chapter_notebooks/chapter_8.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font color=\"#49699E\" size=40>Latent Factors & Components</font>\n",
    "# LEARNING OBJECTIVES\n",
    "# LEARNING MATERIALS\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# INTRODUCTION\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imports and Data Preparation\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "pd.set_option(\"display.notebook_repr_html\", False)\n",
    "import numpy as np\n",
    "from scipy.stats import zscore\n",
    "import random\n",
    "\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "from sklearn.decomposition import PCA\n",
    "from sklearn.cluster import KMeans\n",
    "from sklearn.metrics import silhouette_score, silhouette_samples\n",
    "\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "from dcss.plotting import format_axes_commas, custom_seaborn\n",
    "custom_seaborn()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.read_csv('../data/vdem/V-Dem-CY-Full+Others-v10.csv', low_memory=False) \n",
    "df = df.query('year == 2019').reset_index()\n",
    "df.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "indicators = [\n",
    "    'v2dlreason', 'v2dlcommon', 'v2dlcountr', 'v2dlconslt', 'v2dlengage',\n",
    "    'v2dlencmps', 'v2dlunivl', 'v2cseeorgs', 'v2csreprss', 'v2cscnsult',\n",
    "    'v2csprtcpt', 'v2csgender', 'v2csantimv', 'v2csrlgrep', 'v2csrlgcon',\n",
    "    'v2mecenefm', 'v2mecenefi', 'v2mecenefibin', 'v2mecrit', 'v2merange',\n",
    "    'v2mefemjrn', 'v2meharjrn', 'v2meslfcen', 'v2mebias', 'v2mecorrpt',\n",
    "    'v2exrescon', 'v2exbribe', 'v2exembez', 'v2excrptps', 'v2exthftps',\n",
    "    'v2cldiscm', 'v2cldiscw', 'v2clacfree', 'v2clrelig', 'v2clfmove'\n",
    "]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "countries = df['country_name'].tolist()\n",
    "df = df.set_index('country_name')[indicators]\n",
    "df.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.isna().sum().value_counts()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = df.to_numpy() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# LATENT VARIABLES AND THE CURSE OF DIMENSIONALITY\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Theory First: Measuring Latent Variables with Exploratory Factor Analysis\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conducting a Principal Component Analysis in Sklearn\n",
    "## Standardization\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = StandardScaler().fit_transform(X) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Brief Refresher on Variance, Standard Deviation, and Z-score Normalization\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ABCD = {\n",
    "    'A': [1, 1, 1, 1, 1], # no variance...\n",
    "    'B': [1, 2, 3, 4, 5], # some variance...\n",
    "    'C': [-1, 1, 3, 5, 7], # a bit more variance...\n",
    "    'D': [-10, -9, 3, 4, 4] # still more variance...\n",
    "}\n",
    "\n",
    "for k, v in ABCD.items():\n",
    "    print(f'{k} has a variance of {np.round(np.var(v), 3)}.')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for k, v in ABCD.items():\n",
    "    print(f'{k} has a standard deviation of {np.round(np.std(v), 3)}.')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for k, v in ABCD.items():\n",
    "    print(f'The values in {k} have the following Z-scores: {np.round(zscore(v), 3)}.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Back to PCA!\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pca = PCA()\n",
    "pca_results = pca.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "res = pd.DataFrame(pca_results, index=countries)\n",
    "res.columns=[f'PC {i}' for i in res.columns]\n",
    "\n",
    "res['PC 0'].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evr = pca.explained_variance_ratio_\n",
    "evr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(f'The sum of the array is: {np.round(np.sum(evr), 2)}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.sum(evr[:3]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cve = pd.Series(np.cumsum(evr))\n",
    "cve[:12]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots()\n",
    "sns.lineplot(x=cve.index, y=cve)\n",
    "plt.scatter(x=cve.index, y=cve)\n",
    "ax.set(xlabel='Principal component ID',\n",
    "       ylabel='Proportion of explained variance (cumulative)')\n",
    "ax.set(ylim=(0, 1.1))\n",
    "sns.despine()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix Decomposition: Eigenvalues, Eigenvectors, and Extracting Components\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvalues = pd.Series(pca.explained_variance_)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "sns.lineplot(x=eigenvalues.index, y=eigenvalues, data=eigenvalues)\n",
    "plt.scatter(x=eigenvalues.index, y=eigenvalues)\n",
    "ax.set(xlabel='Principal component ID', ylabel='Eigenvalue')\n",
    "sns.despine()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvalues.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "component_1 = pca_results[:, 0]\n",
    "component_2 = pca_results[:, 1]\n",
    "\n",
    "PC12 = pd.DataFrame(zip(component_1, component_2), columns=['PC1', 'PC2'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "PC12['Country'] = countries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ax = sns.kdeplot(data=PC12, x='PC1', y='PC2', alpha=.8, fill=True)\n",
    "for i, country in enumerate(PC12['Country']):\n",
    "    ax.text(PC12['PC1'][i],\n",
    "            PC12['PC2'][i],\n",
    "            country,\n",
    "            horizontalalignment='left',\n",
    "            size=3,\n",
    "            color='black',\n",
    "            weight='normal')\n",
    "ax.set(xticklabels=[], yticklabels=[])\n",
    "ax.set(\n",
    "    xlabel=\n",
    "    f'$\\longleftarrow$ PC1 (eigenvalue: {np.round(eigenvalues.loc[0], 2)}) $\\longrightarrow$',\n",
    "    ylabel=\n",
    "    f'$\\longleftarrow$ PC2 (eigenvalue: {np.round(eigenvalues.loc[1], 2)}) $\\longrightarrow$'\n",
    ")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CONCLUSION\n",
    "## Key Points \n"
   ]
  }
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