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‎Chapter00.ipynb

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"data": {
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc943adaf10>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc9432001d0>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0c0c3f58d0>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc54b9410>],\n",
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"(<Figure size 432x432 with 1 Axes>,\n",
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" <matplotlib.axes._subplots.AxesSubplot at 0x7fc942fc7e10>)"
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" <matplotlib.axes._subplots.AxesSubplot at 0x7f0bc521ff90>)"
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"metadata": {},
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"source": [
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"$$\n",
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"\\Large error_i = \\hat{y_i} - y_i\n",
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"\\Large \\text{error}_i = \\hat{y_i} - y_i\n",
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"$$"
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]
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"data": {
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"text/plain": [
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"(<Figure size 432x432 with 1 Axes>,\n",
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" <matplotlib.axes._subplots.AxesSubplot at 0x7fc9430fbf90>)"
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" <matplotlib.axes._subplots.AxesSubplot at 0x7f0bc53b5e90>)"
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"$$\n",
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"\\Large\n",
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"\\begin{aligned}\n",
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"MSE &= \\frac{1}{n} \\sum_{i=1}^n{error_i}^2\n",
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"\\text{MSE} &= \\frac{1}{n} \\sum_{i=1}^n{\\text{error}_i}^2\n",
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"\\\\\n",
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"&= \\frac{1}{n} \\sum_{i=1}^n{(\\hat{y_i} - y_i)}^2\n",
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"\\\\\n",
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"data": {
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"text/plain": [
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"(<Figure size 864x432 with 2 Axes>,\n",
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" (<matplotlib.axes._subplots.Axes3DSubplot at 0x7fc942d92fd0>,\n",
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" <matplotlib.axes._subplots.AxesSubplot at 0x7fc9430497d0>))"
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" (<matplotlib.axes._subplots.Axes3DSubplot at 0x7f0bc507e410>,\n",
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" <matplotlib.axes._subplots.AxesSubplot at 0x7f0bc5306450>))"
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"data": {
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"text/plain": [
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc9338ce190>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc941b8a150>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2aef710>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc4266210>],\n",
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" dtype=object))"
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"data": {
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"text/plain": [
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc940063890>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc933959250>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc24c8250>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc249b4d0>],\n",
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" dtype=object))"
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"metadata": {},
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"source": [
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"$$\n",
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"\\large\n",
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"\\Large\n",
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"\\begin{aligned}\n",
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"\\frac{\\partial{MSE}}{\\partial{b}} = \\frac{\\partial{MSE}}{\\partial{\\hat{y_i}}} \\cdot \\frac{\\partial{\\hat{y_i}}}{\\partial{b}} &= \\frac{1}{n} \\sum_{i=1}^n{2(b + w x_i - y_i)} \n",
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"\\frac{\\partial{\\text{MSE}}}{\\partial{b}} = \\frac{\\partial{\\text{MSE}}}{\\partial{\\hat{y_i}}} \\frac{\\partial{\\hat{y_i}}}{\\partial{b}} &= \\frac{1}{n} \\sum_{i=1}^n{2(b + w x_i - y_i)} \n",
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"\\\\\n",
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"&= 2 \\frac{1}{n} \\sum_{i=1}^n{(\\hat{y_i} - y_i)}\n",
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"\\\\\n",
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"\\frac{\\partial{MSE}}{\\partial{w}} = \\frac{\\partial{MSE}}{\\partial{\\hat{y_i}}} \\cdot \\frac{\\partial{\\hat{y_i}}}{\\partial{w}} &= \\frac{1}{n} \\sum_{i=1}^n{2(b + w x_i - y_i) \\cdot x_i} \n",
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"\\frac{\\partial{\\text{MSE}}}{\\partial{w}} = \\frac{\\partial{\\text{MSE}}}{\\partial{\\hat{y_i}}} \\frac{\\partial{\\hat{y_i}}}{\\partial{w}} &= \\frac{1}{n} \\sum_{i=1}^n{2(b + w x_i - y_i) x_i} \n",
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"\\\\\n",
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"&= 2 \\frac{1}{n} \\sum_{i=1}^n{x_i (\\hat{y_i} - y_i)}\n",
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"\\end{aligned}\n",
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc933a59490>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc933a73450>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2ba1fd0>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2bc12d0>],\n",
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" dtype=object))"
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc94306e490>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc94010d390>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2c60650>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2c5a390>],\n",
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"$$\n",
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"\\Large\n",
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"\\begin{aligned}\n",
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"& b = b - \\eta \\frac{\\partial{MSE}}{\\partial{b}}\n",
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"b &= b - \\eta \\frac{\\partial{\\text{MSE}}}{\\partial{b}}\n",
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"\\\\\n",
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"& w = w - \\eta \\frac{\\partial{MSE}}{\\partial{w}}\n",
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"w &= w - \\eta \\frac{\\partial{\\text{MSE}}}{\\partial{w}}\n",
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"\\end{aligned}\n",
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"$$"
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"(<Figure size 432x432 with 1 Axes>,\n",
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" <matplotlib.axes._subplots.AxesSubplot at 0x7fc94200a050>)"
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" <matplotlib.axes._subplots.AxesSubplot at 0x7f0bc42c5750>)"
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"#### Small Learning Rate"
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"#### Low Learning Rate"
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]
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc94014f650>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc9401a00d0>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc42e7810>,\n",
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"cell_type": "markdown",
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"#### Big Learning Rate"
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"#### High Learning Rate"
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]
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc9402c4b50>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2539190>,\n",
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"#### Very Big Learning Rate"
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"#### Very High Learning Rate"
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc94024f890>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc94014f550>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc2cdf710>,\n",
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" <matplotlib.axes._subplots.AxesSubplot object at 0x7fc931f6b410>],\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc0a10510>,\n",
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"(<Figure size 864x432 with 2 Axes>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7fc931e74f90>,\n",
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" array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f0bc094a190>,\n",
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"\\sigma(X) = \\sqrt{\\frac{1}{N}\\sum_{i=1}^N{(x_i - \\overline{X})^2}}\n",
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"\\\\\n",
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"\\Large\n",
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"scaled x_i=\\frac{x_i-\\overline{X}}{\\sigma(X)}\n",
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"\\text{scaled } x_i=\\frac{x_i-\\overline{X}}{\\sigma(X)}\n",
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"$$"
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},

‎Chapter01.ipynb

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"cell_type": "markdown",
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"# A Simple Regression Problem"
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"# A Simple Regression Problem\n",
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"\n",
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"$$\n",
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"\\Large y = b + w x + \\epsilon\n",
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"$$"
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{

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