diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..f73806e --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +.ipynb_checkpoints/ \ No newline at end of file diff --git a/ntebooks/python/estimates.ipynb b/ntebooks/python/estimates.ipynb new file mode 100644 index 0000000..4834b42 --- /dev/null +++ b/ntebooks/python/estimates.ipynb @@ -0,0 +1,15694 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Глава 3. Теория оценок" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## 3.1 Понятие о точечной оценке" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hide_input": false + }, + "source": [ + "Математическая статистика имеет огромное количество разообразнейших применений, но с точки зрения экспериментальной физики (и как следствие студентов, изучающих эту науку) наиболее интересным применением является оценка параметров закономерностей. Пусть есть некоторое явление природы, которое можно описать при помощи модели $M(\\theta)$. Здесь $\\theta$ - это некоторый набор параметров модели, которые могут принимать различные значения. На этом этапе мы не оговариваем, как именно модель описывает процесс и что мы можем принимать в качестве параметров. Положим теперь, что существует некоторый выделенный набор параметров $\\theta_0$, который соответствует некоторому \"истинному\" состоянию природы. Далее мы будем исходить из того предположения, что при попытке предпринять некоторые измерения, мы будем получать результаты, соответствующие нашей модели именно с этим набором параметров.\n", + "\n", + "
\n", + "**Замечание** \n", + "Тут важно заметить, что мы также негласно предполагаем, что природа вообще действует согласно нашей модели, но этот вопрос мы пока оставим за кадром. В какой-то мере мы вернемся к нему, в главе 5, когда будем обсуждать теорию проверки гипотез.\n", + "
\n", + "\n", + "Предаставим теперь, что мы провели некоторую серию экспериментов $X = \\{X_0, X_1,...X_N\\}$, в которых мы тем или иным способом изучаем состояние природы (будем дальше называть результаты этих экспериментов экспериментальной выборкой). Нашей задачей в этой главе будет описание процедуры, при помощи которой можно на основе выборки сделать вывод об истинном состоянии природы $\\theta_0$. Важно понимать, что в общем случае, результаты измерений являются случайными величинами, поэтому полученное нами на основании этих данных состояние природы также будет случайной величиной в противовес истинному состоянию природы $\\theta_0$, которое вообще говоря, истинно случайной величиной не является. Полученную величину будем называть *точечной оценкой* состояния природы $\\hat{\\theta}$ или просто оценкой. Саму процедуру, в процессе которой получена оценка, будем называть оцениванием.\n", + "\n", + "
\n", + "**Пример** \n", + "Положим, что знания студента в области физики являются состоянием природы (а точнее данного конкретного студента). Очевидно, что досконально проверить этот факт не представляется возможным, поэтому для измерения этой величины мы проводим эксперимент - экзамен. То, что по результатам экзамена оказывается в ведомости является оценкой не только с точки зрения деканата, но и с точки зрения математической статистики.\n", + "
\n", + "\n", + "В дальнейшем будем считать, что состояния природы описываются действительным числом или набором действительных чисел. Сама по себе теория этого не требует, но в противном случае довольно сложно сравнивать состояния между собой (требуется определять понятие близости в произвольном пространстве). В этом случае наша процедура оценивания:\n", + "\\begin{equation}\n", + " \\hat{\\theta} = f(X)\n", + "\\end{equation}\n", + "является действительной функцией на пространстве векторов $X$, состоящих из случайных переменных. Такие функции еще называют статистиками. Очевидно, что далеко не людая такая функция будет давать тот результат, которого мы хотим. Поэтому вводятся дополнительные обязательные свойства оценок." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### 3.1.1 Свойства точечных оценок" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Состоятельность\n", + "\n", + "Естественное пожелание к оценщику, заключается в том, что качество оценки должно зависеть от объема выборки, числа измерений $n$ случайных переменных $X$: чем больше выборка, тем качественней оценка $\\hat{\\theta}$. Иными словами, мы хотим, чтобы с ростом объема выборки значение оценки приближалось к истинному значению параметра. При использовании сходимости по вероятности оценку $\\hat{\\theta}$ определяют как состоятельную, если при любых $\\varepsilon > 0$ и $\\eta > 0$, найдется такое $N$, что вероятность $P \\left( \\left| \\hat{\\theta} - \\theta \\right| > \\varepsilon \\right) < \\eta $ при всех $n > N$. Или другими словами, всегда можно выбрать такое количество измерений, для которого вроятность отклонения оценки от истинного значения не превышает наперед заданное число.\n", + "\n", + "
\n", + "**Замечание**\n", + "Нужно заметить, что на практике оценки являются состоятельными только когда при построении оценки не учитывается систематическая ошибка. В противном случае, может наблюдаться сходимость по вероятности не к нулю, а к некоторой фиксированной константе.\n", + "
\n", + "\n", + "#### Несмещенность\n", + "\n", + "Рассмотрим набор измерений, каждое из которых состоит из $n$ наблюдений $X$, характеризуемый функцией плотности вероятности $P(X | \\theta) = P(\\hat\\theta | \\theta)$ при фиксированном $n$ и определим смещение как отклонение среднего по этому набору $\\hat{\\theta_n}$ от истинного\n", + "\\begin{equation}\n", + " b = E[\\hat{\\theta_n}] - \\theta\n", + "\\end{equation}\n", + "Оценка называется несмещенной, если $b = 0$.\n", + "\n", + "Заметим, что смещение не зависит от измеренных величин, но зависит от размера образца, формы оценщика и от истинных (в общем случае неизвестных) свойств ФПВ $f(x)$, включая истинное значение параметра. Если смещение исчезает в пределе $n \\to \\infty$, говорят об асимптотически несмещенной оценке. Заметим, что из состоятельности оценки не следует несмещенность. Это означает, что даже если $\\hat{\\theta}$ сходится к истинной величине $\\theta$ в единичном эксперименте с большим числом измерений, нельзя утверждать, что среднее $\\hat{\\theta}$ по бесконечному числу повторений эксперимента с конечным числом измерений $n$ будет сходится к истинному $\\theta$. Несмещенные оценки пригодны для комбинирования результатов разных экспериментов. В большинстве практических случаев смещение должно быть мало по сравнению со статистической ошибкой и им пренебрегают.\n", + "\n", + "#### Эффективность\n", + "\n", + "Для сравнения разных методов оценки, очень важным свойством является эффективность. Говоря простым языком, эффективность - это величина, обратная разбросу значений $\\hat{\\theta}$ при применении к разным наборам данных. Для того, чтобы хорошо разобраться в этом свойстве, надо вспомнить, что оценка, как случайная величина, распределена с плотностью $P(\\hat\\theta | \\theta)$. Вид этого распределения может быть не известен полностью, но знать его свойства – низшие моменты – необходимо. Среднее по нему суть смещение, а дисперсия $\\sigma_{\\hat\\theta}^2 = \\int{ (\\hat\\theta - \\theta} ) P(\\hat\\theta | \\theta) d\\hat\\theta$ суть мера ошибки в определении оценки. Выбирая между различными методами, мы, естественно, хотим, чтобы ошибка параметра была минимальной из всех доступных нам способов его определения для фиксированного эксперимента. Разные методы обладают разной эффективностью и в общем случае при конечной статистике дисперсия распределения оценки никогда не будет равна нулю. Разумеется, встает вопрос о том, можно ли построить оценку с максимальной возможной эффективностью." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### 3.1.2* Граница Рао-Крамера и информация" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Утверждение\n", + "\n", + "Пусть есть несмещенная оценка $\\hat\\theta$ параметра $\\theta$, тогда всегда выполняется неравенство:\n", + "\n", + "\\begin{equation}\n", + " D(\\hat\\theta) \\geq \\frac{1}{I(\\theta)},\n", + "\\end{equation}\n", + "где $I(\\theta)$ - информация Фишера:\n", + "\n", + "\\begin{equation}\n", + " I(\\theta) = E_X \\left[\\left( \\frac{\\partial \\ln L(X,\\theta)}{\\partial \\theta} \\right)^2 \\right],\n", + "\\end{equation}\n", + "где в свою очередь $L(X, \\theta)$ - функция правдоподобия. $L(X,\\theta) = \\prod{P(X_i,\\theta)}$ - вероятность получить набор данных при фиксированном значении $\\theta$.\n", + "\n", + "#### Доказательство в одномерном случае\n", + "Введем функцию \n", + "\n", + "\\begin{equation}\n", + " W = \\frac{\\partial \\ln L(X,\\theta)} {\\partial \\theta}.\n", + "\\end{equation}\n", + "\n", + "Найдем иатематическое ожидание этой функции:\n", + "\n", + "\\begin{equation}\n", + " E(W) = \\int{L(X, \\theta) W(X, \\theta) dX} = \\int{ L \\frac{1}{L} \\frac{\\partial L}{\\partial \\theta} dX} = \\frac{\\partial}{\\partial \\theta} \\int{L dX} = 0.\n", + "\\end{equation}\n", + "\n", + "Теперь рассмотрим ковариацию $E(\\hat\\theta W)$:\n", + "\n", + "\\begin{equation}\n", + " E(\\hat\\theta W) = \\int{\\hat\\theta \\frac{1}{L} \\frac{\\partial L} {\\partial {\\theta}} L dX} = \\frac{\\partial}{\\partial \\theta} \\int{\\hat\\theta L dX} = \\frac{\\partial}{\\partial \\theta} E(\\hat\\theta).\n", + "\\end{equation}\n", + "\n", + "Для несмещенных оценок последнее выражение упрощается до вида $E(\\hat\\theta W) = {\\partial \\theta}/{\\partial \\theta} = 1$\n", + "Согласно неравенствую [Коши — Буняковского](https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9A%D0%BE%D1%88%D0%B8_%E2%80%94_%D0%91%D1%83%D0%BD%D1%8F%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%BE%D0%B3%D0%BE), получаем: $\\sqrt{D(\\hat\\theta) D(W)} \\geq \\left| E(\\hat\\theta W) \\right| = 1$. Отсюда легко получаем желаемое утверждение.\n", + "\n", + "#### Следствие\n", + "Максимальная эффективность достигается в том случае, если величины $\\hat\\theta$ и $W$ являются скоррелированными. Оценка, максимизирующая функцию $L(X,\\theta)$ является в общем случае состоятельной, несмещенной, кроме того совпадает с оценкой вида $W(\\hat\\theta, X) = 0$. То есть является максимально эффективной." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## 3.2 Интервальные оценки" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "На практике применение точечных оценок сильно затруднено тем, что не известно, на сколько каждая такая оценка точна. Действительно, мы можем спокойно утверждать, что слон весит один килограмм если разброс нашей оценки составляет больше массы слона. Для того, чтобы решить эту проблему есть два пути.\n", + "Первый путь - это на ряду с точечной оценки указывать меру эффективности этой оценки или ее разброс$\\sigma_{\\hat\\theta}$. Но тут любой внимательный слушатель заметит, что для определения эффективности, вообще говоря, надо знать истинное значение парематра $\\theta$, которого мы, разумеется не знаем. Следовательно приходится использовать не эффективность, а оценку этой эффективности, которая сама по себе является случайной величиной. Кроме того, часто случается, что распределение оценки является не симметричным и описать его одним числом не удается.\n", + "\n", + "Более корректным способом является построение интервальной оценки (доверительного интервала). Формально определение интервальной оценки будет отличаться в зависимости от того, какое определение вероятности вы будете использовать.\n", + "\n", + "**Частотная интерпретация**: интервальной оценкой параметра или группы параметров $\\theta$ с уровнем достоверности $\\alpha$ называется такая область на пространстве параметров (в одномерном случае - интервал), которая при многократном повторении эксперимента с вероятностью (частотой) $\\alpha$ перекрывает истинное значение $\\theta$.\n", + "\n", + "**Субъективная интерпретация**: доверительным интервалом для параметров $\\theta$ будем называть такую область в пространстве параметров, в которой интегральная апостериорная вероятность нахождения истинного значения параметра равна $\\alpha$.\n", + "\n", + "Для точного описания результата проведения анализа как правило в качестве результата приводят как точечную оценку, так и интервальную оценку с некоторым уровнем достоверности (в английском варианте Confidence Level или C. L.). В некоторых случаях приводят несколько интервальных оценок с разным уровнем достоверности. В случае, когда речь идет об определении верхней или нижней границы какого-то параметра, точечная оценка как правило не имеет смысла и в качестве результата дается только итнервальная оценка. \n", + "\n", + "
\n", + "**Замечание** \n", + "Точечная оценка также не имеет смысла в случае, когда распределение оценки, скажем имеет вид однородного распределения на отрезке. В этом случае все параметры на этом отрезке равновероятны и не понятно, какой из них называть результатом.\n", + "
\n", + "\n", + "
\n", + "**Замечание** \n", + "Вполне очевидно, что для одних и тех же данных с использованием одного и того же метода оценивания можно построить бесконечное множество интервальных оценок с фиксированным уровнем достоверности. Действительно, мы можем двигать интервал в разные стороны таким образом, чтобы его вероятностное содержание не менялось. Обычно, если не оговорено иначе, используются так называемые центральные доверительные интервалы, в которых вероятностные содержание за границами интервалов с обеих сторон равны.\n", + "
\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## 3.3 Методы построения оценок" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### 3.3.1 Метод максимума правдоподобия" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Определение\n", + "Введем функцию правдоподобия следующим образом: $L(X,\\theta) = \\prod{P(X_i,\\theta)}$, где $P(X_i, \\theta)$ - вероятность получить случайным образом компоненту данных с номером $i$ при истинном значении параметра $\\theta$.\n", + "Будем называеть оценкой максимума правдоподобия такую $\\hat\\theta$, для которой $L(\\hat\\theta, X)$ максимально для заданного набора данных. \n", + "Такая оценка в общем случае является состоятельной и несмещенной. Она имеет очень хорошо понятный физический смысл: $\\hat\\theta$ - это такое значение параметра, для которого вероятность получить набор данных максимальна. Кроме того, как упоминалось в параграфе **3.1.2**, такая оценка кроме всего прочего будет еще и наиболее эффективной из всех возможных (достаточной статистикой).\n", + "#### Ограничения\n", + "Оценка является эффективной только если область измереия $X$ не зависит от $\\theta$.\n", + "#### Интервальная оценка\n", + "Для построения интервальной оценки воспользуемся субъективной вероятностью и теоремой Байеса:\n", + "\\begin{equation}\n", + " P(\\theta | X) = \\frac{L(X|\\theta) \\pi(\\theta)}{\\int{LdX}}\n", + "\\end{equation}\n", + "Здесь $L$ - функция правдоподобия, а $\\pi$ - априорная вероятность для параметра $\\theta$. Если нет никакой дополнительной информации оп параметре, то мы можем положить $\\pi = 1$. Мы получаем, что распределение значения реального параметра повторяет форму функции правдоподобия. Вероятность того, что параметр $\\theta$ лежит в диапазоне от $a$ до $b$ составляет \n", + "\\begin{equation}\n", + " P(a < \\theta < b) = \\int_b^a{L(X | \\theta)}\n", + "\\end{equation}\n", + "#### Интервальная оценка в асимптотическом случае\n", + "Согласно центральной предельной теореме, при достаточно большом количестве данных $X$ (и при разумных предположениях о форме распределения для этих данных), функция правдоподобия будет иметь вид нормального распределения. В этом случае центральный доверительный интервал можно вычислить, пользуясь аналитической формулой. Центральный доверительный интервал с уровнем значимости 68% будет соответствовать диапазону между значениями $\\theta$, такими, что значение функции правдоподобия в них отличается от максимума на 0.5.\n", + "#### Логарифмическое правдоподобие\n", + "В реальных задачах часто работают не с самой функцией правдоподобия, а с ее логарифмом $\\ln L(X|\\theta) = \\sum{\\ln P(X_i | \\theta)}$. Это связано с тем, что при численных манипуляциях проще работать со сложением, чем с умножением. Очевидно, что максимум правдоподобия будет одновременно и максимумом его логарифма и обратно. Для построения интервальных оценок логарифм прадоподобия не годится." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### 3.3.2 Метод минимума $\\chi^2$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Разберем частный, но очень часто встречающийся случай, когда распределение измеренной величины - нормальное (Гауссово):\n", + "\\begin{equation}\n", + " P(X_i | \\theta) = \\frac{1}{\\sqrt {2 \\pi} \\sigma_i} e^{- \\frac{(X_i - \\mu_i(\\theta))^2}{2 \\sigma_i^2}}\n", + "\\end{equation}\n", + "Здесь $\\mu_i(\\theta)$ - модельное значение. Опуская нормировочный множитель, не существенный для нахождения максимума можно записать логарифм функции правдоподобия в виде:\n", + "\\begin{equation}\n", + " \\ln L(X_i | \\theta) = - \\sum{\\frac{(X_i - \\mu_i(\\theta))^2}{2 \\sigma_i^2}}\n", + "\\end{equation}\n", + "Определим величину \n", + "\\begin{equation}\n", + " \\chi^2 = - 2 \\ln L(X_i | \\theta) = \\sum{\\frac{(X_i - \\mu_i(\\theta))^2}{\\sigma_i^2}}.\n", + "\\end{equation}\n", + "Назовем оценкой минимума $\\chi^2$ такое значение параметра, при котором эта величина минимально. Очевидно, что эта оценка эквивалента максимуму правдоподобия для нормально расспределенных измерений. Оценка минимума $\\chi^2$ будет состоятельной и несмещенной и для других распределений измерений, но только для таких измерений она будет обладать максимальной эффективностью.\n", + "\n", + "#### Интервальная оценка\n", + "Также как и в методе максимума правдоподобия, центральный интервал можно получить из аналитической форму функции правдоподобия (для нормально распределенных ошибок, форма правдоподобия будет нормальной всегда, а не только в асимптотике). Из-за множителя 2, мы получаем, что центральный интервал в 68% будет соответствовать отклонению $\\chi^2$ на 1. Отклонение на 2 будет соответствовать уже 95% доверительному интервалу.\n", + "\n", + "#### Проверка качества фита\n", + "Дополнительное полезоное свойство суммы $\\chi^2$ заключается в том, что она позволяет оценить, насколько хорошо данные описываются моделью. В том случае, когда измерения распределены по нормальному закону и независимы между собой, сумма $\\chi^2$ оказывается распределена по [одноименному распределению](https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D1%85%D0%B8-%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82). При хорошем соответствии модели и данных величина $\\chi^2 / n $, где $n$ - так называемое количество степеней свободы (количество точек минус количество параметров), должна в среднем быть равна 1. Значения существенно большие (2 и выше) свидетельствуют о плохом соответствии или заниженных погрешностях. Значения меньше 0.5 как правило свидетельствуют о завышенных ошибках." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### 3.3.3 Метод наименьших квадратов" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "В случа, если все ошибки $\\sigma_i$ одинаковы, множитель $\\frac{1}{\\sigma^2}$ можно вынести за скобку. Для нахождения минимума постоянный множитель не важен, поэтому мы можем назвать оценкой наименьших кавдратов такое значение параметра $\\hat\\theta$, при котором миниальна сумма квадратов:\n", + "\\begin{equation}\n", + " Q = \\sum{(X_i - \\mu_i(\\theta))^2}.\n", + "\\end{equation}\n", + "Эта оценка очевидно сохраняет все свойства оценки минимума $\\chi^2$, правда только в том случае, если все ошибки действительно одинаковы. \n", + "\n", + "Оценка наименьших квадратов удобна в том случае, когда не известны ошибки отдельных измерений. Она является состоятельной и асимптотически несмещенной, хотя ее эффективность оптимальна только для ограниченного количества случаев. В некоторых случаях, ошибку измерения можно оценить по разбросу данных, используя критерий Пирсона.\n", + "\n", + "#### Интервальная оценка\n", + "Оценка методом наименьших квадратов очевидно игнорирует информацию о погрешностях измерений и не позволяет напрямую оценить погрешность результата без дополнительных предположений. Для получения оценки погрешностей надо сделать три предположения:\n", + "\n", + "* Данные описываются предложенной моделью;\n", + "* Отклонения данных от модельной кривой независимы между собой (носят статистический характер);\n", + "* Статистические ошибки для всех точек равны между собой.\n", + "\n", + "При этих предположениях, можно оценить $\\sigma$ для каждой из точек как средне квадратичное отклонение точек от наилучшей модели (той, которая получена минимизацией суммы $Q$). После этого задача получения погрешностей сводится к уже решенной для метода $\\chi^2$.\n", + "\n", + "
\n", + " **Замечание** \n", + " По очевидным причинам оценка погрешностей, проведенная таким образом, не имеет смысла для маленького количества измерений (меньше 8-10 точек). Все будет работать и будет получен какой-то результат, но он будет довольно бессмысленным.\n", + "
\n", + "\n", + "
\n", + " **Замечание** \n", + " Одна из основных проблем, связанных с определением погрешностей методом наименьших квадратов заключается в том, что он дает разумные погрешности даже в том случае, когда данные вообще не соответствуют модели. По этой причине не рекомендуется использовать его в тех случаях, когда погрешности измеренных значений известны. Если других инструментов под рукой нет, то результаты работы метода надо всегда проверять визуально по графику.\n", + "
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### 3.3.4 Другие методы" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Существует довольно большое количество других методов определения параметров зависимости. Как правило они специально заточены под конкретную задачу. В качестве примера можно привести [метод моментов](https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%BE%D0%B2). Методо дает состоятельную и несмещенную, но довольно слабо эффективную оценку, но при этом позволяет очень быстро делать оценки для больших объемов данных. \n", + "Еще один пример - [метод квази-оптимальных весов Ф. В. Ткачева](https://arxiv.org/abs/physics/0604127), напротив обладает высокой эффективностью и стабильностью, но несколько сложнее в реализации." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 3.4* Многопараметрические оценки" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**TBD**" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Практика" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Установка и подготовка программного обеспечения\n", + "При работе с интерактивными примерами предполагается использование интерактивной среды [Jupyter](http://jupyter.org/). Наиболее простым способом подготовки всего, необходимого для работы является установка дистрибутива [Anaocnda](https://www.anaconda.com/download/) (предполагается использование Python версии 3). Для работы необходимы пакеты `numpy`, `pandas` и `matplotlib`, которые входят в поставку Anaconda. Опытные пользователи python могут воспользоваться любым другим способом установки.\n", + "\n", + "Для запуска данного \"блокнота\" в интерактивном режиме необходимо скопировать его в рабочую директорию на свое усмотрение и запустить команду `jupyter notebook` в этой директории." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Инструменты" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "Подготовим несколько инструментов для представления данных и отрисовки. Для начала импортируем нужные библиотеки." + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:55.541135Z", + "start_time": "2018-05-01T10:18:55.531132Z" + }, + "hide_input": false + }, + "outputs": [], + "source": [ + "import numpy as np # Библиотека для численных методов\n", + "import pandas as pd # Библиотека для табличного представления данных\n", + "import matplotlib.pyplot as plt # Библиотека для отрисовки графиков\n", + "# Команда для интерактивной отрисовки графиков\n", + "%matplotlib inline " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Генерация и отрисовка данных" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "#### Генерация данных по нормальному распределениею с заданными средними и стандартными отклонениями" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:55.573131Z", + "start_time": "2018-05-01T10:18:55.544132Z" + }, + "hide_input": false + }, + "outputs": [], + "source": [ + "def generate(x, y, err):\n", + " '''\n", + " Функция в качестве данных берет три массива: средние значения по координате x, по координате y и ошибки по координате y. \n", + " Все массивы должны быть одинаковой длины.\n", + " Возвращает таблицу pandas.DataFrame, содержащую сгенерированные данные.\n", + " '''\n", + " # Проверяем, что среди переданных аргументов нет отрицательных и нулей\n", + " if (err <= 0).any():\n", + " raise Exception\n", + " \n", + " # Генерируем нормально распределенные данные\n", + " generatedY = y + np.random.normal(scale=err)\n", + " \n", + " # Создаем объект типа DataFrame с тремя колонками. Объект в фигурных скобках является словарем\n", + " return pd.DataFrame({'x' : x, 'y' : generatedY, 'err': err}) " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Отрисовка функций" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:55.592132Z", + "start_time": "2018-05-01T10:18:55.576133Z" + } + }, + "outputs": [], + "source": [ + "def plot_function(func, a, b, label=\"\", numPoints = 300):\n", + " '''\n", + " Отрисовать функцию func в интервале (a,b) с заданным количеством точек\n", + " '''\n", + " x = np.linspace(a,b, numPoints, endpoint=True)\n", + " plt.plot(x, func(x), label=label)\n", + " \n", + "\n", + "def plot_data(data, name=\"\"):\n", + " '''\n", + " Отрисовать данные\n", + " '''\n", + " plt.errorbar(data.x, data.y, data.err, label = name, fmt='o')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Параболический апроксиматор##\n", + "Для нахождения оценок методом наименьших квадратов и обобщенным методом наименьших квадратов ($\\chi^2$) потребуется находить минимум функции, которую приближенно можно описать параболической зависимостью. Это можно сделать разными способами, но в одномерном случае это проще всего сделать аппроксимируя параболу по трем произвольным точкам (лучше брать этит точки как можно ближе к минимуму) или используя стандартную библиотечную функцию для полиномиального фита." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:55.729149Z", + "start_time": "2018-05-01T10:18:55.595133Z" + } + }, + "outputs": [], + "source": [ + "class ParabolicFunction:\n", + " def __init__(self, a=None, b=None, c=None):\n", + " \"\"\"\n", + " Конструктор по-умолчанию\n", + " \"\"\"\n", + " self.a = a\n", + " self.b = b\n", + " self.c = c\n", + " \n", + " def from_point(self, x1, x2, x3, y1, y2, y3):\n", + " \"\"\"\n", + " Конструктор по трем точкам \n", + " \"\"\"\n", + " self.a = (y3-(x3*(y2-y1)+x2*y1-x1*y2)/(x2-x1))/(x3*(x3-x1-x2)+x1*x2)\n", + " self.b = (y2-y1)/(x2-x1) - self.a*(x1+x2)\n", + " self.c = (x2*y1-x1*y2)/(x2-x1)+self.a*x1*x2\n", + " return self\n", + " \n", + " def from_function(self, x1,x2,x3, func):\n", + " \"\"\"\n", + " Апроксиматор по трем точкам\n", + " \"\"\"\n", + " return self.from_point(x1,x2,x3, func(x1), func(x2), func(x3))\n", + " \n", + " def from_delta_center(self, centr, delta, func):\n", + " \"\"\"\n", + " Конструктор по центральному значениюи и дельте\n", + " \"\"\"\n", + " return self.from_function(centr-delta, centr, centr+delta, func)\n", + " \n", + " def __call__(self, x):\n", + " return self.a*x*x+self.b*x+self.c\n", + " \n", + " def minX(self):\n", + " \"\"\"\n", + " Метод класса возвращает абсцису минимального значения\n", + " \"\"\"\n", + " return -self.b/2/self.a\n", + " \n", + " def minY(self):\n", + " \"\"\"\n", + " Метод класса возвращает ординату минимального значения\n", + " \"\"\"\n", + " return self(self.minX(self))\n", + "\n", + " def __repr__(self):\n", + " return '{:f} *x^2 + {:f} *x + {:f}\\n'.format(self.a,self.b,self.c)\n", + "\n", + "# Статическая функция, определяющая минимум неизвестной функции, считая ее параболой\n", + "def find_min (est, delta, func): \n", + " return ParabolicFunction().from_delta_center(est, delta, func).minX()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Проверим работу аппроксиматора:" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:55.940160Z", + "start_time": "2018-05-01T10:18:55.732142Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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h93A2J1/pTZH40OrBlTRq0tzqSC7L0SdbvxpYW770AYwxR40xx+2XlwDeIuJf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\nnrCiTRw3DdkYcDVtrryXkPA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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "function = lambda x: x*x + x\n", + "x = np.linspace(-4,4)\n", + "plt.plot(x, function(x), label = \"parabola\")\n", + "plt.plot(x, ParabolicFunction().from_function(-1, 0, 1, function)(x), label = \"parabola-interpolation\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Данные" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Генерируем и запоминаем данные\n", + "В качестве модели берем $y = \\alpha x^2,$\n", + "где $\\alpha$ - параметр модели" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.087163Z", + "start_time": "2018-05-01T10:18:55.942155Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# Запоминаем нашу модель. Функция берет в качестве аргумента число a и возвращает другую функцию\n", + "model = lambda a : lambda x: a*x*x\n", + "\n", + "# Создаем модель с фиксированным параметром\n", + "parabola = model(1.1)\n", + "\n", + "# Генерируем и запоминаем данные\n", + "x = np.arange(1, 10)\n", + "y = parabola(x)\n", + "err = np.sqrt(y)\n", + "data = generate(x, y, err)\n", + "\n", + "#строим график\n", + "plot_data(data)\n", + "plot_function(parabola, 1, 10)" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.100163Z", + "start_time": "2018-05-01T10:18:56.089165Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "
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errxy
01.04880911.841994
12.09761821.461339
23.14642739.005328
34.195235419.136974
45.244044530.266481
56.292853644.855236
67.341662758.983212
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" + ], + "text/plain": [ + " err x y\n", + "0 1.048809 1 1.841994\n", + "1 2.097618 2 1.461339\n", + "2 3.146427 3 9.005328\n", + "3 4.195235 4 19.136974\n", + "4 5.244044 5 30.266481\n", + "5 6.292853 6 44.855236\n", + "6 7.341662 7 58.983212\n", + "7 8.390471 8 70.550984\n", + "8 9.439280 9 80.302089" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#Выводим данные\n", + "data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Метод наименьших квадратов" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Вводим оценку наименьших квадратов. Функция воздействует на модель и набор данных и возвращает функцию одного переменного ( $\\chi^2(\\alpha)$ )." + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.124165Z", + "start_time": "2018-05-01T10:18:56.102165Z" + } + }, + "outputs": [], + "source": [ + "# Фукнкция np.vectorize превращает обычную функцию языка Python в обобщенную функцию библиотеки numpy\n", + "def sum_of_squares(model, data):\n", + " def _sum_of_squares(alpha):\n", + " np_model = np.vectorize(model(alpha))\n", + " return np.sum((data.y - np_model(data.x))**2)\n", + " return np.vectorize(_sum_of_squares)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Строим график для наименьших квадратов" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.418182Z", + "start_time": "2018-05-01T10:18:56.127164Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot_function(sum_of_squares(model, data), 0.5, 1.5, \"Sum of squares\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Находим оценку наименьших квадратов" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.431184Z", + "start_time": "2018-05-01T10:18:56.421182Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.08760757649\n" + ] + } + ], + "source": [ + "minsq = find_min(1, 0.1, sum_of_squares(model, data))\n", + "print(minsq)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Метод $\\chi^2$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Вводим оценку модифицированных наименьших квадратов ($\\chi^2$)" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.454185Z", + "start_time": "2018-05-01T10:18:56.434182Z" + } + }, + "outputs": [], + "source": [ + "def chi2(model, data):\n", + " def _chi2(alpha):\n", + " f = np.vectorize(model(alpha))\n", + " return np.sum(((data.y - f(data.x))/ data.err)**2)\n", + " return np.vectorize(_chi2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Строим график" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.783205Z", + "start_time": "2018-05-01T10:18:56.457184Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot_function(chi2(model, data), 0.5, 1.5, \"chi2\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Находим оценку $\\chi^2$" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:56.799223Z", + "start_time": "2018-05-01T10:18:56.785206Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.11018819489\n" + ] + } + ], + "source": [ + "minchi = find_min(1, 0.1, chi2(model, data))\n", + "print(minchi)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Строим более детальный график" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:57.190229Z", + "start_time": "2018-05-01T10:18:56.803205Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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A4jqTnba9RGQ8teV+9jPs0p8vZ5V7FhBf5/s44HgT0+MamO6sXIhI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\nIC13pZTyQP8fWNwzmiKUClkAAAAASUVORK5CYII=\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot_function(chi2(model, data),minchi - 0.1,minchi + 0.1,\"chi2\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Утверждение!\n", + "Сумма вида $\\chi^2 = \\sum {\\frac{y_i - \\mu(x_i)}{\\sigma_i^2}}$ в достаточно общем случае распределена по распределению $\\chi^2$. \n", + "Доверительный интервал с достоверностью $1\\sigma$ (~68%) можно построить, найдя точки, где эта величина откланяется от минимума на 1.\n", + "Кроме того, при хорошем согласии эксперимента с теорией величина $\\frac{\\chi^2}{n-k}$ должна быть порядка 1." + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:57.213228Z", + "start_time": "2018-05-01T10:18:57.193230Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "259.090909 *x^2 + -575.279337 *x + 324.309668\n", + "\n", + "0.0621260744197\n" + ] + } + ], + "source": [ + "sigma = lambda parabolic : 1/np.sqrt(parabolic.a)\n", + "print(ParabolicFunction().from_delta_center(1, 0.1,chi2(model, data)))\n", + "errchi = sigma(ParabolicFunction().from_delta_center(1,0.1,chi2(model, data)))\n", + "print(errchi)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Сравнение оценок" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Сравниваем оценки" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:18:57.234232Z", + "start_time": "2018-05-01T10:18:57.215229Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Оценка методом наименьших квадратов = 1.088\n", + "Оценка методом хи-квадрат = 1.110 +- 0.062\n" + ] + } + ], + "source": [ + "print('Оценка методом наименьших квадратов = {:.3f}'.format(minsq))\n", + "print('Оценка методом хи-квадрат = {:.3f} +- {:.3f}'.format(minchi,errchi))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Строим распределения оценок. Заодно проверяем, что средняя ошибка методом $\\chi^2$ соответствует разбросу этой оценки" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "metadata": { + "ExecuteTime": { + "end_time": "2018-05-01T10:21:55.257651Z", + "start_time": "2018-05-01T10:20:04.092073Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Стандартное отклонение оценки методом наименьших квадратов: 0.06516846169693595\n", + "Стандартное отклонение оценки методом хи-квадрат: 0.05951027223128111\n", + "Средняя ошибка методом хи-квадрат: 0.0592348877759137\n" + ] + }, + { + "data": { + "image/png": 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dispchi2 += (chi2val-1)**2 / num\n", + "\n", + " # Считаем среднюю ошибку методом хи-квадрат\n", + " errchi += sigma(ParabolicFunction().from_delta_center(1,0.1,chi2(model, data))) / num\n", + "\n", + "print(\"Стандартное отклонение оценки методом наименьших квадратов: {}\".format(np.sqrt(displq)))\n", + "print(\"Стандартное отклонение оценки методом хи-квадрат: {}\".format(np.sqrt(dispchi2)))\n", + "print(\"Средняя ошибка методом хи-квадрат: {}\".format(errchi))\n", + "\n", + "plt.title(\"Сравнение разных методов оценок\");\n", + "plt.hist(lqs, label=\"least squares\", bins=30)\n", + "plt.hist(chi2s, label=\"chi2\", bins = 30)\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + 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+ "metadata": { + "hidden": true + }, + "source": [ + "### Проверка отрисовки графика" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [ + { + "data": { + "image/png": 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XohQsex38Q6Dj/WanqZB0YbDCpysPkp2bx1P9nfCOys5jwbcmLH/D+IXSijW0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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Plot test frame\")\n", + "x = np.linspace(0, 2*math.pi)\n", + "plt.plot(x, np.sin(x), label=\"sin(x)\")\n", + "plt.plot(x, np.cos(x), label=\"cos(x)\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "heading_collapsed": true, + "hidden": true + }, + "source": [ + "## Численное интегрирование" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "### Вариант 1: простой интеграл Римана\n", + "Фиксированное количество точек. Интеграл всегда сходится, но точность результата зависит от функции." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:34+0000", + "start_time": "2017-09-22T06:34+0000" + }, + "hidden": true, + "init_cell": true + }, + "outputs": [], + "source": [ + "def integrateR(func, a, b, numPoints = 400):\n", + " res = 0\n", + " points, step = np.linspace(a, b, numPoints, endpoint=True, retstep=True)\n", + " return np.sum(func(points))*step" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "### Вариант 2: итеративный интеграл Симпсона \n", + "Используется [итеративный численный интегратор из библиотеки SciPy](https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.integrate.simps.html), \n", + "в основе которого лежит [квадратичная формула Симпсона](http://mathworld.wolfram.com/SimpsonsRule.html). Первый аргумент - сама функция, второй и третий - границы интегрирования. \n", + "Итеративная формула гарантирует точность интегрирования, но не гарантирует сходимости." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [], + "source": [ + "# from scipy.integrate import simps\n", + "# def integrateS(func, a, b):\n", + " \n", + "# return simps(func( np.linspace(a,b, 5000, endpoint=True)), np.linspace(a,b, 5000, endpoint=True))\n", + "# # integrateS = {UnivariateFunction func, Number a, Number b -> new SimpsonIntegrator().integrate(5000,func,a,b)}" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "from scipy.integrate import quad\n", + "\n", + "def integrateQ(func, a, b):\n", + " return quad(func, a,b)[0]" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "### Интегратор по умолчанию\n", + "В качестве интегратора по умолчанию выбираем интегратор Римана. Если захотим использовать другой, надо переопределить переменную `integrate,` например вот так:\n", + "\n", + "`integrate = integrateS`" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:34+0000", + "start_time": "2017-09-22T06:34+0000" + }, + "hidden": true, + "init_cell": true + }, + "outputs": [], + "source": [ + "integrate = integrateR" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "### Проверяем, что интегратор работает" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "Итеративный интеграл:" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "2.0" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "integrateQ(np.sin,0, math.pi)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "Интеграл Римана 100 точек:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "1.9998321638939927" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "integrateR(np.sin,0,math.pi,100)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "Интеграл Римана 400 точек:" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "1.9999896675538067" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "integrateR(np.sin,0,math.pi)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "Интеграл Римана 5000 точек:" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "1.9999999341763102" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "integrateR(np.sin,0,math.pi,5000)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "heading_collapsed": true, + "hidden": true + }, + "source": [ + "## Свертка" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "[Сверткой](https://ru.wikipedia.org/wiki/%D0%A1%D0%B2%D1%91%D1%80%D1%82%D0%BA%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7) \n", + "двух функций называется следующее интегральное преобразование:\n", + "$$\n", + " h(x) = f(x) \\otimes g(x) = \\int {f(x-y) g(y) dy }\n", + "$$\n", + "Интегрирование ведется по всей области определения функций. Очевидно, что переменные под интегрированием можно менять местами." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "convolute = lambda f, g, a, b: np.vectorize(lambda x: integrate(lambda y: f(x-y)*g(y), a, b))" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "### Проверяем работу свертки" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "Проверяем работу свертки, взяв в качестве одной функции ступеньку, а в качестве другой Гауссиану. Параметры `wStep` и `sigma` можно менять по своему усмотрению для получения различного результата." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + }, + "hidden": true + }, + "outputs": [ + { + "data": { + "image/png": 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G1+8etTizy5nMPW8uANcvuZ7DJw9XLJDQt15PqWq3jKoLTe4qaByOeg6olpXA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\nwr//B5b+tU6rRwJc3PtiFv5qIWckn8F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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "wStep = 2; # Ширина ступеньки\n", + "sigma = 0.5;# Стандартное отклонение Гауссианы\n", + "\n", + "# ступенька\n", + "f = lambda x: 1/wStep if (x >= -wStep/2 and x<= wStep/2) else 0.0 # нормировка на единицу\n", + "f = np.vectorize(f)\n", + "\n", + "# Функция Гаусса\n", + "g = lambda x : 1/np.sqrt(2*math.pi)/sigma*np.exp(-x*x/2/sigma/sigma)\n", + "\n", + "# Plot plot = new Plot(title: \"Convolution test plot\", showLegend: true)\n", + "\n", + "\n", + "# автоматическое определение окна интегрирования и отрисовки\n", + "wind = max(2*wStep, 4*sigma)\n", + "x = np.linspace(-wind, wind, 300)\n", + "plt.plot(x, f(x), label=\"f(x)\");\n", + "plt.plot(x, g(x), label=\"g(x)\");\n", + "plt.plot(x, convolute(f,g, -wind, wind)(x), label=\"convolution\");\n", + "plt.title( \"Convolution test plot\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "heading_collapsed": true, + "hidden": true + }, + "source": [ + "## Интерполяция" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hidden": true + }, + "source": [ + "При использовании численных методов часто нужно уметь не только вычислить функцию, но и сохранить ее значения в виде таблицы для дальнейших вычислений.\n", + "В данном случае используется кубический сплайн-интерполятор из библиотеки commons-math. Сплайн-интерполяторы характерны тем, что выдают гладкую функцию, проходящую через все заданные точки. \n", + "Легко видно, что для функций с большой степенью гладкости такой интерполятор будет выдавать хорошее приближение даже при сравнительно маленьком количестве узлов." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "from scipy.interpolate import CubicSpline\n", + "\n", + "def interpolate(func, a, b, numPoints=300):\n", + " xs = np.linspace(a,b, numPoints, endpoint=True)\n", + " spline = CubicSpline(xs, func(xs), extrapolate=False)\n", + " \n", + " def my_spline(x):\n", + " if xb:\n", + " return spline(b)\n", + " else:\n", + " return spline(x)\n", + " \n", + " return np.vectorize(my_spline)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "heading_collapsed": true, + "hidden": true + }, + "source": [ + "### Проверка работы интерполятора" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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mWh1HqSxTefRUTA44MmO81VE8ghZ3F7P4zcEEHbvBweZBBFYNsTqOUlnmnkbt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\nyW2HH1DuIzExkejoaK5du2Z1FI+UK1c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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Testing function interpolation (sin)\")\n", + "x = np.linspace(0, math.pi, 300, endpoint=True)\n", + "plt.plot(x, np.sin(x), label=\"original\")\n", + "plt.plot(x, interpolate(np.sin, 0, math.pi, 10)(x), label=\"10 points\")\n", + "plt.plot(x, interpolate(np.sin, 0, math.pi, 100)(x), label=\"100 points\")\n", + "plt.plot(x, interpolate(np.sin, 0, math.pi, 200)(x), label=\"200 points\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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thhH3qMcXc6/ioGh2pl5jMzYxBDYZbEdiI0hKr7K4m7CMfTHiHu34smUcZS2o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\nQwjxhWVivK7jnjtA28E6bHpgS4Qsq4C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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Testing function interpolation (step)\")\n", + "x = np.linspace(-1, 1, 300, endpoint=True)\n", + "# ступенька\n", + "f = lambda x: 1 if (x >= -0.5 and x<= 0.5) else 0.0 # нормировка на единицу\n", + "f = np.vectorize(f)\n", + "\n", + "plt.plot(x, f(x), label=\"original\")\n", + "plt.plot(x, interpolate(f, -1, 1, 10)(x), label=\"10 points\")\n", + "plt.plot(x, interpolate(f, -1, 1, 100)(x), label=\"100 points\")\n", + "plt.plot(x, interpolate(f, -1, 1, 200)(x), label=\"200 points\")\n", + "plt.plot(x, interpolate(f, -1, 1, 300)(x), label=\"300 points\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Распределения" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Равномерное распределение" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Равномерное распределение говорит о том, что величина может принимать любое значение в заданном интервале $(a,b)$ с равной вероятностью.\n", + "\n", + "*Распределение:* \n", + "$$\n", + " f(x) = 1/(b-a), a < x < b\n", + "$$\n", + "\n", + "Равномерное распределение с конечными пределами указывает на то, что величина лежит строго в заданном интервале, но ее точное значение в этом интервале неизвестно.\n", + "В случае бесконечных пределов нормировка равномерного распределения расходится, но его все равно можно использовать.\n", + "\n", + "*Среднее значение:* \n", + "$\n", + " \\mu = (b-a)/2\n", + "$\n", + "\n", + "*Дисперсия:* \n", + "$\n", + " D = (b-a)^2/12\n", + "$" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "uniform = lambda a,b: np.vectorize(lambda x: 1/(b-a) if (x>=a and x<=b) else 0.0)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Демонстрация равномерного распределения" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Uniform distribution\")\n", + "x = np.linspace(-1,1, endpoint=True)\n", + "plt.plot(x, uniform(-0.5, 0.5)(x), label=\"d = 1\")\n", + "x = np.linspace(-1.5,1.5, endpoint=True)\n", + "plt.plot(x, uniform(-1, 1)(x), label=\"d = 2\")\n", + "x = np.linspace(-2,2, endpoint=True)\n", + "plt.plot(x, uniform(-1.5, 1.5)(x), label=\"d = 3\");\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Биномиальное распределение" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Биномиальное распределение описывает вероятность получить число \"успехов\" $k$ при проведении $n$ экспериментов при условии, что вероятность успеха в одном эксперименте равна $p$.\n", + "\n", + "*Распределение:* \n", + "$$\n", + " p(k) = \\frac{n!}{k! (n-k!)} p^k (1-p)^{n-k}\n", + "$$\n", + "\n", + "*Среднее значение:* \n", + "$\n", + " \\mu = np\n", + "$\n", + "\n", + "*Дисперсия:* \n", + "$\n", + " D = np(1-p)\n", + "$\n", + "\n", + "Для вычислений используется [библиотечная функция](https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.binom.html)" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [], + "source": [ + "from scipy.stats import binom" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Демонстрация биномиального распределения" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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I9adRHwxTH2x6GUInsOUzCNVBvyPatl2/IwBjjbuvlHIM1yT9cNjYk6K3r6UP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\ny4FDjDGXisg04AxjzJkichDwHDAO6AP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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Binomial distribution (p = 0.5)\")\n", + "p =0.5\n", + "x = range(0,10)\n", + "plt.plot(x, binom.pmf(x,10,p), label=\"n = 10\")\n", + "x = range(0,50)\n", + "plt.plot(x, binom.pmf(x,50,p), label=\"n = 50\")\n", + "x = range(0,100)\n", + "plt.plot(x, binom.pmf(x,100,p), label=\"n = 100\");\n", + "x = range(0,100)\n", + "plt.plot(x, binom.pmf(x,150,p), label=\"n = 150\");\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Распределение Пуассона" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Распределение Пуассона описывает вероятность насчитать количество событий $k$ при условии, что все события равновероятны, а сренее количество таких событий равно $\\lambda$. \n", + "Это распределение - одно из самых распространенных в физике, поскольку описывает отклонение от среднего в процессе достаточно произвольного вида.\n", + "\n", + "*Распределение:* \n", + "$$\n", + " p(k) = \\frac{\\lambda^k}{k!} e^{-\\lambda}\n", + "$$\n", + "\n", + "*Среднее значение:* \n", + "$\n", + " \\mu = \\lambda\n", + "$\n", + "\n", + "*Дисперсия:* \n", + "$\n", + " D = \\lambda\n", + "$\n", + "\n", + "Для вычислений используется [библиотечная функция](http://commons.apache.org/proper/commons-math/javadocs/api-3.5/org/apache/commons/math3/distribution/PoissonDistribution.html)" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:34+0000", + "start_time": "2017-09-22T06:34+0000" + }, + "init_cell": true + }, + "outputs": [], + "source": [ + "from scipy.stats import poisson" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Демонстрация распределения Пуассона" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + } + }, + "outputs": [ + { + "data": { + "image/png": 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17l7tVF39GemkfxrGV9j0t/TDyOeFnwMwM2umyZFoXRUbG0t5eblO/BamlKK8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\nBQ32kO3eOVR5iHp3PRPSJ5g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y1hM1HSA82r0AnsdthFA3fwC8jfutCWg9CNyL\nx4OEYFoAbgGlBPTtAZ5GL+8CW1LyEbgAvAZeAjeAUgo+AjcJ6wzrhOR0qplvhNLE5RhD84S3kfLQ\nt0Coi2cxc7Xm+krU9wY4mpeHdfOL/F7IbbuH/7r53zA4juMUiE4p7ziO4zgt4EnfcRynQHjSdxzH\nKRCe9B3HcQqEJ33HcZwC4UnfcRynQHjSdxzHKRA/AZEkKMH7TGJ6AAAAAElFTkSuQmCC\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Poisson distribution\")\n", + "for i, mean in zip([15,25,50,70,100,120,150], [5,10,20,35,50,75,100]):\n", + " x = range(0,i)\n", + " plt.plot(x, poisson.pmf(x,i), '-', label=\"lambda = {:d}\".format(i))\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Нормальное распределение" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Нормальное или Гауссово распределение - наиболее часто встречается в физике (и не только). Оно является предельным случаем распределение Пуассона при больших $\\lambda$, \n", + "а также всегда всплывает при усреднеии случайных величин в силу *центральной предельной теоремы*.\n", + "\n", + "*Распределение:* \n", + "$$\n", + " f(x, \\mu, \\sigma) = \\frac {1}{\\sqrt{2\\pi} \\sigma} e^{-\\frac{(x-\\mu)^2}{2 \\sigma^2}}\n", + "$$\n", + "\n", + "*Среднее значение:* \n", + "$\n", + " \\mu \n", + "$\n", + "\n", + "*Дисперсия:* \n", + "$\n", + " D = \\sigma^2\n", + "$\n", + "\n", + "Параметр $\\sigma$ называют стандартным отклонением и именно он в большинстве случаев используется для оценки погрешности эксперимента." + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "# Функция Гаусса\n", + "gauss = lambda mu, sigma: lambda x : 1/np.sqrt(2*math.pi)/sigma*np.exp(-(x-mu)*(x-mu)/2/sigma/sigma)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Демонстрация распределения Гаусса" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + } + }, + "outputs": [ + { + "data": { + "image/png": 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0xuh7IaVyUzKSoqizNVBcVU+/2J6RbdNWYAHwyUzflJEBNhu2Awcw+2B8zU94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\nAUM6DWHl9JVM6jUpuJMNA9b8CIqOmXb8qKZ7wvz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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Gaussian (Normal) distribution\")\n", + "x = np.linspace(-10,10)\n", + "plt.plot(x, gauss(0,1)(x), label=\"sigma = 1\")\n", + "plt.plot(x, gauss(0,2)(x), label=\"sigma = 2\")\n", + "plt.plot(x, gauss(0,3)(x), label=\"sigma = 3\")\n", + "plt.plot(x, gauss(5,1)(x), label=\"sigma = 1, mu = 5\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Распределение Коши" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Распределение Коши (или Лорнца или Брейта-Вигнера) описывает резонансные процессы в физике.\n", + "\n", + "*Распределение:* \n", + "$$\n", + " f(x, \\mu, \\gamma) = \\frac {1}{\\pi} \\frac{\\gamma}{ (x-\\mu)^2 + \\gamma^2}\n", + "$$\n", + "\n", + "*Среднее значение:* не определено\n", + "\n", + "*Дисперсия:* распеделение Коши имеет бесконечную дисперсию (интеграл расходится)." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:34+0000", + "start_time": "2017-09-22T06:34+0000" + }, + "init_cell": true + }, + "outputs": [], + "source": [ + "cauchy = lambda mu,gamma : lambda x : 1/math.pi * gamma/ ((x-mu)*(x-mu)+gamma*gamma)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Демонстрация распределения Коши" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + } + }, + "outputs": [ + { + "data": { + "image/png": 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CV5SOyS+tdcskLvy5wzeQVvAY8/ajje+CxgXPaG5K160baLWqp+8BziT9JCC/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\nJ9QPH3zeUVFjxqBLSFDLN5uhkr7iNVJ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9Wq68D+o6q/r+QwMJvvHkfl49Psq1HXX8zcfW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\nYnNdWi/Ciw2bTbDINFKmY2prdsCMhxI5RlJ5egaTjKQKFEr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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.title(\"Cauchy distribution\")\n", + "x = np.linspace(-10,10)\n", + "plt.plot(x, cauchy(0,1)(x), label=\"gamma = 1\")\n", + "plt.plot(x, cauchy(0,2)(x), label=\"gamma = 2\")\n", + "plt.plot(x, cauchy(0,3)(x), label=\"gamma = 3\")\n", + "plt.plot(x, cauchy(5,1)(x), label=\"gamma = 1, mu = 5\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Сравнение распределений" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + } + }, + "outputs": [ + { + "data": { + "image/png": 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r1103GQ6st4Ob+Ref3frOTnX4aM42Xv9lM13uq5x9OIWYYTD/Tfj9FRj6o3tB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\nYKIxZp2IPC8iNzqzjcbes4kDHgXOaTq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N+9aAaYWscXDFTfbkckA+Dc2tfLbnCB9treST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\nrkJggg==\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "lambda_ = 10\n", + "\n", + "plt.title(\"Comparison of Poisson and Gaussian distribution for lambda = \" + str(lambda_));\n", + "xg = np.linspace(0, 2*lambda_, 300)\n", + "xp = range(2*lambda_ + 1)\n", + "plt.plot(xp, poisson.pmf(xp, lambda_), label=\"Poisson\")\n", + "plt.plot(xg, gauss(lambda_, np.sqrt(lambda_))(xg), label=\"Normal\")\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + } + }, + "outputs": [ + { + "data": { + "image/png": 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PyElcgGHZVtM3sSWUpD8c2OJ3v9Rb1p4rgJf97qeJSImILBGRs7oRozFhtbWq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\n2vQJE6recCdIs4ZGO6TQHXU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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "sigma = 1;\n", + "gamma = 1.75;\n", + "xg = np.linspace(-4*sigma, 4*sigma)\n", + "xc = np.linspace(-6*gamma, 6*gamma)\n", + "plt.title(\"Comparison of Gaussian and Cauchy distribution\");\n", + "plt.plot(xg, gauss(0, sigma)(xg),label=\"Normal\")\n", + "plt.plot(xc, cauchy(0, gamma)(xc),label=\"Cauchy\" )\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Центральная предельная теорема" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Центральная предельная теоремма утверждает, что если некоторая случайная величина $x$ имеет распределение со средним $\\mu$ с конечной дисперсией $D$, то сумма большого количества таких случайных величин\n", + "$\\sum_{i=1}^{n}{x_i}$ будет иметь нормальное распределение со средним $n \\mu$ и дисперсией $n D$ не зависимо от формы исходного распределения.\n", + "\n", + "Центральная предельная теорема в общем случае не говорит о том, каким должно быть $n$, обычно это надо определять экспериментально для каждого типа распределения." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "ExecuteTime": { + "end_time": "2017-09-22T06:29+0000", + "start_time": "2017-09-22T06:29+0000" + } + }, + "outputs": [ + { + "data": { + "image/png": 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7j6BFizCPHdvjLt5JSQRceil177yDtloxGUykBCdxyMur625XGD03MYmRQ5K7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\nZgNx/rVYCosIWryYLaVbCPUOJTUs1bH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