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@tejas-2232
Last active April 26, 2020 13:01
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Created on Skills Network Labs
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Non-Linear regression"
]
},
{
"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. \n",
"<br><br>\n",
"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": [
"### Importing required libraries"
]
},
{
"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 = 2*(x) + 3."
]
},
{
"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",
"# y=m*(x) + c\n",
"\n",
"y = 2*(x) + 3\n",
"\n",
"y_noise = 2 * np.random.normal(size=x.size)\n",
"\n",
"ydata = y + y_noise\n",
"\n",
"#plt.figure(figsize=(8,6))\n",
"\n",
"plt.plot(x, ydata, 'bo') #bo for blue\n",
"plt.plot(x,y, '+') \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": 11,
"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",
"\n",
"y_noise = 20 * np.random.normal(size=x.size)\n",
"\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": 16,
"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": 17,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"data": {
"image/png": 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2+P3aGkaXFTHvqFFRh5KwRJJCqbv/Q9IjERHJIi3tMZZu2MUlp0yiID9zHglLJNLfmtmHkx6JiEgWefaNOprbYnzohIlRh/IXSSQpfJEgMbSYWYOZNZpZQ7IDExHJZL9fW8PI0kLOOjp9J9TpyREvH7n7sFQEIiKSLVo7Yvxx/S4+dOIECjPo0hEkNsyFmdlVZvaP4fpUM5uf/NBERDLT85t209jawYdOzKxLR5DY5aPbgbOAT4brTcBPBlKpmY00s4fNbKOZbTCzs8xstJktMbNN4XvmdNeLiMT53ZoahhcXcPbR6T8A3uESSQpnuPuNQAuAu+8DigZY7w+BJ9x9NnAywTSfNwNL3X0WsDRcFxHJKG0dnSxZX8MFcyZQVJBZl44gsaTQbmb5gAOYWTnQ2d8KzWw48B7glxDM/+zu+4EFwMJwt4XApf2tQ0QkKi+8uZuGlg4+fOKEqEPpl0SSwn8DjwLjzOw7wPPAvw2gzhlAHXC3ma0yszvNrAwY7+47AcL3zHkEUEQktHj1DoYVF6T9XMy9SeTuo/vMbAVwHmDApe6+YYB1ngp83t1fNrMf8hdcKjKz64HrAaZNmzaAMEREBldzawdPrK3h0rmTGFKQH3U4/dJrSyHs+B1tZqOBWuB+4NfArrCsv6qBand/OVx/mCBJ7DKziWHdE8M638Hd73D3ee4+r7y8fABhiIgMrj+sq+Fge4yPzp0SdSj91ldLYQVBP4IB04B94fJIoAro17B/7l5jZtvM7Fh3f52gBbI+fF0D3Ba+L+rP+UVEovLoqu1MGVWSUWMdHa6vobOnA5jZz4DF7v67cP1DwPkDrPfzwH1mVgRsBj5N0Gp50MyuI0g6lw+wDhGRlKmpb+GFyt187v0zycuzqMPpt0QGxDvd3f+ua8Xdf29m3x5Ipe6+GpjXw6bzBnJeEZGoLFq9nU6Hj56auZeOILGksNvMbgXuJbicdBWwJ6lRiYhkmEdXbeeUqSOZPjYzZljrTSK3pF4JlBPclvoYwa2iVyYzKBGRTLJ+RwMbaxr52KmTow5lwBK5JXUvwUipIiLSg0dWVlOQZ1x80qSoQxmwIyYFMzsG+CrvnI7z3OSFJSKSGVraYzyyspoPHj+B0WUDHQEoeon0KTwE/Ay4E03HKSJyiN+v3cn+A+1cOT87HqZNJCl0uPtPkx6JiEgGuv/lbRw1ppR3ZdhkOr1JpKP5cTP7rJlNPOwpZxGRnLZpVyPLtu7lyvnTMvrZhHiJtBSuCd+/FlfmBAPbiYjkrF8vq6Iw37jstMx+NiFeIncf9Ws4CxGRbNbSHuORFUEH89ihQ6IOZ9AkMh1nqZndamZ3hOuzzOzi5IcmIpK+/ve1nTS0dPDJM7Kjg7lLIn0KdwNtwLvC9WrgX5MWkYhImnN3Fr64lRljyzhrRnZ0MHdJJCkc7e7fA9oB3P0gwWipIiI5aflb+3itup7PnDMds+z6OkwkKbSZWQlvT8d5NNCa1KhERNLYnc9tZmRpIR/P8MHvepLI3UffAp4ApprZfcDZwLXJDEpEJF29taeZJ9fv4rPvO5qSosycXa0vidx9tMTMVgJnElw2+qK77056ZCIiaejuF7ZSkGdcfVZF1KEkRSItBYD3AucQXEIqJBgxVUQkp9QfbOfB5dv4yMmTGD+8OOpwkiKRW1JvB/4OWAOsBW4ws58kOzARkXTzwLIqDrTFuO6c7H18K5GWwnuBE9y9q6N5IUGCEBHJGS3tMe58fgvvOnoMx08aEXU4SZPI3UevA/FPZ0wFXktOOCIi6emBZVXUNbbyhfNmRR1KUiXSUhgDbDCzZeH66cCLZrYYwN0vSVZwIiLpoLUjxs+e2cz8itGcmWUPqx0ukaTwzWRUbGb5wHJgu7tfHI68+j8Ek/lsBf7K3fclo24Rkb/EQ8urqWlo4d8vPznqUJLuiJeP3P0Zgi/pwnB5GbDS3Z8J1/vri8CGuPWbgaXuPgtYGq6LiESqraOTnz79JqdOG8nZM7O7lQCJ3X30t8DDwM/DoinAYwOp1MymABcRzObWZQGwMFxeCFw6kDpERAbDo6uq2b7/IF84b1bWDWnRk0Q6mm8keIq5AcDdNwHjBljvD4CbgM64svHuvjOsY+cg1CEiMiAt7TH+e2klJ08ZwXuPKY86nJRIJCm0untb14qZFRCOg9Qf4bDbte6+op/HX29my81seV1dXX/DEBE5ontfeovt+w9y04Wzc6KVAIklhWfM7OtAiZldADwEPD6AOs8GLjGzrcADwLlmdi+wy8wmAoTvtT0d7O53uPs8d59XXp4bmVtEUq/+YDs//lMl7zmmnLNnjo06nJRJJCncDNQRPLB2A/A74Nb+Vujut7j7FHevAK4AnnL3q4DFvD315zXAov7WISIyUD99+k3qD7Zz84Wzow4lpRIZEK/TzB4DHnP3ZF6vuQ140MyuA6qAy5NYl4hIr3bsP8jdL2zho6dMZs6k4VGHk1K9JgULLqB9C/gcweioZmYx4Efu/i+DUbm7Pw08HS7vAc4bjPOKiAzEfy55A3f4+w8cE3UoKdfX5aMvEVz/P93dx7j7aOAM4Gwz+3JKohMRSbFVVft4eEU1nz67gimjSqMOJ+X6SgpXA1e6+5auAnffDFwVbhMRySqxTuebi9YxfvgQPp/lYxz1pq+kUNjTZDphv0Jh8kISEYnG/cuqWLO9nm9cNIehQxKdbia79JUU2vq5TUQk4+xtbuP7f3ids2aM4SMnTYw6nMj0lQpPNrOGHsoNyM4ph0QkZ33viY00t3bwzwuOz5kH1XrSa1Jw9+ybkVpEpAd/rtzNA69s4/r3zOCY8cOiDidSiTy8JiKStZpbO7jpkdeYPraML5+fe7egHi43e1JERELffWIj2/cf5MEbzqKkSBdI1FIQkZz10uY9/L8X3+Lad1VwesXoqMNJC0oKIpKTGlvauenh1zhqTClf++CxUYeTNnT5SERyjrtz62Nrqd53gP+54SxKi/RV2EUtBRHJOQ+vqGbR6h186fxjdNnoMEoKIpJT3qxr4puL1nHmjNHc+P6ZUYeTdpQURCRntLTH+NyvV1FcmMcPPjGX/LzcfUitN7qQJiI5wd35h0deY2NNA3ddczoTRmhghp6opSAiOeEXz21m0eodfOWCY3j/7HFRh5O2lBREJOs980Ydt/1+Ix8+cYL6EY5ASUFEslplbSOf//VKjhk/jO9fdnJOD3aXCCUFEclaNfUtXP3LZRQV5POLq+dRlqNzJPwllBREJCvVH2zn2ruXUX+wnXs+fTpTR+fe1Jr9obQpIlmnpT3GDb9azpt1Tdx17emcMHlE1CFljJS3FMxsqpn9ycw2mNk6M/tiWD7azJaY2abwfVSqYxORzNfaEeOGX63g5S17+f5lJ/PuWeVRh5RRorh81AF8xd2PA84EbjSzOcDNwFJ3nwUsDddFRBLW2hHj/9y7Mrjb6GMncuncyVGHlHFSnhTcfae7rwyXG4ENwGRgAbAw3G0hcGmqYxORzNXW0cmN963kqY21/NtHT+QTp0+LOqSMFGlHs5lVAHOBl4Hx7r4TgsQB9Ph0iZldb2bLzWx5XV1dqkIVkTTW1NrBZ+55hT9uqOXbC47nk2coIfRXZEnBzIYCjwBfcveGRI9z9zvcfZ67zysv17VCkVy3p6mVv/7FS7y4eQ/fv+wkPnVWRdQhZbRI7j4ys0KChHCfu/8mLN5lZhPdfaeZTQRqo4hNRDLHtr0HuObuZWzfd5CfX3Ua588ZH3VIGS+Ku48M+CWwwd3/M27TYuCacPkaYFGqYxORzPHy5j0s+MkL7G5s5d6/OUMJYZBE0VI4G/gUsMbMVodlXwduAx40s+uAKuDyCGITkQzwwLIqbn1sLdPGlHLn1fOYUT406pCyRsqTgrs/D/Q2+Mh5qYxFRDJLS3uMb/92Pfe9XMW7Z43lx588lRElhVGHlVX0RLOIZIQtu5v57H0r2bCzgRveM4OvffBYCvI1Us9gU1IQkbTm7vxm5Xa+uWgthQV53HXtPM6drf6DZFFSEJG0VdfYytcfXcOS9bs4vWIUP7xiLpNGlkQdVlZTUhCRtOPuLH51B/+0eB3NbTG+8eHj+Mw50zWncgooKYhIWnmzrolvLlrLC5V7OHnqSP7j8pOYOW5Y1GHlDCUFEUkLjS3t3P70m/zyuS0MKcwLh6s4Sq2DFFNSEJFItcc6eWBZFT/44yb2NLfxsbmTufnDsxk3rDjq0HKSkoKIRKIj1smi1Tv40VOb2LrnAGdMH83dFx3HSVNGRh1aTlNSEJGUauvoZPGrO/jJnyrZsruZ4yYO5xdXz+P848YRjIIjUVJSEJGUaGhp5/6Xq7j7ha3UNLQwe8IwfnbVaXxgznjy1G+QNpQURCSp1u2o596Xqli0ejsH2mKcPXMMt338RN57TLlaBmlISUFEBl39gXYef20HD6+oZvW2/RQX5nHJyZO4+qwKTpg8IurwpA9KCiIyKFraYzz9ei2Pv7qTJRt20dbRyTHjh3LrRcdx+WlTGVGqgesygZKCiPRb/cF2nnmjjiXrd7F0wy4OtMUYU1bEJ+dP4+OnTuGEycN1iSjDKCmISMLcnY01jTz7Rh3PvFHHsi176eh0xpQVcencyVx84kTmTx+t0UszmJKCiPTK3dm8u5mXNu/h5c17eXHzHuoaWwE4ZvxQ/ubdM7hgznhOmTpSTx5nCSUFEelWf7CdtdvrWb1tP6uq9rGyaj97m9sAGD98CGfNGMM5s8bynlnlTBihJ46zkZKCSA5yd3bUt/B6TQMbdjayfkcD63c2sGV3c/c+M8rLOG/2OE47ahRnzBhDxZhS9Q/kACUFkSx2oK2Dqr0H2Lq7mTfrmtlc10xlXROVuxppbot17zd1dAlzJg7nstOmcNKUEZw4eQQjS4sijFyioqQgkqHcnfqD7eysb6GmvoUd9QfZsf8g1fuC11t7DrC7qfWQY8YNG8LR5UO57LQpzBo/jGMnDGP2hGEMK9btohJIu6RgZhcCPwTygTvd/baIQxJJmbaOTvYfaGPfgXb2HWhjT1Mbe5tb2dPcxu6mVnY3tlHX1EptYwu7Glpp6+g85PiCPGPSyBImjyzh3NnlHDWmjGmjS6kYU8b08jKGDkm7f/KSZtLq/xAzywd+AlwAVAOvmNlid18fbWQivYt1Oi3tMQ60xTjYFuNAewcH2mIcaI3R3NZBc2vwamqN0dTaTlNLB40tHTS0dNDQ0k7DweC1/2A7B+Iu6RxuVGkhY4cOYezQIZw2bRTjhxdTPmwIE0eUMHFkMRNHFDNuWLHuApIBSaukAMwHKt19M4CZPQAsAJQUMoy74w4OdIbLnd1lTmfXemfw3ulOLG6/TofOTifW6W9v76R7PdYZ7B/r2idc7+h0YrHwvdPp6OykIxa8t8ecjlgnHZ3evdwe66S902nv6KQtXG/t6KQtfMWvt3Z00tIeo6UjRmt7uNweHJeo/Dxj6JAChg4pYHhJIcOLC5gyqpSRkwsZWVLIiJJCRpYVMaq0kFGlRYwuK2JMWRGjyooo1L3/kgLplhQmA9vi1quBMwa7ko01DXzu16uOuJ+7H3mfIxb0WXxIHX5Iefyx3nP5IcvvPE/XF3D8vn7IcX5YuR+yj3t4tL+93ePO6/HlhyWATFKUn0dBvlFUkEdRfh6F+XkMKcgL1guC5eLCfIaXFFJcmMeQgvzusuLCfEoK8ykuzKO0KJ+SogJKCvMpHZJPWVEBpUX5DB1SQFmYCIoL83QHj6S1dEsKPf1rOeQrxsyuB64HmDZtWr8qKS7I59jxCc75msC/38N36e0ffW+nit/dDim3Hss5ZH/rPsehx769vXvZ4o7q3t7LvnH1B+e27jrMgm1d+3SV51n8PsFyXvd+QVnXPvld+5qRb5CXZ+H2YFv3fnnBcl6ekW9Gfl5wzvy8uFfX9jyjIM8oyMsjLw8K8/O61wvyg235eUZhQR6FcWX6khZ5W7olhWpgatz6FGBH/A7ufgdwB8C8efP69Zu0YmwZP/nrU/sbo4hI1kq3i5SvALPMbLqZFQFXAIsjjklEJGekVUvB3TvM7HPAHwhuSb3L3ddFHJaISM5Iq6QA4O6/A34XdRwiIrko3S4fiYhIhJQURD9FmqIAAAd0SURBVESkm5KCiIh0U1IQEZFuSgoiItLNEhnKIV2ZWR3wVtRx9MNYYHfUQUQgFz93Ln5myM3PnUmf+Sh3L+9pQ0YnhUxlZsvdfV7UcaRaLn7uXPzMkJufO1s+sy4fiYhINyUFERHppqQQjTuiDiAiufi5c/EzQ25+7qz4zOpTEBGRbmopiIhINyWFiJnZV83MzWxs1LGkgpl938w2mtlrZvaomY2MOqZkMbMLzex1M6s0s5ujjifZzGyqmf3JzDaY2Toz+2LUMaWKmeWb2Soz+23UsQyUkkKEzGwqcAFQFXUsKbQEOMHdTwLeAG6JOJ6kMLN84CfAh4A5wJVmNifaqJKuA/iKux8HnAncmAOfucsXgQ1RBzEYlBSi9V/ATfQ+hXPWcfcn3b0jXH2JYHa9bDQfqHT3ze7eBjwALIg4pqRy953uvjJcbiT4kpwcbVTJZ2ZTgIuAO6OOZTAoKUTEzC4Btrv7q1HHEqHPAL+POogkmQxsi1uvJge+ILuYWQUwF3g52khS4gcEP+46ow5kMKTdJDvZxMz+CEzoYdM3gK8DH0htRKnR1+d290XhPt8guNxwXypjSyHroSwnWoRmNhR4BPiSuzdEHU8ymdnFQK27rzCz90Udz2BQUkgidz+/p3IzOxGYDrxqZhBcQllpZvPdvSaFISZFb5+7i5ldA1wMnOfZe090NTA1bn0KsCOiWFLGzAoJEsJ97v6bqONJgbOBS8zsw0AxMNzM7nX3qyKOq9/0nEIaMLOtwDx3z5TBtPrNzC4E/hN4r7vXRR1PsphZAUFH+nnAduAV4JPZPOe4Bb9wFgJ73f1LUceTamFL4avufnHUsQyE+hQk1X4MDAOWmNlqM/tZ1AElQ9iZ/jngDwQdrg9mc0IInQ18Cjg3/G+7OvwFLRlELQUREemmloKIiHRTUhARkW5KCiIi0k1JQUREuikpiIhINyUFiYyZNf2F+79vsEahNLN/MrOvDtK57jGzy/p57Ck93bZpZmVmtsfMRhxW/piZ/dVfcP5JZvbwEfbp9e9qZltzZQRfCSgpiETrFOAdScHdm4EngUu7ysIEcQ6QUGI0swJ33+Hu/UpYkpuUFCRy4S/Vp83s4XCuhfvCp2O75iTYaGbPAx+LO6bMzO4ys1fCcewXhOXXmtkiM3sinMvgW3HHfCMs+yNwbFz50eH+K8zsOTObHZbfY2b/bWZ/NrPNXa0BC/zYzNab2f8C4+LOdZqZPROe6w9mNjEsf9rMvmtmy8zsDTN7t5kVAf8CfCJ80OsTh/1p7geuiFv/KPCEux8ws/lhXKvC92PjPv9DZvY48KSZVZjZ2nBbRfj5Voavd8Wde7gF81usN7Ofmdk7vhvM7Kow/tVm9nMLhgeXbOPueukVyQtoCt/fB9QTjA+UB7xI8Iu4mGCk0VkEA8w9CPw2PObfgKvC5ZEEQ0qUAdcCO4ExQAmwFpgHnAasAUqB4UAlwZAEAEuBWeHyGcBT4fI9wENhTHMIhsKGIDktAfKBScB+4DKgEPgzUB7u9wngrnD5aeA/wuUPA38Ml68FftzL36cIqAXGhOtPABeFy8OBgnD5fOCRuPNVA6PD9QpgbbhcChSHy7OA5XF//xZgRviZlgCXhdu2AmOB44DHgcKw/Hbg6qj/H9Jr8F8aEE/SxTJ3rwYws9UEX2ZNwBZ33xSW3wtcH+7/AYKByLr6BYqBaeHyEnffEx7zG4IEA/Coux8IyxeH70OBdwEPhY0TgCFxcT3m7p3AejMbH5a9B7jf3WPADjN7Kiw/FjiBYAgPCL5gd8adq2uAuBXh5+uTu7eFcV5mZo8QXGp6Mtw8AlhoZrMIRl8tjDt0ibvv7eGUhcCPzewUIAYcE7dtmbtvBjCz+wn+ZvF9EecRJNZXws9WQpCwJMsoKUi6aI1bjvH2/5u9jcNiwMfd/fVDCs3O6OEYD/fv6Vx5wH53PyWBuOKHw+7pXAasc/ezjnCu+M93JPcDt4bnXuTu7WH5t4E/uftHLZi74Om4Y5p7OdeXgV3AyQSfuyVuW09/s3gGLHT3rJwpT96mPgVJZxuB6WZ2dLh+Zdy2PwCfj+t7mBu37QIzG21mJQQdtS8AzwIfNbMSMxsGfATAg/H+t5jZ5eF5zMxOPkJczwJXWDAv70Tg/WH560C5mZ0VnqvQzI4/wrkaCQYI7M2fCC713EiQILqMIBh9FYJLRokYAewMWz6fImjJdJlvZtPDvoRPAM8fduxSghbLOIDw73tUgvVKBlFSkLTl7i0El4v+N+xofitu87cJLoe8Fnakfjtu2/PAr4DVBNfal3swTeT/dJUBz8Xt/9fAdWb2KrCOI0+b+SiwiaCP4qfAM2G8bQR9C98Nz7Wa4NJUX/4EzOmlo5nwC/wRgj6SZ+M2fQ/4v2b2Aod+ufflduAaM3uJ4NJRfIviReA2gj6YLeFnjI9jPUGL5Ukze42g32FigvVKBtEoqZJVzOxagrkpPhd1LCKZSC0FERHpppaCiIh0U0tBRES6KSmIiEg3JQUREemmpCAiIt2UFEREpJuSgoiIdPv/oejixoSG5Z4AAAAASUVORK5CYII=\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": 18,
"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": 19,
"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 corrensponding 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": 20,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2020-04-26 12:17:09 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": 20,
"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 deaccelerates slightly in the 2010s."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"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",
"\n",
"x_data, y_data = (df[\"Year\"].values, df[\"Value\"].values)\n",
"\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": 22,
"metadata": {
"collapsed": false,
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"data": {
"image/png": 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S2+kf3f1vzTdkZlcBVwEMGzasjXFEOp9bnlnOnup6fvW5CaSqLyEJSTydzmUBV/LRTuf2N1xlS3/VzZuhphG5BnESkA28bWZz3H31h97kPguYBTBlypSPNGUVSURvrinm8Xe3cs0JhzJ6oLqXlvDE09fQ34GBwCnAa0S6lyhv9R0RW4ChMdNDiHRP0Xyd59290t2LgdeBiXFsWyShVdc1csMT75Gfl8M1Jx4adhxJcvEUgkPd/Sag0t3/CpwBjI/jffOBkWaWb2YZwAXA7GbrPAUca2ZpZtaNyKmjFYh0cX96ZQ2bSqr42TnjyErXiGMSrgPpdK7UzMYB24ER+3uTuzeY2TXAC0AqcK+7LzOzq6PLZ7r7CjN7HlgCNAF3u/vSNuyHSMJYuX0Pd72+js9PHqJB6KVTiKcQzDKz3sCNRL7Rdwduimfj7v4c8FyzeTObTf8a+HVcaUUSXFOT84PH36Nndjo3nH5Y2HFEgNb7Ghrg7jvc/e7orNeBgzsmlkjX9OC8Tby7qZTffWEivXMywo4jArR+jWCxmb1kZl+KdkMtIh/DzvIafvX8SmYc0lc9i0qn0lohGAz8BjgWWG1mT5rZ+WamMfNE2uDmZ1ZQ29DELWePUzcS0qnssxC4e6O7vxC9oWwo8BfgbGC9mT3QUQFFuoLXVhfx9OJCvn78oRzcr3vYcUQ+JJ7mo7h7HZGuIVYAe4CxQYYS6Upq6hu56cmlHNwvh6uP12U26XxaLQRmNszMrjWzd4BniDQDPcvdj+iQdCJdwO3/KWBTSRW3nD2OzDTdMyCdT2utht4icp3gEeAqd9cANSIHqGBnBTNfW8u5RwzWPQPSabV2H8EPgNfdXX37iLSBu3PTk0vJTk/lhjN0z4B0Xq1dLH5NRUCk7Z5ctJW31+3i+6eNIa97ZthxRPYprovFInJgyqrqueWZFRwxrBcXTlXX6dK57bcQmFl+PPNE5AO3vrCS3VV13HL2OFI0zoB0cvEcETzWwrxH2zuISFfx7qbdPDhvE5fPyOfwQbopXzq/1loNjSEyGE2umZ0bs6gnMQPUiMgHGhqb+OETS+nfI5PvfHpU2HFE4tJaq6HRwJlAL+AzMfPLgS8HGUokUf3t7Y0s37aH2y86ku6Z8XTuKxK+ff6luvtTwFNmdrS7v92BmUQS0vayGn774io+Oaofp48fGHYckbjF85WlwMxuIDIYzfvrxzFmsUhSufnZ5dQ3OTefdbg6lZOEEk8heAp4A3gZaAw2jkhien11Ec8u2cZ3Th7F8L45YccROSDxFIJu7v79wJOIJKia+kZ+9NRSDs7L4SvHqVM5STzxNB99xsxODzyJSIK689W1bNhVxc3qVE4SVDyF4FtEikGNme0xs3Iz2xN0MJFEsK6ogjtfXctnJw7imEPVqZwkpv2eGnL3Hh0RRCTRuDs3PbWUzPQUbjxTncpJ4oqniwkzs0vM7Kbo9FAzmxZ8NJHObfbiQv5bsIvrTh1D/x66x1ISVzynhu4AjgYuik5XALcHlkgkAZRV1XPzM8uZOLQXF01Tp3KS2OJpNTTd3Y80s3cB3H23mWUEnEukU/v1iyspqazjviumkapO5STBxXNEUG9mqYADmFk/oCnQVCKd2DubdvPA3E18ccYIxg1Wp3KS+OIpBH8CngD6m9nPgDeBnweaSqSTqm9s4obH32NAjyy+++nRYccRaRfxtBp6wMwWAicBBpzt7isCTybSCd375npWbi9n5iWT1amcdBmtdUPdJ2ZyJ/BQ7DJ3LwkymEhns7mkij+8vIZPHTaAUw4fEHYckXbT2leahUSuCxgwDNgdfd0L2ARolDJJGu7Oj2cvwwx+qk7lpItpbfD6fHc/GHgB+Iy757l7XyJjFDzeUQFFOoNn39vGKyt38u1PjWJwr+yw44i0q3guFk919+f2Trj7v4Djgosk0rmUVdXzk9nLGTe4J1ccMyLsOCLtLp5CUGxmN5rZCDMbbmY/BHbFs3EzO9XMVplZgZld38p6U82s0czOize4SEf55fMrKams5ZfnTiAtNZ7/MiKJJZ6/6guBfkSakD4J9I/Oa1X03oPbgdOAscCFZjZ2H+v9isgpKJFOZd76Eh6at4krP5Gvewaky4qn+WgJkR5ID9Q0oMDd1wGY2cPAWcDyZut9A3gMmNqGnyESmNqGRn7w+BIG98rm2ydrIHrpuvZbCMxsFPA9PjpU5Yn7eetgYHPM9BZgerNtDwbOAU6klUJgZlcBVwEMG6Z+XaRj3PZKAWuLKrnviql0y9A9A9J1xfPX/QgwE7ibAxuqsqX2dd5s+g/A9929sbXmeO4+C5gFMGXKlObbEGl3K7bt4c5X13LuEYM5fnT/sOOIBCqeQtDg7ne2YdtbgKEx00OAwmbrTAEejhaBPOB0M2tw9yfb8PNE2kVjk3P9Y0vIzU7npjM/cllLpMuJpxA8bWZfI3KxuHbvzDjuLJ4PjDSzfGArcAEfdGW9dxvv35RmZvcBz6gISNj+8t/1LN5Sxp8uPILeOepoV7q+eArBF6PP18bMc6DVUbrdvcHMriHSGigVuNfdl5nZ1dHlM9uQVyRQG4or+c2LqzhpTH8+M+GgsOOIdIh4Wg21uSuJ6I1ozzWb12IBcPfL2/pzRNpDU5Nz3WNLSE9J4ZZzxqkbCUka8QxV2S16Q9ms6PRIMzsz+GgiHev+uRuZt76Em84cy0G56kZCkkc8N5T9BagDZkSntwC3BJZIJASbS6r45b9W8slR/fj8lCFhxxHpUPEUgkPc/VagHsDdq2m5aahIQmpqcr7/2BJSzPjFueN1SkiSTjyFoM7MsvlgqMpDiGk9JJLoHpi7kbfW7uIHp49Rz6KSlOJpNfRj4HlgqJk9ABwDXB5kKJGOsqG4kp8/FzkldNE03bUuySmeVkMvmdk7wFFETgl9y92LA08mErDGJud7jywmPdW49XMTdEpIkla8HagcB3yCyOmhdCI3l4kktHveXMeCjbv5/fkTGZibFXYckdDE03z0DuBq4D1gKfAVM7s96GAiQVq1vZzfvLiaUw4fwNmTBocdRyRU8RwRHAeMc/e9F4v/SqQoiCSkmvpGvvXwu/TMSuNn56iVkEg8rYZWERm8fq+hwJJg4ogE77cvrmLl9nJ+fd5E8rpnhh1HJHTxHBH0BVaY2bzo9FTgbTObDeDunw0qnEh7+29BMXe9sZ5LjxrOCWPUvbQIxFcIfhR4CpEOUFpVx3f/uZiD++Vww+mHhR1HpNOIp/noa2Y2HBjp7i9Hby5Lc/fy4OOJtA9357pHl7Crspa7LjuG7IzUsCOJdBrxtBr6MvAo8OforCFEBrEXSRj3z9nIi8t3cN0pYxg/RIPQi8SK52Lx14ncTbwHwN3XADq5KgljxbY93PzsCo4b1Y8rP9HmXtVFuqx4CkGtu9ftnTCzND469rBIp1RV18A3HnqX3Ox0fvuFiaSkqKmoSHPxFILXzOwGINvMTiYymP3TwcYS+fjcnRufXMraogp+/4VJaioqsg/xFILrgSIiN5F9hciIYzcGGUqkPfxj/mYef2cr3zxxJJ8YmRd2HJFOK55WQ01m9iTwpLsXdUAmkY9tWWEZP5q9jE8cmsc3TxoZdhyRTm2fRwQW8RMzKwZWAqvMrMjMdF+BdGp7aur52gPv0KdbBn+8YBKpui4g0qrWTg39D5HWQlPdva+79wGmA8eY2bc7JJ3IAWpqcr7zj0Vs3V3NbRcdQV9dFxDZr9YKwWXAhe6+fu8Md18HXBJdJtLp/PHfa3h5xU5uOnMsU0b0CTuOSEJorRCktzQATfQ6QXpwkUTa5qXlO/jjv9dw3uQhXHb08LDjiCSM1gpBXRuXiXS4gp0VfPsfi5gwJJdbzh6nrqVFDkBrrYYmmtmeFuYboOGcpNPYXVnHlX+dT2ZaCjMvmUxWuvoREjkQ+ywE7q7/TdLp1TU0cfX9C9lWWsNDV01nUK/ssCOJJJx4xywW6XTcnR89tZS560v4w/mTmDxcF4dF2iKeO4tFOqW73ljHw/M3c80Jh3L2ERp3WKStVAgkIT21aCs/f24lZ4w/iO+cPCrsOCIJTYVAEs5ba4v53iOLmZbfRz2KirQDFQJJKKu2l/OVvy9kRN8c7rp0iloIibSDQAuBmZ1qZqvMrMDMrm9h+cVmtiT6eMvMJgaZRxLbxl2VXHrPXLplpHLfl6aR2033NYq0h8AKgZmlArcDpwFjgQvNbGyz1dYDx7n7BOBmYFZQeSSxbS+r4ZJ75lLf2MT9V05nsJqJirSbII8IpgEF7r4uOsLZw8BZsSu4+1vuvjs6OYfIeMgiH1JSWccl98ylpKKO+66YxsgBPcKOJNKlBFkIBgObY6a3ROfty5XAv1paYGZXmdkCM1tQVKQhEZJJaVUdl94zl00lVdz9xalMHNor7EgiXU6QhaClphwtjnVsZicQKQTfb2m5u89y9ynuPqVfv37tGFE6s9KqyJHAmh0V/PnSyRx9SN+wI4l0SUHeWbwFGBozPQQobL6SmU0A7gZOc/ddAeaRBFJWVc8l98xl9fYK/nzZZE4Y3T/sSCJdVpBHBPOBkWaWb2YZwAXA7NgVzGwY8DhwqbuvDjCLJJBdFbVcfM+cSBG4VEVAJGiBHRG4e4OZXQO8AKQC97r7MjO7Orp8JvAjoC9wR7Tb4AZ3nxJUJun8tpfVcPHdc9haWs2syyZzvIqASODMvcXT9p3WlClTfMGCBWHHkABs3FXJxXfPpbSqnnsvn8q0fHUiJ9JezGzhvr5oq/dR6RTe21LGFffNp7GpiQe/PJ0JQ9Q6SKSjqIsJCd1/Vu3k/Flvk5mWwiNXH60iINLBdEQgoXp43iZ++ORSxgzswV8un0r/nhr8TqSjqRBIKBoam/jFv1Zyz5vr+eSoftxx8ZF0z9Sfo0gY9D9POlxZVT3XPPQOb6wp5vIZI7jxjMNIS9VZSpGwqBBIh1q5fQ9fvf8dtuyu4pfnjueCacPCjiSS9FQIpMM8unALNz75Hj2y0nnwy0cxdYSah4p0BioEErjqukZ++vQyHp6/maMP7ssfL5xE/x66KCzSWagQSKCWbi3jmw+/y/riSr52/CF85+RRuh4g0smoEEggGpucu95Yx29fXEXfnEweuHI6Mw7NCzuWiLRAhUDaXcHOcq59dAnvbirl1MMH8otzx9M7JyPsWCKyDyoE0m7qG5u46411/OHlNXTLSOWPF0zisxMHEe1QUEQ6KRUCaRfzN5Rw4xNLWbWjnNPGDeR/zxpHvx6ZYccSkTioEMjHsrO8hlufX8WjC7cwuFc2sy6dzKcPHxh2LBE5ACoE0iY19Y3c/cY67nx1LXWNTXz1+EP4xomH0i1Df1IiiUb/a+WANDQ28fi7W/nDS6spLKvhlMMHcP1ph5GflxN2NBFpIxUCiUtTk/PMe9v4w0urWVdcyYQhufzu/EkcdbAGlBdJdCoE0qr6xiZmLyrkjlcLWFtUyegBPfjzpZP59NgBag0k0kWoEEiLKmobeGTBZu5+Yz1bS6sZM7AH/3fhEZw+/iBSU1QARLoSFQL5kA3Flfx9zkb+OX8z5bUNTBnem5vPPpwTRvfXEYBIF6VCINQ1NPHyih08OHcTbxYUk5ZinDnhIK44Jp+JQzVspEhXp0KQpNydpVv38Ng7W5i9uJCSyjoG98rmuyeP4gtThzJAQ0aKJA0VgiSzekc5zyzZxrNLCllbVElGWgonjx3AeUcO4ZOj+un8v0gSUiHo4pqanHc3l/LS8h28tHw7a4sqSTGYnt+XL30inzPHDyK3W3rYMUUkRCoEXVBxRS1vrinmtdVFvLGmiOKKOtJSjOkH9+GLM0Zw6riBGhhGRN6nQtAFlFTWMX9DCXPW7eLttbtYub0cgD45GXxyZB4njOnP8aP7k5utb/4i8lEqBAmmsclZvaOcxZtLWbS5lPkbSlhbVAlAZloKU0f04dpTBvGJQ/MYPziXFJ3zF5H9UCHoxGrqG1mzo4IV2/ewbGsZSwv3sLxwD9X1jQDkZqczeXhvPjd5CFOG92Hi0Fwy01JDTi0iiUaFoBMoq6pn/a5K1hdXsGZHBQU7I48Nuypp8sg6ORmpjB3Uk/OnDmXi0FwmDulFfl6ObvISkY9NhaAD1IMhgRAAAAnsSURBVNQ3UlhazdbSarburmbL7mo2lVS9/yiprHt/3bQUY3jfbowc0J3PTBzEmIE9GD2wByP65ug0j4gEQoWgjdydPTUNlFTWsauilqLyWoqjzzv21LKjvIYde2rZXlbN7qr6D703NcUY1CuLYX26ccrhA8jPy2FE3xzy83IY3jeHjLSUkPZKRJJRoIXAzE4F/gikAne7+y+bLbfo8tOBKuByd38nyEx7uTu1DU1U1jZQWdtIRW0DFbUNlNfUU14Ted5T00BZdT1lVfWUVtexu6qe0qrI8+7KOhr2nreJkWKQ1z2T/j0zGZSbxeThvTgoN5uBPbMY3Dubwb2yGZibRXqqPuxFpHMIrBCYWSpwO3AysAWYb2az3X15zGqnASOjj+nAndHndvfqqp3c/Mxyquoao48G6hs/+kHeXFZ6CrnZ6eRmp9OrWwb5eTkc2S2D3jkZ9M3JoE9OBn27Z5LXPYN+PTLp0y2DNH3Ii0gCCfKIYBpQ4O7rAMzsYeAsILYQnAX8zd0dmGNmvczsIHff1t5hemanM2ZgT7plpEYemWl0z0wjJyOVnMw0emSl0T0zne5ZafTMSqNndjo9stLUCkdEurwgC8FgYHPM9BY++m2/pXUGAx8qBGZ2FXAVwLBhw9oU5shhvTny4t5teq+ISFcW5DmMlpq4ND8XE886uPssd5/i7lP69evXLuFERCQiyEKwBRgaMz0EKGzDOiIiEqAgC8F8YKSZ5ZtZBnABMLvZOrOByyziKKAsiOsDIiKyb4FdI3D3BjO7BniBSPPRe919mZldHV0+E3iOSNPRAiLNR68IKo+IiLQs0PsI3P05Ih/2sfNmxrx24OtBZhARkdapwbuISJJTIRARSXIqBCIiSc4ip+kTh5kVARvDztEGeUBx2CFCkIz7nYz7DMm534m0z8PdvcUbsRKuECQqM1vg7lPCztHRknG/k3GfITn3u6vss04NiYgkORUCEZEkp0LQcWaFHSAkybjfybjPkJz73SX2WdcIRESSnI4IRESSnAqBiEiSUyEIgZl9z8zczPLCzhI0M/u1ma00syVm9oSZ9Qo7U5DM7FQzW2VmBWZ2fdh5gmZmQ83sP2a2wsyWmdm3ws7UUcws1czeNbNnws7ycakQdDAzG0pkHOdNYWfpIC8B49x9ArAa+EHIeQITM073acBY4EIzGxtuqsA1AN9198OAo4CvJ8E+7/UtYEXYIdqDCkHH+z1wHS2MxNYVufuL7t4QnZxDZPChrur9cbrdvQ7YO053l+Xu29z9nejrciIfjIPDTRU8MxsCnAHcHXaW9qBC0IHM7LPAVndfHHaWkHwJ+FfYIQK0rzG4k4KZjQCOAOaGm6RD/IHIF7qmsIO0h0DHI0hGZvYyMLCFRT8EbgA+3bGJgtfaPrv7U9F1fkjkNMIDHZmtg8U1BndXZGbdgceA/3H3PWHnCZKZnQnsdPeFZnZ82HnagwpBO3P3T7U038zGA/nAYjODyCmSd8xsmrtv78CI7W5f+7yXmX0ROBM4ybv2jStJOQa3maUTKQIPuPvjYefpAMcAnzWz04EsoKeZ3e/ul4Scq810Q1lIzGwDMMXdE6XnwjYxs1OB3wHHuXtR2HmCZGZpRC6InwRsJTJu90XuvizUYAGyyLeavwIl7v4/YefpaNEjgu+5+5lhZ/k4dI1AgnYb0AN4ycwWmdnM/b0hUUUviu8dp3sF8M+uXASijgEuBU6M/vsuin5TlgSiIwIRkSSnIwIRkSSnQiAikuRUCEREkpwKgYhIklMhEBFJcioE0qHMrOIA1z++vXp3NLOfmNn32mlb95nZeW1876SWmliaWY6Z7TKz3GbznzSzLxzA9geZ2aP7WWefv1cz25AMPePKB1QIRDreJOAjhcDdK4EXgbP3zosWhU8AcRVDM0tz90J3b1ORkuSkQiChiH4jfdXMHo2OV/BA9C7VvX36rzSzN4FzY96TY2b3mtn8aD/wZ0XnX25mT5nZ89GxAH4c854fRue9DIyOmX9IdP2FZvaGmY2Jzr/PzP5kZm+Z2bq93/ot4jYzW25mzwL9Y7Y12cxei27rBTM7KDr/VTP7lZnNM7PVZnasmWUA/wucH7356vxmv5qHgAtips8Bnnf3KjObFs31bvR5dMz+P2JmTwMvmtkIM1saXTYiun/vRB8zYrbd0yJjRCw3s5lm9pHPAzO7JJp/kZn92SJdbUtX4+566NFhD6Ai+nw8UEakP54U4G0i33yziPTgOZJIJ27/BJ6JvufnwCXR172IdOeQA1wObAP6AtnAUmAKMBl4D+gG9AQKiHQHAPBvYGT09XTglejr+4BHopnGEulWGiIF6SUgFRgElALnAenAW0C/6HrnA/dGX78K/Db6+nTg5ejry4Hb9vH7yQB2An2j088DZ0Rf9wTSoq8/BTwWs70tQJ/o9AhgafR1NyAr+noksCDm918DHBzdp5eA86LLNgB5wGHA00B6dP4dwGVh/w3p0f4PdTonYZrn7lsAzGwRkQ+wCmC9u6+Jzr8fuCq6/qeJdPa19zx/FjAs+vold98Vfc/jRIoKwBPuXhWdPzv63B2YATwSPQgByIzJ9aS7NwHLzWxAdN4ngYfcvREoNLNXovNHA+OIdKEBkQ/VbTHb2tsJ28Lo/rXK3euiOc8zs8eInEZ6Mbo4F/irmY0k0qtpesxbX3L3khY2mQ7cZmaTgEZgVMyyee6+DsDMHiLyO4u9tnASkWI6P7pv2USKlHQxKgQSptqY14188Pe4r35PDPicu6/60Eyz6S28x6Prt7StFKDU3SfFkSu2a+mWtmXAMnc/ej/bit2//XkIuDG67afcvT46/2bgP+5+jkX6/n815j2V+9jWt4EdwEQi+10Ts6yl31ksA/7q7l12VDmJ0DUC6WxWAvlmdkh0+sKYZS8A34i5lnBEzLKTzayPmWUTudj6X+B14BwzyzazHsBnADzSX/56M/t8dDtmZhP3k+t14AKLjFN7EHBCdP4qoJ+ZHR3dVrqZHb6fbZUT6YhvX/5D5DTO14kUhb1yifRqCpHTQfHIBbZFj3AuJXLEstc0M8uPXhs4H3iz2Xv/TeTIpD9A9Pc7PM6fKwlEhUA6FXevIXIq6NnoxeKNMYtvJnKqY0n0YujNMcveBP4OLCJy7nyBR4ZQ/MfeecAbMetfDFxpZouBZex/SMkngDVErjncCbwWzVtH5FrBr6LbWkTktFNr/gOM3cfFYqIf2o8RuebxesyiW4FfmNl/+fAHemvuAL5oZnOInBaKPXJ4G/glkWsq66P7GJtjOZEjkxfNbAmR6wgHxflzJYGo91FJeGZ2OZGxHa4JO4tIItIRgYhIktMRgYhIktMRgYhIklMhEBFJcioEIiJJToVARCTJqRCIiCS5/w88S56JL7VaaAAAAABJRU5ErkJggg==\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": 23,
"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": 27,
"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": [
"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')\n",
"plt.show()"
]
},
{
"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": 28,
"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": 29,
"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",
"\n",
"popt, pcov = curve_fit(sigmoid, xdata, ydata)\n",
"\n",
"#print the final parameters\n",
"\n",
"print(\" beta_1 = %f, beta_2 = %f\" % (popt[0], popt[1]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we plot our resulting regresssion model."
]
},
{
"cell_type": "code",
"execution_count": 30,
"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",
"\n",
"y = sigmoid(x, *popt)\n",
"\n",
"plt.plot(xdata, ydata, 'ro', label='data')\n",
"\n",
"plt.plot(x,y, linewidth=3.0, label='fit')\n",
"\n",
"plt.legend(loc='best')\n",
"\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": 35,
"metadata": {},
"outputs": [],
"source": [
"# write your code here\n",
"\n",
"\n",
"asd= np.random.rand(len(df)) < 0.8\n",
"\n",
"train_x= xdata[asd]\n",
"\n",
"test_x= xdata[~asd]\n",
"\n",
"train_y = ydata[asd]\n",
"\n",
"test_y = ydata[~asd]\n"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"#build model\n",
"mopt,mcov= curve_fit(sigmoid,train_x,train_y)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [],
"source": [
"#predict using test set\n",
"y_hat= sigmoid(test_x,*mopt)"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mean absolute error: 0.04\n",
"Residual sum of squares (MSE): 0.00\n"
]
}
],
"source": [
"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))"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"R2-score: 0.82\n"
]
}
],
"source": [
"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",
"<!-- 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": [
"## Thanks for completing...\n"
]
}
],
"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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