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  "name": "",
  "signature": "sha256:a6c6a1f9f470ca9fefc98912d7c2b3eff35018b73b1523a7fef4667000a89a46"
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<small><i>This notebook was put together by [Jake Vanderplas](http://www.vanderplas.com) for PyCon 2014. Source and license info is on [GitHub](https://github.com/jakevdp/sklearn_pycon2014/).</i></small>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Validation and Model Selection"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This section builds on the previous discussions, and goes into more depth on how to evaluate and improve a the training of a supervised Classification or Regression model.\n",
      "\n",
      "By the end of this section, you will:\n",
      "\n",
      "- Know several metrics by which models can be evaluated\n",
      "- Know the difference between *bias* and *variance*, and why there is a tradeoff\n",
      "- Know how to use *(cross) validation curves* to evaluate & improve your model\n",
      "- Know how to use *learning curves* to further evaluate & improve your model\n",
      "\n",
      "Recall our previous discussion of the importance of splitting data into a *training set* and a *testing set*: we'll be extending that here."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Exploring Validation Metrics"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How can you evaluate the performance of a model?\n",
      "\n",
      "The simplest way might be to count the number of matches and mis-matches. But this is not always sufficient.  For example, imagine you have a situation where you'd like to identify a rare class of event from within a large number of background sources (in my field, an example of this is finding variable stars from the background of non-varying stars).  This \"number of matches\" metric can break down:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Generate an un-balanced 2D dataset\n",
      "np.random.seed(0)\n",
      "X = np.vstack([np.random.normal(0, 1, (950, 2)),\n",
      "               np.random.normal(-1.8, 0.8, (50, 2))])\n",
      "y = np.hstack([np.zeros(950), np.ones(50)])\n",
      "\n",
      "plt.scatter(X[:, 0], X[:, 1], c=y, edgecolors='none',\n",
      "            cmap=plt.cm.Accent);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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i+wtOfIZ8QzgxdCKhjSt3yI23m96G1WeFRqrB0yVPI1116/tlVBmrUJJUgiAd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+lpQb1//x6vX79GbGwsNm3a5DFSt62tDdeuXXPf12g0qKur89u+3/e1D3///BsZKzM8ti8j+h2w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\niFSKBZyISKVYwImIVIoFnIhIpf4H3k4u7riWMx4AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1061f0550>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Quick Exercise #1"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use a *Support Vector Machine*, and see how well the model can classify this data.  You can use what we did above as a template:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.svm import SVC\n",
      "# First instantiate the \"Support Vector Classifier\" (SVC) model\n",
      "\n",
      "\n",
      "# Next split the data (X and y) into a training and test set\n",
      "\n",
      "\n",
      "# fit the model to the training data\n",
      "\n",
      "\n",
      "# compute y_pred, the predicted labels of the test data\n",
      "\n",
      "\n",
      "# Now that this is finished, we'll compute the classification rate\n",
      "# print \"accuracy:\", np.sum(ytest == ypred) * 1. / len(y_test)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Did you do well?  Would you say that this is a good classification scheme for the problem?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run the following to load the solution:\n",
      "#  %load solutions/06-1_svm_class.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "The Problem with Simple Validation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The problem here is that we might not care how well we can classify the **background**, but might instead be concerned with successfully pulling-out an uncontaminated set of **foreground** sources.  We can get at this by computing statistics such as the **precision**, the **recall**, and the **f1 score**:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import train_test_split\n",
      "from sklearn import metrics\n",
      "from sklearn.svm import SVC\n",
      "\n",
      "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)\n",
      "clf = SVC().fit(X_train, y_train)\n",
      "y_pred = clf.predict(X_test)\n",
      "\n",
      "print \"accuracy:\", metrics.accuracy_score(y_test, y_pred)\n",
      "print \"precision:\", metrics.precision_score(y_test, y_pred)\n",
      "print \"recall:\", metrics.recall_score(y_test, y_pred)\n",
      "print \"f1 score:\", metrics.f1_score(y_test, y_pred)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "accuracy: 0.968\n",
        "precision: 0.833333333333\n",
        "recall: 0.625\n",
        "f1 score: 0.714285714286\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "What do these mean?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These are ways of taking into account not just the classification results, but the results **relative to the true category**.\n",
      "\n",
      "$$ {\\rm accuracy} \\equiv \\frac{\\rm correct~labels}{\\rm total~samples} $$\n",
      "\n",
      "$$ {\\rm precision} \\equiv \\frac{\\rm true~positives}{\\rm true~positives + false~positives} $$\n",
      "\n",
      "$$ {\\rm recall} \\equiv \\frac{\\rm true~positives}{\\rm true~positives + false~negatives} $$\n",
      "\n",
      "$$ F_1 \\equiv 2 \\frac{\\rm precision \\cdot recall}{\\rm precision + recall} $$\n",
      "\n",
      "The **accuracy**, **precision**, **recall**, and **f1-score** all range from 0 to 1, with 1 being optimal.\n",
      "Here we've used the following definitions:\n",
      "\n",
      "- *True Positives* are those which are labeled ``1`` which are actually ``1``\n",
      "- *False Positives* are those which are labeled ``1`` which are actually ``0``\n",
      "- *True Negatives* are those which are labeled ``0`` which are actually ``0``\n",
      "- *False Negatives* are those which are labeled ``0`` which are actually ``1``\n",
      "\n",
      "\n",
      "We can quickly compute a summary of these statistics using scikit-learn's provided convenience function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print metrics.classification_report(y_test, y_pred,\n",
      "                                    target_names=['background', 'foreground'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "             precision    recall  f1-score   support\n",
        "\n",
        " background       0.97      0.99      0.98       234\n",
        " foreground       0.83      0.62      0.71        16\n",
        "\n",
        "avg / total       0.97      0.97      0.97       250\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This tells us that, though the overall correct classification rate is 97%, we only correctly identify 67% of the desired samples, and those that we label as positives are only 83% correct!  This is why you should make sure to carefully choose your metric when validating a model."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Quick Exercise #2"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compare the *K Neighbors Classifier*, the *Gaussian Naive Bayes Classifier*, and the *Support Vector Machine Classifier* on this toy dataset, using their default parameters.  Which is most accurate?  Which is most precise?  Which has the best recall?\n",
      "\n",
      "Imagine that you are a medical doctor, and here the features are biometric measurements, and the labels are\n",
      "\n",
      "- 0 = no disease\n",
      "- 1 = potential infection\n",
      "\n",
      "For your patients, which classifier might you choose? (That is, would you be more concerned with accuracy, precision, or recall?)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run the following to load the solution:\n",
      "#  %load solutions/06-2_unbalanced.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Cross-Validation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using the simple train/test split as above can be useful, but there is a disadvantage: **Your fit is ignoring a portion of your dataset**.  One way to address this is to use cross-validation.\n",
      "\n",
      "The simplest cross-validation scheme involves running two trials, where you split the data into two parts, first training on one, then training on the other:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X1, X2, y1, y2 = train_test_split(X, y, test_size=0.5)\n",
      "print X1.shape\n",
      "print X2.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(500, 2)\n",
        "(500, 2)\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y2_pred = SVC().fit(X1, y1).predict(X2)\n",
      "y1_pred = SVC().fit(X2, y2).predict(X1)\n",
      "\n",
      "print np.mean([metrics.precision_score(y1, y1_pred),\n",
      "               metrics.precision_score(y2, y2_pred)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.794117647059\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is known as **two-fold** cross-validation, and is a special case of *K*-fold cross validation.\n",
      "\n",
      "Because it's such a common routine, scikit-learn has a K-fold cross-validation scheme built-in:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import cross_val_score\n",
      "\n",
      "# Let's do a 2-fold cross-validation of the SVC estimator\n",
      "print cross_val_score(SVC(), X, y, cv=2, scoring='precision')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.90909091  0.76190476]\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It's also possible to use ``sklearn.cross_validation.KFold`` and ``sklearn.cross_validation.StratifiedKFold`` directly, as well as other cross-validation models which you can find in the ``cross_validation`` module."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Quick Exercise #3"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The ``SVC`` classifier takes a parameter ``C`` whose default value is ``1``.  Using 5-fold cross-validation, make a plot of the precision as a function of ``C``, for the ``SVC`` estimator on this dataset.  For best results, use a logarithmic spacing of ``C`` between 0.01 and 100."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run the following to load the solution:\n",
      "#  %load solutions/06-3_5fold_crossval.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Grid Search"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This exercise you just completed is an example of a **grid search** for model evaluation.  Again, because this is such a common task, Scikit-learn has a grid search tool built-in, which is used as follows.  Note that ``GridSearchCV`` has a ``fit`` method: it is a meta-estimator: an estimator over estimators!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.grid_search import GridSearchCV\n",
      "\n",
      "clf = SVC()\n",
      "Crange = np.logspace(-2, 2, 40)\n",
      "\n",
      "grid = GridSearchCV(clf, param_grid={'C': Crange},\n",
      "                    scoring='precision', cv=5)\n",
      "grid.fit(X, y)\n",
      "\n",
      "print \"best parameter choice:\", grid.best_params_"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "best parameter choice: {'C': 0.13433993325989002}\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "/Users/jakevdp/anaconda/lib/python2.7/site-packages/sklearn/metrics/metrics.py:1734: UserWarning: The sum of true positives and false positives are equal to zero for some labels. Precision is ill defined for those labels [ 1.]. The precision and recall are equal to zero for some labels. fbeta_score is ill defined for those labels [ 1.]. \n",
        "  average=average)\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "scores = [g[1] for g in grid.grid_scores_]\n",
      "plt.semilogx(Crange, scores);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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OgnINAOdihnwkElFBQcHA80AgoEgkMqRNW1ubli5dKkny+Xwp6CYkZvIAnIsZ8okE9vLl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\nf4QAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10620f190>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Grid search can come in very handy when you're tuning a model for a particular task."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Quick Exercise #4"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use ``GridSearchCV`` with the nearest neighbors estimator (``sklearn.neighbors.KNeighborsClassifier``) to determine the optimal number of neighbors (``n_neighbors``) for classifying digits (use accuracy as your metric)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.neighbors import KNeighborsClassifier\n",
      "from sklearn.datasets import load_digits\n",
      "digits = load_digits()\n",
      "X, y = digits.data, digits.target\n",
      "\n",
      "# construct the K Neighbors classifier\n",
      "\n",
      "\n",
      "# Use GridSearchCV to find the best accuracy given choice of ``n_neighbors``\n",
      "\n",
      "\n",
      "# Plot the accuracy as a function of the number of neighbors.\n",
      "# Does this change significantly if you use more/fewer folds?"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run the following to load the solution:\n",
      "#  %load solutions/06-4_gridsearch.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Overfitting, Underfitting and Model Selection"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now that we've gone over the basics of metrics, validation, and cross-validation, it's time to go into even more depth regarding model selection.\n",
      "\n",
      "The issues associated with validation and \n",
      "cross-validation are some of the most important\n",
      "aspects of the practice of machine learning.  Selecting the optimal model\n",
      "for your data is vital, and is a piece of the problem that is not often\n",
      "appreciated by machine learning practitioners.\n",
      "\n",
      "Of core importance is the following question:\n",
      "\n",
      "**If our estimator is underperforming, how should we move forward?**\n",
      "\n",
      "- Use simpler or more complicated model?\n",
      "- Add more features to each observed data point?\n",
      "- Add more training samples?\n",
      "\n",
      "The answer is often counter-intuitive.  In particular, **Sometimes using a\n",
      "more complicated model will give _worse_ results.**  Also, **Sometimes adding\n",
      "training data will not improve your results.**  The ability to determine\n",
      "what steps will improve your model is what separates the successful machine\n",
      "learning practitioners from the unsuccessful."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Illustration of the Bias-Variance Tradeoff"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For this section, we'll work with a simple 1D regression problem.  This will help us to\n",
      "easily visualize the data and the model, and the results generalize easily to  higher-dimensional\n",
      "datasets.  We'll explore a simple **linear regression** problem.\n",
      "This can be accomplished within scikit-learn with the `sklearn.linear_model` module.\n",
      "\n",
      "We'll create a simple nonlinear function that we'd like to fit"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def test_func(x, err=0.5):\n",
      "    y = 10 - 1. / (x + 0.1)\n",
      "    if err > 0:\n",
      "        y = np.random.normal(y, err)\n",
      "    return y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now let's create a realization of this dataset:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def make_data(N=40, error=1.0, random_seed=1):\n",
      "    # randomly sample the data\n",
      "    np.random.seed(1)\n",
      "    X = np.random.random(N)[:, np.newaxis]\n",
      "    y = test_func(X.ravel(), error)\n",
      "    \n",
      "    return X, y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X, y = make_data(40, error=1)\n",
      "plt.scatter(X.ravel(), y);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ffIXdPg6oBdTGbh/HJ5985e5YInINU6GXUNWqlfDy2nX2uZfXTqpVq+TGRCJy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\nRajQRUQs4v8BBngv2rMlpVwAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1062055d0>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now say we want to perform a regression on this data.  Let's use the built-in linear regression function to compute a fit:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X_test = np.linspace(-0.1, 1.1, 500)[:, None]\n",
      "\n",
      "from sklearn.linear_model import LinearRegression\n",
      "model = LinearRegression()\n",
      "model.fit(X, y)\n",
      "y_test = model.predict(X_test)\n",
      "\n",
      "plt.scatter(X.ravel(), y)\n",
      "plt.plot(X_test.ravel(), y_test)\n",
      "print \"mean squared error:\", metrics.mean_squared_error(model.predict(X), y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "mean squared error: 1.78514645061\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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vsV+0aBHXrtUnM3MlBsNU9PoVjB496ZHnzMnRphs2bQpdukBgIJw5o80nlzAX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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1062406d0>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have fit a straight line to the data, but clearly this model is not a good choice.  We say that this model is **biased**, or that it **under-fits** the data.\n",
      "\n",
      "Let's try to improve this by creating a more complicated model.  We can do this by adding degrees of freedom, and computing a polynomial regression over the inputs.  Let's make this easier by creating a quick PolynomialRegression estimator:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class PolynomialRegression(LinearRegression):\n",
      "    \"\"\"Simple Polynomial Regression to 1D data\"\"\"\n",
      "    def __init__(self, degree=1, **kwargs):\n",
      "        self.degree = degree\n",
      "        LinearRegression.__init__(self, **kwargs)\n",
      "        \n",
      "    def fit(self, X, y):\n",
      "        if X.shape[1] != 1:\n",
      "            raise ValueError(\"Only 1D data valid here\")\n",
      "        Xp = X ** (1 + np.arange(self.degree))\n",
      "        return LinearRegression.fit(self, Xp, y)\n",
      "        \n",
      "    def predict(self, X):\n",
      "        Xp = X ** (1 + np.arange(self.degree))\n",
      "        return LinearRegression.predict(self, Xp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we'll use this to fit a quadratic curve to the data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model = PolynomialRegression(degree=2)\n",
      "model.fit(X, y)\n",
      "y_test = model.predict(X_test)\n",
      "\n",
      "plt.scatter(X.ravel(), y)\n",
      "plt.plot(X_test.ravel(), y_test)\n",
      "print \"mean squared error:\", metrics.mean_squared_error(model.predict(X), y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "mean squared error: 0.919717192242\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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XV7NmfQoV+ghQwEm8vH6gXr16d7Vr06YzdeseJCamJ7/8YsRkygC6oXW1PEGh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\nIYSbkIIuhBBuQgq6EEK4if8H0d69Sjdz9OEAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x106230410>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This reduces the mean squared error, and makes a much better fit.  What happens if we use an even higher-degree polynomial?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model = PolynomialRegression(degree=30)\n",
      "model.fit(X, y)\n",
      "y_test = model.predict(X_test)\n",
      "\n",
      "plt.scatter(X.ravel(), y)\n",
      "plt.plot(X_test.ravel(), y_test)\n",
      "plt.ylim(-4, 14)\n",
      "print \"mean squared error:\", metrics.mean_squared_error(model.predict(X), y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "mean squared error: 0.375570136658\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5eXksWfItOTnfADdSXDwRg2Ebhw9f4OOPje9v/PjnSUzMpqjoLIWFB1m1Kgo/\nv88pLOyCtpY6QE9OnDjk0LbeU089wYsvDqVZs6eJi3uXRYs+Z/XqpTz2mB81aybRoEFXfv99PdVN\nXB65bZvWZlq8GEJCHFbmVU1G7MZJqJsgI3XrzJsH994LxlYh8PDwuHjh0+X1gAyGbEaPTub557Vf\nCFdatWo9ublPAwFAKEpFYDBswNPzByD94j4+oFmzNg69qMpgMDB+/Fh27drEtm1r6d27N76+vsyY\nMZXU1N54eHRn5cpwo6/94w9tGYCPPoLYWIeVeNUqHeYyci9PQt0E6albppQW6qZutuDt7c2IEQ8R\nEHAL8CXe3o9To0Yqo0Z15skntZUEr/zFWa9eGAbD7yWfe3ouoW3blkyaNBYfn2b4+4cREfExS5eW\nndJXmYKDtZN1L7wAM2ZoJ4EBCgrg44+1G17MmqUtBib0JSFumV1TGqsyGalbtmmTtpaHubnX77//\nJtHR77Nq1TIaNKjL5Mm/EhwczPjxWr95+nR49tnS279Kp049ycs7R17ecOrXf41nnkkiJCSERx4Z\nxblz56hbt67TL4Fv1gzWr4eHHoI334ToaPjnH+0q2rVrISbGqeVVaTLYMk9C3QTpqVv2xRfaCobm\nRk8eHh6MGfMoY8Y8WuZxLy/tNmzt2mlBePfd2uPNmzfn11+30717MGPG/M7zz28sWTY4MDDQ4pzw\nyhQZCStXamua7N+vXf5+xXlioTMZqVsmoW6CtF/My8/XQvn33y1va0r9+rBihbbm9z//aNcGHD4M\nw4eHMXIkTJ/eXb+CHcRggOuu0/4TwhVIT90Eab+Yt3w5tGhh/4p7sbGwYQPs3KnNEunaFe65B6ZN\n06dOUfXIYMs8Gamb4OFx+QSYKE+b9VLIyZNnqVWrll0zURo1gm++0dYG8fCQP7GFafKzYZmM1E2Q\nnrppZ85AYmI+jz/egIiIpjRoEM3u3bvt3q+np7xphWVVdeldvUiomyA9ddPefDOd/PwV5OevIS/v\nDOnpj9G3793OLktcBa78pS+DgPIk1E1w5576mTPaSccuXbQpd+np+u7/m2+88fLaDzQDQKnRHDq0\nm5ycHH0PJISoMAl1E9w11A8dgo4dtTvsTJmirV/eoQOkpOiz//374dSpGnh6zgWyLz66mYCAavj5\n+elzECHMkL+gzZMTpSa4Y089Lw/uuguGD4enn9Yei4/X1rG++WZtPZI6dew7xhdfwODBPpw7dwPf\nfx+Lp2d6thJ4AAAbXklEQVQMhYW/Mn/+HLe/F6pwffIjZpndoX7u3DkefPBBUlJSMBgMzJ49mxtu\nuEGP2pzKHXvqL76orQH/1FNlH7/vPu0u9sOGaZe32/rGUAq+/BIWLDDQtu1HbNy4kaNHj3L99W/S\nyN4bTwphJXd7X1Y2u0N9zJgx9O3bl++++47CwkKysrL0qMvp3K39sn+/tu7IX38ZD+2XX4brr9fu\nwGPr2t6//XZ5WQCDwUDnzp3tK1qICpKRumV29dQzMjL45ZdfGD58OABeXl4mlyJ1N+4W6s89p50c\nrVvX+PPe3vDOO/Dkk9rdeGwxdy4MGSJvLOEaZMRunF2hnpqaSu3atRk2bBht2rRh5MiRZNuaGC7G\nnXrqe/Zoa5qPGWN+u/h4aN9eW1mworKy4LvvtFAXwplk6V3z7Ar1wsJCtm3bxujRo9m2bRuBgYFM\nnz5dr9qcyp166jNmwMMPFzN79rsMHDiCl16aavKX64wZ2oj9yJGKHWPhQujUCRx8a1AhzJIQt8yu\nnnp4eDjh4eG0a9cOgAEDBhgN9cmTJ5d8HB8fT3x8vD2HrRTu0n45fFgL3O7dx7By5V9kZw/Cz28V\nP/zQl99+W42XV9l/4oYNtXXMJ02CTz6x/jiffWb5LwEhKoO7DLYqKikpiaSkJPt3pOzUtWtXtXv3\nbqWUUpMmTVJPPfVUmed1OIRTvP++Uv/9r7Or0Lz//keqZs1rVVBQLTVixCMqLy+v5Lknn1Rq5MhM\n5etbQ0Gm0n7ki1RQUAv166+/Gt3f2bNK1a6t1M6d1h1/1y6l6tRRqtRhhXCKRYuUuuMO7WMfH6Vy\nc51bjyPZmp12X3w0a9Ys7rvvPmJjY/nzzz95tvQdD9yYq4zUly1bxvjx0zh9ejGZmduZP38vEya8\nAGhXjs6eDcOHn8PDw5fL9/D0wGCoxptvvkf16nWpWfNaZs58t2SfNWpoN6a4cuqjKW++CQ8/DD4+\n+n5tQgj92R3qsbGxJCcns2PHDhYtWlSlZr9Uxp9527ZB//4wdKh2NeiVFi9eQXb240AcEE5OznSW\nLFkBwHvvQb9+0L59XaKjo/DxeRjYgqfnSyi1nxUr0jh/fiNnzvzIs8/O5Ntv/1ey39GjYfdu7QSr\nOSdOaOumjx6t11cshH2qavtFL7JMgAkeHtpSsI506JB2g4ibb9ZustCpU/lgr137Gry89pZ6ZC81\natTgwgXthOczz2h3F1q9egl33VVEZOTDJCT8RVhYBDk5U4FGQCuysyewcOGKkr34+Ghrlo8fb/4v\nkilTYPBg+69EFUIPcqLUMlkmwARHtV/27NnDW2+9T3Z2Lv/++yKPPhrKQw9pz1WvDn37wubNcPEO\nbowd+xizZ99ARsZ9FBbWxsfnK2bNWsT778NNN0HTptp211xzDfPnf1pynC5d+rBv336gGwCenvup\nXbtGmVoGDIB334W33tLmr5evFebPh1279P4uCGE/GbEbJ6Fugqen/iP1PXv20LZtFzIzH0apVoA3\no0b9CNwKaLNLduzQ+tfz5mmjkjp16pCSksz8+fPJycnh1lvX0aBBc+6+G1avNn2sN96YTM+et5KX\ntwMPjwsEBf3E00//VmYbg0G7oKh9e21Fxw4dLj+Xl6eN0P/v/6B2bX2/D0LYQ+apmyehboKnp/4j\n9VmzPiIz878o9eLFRw4ydeo07r331lLbaCE7Z462MBdAzZo1eeyxx0q2eekl6NYNiov/5LPPkgkP\nD6dXr15lFtTq0KEDW7f+yvfff4+Pz7UMGjSVsLCwcjU1aqQdq18/WLIEbrhBW+HxwQe1e4iWOqwQ\nTichbpmEugmO6Knn5uahVOnr+C+Ql5dXZpvAQO3EZPfu0KZNIStWvMUvv/xOVFQDJk9+lrNnazBz\nJjz33EI6dnwEg+FmYCt9+lzPt9/OLRPsTZs2ZeLEiRbruvVW+PRTuP12aNxY6+t36gRffSVvIuF6\npO1inoS6CY5ovwwZMpD58weQnR0N3Iy//xBGjbq/3HbNm2tXfnbrdoLCwnXk5Azi559/ZtmyPlSr\ntoGJExXPPTeM3NxNQHMglxUr4li3bp3NF3bddhscOADJyRAWBs2a2fOVCuEYMsiwTGa/mOCIUL/2\n2mt5+eWniIjYTFDQb7z00gOMHz/W6La3336WzMyfyMlZAgwiL+8TUlPfoHr14zz44AWKihRaoAP4\n4eERw9GjR+2qLzBQWx9GAl0I9yWhboLeoT5lyms0bRrHSy8t4OjRxgwZEsb48WNN3liiqKgIL69x\npR4x4Om5nyee+JMaNaoTHt4Qg+EtoBjYTFFRUslyDUJUZdJ+MU9C3QQ9Q3379u1MmzaT3NydZGQk\nU1g4gNmzh/LBBx/w888/G31NzZo16dKlC35+9wE/4+n5PLVqvUR8fGcMBgOrVi2mceMv8PDwJTj4\nVhYs+IzGjRvrU7AQLqr0GEjC3TgJdRP0DPXdu3fj6dkJuHSS1IOcnL08+eTv9Ov3EI88Un6SuMFg\n4Mcfv2X48PrExr7I7bf/y5YtSQRdnMAeGRnJnj3byMq6QEbGCfr166dPsUK4OJnSaJ6cKDVBz1CP\njo6mqGgDkA4EA9uBP8jJqQdkMGdONI8++iDR0dFlXhcQEMB7771hdt9ys2dxNZEQt0xG6iboOU89\nNjaWSZMm4OvbEm/v2UAKcGlh8ur4+ERy/PhxfQ4mhLiqSaiboPc89aeeGsfBg7vo1GkIQUEpwDy0\nk5zLKC7eQ8uWLfU7mBBVmPTSzZNQN8ERUxrDwsLYvz+Eb74Zw7XXTsdg8KZOnUdZvnwhNWvW1Pdg\nQlRB0n6xTHrqJjgi1E+ehMxM6NOnKYcO/U1BQQHe3t76HkSIKk5G6ubJSN0ER4R6SgrExFwebUig\nC1ExMqXRMl1CvaioiLi4OG677TY9ducSHBHqf/+tLQEghNCHtGPK0yXUZ86cSfPmzU1eHemOJNSF\ncE0yQjfP7lBPT09n+fLlPPjgg6gq9N12xNK7EupC2KcKjRsdxu5QHzduHDNmzMDDo2q15x2x9K6E\nuhDC0exK4h9//JE6deoQFxdXpUbpYLr9kpWVxejRTxAXF8/AgcM5duyYVfs7fRpycqBePcvbCiFM\nq2JRozu7pjRu3LiRpUuXsnz5cnJzczl//jwPPPAA8+bNK7Pd5MmTSz6Oj4+3ec3vymQs1JVS9O59\nF7//HkJu7gvs3PkTv/3Wg127thIQEGB2f7t2aaN0+fNRCNtV5fdPUlISSUlJdu/HoHQaYq9bt47X\nX3+dH374oewBDAa3HMXv3Qt9+sC+fZcfO3LkCJGRrcjNPcal34fBwTewePEr3HjjjWb39/HHsGkT\nzJ7twKKFqOISE7Ubpf/0E3h5abde9KqiV9vYmp26NsKr+uwXT09PlCoCCi8+ooB8PD09Le5P+ulC\n2E/mqVumW6h3796dpUuX6rU7pzMW6qGhoSQk9MLfvz+wAF/fEYSHe9KxY0eL+5NQF0J/VWgcqZuq\nNWVFR6ZOlC5a9CXPPNONXr0W8fDDtfntt9X4+PhY3J+EuhD6kBG6eVW0G2U/U/PUvb29eeGFZ/j5\nZ2jaFKpXt7yvjAw4dw6uvVb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       "text": [
        "<matplotlib.figure.Figure at 0x106230b90>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When we increase the degree to this extent, it's clear that the resulting fit is no longer reflecting the true underlying distribution, but is more sensitive to the noise in the training data. For this reason, we call it a **high-variance model**, and we say that it **over-fits** the data."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Detecting over-fitting\n",
      "Clearly, computing the error on the training data is not enough (we saw this previously).  But computing this *training error* can help us determine what's going on: in particular, comparing the training error and the validation error can give you an indication of how well your data is being fit.\n",
      "\n",
      "Let's do this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "degrees = np.arange(1, 30)\n",
      "\n",
      "X, y = make_data(100, error=1.0)\n",
      "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)\n",
      "\n",
      "training_error = []\n",
      "test_error = []\n",
      "mse = metrics.mean_squared_error\n",
      "\n",
      "for d in degrees:\n",
      "    model = PolynomialRegression(d).fit(X_train, y_train)\n",
      "    training_error.append(mse(model.predict(X_train), y_train))\n",
      "    test_error.append(mse(model.predict(X_test), y_test))\n",
      "    \n",
      "# note that the test error can also be computed via cross-validation\n",
      "plt.plot(degrees, training_error, label='training')\n",
      "plt.plot(degrees, test_error, label='test')\n",
      "plt.legend()\n",
      "plt.xlabel('degree')\n",
      "plt.ylabel('MSE');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Gjt0cxQ6FSFIu37qM57Y9hy3PbMEY7zFih0MdYBIlBIDzIhDpw536O5icNBkfDPuAyUAC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\nhwmBiIgAAP8PZ6TQCKbg+y8AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x106211950>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is a typical bias/variance plot.\n",
      "\n",
      "On the **Left Side** of the plot, we have a **high-bias** model, characterized by the training and test data showing equally bad performance.  This shows that the model is **under-fitting** the data, because it does equally poorly on both known and unknown values.\n",
      "\n",
      "On the **Right Side** of the plot, we have a **high-variance model**, characterized by a divergence of the training and test data.  Here the model is **over-fitting** the data: in other words, the particular noise distribution of the input data has too much effect on the result.\n",
      "\n",
      "The optimal model here will be around the point where the **test** error is minimized."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Quick Exercise # 5"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's appy this learning curve idea to a **Decision Tree** regression on this dataset. ``sklearn.tree.DecisionTree`` takes a number of hyper-parameters, but one that can really control the over-fitting, under-fitting is the ``max_depth`` parameter especially controls the tradeoff between bias and variance.\n",
      "\n",
      "Generate a dataset like the one above, but containing 500 points, and use a Decision Tree regressor to explore the effect of the ``max_depth`` parameter on the bias/variance tradeoff.  What is the optimal value?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.tree import DecisionTreeRegressor"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run the following to load the solution:\n",
      "#  %load solutions/06-5_decisiontree.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Quick Exercise #6"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "One popular and effective way to address over-fitting is to use **Ensemble methods**.  ``sklearn.ensemble.RandomForestRegressor`` uses multiple randomized decision trees and averages their results.  The ensemble of estimators can often do better than any individual estimator for data that is over-fit.  Repeat the above experiment using the Random Forest regressor, with 10 trees.  What is the best ``max_depth`` for this model?  Does the accuracy improve?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run the following to load the solution:\n",
      "#  %load solutions/06-6_randomforest.py"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Illustration of Learning Curves"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The exact turning-point of the tradeoff between bias and variance is highly dependent on the number of training points used.  Here we'll illustrate the use of *learning curves*, which display this property.\n",
      "\n",
      "The idea is to plot the mean-squared-error for the training and test set as a function of *Number of Training Points*"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X, y = make_data(200, error=1.0)\n",
      "degree = 3\n",
      "\n",
      "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)\n",
      "\n",
      "N_range = np.linspace(15, X_train.shape[0], 20).astype(int)\n",
      "\n",
      "def plot_learning_curve(degree=3):\n",
      "    training_error = []\n",
      "    test_error = []\n",
      "    mse = metrics.mean_squared_error\n",
      "    \n",
      "    for N in N_range:\n",
      "        XN = X_train[:N]\n",
      "        yN = y_train[:N]\n",
      "    \n",
      "        model = PolynomialRegression(degree).fit(XN, yN)\n",
      "        training_error.append(mse(model.predict(XN), yN))\n",
      "        test_error.append(mse(model.predict(X_test), y_test))\n",
      "      \n",
      "    plt.plot(N_range, training_error, label='training')\n",
      "    plt.plot(N_range, test_error, label='test')\n",
      "    plt.plot(N_range, np.ones_like(N_range), ':k')\n",
      "    plt.legend()\n",
      "    plt.title('degree = {0}'.format(degree))\n",
      "    plt.xlabel('num. training points')\n",
      "    plt.ylabel('MSE')\n",
      "    \n",
      "plot_learning_curve(3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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r06+PdB4p/3AIDAzElStXpMbbes43CgeijBgDrl7lwuDIEeD6dSA0lAuECROA\nvn1bzy8WA8nJQHw8UFwMvPsuMH8+YGyskPKJGmMMKC0FrK3V4PIZhKiCqirgxAkuDI4eBYyMuDD4\n7DNgxAhAv4Nz2rS0gKgobjh/nguJv/0NWLgQeOstrumJkM5gDKioAHJygNxc6cfcXMDAoGvroHAg\npAOMcU1CLXsHly4Bw4ZxgbBiBeDmJttyX3gB2L2b+2P+178APz8gPJxrcgoIkO9nIKqpulp6w//k\nuIYG4OzMDU5OQP/+wLhxj58bG3PzyKrdZiUDAwPJ5Syys7Ph4uIimZadnY26ujrZ1/qcqFmJKMqn\nnwKJicDkyVwgjBwJGBrKfz2VlcD33wPr13N/5O+/zzVNafJ6xxWiKLW1QH4+UFDADW2NNzY+3tA/\n+dgy3sZl6KTw0lspNze3wzc6OTnJtEJZUDj0bLdvc7+iBg7s3vXu3g0sWwZcvMgdSO4OjY3ceuPj\ngUePgOhorjmqsZEbmpoej7f32tPPbWyA4cO5YcgQfsLtSTU13PGXa9cAU1Ng+nRAuxvbKOrqgNWr\nAaEQGDQIGDqUGxwcumf9TU3AnTuPN/LtbfgdHLimxJbHp8fNzbv2yx/opq6sZWVlOHXqFPr27YuB\n3fxXSuHQc/30E9ceDwCbN3O/3rvD1avA2LFc76PAwO5Z55MY4zZuKSnchlVX9/Ggo9P55zo6wN27\nwO+/A6dPAxkZgI8PEBLChcWwYYClpew15uZyIZCR8XgoKgK8vAB/f+DWLaCkBPj4Y+Dll/kNCcaA\nAweAd97hmu3mzweuXAHOneMGXV0uJIKDucfAQEBPr+vrzM7mfkCkpXFDRgZgZwc4Ora/8Tcz6/qG\nvzN4CYfw8HCsXbsWPj4+KC4uRmBgIAYNGoTs7GzExMTgvffe61LRz1UkhUOP09DAnRfwyy/cL+mG\nBiAyEvj2W2DaNH7XXVrK/eJcuxaYOZPfdXW3R4+4Ddnp01xgnDsH2NpyQdESGM7O0huuujrgxo3W\nIXDtGteu7efHBUHL4Ob2OARaQi42lguNjz4CXnlF/iGRnQ28/Tbw559AQgIwenTr6Yxx086efRwW\nt25x9bbsWQwdCtjbd7ye+/cfh0BaGvddGhoCgwc/HgYOVJ4eaLyEg7e3N/744w8AwOrVq5GZmYkt\nW7agpqYGwcHBz7yzmzxROPQsOTnAjBncr6xNm4DevbnXr1zh9hzWreM2MHxoauL2GIKDuaYJdScW\nc01ALWGZE4puAAAaf0lEQVRx+jQXDMOHA+7uQGYmFwT5+YCHB7cxbQkDPz+gT5/Or0soBFat4vZk\nPv6Y+zfU6eLJwY8ecSGekAB88AHw3nvcHkJn1NZyG/eWsDh3DujVq3VYNDS0DoOqKu6HQ0sQDBrE\nhauy4iUcAgICcPXqVQDAqFGjEBMTg+joaACAv78/MjIyZCxXhiIpHHqMgweB114D/vpXrnng6V+w\nN29yPTI+/RR4/XX5r3/JEm7jlZzcMw8GM8aF8++/c8d6PD25EHB37/qGvMWpU1xI5ORwexJz58q2\n7EOHuL2FAQO4Ewy7ekyBMe5YQUtQnD/PBc2QIY/DwNVVtf5fdGnbydoRHh7O1q9fz37++WdmamrK\nysvLGWOMPXz4kHl5ebX3Nl50UCZRE01NjH34IWMODoydPdvxvLdvM9a3L2NffinfGjZsYMzDg7HK\nSvkul7Tt1CnGRo9mzMmJse+/Z6yhoXPv+/NPxiIjGXNzYywlhd8aVV1Xtp3tZmBiYiJu3LiBzZs3\nY+fOnZLbd164cAHz58+XLYkIaUNhITBqFNd8kZ7O7c53xNWV+/X5738D//gH94uvq37/HfjkE26P\noaUZi/ArJIQ7oXDbNu64Uv/+wIYNXE+ettTXA3//OxAUxP2av34dGD++e2vuSejCe0ShTpwA5szh\nmnNWrny+XfbiYu74QEQEd3xA1t4fd+9yvVs2baKNjSKdPcs1N2Vmcs2K8+c/7k2UksL1WvPx4ZqQ\nurEnvUrj5ZhDREREuwvW0NDAgQMHZFqhLCgc1E9zM/er/7vvuF+Oo0bJtpyyMm6DPmwY8PXXz98e\nXFfH/YKNjubOaSCKd/48FxJ//MGdDHjqFLdXuX5993VlVhe8hIOlpSUEAgGio6MxZMgQAJCsREND\nAy+++KKM5cpQJIWDWikt5Xqq1NcDO3ZwfcK7orKS22h4enJnGXf2aqeMPe57v2VL9/Q7J5134QLw\nxReAry/w4YcdX7eKtI2XcBCJRDh+/Dh27NiB69evIzw8HNHR0fD29u5SsbKgcOAUFXEbVmNjwMSE\ne+zqSTzd7exZYNYsYPZsrv1YXv3da2u58yBsbLiT5TrT+2XtWmDPHu6XaVcvUkaIMuL9DOmGhgbs\n2LEDy5YtQ2xsbLfeywHo2eFQXQ38/DPX9HLlCnd2ZU0N93pNDfdr98mweHL8yUdLS6691teXOy2/\nuzHGtRWvXctdq2jSJPmv49Ej7lINOjrAzp0dB+eRI0BMDPfrlK6GStQVb+FQX1+Pw4cP46effkJu\nbi4iIyOxYMEC2D/rNEI562nh0NTEHYDbto17HDmSO2gbHi69a93Q8Dgo2nusqeEO3t64wfXw6N2b\n67v+5NC/v/z6sT+tshJYsIC7psyuXfweTGxs5JqKamqAffu4k5qelpnJXV57/366MxtRb7yEw5w5\nc/DHH39g4sSJmDlzJnx9fbtUZFf0hHBgjPsVu20b96vX3Z1rl3/pJcDCQn7raW4G8vK4Sx88Ody9\ny63z6dCwtn52W3xTE3dg+P59rtnr/v3HQ2kp1yMpIoI7s7k7msFEIi6McnO5E6VMTB5Pq6zkukEu\nX87NQ4g64yUcNDU1YdjO5Rs1NDRQXV0t0wploc7hcPs2sH07FwpaWlwgzJ4N9OvXvXXU1XFnHz8Z\nGBkZXE1+flxzlIlJ2wFQU8MFmJXV48HS8vG4pyd3OYbu1NzMdY9NT+duymNuzl0qYtIk7to/69d3\nbz2EKEK3XJVVFgsWLMDhw4dhZWXV7rWY3n77bRw9ehS9evVCUlISAtu4BKa6hUNpKde8sm0bdzGw\nWbO4UAgKUq4eM4xxzVEtYfHwYesAaBnMzJTzkgKMcd1TT5wAjh/nLoN96RLXVMdXExohykRpw+H0\n6dMwMjLC3Llz2wyHI0eOICEhAUeOHMGFCxfwzjvv4Pz589JFqkE4iMVcz5ht27jeMeHhXCCMHUsb\nKj4xxt3CMzGRa9K6eFG+zXSEKLOubDt5vQVHSEhIhzcNOnDgAF599VUAwJAhQ1BZWYmSkhJYW1vz\nWZZC7NrFbaQ++gj48UfluaSvutPQ4O7T3K8fd6yBgoGQzlHoPaQLCwvh8MSlFAUCAQoKCtQyHG7c\n4I4l/C8LSTebN0/RFRCiWhQaDgCkdnk02ml0j42NlYyHhoYiNDSUx6o61lLL8zzu2wd88snzv48e\n6ZEe6bGzj0KhUPK8q9tI3i+8l5ubi4iIiDaPObzxxhsIDQ3FrFmzAAAeHh5ITU2V2nNQh2MOfn5A\nUhJ37XlCCOkOXdl2KrSPSWRkJLZs2QIAOH/+PExNTdWySam5mbuJSP/+iq6EEEI6R5vPhUdHRyM1\nNRVlZWVwcHDAqlWr0NTUBABYtGgRJk6ciCNHjsDV1RWGhobYtGkTn+UoTH4+193TyEjRlRBCSOfQ\n/Ry6wbFjQFwc8Ouviq6EENKTqGyzUk+RlcVdmoIQQlQFhUM3oHAghKgaCoducOsWHYwmhKgWCodu\nQHsOhBBVQwekeVZXx10R9OHDzt++khBC5IEOSCuxO3e46/pQMBBCVAmFA8+oSYkQooooHHhG4UAI\nUUUUDjy7dYvCgRCieigceJaVRd1YCSGqh8KBR4xRsxIhRDVROPDo/n2ul1KfPoquhBBCng+FA4/o\nzGhCiKqicOARNSkRQlQVhQOPKBwIIaqKwoFH1I2VEKKqKBx4RN1YCSGqii68x5OmJsDYGKisBPT1\nFV0NIaQnogvvKaGcHMDOjoKBEKKaKBx4Qt1YCSGqjMKBJ9RTiRCiyigceELhQAhRZRQOPKFurIQQ\nVcZrOKSkpMDDwwNubm5Yu3at1PSysjKEhYUhICAAPj4+SEpK4rOcbkXdWAkhqoy3rqxisRju7u44\nceIE7O3tMWjQIOzYsQOenp6SeWJjY9HQ0IA1a9agrKwM7u7uKCkpgba2dusiVawra3U111OpuhrQ\npH0zQoiCKGVX1rS0NLi6usLJyQk6OjqYNWsWkpOTW81ja2uL6upqAEB1dTUsLCykgkEVZWUBbm4U\nDIQQ1cXblriwsBAODg6S5wKBABcuXGg1T0xMDEaNGgU7OzvU1NRg165dfJXTragbKyFE1fEWDhoa\nGs+cZ/Xq1QgICIBQKER2djbGjh2LjIwMGBsbS80bGxsrGQ8NDUVoaKgcq5Uv6qlECFEEoVAIoVAo\nl2XxFg729vbIz8+XPM/Pz4dAIGg1z9mzZ/HRRx8BAFxcXODs7IysrCwEBQVJLe/JcFB2WVlAZKSi\nqyCE9DRP/3BetWqVzMvirVU8KCgIt2/fRm5uLhobG7Fz505EPrXF9PDwwIkTJwAAJSUlyMrKQr9+\n/fgqqdtQN1ZCiKrjbc9BW1sbCQkJGD9+PMRiMRYuXAhPT09s2LABALBo0SKsXLkS8+fPh7+/P5qb\nm/H555/D3Nycr5K6RXMzHXMghKg+uiqrnOXnA4MHA8XFiq6EENLTKWVX1p6KmpQIIeqAwkHO6Mxo\nQog6oHCQM+rGSghRBxQOckbhQAhRBxQOckbHHAgh6oB6K8lRfT1gagrU1AA6OoquhhDS01FvJSVx\n5w7g5ETBQAhRfRQOckRNSoQQdUHhIEfUjZUQoi4oHOSIeioRQtQFhYMcUTgQQtQFhYMc0TEHQoi6\noHCQk7IyQCwGLC0VXQkhhHQdhYOctDQpdeIGeIQQovQoHOSEmpQIIeqEwkFOqBsrIUSdUDjICfVU\nIoSoEwoHOaFwIISoE7rwnhyIxYChIVBRARgYKLoaQgjh0IX3FCw3F7C2pmAghKgPCgc5oCYlQoi6\noXCQA+rGSghRNxQOckDdWAkh6obXcEhJSYGHhwfc3Nywdu3aNucRCoUIDAyEj48PQkND+SyHN9Ss\nRAhRN7z1VhKLxXB3d8eJEydgb2+PQYMGYceOHfD09JTMU1lZiWHDhuGXX36BQCBAWVkZ+vTpI12k\nkvdWEgiAM2eAvn0VXQkhhDymlL2V0tLS4OrqCicnJ+jo6GDWrFlITk5uNc+PP/6IadOmQSAQAECb\nwaDsamuB8nLAwUHRlRBCiPzwFg6FhYVweGKLKRAIUFhY2Gqe27dvo7y8HCNHjkRQUBC2bt3KVzm8\nuXULcHUFNOnoDSFEjWjztWCNTlyetKmpCenp6Th58iTq6uowdOhQvPDCC3Bzc5OaNzY2VjIeGhqq\nNMcn6HgDIURZCIVCCIVCuSyLt3Cwt7dHfn6+5Hl+fr6k+aiFg4MD+vTpAwMDAxgYGGDEiBHIyMh4\nZjgoE+rGSghRFk//cF61apXMy+KtMSQoKAi3b99Gbm4uGhsbsXPnTkRGRraaZ/Lkyfj9998hFotR\nV1eHCxcuwMvLi6+SeEHdWAkh6oi3PQdtbW0kJCRg/PjxEIvFWLhwITw9PbFhwwYAwKJFi+Dh4YGw\nsDD4+flBU1MTMTExKhkO77yj6CoIIUS+6MJ7XcAYYGIC5OcDpqaKroYQQlpTyq6sPUFxMdCrFwUD\nIUT9UDh0AR1vIISoKwqHLqBurIQQdUXh0AXUjZUQoq4oHLqAmpUIIeqKwqELqFmJEKKuqCurjBob\nuW6s1dWArq6iqyGEEGnUlVUBsrMBR0cKBkKIeqJwkBEdbyCEqDMKBxnR8QZCiDqjcJARdWMlhKgz\nCgcZUbMSIUSdUTjIiJqVCCHqjMJBBuXlQEMDYGOj6EoIIYQfFA4yaDne0Ik7oRJCiEqicJABHW8g\nhKg7CgcZUE8lQoi6o3CQAR2MJoSoOwoHGVCzEiFE3dGF956TWAwYGQFlZYChoaKrIYSQ9tGF97pR\nfj7Qpw8FAyFEvVE4PCc63kAI6Ql4DYeUlBR4eHjAzc0Na9eubXe+ixcvQltbG3v37uWzHLmg4w2E\nkJ6At3AQi8VYunQpUlJScPPmTezYsQP//e9/25xv+fLlCAsLU5rjCh2RpRurUCjkpZbuosr1q3Lt\nANWvaKpef1fwFg5paWlwdXWFk5MTdHR0MGvWLCQnJ0vN980332D69OmwtLTkqxS5srMDBg16vveo\n+n8wVa5flWsHqH5FU/X6u4K3cCgsLISDg4PkuUAgQGFhodQ8ycnJWLx4MQDuyLqyW7kSCA5WdBWE\nEMIv3sKhMxv6d999F3FxcZLuVqrQrEQIIT0C48m5c+fY+PHjJc9Xr17N4uLiWs3j7OzMnJycmJOT\nEzMyMmJWVlYsOTlZalkuLi4MAA000EADDc8xuLi4yLwN5+0kOJFIBHd3d5w8eRJ2dnYYPHgwduzY\nAU9Pzzbnnz9/PiIiIhAVFcVHOYQQQp6DNm8L1tZGQkICxo8fD7FYjIULF8LT0xMbNmwAACxatIiv\nVRNCCOkilbh8BiGEkO6l1GdId/YkOmWRn5+PkSNHwtvbGz4+Pli/fj0AoLy8HGPHjkX//v0xbtw4\nVFZWKrjSjonFYgQGBiIiIgKAatVfWVmJ6dOnw9PTE15eXrhw4YJK1b9mzRp4e3vD19cXL7/8Mhoa\nGpS2/gULFsDa2hq+vr6S1zqqdc2aNXBzc4OHhweOHTumiJJbaav+Dz74AJ6envD390dUVBSqqqok\n01Sh/hbx8fHQ1NREeXm55LXnrl/moxU8E4lEzMXFheXk5LDGxkbm7+/Pbt68qeiyOlRcXMyuXLnC\nGGOspqaG9e/fn928eZN98MEHbO3atYwxxuLi4tjy5csVWeYzxcfHs5dffplFREQwxphK1T937lyW\nmJjIGGOsqamJVVZWqkz9OTk5zNnZmdXX1zPGGJsxYwZLSkpS2vpPnTrF0tPTmY+Pj+S19mr9448/\nmL+/P2tsbGQ5OTnMxcWFicVihdTdoq36jx07Jqlr+fLlKlc/Y4zdvXuXjR8/njk5ObEHDx4wxmSr\nX2nD4ezZs616O61Zs4atWbNGgRU9v8mTJ7Pjx48zd3d3du/ePcYYFyDu7u4Krqx9+fn5bPTo0ezX\nX39lkyZNYowxlam/srKSOTs7S72uKvU/ePCA9e/fn5WXl7OmpiY2adIkduzYMaWuPycnp9XGqb1a\nn+6tOH78eHbu3LnuLbYNT9f/pL1797LZs2czxlSr/unTp7OMjIxW4SBL/UrbrNSZk+iUWW5uLq5c\nuYIhQ4agpKQE1tbWAABra2uUlJQouLr2vffee1i3bh00NR//11CV+nNycmBpaYn58+djwIABiImJ\nwcOHD1WmfnNzc7z//vtwdHSEnZ0dTE1NMXbsWJWpH2j//0pRUREEAoFkPlX4e964cSMmTpwIQHXq\nT05OhkAggJ+fX6vXZalfacNBFc6Wbk9tbS2mTZuGf/3rXzA2Nm41TUNDQ2k/26FDh2BlZYXAwMB2\nT0hU5vpFIhHS09Px5ptvIj09HYaGhoiLi2s1jzLXn52dja+//hq5ubkoKipCbW0ttm3b1moeZa7/\nac+qVZk/xz//+U/o6uri5ZdfbnceZau/rq4Oq1evxqpVqySvtfd3DDy7fqUNB3t7e+Tn50ue5+fn\nt0o+ZdXU1IRp06Zhzpw5mDJlCgDuF9S9e/cAAMXFxbCyslJkie06e/YsDhw4AGdnZ0RHR+PXX3/F\nnDlzVKZ+gUAAgUCAQf+7+NX06dORnp4OGxsblaj/0qVLCA4OhoWFBbS1tREVFYVz586pTP1A+//X\nn/57LigogL29vUJqfJakpCQcOXIE27dvl7ymCvVnZ2cjNzcX/v7+cHZ2RkFBAQYOHIiSkhKZ6lfa\ncAgKCsLt27eRm5uLxsZG7Ny5E5GRkYouq0OMMSxcuBBeXl549913Ja9HRkZi8+bNAIDNmzdLQkPZ\nrF69Gvn5+cjJycFPP/2EUaNGYevWrSpTv42NDRwcHHDr1i0AwIkTJ+Dt7Y2IiAiVqN/DwwPnz5/H\no0ePwBjDiRMn4OXlpTL1A+3/X4+MjMRPP/2ExsZG5OTk4Pbt2xg8eLAiS21TSkoK1q1bh+TkZOjr\n60teV4X6fX19UVJSgpycHOTk5EAgECA9PR3W1tay1S/fwyPydeTIEda/f3/m4uLCVq9erehynun0\n6dNMQ0OD+fv7s4CAABYQEMCOHj3KHjx4wEaPHs3c3NzY2LFjWUVFhaJLfSahUCjpraRK9V+9epUF\nBQUxPz8/NnXqVFZZWalS9a9du5Z5eXkxHx8fNnfuXNbY2Ki09c+aNYvZ2toyHR0dJhAI2MaNGzus\n9Z///CdzcXFh7u7uLCUlRYGVc56uPzExkbm6ujJHR0fJ3+/ixYsl8ytr/bq6upLv/0nOzs6SA9KM\nPX/9dBIcIYQQKUrbrEQIIURxKBwIIYRIoXAghBAihcKBEEKIFAoHQgghUigcCCGESKFwID1GRkYG\njh49+tzvKyoqwksvvfTM+cLDw1FdXS1LaV3WmXVv3rwZxcXF3VQRUXV0ngPpMZKSknD58mV88803\nUtNEIhG0tXm7MaJSGDlyJL744gsMHDhQ0aUQFUB7DkQhcnNz4enpiddffx0+Pj4YP3486uvrAQCh\noaG4fPkyAKCsrAzOzs4AuI37lClTMG7cODg7OyMhIQFffPEFBgwYgKFDh6KioqLd9TU2NuLTTz/F\nzp07ERgYiF27diE2NhZz5szB8OHD8eqrryIvLw8jRozAwIEDMXDgQJw7d05Sa8sNVZKSkhAVFYUJ\nEyagf//+WL58uWQdTk5OKC8v7/CzXbx4EX5+fggMDMQHH3zQ5o1ahEIhRowYgUmTJsHDwwOLFy+W\nXEBtx44d8PPzg6+vL1asWNHpde/ZsweXLl3C7NmzMWDAANTX12PFihXw9vaGv78/PvjgA5n/LYma\nkufp3IR0Vk5ODtPW1mYZGRmMMe7GNtu2bWOMMRYaGsouX77MGGOstLSUOTk5McYY27RpE3N1dWW1\ntbWstLSUmZiYsA0bNjDGGHvvvffY119/3eE6k5KS2FtvvSV5/tlnn7GgoCDJzXXq6uok47du3WJB\nQUGSWluumb9p0ybWr18/Vl1dzerr61nfvn1ZQUEBY4xJrp/f0Wfz9vZm58+fZ4wxtmLFCubr6ytV\n52+//cb09fVZTk4OE4vFbOzYsWzPnj2ssLCQOTo6srKyMiYSidioUaPY/v37O73uJ7/XsrKyVveF\nqKqq6vC7Iz0P7TkQhXF2dpZcd37gwIHIzc195ntGjhwJQ0ND9OnTB6amppJbmfr6+j7z/Yy7uZXk\nuYaGBiIjI6GnpweA27t47bXX4OfnhxkzZuDmzZttLmf06NEwNjaGnp4evLy8kJeX16nPVlVVhdra\nWgwZMgQA8PLLL7d7SeXBgwfDyckJmpqaiI6Oxu+//45Lly4hNDQUFhYW0NLSwuzZs3Hq1KlOrfvJ\n7wAAevfuDX19fSxcuBD79u2DgYFBh98d6XkoHIjCtGyUAUBLSwtisRgAoK2tjebmZgCQNMe09R5N\nTU3Jc01NTYhEog7X19b163v16iUZ/+qrr2Bra4tr167h0qVLaGxs7FTdba23M/O0FwxP18oYa7P2\n9l5v73t9crna2tpIS0vD9OnTcejQIYSFhbVbC+mZKByI0mjZWDo5OeHSpUsAgD179jzXeztibGyM\nmpqadqdXV1fDxsYGALBly5ZWG1V56N27N4yNjZGWlgYA+Omnn9qdNy0tDbm5uWhubsauXbsQEhKC\nwYMHIzU1FQ8ePIBYLMZPP/2EF1988ZnrbflujI2NJT2aHj58iMrKSkyYMAFffvklMjIy5PAJiTqh\ncCAK8/Sv3pbny5Ytw7fffosBAwbgwYMHktefvrPY0+Mtzzds2IANGzZIrW/kyJG4efOm5ID008t4\n8803sXnzZgQEBCArKwtGRkZS6+rsndja+2yJiYmIiYlBYGAg6urq0Lt37zbfO2jQICxduhReXl7o\n168fpk6dChsbG8TFxWHkyJEICAhAUFCQpFmtve/lyefz5s3DG2+8gQEDBqCmpgYRERHw9/dHSEgI\nvvrqq2d+JtKzUFdWQrrRw4cPYWhoCACIi4tDSUmJ1IZZKBQiPj4eBw8eVESJhAAA1LtjNyFK5vDh\nw1izZg1EIhGcnJyQlJQkNY8q3SeaqC/acyCEECKFjjkQQgiRQuFACCFECoUDIYQQKRQOhBBCpFA4\nEEIIkULhQAghRMr/Aw8ag66aaRLUAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108083a10>"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This shows a typical learning curve: for very few training points, there is a large separation between the training and test error, which indicates **over-fitting**.  Given the same model, for a large number of training points, the training and testing errors converge, which indicates potential **under-fitting**.\n",
      "\n",
      "It is easy to see that, in this plot, if you'd like to reduce the MSE down to the nominal value of 1.0 (which is the magnitude of the scatter we put in when constructing the data), then adding more samples will *Never* get you there.  For $d=3$, we've already converged.  What about for $d=1$?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_learning_curve(2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2zMmTJ9mYMWMYY4xdu3aN+fj41FgWj2EqRWUlY/37M7Z3r6ojUa4zZxizsmLs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\nUHIghBAi5/8Baw91WHF7zrEAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10625fb50>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We see that it hasn't converged yet: you still might expect to decrease the test error by adding more points.\n",
      "\n",
      "What about the other extreme?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_learning_curve(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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fQq5TAZlOBQa7eahncjh79iw6deqEt956q97kcPz4cURGRuL48eO4dOkS3nvv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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x106264290>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For $d=5$, it's converged, but to a better value than for $d=3$.\n",
      "\n",
      "Thus you can **bring the two curves closer together** by adding more points, but you can **bring the convergence level down** only by adding complexity to the model.\n",
      "\n",
      "This can be very useful, because it gives you a hint about how to improve upon results which are sub-par.  To make this more concrete, imagine an astronomy project in which the results are not robust enough.  You must think about whether to spend your valuable telescope time observing *more objects* to get a larger training set, or *more attributes of each object* in order to improve the model.  The answer to this question is important, and can be addressed using these metrics."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Final Exercise"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's return to our digits data.  Using what you've learned here, explore several classifiers and find the optimal classifier for this dataset.  What is the best **accuracy** you can attain on a 5-fold cross-validation over the data?  Are there indications that more training data would be helpful in improving this accuracy?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# As this final exercise is fairly open-ended, there is no solution provided!"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    }
   ],
   "metadata": {}
  }
 ]
}