎÏÇÏ ÒÏÄÁ ÏÛÉÂËÉ (ÇÎÕÔÉÅ ÏÓÅÊ É ÎÅÐÅÒÐÅÎÄÉËÕÌÑÒÎÏÓÔØ ÏÓÅÊ É
|
||
Ô.Ð.). õ âôá ÄÁÎÎÙÅ ÏÛÉÂËÉ ×ÙÒÁÖÁÀÔÓÑ ÐÏÌÉÎÏÍÁÍÉ:
|
||
$$dA = K_0 + K_1\frac{1}{\tg Z} + K_2\frac{1}{\sin Z} - K_3\frac{\sin A}{\tg Z}
|
||
+K_4\frac{\cos\delta\cos P}{\sin Z},$$
|
||
$$dZ = K_5 + K_6\sin Z + K_7\cos Z + K_3\cos A + K_4 \cos\phi\sin A.$$
|
||
|
||
úÄÅÓØ:
|
||
\begin{description}
|
||
\item[$dA$, $dZ$] ÐÏÇÒÅÛÎÏÓÔÉ ÎÁ×ÅÄÅÎÉÑ ÐÏ ÁÚÉÍÕÔÕ É ÚÅÎÉÔÎÏÍÕ ÒÁÓÓÔÏÑÎÉÀ;
|
||
\item[$K_x$] ËÏÜÆÆÉÃÉÅÎÔÙ ÏÛÉÂÏË;
|
||
\item[$\phi$] ÛÉÒÏÔÁ ÍÅÓÔÁ ÎÁÂÌÀÄÅÎÉÑ;
|
||
\item[$t$] ÞÁÓÏ×ÏÊ ÕÇÏÌ;
|
||
\item[$P$] ÐÁÒÁÌÌÁËÔÉÞÅÓËÉÊ ÕÇÏÌ.
|
||
\end{description}
|
||
|
||
äÌÑ ÐÏÌÕÞÅÎÉÑ ÜÔÉÈ ËÏÜÆÆÉÃÉÅÎÔÏ× ÎÅÏÂÈÏÄÉÍÏ ÐÒÏ×ÅÓÔÉ ÎÁÂÌÀÄÅÎÉÅ × ÎÅÓËÏÌØËÉÈ ÄÅÓÑÔËÁÈ ÒÁ×ÎÏÍÅÒÎÏ
|
||
ÒÁÓÐÒÅÄÅÌÅÎÎÙÈ ÐÏ ÎÅÂÅÓÎÏÊ ÐÏÌÕÓÆÅÒÅ ÔÏÞÅË (ÚÁ ÉÓËÌÀÞÅÎÉÅÍ ÏÂÌÁÓÔÅÊ ÚÁÐÒÅÔÁ, $Z<5\degr$, É ÏËÏÌÏ
|
||
ÇÏÒÉÚÏÎÔÁ, $Z>70\degr$). äÁÌÅÅ × ËÁÖÄÏÍ ÐÏÌÅ ×ÙÞÉÓÌÑÅÔÓÑ ÁÓÔÒÏÍÅÔÒÉÑ É ÏÐÒÅÄÅÌÑÅÔÓÑ ÐÏÇÒÅÛÎÏÓÔØ
|
||
ÎÁ×ÅÄÅÎÉÑ. óÏÓÔÁ×ÌÑÅÔÓÑ ÔÁÂÌÉÃÁ (ÎÁÐÒÉÍÅÒ, \t{2015\_09\_30\_pf.tab}, ÐÏ ËÏÔÏÒÏÊ É ÎÅÏÂÈÏÄÉÍÏ
|
||
×ÙÞÉÓÌÉÔØ ËÏÜÆÆÉÃÉÅÎÔÙ. ÷ÙÞÉÓÌÑÔØ ËÏÜÆÆÉÃÉÅÎÔÙ ÂÕÄÅÍ ÓÌÅÄÕÀÝÉÍ ÓËÒÉÐÔÏÍ:
|
||
|
||
\verbatiminput{Materials4Pract/03/getSKNcoeff.m}
|
||
|
||
úÁÐÕÓËÁÅÍ: \t{SKN = getSKNcoeff('2015\_09\_30\_pf.tab')}. óÔÒÏÑÔÓÑ ÇÒÁÆÉËÉ ÏÓÔÁÔÏÞÎÙÈ ÎÅ×ÑÚÏË É
|
||
×Ù×ÏÄÑÔÓÑ ÚÎÁÞÅÎÉÑ ×ÓÅÈ ËÏÜÆÆÉÃÉÅÎÔÏ×.
|
||
|
||
éÚ-ÚÁ ÌÉÎÅÊÎÏÊ ÚÁ×ÉÓÉÍÏÓÔÉ ËÏÜÆÆÉÃÉÅÎÔÏ× ÚÁÄÁÞÁ ÉÈ ×ÙÞÉÓÌÅÎÉÑ Ñ×ÌÑÅÔÓÑ ÎÅËÏÒÒÅËÔÎÏÊ, Á ××ÉÄÕ
|
||
ÍÁÌÏÓÔÉ ÏÂßÅÍÁ ÜËÓÐÅÒÉÍÅÎÔÁÌØÎÏÇÏ ÍÁÔÅÒÉÁÌÁ, ÒÅÛÁÔØ ÚÁÄÁÞÕ ÂÕÄÅÍ ÉÔÅÒÁÃÉÑÍÉ, ÎÁ ËÁÖÄÏÍ ÛÁÇÅ
|
||
ÉÚÂÁ×ÌÑÑÓØ ÏÔ ×ÙÂÒÏÓÏ×.
|
||
|
||
\section{úÁÄÁÎÉÑ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ×ÙÐÏÌÎÅÎÉÑ}
|
||
\begin{enumerate}
|
||
\item îÅËÏÔÏÒÁÑ ÓÏ×ÏËÕÐÎÏÓÔØ ÓÏÓÔÏÉÔ ÉÚ ÔÒÅÈ ÇÒÕÐÐ:~$X_1$, $X_2$, É~$X_3$. çÒÕÐÐÙ
|
||
ÉÍÅÀÔ ÓÌÅÄÕÀÝÉÅ ÚÎÁÞÅÎÉÑ:
|
||
\begin{verbatim}
|
||
X1 = 35.04 35.45 35.01 34.94 34.63 35.11 34.41 35.29 35.69
|
||
34.69 35.36 35.53 34.30 34.36 35.23
|
||
X2 = 34.30 34.80 34.86 34.81 35.08 34.79 35.04 33.93 34.48
|
||
34.41 33.74 34.60 34.00
|
||
X3 = 35.17 34.21 34.78 34.65 34.16 33.62 34.53 34.12 34.82
|
||
34.77 35.29 34.81 34.28 34.72 34.12 34.55 34.53 34.55
|
||
\end{verbatim}
|
||
îÁÊÄÉÔÅ: ÇÒÕÐÐÏ×ÙÅ ÓÒÅÄÎÉÅ (35,00, 34.53, 34.54), ÏÂÝÅÅ ÓÒÅÄÎÅÅ (34.69), ÇÒÕÐÐÏ×ÙÅ
|
||
ÄÉÓÐÅÒÓÉÉ (0.19, 0.19, 0.16), ÇÅÎÅÒÁÌØÎÕÀ ÄÉÓÐÅÒÓÉÀ (0.22).
|
||
|
||
\item õÓÏ×ÅÒÛÅÎÓÔ×ÕÊÔÅ ÓËÒÉÐÔ \verb'three_s.m' ÔÁË, ÞÔÏÂÙ ÐÏÍÉÍÏ ÏÓÎÏ×ÎÙÈ
|
||
×ÙÞÉÓÌÅÎÉÊ ÎÁ ÜËÒÁÎÅ ÏÔÏÂÒÁÖÁÌÉÓØ ÓÒÅÄÎÅÅ ÁÒÉÆÍÅÔÉÞÅÓËÏÅ ÚÎÁÞÅÎÉÅ ÍÁÓÓÉ×Á Ó
|
||
ÄÁÎÎÙÍÉ, Á ÔÁËÖÅ 95\%-Ê ÄÏ×ÅÒÉÔÅÌØÎÙÊ ÉÎÔÅÒ×ÁÌ ÐÏ ËÒÉÔÅÒÉÀ óÔØÀÄÅÎÔÁ.
|
||
|
||
\item ïÐÒÅÄÅÌÉÔÅ ÄÁ×ÌÅÎÉÅ × ÃÉÌÉÎÄÒÅ Ó ÇÁÚÏÍ, ÉÓÈÏÄÑ ÉÚ ÚÁËÏÎÁ íÅÎÄÅÌÅÅ×Á--ëÌÁÐÅÊÒÏÎÁ:
|
||
$pV=mRT/\mu$, ÅÓÌÉ ÉÚ×ÅÓÔÎÏ, ÞÔÏ ÍÁÓÓÁ ÇÁÚÁ $m=2\,$ÇÒÁÍÍÁ, $\mu=29\,$Ç/ÍÏÌØ, $R=8.31$, Á ÏÂßÅÍ É
|
||
ÔÅÍÐÅÒÁÔÕÒÕ ÇÁÚÁ ÉÚÍÅÒÑÌÉ × ÔÅÞÅÎÉÅ ÍÉÎÕÔÙ, ÐÏÌÕÞÉ× ÓÌÅÄÕÀÝÉÅ ÚÎÁÞÅÎÉÑ:
|
||
\begin{center}\small
|
||
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|}
|
||
\hline
|
||
÷ÅÌÉÞÉÎÁ &\multicolumn{10}{|c|}{úÎÁÞÅÎÉÅ}\\
|
||
\hline
|
||
$V$, Ì&2.27&2.27&2.26&2.25&2.26&2.27&2.29&2.28&2.25&2.28\\
|
||
$T$, ë&399.4&399.1&399.3&396.8&399.5&400.2&400.6&403.0&399.2&401.3\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{center}
|
||
óÞÉÔÁÊÔÅ, ÞÔÏ ÚÁ ÜÔÏ ×ÒÅÍÑ ÄÁ×ÌÅÎÉÅ ÇÁÚÁ ÎÅ ÕÓÐÅÌÏ ÓËÏÌØ-ÎÉÂÕÄØ ÚÎÁÞÉÔÅÌØÎÏ ÉÚÍÅÎÉÔØÓÑ.
|
||
ïÐÒÅÄÅÌÉÔÅ ÐÏÇÒÅÛÎÏÓÔÉ ÉÚÍÅÒÅÎÉÑ ×ÅÌÉÞÉÎ~$V$ É~$T$. óÞÉÔÁÑ, ÞÔÏ ÏÓÔÁÌØÎÙÅ ×ÅÌÉÞÉÎÙ Ñ×ÌÑÀÔÓÑ
|
||
ÐÏÓÔÏÑÎÎÙÍÉ, ÏÐÒÅÄÅÌÉÔÅ ËÏÓ×ÅÎÎÕÀ ÐÏÇÒÅÛÎÏÓÔØ ÉÚÍÅÒÅÎÉÑ~$p$.
|
||
|
||
äÌÑ ÕÄÏÂÓÔ×Á ×ÙÞÉÓÌÅÎÉÊ\Ë ÓÏÚÄÁÊÔÅ ÓËÒÉÐÔ, ÐÏÚ×ÏÌÑÀÝÉÊ ÄÌÑ ÚÁÄÁÎÎÏÇÏ ÒÑÄÁ ÄÁÎÎÙÈ
|
||
ÐÏÌÕÞÉÔØ ÍÁÔÅÍÁÔÉÞÅÓËÏÅ ÏÖÉÄÁÎÉÅ, ÓÒÅÄÎÅË×ÁÄÒÁÔÉÞÎÏÅ ÏÔËÌÏÎÅÎÉÅ É ÏÔÎÏÓÉÔÅÌØÎÕÀ
|
||
ÏÛÉÂËÕ\Î.
|
||
|
||
úÁÐÉÛÉÔÅ ÒÅÚÕÌØÔÁÔ × ×ÉÄÅ $p=\mean{p}\pm\sigma_p$ ($p=101\pm1\,$ËðÁ).
|
||
|
||
\item äÌÑ ÏÐÒÅÄÅÌÅÎÉÑ ÅÍËÏÓÔÉ~$C$ ÎÅÉÚ×ÅÓÔÎÏÇÏ ËÏÎÄÅÎÓÁÔÏÒÁ ÐÒÉ ÐÏÍÏÝÉ ÏÓÃÉÌÌÏÇÒÁÆÁ ÉÓÓÌÅÄÏ×ÁÌÉ
|
||
ÚÁÔÕÈÁÀÝÉÊ ÉÍÐÕÌØÓ, ×ÏÚÎÉËÁÀÝÉÊ ÐÒÉ ÒÁÚÒÑÄËÅ ËÏÎÄÅÎÓÁÔÏÒÁ ÞÅÒÅÚ ÒÅÚÉÓÔÏÒ~$R=3\,$ËïÍ.
|
||
ðÏ ÐÏËÁÚÁÎÉÑÍ ÏÓÃÉÌÌÏÇÒÁÆÁ ÂÙÌÉ ÚÁÐÉÓÁÎÙ ÓÌÅÄÕÀÝÉÅ ÚÎÁÞÅÎÉÑ ÔÏËÁ:
|
||
\begin{center}\small
|
||
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
|
||
\hline
|
||
$t$, Ó&0.0&0.1&0.2&0.3&0.4&0.5&0.6&0.7&0.8&0.9&1.0\\
|
||
$I$, á&1.00&0.72&0.52&0.37&0.26&0.19&0.14&0.10&0.07&0.04&0.03\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{center}
|
||
éÚ×ÅÓÔÎÏ, ÞÔÏ ÐÏÇÒÅÛÎÏÓÔØ ÁÍÐÅÒÍÅÔÒÁ ÓÏÓÔÁ×ÌÑÅÔ
|
||
$\sigma_I=0.01\,$á. ëÒÏÍÅ ÔÏÇÏ, ÉÚ×ÅÓÔÎÏ ÞÔÏ ÓÏÐÒÏÔÉ×ÌÅÎÉÅ ÒÅÚÉÓÔÏÒÁ ÉÚ×ÅÓÔÎÏ Ó ÔÏÞÎÏÓÔØÀ~5\%.
|
||
éÚ ÆÏÒÍÕÌÙ $I=I_0\exp(-t/[RC])$ ÏÐÒÅÄÅÌÉÔÅ ÐÏÇÒÅÛÎÏÓÔØ ÉÚÍÅÒÅÎÉÑ ÅÍËÏÓÔÉ ËÏÎÄÅÎÓÁÔÏÒÁ.
|
||
|
||
íÅÔÏÄÏÍ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ× ÏÐÒÅÄÅÌÉÔÅ ÚÎÁÞÅÎÉÅ ÅÍËÏÓÔÉ ËÏÎÄÅÎÓÁÔÏÒÁ, ÉÓÈÏÄÑ ÉÚ
|
||
ÕÒÁ×ÎÅÎÉÑ $t=-RC\ln I$ (97\,ÍËæ). úÁÐÉÛÉÔÅ ÏÔ×ÅÔ × ×ÉÄÅ $C=\aver{C}\pm\sigma_C$.
|
||
|
||
äÌÑ Õ×ÅÌÉÞÅÎÉÑ ÔÏÞÎÏÓÔÉ ÜËÓÐÅÒÉÍÅÎÔÁ ÂÙÌÏ ÐÒÏ×ÅÄÅÎÏ ÅÝÅ ÏÄÎÏ ÉÚÍÅÒÅÎÉÅ, ÒÅÚÕÌØÔÁÔÙ
|
||
ËÏÔÏÒÏÇÏ ÎÅÓËÏÌØËÏ ÏÔÌÉÞÁÌÉÓØ ÏÔ ÐÒÅÄÙÄÕÝÉÈ:
|
||
\begin{center}\small
|
||
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
|
||
\hline
|
||
$t$, Ó&0.0&0.1&0.2&0.3&0.4&0.5&0.6&0.7&0.8&0.9&1.0\\
|
||
$I$, á&1.00&0.75&0.56&0.41&0.30&0.23&0.17&0.12&0.10&0.07&0.05\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{center}
|
||
ðÒÏ×ÅÒØÔÅ ÎÕÌÅ×ÕÀ ÇÉÐÏÔÅÚÕ Ï ÒÁ×ÅÎÓÔ×Å ÓÒÅÄÎÉÈ × ÏÂÏÉÈ ÏÐÙÔÁÈ. ïÐÒÅÄÅÌÉÔÅ ×ÅÌÉÞÉÎÕ ÅÍËÏÓÔÉ
|
||
×Ï ×ÔÏÒÏÍ ÓÌÕÞÁÅ (112\,ÍËæ).
|
||
|
||
óÔÏÌØ ÂÏÌØÛÏÅ ÒÁÚÌÉÞÉÅ ÅÍËÏÓÔÅÊ, ÐÏÌÕÞÅÎÎÙÈ × ÒÅÚÕÌØÔÁÔÅ Ä×ÕÈ ÎÅÚÁ×ÉÓÉÍÙÈ ÜËÓÐÅÒÉÍÅÎÔÏ×,
|
||
ÚÁÓÔÁ×ÉÌÏ ÐÒÅÄÐÏÌÏÖÉÔØ, ÞÔÏ × ÒÅÚÕÌØÔÁÔÅ ÄÌÉÔÅÌØÎÏÊ ÜËÓÐÌÕÁÔÁÃÉÉ ÒÅÚÉÓÔÏÒ~$R$ ÎÁÇÒÅÌÓÑ, ÞÔÏ
|
||
×ÙÚ×ÁÌÏ Õ×ÅÌÉÞÅÎÉÅ ÅÇÏ ÓÏÐÒÏÔÉ×ÌÅÎÉÑ. óÞÉÔÁÑ ÅÍËÏÓÔØ ËÏÎÄÅÎÓÁÔÏÒÁ ÐÒÅÖÎÅÊ, ÏÐÒÅÄÅÌÉÔÅ ÓÏÐÒÏÔÉ×ÌÅÎÉÅ
|
||
ÒÅÚÉÓÔÏÒÁ ×Ï ×ÔÏÒÏÍ ÓÌÕÞÁÅ (3.5\,ËïÍ).
|
||
|
||
\item îÁÊÄÉÔÅ\Ë ÏÂÏÉÍÉ ÓÐÏÓÏÂÁÍÉ\Î ËÏÜÆÆÉÃÉÅÎÔÙ $a$ É~$b$ ÄÌÑ ÔÁÂÌÉÞÎÏ ÐÒÅÄÓÔÁ×ÌÅÎÎÏÊ
|
||
ÚÁ×ÉÓÉÍÏÓÔÉ $y(x)$, ÐÒÅÄÐÏÌÁÇÁÑ, ÞÔÏ ÏÎÁ ÉÍÅÅÔ ÌÉÎÅÊÎÙÊ ×ÉÄ. îÁÊÄÉÔÅ ËÏÜÆÆÉÃÉÅÎÔ ËÏÒÒÅÌÑÃÉÉ~$x$
|
||
É~$y$,
|
||
äÁÎÎÙÅ ÐÒÅÄÓÔÁ×ÌÅÎÙ × ÔÁÂÌÉÃÅ:
|
||
% a=5.15, b=2.74
|
||
% ÄÏÌÖÎÏ ÐÏÌÕÞÉÔØÓÑ: a=5.0644, b=3.4020
|
||
\begin{center}
|
||
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
|
||
\hline
|
||
\bf x&1&2&3&4&5&6&7&8&9&10\\
|
||
\hline
|
||
\bf y&7.7 & 13.7 & 22.0 & 23.1 & 23.7 & 36.7 & 35.6 & 47.8 &
|
||
50.2 & 52.1\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{center}
|
||
($a=5.1$, $b=3.4$).
|
||
|
||
\item éÚ×ÅÓÔÎÏ, ÞÔÏ ÎÅËÏÔÏÒÁÑ ÚÁ×ÉÓÉÍÏÓÔØ (ÓÍ. ÔÁÂÌÉÃÕ ÎÉÖÅ) ÉÍÅÅÔ ×ÉÄ
|
||
$y=ax\sin(x)-b\ln(x)$. ïÐÒÅÄÅÌÉÔÅ ËÏÜÆÆÉÃÉÅÎÔÙ~$a$ É~$b$ É ÐÏÓÔÒÏÊÔÅ
|
||
ÄÁÎÎÕÀ ËÒÉ×ÕÀ Ó ÂÏÌÅÅ ÄÅÔÁÌØÎÙÍ ÏÔÏÂÒÁÖÅÎÉÅÍ (ÎÁ ×ÅËÔÏÒÅ \verb'[1:0.05:10]').
|
||
ðÏÄÓËÁÚËÁ: ÓÒÁÚÕ ÖÅ ÚÁÄÁÊÔÅ ×ÅËÔÏÒÁ~$x$ É~$y$ ËÁË ÓÔÏÌÂÃÙ; ÍÁÔÒÉÃÁ~$X$ ÚÁÄÁÅÔÓÑ
|
||
ËÏÍÁÎÄÏÊ \verb'X=[x.*sin(x) -log(x)]'.
|
||
\begin{center}
|
||
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
|
||
\hline
|
||
\bf x&1&2&3&4&5&6&7&8&9&10\\
|
||
\hline
|
||
\bf y&-0.68 & 8.41 & -23.0 & -37.2 & -73.2 & -39.7 & 9.14 & 21.0 &
|
||
7.97 & -72.5\\
|
||
\hline
|
||
\end{tabular}
|
||
\end{center}
|
||
($a=7.72$, $b=14.8$).
|
||
|
||
\item óÏÓÔÁ×ØÔÅ ÍÏÄÅÌØ ÜËÓÐÅÒÉÍÅÎÔÁ ÐÏ ÉÚÍÅÒÅÎÉÀ
|
||
ÁÍÐÌÉÔÕÄÙ ÎÁÐÒÑÖÅÎÉÑ × ËÏÎÔÕÒÅ, ÉÓÐÙÔÙ×ÁÀÝÅÍ ËÏÌÅÂÁÎÉÑ Ó ÏÓÎÏ×ÎÏÊ ÞÁÓÔÏÔÏÊ
|
||
$\Omega=1000\,$çÃ É Ä×ÕÍÑ ÇÁÒÍÏÎÉËÁÍÉ $\Omega\pm\omega$, ÇÄÅ $\omega=74\,$çÃ.
|
||
éÚ×ÅÓÔÎÏ, ÞÔÏ ÓÕÍÍÁÒÎÏÅ ËÏÌÅÂÁÎÉÅ ÏÐÉÓÙ×ÁÅÔÓÑ ÐÒÉÂÌÉÖÅÎÎÏÊ
|
||
ÆÏÒÍÕÌÏÊ $U=a\sin(\Omega t)+b\sin(\omega t)-c\cos(\omega t)$.
|
||
óÏÚÄÁÊÔÅ ÉÎÔÅÒ×ÁÌ ×ÒÅÍÅÎ {\tt t=[0: 0.06: 120]}. äÌÑ ÐÏÌÕÞÅÎÉÑ ÉÄÅÁÌØÎÙÈ
|
||
ÚÎÁÞÅÎÉÊ~$U$ ÐÏÌÏÖÉÔÅ $a=361$, $b=117$, $c=92$. ïÔÎÏÛÅÎÉÅ ÓÉÇÎÁÌ/ÛÕÍ ÐÒÉ
|
||
ÐÏÌÕÞÅÎÉÉ ÚÁÛÕÍÌÅÎÎÏÇÏ ÓÉÇÎÁÌÁ ×ÙÂÅÒÉÔÅ ÒÁ×ÎÙÍ~20\,Äâ.
|
||
|
||
÷ÏÓÓÔÁÎÏ×ÉÔÅ ÚÎÁÞÅÎÉÑ ËÏÜÆÆÉÃÉÅÎÔÏ×~$a$, $b$ É~$c$.
|
||
|
||
|
||
\end{enumerate}
|
||
\end{document}
|