},

"outputs": [],

"source": [

"print(__doc__)\n\n# Author: Vincent Dubourg <vincent.dubourg@gmail.com>\n# Adapted to GaussianProcessClassifier:\n# Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>\n# License: BSD 3 clause\n\nimport numpy as np\n\nfrom matplotlib import pyplot as plt\nfrom matplotlib import cm\n\nfrom sklearn.gaussian_process import GaussianProcessClassifier\nfrom sklearn.gaussian_process.kernels import DotProduct, ConstantKernel as C\n\n# A few constants\nlim = 8\n\n\ndef g(x):\n \"\"\"The function to predict (classification will then consist in predicting\n whether g(x) <= 0 or not)\"\"\"\n return 5. - x[:, 1] - .5 * x[:, 0] ** 2.\n\n# Design of experiments\nX = np.array([[-4.61611719, -6.00099547],\n [4.10469096, 5.32782448],\n [0.00000000, -0.50000000],\n [-6.17289014, -4.6984743],\n [1.3109306, -6.93271427],\n [-5.03823144, 3.10584743],\n [-2.87600388, 6.74310541],\n [5.21301203, 4.26386883]])\n\n# Observations\ny = np.array(g(X) > 0, dtype=int)\n\n# Instanciate and fit Gaussian Process Model\nkernel = C(0.1, (1e-5, np.inf)) * DotProduct(sigma_0=0.1) ** 2\ngp = GaussianProcessClassifier(kernel=kernel)\ngp.fit(X, y)\nprint(\"Learned kernel: %s \" % gp.kernel_)\n\n# Evaluate real function and the predicted probability\nres = 50\nx1, x2 = np.meshgrid(np.linspace(- lim, lim, res),\n np.linspace(- lim, lim, res))\nxx = np.vstack([x1.reshape(x1.size), x2.reshape(x2.size)]).T\n\ny_true = g(xx)\ny_prob = gp.predict_proba(xx)[:, 1]\ny_true = y_true.reshape((res, res))\ny_prob = y_prob.reshape((res, res))\n\n# Plot the probabilistic classification iso-values\nfig = plt.figure(1)\nax = fig.gca()\nax.axes.set_aspect('equal')\nplt.xticks([])\nplt.yticks([])\nax.set_xticklabels([])\nax.set_yticklabels([])\nplt.xlabel('$x_1$')\nplt.ylabel('$x_2$')\n\ncax = plt.imshow(y_prob, cmap=cm.gray_r, alpha=0.8,\n extent=(-lim, lim, -lim, lim))\nnorm = plt.matplotlib.colors.Normalize(vmin=0., vmax=0.9)\ncb = plt.colorbar(cax, ticks=[0., 0.2, 0.4, 0.6, 0.8, 1.], norm=norm)\ncb.set_label('${\\\\rm \\mathbb{P}}\\left[\\widehat{G}(\\mathbf{x}) \\leq 0\\\\right]$')\nplt.clim(0, 1)\n\nplt.plot(X[y <= 0, 0], X[y <= 0, 1], 'r.', markersize=12)\n\nplt.plot(X[y > 0, 0], X[y > 0, 1], 'b.', markersize=12)\n\nplt.contour(x1, x2, y_true, [0.], colors='k', linestyles='dashdot')\n\ncs = plt.contour(x1, x2, y_prob, [0.666], colors='b',\n linestyles='solid')\nplt.clabel(cs, fontsize=11)\n\ncs = plt.contour(x1, x2, y_prob, [0.5], colors='k',\n linestyles='dashed')\nplt.clabel(cs, fontsize=11)\n\ncs = plt.contour(x1, x2, y_prob, [0.334], colors='r',\n linestyles='solid')\nplt.clabel(cs, fontsize=11)\n\nplt.show()"

"print(__doc__)\n\n# Author: Vincent Dubourg <vincent.dubourg@gmail.com>\n# Adapted to GaussianProcessClassifier:\n# Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>\n# License: BSD 3 clause\n\nimport numpy as np\n\nfrom matplotlib import pyplot as plt\nfrom matplotlib import cm\n\nfrom sklearn.gaussian_process import GaussianProcessClassifier\nfrom sklearn.gaussian_process.kernels import DotProduct, ConstantKernel as C\n\n# A few constants\nlim = 8\n\n\ndef g(x):\n \"\"\"The function to predict (classification will then consist in predicting\n whether g(x) <= 0 or not)\"\"\"\n return 5. - x[:, 1] - .5 * x[:, 0] ** 2.\n\n# Design of experiments\nX = np.array([[-4.61611719, -6.00099547],\n [4.10469096, 5.32782448],\n [0.00000000, -0.50000000],\n [-6.17289014, -4.6984743],\n [1.3109306, -6.93271427],\n [-5.03823144, 3.10584743],\n [-2.87600388, 6.74310541],\n [5.21301203, 4.26386883]])\n\n# Observations\ny = np.array(g(X) > 0, dtype=int)\n\n# Instantiate and fit Gaussian Process Model\nkernel = C(0.1, (1e-5, np.inf)) * DotProduct(sigma_0=0.1) ** 2\ngp = GaussianProcessClassifier(kernel=kernel)\ngp.fit(X, y)\nprint(\"Learned kernel: %s \" % gp.kernel_)\n\n# Evaluate real function and the predicted probability\nres = 50\nx1, x2 = np.meshgrid(np.linspace(- lim, lim, res),\n np.linspace(- lim, lim, res))\nxx = np.vstack([x1.reshape(x1.size), x2.reshape(x2.size)]).T\n\ny_true = g(xx)\ny_prob = gp.predict_proba(xx)[:, 1]\ny_true = y_true.reshape((res, res))\ny_prob = y_prob.reshape((res, res))\n\n# Plot the probabilistic classification iso-values\nfig = plt.figure(1)\nax = fig.gca()\nax.axes.set_aspect('equal')\nplt.xticks([])\nplt.yticks([])\nax.set_xticklabels([])\nax.set_yticklabels([])\nplt.xlabel('$x_1$')\nplt.ylabel('$x_2$')\n\ncax = plt.imshow(y_prob, cmap=cm.gray_r, alpha=0.8,\n extent=(-lim, lim, -lim, lim))\nnorm = plt.matplotlib.colors.Normalize(vmin=0., vmax=0.9)\ncb = plt.colorbar(cax, ticks=[0., 0.2, 0.4, 0.6, 0.8, 1.], norm=norm)\ncb.set_label('${\\\\rm \\mathbb{P}}\\left[\\widehat{G}(\\mathbf{x}) \\leq 0\\\\right]$')\nplt.clim(0, 1)\n\nplt.plot(X[y <= 0, 0], X[y <= 0, 1], 'r.', markersize=12)\n\nplt.plot(X[y > 0, 0], X[y > 0, 1], 'b.', markersize=12)\n\nplt.contour(x1, x2, y_true, [0.], colors='k', linestyles='dashdot')\n\ncs = plt.contour(x1, x2, y_prob, [0.666], colors='b',\n linestyles='solid')\nplt.clabel(cs, fontsize=11)\n\ncs = plt.contour(x1, x2, y_prob, [0.5], colors='k',\n linestyles='dashed')\nplt.clabel(cs, fontsize=11)\n\ncs = plt.contour(x1, x2, y_prob, [0.334], colors='r',\n linestyles='solid')\nplt.clabel(cs, fontsize=11)\n\nplt.show()"

]

}

],

y=np.array(g(X)>0,dtype=int)

kernel=C(0.1, (1e-5,np.inf))*DotProduct(sigma_0=0.1)**2

gp=GaussianProcessClassifier(kernel=kernel)

gp.fit(X,y)

},

"outputs": [],

"source": [

"print(__doc__)\n\n# Author: Vincent Dubourg <vincent.dubourg@gmail.com>\n# Jake Vanderplas <vanderplas@astro.washington.edu>\n# Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>s\n# License: BSD 3 clause\n\nimport numpy as np\nfrom matplotlib import pyplot as plt\n\nfrom sklearn.gaussian_process import GaussianProcessRegressor\nfrom sklearn.gaussian_process.kernels import RBF, ConstantKernel as C\n\nnp.random.seed(1)\n\n\ndef f(x):\n \"\"\"The function to predict.\"\"\"\n return x * np.sin(x)\n\n# ----------------------------------------------------------------------\n# First the noiseless case\nX = np.atleast_2d([1., 3., 5., 6., 7., 8.]).T\n\n# Observations\ny = f(X).ravel()\n\n# Mesh the input space for evaluations of the real function, the prediction and\n# its MSE\nx = np.atleast_2d(np.linspace(0, 10, 1000)).T\n\n# Instanciate a Gaussian Process model\nkernel = C(1.0, (1e-3, 1e3)) * RBF(10, (1e-2, 1e2))\ngp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9)\n\n# Fit to data using Maximum Likelihood Estimation of the parameters\ngp.fit(X, y)\n\n# Make the prediction on the meshed x-axis (ask for MSE as well)\ny_pred, sigma = gp.predict(x, return_std=True)\n\n# Plot the function, the prediction and the 95% confidence interval based on\n# the MSE\nplt.figure()\nplt.plot(x, f(x), 'r:', label=u'$f(x) = x\\,\\sin(x)$')\nplt.plot(X, y, 'r.', markersize=10, label=u'Observations')\nplt.plot(x, y_pred, 'b-', label=u'Prediction')\nplt.fill(np.concatenate([x, x[::-1]]),\n np.concatenate([y_pred - 1.9600 * sigma,\n (y_pred + 1.9600 * sigma)[::-1]]),\n alpha=.5, fc='b', ec='None', label='95% confidence interval')\nplt.xlabel('$x$')\nplt.ylabel('$f(x)$')\nplt.ylim(-10, 20)\nplt.legend(loc='upper left')\n\n# ----------------------------------------------------------------------\n# now the noisy case\nX = np.linspace(0.1, 9.9, 20)\nX = np.atleast_2d(X).T\n\n# Observations and noise\ny = f(X).ravel()\ndy = 0.5 + 1.0 * np.random.random(y.shape)\nnoise = np.random.normal(0, dy)\ny += noise\n\n# Instantiate a Gaussian Process model\ngp = GaussianProcessRegressor(kernel=kernel, alpha=dy ** 2,\n n_restarts_optimizer=10)\n\n# Fit to data using Maximum Likelihood Estimation of the parameters\ngp.fit(X, y)\n\n# Make the prediction on the meshed x-axis (ask for MSE as well)\ny_pred, sigma = gp.predict(x, return_std=True)\n\n# Plot the function, the prediction and the 95% confidence interval based on\n# the MSE\nplt.figure()\nplt.plot(x, f(x), 'r:', label=u'$f(x) = x\\,\\sin(x)$')\nplt.errorbar(X.ravel(), y, dy, fmt='r.', markersize=10, label=u'Observations')\nplt.plot(x, y_pred, 'b-', label=u'Prediction')\nplt.fill(np.concatenate([x, x[::-1]]),\n np.concatenate([y_pred - 1.9600 * sigma,\n (y_pred + 1.9600 * sigma)[::-1]]),\n alpha=.5, fc='b', ec='None', label='95% confidence interval')\nplt.xlabel('$x$')\nplt.ylabel('$f(x)$')\nplt.ylim(-10, 20)\nplt.legend(loc='upper left')\n\nplt.show()"

"print(__doc__)\n\n# Author: Vincent Dubourg <vincent.dubourg@gmail.com>\n# Jake Vanderplas <vanderplas@astro.washington.edu>\n# Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>s\n# License: BSD 3 clause\n\nimport numpy as np\nfrom matplotlib import pyplot as plt\n\nfrom sklearn.gaussian_process import GaussianProcessRegressor\nfrom sklearn.gaussian_process.kernels import RBF, ConstantKernel as C\n\nnp.random.seed(1)\n\n\ndef f(x):\n \"\"\"The function to predict.\"\"\"\n return x * np.sin(x)\n\n# ----------------------------------------------------------------------\n# First the noiseless case\nX = np.atleast_2d([1., 3., 5., 6., 7., 8.]).T\n\n# Observations\ny = f(X).ravel()\n\n# Mesh the input space for evaluations of the real function, the prediction and\n# its MSE\nx = np.atleast_2d(np.linspace(0, 10, 1000)).T\n\n# Instantiate a Gaussian Process model\nkernel = C(1.0, (1e-3, 1e3)) * RBF(10, (1e-2, 1e2))\ngp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9)\n\n# Fit to data using Maximum Likelihood Estimation of the parameters\ngp.fit(X, y)\n\n# Make the prediction on the meshed x-axis (ask for MSE as well)\ny_pred, sigma = gp.predict(x, return_std=True)\n\n# Plot the function, the prediction and the 95% confidence interval based on\n# the MSE\nplt.figure()\nplt.plot(x, f(x), 'r:', label=u'$f(x) = x\\,\\sin(x)$')\nplt.plot(X, y, 'r.', markersize=10, label=u'Observations')\nplt.plot(x, y_pred, 'b-', label=u'Prediction')\nplt.fill(np.concatenate([x, x[::-1]]),\n np.concatenate([y_pred - 1.9600 * sigma,\n (y_pred + 1.9600 * sigma)[::-1]]),\n alpha=.5, fc='b', ec='None', label='95% confidence interval')\nplt.xlabel('$x$')\nplt.ylabel('$f(x)$')\nplt.ylim(-10, 20)\nplt.legend(loc='upper left')\n\n# ----------------------------------------------------------------------\n# now the noisy case\nX = np.linspace(0.1, 9.9, 20)\nX = np.atleast_2d(X).T\n\n# Observations and noise\ny = f(X).ravel()\ndy = 0.5 + 1.0 * np.random.random(y.shape)\nnoise = np.random.normal(0, dy)\ny += noise\n\n# Instantiate a Gaussian Process model\ngp = GaussianProcessRegressor(kernel=kernel, alpha=dy ** 2,\n n_restarts_optimizer=10)\n\n# Fit to data using Maximum Likelihood Estimation of the parameters\ngp.fit(X, y)\n\n# Make the prediction on the meshed x-axis (ask for MSE as well)\ny_pred, sigma = gp.predict(x, return_std=True)\n\n# Plot the function, the prediction and the 95% confidence interval based on\n# the MSE\nplt.figure()\nplt.plot(x, f(x), 'r:', label=u'$f(x) = x\\,\\sin(x)$')\nplt.errorbar(X.ravel(), y, dy, fmt='r.', markersize=10, label=u'Observations')\nplt.plot(x, y_pred, 'b-', label=u'Prediction')\nplt.fill(np.concatenate([x, x[::-1]]),\n np.concatenate([y_pred - 1.9600 * sigma,\n (y_pred + 1.9600 * sigma)[::-1]]),\n alpha=.5, fc='b', ec='None', label='95% confidence interval')\nplt.xlabel('$x$')\nplt.ylabel('$f(x)$')\nplt.ylim(-10, 20)\nplt.legend(loc='upper left')\n\nplt.show()"

]

}

],

x=np.atleast_2d(np.linspace(0,10,1000)).T

kernel=C(1.0, (1e-3,1e3))*RBF(10, (1e-2,1e2))

gp=GaussianProcessRegressor(kernel=kernel,n_restarts_optimizer=9)

"cell_type":"markdown",

"metadata": {},

"source": [

"\n# RBF SVM parameters\n\n\nThis example illustrates the effect of the parameters ``gamma`` and ``C`` of\nthe Radial Basis Function (RBF) kernel SVM.\n\nIntuitively, the ``gamma`` parameter defines how far the influence of a single\ntraining example reaches, with low values meaning 'far' and high values meaning\n'close'. The ``gamma`` parameters can be seen as the inverse of the radius of\ninfluence of samples selected by the model as support vectors.\n\nThe ``C`` parameter trades off misclassification of training examples against\nsimplicity of the decision surface. A low ``C`` makes the decision surface\nsmooth, while a high ``C`` aims at classifying all training examples correctly\nby giving the model freedom to select more samples as support vectors.\n\nThe first plot is a visualization of the decision function for a variety of\nparameter values on a simplified classification problem involving only 2 input\nfeatures and 2 possible target classes (binary classification). Note that this\nkind of plot is not possible to do for problems with more features or target\nclasses.\n\nThe second plot is a heatmap of the classifier's cross-validation accuracy as a\nfunction of ``C`` and ``gamma``. For this example we explore a relatively large\ngrid for illustration purposes. In practice, a logarithmic grid from\n$10^{-3}$ to $10^3$ is usually sufficient. If the best parameters\nlie on the boundaries of the grid, it can be extended in that direction in a\nsubsequent search.\n\nNote that the heat map plot has a special colorbar with a midpoint value close\nto the score values of the best performing models so as to make it easy to tell\nthem appart in the blink of an eye.\n\nThe behavior of the model is very sensitive to the ``gamma`` parameter. If\n``gamma`` is too large, the radius of the area of influence of the support\nvectors only includes the support vector itself and no amount of\nregularization with ``C`` will be able to prevent overfitting.\n\nWhen ``gamma`` is very small, the model is too constrained and cannot capture\nthe complexity or \"shape\" of the data. The region of influence of any selected\nsupport vector would include the whole training set. The resulting model will\nbehave similarly to a linear model with a set of hyperplanes that separate the\ncenters of high density of any pair of two classes.\n\nFor intermediate values, we can see on the second plot that good models can\nbe found on a diagonal of ``C`` and ``gamma``. Smooth models (lower ``gamma``\nvalues) can be made more complex by selecting a larger number of support\nvectors (larger ``C`` values) hence the diagonal of good performing models.\n\nFinally one can also observe that for some intermediate values of ``gamma`` we\nget equally performing models when ``C`` becomes very large: it is not\nnecessary to regularize by limiting the number of support vectors. The radius of\nthe RBF kernel alone acts as a good structural regularizer. In practice though\nit might still be interesting to limit the number of support vectors with a\nlower value of ``C`` so as to favor models that use less memory and that are\nfaster to predict.\n\nWe should also note that small differences in scores results from the random\nsplits of the cross-validation procedure. Those spurious variations can be\nsmoothed out by increasing the number of CV iterations ``n_splits`` at the\nexpense of compute time. Increasing the value number of ``C_range`` and\n``gamma_range`` steps will increase the resolution of the hyper-parameter heat\nmap.\n\n\n"

"\n# RBF SVM parameters\n\n\nThis example illustrates the effect of the parameters ``gamma`` and ``C`` of\nthe Radial Basis Function (RBF) kernel SVM.\n\nIntuitively, the ``gamma`` parameter defines how far the influence of a single\ntraining example reaches, with low values meaning 'far' and high values meaning\n'close'. The ``gamma`` parameters can be seen as the inverse of the radius of\ninfluence of samples selected by the model as support vectors.\n\nThe ``C`` parameter trades off misclassification of training examples against\nsimplicity of the decision surface. A low ``C`` makes the decision surface\nsmooth, while a high ``C`` aims at classifying all training examples correctly\nby giving the model freedom to select more samples as support vectors.\n\nThe first plot is a visualization of the decision function for a variety of\nparameter values on a simplified classification problem involving only 2 input\nfeatures and 2 possible target classes (binary classification). Note that this\nkind of plot is not possible to do for problems with more features or target\nclasses.\n\nThe second plot is a heatmap of the classifier's cross-validation accuracy as a\nfunction of ``C`` and ``gamma``. For this example we explore a relatively large\ngrid for illustration purposes. In practice, a logarithmic grid from\n$10^{-3}$ to $10^3$ is usually sufficient. If the best parameters\nlie on the boundaries of the grid, it can be extended in that direction in a\nsubsequent search.\n\nNote that the heat map plot has a special colorbar with a midpoint value close\nto the score values of the best performing models so as to make it easy to tell\nthem apart in the blink of an eye.\n\nThe behavior of the model is very sensitive to the ``gamma`` parameter. If\n``gamma`` is too large, the radius of the area of influence of the support\nvectors only includes the support vector itself and no amount of\nregularization with ``C`` will be able to prevent overfitting.\n\nWhen ``gamma`` is very small, the model is too constrained and cannot capture\nthe complexity or \"shape\" of the data. The region of influence of any selected\nsupport vector would include the whole training set. The resulting model will\nbehave similarly to a linear model with a set of hyperplanes that separate the\ncenters of high density of any pair of two classes.\n\nFor intermediate values, we can see on the second plot that good models can\nbe found on a diagonal of ``C`` and ``gamma``. Smooth models (lower ``gamma``\nvalues) can be made more complex by selecting a larger number of support\nvectors (larger ``C`` values) hence the diagonal of good performing models.\n\nFinally one can also observe that for some intermediate values of ``gamma`` we\nget equally performing models when ``C`` becomes very large: it is not\nnecessary to regularize by limiting the number of support vectors. The radius of\nthe RBF kernel alone acts as a good structural regularizer. In practice though\nit might still be interesting to limit the number of support vectors with a\nlower value of ``C`` so as to favor models that use less memory and that are\nfaster to predict.\n\nWe should also note that small differences in scores results from the random\nsplits of the cross-validation procedure. Those spurious variations can be\nsmoothed out by increasing the number of CV iterations ``n_splits`` at the\nexpense of compute time. Increasing the value number of ``C_range`` and\n``gamma_range`` steps will increase the resolution of the hyper-parameter heat\nmap.\n\n\n"

]

},

{