Created on Skills Network Labs

ML0101EN-Reg-NoneLinearRegression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center>\n",
    "    <img src=\"https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/Logos/organization_logo/organization_logo.png\" width=\"300\" alt=\"cognitiveclass.ai logo\"  />\n",
    "</center>\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": {},
   "source": [
    "If the data shows a curvy trend, then linear regression will not produce very accurate results when compared to a non-linear regression because, as the name implies, linear regression presumes that the data is linear. \n",
    "Let's learn about non linear regressions and apply an example on 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": {},
   "source": [
    "<h2 id=\"importing_libraries\">Importing required libraries</h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Though Linear regression is very good to solve many problems, it cannot be used for all datasets. First recall how linear regression, could model a dataset. It models a linear relation between a dependent variable y and independent variable x. It had a simple equation, of degree 1, for example y = $2x$ + 3.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "source": [
    "Non-linear regressions are a relationship between independent variables $x$ and a dependent variable $y$ which result in a non-linear function modeled data. 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 $x$). \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 others. For example: $$ y = \\log(x)$$\n",
    "\n",
    "Or even, more complicated such as :\n",
    "$$ y = \\log(a x^3 + b x^2 + c x + d)$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at a cubic function's graph.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "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": {},
   "source": [
    "Some other types of non-linear functions are:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quadratic\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ Y = X^2 $$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "source": [
    "### Exponential\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "source": [
    "### Logarithmic\n",
    "\n",
    "The response $y$ is a results of applying logarithmic map from input $x$'s to output variable $y$. 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 polynomial representation of the $x$'s. In general form it would be written as  \n",
    "\\begin{equation}\n",
    "y = \\log(X)\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/jupyterlab/conda/envs/python/lib/python3.6/site-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"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "source": [
    "### Sigmoidal/Logistic\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ Y = a + \\frac{b}{1+ c^{(X-d)}}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "source": [
    "<a id=\"ref2\"></a>\n",
    "\n",
    "# Non-Linear Regression example\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2021-02-14 07:32:33 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"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <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",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "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"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": {},
   "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": {},
   "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 decelerate slightly in the 2010s.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "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": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "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": {},
   "source": [
    "### Building The Model\n",
    "\n",
    "Now, let's build our regression model and initialize its parameters. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "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": {},
   "source": [
    "Lets look at a sample sigmoid line that might fit with the data:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0274631780>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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TTNa0JRw/6WHWbSMZlNoq3CVJMMK4VK4/6qGLhNnq7Yf4wQtfUO6Bv9+uMI8qgZbErYelcv1RoIuEUc7WIib89QuSE+OZc8e59OnYItwlSU2Ecalcf4IKdDMbZ2brzCzfzB7y83pLM/uHma0ws1wzmxj6UkViy+f5+7h+2pe0bdaI1+84l25tm4a7JKmpMC6V648556rewSweWA9cCBQCS4EJzrm8Cvs8DLR0zj1oZu2AdUBH51xpoONmZma67OzsEDRBJPp8vHY3d8xcRreUprxy63DaN28c7pIkSphZjnMu099rwfTQhwP5zrlNvoCeDVxZaR8HNDfvNcnNgCKgrBY1i8Ss91buZPKMHM7q0JzZk0cqzCVkggn0LkBBheeFvm0VPQucDewAVgH3Oec8lQ9kZpPNLNvMsvfu3XuGJYtEr7k5hdz72jIGpbZi1m0jaN00KdwlSQwJJtD9rQRUeZzmYmA50BkYBDxrZqed3XHOTXXOZTrnMtu1a1fDUkWi2ytfbOX+OSsY1aMtM24ZTgutZS4hFkygFwKpFZ53xdsTr2gi8Kbzygc2A31CU6JI9Pvrwk38/O3VXNCnPdNuyqRJki4BkdALJtCXAr3MrJuZJQHXAvMq7bMNuADAzDoAZwGbQlmoSDRyzvHURxt4bP4aLh3QiSk3DKVxom5MIXWj2m6Cc67MzO4BFgDxwHTnXK6Z3eF7fQrwG+AlM1uFd4jmQefcvjqsWyTiOef4/QdreeGTTVw1pCtPXD1AN6aQOhXUPHTn3HznXG/nXA/n3GO+bVN8YY5zbodz7iLnXH/n3DnOuZl1WbRIpPN4HI/Oy+WFTzZx/cg0/qAwj34RtKpiIBrIEwmxco/j4TdX8ffsAm77VjceHn+27jIU7SJsVcVAdOm/SAidLPfw339fzt+zC/jRd3oqzGNFhK2qGIh66CIhcqKsnHtf/Yp/5u3mwXF9uHNsj3CXJKESYasqBqIeukgIHCstZ/KMHP6Zt5tHL++rMI81EbaqYiAKdJFaOnKijIkvfcnCDXt5/Kr+3Dy6W7hLklCLsFUVA1Ggi9TCoWMnufHFJSzdcoAnfziIHw6LrB6bhEiEraoYiMbQRc5Q0dFSbpy+hHW7innuusGMO6dTuEuSupSVFXEBXpkCXeQM7Dl8nCzfzZyn3qCbOUtkUKCL1FDhgRKypi1hb/EJ/jZxGKN6tA13SSKAAl2kRjbvO0rWX7+g+EQZM28dwZC01uEuSeT/KNBFgrRuVzFZ05bgcY7XbhvJOV1ahrskkW9QoIsEYVXhIW6cvoTE+Dheu20kvTo0D3dJIqdRoItU48vNRUx6aSktkxN59bYRpKfoZs4SmRToIlX4z7o93DEzh86tkpl5ywg6t0oOd0kiASnQRQJ4f9VOfjT7K3q1b86MW4bTtlmjcJckUiUFuogfc3MKeWDuCgaltuJvE4fTMln3/5TIp0AXqeTlRVv45bxcRvdMYeoNmTRtpF8TiQ76SRXxcc7x7Mf5/OnD9VzYtwPPTBis+39KVFGgi+AN88feW8O0zzbz/cFdeOLqASTGa+06iS76iZUGr6zcwwNzVzLts83cPCqDP10zUGHekEXBvUMDCeqn1szGmdk6M8s3s4cC7DPWzJabWa6ZfRLaMkXqxomycu559Svm5BRy3wW9+OXlfYnTzZwbrlP3Dt26FZz7+t6hURLq5pyregezeGA9cCFQCCwFJjjn8irs0wpYBIxzzm0zs/bOuT1VHTczM9NlZ2fXsnyRM3f0RBm3v5LDZ/n7+MVlfZl0nm5M0eBlZHhDvLL0dNiypb6r8cvMcpxzmf5eC2YMfTiQ75zb5DvYbOBKIK/CPtcBbzrntgFUF+Yi4XbgaCkTX1rKqu2H+OM1A7l6aNdwlySRIEruHRpIMEMuXYCCCs8Lfdsq6g20NrP/mFmOmd3o70BmNtnMss0se+/evWdWsUgtbT94jKunLCJv52H+kjVEYS5fi5J7hwYSTKD7G1CsPE6TAAwFLgUuBn5uZr1P+yLnpjrnMp1zme3atatxsSK1tWF3MVf/ZRF7ik/wyqThXNSvY7hLkkgSJfcODSSYQC8EUis87wrs8LPPB865o865fcBCYGBoShQJjZytB7jmhcWUeRyv334uI7qnhLskiTRRcu/QQIIJ9KVALzPrZmZJwLXAvEr7vAN8y8wSzKwJMAJYE9pSRc7cv9ftIWvaF7RKTuTNO0dxdqcW4S5JIlVWlvcEqMfj/RwlYQ5BnBR1zpWZ2T3AAiAemO6cyzWzO3yvT3HOrTGzD4CVgAeY5pxbXZeFiwTrzWWFPDB3JWd1bM5LE4fTrrkW2ZLYVO20xbqiaYtS15xz/OWTjTzxwTpG9UjhhRuG0ryxFtmS6FbbaYsiUafc4/jVP3KZsXgrVwzszB+vGUhSgq7+lNimQJeYc/xkOffN/ooFubu5/fzuPDiuj67+lAZBgS4x5WBJKbe+nE3OtgO6+lMaHAW6xIyCohJu/tuXFBQd45kJg7lsQOdwlyRSrxToEhOWFxzk1peXUlrmYcYtwxmpOebSAOkskUS9D1bv4tqpi0lOiufNu0YpzCV4UbxUrj/qoUvUcs4x/fMt/Pa9PAZ2bcW0mzJ1I2cJ3qmlcktKvM9PLZULUXUxUUXqoUtUKvc4Hp2Xy2/ezWNcv47MnjxSYS4188gjX4f5KSUl3u1RSj10iTrFx09y3+zlfLx2D5PP785DmpYoZyLKl8r1R4EuUaWgqIRbX84mf+8Rfvu9c7h+ZHq4S5JolZbm/2YWUbJUrj8acpGosXRLEd977nN2HjrGjEnDFeZSO1G+VK4/CnSJCnNzCsn66xJaJCfy9t2jGd2zbbhLkmgX5Uvl+qNAl4jm8Th+//5a7p+zgmHdWvP2XaPp3q5ZuMuSaBNoemIUL5Xrj8bQJWIdPn6SH/tOfmaNSOPRK/qRGK8+iNRQDE5PDES/HRKR8vcc4XvPfs7C9Xv59ZX9+O33zlGYy5mJwemJgaiHLhHno7zd/Pjvy2mUEMesW0foVnFSOzE4PTEQBbpEDI/H8ey/8/nzh+vp36UlL9wwlM6tksNdlkS7GJyeGIjew0pEKD5+kjtn5fDnD9fzX4O7MOeOcxXmEhoxOD0xEPXQJezW7jrMnTOXsa2ohJ9f1pdJozMw05WfEiKnTnw+8oh3mCUtzRvmMXZCFBToEmZvLivk4bdW0aJxIq/dNpLh3dqEuySJRVlZMRnglQU15GJm48xsnZnlm9lDVew3zMzKzezq0JUosehEWTmPvLWK//f6CgZ2bcW7PzpPYS5SS9X20M0sHngOuBAoBJaa2TznXJ6f/R4HFtRFoRI7Cg+UcPesZawoPMQdY3pw/0W9SdCURJFaC2bIZTiQ75zbBGBms4ErgbxK+90LvAEMC2mFElM+WL2LB+auwDl44YahXNyvY7hLEokZwQR6F6CgwvNCYETFHcysC/B94DtUEehmNhmYDJAWg1OGJLDjJ8v53fw1zFi8lQFdW/LMhMGkpzQNd1kiMSWYQPc33cBVev4k8KBzrryq2QnOuanAVIDMzMzKx5AYtXHvEe559SvW7DzMred144FxfUhK0BCLSKgFE+iFQGqF512BHZX2yQRm+8K8LTDezMqcc2+HokiJXm/kFPLzd1bTKCGO6Tdn8p0+HcJdkkjMCibQlwK9zKwbsB24Friu4g7OuW6nHpvZS8C7CvOG7VDJSX72zmr+sWIHI7q14alrB9OxZeNwlyUS06p93+ucKwPuwTt7ZQ3wunMu18zuMLM76rpAiT6L8vcx7qmFvL9qJ/9zYW9evW2kwlzqR6BlchuIoC4scs7NB+ZX2jYlwL43174siUYnysr544J1/PXTzXRv25Q37xrFgK6twl2WNBQNaJncQMy58JybzMzMdNnZ2WH53hJ6a3cd5sezl7N2VzHXj0zj4fFn0yRJFyJLPcrI8L8IV3q69+YVMcLMcpxzmf5e02+c1EpZuYcXFm7iqY820CI5QSc+JXwa0DK5gSjQ5Yyt3XWYn8xZyarth7h0QCd+fUU/Upo1CndZ0lA1oGVyA1GgS42dLPcw5T8befrjDbRonMjzWUMY379TuMuShu6xx745hg4xu0xuIAp0qZHcHYd4YO5Kcncc5vKBnfnVFf1o0zQp3GWJNKhlcgNRoEtQSkrLePKjDbz42WZaN0lkyvVDGHeOeuUSYRrIMrmBKNClWv9eu4efvb2a7QePMWF4Kg+O60OrJuqVi0QaBboEtOfwcX71bh7vrdxJz/bNeP32c7VmuUgEU6DLacrKPcxYvJX//Wg9J8o8/M+Fvbl9TA8tqCUS4fQbKt+weON+Ln36M379bh6DUlvxwX3f4t4LeinMJfI08Mv8/VEPXQDYcfAYj81fw3srd9K1dTIv3DCUi/p20M2aJTLpMn+/dOl/A3estJxpn27i+f9sxOMcd43tye1jutM4MT7cpYkE1kAu8/dHl/7Laco9jre+2s4fF6xj1+HjjOvXkUcuPZvUNk3CXZpI9XSZv18K9Abosw37+N38NeTtPMzAri15esJgzV6R6KLL/P1SoDcga3Ye5vEP1vKfdXvp0iqZpycM5rL+nYiL0zi5RBld5u+XAr0B2LT3CP/70Qb+sWIHzRsn8PD4Ptx4bobGySV66TJ/vxToMazwQAlP/2sDbyzbTlJ8HHeN7cHk87vrKk+JLrNm+Q/uBn6Zvz8K9Bi04+Axpnyykde+3IaZcdO5Gdw5tgftmmtpW4kymp5YI5q2GEO27S/hL5/kMzenEOfgmsxU7v1OTzq3Sg53aSJnpgFPTwxE0xZjXP6eYp7/90beWbGD+Djj2mFp3D6mO11bawqiRDlNT6yRoALdzMYBTwHxwDTn3O8rvZ4FPOh7egS40zm3IpSFyulyth5g2qeb+CB3F40T4pk0OoPbvtWd9i0ah7s0kdDQ9MQaqTbQzSweeA64ECgElprZPOdcXoXdNgNjnHMHzOwSYCowoi4KbujKPY4P83YxdeEmlm07SMvkRO4e25NJ53XTjSYkuvk7+anpiTUSTA99OJDvnNsEYGazgSuB/wt059yiCvt/AXQNZZECR0+UMTenkOmfb2br/hJS2yTzqyv6cfXQrjRtpJEziXKBTn5Oner90PTEoASTBF2AggrPC6m6930L8L6/F8xsMjAZIE1vmYKSv+cIryzewhvLtnPkRBmD01rx0Lg+XNSvI/G6IEhixSOPfLMXDt7njzziPfmpAA9KMIHuLzX8To0xs2/jDfTz/L3unJuKdziGzMzM8EyviQJl5R4+WrOHV77Ywuf5+0mKj+PSAZ244dx0hqS1Dnd5IqGnk58hEUygFwKpFZ53BXZU3snMBgDTgEucc/tDU17DUlBUwuvZBczNKWTnoeN0btmYn1x8Fj8clkrbZppDLjGk8nh5mzaw309s6J18jQQT6EuBXmbWDdgOXAtcV3EHM0sD3gRucM6tD3mVMez4yXIW5O7i9ewCPs/fjxmc36sdj17Rjwv6tCchXjeWkBjjb7w8MRGSkqC09Ov9dPKzxqoNdOdcmZndAyzAO21xunMu18zu8L0+BfgFkAI877shQlmgie8CzjmWbTvIO8u3887yHRw6dpIurZL5fxf25uqhXXUhkMQ2f+PlJ09CSgo0a6aTn7UQ1PQI59x8YH6lbVMqPL4VuDW0pcWeTXuP8PbyHbyzfDtb95fQKCGOi/p15IeZqYzqkaJVDyX2+JuKGGhcvKgI9u2r3/pijOa71bHCAyW8v2oX767cwYrCQ5jB6B5tufc7vbi4XweaN04Md4kioVE5vMePh5dfPn0qosbL64wCvQ4UFJUwf9VO5q/ayYrCQwCc06UFj4w/m8sHdqZjS13JKTHG37j4lClQea2okhJITvaOj+tioZBToIeAc468nYf5KG8PH67ZxerthwHo36UlD47rw/j+HUlPaRrmKkVCxN8wir9x8UAL/xUVwSuv6GKhOqDVFs/QibJylmwq4qM1u/kobzc7Dh3HDAantuLifh0Z37+T7s8psadyTxxO721XpwGvlBgKWm0xRLbuP8on6/fyybq9LNq4n2Mny2mcGMe3erXjx9/tzbf7tNea4xI7gu2Jl5RAfDyUl59+DLNv9tQ1tFKnFOhVOFhSyhebili0cR8L1+9ly37vD3JamyZck9mVMb3bMbpnW93KTaJfsCc0A/XEy8v9j4vfdBPMn6+hlXqiQK/gyIkylm4pYvHG/SzauI/cHYdxDpIT4zm3RwoTR3djTO92ZLTVeLhEKX+9bgj+hGagnnh6+tc9eIV32DToMfS9xSfI3lLE0i0HWLqliLydhyn3OJLi4xic1opRPdoyqmcKA7u2IilBV2xKFAkmuMHbi05O9j+NMBB/PfGpUxXe9URj6HgXvFq7q5ivCg6yfNtBvtp2gE37jgLQKCGOQamtuGtsD4Z3a0NmehuSkzSMIlEq0FK0ycn+x79rekJTPfGIFZOB7vE4thaVsGr7IVZvP8TybQdZuf0gx096AEhpmsTgtFb8YFgqwzLa0L9LS/XAJXpV7o0fOVL74IbAJzSzshTgESrqA/1kuYeNe4+wZudhcrcfZtX2Q+TtOEzxiTIAkuLj6NelBROGpzEotRVD0lrTtXUyvjVnRKKbv954TaWkwLFjOqEZA6Iu0LfsO8qHebtZs/Mwa3YVk7+nmJPl3l5EUkIcZ3dqwZWDO9O/S0vO6dKS3h2ak6gVCyUWBDuNMJBAwf3UU97HGkaJelEX6Gt3FfPY/DV0aNGIPh1bMKZ3O87u1JyzO7WgW9umCm+JTYHGxYMN82CCWwEe9aJulsux0nKOnSzXDZElNgSajeKvJ+5vOCXQNEItRRuzqprlEnWBLhLxgg1pOL2XnZjoPRlZ+UYPVfXENY2wQakq0DU+IbFt1izIyIC4OO/nWbOq3l7bY5waGtm61TtDZOtWmDgRJk365rbJk+G++/zf6KFimMPXF/T4k57uDe/0dO8fglPPFeYNk3MuLB9Dhw51Imdk5kzn0tOdM/N+njnT//Y773SuSRPnvDHq/WjSJPD2mTNrf4yUlG9uC+VHoJqlQQGyXYBcVaA3dMGGo7+wq2rfUBwj0LZgA9bMfzDGx/vfnpJS+2PU1cep9vv7t5MGRYEeaqH4xYqEIK1JOCYmOpeUFNy+oThGoH0D9YDrO2BD8eHvD0igdiu8xafWgQ6MA9YB+cBDfl434Gnf6yuBIdUd84wCvb57iP621+TteiiGAuoySEMRjoH2DcUx6uojFN+vJr38qkK6Jj+jIs7VLtCBeGAj0B1IAlYAfSvtMx543xfsI4El1R23xoFek95kXYZjoLfg/n6RQzUUUFdBGksfgdpd+d/6TMa/a3IMhbTUsdoG+rnAggrPfwr8tNI+LwATKjxfB3Sq6rg1DvT09LoLtroMx2gM2EjuodfkD+ep7TV591bbY4jUsdoG+tXAtArPbwCerbTPu8B5FZ7/C8is6rg1DvRAvdpY+qjvIK1JOEbKGPqZDJvVhEJaIlxtA/0aP4H+TKV93vMT6EP9HGsykA1kp6Wl1awVkdRD9/cWvKbj0cG+ja/LIA3FrJNA+4biGApXkdPExpBLpIyhB3oLfib1RUKQikhUqW2gJwCbgG4VTor2q7TPpZVOin5Z3XGjdpZLVUFYl0MBIiKu6kAPai0XMxsPPOmb8TLdOfeYmd0B4JybYt7FxZ/FO72xBJjonKtyoRat5SIiUnO1vgWdc24+ML/StikVHjvg7toUKSIitaPFuUREYoQCXUQkRijQRURihAJdRCRGhO2ORWa2FziDW5QD0BbYF8JyIlGstzHW2wex30a1LzzSnXPt/L0QtkCvDTPLDjRtJ1bEehtjvX0Q+21U+yKPhlxERGKEAl1EJEZEa6BPDXcB9SDW2xjr7YPYb6PaF2GicgxdREROF609dBERqUSBLiISIyIm0M1supntMbPVFbYNNLPFZrbKzP5hZi0qvDbA91qu7/XGvu1Dfc/zzexp30qQYVeT9plZlpktr/DhMbNBvtcisn1Q4zYmmtnLvu1rzOynFb4mIttYw/YlmdnffNtXmNnYCl8Tqe1LNbN/+/4/cs3sPt/2Nmb2oZlt8H1uXeFrfuprxzozu7jC9ohrY03bZ2Ypvv2PmNmzlY4Vce0Dql8Pvb4+gPOBIcDqCtuWAmN8jycBv3Ffr9G+Ehjoe54CxPsef4n3phyGd432S8Ldtpq2r9LX9Qc2VXgeke07g//D64DZvsdNgC1ARiS3sYbtuxv4m+9xeyAHiIvw9nUChvgeNwfWA32BJ4CHfNsfAh73Pe6L9/4IjfDeL2FjJP8enkH7mgLnAXdw+m03I659zrnI6aE75xYCRZU2nwUs9D3+ELjK9/giYKVzboXva/c758rNrBPQwjm32Hn/1WcA36vz4oNQw/ZVNAF4DSCS2wc1bqMDmppZApAMlAKHI7mNNWxfX7y3YsQ5twc4CGRGePt2OueW+R4XA2uALsCVwMu+3V7m63qvxPtH+YRzbjOQDwyP1DbWtH3OuaPOuc+A4xWPE6ntgwgacglgNXCF7/E1QKrvcW/AmdkCM1tmZg/4tncBCit8faFvW6QK1L6Kfogv0Im+9kHgNs4FjgI7gW3AH51zRURfGwO1bwVwpZklmFk3YKjvtahon5llAIOBJUAH59xO8IYi3ncc4K27oMKXnWpLxLcxyPYFErHti/RAnwTcbWY5eN8ilfq2J+B9K5Tl+/x9M7sA79ufyiJ5Xmag9gFgZiOAEufcqTHbaGsfBG7jcKAc6Iz37fr/mFl3oq+Ngdo3He8vejbeu30tAsqIgvaZWTPgDeDHzrnDVe3qZ5urYntEqEH7Ah7Cz7aIaF9QdywKF+fcWrzDK5hZb7z3LgXvL8onzrl9vtfm4x3bnAl0rXCIrsCOeiu4hqpo3ynX8nXvHLztjpr2QZVtvA74wDl3EthjZp8DmcCnRFEbA7XPOVcG/Pep/cxsEbABOEAEt8/MEvGG3Szn3Ju+zbvNrJNzbqdvuGGPb3sh33xXeaotEftzWsP2BRKx7YvoHrqZtfd9jgN+Bpy67d0CYICZNfGNwY4B8nxvl4rNbKTvrPONwDthKD0oVbTv1LZrgNmntkVb+6DKNm4DvmNeTfHeXHxttLUxUPt8P5tNfY8vBMqccxH9M+qr50VgjXPuzxVemgfc5Ht8E1/XOw+41swa+YaVeuG9QXxEtvEM2udXpLYPiKhZLq/hHU89ifcv4C3AfXjPRK8Hfo/vylbf/tcDuXjHMJ+osD3Tt20j3htXW323JUTtGwt84ec4Edm+mrYRaAbM8f0f5gE/ifQ21rB9GcA6vCfePsK75Gmkt+88vEMHK4Hlvo/xeGeR/QvvO4x/AW0qfM0jvnaso8JMj0hs4xm2bwveE+FHfP/nfSO1fc45XfovIhIrInrIRUREgqdAFxGJEQp0EZEYoUAXEYkRCnQRkRihQBcRiREKdBGRGPH/ARnGfxuWiNipAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "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": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Lets normalize our data\n",
    "xdata =x_data/max(x_data)\n",
    "ydata =y_data/max(y_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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. Optimal 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": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "source": [
    "Now we plot our resulting regression model.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "source": [
    "## Practice\n",
    "\n",
    "Can you calculate what is the accuracy of our model?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean absolute error: 1.74\n",
      "Residual sum of squares (MSE): 4.39\n",
      "R2-score: 0.92\n"
     ]
    }
   ],
   "source": [
    "# write your code here\n",
    "\n",
    "from sklearn.metrics import r2_score\n",
    "\n",
    "print(\"Mean absolute error: %.2f\" % np.mean(np.absolute(ydata - y)))\n",
    "print(\"Residual sum of squares (MSE): %.2f\" % np.mean((ydata - y) ** 2))\n",
    "print(\"R2-score: %.2f\" % r2_score(y,ydata) )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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(y_hat , test_y) )\n",
    "\n",
    "```\n",
    "\n",
    "</details>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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\">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\">Watson Studio</a>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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/\" 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"
   ]
  }
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