Liner regression update

Notebooks/Spark-Example-18-Linear-Regression-Gradient Descent.ipynb

@@ -430,15 +430,15 @@
     "\n",
     "Here, $\\theta_0$ is the intercept term, representing the value of $y$ when all input variables are zero.\n",
     "\n",
-    "### Modified Cost Function\n",
+    "### Cost Function with Intercept\n",
     "\n",
     "The mean squared error (MSE) for the cost function now becomes:\n",
     "\n",
     "$$\n",
     "L(\\theta_0, \\theta_1, \\theta_2, \\theta_3) = \\frac{1}{2n} \\sum_{i=1}^{n} \\left( y_i - (\\theta_0 + \\theta_1 x_{1i} + \\theta_2 x_{2i} + \\theta_3 x_{3i}) \\right)^2\n",
     "$$\n",
     "\n",
-    "### Modified Gradient Descent\n",
+    "### Gradient Descent with Intercept\n",
     "\n",
     "The steps in the gradient descent algorithm would be as follows:\n",
     "\n",
@@ -463,14 +463,22 @@
     "\\theta_j = \\theta_j - \\alpha \\frac{\\partial L}{\\partial \\theta_j} , \\quad \\text{for} \\; j=0,1,2,3\n",
     "$$\n",
     "\n",
-    "### Interpretation of Modified Coefficients\n",
+    "### Interpretation of Coefficients\n",
     "\n",
     "- $\\theta_0$ is the estimated value of the dependent variable when all the predictors are zero.\n",
     "- $\\theta_1$, $\\theta_2$, $\\theta_3$ have similar interpretations as before, but they are adjusted for the presence of $\\theta_0$ and other variables.\n",
     "\n",
     "The introduction of the intercept term $\\theta_0$ allows the model to better capture the relationship between the dependent variable and the predictors, thereby enhancing its predictive accuracy."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Generate Data.\n",
+    "Prepare data for the linear regression model with an intercept term."
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 19,
@@ -531,6 +539,9 @@
     "$$\n",
     "X_i = (1, x_1, x_2, x_3)\n",
     "$$\n",
+    "$$\n",
+    "\\theta = (\\theta_0, \\theta_1, \\theta_2, \\theta_3)\n",
+    "$$\n",
     "\n",
     "The mean squared error (MSE) for the cost function now becomes:\n",
     "\n",