fixed negative sign

Author: jonkrohn

notebooks/8-optimization.ipynb

@@ -107,7 +107,7 @@
         "import numpy as np\n",
         "import matplotlib.pyplot as plt"
       ],
-      "execution_count": null,
+      "execution_count": 18,
       "outputs": []
     },
     {
@@ -215,7 +215,7 @@
         "Let's use the contrived values again to show how MSE is calculated. As covered in [*Calculus II*](https://github.com/jonkrohn/ML-foundations/blob/master/notebooks/4-calculus-ii.ipynb), the individual-instance variant of MSE is *squared error* or *quadratic cost*, which -- as with absolute error -- dictates that the cost for each instance $i$ must be $\\geq 0$: \n",
         "\n",
         "* $(\\hat{y}_1 - y_1)^2 = (7 - 2)^2 = 5^2 = 25$\n",
-        "* $(\\hat{y}_2 - y_2)^2 = (3 - 8)^2 = 5^2 = 25$\n",
+        "* $(\\hat{y}_2 - y_2)^2 = (3 - 8)^2 = (-5)^2 = 25$\n",
         "\n",
         "As suggested by its name, MSE is the average: $$C = \\frac{1}{n} \\sum_{i=1}^n (\\hat{y_i}-y_i)^2 = \\frac{1}{2} (25 + 25) = \\frac{50}{2} = 25 $$\n",
         "\n",
@@ -233,7 +233,7 @@
       "source": [
         "delta_y = np.linspace(-5, 5, 1000)"
       ],
-      "execution_count": null,
+      "execution_count": 19,
       "outputs": []
     },
     {
@@ -244,7 +244,7 @@
       "source": [
         "abs_error = np.abs(delta_y)"
       ],
-      "execution_count": null,
+      "execution_count": 20,
       "outputs": []
     },
     {
@@ -255,7 +255,7 @@
       "source": [
         "sq_error = delta_y**2"
       ],
-      "execution_count": null,
+      "execution_count": 21,
       "outputs": []
     },
     {
@@ -266,7 +266,7 @@
           "base_uri": "https://localhost:8080/",
           "height": 279
         },
-        "outputId": "09fd0964-4718-4243-8966-45f47e8028c0"
+        "outputId": "ca63427c-ca25-4fff-9445-a77d6e172e58"
       },
       "source": [
         "fig, ax = plt.subplots()\n",
@@ -282,7 +282,7 @@
         "ax.set_ylim(-1, 17)\n",
         "_ = ax.legend(['Absolute', 'Squared'])"
       ],
-      "execution_count": null,
+      "execution_count": 22,
       "outputs": [
         {
           "output_type": "display_data",
@@ -361,7 +361,7 @@
       "source": [
         "x = np.linspace(-10, 10, 1000)"
       ],
-      "execution_count": null,
+      "execution_count": 23,
       "outputs": []
     },
     {
@@ -381,7 +381,7 @@
       "source": [
         "y_min = x**2 + 3*x + 4"
       ],
-      "execution_count": null,
+      "execution_count": 24,
       "outputs": []
     },
     {
@@ -411,7 +411,7 @@
           "base_uri": "https://localhost:8080/",
           "height": 265
         },
-        "outputId": "9b8a6556-37e6-41f8-bac7-e010948a5b66"
+        "outputId": "b5d76c63-d12c-4237-92b9-53509f1d9711"
       },
       "source": [
         "fig, ax = plt.subplots()\n",
@@ -427,7 +427,7 @@
         "\n",
         "_ = ax.plot(x, y_min)"
       ],
-      "execution_count": null,
+      "execution_count": 25,
       "outputs": [
         {
           "output_type": "display_data",
@@ -470,7 +470,7 @@
       "source": [
         "y_max = -x**2 + 3*x + 4"
       ],
-      "execution_count": null,
+      "execution_count": 26,
       "outputs": []
     },
     {
@@ -500,7 +500,7 @@
           "base_uri": "https://localhost:8080/",
           "height": 269
         },
-        "outputId": "58e3dd67-6466-497f-a2b9-d1d8f4184146"
+        "outputId": "5c77fa62-fa1d-4bfc-ac39-74eabc9781fa"
       },
       "source": [
         "fig, ax = plt.subplots()\n",
@@ -516,7 +516,7 @@
         "\n",
         "_ = ax.plot(x, y_max)"
       ],
-      "execution_count": null,
+      "execution_count": 27,
       "outputs": [
         {
           "output_type": "display_data",
@@ -559,7 +559,7 @@
       "source": [
         "y_sp = x**3 + 6"
       ],
-      "execution_count": null,
+      "execution_count": 28,
       "outputs": []
     },
     {
@@ -589,7 +589,7 @@
           "base_uri": "https://localhost:8080/",
           "height": 269
         },
-        "outputId": "ace66e80-8afc-463e-c3bf-971f1e56dc33"
+        "outputId": "7680e12d-4d8a-4f8b-a371-d285c6b2b704"
       },
       "source": [
         "fig, ax = plt.subplots()\n",
@@ -605,7 +605,7 @@
         "\n",
         "_ = ax.plot(x, y_sp)"
       ],
-      "execution_count": null,
+      "execution_count": 29,
       "outputs": [
         {
           "output_type": "display_data",