Machine Learning/Linear-Regression-Check-Output.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e4fd1af1",
   "metadata": {},
   "source": [
    "# Daily-Dose-of-Data-Science\n",
    "\n",
    "[Daily Dose of Data Science](https://avichawla.substack.com) is a publication on Substack that brings together intriguing frameworks, libraries, technologies, and tips that make the life cycle of a Data Science project effortless. \n",
    "\n",
    "Author: Avi Chawla\n",
    "\n",
    "[Medium](https://medium.com/@avi_chawla) | [LinkedIn](https://www.linkedin.com/in/avi-chawla/)\n",
    "\n",
    "# Always Validate Your Output Variable Before Using Linear Regression\n",
    "\n",
    "Post Link: [Substack](https://avichawla.substack.com/p/always-validate-your-output-variable)\n",
    "\n",
    "LinkedIn Post: [LinkedIn](https://www.linkedin.com/feed/update/urn:li:activity:7043533853507862528/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "eebc4fae",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import seaborn as sns\n",
    "sns.set()\n",
    "\n",
    "plt.style.use([\"seaborn-darkgrid\"])\n",
    "c_list = ['#46C7C7', '#00B9FF', '#FF7051']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "63e0b90d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(123)\n",
    "\n",
    "# Generate random X values\n",
    "X = np.random.uniform(low=-5, high=5, size=(100, 1))\n",
    "\n",
    "# Generate dependent variable Y with a skewed distribution\n",
    "Y = 10 + X + np.random.normal(loc=0, scale=1, size=(100, 1))\n",
    "Y = np.exp(Y)\n",
    "\n",
    "# Plot the data\n",
    "fig, ax = plt.subplots()\n",
    "plt.scatter(X, Y, c = c_list[2])\n",
    "plt.xlabel('X')\n",
    "plt.ylabel('Y')\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "a2fecdaa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import matplotlib as mpl\\nimport pandas as pd'); }\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAESCAYAAADg0F5TAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/NK7nSAAAACXBIWXMAAAsTAAALEwEAmpwYAAAXkUlEQVR4nO3de3BU5f3H8c/mRhJIIYEVgkVpqEaoQBNSK9Aqt8Bwx8jNNkktCA2MyIA3VATKlJbLMAgBpU4FJNI2DFARpSPlZluGAkILGIhSoA4gpJFEmrAkwGZ/f/DjTBaS3ZOQzeYh79cMM7vnss9zvmE+e/I855w4PB6PRwAAY4UEuwMAgDtDkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gv8tlZ2crMTHxtn9dunTRY489pmeffVaffvrpbfudPXvW2nby5Ml31If8/Hxt2bKlRvv06dNHiYmJSklJsZbt27fP6tO8efPuqE++XL16VW+//bYuX75c720H0uHDh5Wenq6kpCQlJSVp+PDhOnHihNc2brdbI0aMsI51wYIFVX5Wbm6utU2/fv1UXl5eH4eAahDkjVR5ebkKCgr0l7/8Renp6VqzZk2dt1FaWqq5c+cqLS1NR44cqfPPD4Q9e/Zo6NChWrx4sdxud7C7U2fKy8s1YcIEHThwQC6XSy6XS/n5+WrWrJnXdqGhoZo1a5YcDockKScnR2fOnPHaxuVyKTs723o/c+ZMNWnSJPAHgWqFBbsDqD8ZGRl69NFHdf36dZWWlmr//v364IMP5PF4tGDBAj300EN69NFHJUktW7bUihUrJElOp7NW7eXl5WndunW12nfu3LkqKytTWFj9/hdduXKl/vOf/9y2/IEHHrDq0a5du3rtU13497//rUuXLkmS2rdvrylTpqisrEzx8fG3bZucnKy0tDRt3LhR165d06JFi7Rs2TJr/TvvvKPCwkJJUt++fdWrV696OQZUjyBvRDp16qR+/fpZ70eOHKnvf//7+uUvf6mKigotXbrUCvKoqCivbevbj370o6C1XZW4uLig1uNOVR4m+vGPf6whQ4b43P6FF17Q9u3bdenSJX388cf69NNPlZKSosLCQq1atUrSjf8jr732WkD7DXsYWmnkxo4da52VHTp0SOfPn5dU/Ri5x+PR+vXrNXr0aCUnJ6tTp07q3r27srKy9K9//cvaLjs7W5mZmdb7tWvXKjExUZs2bZJ047eDxMREpaamau/everXr58efvhhjR07VlLVY+S32r17t9LS0tS5c2f17dtXy5cv17Vr17y2uXkMw4cP91q+Zs0aa93NPiUmJmr//v3WNj/4wQ/Up08fSf7HyDdv3qz09HQ98sgj6tq1q4YPH67f/e53t40dV56zuHDhgnJzczV48GB17txZ/fv316pVq2T3qRnXr1/XunXrNHLkSHXr1k1JSUkaPXq01q9fr4qKCmu7jIwMZWRkWO9zcnKUmJioGTNmVPvZcXFxmjZtmvV+/vz58ng8WrZsmVwulyQpKytL9957r62+IrA4I2/kQkJC1K1bN3344YeSbkyIVfXr9k1Lly7VW2+95bWsqKhIu3bt0t69e7V27Vp17drVdvvFxcWaPHmyFQ7f+c53bO3397//Xe+9954VWGfPnlV2draOHz9uDYHUB4/Ho+nTp2vr1q1ey/Pz85Wfn69t27Zp1apVt41FS9KiRYusukvSl19+qQULFsjtdmvChAk+2y0rK9OECRO8vnikGz+/w4cPa+fOnVq+fPkdDU2NGTNGGzZs0GeffaajR4/qjTfe0MaNGyXdGJ4ZN25crT8bdYszcuiee+6xXv/3v/+tdju3221NirZv315z5szRkiVLlJmZKYfDobKyMi1dulSSNGjQIE2dOtXat3fv3lqxYoU1dHNTSUmJIiIiNGfOHD3//PNKS0uz1edTp06pY8eOmjdvnqZOnaqoqChJ0vbt27V9+3Zbn3GrFStW6IEHHrDeL1q0SHPnzvW5z7vvvmuFuNPp1CuvvKK5c+eqY8eOkm4Ea3Wf8eGHH2rkyJFavHixUlNTreU3f0PwZfHixVaIJyQkaPbs2Xr99df17W9/W5K0a9cu6wtt6tSpVf4sKp+lVyUkJESzZ89WSMiNmFi5cqU1ATxr1ixFRET47SfqB2fk8LrioPJY6q3cbrc1VHDfffdpwIABiouL06BBg5ScnKyYmBgrCDt06KBu3bpZ+7Zr167aMeYpU6boqaeeqlGfW7RoobVr11pnum3atNErr7wiSdq6dWutxrP79eund99913rfq1cvfetb36p2+4qKCq1evVqSFBERoT/84Q/WROiwYcM0ePBgnTt3Tlu2bNH06dPVpk0br/0HDRpkDdP0799fP/zhD+VyuXTu3Dmf/SwpKdH69esl3RgCyc3Ntfo5YMAADRgwQJcvX9aaNWs0adIkpaSkeF2B4+tncasuXbpo1KhRys3NtZYNHDhQPXv2tLU/6gdn5PAax715ZluViIgI68zxr3/9q3r27KknnnhCv/rVrxQZGalHHnlErVu3rnH7vsbBq9OjRw+v4Yr+/ftbr0+ePFnjz6uNL7/8UhcuXJAk9ezZ0+tqlqioKA0bNkzSjcA/ePDgbfv36NHDeh0REWEFvb9rsg8fPqyysjJJN74MKn/ZOJ1O9e3bV9KNywSPHTtWm0PzMn36dDVt2lSSFB4ebn1houEgyGFdSiZJrVq18rnt/PnzlZGRoaZNm6qiokLHjh1TTk6OsrKy1Lt3b+3cubPG7cfGxtZ4n+bNm3u9b9asmfWbxZUrV27b/tYJxLq4Rvybb76xXlcenrqp8pda5W1vuvVs3+5QReXPquqLs/Kym5cc3okWLVqoRYsWkqTIyMhafVkjsAhy6OjRo9ZrfxOV0dHRmjlzpv7xj39o1apVysrKUkpKihwOh77++mtNnz5dRUVFNWq/NjeT3DqW73K5rDPZqiYWr1+/7vXe1xCSXS1btqy2P5JUUFBgvY6Li7ttfWho6B23W7mNqpbV5ksS5iHIG7ndu3dbN8AkJyf7vNnl9OnTWrx4sSZNmqTNmzerZ8+emjZtmtatW2ddNnjlyhXl5+dLkjVJJt1+RlxZ5e3s2rt3r9cXxo4dO6zXDz30kPU6OjpaknTx4kWvs/Dqhhxu3tHor8+SdO+991o3S+3Zs0dnz5611pWXl+uDDz6QdCOwK88X3KlOnTopPDxckvTnP/9ZJSUl1rqvv/7aqkVMTIw16Yq7G5OdjcixY8esX+cvX76szz77zJrEcjgceu6553zu36xZM61evVrXrl3T3r17VVBQoISEBH311Vf6+OOPre1uXlscGRlpLdu3b5/ef/99JSQkqEuXLnd8LC6XS5mZmXr66adVXFzsdUnk0KFDrdft2rXT559/rm+++UYLFy7U4MGDtWPHDu3atavKz608R5CTk6OWLVtWOxEbGhqqsWPHKjs7W1evXtVPfvITjR8/XlFRUfrjH/9oTVqmpaVVOfRSW82bN9fQoUO1adMmXbx4UWPGjFFGRoY1+Xrzt43x48dbgY+7G0HeiOTk5CgnJ+e25Q6HQy+++KK6d+/uc3+n06lZs2bp9ddf15UrV7yet3FTZmam7r//fkk3Lotr0qSJysvL9cUXX+jll1/WtGnT6iTIU1NTtX379tvuLExLS/M6jieffFK//vWvJd24Cejm5ZP9+vWr8jLFjh07avfu3ZJu3LwTERFh/bZRlaysLOXl5Wnnzp0qKCiw2rqpW7duevXVV2tziD699tprOnnypA4fPqyTJ09qzpw5XusHDBigiRMn1nm7aJgI8kYqPDxcrVq1UlJSkjIyMpScnGxrv9GjRyshIUGrVq3S8ePHVVhYqMjISCUmJmrUqFFed1A2bdpU8+fP15IlS3T+/Hm1atWq1s9tuVVqaqrGjBmjZcuWKT8/X61bt9bo0aP1zDPPeG2XmZkpt9utdevWqbCwUN/97nf1zDPP6J577qkyyMeNG6dTp07pb3/7m6Qbwe7rKpKwsDC99dZb2rx5szZu3Kj8/HyVl5erffv2GjFihH76058G5HrrZs2a6fe//71yc3O1efNmnTx5UhUVFXrwwQc1atQoPfnkk17DRLi7OTx27wcGADRITHYCgOEIcgAwHEEOAIYjyAHAcAQ5ABguKJcfFhaW+N+oGrGx0SoudtVhb+4u1Mc36uMfNfItmPVxOmOqXG7cGXlYWO2eT9FYUB/fqI9/1Mi3hlgf44IcAOCNIAcAwxHkAGA4ghwADEeQA4DhCHIAMBxBDgCGI8gBwHDG/WGJq9fc1d7dVFlZ+XWV/O/2v6YOAHcb44I8IjxUQ5/f7He7LYuHq/YPAgAAczC0AgCGI8gBwHAEOQAYjiAHAMMR5ABgOIIcAAxHkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gBwDDEeQAYDiCHAAMR5ADgOGC8jzy2NhohYWFBrwdO3+A4m7UWI/bLurjHzXyraHVJyhBXlzsqvW+NSlgYWHj+9MSTmdMozxuu6iPf9TIt2DWp7r8Y2gFAAxHkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gBwDDEeQAYDiCHAAMR5ADgOEIcgAwHEEOAIYjyAHAcAQ5ABiOIAcAwxHkAGA4ghwADEeQA4DhCHIAMBxBDgCGI8gBwHAEOQAYjiAHAMMR5ABgOIIcAAwXFoxGY2OjFRYWGvB2nM6YgLfREDXW47aL+vhHjXxraPUJSpAXF7tqvW9NClhYWFLrdkzldMY0yuO2i/r4R418C2Z9qss/hlYAwHAEOQAYjiAHAMMR5ABgOIIcAAxHkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gBwDDEeQAYDiCHAAMR5ADgOEIcgAwHEEOAIYjyAHAcAQ5ABiOIAcAwxHkAGA4ghwADEeQA4DhCHIAMBxBDgCGI8gBwHBhwWg0NjZaYWGhAW/H6YwJeBsNUWM9bruoj3/UyLeGVp+gBHlxsavW+9akgIWFJbVux1ROZ0yjPG67qI9/1Mi3YNanuvxjaAUADEeQA4DhCHIAMBxBDgCGI8gBwHAEOQAYjiAHAMPZDvK+fftq9uzZty2fOHGiBg0aVKedAgDY5/OGoB07duj48eOSpHPnzunAgQNavny5td7j8SgvL0+lpaWB7SUAoFo+gzw+Pl5TpkyRx+ORw+HQ6dOntWLFCq9tPB6PunbtGtBOAgCq5zPIO3XqpJdfflknTpzQhg0b1LZtW/Xo0cNaHxISotjYWI0aNSrgHQUAVM3vs1Z+9rOfSZLatGmjhIQExsMBoIGx/dCsZ599Vi6XS3l5ebp69ao8Ho/X+uTk5DrvHADAP9tBvmPHDs2YMaPKiU2Hw6Fjx47VaccAAPbYDvKFCxeqpKREYWFhatmypUJDA/88cQCAf7aD/Pz584qPj9fGjRsVFxcXyD4BAGrAdpB37dpV169fJ8QBoIGxHeRTp07V5MmTtXDhQvXo0UNRUVFyOBzWeiY7ASA4bAd5enq6HA6HVq9erdWrV3utY7ITAILHdpC3bds2kP0AANSS7SDfuXNnIPsBAKgl20FeUFDgc33r1q3vuDMAgJqzHeS9evWqdh1j5AAQPLaD/NZb8m+KiopSmzZt6qxDAICasR3k+fn5Xu/Ly8t19OhRPffcc3rxxRfrvGMAAHtsB/mtmjRpopSUFD3++ON644031KdPH9v7xsZGKyws8Lf4O50xAW+jIWqsx20X9fGPGvnW0OpT68nOiooKnT9/Xvv27dPFixdr1GhxsatG21dWkwIWFpbUuh1TOZ0xjfK47aI+/lEj34JZn+ry744nOz0ej5KSkmrVKQDAnbujyc7IyEh17txZc+fOrdNOAQDsq/VkJwCgYajxZOeRI0d08OBBORwOpaSk6OGHHw5EvwAANtkOcrfbrZdeeklbt271Wj58+HD95je/8XoSIgCg/oTY3fC3v/2tPvroI0VHRys1NVWpqamKjo7W5s2b9fbbbweyjwAAH2yfkW/atEnNmzfX+++/r/j4eEnSV199pREjRmjDhg36xS9+EbBOAgCqZ/uMvKCgQJ06dbJCXLrxaNvvfe97unDhQkA6BwDwz3aQx8fH69ixY143Bp0/f155eXk8qxwAgsj20MqwYcO0fPlyDRkyRN27d5ck7d27V6WlpcrMzAxYBwEAvtkO8qysLH3xxRfatm2btm3bZi1/7LHHNHHixIB0DgDgn+0gDwsL07Jly3TkyBEdOnRIbrdbSUlJ/NFlAAgyv2Pkn3zyifr3769Tp05Jkrp06aKnn35aRUVFWrJkiT7//POAdxIAUD2fQX7o0CFNnjxZZ86c0f79+73Wbd++XQcOHFB6erpOnjwZ0E4CAKrnM8hXrlwpt9utJ554Qv379/da9+abb6p3794qKSnRm2++GdBOAgCq5zPI//nPf+q+++7TvHnzFBcX57WuQ4cOWrZsmeLj43Xw4MGAdhIAUD2fQV5eXq62bdtW+xyV8PBw3X///SoqKgpI5wAA/vkM8nbt2ikvL6/aoC4qKlJeXp7X3Z4AgPrlM8gHDhyokpISTZgwwbr5R5JKSkq0Z88eTZgwQaWlpRo4cGC9dBYAcDuf15GPHz9eO3fuVF5ensaNG3fbeo/Ho8TERG4IAoAg8nlGHhUVpffee0/p6emKjo6Wx+Ox/jVp0kRjxozRunXrFB0dXV/9BQDcwu+dndHR0Zo5c6ZeffVVnT59WpcuXVLTpk2VkJCg8PDw+ugjAMAH27foh4SEqEOHDoHsCwCgFmw/xhYA0DAR5ABgOIIcAAxne4y8LsXGRissLDTg7TidMQFvoyFqrMdtF/Xxjxr51tDqE5QgLy521XrfmhSwsLCk1u2YyumMaZTHbRf18Y8a+RbM+lSXfwytAIDhCHIAMBxBDgCGI8gBwHAEOQAYjiAHAMMR5ABgOIIcAAxHkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gBwDDEeQAYDiCHAAMR5ADgOEIcgAwHEEOAIYjyAHAcAQ5ABiOIAcAwxHkAGA4ghwADEeQA4DhwoLRaGxstMLCQgPejtMZE/A2GqLGetx2UR//qJFvDa0+QQny4mJXrfetSQELC0tq3Y6pnM6YRnncdlEf/6iRb8GsT3X5x9AKABiOIAcAwxHkAGA4ghwADEeQA4DhCHIAMBxBDgCGI8gBwHAEOQAYjiAHAMMR5ABgOIIcAAxHkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gBwDDEeQAYDiCHAAMR5ADgOEIcgAwHEEOAIYjyAHAcAQ5ABguLBiNxsZGKywsNODtOJ0xAW+jIWqsx20X9fGPGvnW0OoTlCAvLnbVet+aFLCwsKTW7ZjK6YxplMdtF/Xxjxr5Fsz6VJd/DK0AgOEIcgAwHEEOAIYjyAHAcEGZ7KwPV6+5bU2MlpVfV8n/rtRDjwAgMO7aII8ID9XQ5zf73W7L4uFifh6AyRhaAQDDEeQAYDiCHAAMR5ADgOEIcgAwHEEOAIYjyAHAcAQ5ABjurr0hyC7uAAVgukYf5NwBCsB0DK0AgOEIcgAwHEEOAIYjyAHAcI1+sjOYYr4Vpcgm/n8E5VfdahIR6ne7svLrddEti93+cUUPEFwEeRBFNgmzfcWM3e3qUk36xxU9QPAwtAIAhuOMHHfM7k1VUt0PwzD8U3+CWeu6bruhf15NEeS4Y3ZvqpLqfhiG4Z/6E8xa13XbDf3zasrh8Xg8AfhcAEA9YYwcAAxHkAOA4QhyADAcQQ4AhiPIAcBwBDkAGI4gBwDDEeRo1DIyMpSYmKinnnrKa7nb7daQIUOUmJioGTNmBKl3gD0EORq1OXPmKDw8XIcOHdLWrVut5bm5uTpx4oRatGihl156KYg9BPwjyNGodejQQePHj5ckLVq0SGVlZSotLVV2drYk6YUXXlBcXFwwuwj4xbNW0OhNmjRJH330kc6cOaN33nlHLpdLRUVF6tatm0aOHBns7gF+8awVQNInn3yiiRMnKioqSm63WxUVFfrTn/6kBx98MNhdA/xiaAWQ9Pjjj2vAgAG6cuWKrl69qp///OeEOIzBGTnw/06cOKEhQ4YoJCREhw4dUlRUVLC7BNjCGTnw/24Gt8PhIMRhFIIcAAxHkAOA4QhyADAck50AYDjOyAHAcAQ5ABiOIAcAwxHkAGA4ghwADEeQA4DhCHIAMBxBDgCGI8gBwHD/ByXK7lKM/RBXAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "plt.hist(Y, bins=30)\n",
    "plt.ylabel('Count', fontsize=15, fontweight=\"bold\")\n",
    "plt.xlabel('Y', fontsize=15, fontweight=\"bold\")\n",
    "plt.title('Distribution of Y', fontsize = 20, fontweight=\"bold\")\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "41563c07",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot data\n",
    "fig, ax = plt.subplots()\n",
    "plt.hist(np.log(Y), bins=30)\n",
    "plt.ylabel('Count', fontsize=15, fontweight=\"bold\")\n",
    "plt.xlabel('log(Y)', fontsize=15, fontweight=\"bold\")\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "plt.title('Distribution of log(Y)', fontsize = 20, fontweight=\"bold\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3552bbb4",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "# Train a linear regression model without addressing the skewness\n",
    "model1 = LinearRegression().fit(X, Y)\n",
    "\n",
    "# Train a linear regression model with a log transformation to address the skewness\n",
    "model2 = LinearRegression().fit(X, np.log(Y))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "0c8accf9",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generate X values for plotting\n",
    "X_plot = np.linspace(-5, 5, 100).reshape(-1, 1)\n",
    "\n",
    "# Get the predictions from each model\n",
    "Y_pred1 = model1.predict(X_plot)\n",
    "Y_pred2 = model2.predict(X_plot)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "76627dc0",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "2023-03-20 16:54:30,505 [24224] WARNING  matplotlib.legend:1217: [JupyterRequire] No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the data and predictions from each model\n",
    "fig, ax = plt.subplots()\n",
    "plt.scatter(X, Y)\n",
    "plt.plot(X_plot, Y_pred1, c = c_list[2], linewidth = 3)\n",
    "plt.xlabel('X', fontsize=15, fontweight=\"bold\")\n",
    "plt.ylabel('Y', fontsize=15, fontweight=\"bold\")\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "plt.legend(fontsize = 15)\n",
    "plt.title('Regression fit on Y', fontsize = 20, fontweight=\"bold\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "8c7486a8",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "2023-03-20 16:54:36,663 [24224] WARNING  matplotlib.legend:1217: [JupyterRequire] No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the data and predictions from each model\n",
    "fig, ax = plt.subplots()\n",
    "plt.scatter(X, np.log(Y))\n",
    "plt.plot(X_plot, Y_pred2, c = c_list[2], linewidth = 3)\n",
    "plt.xlabel('X', fontsize=15, fontweight=\"bold\")\n",
    "plt.ylabel('log(y)', fontsize=15, fontweight=\"bold\")\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "plt.legend(fontsize = 15)\n",
    "plt.title('Regression fit on log(Y)', fontsize = 20, fontweight=\"bold\")\n",
    "\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.12"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}