Lecture 08_Penalized Regression_Elastic Net.ipynb

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   "cell_type": "markdown",
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   "source": [
    "## <center><font color=navy>Big Data Economics</font></center>\n",
    "### <center>Elastic Net Regularization</center>\n",
    "#### <center>Ali Habibnia</center>\n",
    "    \n",
    "<center> Assistant Professor, Department of Economics, </center>\n",
    "<center> and Division of Computational Modeling & Data Analytics at Virginia Tech</center>\n",
    "<center> habibnia@vt.edu </center> "
   ]
  },
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   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "\n",
    "> Elastic net (similar to ridge and lasso) is a penalty mechanism which applies shrinkage (driving them closer to zero) of coefficient to build a more robust and parsimonious model. This add regularization terms in the model which are combination of both L1 and L2 norm. \n",
    "\n",
    "<img src=\"images/elastic1.png\"  width=\"600\">\n",
    "\n",
    "\n",
    "- Ridge regression; Hoerl and Kennard (1970)     $$\n",
    "\\hat{\\beta}_{Ridge} = \\underset{\\beta}{\\text{argmin}} \\left\\{ \\frac{1}{2n} \\| y - X\\beta \\|^2_2 + \\lambda \\|\\beta\\|^2_2 \\right\\}\n",
    "$$\n",
    "- Lasso; Tibshirani (1996)     $$\n",
    "\\hat{\\beta}_{LASSO} = \\underset{\\beta}{\\text{argmin}} \\left\\{ \\frac{1}{2n} \\| y - X\\beta \\|^2_2 + \\lambda \\|\\beta\\|_1 \\right\\}\n",
    "$$\n",
    "- Elastic net; Zou and Hastie (2004)     $$\n",
    "\\hat{\\beta}_{ElasticNet} = \\underset{\\beta}{\\text{argmin}} \\left\\{ \\frac{1}{2n} \\| y - X\\beta \\|^2_2 + \\lambda_1 \\|\\beta\\|_1 + \\frac{\\lambda_2}{2} \\|\\beta\\|^2_2 \\right\\}\n",
    "$$\n",
    "- Least Angle Regression (LARS); Efron, Hastie, Johnstone and Tibshirani (2004)\n",
    "\n",
    "\n",
    "</div>"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Readings:\n",
    "\n",
    "1. ***Chapter 3.4,*** [The Elements of Statistical Learning: Data Mining, Inference, and Prediction](https://web.stanford.edu/~hastie/ElemStatLearn/printings/ESLII_print12.pdf). \n",
    "2. [Regularization and Variable Selection via the Elastic Net](https://web.stanford.edu/~hastie/TALKS/enet_talk.pdf)"
   ]
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    "As we have seen before, parameter instability in high-dimensional settings often arises when dealing with the \"curse of dimensionality,\" where the number of predictors ($p$) is large relative to the number of observations ($n$). This complicates traditional model estimation processes, like ordinary least squares (OLS) in linear regression, leading to unstable and unreliable parameter estimates.\n",
    "\n",
    "### Mathematical Definition of Parameter Instability:\n",
    "\n",
    "In a linear regression model, we aim to predict an outcome $y$ based on predictors $X$ using the relationship $y = X\\beta + \\epsilon$, where $\\beta$ represents the coefficients, and $\\epsilon$ is the error term.\n",
    "\n",
    "The OLS estimator for $\\beta$ is given by:\n",
    "\n",
    "$$\n",
    "\\hat{\\beta}_{OLS} = (X^TX)^{-1}X^Ty\n",
    "$$\n",
    "\n",
    "However, in high-dimensional settings where $p \\approx n$ or $p > n$, several issues arise:\n",
    "\n",
    "1. **Singular $X^TX$ Matrix**: When $p > n$, $X^TX$ is singular, meaning it does not have an inverse, making $\\hat{\\beta}_{OLS}$ undefined.\n",
    "\n",
    "2. **Near-singularity and High Variance**: Even when $p \\approx n$, $X^TX$ can be near-singular, leading to extremely large values in its inverse, which greatly amplifies the variance of $\\hat{\\beta}_{OLS}$.\n",
    "\n",
    "3. **Multicollinearity**: High correlation among predictors exacerbates these problems, further inflating the variance of the coefficient estimates.\n",
    "\n",
    "\n",
    "Regularization techniques effectively address the instability of parameter estimates in high-dimensional settings by incorporating additional information (penalties on coefficient sizes) into the the OLS objective, leading to more robust and interpretable models.\n",
    "\n",
    "- **Shrinkage**: The penalties in LASSO and Ridge reduce the magnitude of coefficients, decreasing their variance and increasing model stability.\n",
    "\n",
    "- **Variable Selection (LASSO)**: By setting some coefficients to zero, LASSO helps in identifying a smaller, more relevant set of predictors, reducing model complexity and further stabilizing parameter estimates.\n",
    "\n",
    "- **Mitigating Multicollinearity**: Regularization, particularly Ridge, can handle multicollinearity by allowing correlated predictors to have non-zero coefficients that are shrunk towards zero, thus stabilizing the estimates.\n",
    "\n",
    "- **Bias-Variance Trade-off**: Introducing bias through regularization (shrinking coefficients) results in a significant reduction in variance, leading to more stable and reliable models, especially in prediction tasks.\n"
   ]
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   "metadata": {},
   "source": [
    "### Goals\n",
    "\n",
    "A regularization technique helps in the following main ways:\n",
    "\n",
    "1. Doesn’t assume any particular distribution of the dependent variable \n",
    "2. Address Variance-Bias Tradeoffs. Generally will lower the variance from the model\n",
    "3. More robust to handle multicollinearity\n",
    "4. Better sparse data (observations < features) handling\n",
    "5. Natural feature selection\n",
    "6. More accurate prediction on new data as it minimizes overfitting on the train data\n",
    "7. Easier interpretation of the output\n",
    "\n",
    "And so on\n",
    "\n",
    "\n",
    "> In mathematics, **sparse** and **dense** often refer to the number of zero vs. non-zero elements in an array (e.g. vector or matrix). A sparse array is one that contains mostly zeros and few non-zero entries, and a dense array contains mostly non-zeros.\n",
    "\n",
    "> LASSO and Ridge encourage sparse and dense model, respectively, but since it never be that clear how the true model looks like, it’s typical to apply both methods and determine the best model."
   ]
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   "cell_type": "markdown",
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   "source": [
    "<img src=\"images/l1l2_1.png\"  width=\"400\">\n",
    "<img src=\"images/l1l2_2.png\"  width=\"400\">"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### Elastic Net (L1 + L2)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "- The elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.\n",
    "\n",
    "- Elastic net penalizes large coeficients by linearly combines the squared magnitude and absolute value of magnitude to the cost function. \n",
    "\n",
    "- This combination allows for learning a sparse model where few of the coeficients are non-zero like Lasso, while still maintaining the regularization properties of Ridge. We control the convex combination of L1 and L2 using the a ratio parameter $\\alpha$.\n",
    "\n",
    "This can be mathematically represented as\n",
    "\n",
    "<br>\n",
    "<center>  $L_{elastic\\_net}(\\hat\\beta) = \\sum_{I=1}^n(y_i-x_i’\\hat\\beta)^2 + \\lambda_2 \\sum_{j=1}^p\\hat\\beta_j^2 + \\lambda_1 \\sum_{j=1}^p|\\hat\\beta_j| $   </center>\n",
    "\n",
    "Or we can also represent the objective function for elastic net based on a ratio parametre $\\alpha$\n",
    "\n",
    "<br>\n",
    "<img src=\"images/elasticbeta.png\"  width=\"400\">\n",
    "\n",
    "In addition to setting and choosing a lambda value elastic net also allows us to tune the alpha parameter where 𝞪 = 0 corresponds to ridge and 𝞪 = 1 to lasso. Simply put, if you plug in 0 for alpha, the penalty function reduces to the L1 (ridge) term and if we set alpha to 1 we get the L2 (lasso) term. Therefore we can choose an alpha value between 0 and 1 to optimize the elastic net. Effectively this will shrink some coefficients and set some to 0 for sparse selection."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get a visual understanding we can look at the geometry of the two models\n",
    "\n",
    "<img src=\"images/elastic2.png\"  width=\"600\">\n",
    "\n",
    "Elastic-net is useful when there are multiple features which are correlated with one another. Lasso is likely to pick one of these at random, while elastic-net is likely to pick both."
   ]
  },
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   "source": [
    "### Example: Elastic net on Boston Housing Dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how Elastic net regularization works with the Housing dataset.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# dataframe and plot\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.ticker as ticker\n",
    "\n",
    "# dataset\n",
    "from sklearn.datasets import load_boston\n",
    "# scaling and dataset split\n",
    "from sklearn import preprocessing\n",
    "from sklearn.model_selection import train_test_split\n",
    "# OLS, Ridge, LASSO, Elastic Net\n",
    "from sklearn.linear_model import LinearRegression, Ridge, Lasso, ElasticNet\n",
    "# model evaluation\n",
    "from sklearn.metrics import r2_score, mean_squared_error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "      <th>PRICE</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00632</td>\n",
       "      <td>18.0</td>\n",
       "      <td>2.31</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.538</td>\n",
       "      <td>6.575</td>\n",
       "      <td>65.2</td>\n",
       "      <td>4.0900</td>\n",
       "      <td>1.0</td>\n",
       "      <td>296.0</td>\n",
       "      <td>15.3</td>\n",
       "      <td>396.90</td>\n",
       "      <td>4.98</td>\n",
       "      <td>24.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.02731</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>6.421</td>\n",
       "      <td>78.9</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>396.90</td>\n",
       "      <td>9.14</td>\n",
       "      <td>21.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.02729</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>7.185</td>\n",
       "      <td>61.1</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>392.83</td>\n",
       "      <td>4.03</td>\n",
       "      <td>34.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.03237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>6.998</td>\n",
       "      <td>45.8</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>394.63</td>\n",
       "      <td>2.94</td>\n",
       "      <td>33.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.06905</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>7.147</td>\n",
       "      <td>54.2</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>396.90</td>\n",
       "      <td>5.33</td>\n",
       "      <td>36.2</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
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      "text/plain": [
       "      CRIM    ZN  INDUS  CHAS    NOX     RM   AGE     DIS  RAD    TAX  \\\n",
       "0  0.00632  18.0   2.31   0.0  0.538  6.575  65.2  4.0900  1.0  296.0   \n",
       "1  0.02731   0.0   7.07   0.0  0.469  6.421  78.9  4.9671  2.0  242.0   \n",
       "2  0.02729   0.0   7.07   0.0  0.469  7.185  61.1  4.9671  2.0  242.0   \n",
       "3  0.03237   0.0   2.18   0.0  0.458  6.998  45.8  6.0622  3.0  222.0   \n",
       "4  0.06905   0.0   2.18   0.0  0.458  7.147  54.2  6.0622  3.0  222.0   \n",
       "\n",
       "   PTRATIO       B  LSTAT  PRICE  \n",
       "0     15.3  396.90   4.98   24.0  \n",
       "1     17.8  396.90   9.14   21.6  \n",
       "2     17.8  392.83   4.03   34.7  \n",
       "3     18.7  394.63   2.94   33.4  \n",
       "4     18.7  396.90   5.33   36.2  "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# load dataset\n",
    "house_price = load_boston()\n",
    "df = pd.DataFrame(house_price.data, columns=house_price.feature_names)\n",
    "df['PRICE'] = house_price.target\n",
    "\n",
    "# standardize and train/test split\n",
    "house_price.data = preprocessing.scale(house_price.data)\n",
    "X_train, X_test, y_train, y_test = train_test_split(\n",
    "    house_price.data, house_price.target, test_size=0.3, random_state=10)\n",
    "\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,\n",
      "         normalize=False)\n",
      "Lasso(alpha=1.0, copy_X=True, fit_intercept=True, max_iter=1000,\n",
      "   normalize=False, positive=False, precompute=False, random_state=None,\n",
      "   selection='cyclic', tol=0.0001, warm_start=False)\n",
      "Ridge(alpha=1.0, copy_X=True, fit_intercept=True, max_iter=None,\n",
      "   normalize=False, random_state=None, solver='auto', tol=0.001)\n",
      "ElasticNet(alpha=1.0, copy_X=True, fit_intercept=True, l1_ratio=0.5,\n",
      "      max_iter=1000, normalize=False, positive=False, precompute=False,\n",
      "      random_state=None, selection='cyclic', tol=0.0001, warm_start=False)\n"
     ]
    }
   ],
   "source": [
    "# Define Models\n",
    "\n",
    "lm = LinearRegression()\n",
    "lm_lasso = Lasso()\n",
    "lm_ridge = Ridge()\n",
    "lm_elastic = ElasticNet()\n",
    "\n",
    "print(lm)\n",
    "print(lm_lasso)\n",
    "print(lm_ridge)\n",
    "print(lm_elastic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ElasticNet(alpha=1.0, copy_X=True, fit_intercept=True, l1_ratio=0.5,\n",
       "      max_iter=1000, normalize=False, positive=False, precompute=False,\n",
       "      random_state=None, selection='cyclic', tol=0.0001, warm_start=False)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fit Models (on train data)\n",
    "\n",
    "lm.fit(X_train, y_train)\n",
    "lm_lasso.fit(X_train, y_train)\n",
    "lm_ridge.fit(X_train, y_train)\n",
    "lm_elastic.fit(X_train, y_train)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Coeficients of the linear regression\n",
    "\n",
    "plt.figure(figsize=(15,10))\n",
    "ft_importance_lm = pd.Series(lm.coef_, index = house_price.feature_names)\n",
    "ft_importance_lm.plot(kind='barh')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Coeficients of lasso\n",
    "\n",
    "plt.figure(figsize=(15,10))\n",
    "ft_importance_lm_lasso = pd.Series(lm_lasso.coef_, index = house_price.feature_names)\n",
    "ft_importance_lm_lasso.plot(kind='barh')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Coeficients of Elastic net\n",
    "\n",
    "plt.figure(figsize=(15,10))\n",
    "ft_importance_lm_elastic = pd.Series(lm_elastic.coef_, index = house_price.feature_names)\n",
    "ft_importance_lm_elastic.plot(kind='barh')\n",
    "plt.show();"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}