{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Learning Objectives\n",
    "\n",
    "Today we learn about the second most important tradeoff: the bias variance tradeoff."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Another way to look at it\n",
    "\n",
    "Some might say that this class is redundant, we are simply going to rephrase the results that we got last class. That being said we are going to do this for two reasons:\n",
    "\n",
    "1. This is the common way of looking at the approximation generalization tradeoff, even though in my opinion it should be the other way around\n",
    "2. The more ways that you can independently verify your conclusion (a classic probablist/data science way of thinking) the stronger you should feel about it\n",
    "\n",
    "So let's begin in the same way that we did last time, let's make a sine wave:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "\n",
    "X = np.random.uniform(-1, 1, size=1000)\n",
    "y = np.sin(np.pi * X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/nate/Desktop/core-data-science/env/lib/python2.7/site-packages/matplotlib/font_manager.py:280: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.\n",
      "  'Matplotlib is building the font cache using fc-list. '\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10c4a6610>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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SHODF1alRRscRcQoDwv2ZMKwvJ8vq2X2opEcfW8UuIp3y/rYCWtvs3DQpFg83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\n7xQfbC8iItiHB+cPx8OqHw8RcT36zeW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zZs2ivr6+z4dPe2Ip34gRI4iPj1euv9DQUKqq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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106ff5bd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.tsplot(y, X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Wonderful right. This will be the population distribution. Again we are going to be naughty and skip the splitting of training and test and we are just going to get some training data. \n",
    "\n",
    "Again we are going to be testing our two hypotheses below:\n",
    "\n",
    "1. f(X) = w_0 = y\n",
    "2. f(X) = w_0 + w_1 * x_1 = y\n",
    "\n",
    "But we are going to be looking at different quantities this time, we are going to be looking at the bias and the variance. So let's start off by defining them:\n",
    "\n",
    "* Bias: the error of the best hypothesis in your hypothesis set\n",
    "* Variance: the average error between a hypothesis generated on test data points and the best hypothesis\n",
    "\n",
    "Then if you add the bias to the variance you will get a measure of what your test error will be. Take the best and then look to see on average how far are you from the best.\n",
    "\n",
    "*Note: these definitions are contextual*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Bias\n",
    "\n",
    "You might notice some strange things about the bias. First it requires us to know the best hypothesis in our set. This is often impossible because we don't know the true distribution. But the second thing is that the bias does not depend on the number data, it only depends on the size of the hypothesis set. Thus once we know the hypothesis set, we know what our bias is.\n",
    "\n",
    "Fortunately for us I know what the bias is for this data set and for both our hypotheses, but let's show you how you could approximate it.\n",
    "\n",
    "Once again, it is as easy as training a model on as much data as we can provide it (around 10k will be more than enough). Let's go ahead:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.preprocessing import add_dummy_feature\n",
    "from sklearn.metrics import mean_squared_error\n",
    "\n",
    "bias0 = 0\n",
    "bias1 = 0\n",
    "\n",
    "X = np.random.uniform(-1, 1, size=10000)\n",
    "y = np.sin(np.pi * X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.504649692562\n",
      "0.196324511153\n"
     ]
    }
   ],
   "source": [
    "best_h0 = LinearRegression(fit_intercept=False)\n",
    "best_h1 = LinearRegression(fit_intercept=False)\n",
    "\n",
    "# this hypothesis ignores the X variable\n",
    "best_h0.fit(np.ones((10000, 1)), y)\n",
    "# this hypothesis can use the X variable as well as an intercept\n",
    "best_h1.fit(add_dummy_feature(X[:, None]), y)\n",
    "\n",
    "# now we score both hypotheses on the traning data\n",
    "preds = reg0.predict(np.ones((10000, 1)))\n",
    "bias0 = mean_squared_error(preds, y)\n",
    "preds = reg1.predict(add_dummy_feature(X[:, None]))\n",
    "bias1 = mean_squared_error(preds, y)\n",
    "\n",
    "print bias0\n",
    "print bias1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In truth these numbers should be .5 and .21. So the more complex hypothesis has a lower bias. This will hold true universally.\n",
    "\n",
    "And just for fun, what do you think these lines look like?\n",
    "\n",
    "Well the first will be a straight line through 0:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.01104962])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "best_h0.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10f306550>"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DRdQYm7t9+1LchRCdtuZyt5tzp8RK/xVCAB4GPQ9OiMJktvLVnvxu374UdyFE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\n5v9oDyQuJc4iygAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f4a5fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.random.uniform(-1, 1, size=1000)\n",
    "y0 = np.sin(np.pi * X)\n",
    "y1 = best_h0.coef_ * X\n",
    "\n",
    "sns.tsplot(np.array([y0[:, None], y1[:, None]]).T, X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "And the other one would go through zero and almost have a slope of 1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.01213531,  0.95799081])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "best_h1.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10f58f210>"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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YifGKIiN+DWEeIZYOza4smxHO6cI7HMurJS0phGBf2z5pkuIuhLAq/Xojmdll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\nWBkp7kIIIYSVkeIuhBBCWJn/BzosAZzgyb+QAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f5a4650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y2 = best_h1.coef_[1] * X + best_h1.coef_[0]\n",
    "\n",
    "sns.tsplot(np.array([y0[:, None], y2[:, None]]).T, X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Those are the biases that these two hypothesis sets will have for this particular data set, for ever. They just can't get any better than this. There really isn't any randomness here. \n",
    "\n",
    "So the question then becomes why not just use the bigger hypothesis set always? \n",
    "\n",
    "Well remember, while the bigger hypothesis does have the ability to make a better hypothesis, the training process is random. Sometimes we will get a hypothesis near the best, and sometimes not. Variance is where the randomness comes in!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Variance\n",
    "\n",
    "Variance is different, it is incredibly dependent on the data that you get. If you always are in a data rich environment, then you will have very low variance, but if you don't have enough data, then you will get much higher variance. This is the moving quantity!\n",
    "\n",
    "Let's start off by exploring what the variance of these two models would be if you only have 2 data points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "h0_scores = []\n",
    "h1_scores = []\n",
    "X_real = np.random.uniform(-1, 1, size=1000)\n",
    "\n",
    "for _ in range(1000):\n",
    "    X = np.random.uniform(-1, 1, size=2)\n",
    "    y = np.sin(np.pi * X)\n",
    "    y0 = best_h0.coef_ * X_real\n",
    "    y1 = best_h1.coef_[1] * X_real + best_h1.coef_[0]\n",
    "    \n",
    "    reg0 = LinearRegression(fit_intercept=False)\n",
    "    reg1 = LinearRegression(fit_intercept=False)\n",
    "    \n",
    "    # this hypothesis ignores the X variable\n",
    "    reg0.fit(np.ones((2, 1)), y)\n",
    "    # this hypothesis can use the X variable as well as an intercept\n",
    "    reg1.fit(add_dummy_feature(X[:, None]), y)\n",
    "    \n",
    "    # now we score both hypotheses on the traning data\n",
    "    preds = reg0.predict(np.ones((1000, 1)))\n",
    "    h0_scores.append(mean_squared_error(preds, y0))\n",
    "    preds = reg1.predict(add_dummy_feature(X_real[:, None]))\n",
    "    h1_scores.append(mean_squared_error(preds, y1))\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x11365c910>"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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AAGOIMwAAxhBnAACMIc4AABhDnAEAMMYT74RoNKrq6modPXpUGRkZqqmp0eLFi2PHa2pq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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c6ca590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.pointplot(data=[h0_scores, h1_scores], join=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Wow! Look at the difference between the variances. The first hypothesis set does not have too high a variance, whereas the second hypothesis set has quite a high one. Mathematically the variance of the first is .25 and the variance of the second is 1.69.\n",
    "\n",
    "So you can see that our estimate of the error of the first hypothesis set training on two data points is:\n",
    "\n",
    "$$ BIAS + VARIANCE = .5 + .25 = .75$$\n",
    "\n",
    "Now you might not remember last time, but that is percisely what our out of sample error was! Remember it does not always have to be this, it could be better or worse, but on average it is around this much. \n",
    "\n",
    "Whereas our estimate of how the second hypothesis set will do on this data with two data points is:\n",
    "\n",
    "$$ BIAS + VARIANCE = .21 + 1.69 = 2$$\n",
    "\n",
    "And again, you might be thinking to yourself, wow that is around what we had gotten before! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The trade off\n",
    "\n",
    "This should make the trade off very clear. the bigger the hypothesis set (the more complex it is/the number of infinite parameters) the lower the bias, but also the higher the variance.\n",
    "\n",
    "As always there is one saving grace here. The more data points, the lower your variance will be over all.\n",
    "\n",
    "As we showed before let's check out how the bias and the variance change as we add more data points to verify what we said above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "h0_var = []\n",
    "h1_var = []\n",
    "X_real = np.random.uniform(-1, 1, size=1000)\n",
    "\n",
    "for num_points in range(2, 40):\n",
    "\n",
    "    h0_scores = []\n",
    "    h1_scores = []\n",
    "    for _ in range(1000):\n",
    "        X = np.random.uniform(-1, 1, size=num_points)\n",
    "        y = np.sin(np.pi * X)\n",
    "        y0 = best_h0.coef_ * X_real\n",
    "        y1 = best_h1.coef_[1] * X_real + best_h1.coef_[0]\n",
    "\n",
    "        reg0 = LinearRegression(fit_intercept=False)\n",
    "        reg1 = LinearRegression(fit_intercept=False)\n",
    "\n",
    "        # this hypothesis ignores the X variable\n",
    "        reg0.fit(np.ones((num_points, 1)), y)\n",
    "        # this hypothesis can use the X variable as well as an intercept\n",
    "        reg1.fit(add_dummy_feature(X[:, None]), y)\n",
    "\n",
    "        # now we score both hypotheses on the traning data\n",
    "        preds = reg0.predict(np.ones((1000, 1)))\n",
    "        h0_scores.append(mean_squared_error(preds, y0))\n",
    "        preds = reg1.predict(add_dummy_feature(X_real[:, None]))\n",
    "        h1_scores.append(mean_squared_error(preds, y1))\n",
    "        \n",
    "    h0_var.append(h0_scores)\n",
    "    h1_var.append(h1_scores)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x113894f50>"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hVE0Dr7e/wb5UGw1U51CJiIh4n+G6rlvuRgC0tXXx9t5OvvXkq6QzDv94zVSm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\nIuIxCmcRERGPUTiLiIh4zP8P3aXU47SyoK0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1138a7a50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# now we plot them :)\n",
    "sns.tsplot(np.array([h0_var, h1_var]).T, range(2, 40))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Notice that the variance drops a ton for the second hypothesis, and becomes near 0 as we get 40 points. But remember, above is just the variance. While it is nice to see that go down, we are looking for our approximation of what the actual error will be. In this case we need to add the bias to the above estimates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x113ae8150>"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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TiYiIT1REOEftGlKZFK7rlrspIiIiA64ywjlQQ9bNkXYy5W6KiIjIgKuIcK4J1ACQSCfK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\nziIiIh6jcBYREfEYhbOIiIjH/P935tjKzEEUzAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11024f410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "h0_error = np.array(h0_var) + .5\n",
    "h1_error = np.array(h1_var) + .21\n",
    "\n",
    "sns.tsplot(np.array([h0_error, h1_error]).T, range(2, 40))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "And now we see something magical. There is a point where the errors cross each other! Yes if we have too few data points then the simpler hypothesis set is better. This is because the variance is so high for the first hypothesis set. But after we get to a reasonable number of data points, the more complex hypothesis is better because it has lower bias!\n",
    "\n",
    "Notice as well that both errors approach the bias. \n",
    "\n",
    "As one final note, the above curves are approximations as to what the real ones will look like, but they are pretty good approximations if I can say so myself :)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Putting it all together\n",
    "\n",
    "Let's finally put it all together.  Let's see what happens as I change the size of the hypothesis set and the number of training examples. And let's check what happens to the bias, the variance and the estimated test error as we change the above.\n",
    "\n",
    "We will start off with the bias, this is a little simple because it will not change as the number of points do:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "biases = []\n",
    "best_models = []\n",
    "\n",
    "for model_complexity in range(5):\n",
    "    X = np.random.uniform(-1, 1, size=10000)\n",
    "    y = np.sin(np.pi * X)\n",
    "    poly = PolynomialFeatures(model_complexity)\n",
    "    X = poly.fit_transform(X[:, None])\n",
    "\n",
    "    reg = LinearRegression(fit_intercept=False)\n",
    "\n",
    "    reg.fit(X, y)\n",
    "\n",
    "    preds = reg.predict(X)\n",
    "    \n",
    "    best_models.append(reg)\n",
    "    biases.append([mean_squared_error(preds, y)])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x106d1d890>"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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JWoMbAUB0CRjC+fn58ng8ysvL06OPPqply5ZZx06ePKlXX31Vf/nLX/TSSy/p6aefpvwh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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115d88090>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.pointplot(data=biases)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "As the model complexity increases, the bias decreases (notice that it might go down in fits and starts). \n",
    "\n",
    "Let's see what happens to the variance:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "X_real = np.random.uniform(-1, 1, size=1000)\n",
    "\n",
    "model_vars = []\n",
    "for model_complexity in range(5):\n",
    "    variances = []\n",
    "    for num_points in range(2, 40):\n",
    "\n",
    "        var = 0\n",
    "        for _ in range(100):\n",
    "            poly = PolynomialFeatures(model_complexity)\n",
    "\n",
    "            X_real_tran = poly.fit_transform(X_real[:, None])\n",
    "            y_best = best_models[model_complexity].predict(X_real_tran)\n",
    "            X = np.random.uniform(-1, 1, size=num_points)\n",
    "            y = np.sin(np.pi * X)\n",
    "\n",
    "            X = poly.fit_transform(X[:, None])\n",
    "            reg = LinearRegression(fit_intercept=False)\n",
    "            reg.fit(X, y)\n",
    "\n",
    "            preds = reg.predict(X_real_tran)\n",
    "            var += mean_squared_error(preds, y_best)\n",
    "\n",
    "        var /= 100\n",
    "        variances.append(var)\n",
    "        \n",
    "    model_vars.append(variances)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x116716dd0>"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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heT/11FNtH1943u2OF573L7/80hJjl3e75/rZZ58dkndjTIOvB3bjvPjii5a8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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115f3ff90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.heatmap(model_vars[:3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Again the variance increases as we get more complex models, but it then decreases later on as we get more data points. This is what we expect. Finally we show off the bias variance combination:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x116852e50>"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1168f9bd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.heatmap(np.array(model_vars) + np.array(biases))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Conclusion\n",
    "\n",
    "There you have it. Notice that there is a time when the second hypothesis set does better, but as the data increases, the fourth hypothesis wins the day. And then finally as we get even more data, the final and most complex hypothesis will win out. \n",
    "\n",
    "This is the fundamental nature of things here, complexity is good, but only when you have the data.\n",
    "\n",
    "Next time we will show this off one final time with an analysis of why this fundamentally happens. But until then, remember either one of the above: the bias variance trade off of the approximation generalization trade off."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Learning Objectives\n",
    "\n",
    "Today we learn about the second most important tradeoff: the bias variance tradeoff."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Comprehension Questions\n",
    "\n",
    "1.\tWhere does the randomness come in training an ML algorithm (say linear regression)?\n",
    "2.\tDo we ever really know the bias of a model if you only have a sample from a population?\n",
    "3.\tDoes bias depend on your data sample?\n",
    "4.\tWhy does bias + variance = test error?\n",
    "5.\tWhen you have a lot of data points are low bias models better than high bias ones?\n",
    "6.\tWhy does variance always seem to work against us?\n",
    "7.\tWhat is the lowest bias model?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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