{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
This is a data simulation that is meant to show several things:
\n",
"
First we import necessary libraries and set up functions that can generate a random bernoulli dataset.
\n",
"We input the parameters of the linear hyperplane and get a dataset with $X=
Generally when we have features X and a target variable Y, our goal is to understand understand how Y varies with X.\n", "\n", "We can start by just plotting Y as a function of X. Since Y is binary we bin the X features and measure $E[Y|X_{bin}]$.\n", "
" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "The nice thing about simulations is we know the truth and we can compare our learned model to the truth. This is incredibly useful \n", "\n", "way to study the phenomenon of overfitting. In the above example we can compare the truth (that we created of course) to best answer\n", "\n", "given by the data. Sometimes the fitted vs. truth lines don't match exactly. In extreme cases they are no where near each other.\n", "\n", "Let's see how the number of records affects this phenomenon.\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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SJGXiAkuSJEmSMnGBJUmSJEmZuMCSJEmSpExcYEmSJElSJuMssI4D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\nFqGv/AIiZwAAAABJRU5ErkJggg==\n", "text/plain": [ "Look at how much the black (fitted) line varies from the green (truth) line as we vary the size of the sample. With n=10, the fitted separating plane is nearly perpendicular to the truth. We only visibly fit the truth exactly when n=1 MM (though we come very close with a few orders of magnitude lower). The lesson here is that with more data our fitted curve is expected to look more like the actual truth. Each realization of the green curve is a single estimate from one subset of some theoretically infinite data set. The fitted curve is essentially a function of the data. If we expect randomness across datasets (an artifact of sampling), we also should expect randomness in the curves we fit. The plot below illustrates this.
\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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XQPw1xEUQt48bCJfb1O6nj+lyPze1qU3Nm9YjkT9nu9zV13zuNX8B0jSVcB9T\nldempw2WC81BXtosV1Xef6+rY/GqfMz861wp19rsezb6HMS3u+D+ibfQDX57DN1guGMhXgDx+xCf\n6LrTj5kbHic/hnXMQ2FqbXIo1+6Vj1kxb1qX+6BSN8jXleZzr/kLkMbKQV725aq8Nj2t/4/8VTnI\nS5sxi8YcaFU+uQ+ev0gMgry0GVSrafv6SptcBV+Y2+yAHZdC/H8Qr4V4Zhfcm+4F8TSIV0P8Gez9\nHMRNC4dyfl1SaZO/AIzpSag9XW7MvPsdOchLG8N8bWo+95q/AOlA5DDPQV72TZZCnTwtbRDkpU0e\nWV1bNCY/53ypVXltKddrcpBPOU4/qGqD5fJyr2MesboDYCvEuRD/D8QrYO5JEHeHOBriJ7rK/Q8+\nDPGvELu738vLtEblvXLFPWZt+VqbPE6g1uU+7SE1/d6Os2tBbrg3qfnca/4CpOWUg7zsOzq6ldqO\nLtuDIC/7dvWDOodradMPgmtykJc2Cw2oqy3lOm3RmFyVL9blnqew1ary3DX9hUqbzQGxF+IKiL/r\n7nG/6acgHkE3uv1xMPcLEP83xD9CbIKbmT/3fEz3/hdHtMkV95gu9y2VNntykJc2drm3p/nca/4C\npJWWg7zs+2z5x/uS6MJ/EOSlzUKjyieBm6vy2lKuS100ZlpV3g+mc3KYVdrkqrwWlLl6HYTgbRAX\nwJ73Q/w6xAb2PYRly49A/HeIv4T4ItyxffH3yhX36yptxkypG9PlnttMuy9vl/uhr/nca/4CpENB\nzO9yHwR52be17NseXfjXRpXnqnzWRWMGVeaUcO+/19WpzQmV44ytTPvbtZHwF/S393bV/YM+DfF2\niJdAPBbm7gLxSIifhjilu5/+X6/s2k97r6i81yVpu9ZVno9TazOmtyE/gGZal7vPPV9dzede8xcg\nHYpykJd9R8ewy30Q5GXfmKp8zFKuuet+EOSV9zo7hiPqa/fuT5hynIXCqzbXOz/0pDqgbhfEZRAf\ngngNbAdOPwriHhBPhvgViHdBnAdxY/c7N1WOk183VNrkLvfadeXBcrWqfOwXm9psgm3lz+4of6dW\n6QdP87nX/AVIrcpBXvbNUpVPumfnYv998KV23ed75ZN7uIMwj8XvleeFU7ZX2uQu9zFt9lXuN0L8\nE8Sfwq2/UkL8HhBHdf+efer1EP+zhP4dw+eJR1CtynPFXaum82pvOyuf4YtHhHvucp82yr02uNFK\nfXk0n3vlvZt3AAAUsUlEQVTNX4C0luQwz0Fe9uWqvDby/Kayf1LB1bruxwyoW6gqnwRKXsSmH0KT\nedMLVdxjpqfVFmk5a/LzXoh/76rp578Z4mcgfoBu7vlxEC+GeBvEp2AXcGSae54fjVpbpz0H7pgu\n92lPT+tvT5t7XvvSkqcKGuSzaT73mr8AaS3LQV72LVSV16aD3Z6DvLS5uvz5ZNGYMVV5bZDbIMyj\nvojNIMxzkFfavDKHYqXNlnR+G3ObW+H8L0N8AOI3IZ7R3Svfch+6gXG/DvHnsOuLdAPnJkFeea88\nyr32ZeP8tD3m4TK1qnzacq/5OINnyZc2drkvrvnca/4CpPUmh3kvMC6L/dX0mPng/cr39Kh3uY+p\nyvPUs/72ZGT1Hem9BkFe2vT37U7bte7rMeuib8yhCHAtxGfopqS9FPY8nm6q2sMhfrKbwvbmv4f4\nd/Y9iCVX5dMesDLvvdK+POq+1uWep7CNmXs+bZS7Xe7zNZ97zV+AtN5FfSDcIMxzkJc2Y7rcB2Ge\ng7xynGtSyOU12CO6+c61qjwPhKudzyDMpwR3DvfBfehKm2uDbnGYf4P4UNed/vs/CXEM3YNYHg23\nvgLiHRCfhbh+XFVeWxAmV9xj1mCvhfKYUe7TutzzGIn1FuTN517zFyBpvhzmOcjLvn4Inz4i3G/P\nQV45zmlRH8A2CPMc5KXNjnQ+tcVnFgqv6pPRcrhWrisPKhtMYbsZ4ny44z0QvwrxdIh7dsfZ9EyI\n/wHxQYiLYe/O4XGicj4LPmO9vM4dGcqLfbEZ8/jUzTGiy73sWyvd7s3nXvMXIGlxOcjLvjzyfNYu\n91yVj3kKW60qz5V7P5hrVXkeCFcLr9xVXgv3uXTcWptB9/XeLuCO/iTEH0C8EOIHYPddIB4N8XMQ\nJ3e3En7i23QPbum912LnvDW1q40BmLZM60JfEmat7ifHzrdWWq3Um8+95i9A0mxymOftsm+pXe7T\nqvJ8j7tWlY9ZynUSHpM57LXziXQ+tWlu/TYXpu0x969rlfK1O+gegfo3EL8Fu4F/fCDE99A9MvUl\nsPm9dI9SvaV8RpXj5NdJU66rv31ipU3ubahd1z+n7Wp1X9mXK/XJ/fwWnoXefO41fwGSDp4c5jnI\ny74xVfmYe9x5KddBKFaOc3UO8inns1iFm6enzXsyWswfCFcLwXzPfd+XhC0Q50D8Mex6OcQT6e6d\nH9kNTvvEGyE+CvE1iF3jHsLy5Smfc79NXn9+KQP8+vvGPAe+9gCca9NnO/l7Xu0wbz73mr8ASSsn\nB3nZt9SqPE8rm9zj7t+TPTXqA+EGYZ6DvLRZalWep6ftrrTJ09NqYTamy33fVK89XVDvBH7mdyFO\nhHgYxF1h7gkQvwjxJxCfhluAB+QwT6+LK+/1uVi8y33UgjCVfXm1uVoPRO4VmHxm/RkDZ65CkDef\ne81fgKTVlcM8B3nZt9BAuNq0sk9HmmdeOc60pVzHVOX9FdXys9Fr67Tn6Wm1rul8P33MQLh5U8Zu\nhs1fgPhziP8G8dRulPu2IyB+DOK3IP4CbrkIYkfv9yrnk6viV1ba5HDdWGkz65r0Yx4VmwfZTaYG\nHswu9zjA3191zV+ApENLjBsINwjy0mbMM83zojFjlmCtVeX5KWwX9dp8tHI+eZ32SVW+d4E2YwbC\n1aan5dHg1wGHfRvmPgXxVohnw7WPoVsZ7thu+wbgdz4B8U32DYbL4Vq7n54r59oAttrz28cu3brY\nsccMjtscy9vlHjP8Z31Iaf4CJB36YtzT08ZU5YMwj6UvwTqtKq89hW0Q5jGuKl9sIFweDV4L0zyo\nrBZ41wbsuQPiUoh3wKXA254FcSTEPSEeCze/GuLPIM6H2FZ/4Ese5V4L4MkKcINu93xdld8bM/Ut\nrzZXm/ueeztGd7nX9rEGcq/5C5DUnli+qnzMEqyzVuX5KWy1qrx/nLPTdm3AVm4zJvCmjequfSGZ\nC4gbID4BF78T4pchjoe4S1fNfvu5EG+A+DDEv8C23cPj1MK91sV94og2+dbBmKp81uelT+ty75/D\nRwPexxrIveYvQNLaELNV5f2qa/Io0rwE61Kr8s2V40yryifTqf41pg/MG4ROLF6VnzflfBYKvGkP\natn3Xnu6+9cP/3uYO4XuOeffD7vvCvE4iJdCvK27BXDCNQzmno9Zua22vGv+crGx0ibfBx9zrbU2\n06a+9T/rE6O7vuZzr/kLkLQ25SAv+3JVPvmH/7bYP4BtEOY5yEubPBCu/w/8pOLOIVSryr8e3ZeJ\n66O75z4ZNX5L1J/mNmZ6Wn7ISPV55VOCqr9v2oNaBmF2G8SFEH8J8RL4DvDZ+0LcD+JHIH61G43+\n8i9BbB9+Kegfe8ySq7XAzV8AxgT3mBHstS73uegyr/nca/4CJK0fOcyj/vS0QZjnIC9tpnW59xeN\nGYT5lIDJVflgSltpMwjzHOSlTf/Yef51LdzzvfJa4I15UEu+L38WwF6YuwbiTIjXwhXABx9H9yCW\nR0I8D3aeAvExiCsh9oy7V1/rBs/d6bXR6WO63HPlXmvzJYNbkg5BOcxzkJd9ucu9vz2ppgdhnoO8\ntMlV8LwQjLR0aw7yKefTP26tO33M9LRclc86j3vf/fNdEF+FeDdcdjLEcyEeAnG3rpq/4JUQ74LY\nCLF13P3rPIJ+TJd7bVxArrinVuWsgdxr/gIkaSE5yMu+MQPhBmGeg7y0WagqnxZUg9HQpc2kyp5U\n5f155qdW3mva9LRBmI8IxTw6vDoQrrJvMKBvc/fl4unvhngVxFMg7gFzR0E8G+J1EH8FNwOPvSO9\nX7qu8yvvlUO5do65B6J2rVsMbklqVMw2EG6pVflkRbgxIZSnNl1Vfu4vPjPL9LRBkFfajBlANqZr\netLFve+cdsAzvwFzn4D4PYjnd13+l98F4jiIF0G8BXafAXE1++ee5+PE/EVrrhnxuda63K8zuCVp\njchBXvbNUpUPFnYpbfr3wZ86IoR25CAvbb5Ttm+Kripf6vS0WlW+MYVk7Xw2VvaNWeM8j6A/EWA7\nzH0Z4q8gXgLf/FGIB0B8L8TTulXi3vk+iAsgbt3/XostWnNW2q5V3DsMbklaw3KY5yAv+8ZU5f1B\nWqfG/oep7OmF+yDMp4Rifwrb2TnIS5s8PW0Q5JX3yl3lGyttBnPEJyGcQ7rye7kqn7ba3K6AuA7i\nI90Aut96GcQT6AbDPRT2nAjxuxB/D3F5NxDuTul88heQc6d9rqyB3Gv+AiRpJeQgL/vGVOU7e/tO\ni2H39KmpzZ6Y/2S0WhW8Jwd5afPX5YvCZ8r5DIK8tFloelptANuk27n/fvmxq7Wu+vylYNrTyfZW\njrMrIHZDXADnfgTijRDPg3hI152+/T9DvBziHRDnwN7r61928riAOdZA7jV/AZK0WnKY5yAv+27s\nB3UO8tImjzxfrCrP3eCTe8O5Kh8EeWkz5mlu/WPX1iqfNhp8oQF0tS8FuSqftnDKtt6XgaOBe30e\n4r0Q/wfEk2HX90B8H8QJECd17/XSiyF29sLc4JYkDeQgL/s+U4LjK+XPB0Fe2uRFYwZBXtpMwvLW\n6O5xD4K8tJmlKl/0aW6VY4+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"text/plain": [ "The black line here shows the truth. Each red line is a line fitted against a single realization of the data. Most are close to the\n", "\n", "truth, and it looks like E[red line]=black line. In real world situtations though we only get one data set. When your data is small\n", "\n", "(and model too complex), you might get unlucky and it up with a line that is very different from the theoretical truth. The red lines\n", "\n", "that are far off from the black lines are examples of unlucky overfitting. In real applications we'll likely never know the true \n", "\n", "underlying data distribution, but with the techniques taught in this course, we can at least be confident that overfitting can be \n", "\n", "avoided.
\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }