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@ -3,7 +3,7 @@
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\input{settings/common-preamble}
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\input{settings/bmstu-preamble}
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\input{settings/fancy-listings-preamble}
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\author{Гребенюк Елена Алексеевна (lgrebenuk12@yandex.ru)}
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\author{Гребенюк Елена Алексеевна}
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\title{Технологии разработки информационных систем для анализа больших объёмов информации}
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\date{2023-02-08}
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@ -637,17 +637,15 @@ $Q(a,x)=\frac{1}{l}\sum^l_{i=1}(a(x_i)-y_i)^2$
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\item Исправленный коэффициент детерминации растет при включении переменной
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\item Другие коэффициенты испытывают значительное смещение при включении новой переменной
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\end{enumerate}
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% \subsection {Пример решения задачи регрессии с ис}
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\subsection {Пример решения задачи регрессии с ис}
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\section{Линейная регрессия}
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$g(X_i,w)=\sum^r_{j=1}(w_jx_{ij}+w_0)\rightarrow Y_i$ , модель прогнозирования, $X_i-r$–мерный вектор. Функция потерь алгоритма: $L(a,y)=(g(X_i,w)-Y_i)^2$. Методы обучения:
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\begin{itemize}
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\item наименьших квадратов;
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\item градиентного спуска:
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\[L(a,y)=\frac{1}{l}\sum^l_{i=1}(g(X_i,w)-Y_i)^2\to \underset{w}{\min}\]
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\end{itemize}
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Мера качества. Средняя квадратичная ошибка (MSE): $Q(a,x)=\frac{1}{l}\sum^l_{i=1}(g(X_i,\omega)-Y_i)^2$
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$g(X_i,w)=\sum^r_{j=1}(w_jx_{ij}+w_0)\rightarrow Y_i$ , модель прогнозирования, $X_i-r$–мерный вектор
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Функция потерь алгоритма: $L(a,y)=(g(X_i,w)-Y_i)^2$
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Методы обучения: наименьших квадратов; градиентного спуска:
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$$L(a,y)=\frac{1}{l}\sum^l_{i=1}(g(X_i,w)-Y_i)^2\rightarrow min \underset{w}{min}$$
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Мера качества. Средняя квадратичная ошибка (MSE): $Q(a,x)=\frac{1}{l}\sum^l_{i=1}(g(X_i,w)-Y_i)^2$
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% lgrebenuk12@yandex.ru
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\begin{equation*}
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\begin{gathered}
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R^2 = 1-\frac{\sum_{i=1}^l(a(x_i)-y_i)}{\sum_{i=1}^l(y_i-\overline{y})^2}\\
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@ -655,7 +653,7 @@ $g(X_i,w)=\sum^r_{j=1}(w_jx_{ij}+w_0)\rightarrow Y_i$ , модель прогн
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MSE = \frac{1}{n}\sum_{i=1}^n(y_i - \overline{y})^2 = \sigma^2_r
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\end{gathered}
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\end{equation*}
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числитель -- среднеквадратичная ошибка, знаменатель -- простое среднее. Хорошая модель -- где ошибка классификатора минимальна. В модель нежелательно включать лишние регрессоры (штрафы по критериям Aкаике и Iварца).
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числитель -- среднеквадратичная ошибка, знаменатель -- простое среднее. Хорошая модель - где ошибка классификатора минимальна. r - число регрессоров. В модель нежелательно включать лишние регрессоры (штрафы по критериям акаике и шварца).
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Критерии для включения переменной
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\begin{enumerate}
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@ -678,159 +676,20 @@ $g(X_i,w)=\sum^r_{j=1}(w_jx_{ij}+w_0)\rightarrow Y_i$ , модель прогн
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\[g(X,w)\sum^p_{j=1}w_j\varphi_j(X)+w_0=\sum^p_{j=0}w_j\varphi_j(X),\varphi_0(X)=1'\]
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Число базисных функций может отличаться от числа признаков $j$.
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\section{Метод опорных векторов}
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\includegraphics[width=0.5\textwidth]{pics/04-bdaisdt-ROC-11.png}
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\begin{itemize}
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\item Решает задачу бинарной классификации $Y=\left\{1,-1\right\}$
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\item Классификатор линейный -- $\langle X_i, w \rangle $,
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\item Функция потерь $L(\mathcal{L},a(x))=\sum^N_{i=1}(0,1-Y_i\langle X_i,w\rangle)\rightarrow \underset{w}{\min}$
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\item Алгоритм максимизирует отступ классификатора (расстояние до ближайшего объекта), при условии минимизации числа ошибок.
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\end{itemize}
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\subsection{Линейно разделимый случай}
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Мы можем найти такие параметры, при которых классификатор не допускает ни одной ошибки
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Пусть существуют такие параметры $w^*$ и $b^*$, что классификатор $a(X_i,w)=sign(\langle w^*, X_i\rangle+b^*)$ не допускает ни одной ошибки на обучающей выборке, т.е. выборка является линейно разделимой. Тогда строим алгоритм такой, что
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\begin{enumerate}
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\item $Y_i(\langle X_i, w\rangle+w_0)>0, i=1,2,...,N;$
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\item Отступ классификатора -- расстояние от границы до ближайшего объекта максимальное.
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\end{enumerate}
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Отступ классификатора
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\includegraphics[width=0.25\textwidth]{pics/04-bdaisdt-ROC-12.png}
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...
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Величина отступа или функция потерь алгоритма на обучающей выборке :
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Вычисление ширины разделяющей полосы классификатора
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\[D(x,a)=\frac{1}{l}\sum^l_{i=1}[sign(w, x_i)*y_i<0]\]
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...
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где $sign(w, x_i)*y_i$ называется отступом. Положительный знак отступа соответствует правильному ответу, отрицательный -- неправильному, величина отступа -- среднее расстояние от разделяющей гиперплоскости до объектов. Метод обучения -- минимизация эмпирического риска. Функционал
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Метод опорных векторов
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\[D(x,a)=\frac{1}{l}\sum^l_{i=1}[sign(w, x_i)*y_i<0]\to \underset{w}{min}\]
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не допускает использование градиентных методов поиска экстремума.
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\subsection{Отступ классификатора}
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Пусть $(sign\langle \hat{w},X_i \rangle+\hat{b})$ -- построенный линейный классификатор. Тогда расстояние от любой точки $X_i\in{X_t,Y_t},t=1,2,...,N$ обучающей выборки до гиперплоскости, определяемой уравнением,$\hat{w}X+\hat{b}=0$ равно:
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\[ \rho_0(x,a)=\frac{|\hat{w}X_i+\hat{b}|}{||\hat{w}||}\]
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оно совпадает с отступом объекта $X_i$, следовательно, отступ классификатора равен
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\[\underbrace{\min}_{i=1,2,...,N}\frac{|\hat{w}X_i+\hat{b}|}{||\hat{w}||}\]
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\textbf{Вычисление ширины разделяющей полосы классификатора}
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Ищем $\alpha$ такое, что для $w^*=\alpha\hat{w}$ и $b^*=\alpha\hat{b}$
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\[\underbrace{\min}_{i=1,2,...,N}|w^*X_i+b^*|=1\]
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Тогда расстояние от гиперплоскости до ближайшего объекта обучающей выборки равно $\underbrace{\min}_{i=1,2,...,N}\frac{w^*X_i+b^*}{||w*||}=\frac{1}{||w^*||}$, т.е. ширина разделяющей полосы (отступ классификатора) равна $\frac{2}{||w^*||}$
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Строим классификатор, без ошибок разделяющий обучающую выборку и имеющий максимальный отступ.
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\[ \begin{cases}
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\begin{gathered}
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y_i(\langle w^*,X_i\rangle + b^*)\geq 1, i = 1, 2, ..., l\\
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||w^*||^2/2 \to \underbrace{\min}_{b^*,w^*}
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\end{gathered}
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\end{cases}\]
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Первое условие означает отсутствие ошибок классификации для линейно разделимой выборки.
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\textbf{Неразделимый случай.} Если в выборке есть $X_i$, такие что при любых $w^*$ и $b^*$, не существует решения задачи, то для них рассматриваются ограничения
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\[y_i(\langle w^*,X_i\rangle+b^*)\geq 1-\xi_i\]
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где $\xi_i>0$ -- штраф за нарушение ограничений. Если отступ объекта лежит между 0 и 1, то объект верно классифицируется, но штрафуется за попадание внутрь разделяющей полосы.
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\subsection{Метод опорных векторов для линейно неразделимой выборки}
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Найти такие $w,b,\xi_i$, для которых задача
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\[ \begin{cases}
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\begin{gathered}
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||\hat{w}||^2/2+C\sum^l_{i=1}\xi_i\to \underbrace{min}_{b,w,\xi}\\
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y_i(\langle \hat{w},X_i\rangle + \hat{b})\geq 1-\xi_i, i = 1, 2, ..., l\\
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\xi_i \geq 0, i = 1, 2, ..., l\\
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\end{gathered}
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\end{cases}\]
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Имеет допустимое решение, где $С>0$ -- гипер-параметр
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\textbf{Замена задачи условной оптимизации задачей безусловной оптимизации}
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Задача условной оптимизации
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\[ \begin{cases}
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\begin{gathered}
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||\hat{w}||^2/2+C\sum^l_{i=1}\xi_i\to \underbrace{min}_{b,w,\xi}\\
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y_i(\langle \hat{w},X_i\rangle + \hat{b})\geq 1-\xi_i, i = 1, 2, ..., l\\
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\xi_i \geq 0, i = 1, 2, ..., l\\
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\end{gathered}
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\end{cases}\]
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Перепишем второе и третье условия в виде:
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\[ \begin{cases}
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\begin{gathered}
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\xi_i\gg 1 - y_i(\langle \hat{w},X_i\rangle + \hat{b})\\
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\xi_i \geq 0
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\end{gathered}
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\to \xi_i \geq max(0,1-y_i(\langle \hat{w},X_i\rangle + \hat{b}))
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\end{cases}\]
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Отступ $М$ по определению равен $M=y_i(\langle \hat{w},X_i \rangle+\hat{b})$, тогда $\max(0,1-(\langle \hat{w},X_i \rangle+\hat{b}))=\max(0,1-M)$. И задача безусловной оптимизации имеет вид
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\[||\hat{w}||^2/2+С\sum^l_{i=1}max(0,1-M)\rightarrow \underbrace{\min}_{\hat{w},M}\]
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\subsection{Сравнение логистической регрессии и SVM}
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Логистическая регрессия требует, чтобы отступ уходил в $+\infty$ на каждом объекте -- важно для формирования корректных вероятностей, SVM достаточно отступа равного 1.
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\includegraphics[width=0.5\textwidth]{pics/04-bdaisdt-ROC-13.png}
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\subsection{Логистическая регрессия -- задание параметров}
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\begin{verbatim}
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LogisticRegression (penalty='l2', *, dual=False, tol=0.0001, C=1.0,
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fit_intercept=True, intercept_scaling=1, class_weight=None,
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random_state=None, solver='lbfgs', max_iter=100, multi_class='auto',
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verbose=0, warm_start=False, n_jobs=None, l1_ratio=None) [source]
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\end{verbatim}
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Регуляризация применяется по умолчанию, Solver -- \code{liblinear, bfgs, newton-cg, sag} -- алгоритмы минимизации эмпирического риска.
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\begin{enumerate}
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\item Алгоритм Бройдена-Флетчера-Голдфарба-Шанно -- \code{bfgs}
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\item Алгоритм сопряженных градиентов (Ньютона) -- \code{newton-cg}
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\item Стохастический алгоритм усредненного градиента -- \code{sag}
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\end{enumerate}
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Решатели «newton-cg», «sag» и «lbfgs» поддерживают только регуляризацию L2. По умолчанию классификатор содержит константу. \code{Liblinear} для небольших размерностей данных, \code{sag} -- для больших. С -- гиперпараметр, не может определяться по обучающей выборке.
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\code{class_weight}: \code{dict} or \code{‘balanced’} (default=\code{None}) -- веса классов по умолчанию -1, \code{’balanced’} -- веса устанавливаются обратно пропорционально частотам класса во входных данных.
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\code{max_iter}: int(default=100) -- используется только для \code{newton-cg}, \code{sag} и \code{lbfgs}.
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\subsection{svm.LinearSVC}
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\begin{verbatim}
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svm.LinearSVC (penalty='l2', loss='squared_hinge',*, dual=True,
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tol=0.0001, C=1.0, multi_class='ovr', fit_intercept=True,
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intercept_scaling=1, class_weight=None, verbose=0,
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random_state=None, max_iter=1000)
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\end{verbatim}
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использование \code{penalty=‘l1'}, и функции потерь \code{loss='squared_hinge’} не поддерживается, $С >0$ -- гиперпараметр.
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\begin{verbatim}
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svm.SVC (*, C=1.0, kernel='rbf', degree=3, gamma='scale', coef0=0.0,
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shrinking=True, probability=False, tol=0.001, cache_size=200,
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class_weight=None, verbose=False, max_iter=- 1,
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decision_function_shape='ovr', break_ties=False, random_state=None)
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\end{verbatim}
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\subsection{Оценка качества решения задачи. Режимы тестирования}
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\begin{itemize}
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\item \textbf{Отложенная выборка:} выделение обучающей, валидационной, и тестовой выборок, оценка качества работы алгоритма на обеих выборках.
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\item \textbf{Полная кросс-валидация}. Выбирается значение $t$, затем выборка разбивается всеми возможными способами на две части: $T^l=T^t\cup T_{l-t}$, вычисляется средняя оценка качества по всем разбиениям.
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\item \textbf{K-fold кросс-валидация}. Обучающая выборка разбивается на $k$ непересекающихся одинаковых по объему частей. Выполняется $k$ итераций. На каждой итерации из выборки удаляется одна из частей и модель обучается по оставшимся частям выборки. Модель тестируется на части обучающей выборки, которая не участвовала в обучении. Вычисляется средняя оценка качества по всем разбиениям
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\end{itemize}
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% \section{Домашнее задание}
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\section{Домашнее задание}
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\section{Задания на РК}
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В таблице показаны прогнозы, сделанные по двум регрессионным моделям. Для каждой модели рассчитайте сумму квадратов ошибки и оцените качество моделей.
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\begin{table}[H]
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@ -882,42 +741,4 @@ decision_function_shape='ovr', break_ties=False, random_state=None)
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\subsection{Листовые вершины}
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В каждой листовой вершине дерево будет выдавать константу или вектор вероятностей.
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\subsection{Жадный алгоритм построения бинарного решающего дерева}
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Конкретный метод построения решающего дерева определяется
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\begin{enumerate}
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\item Видом предикатов в вершинах
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\item функционалом качества
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\item Критерием останова
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\item Методом обработки пропущенных значений
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\item Методом стрижки удаление некоторых вершин с целью понижения сложности и способности к переобучения
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\end{enumerate}
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Обработка пропусков. Задачи регрессии и бинарной классификации.
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\textbf{Регрессия:} Заменяем пропущенные значения средними, спрогнозированными другими признаками. \textbf{Решающие деревья.} При построении: выбираем предикат в вершине, отправляем объекты с пропусками в оба поддерева. \textbf{Суррогатные предикаты}.
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\subsection{Преимущества и ограничения}
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\subsection{Ансамбль алгоритмов}
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Если ответы регрессоров на объекте -- независимые случайные величины с одинаковым матожиданием и дисперсией, то выполняются следующие свойства:
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\begin{equation*}
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\begin{gathered}
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\xi = \frac{1}{n}(\xi_1+\xi_2+...+\xi_n)\\
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E\xi = \frac{1}{n}(E\xi_1+E\xi_2+...+E\xi_n) = \frac{nE\xi_1}{n} = E\xi_1\\
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D\xi = \frac{1}{n^2}(D\xi_1+D\xi_2+...+D\xi_n) = \frac{D\xi_1}{n}
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\end{gathered}
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\end{equation*}
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Пусть имеется 3 независимых классификатора с вероятностью ошибки $p$ и решение принимается голосованием по большинству. Какова вероятность ошибки трёх классификаторов?
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\begin{equation*}
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\begin{gathered}
|
||||
1-p, p, p; p, 1-p, p; p, p, 1-p;\\
|
||||
p^3 + 3p^2(1-p)\\
|
||||
(p+(1-p))^3\\
|
||||
c_n^0p^0+c_n^1p(1-p)^{n-1}+c_n^2p^2(1-p)^{n-2}+...
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
\subsection{}
|
||||
|
||||
|
||||
\end{document}
|
||||
|
|
@ -99,7 +99,6 @@ $x(t)$ -- это входной непрерывный сигнал, умнож
|
|||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-signal-discretization.svg}
|
||||
\caption{Дискретизация сигнала}
|
||||
\label{pic:sig-discr}
|
||||
\end{figure}
|
||||
|
||||
Дискретный сигнал в частотной области -- бесконечное число повторяющихся копий дискретного представления сигнала. В цифровом вычислительном блоке мы всегда работаем с дискретным сигналом. Важно на каком расстоянии стоят частоты дискретного сигнала (виртуальные образы цифрового сигнана). Чтобы они не накладывались друг на друга нужна предварительная фильтрация (производимая ФПО).
|
||||
|
|
@ -161,7 +160,6 @@ $\frac{\sin{X}}{X}$ -- первый замечательный предел
|
|||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-sampling-afc.svg}
|
||||
\caption{Завал из-за усреднения напряжения на УВХ}
|
||||
\label{pic:sig-discr-uvh}
|
||||
\end{figure}
|
||||
|
||||
В предельном случае -- импульсная характеристика идеального ФНЧ, а в реальности это простая фильтрация. то есть сам АЦП выступает в роли фильтра.
|
||||
|
|
@ -240,18 +238,13 @@ $\frac{\sin{X}}{X}$ -- первый замечательный предел
|
|||
\[X(t) = \sum_{k=-\infty}^\infty x(t_k) \frac{sin 2\pi f_a(t-t_k)}{2\pi f_a(t-t_k)}\]
|
||||
где $t_k = \frac{k}{2f_a}$, $k=0, \pm1, \pm2, \pm3...$.
|
||||
|
||||
$\frac{sin 2\pi f_a(t-t_k)}{2\pi f_a(t-t_k)}$ -- импульсная характеристика идеального фильтра низких частот с прямоугольной АЧХ. То есть это указание о том, как именно восстанавливать сигнал -- пропустить через идеальный ФНЧ. Важно в формулировке теоремы -- ширина спектра сигнала. а не максимальная частота.
|
||||
$\frac{sin 2\pi f_a(t-t_k)}{2\pi f_a(t-t_k)}$ -- импульсная характеристика идеального фильтра низких частот с прямоугольной АЧХ. То есть это указание о том, как именно восстанавливать сигнал -- пропустить через идеальный ФНЧ. Важно в теореме -- ширина спектра сигнала. а не максимальная частота.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-adc-out.svg}
|
||||
\caption{На выходе АЦП}
|
||||
\end{figure}
|
||||
(1)
|
||||
|
||||
Вводить для 8-битной системы мощные средства передискретизации -- избыточно.
|
||||
|
||||
Классический случай дискретизации -- когда исходный сигнал находится в первой зоне (рис. \hrf{pic:sig-discr}, \hrf{pic:sig-discr-uvh}). Другие сигналы называются полосовые сигналы. Для полосового сигнала спектр окажется такой же
|
||||
Классический случай дискретизации -- когда исходный сигнал находится в первой зоне (рис2, рис4). Другие сигналы называются полосовые сигналы. Для полосового сигнала спектр окажется такой же
|
||||
|
||||
Субдискретизация -- (англ. under-sampling) дискретизация полезного сигнала, располагающегося вне основной полосы, т.е. лежащего в зоне с номером больше, чем 1 (полосовая дискретизация).
|
||||
|
||||
|
|
@ -263,53 +256,25 @@ $\frac{sin 2\pi f_a(t-t_k)}{2\pi f_a(t-t_k)}$ -- импульсная харак
|
|||
\item Фазовая манипуляция
|
||||
\end{itemize}
|
||||
|
||||
Если применить теорему Котельникова в лоб -- дискретизация должна быть на 400+ МГЦ.
|
||||
Котельников в лоб -- дискретизация должна быть на 400+ МГЦ.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-freq-move.svg}
|
||||
\caption{Перенос частоты}
|
||||
\end{figure}
|
||||
(2)
|
||||
|
||||
Применяя методику субдискретизации можно существенно уменьшить требования к параметрам приёмника. Но требования к АЦП остаются такими же.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}[b]{0.9\textwidth}
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-freq-z1.svg}
|
||||
\caption{ }
|
||||
\end{subfigure}
|
||||
|
||||
\begin{subfigure}[b]{0.9\textwidth}
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-freq-z2.svg}
|
||||
\caption{ }
|
||||
\end{subfigure}
|
||||
|
||||
\begin{subfigure}[b]{0.9\textwidth}
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-freq-z3.svg}
|
||||
\caption{ }
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
|
||||
(3)
|
||||
Когда сигнал лежит в зоне нечётным номером -- это полный образ, если с чётным -- инверсия. В любом случае в интересующей полосе образ.
|
||||
|
||||
Фильтр для классической дискретизации -- это ФНЧ. для субдискретизации -- полосовой фильтр.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-freq-z-dd.svg}
|
||||
\caption{Полосовой фильтр}
|
||||
\end{figure}
|
||||
(4)
|
||||
|
||||
Полоса пропускания $f_1...f_2$, переходная полоса справа $f_2...2f_s - f2$ слева $f_1...f_s-f_1$, полоса задержания меньше $f_s-f_1$ и больше $2f_s-f_2$. Если большой ДД -- нужен большой порядок фильтра. Снизить требование поможет также передискретизация -- плис -- цифровая фильтрация с прореживанием.
|
||||
полосовой фильтру
|
||||
полоса пропускания f1-f2
|
||||
переходная полоса справа f2...2fs-f2 слева f1...fs-f1
|
||||
полоса задержания <fs-f1 >2fs-f2
|
||||
|
||||
Если большой ДД -- нужен большой порядок фильтра. Снизить требование поможет также передискретизация -- плис -- цифровая фильтрация с прореживанием.
|
||||
|
||||
Методика расчёта чатоты дискретизации при субдискретизации (Как сделать, чтобы сигнал лежал строго в середине зоны)
|
||||
|
||||
|
|
@ -317,41 +282,33 @@ $f_c$ -- центральная частота полосового сигнал
|
|||
|
||||
\[f_s = \frac{4f_c}{2z-1}\]
|
||||
|
||||
$z$ -- номер зоны. Пусть ширина полосы = 4МГц, центральная частота 71МГц, по теореме К $f_s = 8$МГц. зона будет равна $18.25$, зона не может быть дробным, округялем до ближайшего целого, снова подставляем в эту же формулу, частота дискретизации будет $8,1143$МГц. Если хотим запас фильтрации больше -- $f_s=10$МГц, подставляем в выражение -- зона $14,7$, округляем до $14$, частота дискретизации = $10,519$МГц.
|
||||
$z$ -- номер зоны. Пусть ширина полосы = 4МГц, центральная частота 71МГц, по теореме К $f_s = 8$МГц. зона будет равна 18.25, зона не может быть дробным, округялем до ближайшего целого, снова подставляем в эту же формулу, частота дискретизации будет 8,1143МГц. Если хотим запас фильтрации больше -- фс=10МГц, подставляем в выражение - зона 14,7, округляем до 14, частота дискретизации = 10,519МГц.
|
||||
|
||||
\section{Квантование}
|
||||
Дискретный сигнал -- это сигнал с конечным количеством отсчётов, но разрядность пока бесконечна, поэтому возможно применить разрядность АЦП. Передаточная характеристика идеального квантователя
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-sig-sampling-err.svg}
|
||||
\caption{Квантование}
|
||||
\end{figure}
|
||||
(5)
|
||||
|
||||
Явно теряем в точности. Основной параметр -- Signal-Noise-Ratio
|
||||
явно теряем в точности.
|
||||
|
||||
Signal-Noise-Ratio
|
||||
|
||||
\[SNR = 6,02N + 1,76dB\]
|
||||
|
||||
N -- разрядность. Сигнал -- нестационарный процесс, имеет равномерное распределение от $0$ до $f_s/2$, не коррелирует со входным сигналом, матмодель -- входной сигнал умножается на шум квантования. Если аналоговая частота больше -- есть более общий вариант
|
||||
N -- разрядность. Сигнал -- нестационарный процесс, имеет равномерное распределение от 0 до фс/2, не коррелирует со входным сигналом, матмодель - входной сигнал умножается на шум квантования.
|
||||
|
||||
Если аналоговая частота больше -- есть более общий вариант
|
||||
|
||||
\[SNR = 6,02N + 1,76dB +10\log_{10}(\frac{f_s}{2f_a})\]
|
||||
|
||||
Если частота дискретизации много больше -- частота увеличилась, а разрядность не увеличитася, тогда шум уменьшается. Шум всегда будет коррелировать с сигналом, это можно использовать, подав собственный шум на низкой частоте.
|
||||
Если частота дискретизации много больше - частота увеличилась, а разрядность не увеличитася, тогда шум уменьшается. Шум всегда будет коррелировать с сигналом, это можно использовать, подав собственный шум на низкой частоте.
|
||||
|
||||
\section{Характеристики АЦП}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-sig-sampling-check.svg}
|
||||
\caption{}
|
||||
\end{figure}
|
||||
(6)
|
||||
память+пэвм -- способ проверить АЦП для своей системы.
|
||||
|
||||
Характеристики могут быть статические и динамические.
|
||||
|
||||
\subsection{Статические}
|
||||
Характеристики могут быть статические и динамические. Статические:
|
||||
\begin{itemize}
|
||||
\item Дифференциальная нелинейность DNL
|
||||
\item Интегральная нелинейность INL
|
||||
|
|
@ -360,195 +317,34 @@ N -- разрядность. Сигнал -- нестационарный про
|
|||
\item ошибка смещения offset error
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-perfect-tc.svg}
|
||||
\caption{Передаточная характеристика идеального АЦП}
|
||||
\end{figure}
|
||||
ПХ идеального АЦП -- все центры кода лежат на прямой линии симметрии
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-real-tc.svg}
|
||||
\caption{Пример ПХ реального АЦП}
|
||||
\end{figure}
|
||||
(7) ПХ идеального АЦП, все центры кода лежат на прямой
|
||||
(8) пример ПХ реального АЦП.
|
||||
|
||||
Дифференциальная нелинейность -- наибольшее отклонение ширины кода от идеального значения в МЗР или процентах от полной шкалы.
|
||||
|
||||
Интегральная нелинейность -- наихудшее отклонение центра кода от прямой, также измеряется в МЗР или процентах от полной шкалы.
|
||||
|
||||
Пропущенные коды формируется из первых двух. в примере 8 кода 011 не будет, в даташите будет написано no missing code если например ($N=14$) то на младших разрядах можем уже не видеть биты. Эти ошибки невозможно исправить
|
||||
Пропущенные коды формируется из первых двух. в примере 8 кода 011 не будет, в даташите будет написано no missing code если например (N=14) то на младших разрядах можем уже не видеть биты.
|
||||
|
||||
Ошибка смещения -- это аддитивная добавка напряжения на входе проценты от шкалы.
|
||||
Эти ошибки невозможно исправить
|
||||
|
||||
Ошибка усиления -- угол начального наклона (мультипликативная ошибка). Эти ошибки возможно исправить программно или внешними аналоговыми цепями.
|
||||
Ошибка смещения - это аддитивная добавка напряжения на входе проценты от шкалы
|
||||
|
||||
\subsection{Динамические характеристики}
|
||||
ошибка усиления - угол начального наклона (мультипликативная ошибка)
|
||||
|
||||
Эти ошибки возможно исправить программно или внешними аналоговыми цепями.
|
||||
|
||||
Динамические характеристики
|
||||
\begin{itemize}
|
||||
\item Реальное отношение сигнал-шум ($SNR_{real}$). Реальная характеристика точно будет отличаться. Точность разрядности возможно посчитать по формуле
|
||||
\[\frac{1}{LSB} \cdot 100\%\]
|
||||
|
||||
для $N=8$ это $0.4\%$. $SNR = 48dB(+1,76dB)$. Младшие биты АЦП всегда шумят, тест нужно делать на постоянном токе (например, замкнуть на землю, при условии, что земля не дрожжит).
|
||||
|
||||
$N=8$, $SNR=49,7dB$. реальный может быть 48,1 или 47,1 (первый лучше) зависит от частоты
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-typical-snr.svg}
|
||||
\caption{Типовой график}
|
||||
\end{figure}
|
||||
|
||||
\item Реальное отношение сигнал-шум ($SNR_{real}$)
|
||||
для н-разрядного АЦП возможно посчитать теорию. Реальная характеристика точно будет отличаться.
|
||||
N=8, SNR=49,7dB. реальный может быть 48,1 или 47,1 (первый лучше) зависит от частоты типовой график
|
||||
(9)
|
||||
обратный график - эффективное число бит
|
||||
\[ENoB = \frac{SNR_{real} - 1,76dB}{6,02}\]
|
||||
|
||||
\item Коэффициент гармонических искажений Total Harmonic Distortion -- отражает качество и линейность кармоник. Чем меньше брать в расчёт гармоник - тем легче продать. $THD=\frac{A_1}{\sqrt{A_2^2+A_3^2...}}\%$. AD -- считает по 5 гармоникам, TI -- по 7.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-typical-thd.svg}
|
||||
\caption{Типовой график}
|
||||
\end{figure}
|
||||
|
||||
\item Коэффициент гармонических искажений Total Harmonic Distortion -- отражает качество и линейность кармоник. Чем меньше брать в расчёт гармоник - тем легче продать. $THD=\sqrt{\frac{A_2+A_3...}{A_1}}\%$.
|
||||
\item сигнал шум и искажение (SINAD) типовая схема 4096 отсчётов. более качественный параметр.
|
||||
\item свободный динамический диапазон (SFDR spurious free dynamic range) -- свободный от наиболее мешающих компонент. иногда указывают самый мешающий компонент peak spurious (dB). эта характеристика всегда больше СШ. ДД замеряется по самой высокой гармонике.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-sfdr.svg}
|
||||
\end{figure}
|
||||
|
||||
\item частотная характеристика: полномощная полоса пропускания (full-power bandwidth) -- частота на которой амплитуда реконструируемой синусоиды отличается на $-3$дБ.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-fpb.svg}
|
||||
\end{figure}
|
||||
|
||||
\item частотная характеристика: полнолинейная полоса (Full linear bandwidth) срез $-0,1$дБ, показывает где АЧХ максимально плоская.
|
||||
\item при интермодуляции в общем случае появляются гармоники каждой частоты (кубические составляющие).
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-imd.svg}
|
||||
\caption{Продукт интермодуляционных искажений}
|
||||
\end{figure}
|
||||
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
2: f_2-f_1, f_2+f_1;\\
|
||||
3: 2f_1+f_2, 2f_1-f_2, f_1+2f_2, f_1-2f_2;\\
|
||||
...
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
Intermodulation distortion (IMD). обычно используется сигнальный тест на два компонента и используются продукты второго и третьего порядка.
|
||||
|
||||
\[ IMD = \frac{\sqrt{a_{f_1}^2 + a_{f_2}^2}}{\sqrt{\sum(\text{коэффициенты интермодуляции})^2}} \%\]
|
||||
|
||||
В ВЧ и радиочастотных цепях -- это критическая характеристика. Измеряется также через БПФ.
|
||||
\item время восстановления после перенапряжения. overvoltage recovery.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-ovr.svg}
|
||||
\end{figure}
|
||||
время восстановления измерения с приемлемой точностью (1МЗР). Может возникнуть от помех или наводок, индустриальные помехи, итд.
|
||||
\item время установления выходного напряжения (setting time, $t_{set}$) -- фактически, переходная характеристика.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-set.svg}
|
||||
\end{figure}
|
||||
характеристика может быть критична для многоканальных систем (мультиплексор перед АЦП) частота переключения каналов.
|
||||
\item апертурная неопределённость (апертурный джиттер, aperture jitter). дрожание фазы
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-jitter.svg}
|
||||
\end{figure}
|
||||
|
||||
Если это тактовый сигнал который подаётся на АЦП, даётся среднеквадратичное значение.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\fontsize{12}{1}\selectfont
|
||||
\includesvg[scale=1.01]{pics/04-cedd-00-slope.svg}
|
||||
\end{figure}
|
||||
|
||||
влияет на динамический диапазон (напрямую зависит от скорости нарастания сигнала (вольт/мкс)). Если величина этой ошибки превышает 1МЗР -- она становится определяющей. Тогда это нужно определять выходную частоту.
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
\left.\frac{dv}{dt}\right|_{t_0} = U_a\omega_{\max}\\
|
||||
\Delta v = u_a\omega_{\max} \Delta t_a\\
|
||||
\Delta v = \frac{1}{2}LSB = \frac{2u_a}{2^{N+1}}\\
|
||||
\omega_{max} = \frac{1}{2\pi\Delta t_a 2^{N+1}}
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
где N -- разрядность АЦП. Фазовые шумы ТИ накладываются на внутренние фазовые шумы АЦП и так теряются разряды. ТИ должны идти в ту же сторону, что и распространение сигнала. Обязательно через внешние драйверы ТИ с нулевой задержкой. Эмиттерно-селективная логика, дифференциальные сигналы. Снизить шумы позволяют clock-cleaner на основе ФАПЧ.
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Тестирование АЦП}
|
||||
ДПФ (есть оптимизации -- БПФ) $N = 1024$ отсчётов
|
||||
\[X(k) = \frac{1}{N}\sum_{n=0}^{N-1} x(nT) e^{\frac{-j2\pi nk}{N}}\]
|
||||
Получим 1024 градаций спектра.
|
||||
(1)
|
||||
|
||||
подвох --некогерентная дискретизация:
|
||||
(2)
|
||||
Окно (на неполном или более чем единицном периоде) формирует фазовый сдвиг. Эффект Гибса. Каждый скачок -- это гармоники. Точно рассчитать спектр вообще невозможно. Добавляют умножение на весовую функцию на этапе ДПФ -- операцию взвешивания. Фильтр RAMP, Хэмминга, Хана.
|
||||
(3)
|
||||
\[X(k) = \frac{1}{N}\sum_{n=0}^{N-1} x(nT) x(W) e^{\frac{-j2\pi nk}{N}}\]
|
||||
Тем самым мы достаточно сильно повлияли на входной сигнал.
|
||||
|
||||
Для тестирования АЦП необходимо применить когерентную дискретизацию -- один или несколько периоддов дискретизации точно укладываются в окно наблюдения.
|
||||
(4)
|
||||
|
||||
\section{Характеристики ЦАП}
|
||||
Основные характеристики
|
||||
\begin{itemize}
|
||||
\item Разрядность;
|
||||
\item Частота обновления данных.
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Статические характеристики}
|
||||
\begin{itemize}
|
||||
\item дифференциальная нелинейность -- отклонение от идеального напряжения в МЗР формирует нелинейность (по горизонтали).
|
||||
\item интегральная нелинейность -- отклонение вых значения от идеальной прямой (по вертикали)
|
||||
\item немонотонность -- следствие первых двух
|
||||
\item ошибка смещения offset error -- подали 0, на выходе не 0 напряжение (аддитивная ошибка)
|
||||
\item ошибка усиления gain error -- наклон шкалы (мультипликативная)
|
||||
\end{itemize}
|
||||
(5)
|
||||
Идеальная характеристика - каждому отсчёту соответствует точка равная определённому разному напряжению. Пропущенных кодов быть не может.
|
||||
|
||||
\subsection{Динамические характеристики ЦАП}
|
||||
\begin{itemize}
|
||||
\item Время установления выходного напряжения (setting time). -- классический аналоговый параметр. Меняется код и на выходе переходной процесс
|
||||
(6)
|
||||
Время установления является фактической ответной частью частотой дискретизации.
|
||||
\item Форма переходного процесса может отличаться. Эмпирический показатель Область глитч импульса glitch impulse area. Максимальный переходной процесс -- в середине шкалы
|
||||
(7)
|
||||
Измеряется в $V\cdot pS$. Иногда измеряют не только площади, но разницы площадей. Проблему решает деглитчер (УВХ).
|
||||
(8)
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Архитектуры АЦП}
|
||||
\begin{itemize}
|
||||
\item последовательного приближения
|
||||
\item дельта-сигма
|
||||
\item параллельные
|
||||
\end{itemize}
|
||||
|
||||
Основной элемент -- это компаратор. Это однобитный квантователь, перед ним стоит УВХ.
|
||||
(10) получаем 255 уровней квантования. на выходе 255-разрядный (позиционный, температурный) код.
|
||||
Приоритетный шифратор формирует N-разрядный код. Слишком много компараторов (дорого, размер, потребление), зависит от точности резисторов (интегральная нелинейность), разброс задержек компараторов (решается цифровыми средствами), паразитные ёмкости.
|
||||
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@
|
|||
\input{settings/common-preamble}
|
||||
\input{settings/bmstu-preamble}
|
||||
\input{settings/fancy-listings-preamble}
|
||||
\author{Сидякин Иван Михайлович}
|
||||
\author{Сидякин И. М.}
|
||||
\title{Программное обеспечение телекоммуникационных систем}
|
||||
\date{2023-02-09}
|
||||
|
||||
|
|
|
|||
|
|
@ -243,12 +243,11 @@ DevOps-инженер -- высококвалифицированный спец
|
|||
|
||||
Образ контейнера это переносимый пакет, содержащий программное обеспечение. При запуске он становится контейнером. Образ контейнера неизменен. После сборки образа внести в него изменения нельзя. Единственный способ изменить образ создать новый. Эта особенность дает гарантию того, что образ, используемый в рабочей среде, полностью совпадает с образом, который применялся при разработке и контроле качества.
|
||||
|
||||
Образы Docker - это файлы большого размера, которые изначально сохраняются на компьютере. Для управления ими нужны специальные средства. Интерфейс командной строки (CLI) Docker позволяет управлять образами:
|
||||
\begin{itemize}
|
||||
\item выполнять их сборку
|
||||
\item получать их список
|
||||
\item удалять и запускать образы
|
||||
\end{itemize}
|
||||
\subsection{Когда следует использовать контейнеры Docker?}
|
||||
|
||||
Управление средами размещения. Среда для приложения настраивается внутри контейнера. Благодаря этому команда эксплуатации может более гибко управлять средой приложения. Можно также управлять тем, какие приложения следует установить, обновить или удалить, не затрагивая другие контейнеры. Каждый контейнер изолирован. Облачные развертывания. Контейнеры Docker поддерживаются на многих облачных платформах.
|
||||
|
||||
Переносимость приложений. Контейнеры могут выполняться практически везде: на Настольных компьютерах, физических серверах, в виртуальных машинах и в облаке. Такая совместимость со средами выполнения позволяет легко перемещать контейнерные приложения между разными средами. Так как контейнеры требуют меньше ресурсов, увеличение времени запуска или завершения работы сказывается на виртуальных машинах. Благодаря этому упрощается и ускоряется повторное развертывание.
|
||||
|
||||
Dockerfile -- это текстовый файл с инструкциями для сборки и запуска образа Docker.
|
||||
\begin{itemize}
|
||||
|
|
@ -266,6 +265,13 @@ Dockerfile -- это текстовый файл с инструкциями д
|
|||
|
||||
Оба типа образов позволяют создавать многократно используемые образы. Однако базовые образы дают больше возможностей для содержимого итогового образа. Помним, что образ является неизменяемым. В него можно добавлять компоненты, но удалять из него ничего нельзя.
|
||||
|
||||
Образы Docker - это файлы большого размера, которые изначально сохраняются на компьютере. Для управления ими нужны специальные средства. Интерфейс командной строки (CLI) Docker позволяет управлять образами:
|
||||
\begin{itemize}
|
||||
\item выполнять их сборку
|
||||
\item получать их список
|
||||
\item удалять и запускать образы
|
||||
\end{itemize}
|
||||
|
||||
Пример Dockerfile для создания образа для веб-сайта.
|
||||
|
||||
\begin{itemize}
|
||||
|
|
@ -422,21 +428,16 @@ docker run -d tmp-ubuntu
|
|||
|
||||
\begin{verbatim}
|
||||
-v /home/docker_data:/usr/share/nginx/html
|
||||
-mount type=bind, \
|
||||
source=/home/docker_data, \
|
||||
destination=/usr/share/nginx/html
|
||||
-mount type=bind,source=/home/docker_data,destination=/usr/share/nginx/html
|
||||
\end{verbatim}
|
||||
|
||||
Так выглядит запуск контейнера с проброшенной папкой:
|
||||
|
||||
\begin{verbatim}
|
||||
docker run -d--name nginx_vol1
|
||||
-v /home/docker_data:/usr/share/nginx/html:ro nginx
|
||||
docker run -d--name nginx_vol1 -v /home/docker_data:/usr/share/nginx/html:ro nginx
|
||||
|
||||
docker run -d--name nginx_vol2 \
|
||||
-mount type=bind, \
|
||||
source=/home/docker_data, \
|
||||
destination=/usr/share/nginx/html,ro nginx
|
||||
-mount type=bind,source=/home/docker_data,destination=/usr/share/nginx/html,ro nginx
|
||||
\end{verbatim}
|
||||
|
||||
B mount мы используем следующие параметры:
|
||||
|
|
@ -474,67 +475,4 @@ Macvlan. Контейнер подключается при помощи вир
|
|||
|
||||
Overlay. Использование этого драйвера позволяет строить сети на нескольких хостах с Docker (обычно на Docker Swarm кластере). У контейнеров также есть свои адреса сети и подсети, и они могут напрямую обмениваться данными, даже если они располагаются физически на разных хостах.
|
||||
|
||||
\section{Когда следует использовать контейнеры Docker?}
|
||||
Переносимость приложений. Контейнеры могут выполняться практически везде: на Настольных компьютерах, физических серверах, в виртуальных машинах и в облаке. Такая совместимость со средами выполнения позволяет легко перемещать контейнерные приложения между разными средами. Так как контейнеры требуют меньше ресурсов, увеличение времени запуска или завершения работы сказывается на виртуальных машинах. Благодаря этому упрощается и ускоряется повторное развертывание.
|
||||
|
||||
Доставка приложений при использовании докер единицей распространения становится контейнер, благодаря этому происходит стандартизация.
|
||||
|
||||
Управление средами размещения. Среда для приложения настраивается внутри контейнера. Благодаря этому команда эксплуатации может более гибко управлять средой приложения. Можно также управлять тем, какие приложения следует установить, обновить или удалить, не затрагивая другие контейнеры. Каждый контейнер изолирован. Облачные развертывания.
|
||||
|
||||
Контейнеры Docker поддерживаются на многих облачных платформах.
|
||||
|
||||
\section{Когда НЕ следует использовать контейнеры Docker?}
|
||||
использование ядро ОС хоста, которое может стать объектом атаки. для обеспечения безопасности нужно принимать во внимание конфигурацию хранения и сети, в некоторых случаях целесообразнее использовать виртуальные машины.
|
||||
|
||||
Управление контейнерами сложнее, чем ВМ, можем узнать состояние, но получить журнал за определённое время - не очень просто (\code{docker stats} это поток данных, поэтому их обработка и хранение не производится).
|
||||
|
||||
\section{docker-compose}
|
||||
Надстройка над docker, применяется для многоконтейнерных приложений. позволяет запускать множество сервисов при помощи одной команды. docker применяется для управления отдельными контейнерами или сервисами. compose -- для одновременного управления несколькими контейнерами.
|
||||
|
||||
compose написан на python, конфигурация в yaml-файлах.
|
||||
|
||||
в составе docker desktop (Windows, mac) уже есть, на linux надо доставить руками.
|
||||
|
||||
докер-композ
|
||||
нгинкс
|
||||
докерфайл
|
||||
пхп
|
||||
докерфайл
|
||||
ввв
|
||||
индекс.хтмл
|
||||
|
||||
подменяем нгинкс.конф
|
||||
|
||||
докер-композ ямл - описывает процесс загрузки и настройки контейнеров.
|
||||
|
||||
Основные команды
|
||||
\begin{itemize}
|
||||
\item build -- сборка
|
||||
\item up -- запуск
|
||||
\item up -d -- запуск в фоне
|
||||
\item down -- остановка и удаление
|
||||
\item stop -- остановка сервисов
|
||||
\item restart -- перезапуск
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\section{Оркестрация контейнеров}
|
||||
Оркестраторы управляют ЖЦ контейнеров микросервисных приложений. Задачи оркестратора:
|
||||
\begin{itemize}
|
||||
\item подготовка инфраструктуры и развёртывание -- установка приложения и его зависимостей на сервер, включая подготовку этого сервера -- установку библиотек, служб, и так далее.
|
||||
\item Разделение ресурсов -- Для развёртывания в кластере требуется...
|
||||
\item Масштабирование контейнеров на основе рабочих нагрузок -- простой излишних ресурсов приводит к издержкам, а недостаток -- к нестабильно йработе приложения. Регулировать объёмы используемых ресурсов позволяет масштабирование, основанное на анализе нагрузок. Может быть ручным или автоматизированным.
|
||||
\item Балансировка нагрузок -- автоматический анализ и распределение рабочих нагрузок на сервер целиком и контейнеры в частности. Балансировка оптимизирует использование ресурсов, увеличивает пропускную способность каналов связи,...
|
||||
\item Маршрутизация трафика
|
||||
\item Мониторинг состояния контейнеров -- позволяет видеть какие контейнеры и образы запущены
|
||||
\item Обеспечение безопасного взаимодействия между контейнерами -- на основе непрерывной оценки кластеров, узлов и реестра контейнеров...
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Kubernetes}
|
||||
Считается отраслевым стандартом
|
||||
\begin{itemize}
|
||||
\item мониторинг сервисов
|
||||
\item
|
||||
\end{itemize}
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
|
|
@ -17,7 +17,7 @@
|
|||
|
||||
\href{https://jino.cloud/s/GGZgntaAqMRQbK2}{Вентцель -- Теория вероятностей}
|
||||
|
||||
\href{https://jino.cloud/s/8qNSXycHpkmmmZb}{Гмурман -- Теория вероятностей и математическая статистика}
|
||||
\href{https://jino.cloud/s/8qNSXycHpkmmmZb}{Гмурман -- Ьеория вероятностей и математическая статистика}
|
||||
|
||||
\subsection{Содержание курса}
|
||||
\begin{enumerate}
|
||||
|
|
@ -516,31 +516,7 @@ k=\frac{1}{\alpha_1}\\
|
|||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
TS-Процесс.
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
y_t = 0,5y_{t-1}+\xi_t+0,5t\\
|
||||
\varphi(0) = \frac{\delta y_t}{\delta\xi_t} = 1\\
|
||||
y_{t+1} = 0,5(t-1)+\xi_{t+1}+0,5y_t\\
|
||||
\varphi(1) = \frac{2y_{t+1}}{\delta\xi_t} = 0,5\\
|
||||
\varphi(2) = 0,25\\
|
||||
\varphi(k) = 0,5^k
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
Взятие разностей TS-процесса (нельзя брать такие разности)
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
y_t = 1+t+\xi_t-0,5\xi_{t-1}, \sigma_y^2=1,25\sigma_\xi^2\\
|
||||
Var(y_t) = Var(1+t + \xi_t -0,5\xi_{t-1}) = \\
|
||||
0 + \sigma^2 - 0,25\sigma^2\xi \\
|
||||
y_t-y_{t-1} = 1+t+\xi_t-0,5\xi_{t-1} - (1+t-1+\xi_{t-1}-0,5\xi_{t-2})\\
|
||||
1+\xi_t-1,5\xi_{t-1}+0,5\xi_{t-2}=\\
|
||||
Var(z) = (1+2,25+0,25)\sigma_\xi^2
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
Для того чтобы проверить распределение Дики-Фуллера нужно построить константы и тренд и понять значимый ли тренд
|
||||
|
||||
\appendix
|
||||
\setcounter{secnumdepth}{0}
|
||||
|
|
@ -610,7 +586,5 @@ r = число параметров модели, N - объём выборки.
|
|||
|
||||
SARIMA(p,d,q)(P,D,Q,S) -- учёт сезонности.
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
|
|
@ -1,77 +0,0 @@
|
|||
\documentclass[a4paper,fontsize=14bp]{article}
|
||||
|
||||
\input{settings/common-preamble}
|
||||
\input{settings/fancy-listings-preamble}
|
||||
\input{settings/bmstu-preamble}
|
||||
\numerationTop
|
||||
|
||||
\begin{document}
|
||||
\thispagestyle{empty}
|
||||
\makeBMSTUHeader
|
||||
|
||||
\makeReportTitle{домашней}{№ 1}{Введение}{Анализ и прогнозирование временн\'{ы}х рядов}{а}{Е.А.Гребенюк}
|
||||
\newpage
|
||||
\sloppy
|
||||
\pagestyle{fancy}
|
||||
\section{Задание}
|
||||
Рассмотрим процесс
|
||||
\[y_t = \xi_t - 2.5\xi_{t-1}+\xi_{t-2}, \xi_t \sim N(0,1)\]
|
||||
\begin{enumerate}
|
||||
\item Является ли процесс $y_t$ обратимым и стационарным?
|
||||
\item Найти автоковариационную функцию процесса $y_t$.
|
||||
\item Вычислить дисперсию процесса $y_t$.
|
||||
\item Рассматривается процесс ARMA(1, 1):
|
||||
|
||||
$1-\alpha L y_t = (1-0.5L)\xi_t$, где $\alpha$ -- некоторое действительное число и $\xi_t = N(0,\sigma_\xi^2)$. Найти
|
||||
\begin{itemize}
|
||||
\item все $\alpha \in \mathbb{R}$ для которых процесс является стационарным;
|
||||
\item все $\alpha \in \mathbb{R}$ для которых процесс является обратимым
|
||||
\end{itemize}
|
||||
\end{enumerate}
|
||||
|
||||
\section{Выполнение}
|
||||
\subsection{Обратимость и стационарность}
|
||||
Для процесса возможно построить характеристическое уравнение. Если корни характеристического уравнения по модулю больше 1, то процесс обратим. Опишем в с помощью оператора сдвига
|
||||
\[y_t = 1-2.5L+1L^2\]
|
||||
и решим квадратное уравнение
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
1-2.5z+z^2=0\\
|
||||
z = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
|
||||
z_1 = \frac{-2.5-\sqrt{-2.5^2-4}}{2}\\
|
||||
z_1 = 1.25 + \sqrt{1.5625-1} \approx 2\\
|
||||
z_2 = \frac{-2.5+\sqrt{-2.5^2-4}}{2}\\
|
||||
z_2 = 1.25 - \sqrt{1.5625-1} \approx 0.5
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
Только один корень уравнения по модулю больше, поэтому процесс \textbf{не является обратимым}.
|
||||
|
||||
Процесс \textbf{является стационарным} по теореме Вольда.
|
||||
|
||||
\subsection{Автоковариационная функция}
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
\gamma(0) = Var(y_t) = 8.25\\
|
||||
\gamma(1) = cov(\xi_t - 2.5\xi_{t-1}+\xi_{t-2}, \xi_{t-1} - 2.5\xi_{t-2}+\xi_{t-3})=\\
|
||||
= E[\xi_t - 2.5\xi_{t-1}+\xi_{t-2}, \xi_{t-1} - 2.5\xi_{t-2}+\xi_{t-3}] = \\
|
||||
-2.5 \cdot 1 \cdot 1 + -2.5 \cdot 1 \cdot 1 = -5\\
|
||||
\gamma(2) = E[\xi_t - 2.5\xi_{t-1}+\xi_{t-2}, \xi_{t-2} - 2.5\xi_{t-3}+\xi_{t-4}] = 1 \cdot 1 = 1\\
|
||||
\gamma(3) = 0
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
\subsection{Дисперсия процесса}
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
Var(y_t) = ?\\
|
||||
Var(y_t) = Var(\xi_t - 2.5\xi_{t-1}+\xi_{t-2})=\\
|
||||
= Var(\xi_t) +Var(-2.5\xi_{t-1})+Var(\xi_{t-2}))=\\
|
||||
=Var(\xi_t) + 6.25(\xi_{t-1}) + Var(\xi_t) = \\
|
||||
8.25 \cdot Var(\xi_t) = 8.25
|
||||
\end{gathered}
|
||||
\end{equation*}
|
||||
|
||||
\subsection{Процесс ARMA(1, 1)}
|
||||
|
||||
|
||||
\end{document}
|
||||
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Before Width: | Height: | Size: 16 KiB |
Loading…
Reference in New Issue