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Created on Skills Network Labs
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a href=\"https://www.bigdatauniversity.com\"><img src = \"https://ibm.box.com/shared/static/cw2c7r3o20w9zn8gkecaeyjhgw3xdgbj.png\" width=\"400\" align=\"center\"></a>\n",
"\n",
"<h1><center>Non Linear Regression Analysis</center></h1>"
]
},
{
"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."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h2 id=\"importing_libraries\">Importing required libraries</h2>"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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."
]
},
{
"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('Indepdendent 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",
"$$ y = \\log(a x^3 + b x^2 + c x + d)$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's take a look at a cubic function's graph."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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('Indepdendent 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."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Some other types of non-linear functions are:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Quadratic"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ Y = X^2 $$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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('Indepdendent Variable')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exponential"
]
},
{
"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",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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('Indepdendent 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}"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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('Indepdendent Variable')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Sigmoidal/Logistic"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$ Y = a + \\frac{b}{1+ c^{(X-d)}}$$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"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('Indepdendent Variable')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<a id=\"ref2\"></a>\n",
"# Non-Linear Regression example"
]
},
{
"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. "
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2020-05-14 06:39:10 URL:https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/ML0101ENv3/labs/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": 8,
"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://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/ML0101ENv3/labs/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)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Plotting the Dataset ###\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."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"data": {
"image/png": 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7vgBFkrT1Oh2Yn4eVlWL98OFiHXzpyRayJy5J2nr79h0L8FUrK0W5towhLknaektL6yvXhhjikqStNz29vnJtiCEuSdp6CwswNXV82dRUUa4tY4hLkrZeqwXtNszMQESxbLed1LbFnJ0uSapGq2VoV8yeuCRJDWWIS5LUUIa4JEkNZYhLkjbOR6vWyoltkqSN8dGqtbMnLknaGB+tWjtDXJK0MT5atXaGuCRpY3y0au0McUnSxvho1dpVGuIRcX5E3B0RhyLi8j7bWxFxW/n5UkScXWV9JElbyEer1q6y2ekRcQLwPuAVwDJwU0Rcm5l3du32LeBlmfm9iLgAaAMvqKpOkqQt5qNVa1VlT/xc4FBm3pOZDwNXA3u7d8jML2Xm98rVrwCnVVgfSdJ6eR/4jlblfeKnAvd2rS8zuJd9CfDZCusjSVoP7wPf8arsiUefsuy7Y8SvUIT4W9fYPh8RixGxeOTIkS2soiRpTd4HvuNVGeLLwOld66cB9/XuFBHPAT4I7M3M7/Y7UWa2M3MuM+f27NlTSWUlST28D3zHqzLEbwLOjIgzIuJE4ELg2u4dImIa+BTwW5n5jQrrIklaL+8D3/EqC/HMPApcBlwP3AX8r8w8GBGXRsSl5W7/AXgK8P6IuCUiFquqjyRpnbwPfMeLzL6XqXesubm5XFw06yVpW3Q6xTXwpaWiB76w4KS2GkTEgcyc6y33iW2SNOkG3UbWasG3vw2PPFIsDfAdxVeRStIk8zayRrMnLkmTzNvIGs0Ql6RJ5m1kjWaIS9Ik8zayRjPEJWncDZq45m1kjWaIS9I4W524dvgwZB6buLYa5L5OtNG8T1ySxtnsbBHcvWZmilvG1AjeJy5J42ytIXMnro017xOXpKYbdK/39HT/nrgT18aCPXFJarpB93o7cW2sGeKS1ASDZpgPGjJ34tpYczhdkna6YY9GHTZk3moZ2mPKnrgk7RRr9baHPRrVIfOJZU9cknaCQb3tYTPMV3vZvjJ04tgTl6TtMui69qDe9iiPRvWVoRPJEJek7TDsyWmDetsOl2sNhrgkbaWNXtce1Nt2hrnW4DVxSdoqm7muvbBw/LFwfG/bGebqw564JK1HVde17W1rA+yJS9Koht2vPai3/Sd/MrinvXoOQ1vrYE9ckkbldW3tMIa4JPXa6BvBhs0i9zYwbTGH0yWp22beCOZDV7TNIjPrrsO6zM3N5eLiYt3VkDSuZmf7B/XMzNozyB0WV8Ui4kBmzvWWO5wuSd18I5gaxBCXNHkG3SY2yq1gXtfWDmGIS5oswx5/6iNO1SCGuKTJMuw2MYfM1SCGuKRmGjQkPmjbsNvEwCFzNYYhLmlnGhbSaw2JDxsuH+W1nlJDGOKSqjMoiAdtHxbEg4bEhw2Xe81b4yQzG/U555xzUtI6XXVV5sxMZkSxvOqq9W3fyLmvuipzaiqziOHiMzU12vaZmePLVz8zM8WxEf23RwzethU/r1QDYDH7ZGLtobzejyEuraGKMB103mHHDgviQduHBfGgY4f9uVIDGeJS09URpsMCfjNBPGj7sDoPqtewOksNZIhL3aoYPh5l+0aP3alhupljN/OPh822s9Qwhrh2pp0YiJs5djM9xKquEdc1rF3lML40YQxxbU4dYVpXIFY1NN3EMN1MW23md0PScQzxXpv9y6WKUNupx9YVpjsxEDczNF3XNeLN/G5sdlhb0paoJcSB84G7gUPA5X22B/DecvttwPOGnXNLQnwrhvmqCLWdemxdYVpXIFY1NF3nNeLNMKSl2m17iAMnAN8EngGcCNwKnNWzz6uAz5Zh/kLgq8POuyUhvpm/TIdtH8dj6wrTugKxqqFprxFL2qA6QvyXgOu71t8GvK1nnw8AF3Wt3w2cMui8WxLimwmeYdvH8di6wrSuQKzyOq/XiCVtQB0h/jrgg13rvwVc0bPPZ4AXd61/AZgbdF574jUcW2fvsq5ANEwl7SB1hPjr+4T4H/Xs8+d9QvycPueaBxaBxenp6c23htfE13fs6vH2LiWpFg6n99ps8NQxS7zOYyVJtVkrxKPYtvUiYjfwDeDlwN8BNwEXZ+bBrn1eDVxGMcHtBcB7M/PcQeedm5vLxcXFSuosSdJOFBEHMnOut3x3VX9gZh6NiMuA6ylmqu/PzIMRcWm5/UrgOooAPwSsAG+qqj6SJI2bykIcIDOvowjq7rIru74n8OYq6yBJ0rjaVXcFJEnSxhjikiQ1lCEuSVJDGeKSJDWUIS5JUkMZ4pIkNVRlD3upSkQcAQ5v4SlPBr6zhecbd7bX6Gyr0dlWo7OtRjdObTWTmXt6CxsX4lstIhb7PQVH/dleo7OtRmdbjc62Gt0ktJXD6ZIkNZQhLklSQxni0K67Ag1je43OthqdbTU622p0Y99WE39NXJKkprInLklSQ41liEfE/oh4ICLu6Co7OyK+HBG3R8SfRcQTu7Y9p9x2sNz+M2X5OeX6oYh4b0REHT9PldbTVhHRiohbuj6PRMRzy2221fFt9ZiI+EhZfldEvK3rGNvq+LY6MSI+VJbfGhHndR0zCW11ekT8Vfl7cjAi3lKW/2xE/EVE/G25fHLXMW8r2+TuiHhlV/lYt9d62yoinlLu/6OIuKLnXOPRVpk5dh/gpcDzgDu6ym4CXlZ+/23gHeX33cBtwNnl+lOAE8rvXwN+CQjgs8AFdf9sdbZVz3H/GLina922Ov736mLg6vL7FPBtYNa26ttWbwY+VH5/KnAA2DVBbXUK8Lzy+xOAbwBnAe8CLi/LLwfeWX4/C7gVeCxwBvDNSfk7awNtdRLwYuBS4Iqec41FW41lTzwzbwQe7Cl+FnBj+f0vgN8ov/8acFtm3loe+93M/GlEnAI8MTO/nMV/8Y8Cr6m+9ttrnW3V7SLgYwC2Vd+2SuCkiNgNPA54GPiBbdW3rc4CvlAe9wDwfWBugtrq/sy8ufz+Q+Au4FRgL/CRcrePcOxn30vxD8QfZ+a3gEPAuZPQXuttq8x8KDO/CPzf7vOMU1uNZYiv4Q7gn5bfXw+cXn5/JpARcX1E3BwRv1+Wnwosdx2/XJZNgrXaqts/owxxbKt+bXUN8BBwP7AEvDszH8S26tdWtwJ7I2J3RJwBnFNum7i2iohZ4BeBrwJPy8z7oQgvilEKKNrg3q7DVttlotprxLZay9i01SSF+G8Db46IAxTDMA+X5bsphlta5fK1EfFyiiGWXpMylX+ttgIgIl4ArGTm6vVO2+rRbXUu8FPg6RRDnv82Ip6BbdWvrfZT/CW6CLwH+BJwlAlrq4h4PPBJ4F9n5g8G7dqnLAeUj511tNWap+hT1si22l13BbZLZn6dYuiciHgm8Opy0zJwQ2Z+p9x2HcW1vKuA07pOcRpw37ZVuEYD2mrVhRzrhUPRhrbV8W11MfC5zPwJ8EBE/A0wB/w1ttVxbZWZR4HfXd0vIr4E/C3wPSakrSLiMRSh1MnMT5XFfx8Rp2Tm/eXw7wNl+TLHj46ttstE/H+4zrZay9i01cT0xCPiqeVyF/DvgSvLTdcDz4mIqfL65cuAO8shmR9GxAvLWYtvAP60hqpvuwFttVr2euDq1TLbqm9bLQG/GoWTgBcCX7etHt1W5f97J5XfXwEczcyJ+X+w/Nn+GLgrM/+wa9O1wBvL72/k2M9+LXBhRDy2vPxwJvC1SWivDbRVX2PVVnXPrKviQ9FLvB/4CcW/uC4B3kIxk/EbwH+hfNBNuf8/Bw5SXLN7V1f5XFn2TeCK7mPG5bOBtjoP+Eqf89hWXW0FPB74RPl7dSfwe7bVmm01C9xNMUnp8xRva5qktnoxxVDubcAt5edVFHfKfIFiVOILwM92HbOvbJO76ZpVPe7ttcG2+jbFJMsflb+LZ41TW/nENkmSGmpihtMlSRo3hrgkSQ1liEuS1FCGuCRJDWWIS5LUUIa4NMHK+9i/GBEXdJX9ZkR8rs56SRqNt5hJEy4ink1xT/svAidQ3Ht7fmZ+cxPn3J3Fk9gkVcgQl0REvIvihS0nAT/MzHdExBspXhN6IsXzzC/LzEciok3xaOLHAR/PzP9UnmMZ+ABwPvCezPxEDT+KNFEm5tnpkgZ6O3AzxUtJ5sre+WuBF2Xm0TK4LwT+J8V7mx8sH1P8VxFxTWbeWZ7nocz85Tp+AGkSGeKSyMyHIuLjwI8y88cR8U+A5wOLxaOleRzHXn95UURcQvH3x9Mp3ge+GuIf396aS5PNEJe06pHyA8WrGvdn5h907xARZ1I8A/3czPx+RFwF/EzXLg9tS00lAc5Ol9Tf54HfjIiTASLiKRExDTwR+CHwg/KVj6+ssY7SxLMnLulRMvP2iHg78Pny1aE/AS4FFimGzu8A7gH+pr5aSnJ2uiRJDeVwuiRJDWWIS5LUUIa4JEkNZYhLktRQhrgkSQ1liEuS1FCGuCRJDWWIS5LUUP8PFAbnHtRmRXgAAAAASUVORK5CYII=\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:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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('Indepdendent Variable')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"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."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Building The Model ###\n",
"Now, let's build our regression model and initialize its parameters. "
]
},
{
"cell_type": "code",
"execution_count": 11,
"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:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"data": {
"text/plain": [
"[<matplotlib.lines.Line2D at 0x7feb5d3c7748>]"
]
},
"execution_count": 12,
"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:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"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",
"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."
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" beta_1 = 690.447527, 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."
]
},
{
"cell_type": "code",
"execution_count": 15,
"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",
"Can you calculate what is the accuracy of our model?"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mean absolute error: 0.04\n",
"Residual sum of squares (MSE): 0.00\n",
"R2 Score:0.98\n"
]
}
],
"source": [
"# write your code here\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",
"popt,pcov = curve_fit(sigmoid,train_x,train_y)\n",
"y_hat = sigmoid(test_x,*popt)\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": [
"Double-click __here__ for the solution.\n",
"\n",
"<!-- Your answer is below:\n",
" \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",
"-->"
]
},
{
"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=\"http://cocl.us/ML0101EN-SPSSModeler\">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://cocl.us/ML0101EN_DSX\">Watson Studio</a>\n",
"\n",
"<h3>Thanks for completing this lesson!</h3>\n",
"\n",
"<h4>Author: <a href=\"https://ca.linkedin.com/in/saeedaghabozorgi\">Saeed Aghabozorgi</a></h4>\n",
"<p><a href=\"https://ca.linkedin.com/in/saeedaghabozorgi\">Saeed Aghabozorgi</a>, PhD is a Data Scientist in IBM with a track record of developing enterprise level applications that substantially increases clients’ ability to turn data into actionable knowledge. He is a researcher in data mining field and expert in developing advanced analytic methods like machine learning and statistical modelling on large datasets.</p>\n",
"\n",
"<hr>\n",
"\n",
"<p>Copyright &copy; 2018 <a href=\"https://cocl.us/DX0108EN_CC\">Cognitive Class</a>. This notebook and its source code are released under the terms of the <a href=\"https://bigdatauniversity.com/mit-license/\">MIT License</a>.</p>"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python",
"language": "python",
"name": "conda-env-python-py"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.10"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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