scikit-learn-validation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Validation and Model Selection"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import print_function, division\n",
    "\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Use seaborn for plotting defaults\n",
    "import seaborn as sns; sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Validating Models\n",
    "\n",
    "One of the most important pieces of machine learning is **model validation**: that is, checking how well your model fits a given dataset. But there are some pitfalls you need to watch out for.\n",
    "\n",
    "Consider the digits example we've been looking at previously. How might we check how well our model fits the data?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.datasets import load_digits\n",
    "digits = load_digits()\n",
    "X = digits.data\n",
    "y = digits.target"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's fit a K-neighbors classifier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',\n",
       "           metric_params=None, n_jobs=1, n_neighbors=1, p=2,\n",
       "           weights='uniform')"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.neighbors import KNeighborsClassifier\n",
    "knn = KNeighborsClassifier(n_neighbors=1)\n",
    "knn.fit(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we'll use this classifier to *predict* labels for the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_pred = knn.predict(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can check how well our prediction did:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1797 / 1797 correct\n"
     ]
    }
   ],
   "source": [
    "print(\"{0} / {1} correct\".format(np.sum(y == y_pred), len(y)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems we have a perfect classifier!\n",
    "\n",
    "**Question: what's wrong with this?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Validation Sets\n",
    "\n",
    "Above we made the mistake of testing our data on the same set of data that was used for training. **This is not generally a good idea**. If we optimize our estimator this way, we will tend to **over-fit** the data: that is, we learn the noise.\n",
    "\n",
    "A better way to test a model is to use a hold-out set which doesn't enter the training. We've seen this before using scikit-learn's train/test split utility:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\reshma\\Anaconda2\\lib\\site-packages\\sklearn\\cross_validation.py:41: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.\n",
      "  \"This module will be removed in 0.20.\", DeprecationWarning)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "((1347L, 64L), (450L, 64L))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.cross_validation import train_test_split\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y)\n",
    "X_train.shape, X_test.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we train on the training data, and validate on the test data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "443 / 450 correct\n"
     ]
    }
   ],
   "source": [
    "knn = KNeighborsClassifier(n_neighbors=1)\n",
    "knn.fit(X_train, y_train)\n",
    "y_pred = knn.predict(X_test)\n",
    "print(\"{0} / {1} correct\".format(np.sum(y_test == y_pred), len(y_test)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This gives us a more reliable estimate of how our model is doing.\n",
    "\n",
    "The metric we're using here, comparing the number of matches to the total number of samples, is known as the **accuracy score**, and can be computed using the following routine:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.98444444444444446"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.metrics import accuracy_score\n",
    "accuracy_score(y_test, y_pred)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This can also be computed directly from the ``model.score`` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.98444444444444446"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "knn.score(X_test, y_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using this, we can ask how this changes as we change the model parameters, in this case the number of neighbors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 0.984444444444\n",
      "5 0.986666666667\n",
      "10 0.971111111111\n",
      "20 0.964444444444\n",
      "30 0.962222222222\n"
     ]
    }
   ],
   "source": [
    "for n_neighbors in [1, 5, 10, 20, 30]:\n",
    "    knn = KNeighborsClassifier(n_neighbors)\n",
    "    knn.fit(X_train, y_train)\n",
    "    print(n_neighbors, knn.score(X_test, y_test))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that in this case, a small number of neighbors seems to be the best option."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cross-Validation\n",
    "\n",
    "One problem with validation sets is that you \"lose\" some of the data. Above, we've only used 3/4 of the data for the training, and used 1/4 for the validation. Another option is to use **2-fold cross-validation**, where we split the sample in half and perform the validation twice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((898L, 64L), (899L, 64L))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1, X2, y1, y2 = train_test_split(X, y, test_size=0.5, random_state=0)\n",
    "X1.shape, X2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.983296213808\n",
      "0.982202447164\n"
     ]
    }
   ],
   "source": [
    "print(KNeighborsClassifier(1).fit(X2, y2).score(X1, y1))\n",
    "print(KNeighborsClassifier(1).fit(X1, y1).score(X2, y2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus a two-fold cross-validation gives us two estimates of the score for that parameter.\n",
    "\n",
    "Because this is a bit of a pain to do by hand, scikit-learn has a utility routine to help:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.97614938602520218"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.cross_validation import cross_val_score\n",
    "cv = cross_val_score(KNeighborsClassifier(1), X, y, cv=10)\n",
    "cv.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### K-fold Cross-Validation\n",
    "\n",
    "Here we've used 2-fold cross-validation. This is just one specialization of $K$-fold cross-validation, where we split the data into $K$ chunks and perform $K$ fits, where each chunk gets a turn as the validation set.\n",
    "We can do this by changing the ``cv`` parameter above. Let's do 10-fold cross-validation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.93513514,  0.99453552,  0.97237569,  0.98888889,  0.96089385,\n",
       "        0.98882682,  0.99441341,  0.98876404,  0.97175141,  0.96590909])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cross_val_score(KNeighborsClassifier(1), X, y, cv=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This gives us an even better idea of how well our model is doing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overfitting, Underfitting and Model Selection"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's create a realization of this dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0xf840c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X, y = make_data(40, error=1)\n",
    "plt.scatter(X.ravel(), y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Underfit"
   ]
  },
  {
   "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",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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I0Z2EtxgQj0eROiaKkxfqqG2y43B5mDAumpXWVOZdO4YQg3SNCDEcJLxFv9Q3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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xfaf4b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_test = np.linspace(-0.1, 1.1, 500)[:, None]\n",
    "\n",
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.metrics import mean_squared_error\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",
    "plt.title(\"mean squared error: {0:.3g}\".format(mean_squared_error(model.predict(X), y)));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Perfect fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.pipeline import make_pipeline\n",
    "\n",
    "def PolynomialRegression(degree=2, **kwargs):\n",
    "    return make_pipeline(PolynomialFeatures(degree),\n",
    "                         LinearRegression(**kwargs))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Y+255SQjpGYU36VVVfSve2n0BN+paETvQH48vGo1gf7GzyyLEq1F4E7POF9dh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\nH0qcLlebD1SqAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xedf02e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model = PolynomialRegression(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",
    "plt.title(\"mean squared error: {0:.3g}\".format(mean_squared_error(model.predict(X), y)));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overfit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Es+dGT0ds+ROZVIJxg1NR22jE5cqmgL1ObWMrPtpTgjilFD+e6/qLz5uJNQlx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\nQuJNEAQRgZB4EwRBRCAk3gRBEBHI/wcIU5L1jAj/nAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xf5a1cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model = PolynomialRegression(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.title(\"mean squared error: {0:.3g}\".format(mean_squared_error(model.predict(X), y)))\n",
    "plt.ylim(-4, 14);"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.14"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}