Tuto-GUDHI-representations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Machine Learning and Topological Data Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mathieu Carrière, https://mathieucarriere.github.io/website/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import gudhi as gd\n",
    "import gudhi.representations\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we will see how to efficiently combine machine learning and topological data analysis with the Gudhi library and its [representations](https://gudhi.inria.fr/python/3.1.0.rc1/representations.html) module. We will see how to compute the various Hilbert representations of persistence diagrams and how to use them in order to classify a set of persistence diagrams! Ready? Let's go!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we will generate persistence diagrams with orbits of dynamical systems. This dataset is very common in TDA and was introduced in the [persistence image](https://arxiv.org/abs/1507.06217) paper. We use the following system, which depends on a parameter $r>0$:\n",
    "\n",
    "$$x_{n+1}=x_n+ry_n(1-y_n)\\ \\ \\ \\ \\text{(mod 1)}$$\n",
    "$$y_{n+1}=y_n+rx_{n+1}(1-x_{n+1})\\ \\ \\ \\ \\text{(mod 1)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's first see what the point cloud looks like for a given choice of $r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_pts = 1000\n",
    "r       = 3.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we generate the point cloud."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.empty([num_pts,2])\n",
    "x, y = np.random.uniform(), np.random.uniform()\n",
    "for i in range(num_pts):\n",
    "    X[i,:] = [x, y]\n",
    "    x = (X[i,0] + r * X[i,1] * (1-X[i,1])) % 1.\n",
    "    y = (X[i,1] + r * x * (1-x)) % 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now, we visualize it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1], s=3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mmmh, looks like a random point cloud... We will see later on that interesting topology can appear for some specific values of $r$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An easy way to generate persistence diagrams from this cloud is by using alpha filtrations. This can be done in two lines with Gudhi:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "acX = gd.AlphaComplex(points=X).create_simplex_tree()\n",
    "dgmX = acX.persistence()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also easily visualize the persistence diagram: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x7fb72824ceb8>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "gd.plot_persistence_diagram(dgmX)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's see what Gudhi has to offer to turn this diagram into a vector in a [scikit-learn](https://scikit-learn.org/stable/) fashion, that is, with estimators that have fit(), transform(), and fit_transform() methods!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first method that was introduced historically is the [persistence landscape]( http://jmlr.org/papers/v16/bubenik15a.html). A persistence landscape is basically obtained by rotating the persistence diagram by $-\\pi/4$ (so that the diagonal becomes the $x$-axis), and then putting tent functions on each point. The $k$th landscape is then defined as the $k$th largest value among all these tent functions. It is eventually turned into a vector by evaluating it on a bunch of uniformly sampled points on the x-axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "LS = gd.representations.Landscape(resolution=1000)\n",
    "L = LS.fit_transform([acX.persistence_intervals_in_dimension(1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(L[0][:1000])\n",
    "plt.plot(L[0][1000:2000])\n",
    "plt.plot(L[0][2000:3000])\n",
    "plt.title(\"Landscape\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A variation, called the [silhouette](https://arxiv.org/abs/1312.0308), takes a weighted average of these tent functions instead. Here, we weight each tent function by the distance of the corresponding point to the diagonal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "SH = gd.representations.Silhouette(resolution=1000, weight=lambda x: np.power(x[1]-x[0],1))\n",
    "sh = SH.fit_transform([acX.persistence_intervals_in_dimension(1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Silhouette')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(sh[0])\n",
    "plt.title(\"Silhouette\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second method is the [persistence image](http://jmlr.org/papers/v18/16-337.html). A persistence image is obtained by rotating by $-\\pi/4$, centering Gaussian functions on all diagram points (usually weighted by a parameter function---here we consider the squared distance to the diagonal) and summing all these Gaussians. This gives a 2D function, that is pixelized into an image."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "PI = gd.representations.PersistenceImage(bandwidth=1e-4, weight=lambda x: x[1]**2, \\\n",
    "                                         im_range=[0,.004,0,.004], resolution=[100,100])\n",
    "pi = PI.fit_transform([acX.persistence_intervals_in_dimension(1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Persistence Image')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(np.flip(np.reshape(pi[0], [100,100]), 0))\n",
    "plt.title(\"Persistence Image\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Neat, right? Gudhi also contains implementations of less common vectorization methods, such as the [Betti curve](https://www.researchgate.net/publication/316604237_Time_Series_Classification_via_Topological_Data_Analysis), the [complex polynomial]( https://link.springer.com/chapter/10.1007%2F978-3-319-23231-7_27), or the [topological vector](https://diglib.eg.org/handle/10.1111/cgf12692). You can check the [representations](https://gudhi.inria.fr/python/3.1.0.rc1/representations.html) module to check everything that is available."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gudhi also contain implementations of various kernels, i.e., scalar products for persistence diagrams. More precisely a kernel is a function $k$ that takes a pair of diagrams as inputs and outputs a real value such that:\n",
    "$$k(D,D')=\\langle \\Phi(D), \\Phi(D')\\rangle_{\\mathcal{H}},$$\n",
    "for some implicit Hilbert space $\\mathcal{H}$ and continuous function $\\Phi:\\mathcal{D}\\rightarrow\\mathcal{H}$. Many algorithms, such as SVM or PCA, only require pairwise scalar products to be able to run on data. With Gudhi, you can compute the [persistence scale space kernel](https://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Reininghaus_A_Stable_Multi-Scale_2015_CVPR_paper.pdf), [persistence weighted Gaussian kernel](http://proceedings.mlr.press/v48/kusano16.html), [persistence Fisher kernel](https://papers.nips.cc/paper/8205-persistence-fisher-kernel-a-riemannian-manifold-kernel-for-persistence-diagrams) and [sliced Wasserstein kernel]( http://proceedings.mlr.press/v70/carriere17a.html). Of course, you can also directly compute the bottleneck and Wasserstein distances ;-)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to test these functions, we need a second point cloud and corresponding persistence diagram. Let's pick another $r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = 4.1\n",
    "Y = np.empty([num_pts,2])\n",
    "x, y = np.random.uniform(), np.random.uniform()\n",
    "for i in range(num_pts):\n",
    "    Y[i,:] = [x, y]\n",
    "    x = (Y[i,0] + r * Y[i,1] * (1-Y[i,1])) % 1.\n",
    "    y = (Y[i,1] + r * x * (1-x)) % 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "acY = gd.AlphaComplex(points=Y).create_simplex_tree()\n",
    "dgmY = acY.persistence()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x7fb716b90080>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "gd.plot_persistence_diagram(dgmY)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks like this one has interesting homology! Let's plot it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(Y[:,0], Y[:,1], s=3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, there is a hole in the middle now for some reasons..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now check all pairwise kernels and metrics!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PWG kernel is 12.157801963480544\n",
      "PSS kernel is 0.004948290304369607\n",
      "PF kernel is 0.4920080594644995\n",
      "SW kernel is 0.8447691457828302\n",
      "Bottleneck distance is 0.009846354643030016\n",
      "Wasserstein distance is 0.013453246715383761\n"
     ]
    }
   ],
   "source": [
    "PWG = gd.representations.PersistenceWeightedGaussianKernel(bandwidth=0.01, kernel_approx=None,\\\n",
    "                                        weight=lambda x: np.arctan(np.power(x[1], 1)))\n",
    "PWG.fit([acX.persistence_intervals_in_dimension(1)])\n",
    "pwg = PWG.transform([acY.persistence_intervals_in_dimension(1)])\n",
    "print(\"PWG kernel is \" + str(pwg[0][0]))\n",
    "\n",
    "PSS = gd.representations.PersistenceScaleSpaceKernel(bandwidth=1.)\n",
    "PSS.fit([acX.persistence_intervals_in_dimension(1)])\n",
    "pss = PSS.transform([acY.persistence_intervals_in_dimension(1)])\n",
    "print(\"PSS kernel is \" + str(pss[0][0]))\n",
    "\n",
    "PF = gd.representations.PersistenceFisherKernel(bandwidth_fisher=.001, bandwidth=.001, kernel_approx=None)\n",
    "PF.fit([acX.persistence_intervals_in_dimension(1)])\n",
    "pf = PF.transform([acY.persistence_intervals_in_dimension(1)])\n",
    "print(\"PF kernel is \" + str(pf[0][0]))\n",
    "\n",
    "SW = gd.representations.SlicedWassersteinKernel(bandwidth=.1, num_directions=100)\n",
    "SW.fit([acX.persistence_intervals_in_dimension(1)])\n",
    "sw = SW.transform([acY.persistence_intervals_in_dimension(1)])\n",
    "print(\"SW kernel is \" + str(sw[0][0]))\n",
    "\n",
    "BD = gd.representations.BottleneckDistance(epsilon=.001)\n",
    "BD.fit([acX.persistence_intervals_in_dimension(1)])\n",
    "bd = BD.transform([acY.persistence_intervals_in_dimension(1)])\n",
    "print(\"Bottleneck distance is \" + str(bd[0][0]))\n",
    "\n",
    "WD = gd.representations.WassersteinDistance(internal_p=2, order=2)\n",
    "WD.fit([acX.persistence_intervals_in_dimension(1)])\n",
    "wd = WD.transform([acY.persistence_intervals_in_dimension(1)])\n",
    "print(\"Wasserstein distance is \" + str(wd[0][0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cool! However, you have probably noticed that there are quite a lot of parameters to choose. In practice, it is better to cross-validate among a bunch of them and pick the best ones. We will see shortly that it is actually very easy to do this with Gudhi!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's generate a complete dataset. We will generate point clouds and corresponding persistence diagrams for various radii $r$. Of course, you can increase the size of the dataset by modifying the num_diag_per_class variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_diag_per_class = 10\n",
    "\n",
    "dgms, labs = [], []\n",
    "for idx, radius in enumerate([2.5, 3.5, 4., 4.1, 4.3]):\n",
    "    for _ in range(num_diag_per_class):\n",
    "        labs.append(idx)\n",
    "        X = np.empty([num_pts,2])\n",
    "        x, y = np.random.uniform(), np.random.uniform()\n",
    "        for i in range(num_pts):\n",
    "            X[i,:] = [x, y]\n",
    "            x = (X[i,0] + radius * X[i,1] * (1-X[i,1])) % 1.\n",
    "            y = (X[i,1] + radius * x * (1-x)) % 1.\n",
    "        ac = gd.AlphaComplex(points=X).create_simplex_tree(max_alpha_square=1e12)\n",
    "        dgm = ac.persistence()\n",
    "        dgms.append(ac.persistence_intervals_in_dimension(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we shuffle the data and create a 80/20 split for train and test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "test_size            = 0.2\n",
    "perm                 = np.random.permutation(len(labs))\n",
    "limit                = np.int(test_size * len(labs))\n",
    "test_sub, train_sub  = perm[:limit], perm[limit:]\n",
    "train_labs           = np.array(labs)[train_sub]\n",
    "test_labs            = np.array(labs)[test_sub]\n",
    "train_dgms           = [dgms[i] for i in train_sub]\n",
    "test_dgms            = [dgms[i] for i in test_sub]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to cross-validate among all methods and parameters available, we create a scikit-learn pipeline for processing the diagrams. The pipeline will:\n",
    "1. extract the points of the persistence diagrams with finite coordinates (i.e. the non essential points)\n",
    "2. scale or not the diagrams in the unit square\n",
    "3. handle diagrams with vectorization or kernel methods with Gudhi\n",
    "4. train a classifier from the scikit-learn package\n",
    "\n",
    "As you can see from the code below, it is quite simple! Here, we cross validate among a kernel-SVM with sliced Wasserstein and persistence weighted Gaussian kernels, C-SVM on persistence images, random forests on landscapes, and $k$-nearest neighbors on bottleneck distances. We also try uniformly scaling the diagrams to the unit square."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing   import MinMaxScaler\n",
    "from sklearn.pipeline        import Pipeline\n",
    "from sklearn.svm             import SVC\n",
    "from sklearn.ensemble        import RandomForestClassifier\n",
    "from sklearn.neighbors       import KNeighborsClassifier\n",
    "\n",
    "# Definition of pipeline\n",
    "pipe = Pipeline([(\"Separator\", gd.representations.DiagramSelector(limit=np.inf, point_type=\"finite\")),\n",
    "                 (\"Scaler\",    gd.representations.DiagramScaler(scalers=[([0,1], MinMaxScaler())])),\n",
    "                 (\"TDA\",       gd.representations.PersistenceImage()),\n",
    "                 (\"Estimator\", SVC())])\n",
    "\n",
    "# Parameters of pipeline. This is the place where you specify the methods you want to use to handle diagrams\n",
    "param =    [{\"Scaler__use\":         [False],\n",
    "             \"TDA\":                 [gd.representations.SlicedWassersteinKernel()], \n",
    "             \"TDA__bandwidth\":      [0.1, 1.0],\n",
    "             \"TDA__num_directions\": [20],\n",
    "             \"Estimator\":           [SVC(kernel=\"precomputed\", gamma=\"auto\")]},\n",
    "            \n",
    "            {\"Scaler__use\":         [False],\n",
    "             \"TDA\":                 [gd.representations.PersistenceWeightedGaussianKernel()], \n",
    "             \"TDA__bandwidth\":      [0.1, 0.01],\n",
    "             \"TDA__weight\":         [lambda x: np.arctan(x[1]-x[0])], \n",
    "             \"Estimator\":           [SVC(kernel=\"precomputed\", gamma=\"auto\")]},\n",
    "            \n",
    "            {\"Scaler__use\":         [True],\n",
    "             \"TDA\":                 [gd.representations.PersistenceImage()], \n",
    "             \"TDA__resolution\":     [ [5,5], [6,6] ],\n",
    "             \"TDA__bandwidth\":      [0.01, 0.1, 1.0, 10.0],\n",
    "             \"Estimator\":           [SVC()]},\n",
    "            \n",
    "            {\"Scaler__use\":         [True],\n",
    "             \"TDA\":                 [gd.representations.Landscape()], \n",
    "             \"TDA__resolution\":     [100],\n",
    "             \"Estimator\":           [RandomForestClassifier()]},\n",
    "           \n",
    "            {\"Scaler__use\":         [False],\n",
    "             \"TDA\":                 [gd.representations.BottleneckDistance()], \n",
    "             \"TDA__epsilon\":        [0.1], \n",
    "             \"Estimator\":           [KNeighborsClassifier(metric=\"precomputed\")]}\n",
    "           ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our final model is the best estimator found after 3-fold cross-validation of our pipeline."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.model_selection import GridSearchCV\n",
    "\n",
    "model = GridSearchCV(pipe, param, cv=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now is time to train the model. Since we perform cross-validation, the computation can be quite long, especially if using k-NN with bottleneck distances, which is quite time-consuming. You may consider grabbing a cup of coffee at this point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = model.fit(train_dgms, train_labs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Training is finally over! Let us check what is the best method for persistence diagrams with respect to this classification problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'Estimator': RandomForestClassifier(bootstrap=True, ccp_alpha=0.0, class_weight=None,\n",
      "                       criterion='gini', max_depth=None, max_features='auto',\n",
      "                       max_leaf_nodes=None, max_samples=None,\n",
      "                       min_impurity_decrease=0.0, min_impurity_split=None,\n",
      "                       min_samples_leaf=1, min_samples_split=2,\n",
      "                       min_weight_fraction_leaf=0.0, n_estimators=100,\n",
      "                       n_jobs=None, oob_score=False, random_state=None,\n",
      "                       verbose=0, warm_start=False), 'Scaler__use': True, 'TDA': Landscape(num_landscapes=5, resolution=100, sample_range=[nan, nan]), 'TDA__resolution': 100}\n"
     ]
    }
   ],
   "source": [
    "print(model.best_params_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks like random forests and landscapes did the best for this small dataset! Let's see our model accuracy on the test set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Train accuracy = 1.0\n",
      "Test accuracy  = 0.5\n"
     ]
    }
   ],
   "source": [
    "print(\"Train accuracy = \" + str(model.score(train_dgms, train_labs)))\n",
    "print(\"Test accuracy  = \" + str(model.score(test_dgms,  test_labs)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "70%, not so bad for such a small dataset! The accuracy you get can even improve for bigger datasets and more parameters in the cross-validation (but training time will increase as well ;-)). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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