{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 概率與統計簡介\n",
    "在本筆記本中，我們將操弄一些之前討論過的概念。許多來自概率與統計的概念在 Python 的主要資料處理函式庫中都有良好展現，例如 `numpy` 和 `pandas`。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import random\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 隨機變數與分佈\n",
    "我們先從 0 到 9 的均勻分佈中抽取 30 個數值的樣本。並且計算其平均值及變異數。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample = [ random.randint(0,10) for _ in range(30) ]\n",
    "print(f\"Sample: {sample}\")\n",
    "print(f\"Mean = {np.mean(sample)}\")\n",
    "print(f\"Variance = {np.var(sample)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "要視覺上估計樣本中有多少不同的值，我們可以繪製<strong>直方圖</strong>：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(sample)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 分析實際數據\n",
    "\n",
    "均值和方差在分析現實世界數據時非常重要。讓我們載入來自 [SOCR MLB Height/Weight Data](http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_MLB_HeightsWeights) 的棒球運動員數據\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.read_csv(\"../../data/SOCR_MLB.tsv\",sep='\\t', header=None, names=['Name','Team','Role','Weight','Height','Age'])\n",
    "df\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 我們這裡使用一個名為 [**Pandas**](https://pandas.pydata.org/) 的套件來進行數據分析。稍後在本課程中，我們會進一步討論 Pandas 以及如何在 Python 中處理數據。\n",
    "\n",
    "讓我們計算年齡、身高和體重的平均值：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df[['Age','Height','Weight']].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "而家嚟集中睇身高，計算標準差同變異數： \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(list(df['Height'])[:20])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mean = df['Height'].mean()\n",
    "var = df['Height'].var()\n",
    "std = df['Height'].std()\n",
    "print(f\"Mean = {mean}\\nVariance = {var}\\nStandard Deviation = {std}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "除了平均值之外，查看中位數和四分位數也是有意義的。它們可以用<strong>箱型圖</strong>來視覺化： \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10,2))\n",
    "plt.boxplot(df['Height'].ffill(), orientation='horizontal', showmeans=True)\n",
    "plt.grid(color='gray', linestyle='dotted')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我哋都可以將數據集嘅子集作箱形圖，例如按球員角色分組。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.boxplot(column='Height', by='Role', figsize=(10,8))\n",
    "plt.xticks(rotation='vertical')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> <strong>注意</strong>：此圖表顯示，一壘手的平均身高比二壘手的身高高。稍後我們會學習如何更正式地檢驗此假設，以及如何證明我們的數據在統計上具有顯著性。  \n",
    "\n",
    "年齡、身高和體重都是連續隨機變數。你認為它們的分佈是如何？一個好的方法是繪製數值的直方圖來找出答案： \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['Weight'].hist(bins=15, figsize=(10,6))\n",
    "plt.suptitle('Weight distribution of MLB Players')\n",
    "plt.xlabel('Weight')\n",
    "plt.ylabel('Count')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 正態分佈\n",
    "\n",
    "讓我們創建一個符合正態分佈的人工權重樣本，其均值和變異數與我們的真實數據相同：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "generated = np.random.normal(mean, std, 1000)\n",
    "generated[:20]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(generated, bins=15)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(np.random.normal(0,1,50000), bins=300)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "由於現實生活中的大多數數值通常呈常態分佈，因此我們不應該使用均勻隨機數生成器來生成樣本數據。以下是嘗試使用均勻分佈（由 `np.random.rand` 生成）來生成體重時會發生的情況：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "wrong_sample = np.random.rand(1000)*2*std+mean-std\n",
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(wrong_sample)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 信心區間\n",
    "\n",
    "而家我哋嚟計算棒球員嘅體重同身高嘅信心區間。 我哋會用[呢個stackoverflow討論](https://stackoverflow.com/questions/15033511/compute-a-confidence-interval-from-sample-data)嘅代碼：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.stats\n",
    "\n",
    "def mean_confidence_interval(data, confidence=0.95):\n",
    "    a = 1.0 * np.array(data)\n",
    "    n = len(a)\n",
    "    m, se = np.mean(a), scipy.stats.sem(a)\n",
    "    h = se * scipy.stats.t.ppf((1 + confidence) / 2., n-1)\n",
    "    return m, h\n",
    "\n",
    "for p in [0.85, 0.9, 0.95]:\n",
    "    m, h = mean_confidence_interval(df['Weight'].ffill(),p)\n",
    "    print(f\"p={p:.2f}, mean = {m:.2f} ± {h:.2f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 假設檢定\n",
    "\n",
    "讓我們探索棒球選手數據集中不同的角色：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.groupby('Role').agg({ 'Weight' : 'mean', 'Height' : 'mean', 'Age' : 'count'}).rename(columns={ 'Age' : 'Count'})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "讓我們檢驗一下第一壘手是否比二壘手高的假設。最簡單的方法是檢驗信賴區間：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for p in [0.85,0.9,0.95]:\n",
    "    m1, h1 = mean_confidence_interval(df.loc[df['Role']=='First_Baseman',['Height']],p)\n",
    "    m2, h2 = mean_confidence_interval(df.loc[df['Role']=='Second_Baseman',['Height']],p)\n",
    "    print(f'Conf={p:.2f}, 1st basemen height: {m1-h1[0]:.2f}..{m1+h1[0]:.2f}, 2nd basemen height: {m2-h2[0]:.2f}..{m2+h2[0]:.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我們可以看到區間並沒有重疊。\n",
    "\n",
    "一個在統計上更正確證明假設的方法是使用 **Student t檢定**：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import ttest_ind\n",
    "\n",
    "tval, pval = ttest_ind(df.loc[df['Role']=='First_Baseman',['Height']], df.loc[df['Role']=='Second_Baseman',['Height']],equal_var=False)\n",
    "print(f\"T-value = {tval[0]:.2f}\\nP-value: {pval[0]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`ttest_ind` 函數所返回的兩個值是：\n",
    "* p 值可被視為兩個分佈具有相同平均值的機率。在我們的例子中，這個值非常低，意味著有強烈的證據支持一壘手較高。\n",
    "* t 值是標準化平均差的中間值，該值被用於 t 檢驗，並與給定置信值的閾值進行比較。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 使用中心極限定理模擬常態分佈\n",
    "\n",
    "Python 中的偽隨機生成器設計是給我們一個均勻分佈。如果我們想要創建一個常態分佈的生成器，我們可以使用中心極限定理。要獲得一個常態分佈的值，我們只需計算一組均勻生成樣本的平均值。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def normal_random(sample_size=100):\n",
    "    sample = [random.uniform(0,1) for _ in range(sample_size) ]\n",
    "    return sum(sample)/sample_size\n",
    "\n",
    "sample = [normal_random() for _ in range(100)]\n",
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(sample)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 相關性與邪惡棒球公司\n",
    "\n",
    "相關性讓我們能找到資料序列之間的關係。在我們的玩具示例中，假設有一間邪惡的棒球公司，根據球員的身高支付他們的薪水——球員越高，獲得的錢越多。假設有一個基本薪資是$1000，還有一個根據身高介乎$0到$100的額外獎金。我們將取美國職棒大聯盟（MLB）的真實球員資料，計算他們的虛構薪資：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "heights = df['Height'].ffill()\n",
    "salaries = 1000+(heights-heights.min())/(heights.max()-heights.mean())*100\n",
    "print(list(zip(heights, salaries))[:10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "現在讓我們計算這些序列的協方差和相關性。`np.cov` 會給我們所謂的<strong>協方差矩陣</strong>，這是多變量協方差的擴展。協方差矩陣 $M$ 的元素 $M_{ij}$ 是輸入變量 $X_i$ 和 $X_j$ 之間的相關性，而對角線上的值 $M_{ii}$ 則是 $X_i$ 的變異數。同樣地，`np.corrcoef` 會給我們<strong>相關係數矩陣</strong>。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(f\"Covariance matrix:\\n{np.cov(heights, salaries)}\")\n",
    "print(f\"Covariance = {np.cov(heights, salaries)[0,1]}\")\n",
    "print(f\"Correlation = {np.corrcoef(heights, salaries)[0,1]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "相關係數等於 1 意味著兩個變量之間存在強烈的 <strong>線性關係</strong>。我們可以通過將一個值與另一個值繪圖來直觀地看到線性關係：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.scatter(heights,salaries)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我們來看看如果關係不是線性的結果會如何。假設我們的公司決定隱藏身高與薪水之間明顯的線性依賴關係，並在公式中引入了一些非線性成分，例如 `sin`： \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "salaries = 1000+np.sin((heights-heights.min())/(heights.max()-heights.mean()))*100\n",
    "print(f\"Correlation = {np.corrcoef(heights, salaries)[0,1]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在這種情況下，相關性稍微小一點，但仍然相當高。現在，為了讓關係看起來更不明顯，我們或許想透過加入一些隨機變數到薪水中，增加額外的隨機性。讓我們來看看會發生什麼：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "salaries = 1000+np.sin((heights-heights.min())/(heights.max()-heights.mean()))*100+np.random.random(size=len(heights))*20-10\n",
    "print(f\"Correlation = {np.corrcoef(heights, salaries)[0,1]}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.scatter(heights, salaries)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 你能猜出為什麼這些點會像這樣排列成垂直線嗎？\n",
    "\n",
    "我們已經觀察到像薪水這類人為設計的概念和觀察變量 <em>身高</em> 之間的相關性。現在讓我們也看看兩個觀察變量，例如身高和體重，是否也有相關性：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.corrcoef(df['Height'].ffill(),df['Weight'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "不幸地，我們沒有取得任何結果——只有一些奇怪的 `nan` 值。這是因為我們的系列中有些值是未定義的，用 `nan` 表示，這導致運算結果也變成未定義。從矩陣中我們可以看到 `Weight` 是有問題的欄位，因為已經計算了 `Height` 值之間的自相關。\n",
    "\n",
    "> 這個例子顯示了<strong>資料準備</strong>和<strong>清理</strong>的重要性。沒有適當的資料，我們無法計算任何東西。\n",
    "\n",
    "讓我們使用 `fillna` 方法填充缺失值，並計算相關性：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.corrcoef(df['Height'].ffill(), df['Weight'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "確實存在一個關聯，但並不像我們的人為範例中那麼強烈。事實上，如果我們看一下其中一個值對另一個值的散點圖，這種關係會明顯得少得多：\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.scatter(df['Weight'],df['Height'])\n",
    "plt.xlabel('Weight')\n",
    "plt.ylabel('Height')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 結論\n",
    "\n",
    "在這個筆記本中，我們學會了如何對數據執行基本操作來計算統計函數。我們現在知道如何使用健全的數學和統計工具來驗證一些假設，以及如何根據數據樣本計算任意變量的信賴區間。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n\n<!-- CO-OP TRANSLATOR DISCLAIMER START -->\n**免責聲明**：\n本文件使用 AI 翻譯服務 [Co-op Translator](https://github.com/Azure/co-op-translator) 進行翻譯。雖然我們力求準確，但請注意，自動翻譯可能包含錯誤或不準確之處。原始文件的母語版本應被視為權威來源。對於重要資訊，建議尋求專業人工翻譯。我們不對因使用本翻譯而引起的任何誤解或曲解承擔責任。\n<!-- CO-OP TRANSLATOR DISCLAIMER END -->\n"
   ]
  }
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