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ML-For-Beginners/translations/zh/5-Clustering/2-K-Means/solution/R/lesson_15-R.ipynb

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},
"cells": [
{
"cell_type": "markdown",
"metadata": {
"id": "GULATlQXLXyR"
},
"source": [
"## 使用 R 和 Tidy 数据原则探索 K-Means 聚类\n",
"\n",
"### [**课前测验**](https://gray-sand-07a10f403.1.azurestaticapps.net/quiz/29/)\n",
"\n",
"在本课中，您将学习如何使用 Tidymodels 包以及 R 生态系统中的其他包（我们称它们为朋友 🧑‍🤝‍🧑）创建聚类，并使用您之前导入的尼日利亚音乐数据集。我们将介绍 K-Means 聚类的基础知识。请记住，正如您在之前的课程中学到的那样，有许多方法可以处理聚类，您使用的方法取决于您的数据。我们将尝试 K-Means，因为它是最常见的聚类技术。让我们开始吧！\n",
"\n",
"您将学习的术语：\n",
"\n",
"- Silhouette评分\n",
"\n",
"- 肘部法则\n",
"\n",
"- 惯性\n",
"\n",
"- 方差\n",
"\n",
"### **简介**\n",
"\n",
"[K-Means 聚类](https://wikipedia.org/wiki/K-means_clustering) 是一种源自信号处理领域的方法。它用于根据特征的相似性将数据分成 `k 个聚类`。\n",
"\n",
"这些聚类可以通过 [Voronoi 图](https://wikipedia.org/wiki/Voronoi_diagram) 可视化，其中包括一个点（或“种子”）及其对应的区域。\n",
"\n",
"<p >\n",
" <img src=\"../../images/voronoi.png\"\n",
" width=\"500\"/>\n",
" <figcaption>Jen Looper 制作的信息图</figcaption>\n",
"\n",
"K-Means 聚类的步骤如下：\n",
"\n",
"1. 数据科学家首先指定要创建的聚类数量。\n",
"\n",
"2. 接下来，算法从数据集中随机选择 K 个观测值作为聚类的初始中心（即质心）。\n",
"\n",
"3. 然后，将其余的观测值分配到距离最近的质心。\n",
"\n",
"4. 接下来，计算每个聚类的新均值，并将质心移动到均值位置。\n",
"\n",
"5. 现在质心已经重新计算，每个观测值再次被检查是否更接近其他聚类。所有对象再次使用更新后的聚类均值重新分配。聚类分配和质心更新步骤会迭代重复，直到聚类分配不再变化（即达到收敛）。通常，当每次新迭代导致质心的移动微乎其微且聚类变得静态时，算法会终止。\n",
"\n",
"<div>\n",
"\n",
"> 请注意，由于初始 k 个观测值的随机化，作为起始质心，每次应用该过程时可能会得到略有不同的结果。因此，大多数算法会使用多个 *随机起点* 并选择具有最低 WCSS 的迭代。因此，强烈建议始终使用多个 *nstart* 值运行 K-Means，以避免 *不理想的局部最优解*。\n",
"\n",
"</div>\n",
"\n",
"以下短动画使用 Allison Horst 的 [插画](https://github.com/allisonhorst/stats-illustrations) 解释了聚类过程：\n",
"\n",
"<p >\n",
" <img src=\"../../images/kmeans.gif\"\n",
" width=\"550\"/>\n",
" <figcaption>@allison_horst 的插画</figcaption>\n",
"\n",
"聚类中一个基本问题是：如何确定将数据分成多少个聚类？使用 K-Means 的一个缺点是您需要确定 `k`，即 `质心` 的数量。幸运的是，`肘部法则` 可以帮助估算一个好的起始值。您马上就会尝试。\n",
"\n",
"### \n",
"\n",
"**前提条件**\n",
"\n",
"我们将从 [上一课](https://github.com/microsoft/ML-For-Beginners/blob/main/5-Clustering/1-Visualize/solution/R/lesson_14-R.ipynb) 停止的地方继续，在那里我们分析了数据集，进行了大量可视化，并过滤了感兴趣的观测值。一定要查看！\n",
"\n",
"我们需要一些包来完成本模块。您可以通过以下方式安装它们：`install.packages(c('tidyverse', 'tidymodels', 'cluster', 'summarytools', 'plotly', 'paletteer', 'factoextra', 'patchwork'))`\n",
"\n",
"或者，下面的脚本会检查您是否拥有完成本模块所需的包，并在缺少某些包时为您安装它们。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "ah_tBi58LXyi"
},
"source": [
"suppressWarnings(if(!require(\"pacman\")) install.packages(\"pacman\"))\n",
"\n",
"pacman::p_load('tidyverse', 'tidymodels', 'cluster', 'summarytools', 'plotly', 'paletteer', 'factoextra', 'patchwork')\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "7e--UCUTLXym"
},
"source": [
"让我们开始吧！\n",
"\n",
"## 1. 与数据共舞：缩小到最受欢迎的三个音乐类型\n",
"\n",
"这是我们上一节课所做内容的回顾。让我们来分析一些数据吧！\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "Ycamx7GGLXyn"
},
"source": [
"# Load the core tidyverse and make it available in your current R session\n",
"library(tidyverse)\n",
"\n",
"# Import the data into a tibble\n",
"df <- read_csv(file = \"https://raw.githubusercontent.com/microsoft/ML-For-Beginners/main/5-Clustering/data/nigerian-songs.csv\", show_col_types = FALSE)\n",
"\n",
"# Narrow down to top 3 popular genres\n",
"nigerian_songs <- df %>% \n",
" # Concentrate on top 3 genres\n",
" filter(artist_top_genre %in% c(\"afro dancehall\", \"afropop\",\"nigerian pop\")) %>% \n",
" # Remove unclassified observations\n",
" filter(popularity != 0)\n",
"\n",
"\n",
"\n",
"# Visualize popular genres using bar plots\n",
"theme_set(theme_light())\n",
"nigerian_songs %>%\n",
" count(artist_top_genre) %>%\n",
" ggplot(mapping = aes(x = artist_top_genre, y = n,\n",
" fill = artist_top_genre)) +\n",
" geom_col(alpha = 0.8) +\n",
" paletteer::scale_fill_paletteer_d(\"ggsci::category10_d3\") +\n",
" ggtitle(\"Top genres\") +\n",
" theme(plot.title = element_text(hjust = 0.5))\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "b5h5zmkPLXyp"
},
"source": [
"🤩 这进展得很顺利！\n",
"\n",
"## 2. 更多数据探索\n",
"\n",
"这些数据有多干净？让我们使用箱线图检查异常值。我们将专注于异常值较少的数值列（尽管你也可以清理异常值）。箱线图可以显示数据的范围，并帮助选择要使用的列。注意，箱线图并不显示方差，而方差是良好可聚类数据的重要元素。请参阅[这个讨论](https://stats.stackexchange.com/questions/91536/deduce-variance-from-boxplot)以了解更多信息。\n",
"\n",
"[箱线图](https://en.wikipedia.org/wiki/Box_plot)用于以图形方式描述`数值`数据的分布，因此我们先从*选择*所有数值列以及流行音乐流派开始。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "HhNreJKLLXyq"
},
"source": [
"# Select top genre column and all other numeric columns\n",
"df_numeric <- nigerian_songs %>% \n",
" select(artist_top_genre, where(is.numeric)) \n",
"\n",
"# Display the data\n",
"df_numeric %>% \n",
" slice_head(n = 5)\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "uYXrwJRaLXyq"
},
"source": [
"看看选择助手 `where` 是如何让这一切变得简单的 💁？可以在[这里](https://tidyselect.r-lib.org/)探索其他类似的函数。\n",
"\n",
"由于我们将为每个数值特征制作箱线图，并且希望避免使用循环，让我们将数据重新格式化为*更长*的格式，这样就可以利用 `facets`——每个子图分别显示数据的一个子集。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "gd5bR3f8LXys"
},
"source": [
"# Pivot data from wide to long\n",
"df_numeric_long <- df_numeric %>% \n",
" pivot_longer(!artist_top_genre, names_to = \"feature_names\", values_to = \"values\") \n",
"\n",
"# Print out data\n",
"df_numeric_long %>% \n",
" slice_head(n = 15)\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "-7tE1swnLXyv"
},
"source": [
"更长了！现在是时候使用一些 `ggplots` 了！那么我们会用什么 `geom` 呢？\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "r88bIsyuLXyy"
},
"source": [
"# Make a box plot\n",
"df_numeric_long %>% \n",
" ggplot(mapping = aes(x = feature_names, y = values, fill = feature_names)) +\n",
" geom_boxplot() +\n",
" facet_wrap(~ feature_names, ncol = 4, scales = \"free\") +\n",
" theme(legend.position = \"none\")\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "EYVyKIUELXyz"
},
"source": [
"现在我们可以看到这些数据有些杂乱：通过观察每一列的箱线图，可以发现存在异常值。你可以遍历整个数据集并移除这些异常值，但这样会使数据变得非常少。\n",
"\n",
"目前，我们来选择用于聚类练习的列。我们选择范围相似的数值列。我们可以将 `artist_top_genre` 编码为数值，但暂时先舍弃它。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "-wkpINyZLXy0"
},
"source": [
"# Select variables with similar ranges\n",
"df_numeric_select <- df_numeric %>% \n",
" select(popularity, danceability, acousticness, loudness, energy) \n",
"\n",
"# Normalize data\n",
"# df_numeric_select <- scale(df_numeric_select)\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "D7dLzgpqLXy1"
},
"source": [
"## 3. 在 R 中计算 k-means 聚类\n",
"\n",
"我们可以使用 R 中内置的 `kmeans` 函数计算 k-means，参见 `help(\"kmeans()\")`。`kmeans()` 函数的主要参数是一个包含所有数值型列的数据框。\n",
"\n",
"使用 k-means 聚类的第一步是指定最终解决方案中要生成的聚类数量（k）。我们知道从数据集中划分出了 3 种歌曲类型，因此我们可以尝试设置为 3：\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "uC4EQ5w7LXy5"
},
"source": [
"set.seed(2056)\n",
"# Kmeans clustering for 3 clusters\n",
"kclust <- kmeans(\n",
" df_numeric_select,\n",
" # Specify the number of clusters\n",
" centers = 3,\n",
" # How many random initial configurations\n",
" nstart = 25\n",
")\n",
"\n",
"# Display clustering object\n",
"kclust\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "hzfhscWrLXy-"
},
"source": [
"kmeans对象包含了许多信息，这些信息在`help(\"kmeans()\")`中有详细说明。现在，我们先关注几个关键点。我们可以看到数据被分成了3个簇，分别包含65、110和111个样本。输出还包括了这3个簇在5个变量上的簇中心（均值）。\n",
"\n",
"聚类向量是每个观测值的簇分配。我们可以使用`augment`函数将簇分配添加到原始数据集中。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "0XwwpFGQLXy_"
},
"source": [
"# Add predicted cluster assignment to data set\n",
"augment(kclust, df_numeric_select) %>% \n",
" relocate(.cluster) %>% \n",
" slice_head(n = 10)\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "NXIVXXACLXzA"
},
"source": [
"太好了，我们刚刚将数据集划分成了三个组。那么我们的聚类效果如何呢🤷？让我们来看看 `Silhouette score`。\n",
"\n",
"### **轮廓系数**\n",
"\n",
"[轮廓分析](https://en.wikipedia.org/wiki/Silhouette_(clustering))可以用来研究生成的聚类之间的分离距离。这个分数范围从 -1 到 1，如果分数接近 1，说明聚类紧密且与其他聚类分离良好。接近 0 的值表示聚类之间有重叠，样本非常接近邻近聚类的决策边界。[来源](https://dzone.com/articles/kmeans-silhouette-score-explained-with-python-exam)。\n",
"\n",
"平均轮廓方法计算不同 *k* 值下观测点的平均轮廓分数。较高的平均轮廓分数表明聚类效果较好。\n",
"\n",
"使用 cluster 包中的 `silhouette` 函数可以计算平均轮廓宽度。\n",
"\n",
"> 轮廓分数可以使用任何[距离](https://en.wikipedia.org/wiki/Distance \"Distance\")度量来计算，例如我们在[上一课](https://github.com/microsoft/ML-For-Beginners/blob/main/5-Clustering/1-Visualize/solution/R/lesson_14-R.ipynb)中讨论过的[欧几里得距离](https://en.wikipedia.org/wiki/Euclidean_distance \"Euclidean distance\")或[曼哈顿距离](https://en.wikipedia.org/wiki/Manhattan_distance \"Manhattan distance\")。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "Jn0McL28LXzB"
},
"source": [
"# Load cluster package\n",
"library(cluster)\n",
"\n",
"# Compute average silhouette score\n",
"ss <- silhouette(kclust$cluster,\n",
" # Compute euclidean distance\n",
" dist = dist(df_numeric_select))\n",
"mean(ss[, 3])\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "QyQRn97nLXzC"
},
"source": [
"我们的得分是 **.549**，正好处于中间位置。这表明我们的数据并不特别适合这种类型的聚类。让我们看看是否可以通过可视化来验证这个猜测。[factoextra 包](https://rpkgs.datanovia.com/factoextra/index.html) 提供了用于可视化聚类的函数（`fviz_cluster()`）。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "7a6Km1_FLXzD"
},
"source": [
"library(factoextra)\n",
"\n",
"# Visualize clustering results\n",
"fviz_cluster(kclust, df_numeric_select)\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "IBwCWt-0LXzD"
},
"source": [
"聚类之间的重叠表明我们的数据并不特别适合这种类型的聚类，但我们继续进行。\n",
"\n",
"## 4. 确定最佳聚类数\n",
"\n",
"在 K-Means 聚类中经常出现的一个基本问题是——在没有已知类别标签的情况下，如何确定将数据分成多少个聚类？\n",
"\n",
"我们可以尝试的一种方法是使用一个数据样本来`创建一系列聚类模型`，逐步增加聚类的数量（例如从 1 到 10），并评估聚类指标，例如 **Silhouette 分数**。\n",
"\n",
"让我们通过对不同的 *k* 值计算聚类算法，并评估 **聚类内平方和**（WCSS），来确定最佳的聚类数。聚类内平方和（WCSS）总量衡量了聚类的紧凑性，我们希望它尽可能小，较低的值意味着数据点更接近。\n",
"\n",
"让我们探索不同的 `k` 值（从 1 到 10）对聚类结果的影响。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "hSeIiylDLXzE"
},
"source": [
"# Create a series of clustering models\n",
"kclusts <- tibble(k = 1:10) %>% \n",
" # Perform kmeans clustering for 1,2,3 ... ,10 clusters\n",
" mutate(model = map(k, ~ kmeans(df_numeric_select, centers = .x, nstart = 25)),\n",
" # Farm out clustering metrics eg WCSS\n",
" glanced = map(model, ~ glance(.x))) %>% \n",
" unnest(cols = glanced)\n",
" \n",
"\n",
"# View clustering rsulsts\n",
"kclusts\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "m7rS2U1eLXzE"
},
"source": [
"现在我们已经获得了每个聚类算法在中心 *k* 时的总簇内平方和 (tot.withinss)，接下来我们使用[肘部法则](https://en.wikipedia.org/wiki/Elbow_method_(clustering))来确定最佳的聚类数量。该方法的核心是将簇内平方和 (WCSS) 作为聚类数量的函数进行绘图，并选择[曲线的肘部](https://en.wikipedia.org/wiki/Elbow_of_the_curve \"曲线的肘部\")作为要使用的聚类数量。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "o_DjHGItLXzF"
},
"source": [
"set.seed(2056)\n",
"# Use elbow method to determine optimum number of clusters\n",
"kclusts %>% \n",
" ggplot(mapping = aes(x = k, y = tot.withinss)) +\n",
" geom_line(size = 1.2, alpha = 0.8, color = \"#FF7F0EFF\") +\n",
" geom_point(size = 2, color = \"#FF7F0EFF\")\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "pLYyt5XSLXzG"
},
"source": [
"图表显示，当聚类数量从一个增加到两个时，WCSS显著减少（即更高的*紧密度*），从两个增加到三个聚类时也有进一步明显的减少。之后，减少的幅度变得不那么显著，在图表中大约三个聚类处形成一个`肘部` 💪。这表明数据点可以合理地分为两到三个较为独立的聚类。\n",
"\n",
"现在我们可以继续提取聚类模型，其中`k = 3`：\n",
"\n",
"> `pull()`: 用于提取单列\n",
">\n",
"> `pluck()`: 用于索引数据结构，例如列表\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "JP_JPKBILXzG"
},
"source": [
"# Extract k = 3 clustering\n",
"final_kmeans <- kclusts %>% \n",
" filter(k == 3) %>% \n",
" pull(model) %>% \n",
" pluck(1)\n",
"\n",
"\n",
"final_kmeans\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "l_PDTu8tLXzI"
},
"source": [
"太好了！让我们来可视化获得的聚类。想用 `plotly` 增加一些互动性吗？\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "dNcleFe-LXzJ"
},
"source": [
"# Add predicted cluster assignment to data set\n",
"results <- augment(final_kmeans, df_numeric_select) %>% \n",
" bind_cols(df_numeric %>% select(artist_top_genre)) \n",
"\n",
"# Plot cluster assignments\n",
"clust_plt <- results %>% \n",
" ggplot(mapping = aes(x = popularity, y = danceability, color = .cluster, shape = artist_top_genre)) +\n",
" geom_point(size = 2, alpha = 0.8) +\n",
" paletteer::scale_color_paletteer_d(\"ggthemes::Tableau_10\")\n",
"\n",
"ggplotly(clust_plt)\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "6JUM_51VLXzK"
},
"source": [
"也许我们会预期每个聚类（用不同颜色表示）会有明显不同的类型（用不同形状表示）。\n",
"\n",
"让我们来看看模型的准确性。\n"
]
},
{
"cell_type": "code",
"metadata": {
"id": "HdIMUGq7LXzL"
},
"source": [
"# Assign genres to predefined integers\n",
"label_count <- results %>% \n",
" group_by(artist_top_genre) %>% \n",
" mutate(id = cur_group_id()) %>% \n",
" ungroup() %>% \n",
" summarise(correct_labels = sum(.cluster == id))\n",
"\n",
"\n",
"# Print results \n",
"cat(\"Result:\", label_count$correct_labels, \"out of\", nrow(results), \"samples were correctly labeled.\")\n",
"\n",
"cat(\"\\nAccuracy score:\", label_count$correct_labels/nrow(results))\n"
],
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "C50wvaAOLXzM"
},
"source": [
"这个模型的准确性还可以，但并不算优秀。这可能是因为数据本身不太适合使用 K-Means 聚类。这些数据过于不平衡，相关性较低，并且列值之间的差异太大，导致聚类效果不佳。实际上，形成的聚类可能会受到我们之前定义的三个类别的强烈影响或偏斜。\n",
"\n",
"尽管如此，这仍然是一个很好的学习过程！\n",
"\n",
"在 Scikit-learn 的文档中，你可以看到像这样的模型，聚类边界不太清晰，存在“方差”问题：\n",
"\n",
"<p >\n",
" <img src=\"../../images/problems.png\"\n",
" width=\"500\"/>\n",
" <figcaption>来自 Scikit-learn 的信息图</figcaption>\n",
"\n",
"\n",
"\n",
"## **方差**\n",
"\n",
"方差被定义为“与平均值的平方差的平均值” [来源](https://www.mathsisfun.com/data/standard-deviation.html)。在这个聚类问题的背景下，它指的是数据集中数值偏离平均值的程度过大。\n",
"\n",
"✅ 这是一个很好的时机来思考如何解决这个问题。稍微调整数据？使用不同的列？尝试不同的算法？提示：试试[对数据进行缩放](https://www.mygreatlearning.com/blog/learning-data-science-with-k-means-clustering/)以进行归一化，并测试其他列。\n",
"\n",
"> 试试这个‘[方差计算器](https://www.calculatorsoup.com/calculators/statistics/variance-calculator.php)’来更好地理解这个概念。\n",
"\n",
"------------------------------------------------------------------------\n",
"\n",
"## **🚀挑战**\n",
"\n",
"花些时间研究这个笔记本，调整参数。通过进一步清理数据（例如删除异常值），你能否提高模型的准确性？你可以使用权重为某些数据样本赋予更大的权重。还有什么方法可以创建更好的聚类？\n",
"\n",
"提示：试试对数据进行缩放。笔记本中有注释代码，可以添加标准化缩放，使数据列在范围上更接近。你会发现虽然轮廓分数下降了，但肘部图中的“折点”变得更加平滑。这是因为未缩放的数据允许方差较小的数据权重更大。可以在[这里](https://stats.stackexchange.com/questions/21222/are-mean-normalization-and-feature-scaling-needed-for-k-means-clustering/21226#21226)阅读更多相关问题。\n",
"\n",
"## [**课后测验**](https://gray-sand-07a10f403.1.azurestaticapps.net/quiz/30/)\n",
"\n",
"## **复习与自学**\n",
"\n",
"- 看看一个 K-Means 模拟器 [例如这个](https://user.ceng.metu.edu.tr/~akifakkus/courses/ceng574/k-means/)。你可以使用这个工具可视化样本数据点并确定其质心。你可以编辑数据的随机性、聚类数量和质心数量。这是否帮助你更好地理解数据如何分组？\n",
"\n",
"- 另外，看看斯坦福的[这份 K-Means 手册](https://stanford.edu/~cpiech/cs221/handouts/kmeans.html)。\n",
"\n",
"想尝试将你新学到的聚类技能应用到适合 K-Means 聚类的数据集上？请参考以下内容：\n",
"\n",
"- [训练和评估聚类模型](https://rpubs.com/eR_ic/clustering)，使用 Tidymodels 和相关工具\n",
"\n",
"- [K-Means 聚类分析](https://uc-r.github.io/kmeans_clustering)，UC 商业分析 R 编程指南\n",
"\n",
"- [使用整洁数据原则进行 K-Means 聚类](https://www.tidymodels.org/learn/statistics/k-means/)\n",
"\n",
"## **作业**\n",
"\n",
"[尝试不同的聚类方法](https://github.com/microsoft/ML-For-Beginners/blob/main/5-Clustering/2-K-Means/assignment.md)\n",
"\n",
"## 特别感谢：\n",
"\n",
"[Jen Looper](https://www.twitter.com/jenlooper) 创建了这个模块的原始 Python 版本 ♥️\n",
"\n",
"[`Allison Horst`](https://twitter.com/allison_horst/) 创作了令人惊叹的插图，使 R 更加友好和吸引人。可以在她的[画廊](https://www.google.com/url?q=https://github.com/allisonhorst/stats-illustrations&sa=D&source=editors&ust=1626380772530000&usg=AOvVaw3zcfyCizFQZpkSLzxiiQEM)中找到更多插图。\n",
"\n",
"祝学习愉快，\n",
"\n",
"[Eric](https://twitter.com/ericntay)，Gold Microsoft Learn 学生大使。\n",
"\n",
"<p >\n",
" <img src=\"../../images/r_learners_sm.jpeg\"\n",
" width=\"500\"/>\n",
" <figcaption>由 @allison_horst 创作的艺术作品</figcaption>\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n---\n\n**免责声明**： \n本文档使用AI翻译服务[Co-op Translator](https://github.com/Azure/co-op-translator)进行翻译。尽管我们努力确保翻译的准确性，但请注意，自动翻译可能包含错误或不准确之处。原始语言的文档应被视为权威来源。对于关键信息，建议使用专业人工翻译。我们不对因使用此翻译而产生的任何误解或误读承担责任。\n"
]
}
]
}
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