NonLinearRegression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center>\n",
    "    <img src=\"https://cf-courses-data.s3.us.cloud-object-storage.appdomain.cloud/IBMDeveloperSkillsNetwork-ML0101EN-SkillsNetwork/labs/Module%202/images/IDSNlogo.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 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": {},
   "source": [
    "<h2 id=\"importing_libraries\">Importing required libraries</h2>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": {},
   "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 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": {},
   "source": [
    "Let's take a look at a cubic function's graph.\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 = 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": 4,
   "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": 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.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 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": null,
   "metadata": {},
   "outputs": [],
   "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": 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",
    "\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": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2021-07-13 06:28:53 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": 7,
     "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": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(6,6))\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": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(5,5))\n",
    "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": 22,
   "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": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7ffb71f26208>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "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": 27,
   "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": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " beta_1 = 690.451715, 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": {},
   "source": [
    "Now we plot our resulting regression model.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "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": [
    "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": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean absolute error: 0.15\n",
      "Residual sum of squares (MSE): 0.10\n",
      "R2-score: -3592588248904261089361920.00\n"
     ]
    }
   ],
   "source": [
    "# write your code here\n",
    "\n",
    "\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) )"
   ]
  },
  {
   "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?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2021-01-01\">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-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2021-01-01\">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/?utm_medium=Exinfluencer&utm_source=Exinfluencer&utm_content=000026UJ&utm_term=10006555&utm_id=NA-SkillsNetwork-Channel-SkillsNetworkCoursesIBMDeveloperSkillsNetworkML0101ENSkillsNetwork20718538-2021-01-01\" 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"
   ]
  }
 ],
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