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LinearRegression is based on SVD, not the Normal Equation (fixes #184), also fixes #179 (mini-batch gradient descent), and updates matplotlib code to latest version.

main
Aurélien Geron 2018-03-15 18:38:58 +01:00
parent fb29c3b386
commit d9fbf7dd4c
1 changed files with 136 additions and 117 deletions
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04_training_linear_models.ipynb
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@ -61,7 +61,11 @@
" print(\"Saving figure\", fig_id)\n", " print(\"Saving figure\", fig_id)\n",
" if tight_layout:\n", " if tight_layout:\n",
" plt.tight_layout()\n", " plt.tight_layout()\n",
" plt.savefig(path, format='png', dpi=300)\n" " plt.savefig(path, format='png', dpi=300)\n",
"\n",
"# Ignore useless warnings (see SciPy issue #5998)\n",
"import warnings\n",
"warnings.filterwarnings(action=\"ignore\", module=\"scipy\", message=\"^internal gelsd\")"
] ]
}, },
{ {
@ -188,7 +192,7 @@
"cell_type": "markdown", "cell_type": "markdown",
"metadata": {}, "metadata": {},
"source": [ "source": [
"# Linear regression using batch gradient descent" "The `LinearRegression` class is based on the `scipy.linalg.lstsq()` function (the name stands for \"least squares\"), which you could call directly:"
] ]
}, },
{ {
@ -196,6 +200,46 @@
"execution_count": 11, "execution_count": 11,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [
"theta_best_svd, residuals, rank, s = np.linalg.lstsq(X_b, y, rcond=1e-6)\n",
"theta_best_svd"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This function computes $\\mathbf{X}^+\\mathbf{y}$, where $\\mathbf{X}^{+}$ is the _pseudoinverse_ of $\\mathbf{X}$ (specifically the Moore-Penrose inverse). You can use `np.linalg.pinv()` to compute the pseudoinverse directly:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"np.linalg.pinv(X_b).dot(y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Note**: the first releases of the book implied that the `LinearRegression` class was based on the Normal Equation. This was an error, my apologies: as explained above, it is based on the pseudoinverse, which ultimately relies on the SVD matrix decomposition of $\\mathbf{X}$ (see chapter 8 for details about the SVD decomposition). Its time complexity is $O(n^2)$ and it works even when $m < n$ or when some features are linear combinations of other features (in these cases, $\\mathbf{X}^T \\mathbf{X}$ is not invertible so the Normal Equation fails), see [issue #184](https://github.com/ageron/handson-ml/issues/184) for more details. However, this does not change the rest of the description of the `LinearRegression` class, in particular, it is based on an analytical solution, it does not scale well with the number of features, it scales linearly with the number of instances, all the data must fit in memory, it does not require feature scaling and the order of the instances in the training set does not matter."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Linear regression using batch gradient descent"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": [ "source": [
"eta = 0.1\n", "eta = 0.1\n",
"n_iterations = 1000\n", "n_iterations = 1000\n",
@ -209,7 +253,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 12, "execution_count": 14,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -218,7 +262,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 13, "execution_count": 15,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -227,7 +271,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 14, "execution_count": 16,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -253,7 +297,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 15, "execution_count": 17,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -279,7 +323,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 16, "execution_count": 18,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -290,7 +334,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 17, "execution_count": 19,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -326,7 +370,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 18, "execution_count": 20,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -335,7 +379,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 19, "execution_count": 21,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -346,7 +390,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 20, "execution_count": 22,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -362,7 +406,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 21, "execution_count": 23,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -374,7 +418,7 @@
"np.random.seed(42)\n", "np.random.seed(42)\n",
"theta = np.random.randn(2,1) # random initialization\n", "theta = np.random.randn(2,1) # random initialization\n",
"\n", "\n",
"t0, t1 = 10, 1000\n", "t0, t1 = 200, 1000\n",
"def learning_schedule(t):\n", "def learning_schedule(t):\n",
" return t0 / (t + t1)\n", " return t0 / (t + t1)\n",
"\n", "\n",
@ -387,7 +431,7 @@
" t += 1\n", " t += 1\n",
" xi = X_b_shuffled[i:i+minibatch_size]\n", " xi = X_b_shuffled[i:i+minibatch_size]\n",
" yi = y_shuffled[i:i+minibatch_size]\n", " yi = y_shuffled[i:i+minibatch_size]\n",
" gradients = 2 * xi.T.dot(xi.dot(theta) - yi)\n", " gradients = 2/minibatch_size * xi.T.dot(xi.dot(theta) - yi)\n",
" eta = learning_schedule(t)\n", " eta = learning_schedule(t)\n",
" theta = theta - eta * gradients\n", " theta = theta - eta * gradients\n",
" theta_path_mgd.append(theta)" " theta_path_mgd.append(theta)"
@ -395,7 +439,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 22, "execution_count": 24,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -404,7 +448,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 23, "execution_count": 25,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -415,7 +459,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 24, "execution_count": 26,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -440,10 +484,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 25, "execution_count": 27,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"import numpy as np\n", "import numpy as np\n",
@ -454,10 +496,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 26, "execution_count": 28,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"m = 100\n", "m = 100\n",
@ -467,7 +507,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 27, "execution_count": 29,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -481,7 +521,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 28, "execution_count": 30,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -493,7 +533,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 29, "execution_count": 31,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -502,7 +542,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 30, "execution_count": 32,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -513,7 +553,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 31, "execution_count": 33,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -532,7 +572,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 32, "execution_count": 34,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -563,7 +603,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 33, "execution_count": 35,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -589,7 +629,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 34, "execution_count": 36,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -602,7 +642,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 35, "execution_count": 37,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -628,7 +668,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 36, "execution_count": 38,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -671,7 +711,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 37, "execution_count": 39,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -683,7 +723,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 38, "execution_count": 40,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -694,7 +734,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 39, "execution_count": 41,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -705,7 +745,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 40, "execution_count": 42,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -724,7 +764,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 41, "execution_count": 43,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -736,7 +776,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 42, "execution_count": 44,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -748,7 +788,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 43, "execution_count": 45,
"metadata": { "metadata": {
"scrolled": true "scrolled": true
}, },
@ -809,7 +849,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 44, "execution_count": 46,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -832,7 +872,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 45, "execution_count": 47,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -841,7 +881,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 46, "execution_count": 48,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -852,7 +892,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 47, "execution_count": 49,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -879,7 +919,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 48, "execution_count": 50,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -918,6 +958,9 @@
" plt.plot(t1_min, t2_min, \"rs\")\n", " plt.plot(t1_min, t2_min, \"rs\")\n",
" plt.title(r\"$\\ell_{}$ penalty\".format(i + 1), fontsize=16)\n", " plt.title(r\"$\\ell_{}$ penalty\".format(i + 1), fontsize=16)\n",
" plt.axis([t1a, t1b, t2a, t2b])\n", " plt.axis([t1a, t1b, t2a, t2b])\n",
" if i == 1:\n",
" plt.xlabel(r\"$\\theta_1$\", fontsize=20)\n",
" plt.ylabel(r\"$\\theta_2$\", fontsize=20, rotation=0)\n",
"\n", "\n",
" plt.subplot(222 + i * 2)\n", " plt.subplot(222 + i * 2)\n",
" plt.grid(True)\n", " plt.grid(True)\n",
@ -928,14 +971,8 @@
" plt.plot(t1r_min, t2r_min, \"rs\")\n", " plt.plot(t1r_min, t2r_min, \"rs\")\n",
" plt.title(title, fontsize=16)\n", " plt.title(title, fontsize=16)\n",
" plt.axis([t1a, t1b, t2a, t2b])\n", " plt.axis([t1a, t1b, t2a, t2b])\n",
"\n", " if i == 1:\n",
"for subplot in (221, 223):\n", " plt.xlabel(r\"$\\theta_1$\", fontsize=20)\n",
" plt.subplot(subplot)\n",
" plt.ylabel(r\"$\\theta_2$\", fontsize=20, rotation=0)\n",
"\n",
"for subplot in (223, 224):\n",
" plt.subplot(subplot)\n",
" plt.xlabel(r\"$\\theta_1$\", fontsize=20)\n",
"\n", "\n",
"save_fig(\"lasso_vs_ridge_plot\")\n", "save_fig(\"lasso_vs_ridge_plot\")\n",
"plt.show()" "plt.show()"
@ -950,7 +987,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 49, "execution_count": 51,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -971,7 +1008,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 50, "execution_count": 52,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -982,7 +1019,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 51, "execution_count": 53,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -991,10 +1028,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 52, "execution_count": 54,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"X = iris[\"data\"][:, 3:] # petal width\n", "X = iris[\"data\"][:, 3:] # petal width\n",
@ -1003,7 +1038,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 53, "execution_count": 55,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1014,7 +1049,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 54, "execution_count": 56,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1034,7 +1069,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 55, "execution_count": 57,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1061,7 +1096,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 56, "execution_count": 58,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1070,7 +1105,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 57, "execution_count": 59,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1079,7 +1114,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 58, "execution_count": 60,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1123,7 +1158,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 59, "execution_count": 61,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1136,7 +1171,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 60, "execution_count": 62,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1161,7 +1196,7 @@
"from matplotlib.colors import ListedColormap\n", "from matplotlib.colors import ListedColormap\n",
"custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n", "custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n",
"\n", "\n",
"plt.contourf(x0, x1, zz, cmap=custom_cmap, linewidth=5)\n", "plt.contourf(x0, x1, zz, cmap=custom_cmap)\n",
"contour = plt.contour(x0, x1, zz1, cmap=plt.cm.brg)\n", "contour = plt.contour(x0, x1, zz1, cmap=plt.cm.brg)\n",
"plt.clabel(contour, inline=1, fontsize=12)\n", "plt.clabel(contour, inline=1, fontsize=12)\n",
"plt.xlabel(\"Petal length\", fontsize=14)\n", "plt.xlabel(\"Petal length\", fontsize=14)\n",
@ -1174,7 +1209,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 61, "execution_count": 63,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1183,7 +1218,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 62, "execution_count": 64,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1228,10 +1263,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 63, "execution_count": 65,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n", "X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
@ -1247,10 +1280,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 64, "execution_count": 66,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"X_with_bias = np.c_[np.ones([len(X), 1]), X]" "X_with_bias = np.c_[np.ones([len(X), 1]), X]"
@ -1265,10 +1296,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 65, "execution_count": 67,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"np.random.seed(2042)" "np.random.seed(2042)"
@ -1283,7 +1312,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 66, "execution_count": 68,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1314,10 +1343,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 67, "execution_count": 69,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"def to_one_hot(y):\n", "def to_one_hot(y):\n",
@ -1337,7 +1364,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 68, "execution_count": 70,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1346,7 +1373,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 69, "execution_count": 71,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1362,10 +1389,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 70, "execution_count": 72,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"Y_train_one_hot = to_one_hot(y_train)\n", "Y_train_one_hot = to_one_hot(y_train)\n",
@ -1384,10 +1409,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 71, "execution_count": 73,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"def softmax(logits):\n", "def softmax(logits):\n",
@ -1405,10 +1428,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 72, "execution_count": 74,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [ "source": [
"n_inputs = X_train.shape[1] # == 3 (2 features plus the bias term)\n", "n_inputs = X_train.shape[1] # == 3 (2 features plus the bias term)\n",
@ -1435,7 +1456,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 73, "execution_count": 75,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1466,7 +1487,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 74, "execution_count": 76,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1482,7 +1503,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 75, "execution_count": 77,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1503,7 +1524,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 76, "execution_count": 78,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1537,7 +1558,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 77, "execution_count": 79,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1565,7 +1586,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 78, "execution_count": 80,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1605,7 +1626,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 79, "execution_count": 81,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1633,7 +1654,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 80, "execution_count": 82,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1659,7 +1680,7 @@
"from matplotlib.colors import ListedColormap\n", "from matplotlib.colors import ListedColormap\n",
"custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n", "custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n",
"\n", "\n",
"plt.contourf(x0, x1, zz, cmap=custom_cmap, linewidth=5)\n", "plt.contourf(x0, x1, zz, cmap=custom_cmap)\n",
"contour = plt.contour(x0, x1, zz1, cmap=plt.cm.brg)\n", "contour = plt.contour(x0, x1, zz1, cmap=plt.cm.brg)\n",
"plt.clabel(contour, inline=1, fontsize=12)\n", "plt.clabel(contour, inline=1, fontsize=12)\n",
"plt.xlabel(\"Petal length\", fontsize=14)\n", "plt.xlabel(\"Petal length\", fontsize=14)\n",
@ -1678,7 +1699,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 81, "execution_count": 83,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -1700,9 +1721,7 @@
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": null,
"metadata": { "metadata": {},
"collapsed": true
},
"outputs": [], "outputs": [],
"source": [] "source": []
} }