NE-BGD/main.ipynb

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    "# Linear regression\n",
    "\n",
    "The linear regression is a training procedure based on a linear model. The model makes a prediction by simply computing a weighted sum of the input features, plus a constant term called the bias term (also called the intercept term):\n",
    "\n",
    "$$ \\hat{y}=\\theta_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\cdots + \\theta_n x_n$$\n",
    "\n",
    "This can be writen more easy by using vector notation form for $m$ values. Therefore, the model will become:\n",
    "\n",
    "$$ \n",
    "    \\begin{bmatrix}\n",
    "    \\hat{y}^0 \\\\ \n",
    "    \\hat{y}^1\\\\\n",
    "    \\hat{y}^2\\\\\n",
    "    \\vdots \\\\\n",
    "    \\hat{y}^m\n",
    "    \\end{bmatrix}\n",
    "    =\n",
    "    \\begin{bmatrix}\n",
    "    1   & x_1^0 & x_2^0 & \\cdots &x_n^0\\\\\n",
    "    1   & x_1^1 & x_2^1 & \\cdots & x_n^1\\\\\n",
    "    \\vdots & \\vdots &\\vdots & \\cdots & \\vdots\\\\\n",
    "    1   &  x_1^m & x_2^m & \\cdots & x_n^m\n",
    "    \\end{bmatrix}\n",
    "\n",
    "    \\begin{bmatrix}\n",
    "    \\theta_0 \\\\\n",
    "    \\theta_1 \\\\\n",
    "     \\theta_2 \\\\\n",
    "    \\vdots \\\\\n",
    "    \\theta_n\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Resulting:\n",
    "\n",
    "$$\\hat{y}= h_\\theta(x) = x \\theta $$\n",
    "\n",
    "**Now that we have our mode, how do we train it?**\n",
    "\n",
    "Please, consider that training the model means adjusting the parameters to reduce the error or minimizing the cost function. The most common performance measure of a regression model is the Mean Square Error (MSE). Therefore, to train a Linear Regression model, you need to find the value of θ that minimizes the MSE:\n",
    "\n",
    "$$ MSE(X,h_\\theta) = \\frac{1}{m} \\sum_{i=1}^{m} \\left(\\hat{y}^{(i)}-y^{(i)} \\right)^2$$\n",
    "\n",
    "\n",
    "$$ MSE(X,h_\\theta) = \\frac{1}{m} \\sum_{i=1}^{m} \\left(  x^{(i)}\\theta-y^{(i)} \\right)^2$$\n",
    "\n",
    "$$ MSE(X,h_\\theta) = \\frac{1}{m}  \\left(  x\\theta-y \\right)^T \\left(  x\\theta-y \\right)$$\n"
   ]
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    "\n",
    "# The normal equation\n",
    "\n",
    "To find the value of $\\theta$ that minimizes the cost function, there is a closed-form solution that gives the result directly. This is called the **Normal Equation**; and can be find it by derivating the *MSE* equation as a function of $\\theta$ and making it equals to zero:\n",
    "\n",
    "\n",
    "$$\\hat{\\theta} = (X^T X)^{-1} X^{T} y $$\n",
    "\n",
    "$$ Temp = \\theta_0 + \\theta_1 * t $$\n"
   ]
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       "          0\n",
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       "\n",
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     "execution_count": 1,
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   ],
   "source": [
    "import pandas as pd\n",
    "df = pd.read_csv('data.csv')\n",
    "df"
   ]
  },
  {
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    {
     "ename": "NameError",
     "evalue": "name 'df' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[1], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[43mdf\u001b[49m)\n",
      "\u001b[0;31mNameError\u001b[0m: name 'df' is not defined"
     ]
    }
   ],
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  },
  {
   "cell_type": "code",
   "execution_count": null,
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