Reg_NoneLinearRegression.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/OseiasBeu/Data-Science/blob/master/Reg_NoneLinearRegression.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mu1phPJDdA_m"
      },
      "source": [
        "<p style=\"text-align:center\">\n",
        "    <a href=\"https://skills.network/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2022-01-01\" target=\"_blank\">\n",
        "    <img src=\"https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/assets/logos/SN_web_lightmode.png\" width=\"200\" alt=\"Skills Network Logo\"  />\n",
        "    </a>\n",
        "</p>\n",
        "\n",
        "# Non Linear Regression Analysis\n",
        "\n",
        "Estimated time needed: **20** minutes\n",
        "\n",
        "## Objectives\n",
        "\n",
        "After completing this lab you will be able to:\n",
        "\n",
        "*   Differentiate between linear and non-linear regression\n",
        "*   Use non-linear regression model in Python\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "50NceYr-dA_o"
      },
      "source": [
        "If the data shows a curvy trend, then linear regression will not produce very accurate results when compared to a non-linear regression since linear regression presumes that the data is linear.\n",
        "Let's learn about non linear regressions and apply an example in python. In this notebook, we fit a non-linear model to the datapoints corrensponding to China's GDP from 1960 to 2014.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PeUgqFKDdA_p"
      },
      "source": [
        "<h2 id=\"importing_libraries\">Importing required libraries</h2>\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "04TH9_vTdA_q"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oDosnG-UdA_r"
      },
      "source": [
        "Although linear regression can do a great job at modeling some datasets, it cannot be used for all datasets. First recall how linear regression, models a dataset. It models the linear relationship between a dependent variable y and the independent variables x. It has a simple equation, of degree 1, for example y = $2x$ + 3.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "9LW0Mi--dA_r",
        "outputId": "9001f873-652f-4c74-da22-f599c24f451e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "x = np.arange(-5.0, 5.0, 0.1)\n",
        "\n",
        "##You can adjust the slope and intercept to verify the changes in the graph\n",
        "y = 2*(x) + 3\n",
        "y_noise = 2 * np.random.normal(size=x.size)\n",
        "ydata = y + y_noise\n",
        "#plt.figure(figsize=(8,6))\n",
        "plt.plot(x, ydata,  'bo')\n",
        "plt.plot(x,y, 'r') \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sbg9MjkCdA_s"
      },
      "source": [
        "Non-linear regression is a method to model the non-linear relationship between the independent variables $x$ and the dependent variable $y$. Essentially any relationship that is not linear can be termed as non-linear, and is usually represented by the polynomial of $k$ degrees (maximum power of $x$).  For example:\n",
        "\n",
        "$$ \\ y = a x^3 + b x^2 + c x + d \\ $$\n",
        "\n",
        "Non-linear functions can have elements like exponentials, logarithms, fractions, and so on. For example: $$ y = \\log(x)$$\n",
        "\n",
        "We can have a function that's even more complicated such as :\n",
        "$$ y = \\log(a x^3 + b x^2 + c x + d)$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_rjw4XijdA_t"
      },
      "source": [
        "Let's take a look at a cubic function's graph.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "id": "i-Icem_odA_t",
        "outputId": "4675db18-4f7f-46ae-c7a4-cce5b4c01a89",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "x = np.arange(-5.0, 5.0, 0.1)\n",
        "\n",
        "##You can adjust the slope and intercept to verify the changes in the graph\n",
        "y = 1*(x**3) + 1*(x**2) + 1*x + 3\n",
        "y_noise = 20 * np.random.normal(size=x.size)\n",
        "ydata = y + y_noise\n",
        "plt.plot(x, ydata,  'bo')\n",
        "plt.plot(x,y, 'r') \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ngxx9wwzdA_u"
      },
      "source": [
        "As you can see, this function has $x^3$ and $x^2$ as independent variables. Also, the graphic of this function is not a straight line over the 2D plane. So this is a non-linear function.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "glORULcmdA_u"
      },
      "source": [
        "Some other types of non-linear functions are:\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LL34-oyhdA_v"
      },
      "source": [
        "### Quadratic\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "d4CQjDHedA_v"
      },
      "source": [
        "$$ Y = X^2 $$\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "id": "RXnRBeJXdA_v",
        "outputId": "8e25c6cf-51f0-403c-fa06-aa5e9fc5f9cd",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "x = np.arange(-5.0, 5.0, 0.1)\n",
        "\n",
        "##You can adjust the slope and intercept to verify the changes in the graph\n",
        "\n",
        "y = np.power(x,2)\n",
        "y_noise = 2 * np.random.normal(size=x.size)\n",
        "ydata = y + y_noise\n",
        "plt.plot(x, ydata,  'bo')\n",
        "plt.plot(x,y, 'r') \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ycSZHwK-dA_w"
      },
      "source": [
        "### Exponential\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1UHwwd-gdA_w"
      },
      "source": [
        "An exponential function with base c is defined by $$ Y = a + b c^X$$ where b ≠0, c > 0 , c ≠1, and x is any real number. The base, c, is constant and the exponent, x, is a variable.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "id": "dHgH7XIHdA_w",
        "outputId": "5a942f3e-b03f-4ba2-a6aa-e46aa276f5ac",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "X = np.arange(-5.0, 5.0, 0.1)\n",
        "\n",
        "##You can adjust the slope and intercept to verify the changes in the graph\n",
        "\n",
        "Y= np.exp(X)\n",
        "\n",
        "plt.plot(X,Y) \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "krTn-Yo-dA_x"
      },
      "source": [
        "### Logarithmic\n",
        "\n",
        "The response $y$ is a results of applying the logarithmic map from the input $x$ to the output $y$. It is one of the simplest form of **log()**: i.e. $$ y = \\log(x)$$\n",
        "\n",
        "Please consider that instead of $x$, we can use $X$, which can be a polynomial representation of the $x$ values. In general form it would be written as\\\n",
        "\\begin{equation}\n",
        "y = \\log(X)\n",
        "\\end{equation}\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "id": "LrO-rcLJdA_x",
        "outputId": "31184bfa-4ce8-4f4e-effa-5a8cd3e07690",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 314
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/ipykernel_launcher.py:3: RuntimeWarning: invalid value encountered in log\n",
            "  This is separate from the ipykernel package so we can avoid doing imports until\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "X = np.arange(-5.0, 5.0, 0.1)\n",
        "\n",
        "Y = np.log(X)\n",
        "\n",
        "plt.plot(X,Y) \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KwRk64PedA_x"
      },
      "source": [
        "### Sigmoidal/Logistic\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cPT75c4PdA_x"
      },
      "source": [
        "$$ Y = a + \\frac{b}{1+ c^{(X-d)}}$$\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "id": "yCq8llkldA_x",
        "outputId": "00ea424e-a38e-4028-81e8-875f5b5d372f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "X = np.arange(-5.0, 5.0, 0.1)\n",
        "\n",
        "\n",
        "Y = 1-4/(1+np.power(3, X-2))\n",
        "\n",
        "plt.plot(X,Y) \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-kEl1p_BdA_y"
      },
      "source": [
        "<a id=\"ref2\"></a>\n",
        "\n",
        "# Non-Linear Regression example\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "h0vKtiGIdA_y"
      },
      "source": [
        "For an example, we're going to try and fit a non-linear model to the datapoints corresponding to China's GDP from 1960 to 2014. We download a dataset with two columns, the first, a year between 1960 and 2014, the second, China's corresponding annual gross domestic income in US dollars for that year.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "id": "2oi31RPadA_y",
        "outputId": "3b220eb9-97bb-46df-8a3a-765f51801a6f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 400
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2022-08-12 00:29:01 URL:https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-ML0101EN-SkillsNetwork/labs/Module%202/data/china_gdp.csv [1218/1218] -> \"china_gdp.csv\" [1]\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "   Year         Value\n",
              "0  1960  5.918412e+10\n",
              "1  1961  4.955705e+10\n",
              "2  1962  4.668518e+10\n",
              "3  1963  5.009730e+10\n",
              "4  1964  5.906225e+10\n",
              "5  1965  6.970915e+10\n",
              "6  1966  7.587943e+10\n",
              "7  1967  7.205703e+10\n",
              "8  1968  6.999350e+10\n",
              "9  1969  7.871882e+10"
            ],
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              "\n",
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              "      <th></th>\n",
              "      <th>Year</th>\n",
              "      <th>Value</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1960</td>\n",
              "      <td>5.918412e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1961</td>\n",
              "      <td>4.955705e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>1962</td>\n",
              "      <td>4.668518e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1963</td>\n",
              "      <td>5.009730e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1964</td>\n",
              "      <td>5.906225e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>1965</td>\n",
              "      <td>6.970915e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>1966</td>\n",
              "      <td>7.587943e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>1967</td>\n",
              "      <td>7.205703e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>1968</td>\n",
              "      <td>6.999350e+10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>1969</td>\n",
              "      <td>7.871882e+10</td>\n",
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              "              title=\"Convert this dataframe to an interactive table.\"\n",
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              "          document.querySelector('#df-7f8d55d1-3d9e-4a7a-ad6a-83f60f05c939 button.colab-df-convert');\n",
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              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
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              "          const element = document.querySelector('#df-7f8d55d1-3d9e-4a7a-ad6a-83f60f05c939');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
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              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "\n",
        "#downloading dataset\n",
        "!wget -nv -O china_gdp.csv https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-ML0101EN-SkillsNetwork/labs/Module%202/data/china_gdp.csv\n",
        "    \n",
        "df = pd.read_csv(\"china_gdp.csv\")\n",
        "df.head(10)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wy4MCd1SdA_y"
      },
      "source": [
        "**Did you know?** When it comes to Machine Learning, you will likely be working with large datasets. As a business, where can you host your data? IBM is offering a unique opportunity for businesses, with 10 Tb of IBM Cloud Object Storage: [Sign up now for free](http://cocl.us/ML0101EN-IBM-Offer-CC)\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "W-Sf8lkqdA_z"
      },
      "source": [
        "### Plotting the Dataset\n",
        "\n",
        "This is what the datapoints look like. It kind of looks like an either logistic or exponential function. The growth starts off slow, then from 2005 on forward, the growth is very significant. And finally, it decelerates slightly in the 2010s.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "id": "JQv2suZxdA_z",
        "outputId": "418a8752-80c2-451c-ecf2-1d3c98939c22",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 345
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 576x360 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "plt.figure(figsize=(8,5))\n",
        "x_data, y_data = (df[\"Year\"].values, df[\"Value\"].values)\n",
        "plt.plot(x_data, y_data, 'ro')\n",
        "plt.ylabel('GDP')\n",
        "plt.xlabel('Year')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "W-W5E8gMdA_z"
      },
      "source": [
        "### Choosing a model\n",
        "\n",
        "From an initial look at the plot, we determine that the logistic function could be a good approximation,\n",
        "since it has the property of starting with a slow growth, increasing growth in the middle, and then decreasing again at the end; as illustrated below:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "id": "95q1Ii8hdA_z",
        "outputId": "bd59f451-0643-4061-cd7f-fe2259e9ead7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "X = np.arange(-5.0, 5.0, 0.1)\n",
        "Y = 1.0 / (1.0 + np.exp(-X))\n",
        "\n",
        "plt.plot(X,Y) \n",
        "plt.ylabel('Dependent Variable')\n",
        "plt.xlabel('Independent Variable')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0119wVgOdA_z"
      },
      "source": [
        "The formula for the logistic function is the following:\n",
        "\n",
        "$$ \\hat{Y} = \\frac1{1+e^{-\\beta\\_1(X-\\beta\\_2)}}$$\n",
        "\n",
        "$\\beta\\_1$: Controls the curve's steepness,\n",
        "\n",
        "$\\beta\\_2$: Slides the curve on the x-axis.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8H2SenAPdA_z"
      },
      "source": [
        "### Building The Model\n",
        "\n",
        "Now, let's build our regression model and initialize its parameters.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "id": "1RLOFy-jdA_0"
      },
      "outputs": [],
      "source": [
        "def sigmoid(x, Beta_1, Beta_2):\n",
        "     y = 1 / (1 + np.exp(-Beta_1*(x-Beta_2)))\n",
        "     return y"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FM8djNBidA_0"
      },
      "source": [
        "Lets look at a sample sigmoid line that might fit with the data:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "id": "XTF9p20NdA_0",
        "outputId": "5c201e24-95ca-47cb-9bf0-84ac9996ce3e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 293
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f0476e37750>]"
            ]
          },
          "metadata": {},
          "execution_count": 12
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "beta_1 = 0.10\n",
        "beta_2 = 1990.0\n",
        "\n",
        "#logistic function\n",
        "Y_pred = sigmoid(x_data, beta_1 , beta_2)\n",
        "\n",
        "#plot initial prediction against datapoints\n",
        "plt.plot(x_data, Y_pred*15000000000000.)\n",
        "plt.plot(x_data, y_data, 'ro')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zOlNTQe9dA_0"
      },
      "source": [
        "Our task here is to find the best parameters for our model. Lets first normalize our x and y:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "id": "ZKiWm27CdA_0"
      },
      "outputs": [],
      "source": [
        "# Lets normalize our data\n",
        "xdata =x_data/max(x_data)\n",
        "ydata =y_data/max(y_data)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kkhb9UVBdA_1"
      },
      "source": [
        "#### How we find the best parameters for our fit line?\n",
        "\n",
        "we can use **curve_fit** which uses non-linear least squares to fit our sigmoid function, to data. Optimize values for the parameters so that the sum of the squared residuals of sigmoid(xdata, \\*popt) - ydata is minimized.\n",
        "\n",
        "popt are our optimized parameters.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {
        "id": "cmik_hEHdA_1",
        "outputId": "5b943762-2971-4190-be49-692ea0a16604",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            " beta_1 = 690.453019, beta_2 = 0.997207\n"
          ]
        }
      ],
      "source": [
        "from scipy.optimize import curve_fit\n",
        "popt, pcov = curve_fit(sigmoid, xdata, ydata)\n",
        "#print the final parameters\n",
        "print(\" beta_1 = %f, beta_2 = %f\" % (popt[0], popt[1]))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TZdd5PXcdA_1"
      },
      "source": [
        "Now we plot our resulting regression model.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {
        "id": "F5n5sBmxdA_1",
        "outputId": "816b1583-4d86-4165-d269-f9aa65be64e5",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 334
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 576x360 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "x = np.linspace(1960, 2015, 55)\n",
        "x = x/max(x)\n",
        "plt.figure(figsize=(8,5))\n",
        "y = sigmoid(x, *popt)\n",
        "plt.plot(xdata, ydata, 'ro', label='data')\n",
        "plt.plot(x,y, linewidth=3.0, label='fit')\n",
        "plt.legend(loc='best')\n",
        "plt.ylabel('GDP')\n",
        "plt.xlabel('Year')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZkWlEPqQdA_1"
      },
      "source": [
        "## Practice\n",
        "\n",
        "Can you calculate what is the accuracy of our model?\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {
        "id": "1T8EwFQQdA_1"
      },
      "outputs": [],
      "source": [
        "# write your code here\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bK9RtJjcdA_2"
      },
      "source": [
        "<details><summary>Click here for the solution</summary>\n",
        "\n",
        "```python\n",
        "# split data into train/test\n",
        "msk = np.random.rand(len(df)) < 0.8\n",
        "train_x = xdata[msk]\n",
        "test_x = xdata[~msk]\n",
        "train_y = ydata[msk]\n",
        "test_y = ydata[~msk]\n",
        "\n",
        "# build the model using train set\n",
        "popt, pcov = curve_fit(sigmoid, train_x, train_y)\n",
        "\n",
        "# predict using test set\n",
        "y_hat = sigmoid(test_x, *popt)\n",
        "\n",
        "# evaluation\n",
        "print(\"Mean absolute error: %.2f\" % np.mean(np.absolute(y_hat - test_y)))\n",
        "print(\"Residual sum of squares (MSE): %.2f\" % np.mean((y_hat - test_y) ** 2))\n",
        "from sklearn.metrics import r2_score\n",
        "print(\"R2-score: %.2f\" % r2_score(test_y,y_hat) )\n",
        "\n",
        "```\n",
        "\n",
        "</details>\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TY_XAsFFdA_2"
      },
      "source": [
        "<h2>Want to learn more?</h2>\n",
        "\n",
        "IBM SPSS Modeler is a comprehensive analytics platform that has many machine learning algorithms. It has been designed to bring predictive intelligence to decisions made by individuals, by groups, by systems – by your enterprise as a whole. A free trial is available through this course, available here: <a href=\"https://www.ibm.com/analytics/spss-statistics-software?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2022-01-01\">SPSS Modeler</a>\n",
        "\n",
        "Also, you can use Watson Studio to run these notebooks faster with bigger datasets. Watson Studio is IBM's leading cloud solution for data scientists, built by data scientists. With Jupyter notebooks, RStudio, Apache Spark and popular libraries pre-packaged in the cloud, Watson Studio enables data scientists to collaborate on their projects without having to install anything. Join the fast-growing community of Watson Studio users today with a free account at <a href=\"https://www.ibm.com/cloud/watson-studio?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2022-01-01\">Watson Studio</a>\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "51bbWbz3dA_2"
      },
      "source": [
        "### Thank you for completing this lab!\n",
        "\n",
        "## Author\n",
        "\n",
        "Saeed Aghabozorgi\n",
        "\n",
        "### Other Contributors\n",
        "\n",
        "<a href=\"https://www.linkedin.com/in/joseph-s-50398b136/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2022-01-01\" target=\"_blank\">Joseph Santarcangelo</a>\n",
        "\n",
        "## Change Log\n",
        "\n",
        "| Date (YYYY-MM-DD) | Version | Changed By | Change Description                 |\n",
        "| ----------------- | ------- | ---------- | ---------------------------------- |\n",
        "| 2020-11-03        | 2.1     | Lakshmi    | Made changes in URL                |\n",
        "| 2020-08-27        | 2.0     | Lavanya    | Moved lab to course repo in GitLab |\n",
        "|                   |         |            |                                    |\n",
        "|                   |         |            |                                    |\n",
        "\n",
        "## <h3 align=\"center\"> © IBM Corporation 2020. All rights reserved. <h3/>\n"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3 (ipykernel)",
      "language": "python",
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