07_cross_validation.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Cross-validation for parameter tuning, model selection, and feature selection\n",
      "*From the video series: [Introduction to machine learning with scikit-learn](https://github.com/justmarkham/scikit-learn-videos)*"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Agenda\n",
      "\n",
      "- What is the drawback of using the **train/test split** procedure for model evaluation?\n",
      "- How does **K-fold cross-validation** overcome this limitation?\n",
      "- How can cross-validation be used for selecting **tuning parameters**, choosing between **models**, and selecting **features**?\n",
      "- What are some possible **improvements** to cross-validation?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Review of model evaluation procedures"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Motivation:** Need a way to choose between machine learning models\n",
      "\n",
      "- Goal is to estimate likely performance of a model on **out-of-sample data**\n",
      "\n",
      "**Initial idea:** Train and test on the same data\n",
      "\n",
      "- But, maximizing **training accuracy** rewards overly complex models which **overfit** the training data\n",
      "\n",
      "**Alternative idea:** Train/test split\n",
      "\n",
      "- Split the dataset into two pieces, so that the model can be trained and tested on **different data**\n",
      "- **Testing accuracy** is a better estimate than training accuracy of out-of-sample performance\n",
      "- But, it provides a **high variance** estimate since changing which observations happen to be in the testing set can significantly change testing accuracy"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.datasets import load_iris\n",
      "from sklearn.cross_validation import train_test_split\n",
      "from sklearn.neighbors import KNeighborsClassifier\n",
      "from sklearn import metrics"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# read in the iris data\n",
      "iris = load_iris()\n",
      "\n",
      "# create X (features) and y (response)\n",
      "X = iris.data\n",
      "y = iris.target"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# use train/test split with different random_state values\n",
      "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=4)\n",
      "\n",
      "# check classification accuracy of KNN with K=5\n",
      "knn = KNeighborsClassifier(n_neighbors=5)\n",
      "knn.fit(X_train, y_train)\n",
      "y_pred = knn.predict(X_test)\n",
      "print metrics.accuracy_score(y_test, y_pred)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.973684210526\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question:** What if we created a bunch of train/test splits, calculated the testing accuracy for each, and averaged the results together?\n",
      "\n",
      "**Answer:** That's the essense of cross-validation!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Steps for K-fold cross-validation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "1. Split the dataset into K **equal** partitions (or \"folds\").\n",
      "2. Use fold 1 as the **testing set** and the union of the other folds as the **training set**.\n",
      "3. Calculate **testing accuracy**.\n",
      "4. Repeat steps 2 and 3 K times, using a **different fold** as the testing set each time.\n",
      "5. Use the **average testing accuracy** as the estimate of out-of-sample accuracy."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Diagram of **5-fold cross-validation:**\n",
      "\n",
      "![5-fold cross-validation](images/07_cross_validation_diagram.png)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# simulate splitting a dataset of 25 observations into 5 folds\n",
      "from sklearn.cross_validation import KFold\n",
      "kf = KFold(25, n_folds=5, shuffle=False)\n",
      "\n",
      "# print the contents of each training and testing set\n",
      "print '{} {:^61} {}'.format('Iteration', 'Training set observations', 'Testing set observations')\n",
      "for iteration, data in enumerate(kf, start=1):\n",
      "    print '{:^9} {} {:^25}'.format(iteration, data[0], data[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Iteration                   Training set observations                   Testing set observations\n",
        "    1     [ 5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24]        [0 1 2 3 4]       \n",
        "    2     [ 0  1  2  3  4 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24]        [5 6 7 8 9]       \n",
        "    3     [ 0  1  2  3  4  5  6  7  8  9 15 16 17 18 19 20 21 22 23 24]     [10 11 12 13 14]     \n",
        "    4     [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 20 21 22 23 24]     [15 16 17 18 19]     \n",
        "    5     [ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19]     [20 21 22 23 24]     \n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "- Dataset contains **25 observations** (numbered 0 through 24)\n",
      "- 5-fold cross-validation, thus it runs for **5 iterations**\n",
      "- For each iteration, every observation is either in the training set or the testing set, **but not both**\n",
      "- Every observation is in the testing set **exactly once**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Comparing cross-validation to train/test split"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Advantages of **cross-validation:**\n",
      "\n",
      "- More accurate estimate of out-of-sample accuracy\n",
      "- More \"efficient\" use of data (every observation is used for both training and testing)\n",
      "\n",
      "Advantages of **train/test split:**\n",
      "\n",
      "- Runs K times faster than K-fold cross-validation\n",
      "- Simpler to examine the detailed results of the testing process"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Cross-validation recommendations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "1. K can be any number, but **K=10** is generally recommended\n",
      "2. For classification problems, **stratified sampling** is recommended for creating the folds\n",
      "    - Each response class should be represented with equal proportions in each of the K folds\n",
      "    - scikit-learn's `cross_val_score` function does this by default"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Cross-validation example: parameter tuning"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Goal:** Select the best tuning parameters (aka \"hyperparameters\") for KNN on the iris dataset"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import cross_val_score"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 10-fold cross-validation with K=5 for KNN (the n_neighbors parameter)\n",
      "knn = KNeighborsClassifier(n_neighbors=5)\n",
      "scores = cross_val_score(knn, X, y, cv=10, scoring='accuracy')\n",
      "print scores"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 1.          0.93333333  1.          1.          0.86666667  0.93333333\n",
        "  0.93333333  1.          1.          1.        ]\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# use average accuracy as an estimate of out-of-sample accuracy\n",
      "print scores.mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.966666666667\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# search for an optimal value of K for KNN\n",
      "k_range = range(1, 31)\n",
      "k_scores = []\n",
      "for k in k_range:\n",
      "    knn = KNeighborsClassifier(n_neighbors=k)\n",
      "    scores = cross_val_score(knn, X, y, cv=10, scoring='accuracy')\n",
      "    k_scores.append(scores.mean())\n",
      "print k_scores"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[0.95999999999999996, 0.95333333333333337, 0.96666666666666656, 0.96666666666666656, 0.96666666666666679, 0.96666666666666679, 0.96666666666666679, 0.96666666666666679, 0.97333333333333338, 0.96666666666666679, 0.96666666666666679, 0.97333333333333338, 0.98000000000000009, 0.97333333333333338, 0.97333333333333338, 0.97333333333333338, 0.97333333333333338, 0.98000000000000009, 0.97333333333333338, 0.98000000000000009, 0.96666666666666656, 0.96666666666666656, 0.97333333333333338, 0.95999999999999996, 0.96666666666666656, 0.95999999999999996, 0.96666666666666656, 0.95333333333333337, 0.95333333333333337, 0.95333333333333337]\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline\n",
      "\n",
      "# plot the value of K for KNN (x-axis) versus the cross-validated accuracy (y-axis)\n",
      "plt.plot(k_range, k_scores)\n",
      "plt.xlabel('Value of K for KNN')\n",
      "plt.ylabel('Cross-Validated Accuracy')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "<matplotlib.text.Text at 0x168cc3c8>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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ZOqik4VOu0gYODDMGpDl2I/v/m3dnVU+cIPJVYDfwKPBrYAfwl2lWyjW+LVvg5ZfDwk0Z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\nRNvbJU1Nq/6Fgogn1p1z1TZ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       "text": [
        "<matplotlib.figure.Figure at 0x168a7710>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Cross-validation example: model selection"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Goal:** Compare the best KNN model with logistic regression on the iris dataset"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 10-fold cross-validation with the best KNN model\n",
      "knn = KNeighborsClassifier(n_neighbors=20)\n",
      "print cross_val_score(knn, X, y, cv=10, scoring='accuracy').mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.98\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 10-fold cross-validation with logistic regression\n",
      "from sklearn.linear_model import LogisticRegression\n",
      "logreg = LogisticRegression()\n",
      "print cross_val_score(logreg, X, y, cv=10, scoring='accuracy').mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.953333333333\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Cross-validation example: feature selection"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Goal**: Select whether the Newspaper feature should be included in the linear regression model on the advertising dataset"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pandas as pd\n",
      "import numpy as np\n",
      "from sklearn.linear_model import LinearRegression"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# read in the advertising dataset\n",
      "data = pd.read_csv('http://www-bcf.usc.edu/~gareth/ISL/Advertising.csv', index_col=0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# create a Python list of three feature names\n",
      "feature_cols = ['TV', 'Radio', 'Newspaper']\n",
      "\n",
      "# use the list to select a subset of the DataFrame (X)\n",
      "X = data[feature_cols]\n",
      "\n",
      "# select the Sales column as the response (y)\n",
      "y = data.Sales"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 10-fold cross-validation with all three features\n",
      "lm = LinearRegression()\n",
      "scores = cross_val_score(lm, X, y, cv=10, scoring='mean_squared_error')\n",
      "print scores"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-3.56038438 -3.29767522 -2.08943356 -2.82474283 -1.3027754  -1.74163618\n",
        " -8.17338214 -2.11409746 -3.04273109 -2.45281793]\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# fix the sign of MSE scores\n",
      "mse_scores = -scores\n",
      "print mse_scores"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 3.56038438  3.29767522  2.08943356  2.82474283  1.3027754   1.74163618\n",
        "  8.17338214  2.11409746  3.04273109  2.45281793]\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# convert from MSE to RMSE\n",
      "rmse_scores = np.sqrt(mse_scores)\n",
      "print rmse_scores"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 1.88689808  1.81595022  1.44548731  1.68069713  1.14139187  1.31971064\n",
        "  2.85891276  1.45399362  1.7443426   1.56614748]\n"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# calculate the average RMSE\n",
      "print rmse_scores.mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.69135317081\n"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 10-fold cross-validation with two features (excluding Newspaper)\n",
      "feature_cols = ['TV', 'Radio']\n",
      "X = data[feature_cols]\n",
      "print np.sqrt(-cross_val_score(lm, X, y, cv=10, scoring='mean_squared_error')).mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.67967484191\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Improvements to cross-validation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Repeated cross-validation**\n",
      "\n",
      "- Repeat cross-validation multiple times (with **different random splits** of the data) and average the results\n",
      "- More reliable estimate of out-of-sample performance by **reducing the variance** associated with a single trial of cross-validation\n",
      "\n",
      "**Creating a hold-out set**\n",
      "\n",
      "- \"Hold out\" a portion of the data **before** beginning the model building process\n",
      "- Locate the best model using cross-validation on the remaining data, and test it **using the hold-out set**\n",
      "- More reliable estimate of out-of-sample performance since hold-out set is **truly out-of-sample**\n",
      "\n",
      "**Feature engineering and selection within cross-validation iterations**\n",
      "\n",
      "- Normally, feature engineering and selection occurs **before** cross-validation\n",
      "- Instead, perform all feature engineering and selection **within each cross-validation iteration**\n",
      "- More reliable estimate of out-of-sample performance since it **better mimics** the application of the model to out-of-sample data"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Resources\n",
      "\n",
      "- scikit-learn documentation: [Cross-validation](http://scikit-learn.org/stable/modules/cross_validation.html), [Model evaluation](http://scikit-learn.org/stable/modules/model_evaluation.html)\n",
      "- scikit-learn issue on GitHub: [MSE is negative when returned by cross_val_score](https://github.com/scikit-learn/scikit-learn/issues/2439)\n",
      "- Section 5.1 of [An Introduction to Statistical Learning](http://www-bcf.usc.edu/~gareth/ISL/) (11 pages) and related videos: [K-fold and leave-one-out cross-validation](https://www.youtube.com/watch?v=nZAM5OXrktY) (14 minutes), [Cross-validation the right and wrong ways](https://www.youtube.com/watch?v=S06JpVoNaA0) (10 minutes)\n",
      "- Scott Fortmann-Roe: [Accurately Measuring Model Prediction Error](http://scott.fortmann-roe.com/docs/MeasuringError.html)\n",
      "- Machine Learning Mastery: [An Introduction to Feature Selection](http://machinelearningmastery.com/an-introduction-to-feature-selection/)\n",
      "- Harvard CS109: [Cross-Validation: The Right and Wrong Way](https://github.com/cs109/content/blob/master/lec_10_cross_val.ipynb)\n",
      "- Journal of Cheminformatics: [Cross-validation pitfalls when selecting and assessing regression and classification models](http://www.jcheminf.com/content/pdf/1758-2946-6-10.pdf)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Comments or Questions?\n",
      "\n",
      "- Email: <kevin@dataschool.io>\n",
      "- Website: http://dataschool.io\n",
      "- Twitter: [@justmarkham](https://twitter.com/justmarkham)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.core.display import HTML\n",
      "def css_styling():\n",
      "    styles = open(\"styles/custom.css\", \"r\").read()\n",
      "    return HTML(styles)\n",
      "css_styling()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<style>\n",
        "    @font-face {\n",
        "        font-family: \"Computer Modern\";\n",
        "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
        "    }\n",
        "    div.cell{\n",
        "        width: 90%;\n",
        "/*        margin-left:auto;*/\n",
        "/*        margin-right:auto;*/\n",
        "    }\n",
        "    ul {\n",
        "        line-height: 145%;\n",
        "        font-size: 90%;\n",
        "    }\n",
        "    li {\n",
        "        margin-bottom: 1em;\n",
        "    }\n",
        "    h1 {\n",
        "        font-family: Helvetica, serif;\n",
        "    }\n",
        "    h4{\n",
        "        margin-top: 12px;\n",
        "        margin-bottom: 3px;\n",
        "       }\n",
        "    div.text_cell_render{\n",
        "        font-family: Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
        "        line-height: 145%;\n",
        "        font-size: 130%;\n",
        "        width: 90%;\n",
        "        margin-left:auto;\n",
        "        margin-right:auto;\n",
        "    }\n",
        "    .CodeMirror{\n",
        "            font-family: \"Source Code Pro\", source-code-pro,Consolas, monospace;\n",
        "    }\n",
        "/*    .prompt{\n",
        "        display: None;\n",
        "    }*/\n",
        "    .text_cell_render h5 {\n",
        "        font-weight: 300;\n",
        "        font-size: 16pt;\n",
        "        color: #4057A1;\n",
        "        font-style: italic;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 0.5em;\n",
        "        display: block;\n",
        "    }\n",
        "\n",
        "    .warning{\n",
        "        color: rgb( 240, 20, 20 )\n",
        "        }\n",
        "</style>\n",
        "<script>\n",
        "    MathJax.Hub.Config({\n",
        "                        TeX: {\n",
        "                           extensions: [\"AMSmath.js\"]\n",
        "                           },\n",
        "                tex2jax: {\n",
        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
        "</script>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "<IPython.core.display.HTML at 0x3ddd128>"
       ]
      }
     ],
     "prompt_number": 1
    }
   ],
   "metadata": {}
  }
 ]
}