exercise-mutual-information.ipynb 24.4 KB
 Klaus Strohmenger committed Nov 26, 2018 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 { "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# ML-Fundamentals - Exercise - Mutual Information for Feature Selection" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Table of Contents\n", "* [Introduction](#Introduction)\n", "* [Requirements](#Requirements) \n", " * [Knowledge](#Knowledge)\n", " * [Modules](#Python-Modules)\n", "* [Teaching Content](#Teaching-Content)\n", " * [Expected Mutual Information and Kullback Leibler Divergence](#Expected-Mutual-Information-and-Kullback-Leibler-Divergence)\n", " * [Information Gain](#Information-Gain)\n", "* [Preparing the Documents](#Preparing-the-Documents)\n", " * [Loading the Dataset](#Loading-the-Dataset)\n", " * [Vectorization](#From-Documents-to-Feature-Vectors)\n", " * [Review](#Review)\n", "* [Exercise - Mutual Information](#Exercise---Mutual-Information)\n", "* [Training a Feature Selector](#Training-a-Feature-Selector)\n", " * [Testing our Selection](#Testing-our-Selection)\n", "* [Literature](#Literature)\n", "* [Licenses](#Licenses)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Introduction\n", "\n", "In this notebook you will be dealing with feature selection. You will be presented with a binary classification problem in which documents belong to one of two categories. We will be using the presence or absence of terms from a document to predict its category. \n", "\n", "Our aim is to avoid checking for the presence or absence of each possible term in a given document. Some features carry more weight in making a prediction that others, so ideally using only a selected subset of terms in our classification should suffice.\n", "\n", "This process is called **feature selection**. One of the ways to measure how much a feature contributes to the classification is **mutual information**.\n", "\n", "Your task will be to work out the terms we ought to select when using only a limited number of features." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Requirements\n", "\n", "### Knowledge\n", "\n", "To complete this exercise notebook you should possess knowledge about the following topics.\n", "* Feature selection\n", "* Mutual information\n", "* optional: Kullback Leibler Divergence\n", "\n", "The following literature can help you to acquire this knowledge:\n", "* Chapter 13.5 \"Feature Selection\" of *Introduction to Information Retrieval* [[IIR]](#IIR). The introduction defines and discusses the motivation behind feature selection. 13.5.1 in particular explains the approach of mutual information.\n", "* For a review of the state-of-the-art of feature selection methods based\n", "on mutual information see [[VER14]](#VER14).\n", "* Later in the notebook we use Bernoulli Naive Bayes for classification from scikit learn. If you want to learn \n", "more about this kind of classifier study Chapter 13.3 \"The Bernoulli model\" of *Introduction to Information Retrieval* [[IIR]](#IIR)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Python Modules" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [], "source": [ "# External Modules\n", "import numpy as np\n", "from sklearn.datasets import fetch_20newsgroups\n", "from sklearn.feature_extraction.text import CountVectorizer\n", "\n", "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "\n", "from matplotlib.ticker import ScalarFormatter\n", "from sklearn.naive_bayes import MultinomialNB, BernoulliNB\n", "from sklearn.pipeline import Pipeline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Teaching Content" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Expected Mutual Information and Kullback Leibler Divergence\n", "\n", "The Kullback Leibler Divergence between two probability distributions $p$ and $q$ is\n", "\n", "$$\n", " \\mathcal D_{KL} (p({\\bf z}) \\mid\\mid q({\\bf z}))= \\sum_{{\\bf z} \\in \\mathcal Z} p({\\bf z}) \\log\\frac{p({\\bf z})}{q({\\bf z})}\n", "$$\n", "\n", "\n", "The _expected_ __mutual information (eMI)__ is the Kullback Leiber Divergence \n", "between the joint probability $p(x,y)$ and the product of the marginal distributions $p(x)p(y)$, i.e.\n", "- ${\\bf z} = (x,y)$\n", "- $p({\\bf z}) = p(x, y)$\n", "- $q({\\bf z}) = p(x)p(y)$ \n", "\n", "$$\n", " eMI(X; Y) := \\mathcal D_{KL} (p(X,Y) \\mid\\mid p(X)p(Y))= \\sum_{{x,y} \\in \\mathcal{X,Y}} p(x,y) \\log\\frac{p(x,y)}{p(x)p(y)}\n", "$$\n", "\n", "So if the two random variables $x$ and $y$ are statistically independent the eMI is zero.\n", "\n", "The stronger the difference between $p(x\\mid y)$ and $p(x)$ the larger the MI, which can be\n", "easily seen from $p(x,y) = p(x \\mid y) p(y)$. \n",  Klaus Strohmenger committed Jan 18, 2019 153 154 155 156 157 158  "So it's a measure about the information which $y$ has about $x$ (and vice versa). \n", "\n", "**Note:**\n", "$$\n", "\\sum_{{x,y} \\in \\mathcal{X,Y}} = \\sum_{{x} \\in \\mathcal{X}} \\sum_{{y} \\in \\mathcal{Y}}\n", "$$"  Klaus Strohmenger committed Nov 26, 2018 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173  ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Information Gain\n", "\n", "Sometimes you find the definition of Information Gain as $I(X; Y) := H(Y) - H(Y \\mid X)$ \n", "with the entropy $H(Y)$ and the conditional entropy $H(Y\\mid X)$, so we have\n", "\n", "\n", "\\begin{align}\n", "I(X; Y) &= H(Y) - H(Y \\mid X)\\\\\n", "&= - \\sum_y p(y) \\log p(y) + \\sum_{x,y} p(x) p(y\\mid x) \\log p(y\\mid x)\\\\\n",  Klaus Strohmenger committed Jan 18, 2019 174 175  "&= \\sum_{x,y} p(x, y) \\log{p(y\\mid x)} - \\sum_{y}\\left(\\sum_x p(x,y)\\right) \\log p(y)\\\\\n", "&= \\sum_{x,y} p(x, y) \\log{p(y\\mid x)} - \\sum_{x,y}p(x,y) \\log p(y)\\\\\n",  Klaus Strohmenger committed Nov 26, 2018 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236  "&= \\sum_{x,y} p(x, y) \\log \\frac{p(y\\mid x)}{p(y)}\\\\\n", "&= \\sum_{x,y} p(x, y) \\log \\frac{p(y\\mid x)p(x)}{p(y)p(x)}\\\\\n", "&= \\sum_{x,y} p(x, y) \\log \\frac{p(x, y)}{p(y)p(x)}\\\\\n", "&= eMI(X; Y)\n", "\\end{align}\n", "\n", "\n", "\n", "With the definitions above the expected Mutual Information and the Information Gain are the same.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Preparing the documents\n", "\n", "### Loading the dataset\n", "Scikit Learn[SL](#SL) provides [Working With Text Data](https://scikit-learn.org/stable/tutorial/text_analytics/working_with_text_data.html) as a guide on how to load, preprocess and analyse text data.\n", "\n", "To start things off, we fetch a training set of text documents from the 20 Newsgroup dataset. We only include two of the twenty categories which reduces our problem to binary classification." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "categories = ['alt.atheism', 'sci.med' ]\n", "twenty_train = fetch_20newsgroups('./newsgroups_dataset',\n", " subset='train',\n", " categories=categories,\n", " shuffle=True,\n", " random_state=42)\n", "print('(Down)load complete!')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### From Documents to Feature Vectors\n", "We use a [CountVectorizer](https://scikit-learn.org/stable/modules/generated/sklearn.feature_extraction.text.CountVectorizer.html) to transform text documents into numerical feature vectors that we can analyse. The transformation comprises these steps:\n", "1. Extract the lexicon. The vectorizer identifies each term that occurs anywhere in the documents and gives it a fixed integer id.\n", "2. Create a document-term matrix. In this matrix, each row represents a document and each column represents a term. Each cell indicates whether or not a given term is present in a given document.\n", "\n", "Remark: For other tasks, we may have to store more information in the document-term matrix, e.g. the number of occurrences." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# MultinomialNB works with occurrence counts, BernoulliNB is designed for binary/boolean features.\n", "# see [IIR] for details. Here we use binary features:\n", "binary = True; classifier = BernoulliNB()\n", "\n", "# ignore words with lower document frequency -> against overfitting\n",  237  "min_df=5\n",  Klaus Strohmenger committed Nov 26, 2018 238 239 240  "# remove english stop words (words that most likely do not have \n", "# anything to do with the document class because they occur everywhere, e.g. 'and') \n", "stop_words=\"english\" \n",  241  "stop_words=None\n",  Klaus Strohmenger committed Nov 26, 2018 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332  "vectorizer = CountVectorizer(binary=binary, stop_words=stop_words, min_df=min_df)\n", "X_train = vectorizer.fit_transform(twenty_train.data)\n", "y_train = np.array(twenty_train.target).reshape((-1,))\n", "lexicon = vectorizer.get_feature_names()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Review\n", "At this point we've prepared all the data we need to calculate the mutual information of each term.\n", "\n", "* **X_train : *sparse matrix, shape = (n_documents, n_features)***\n", "
Indicates whether or not a term is present in a document.
\n", "
X_train[i,j] is $1$ if $document_i$ contains $term_j$, otherwise it is $0$\n", " \n", " \n", "* **y_train : *array, shape = (n_documents,)***\n", "
The target vector indicating the category of each document. There are two distinct categories, $0$ and $1$.
\n", "
y_train[i] is the category of $document_i$\n", " \n", "
\n", " \n", "* **lexicon : *list, \\_\\_len\\_\\_ = n_features***\n", "
A list of strings that stores the actual term corresponding to each term id. Useful if you've identified some term ids as significant and want to understand what they actually mean.
\n", "
E.g. lexicon[5247] == \"chicken\"\n", " \n", "The following code snippet provides an example of obtaining information about a single document." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def doc_info(idx):\n", " doc = X_train[idx,:].toarray().reshape((-1,))\n", " cat_idx = y_train[idx]\n", " cat_name = twenty_train.target_names[cat_idx]\n", " term_count = np.array(lexicon)[np.where(doc>0)].shape[0]\n", " return locals()\n", "doc_info(123)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "## Exercise - Mutual Information\n", "Calculate the mutual information of each term and return a dict that maps each term id to the mutual information of the term. Later, this function will be invoked with X_train and y_train as arguments. Refer to the docstring for details.\n", "\n", "**Task:**\n", "\n", "Implement the following python function.\n", "\n", "**Hint:**\n", "\n", "When dealing with text classifiaction and binary features, here is an example with concrete values and their meaning:\n", "- $p(y=0)$: The probability that a document is in class $0$. Computed with number of documents in class $0$ divided by the total number of documents.\n", "- $p(x=1)$: The probability that a document contains term $x$.\n", "- $p(x=1, y=0)$: The probability a document contains term $x$ and is in class $0$.\n", "\n", "Further: To avoid division by $0$ when calculating the mutual information, it is common practice to always add $1$ when counting the number of any documents." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def select_features(X, y):\n", " \"\"\"Calculate the mutual information of all terms.\n", " Assumes y comprises two distinct categories 0 and 1.\n", "\n", " :param X: A document-term matrix\n", " :param y: A target vector\n", " :return: A dictionary containg n_features items. Each entry maps\n", " the term id (int) to the mutual information of the term (float or\n", " float-like)\n", " \"\"\"\n", " raise NotImplementedError()" ] },  Klaus Strohmenger committed Jan 18, 2019 333 334 335 336 337 338  { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [  339  "X_train[:,3].toarray().flatten()"  Klaus Strohmenger committed Jan 18, 2019 340 341 342 343 344 345 346 347 348 349 350 351  ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "a = np.array([[0,1,1],[1,0,0]])\n", "a[a == 0]\n" ] },  Klaus Strohmenger committed Nov 26, 2018 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621  { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "ex_mi = select_features(X_train, y_train)\n", "best_terms = sorted(ex_mi.items(), key=lambda kv : kv[1], reverse=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If your implementation is correct, the output of the cell below should look similar to the following (depending on what base for the logarithm is used, numbers may vary, but the order should be the same):\n", "\n", "\n", "Terms with the greatest mutual information\n", "god................. 0.19173825649655832\n", "atheists............ 0.17111409529128105\n", "keith............... 0.11713774078039724\n", "cco................. 0.10858678795625454\n", "atheism............. 0.10115308769302479\n", "schneider........... 0.0986931815614237\n", "pitt................ 0.09657796740920438\n", "religion............ 0.0962422171570431\n", "allan............... 0.09502007881084548\n", "gordon.............. 0.08813798072387438\n", "" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "print('Terms with the greatest mutual information')\n", "for (term_idx, score) in best_terms[:10]:\n", " print('{} {}'.format(lexicon[term_idx].ljust(20,\".\"), score))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Training a feature selector\n", "This segment defines a feature selector class which implements two methods:\n", "* **fit(X,y)**\n", "
Takes a document-term matrix and a target vector. It learns the mutual information of each term using the function you just implemented.\n", " \n", "* **transform(X)**\n", "
Takes a document-term matrix and returns a subset of it. The shape of the subset is (n_samples, self.k_features) and only represents the k best features in the column vectors." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "from sklearn.base import BaseEstimator\n", "\n", "class MutualInformation(BaseEstimator):\n", " cached_best_indices = None\n", "\n", " def __init__(self, k_features=10):\n", " self.k_features=k_features\n", "\n", " def fit(self, X, y=None):\n", " \"\"\"Upon fitting, calculate the best features. The result is cached in a\n", " static map.\"\"\"\n", " if MutualInformation.cached_best_indices is None:\n", " mi = select_features(X,y)\n", " mi_sorted = sorted(mi.keys(), key=mi.__getitem__, reverse=True)\n", " MutualInformation.cached_best_indices = mi_sorted\n", " return self\n", "\n", " def transform(self, X):\n", " \"\"\"Return a subset of X which contains only the best columns.\"\"\"\n", " indices = MutualInformation.cached_best_indices\n", " selected_terms = indices[:self.k_features]\n", " subset = X[:,selected_terms]\n", " return subset\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Testing our Selection\n", "Now that we have a way to identify the k best features, the question arises what value k should be.\n", "In the concluding part of this notebook, we set up a [pipeline](https://scikit-learn.org/stable/modules/compose.html) to streamline the task of Vectorization → Feature selection → Classification and predict the categories of the training set. We increase the value of k in each iteration and observe how the number of features selected affects the accuracy of the prediction.\n", "\n", "As classifier we use Multinominal Naive Bayes from the scikit learn library." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# First we download the test set\n", "twenty_test = fetch_20newsgroups('./newsgroups_dataset',\n", " subset='test',\n", " categories=categories,\n", " shuffle=True,\n", " random_state=42)\n", "print('(Down)load complete!')" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "min_zoom = 600\n", "accuracies = []\n", "\n", "# log distribuition from 1 to n_features\n", "k_values = set(np.geomspace(1,len(lexicon), dtype=np.int))\n", "k_values = sorted(k_values.union(set(np.geomspace(min_zoom,len(lexicon), dtype=np.int))))\n", "\n", "for k in k_values:\n", " pipe = Pipeline([\n", " ('vectorizer', CountVectorizer(binary=binary, stop_words=stop_words, min_df=min_df)),\n", " #('class_mapper', ),\n", " ('feature_selector', MutualInformation(k_features = k)),\n", " ('classifier', classifier) # \n", " ])\n", " pipe.fit(twenty_train.data, twenty_train.target)\n", " prediction = pipe.predict(twenty_test.data)\n", " accuracy = np.mean(prediction == twenty_test.target)# or use sklearn.metrics.accuracy_score(twenty_test.target, prediction)\n", " accuracies.append(accuracy)\n", " \n", "k_values = np.array(k_values)\n", "accuracies = np.array(accuracies)\n", "\n", "print('highest accuracy of {} achieved with top {} features used'.format(accuracies.max(), k_values[accuracies.argmax()]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "When the calculation is done, draw the plot." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "plt.figure(figsize=(10,8))\n", "plt.plot(k_values, accuracies, marker='.')\n", "plt.xscale('log')\n", "plt.xlabel('k features used')\n", "plt.minorticks_off()\n", "plt.gca().get_xaxis().set_major_formatter(ScalarFormatter())\n", "plt.ylabel('Accuracy')\n", "using_all = accuracies[-1]\n", "plt.plot(k_values,using_all*np.ones_like(k_values), color='orange', label='Using all features')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "plt.figure(figsize=(10,8))\n", "plt.plot(k_values[k_values>min_zoom], accuracies[k_values>min_zoom], marker='.')\n", "plt.xscale('log')\n", "plt.xlabel('k features used')\n", "plt.minorticks_off()\n", "plt.gca().get_xaxis().set_major_formatter(ScalarFormatter())\n", "plt.ylabel('Accuracy')\n", "using_all = accuracies[-1]\n", "plt.plot(k_values[k_values>min_zoom], using_all*np.ones_like(k_values[k_values>min_zoom]), color='orange', label='Using all features')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Literature\n", "\n", "\n", "
\n", " [IIR]\n", " \n", " Introduction To Information Retrieval\n", "Christopher Manning-Prabhakar Raghavan-Hinrich Schütze - Cambridge University Press - 2009
Online version: https://nlp.stanford.edu/IR-book/information-retrieval-book.html\n", "
\n", " [SL]\n", " \n", " Scikit-learn: Machine Learning in Python\n", "Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, David Cournapeau, Matthieu Brucher, Matthieu Perrot, Édouard Duchesnay; 12(Oct):2825−2830, 2011.\n", "
Website: https://scikit-learn.org\n", "
\n", " [VER14]\n", " \n", " Jorge R. Vergara, Pablo A. Estevez: A review of feature selection methods based on mutual information, Neural Comput & Applic (2014) 24:175–186\n", "
Website: http://repositorio.uchile.cl/bitstream/handle/2250/126533/A-review-of-feature-selection-methods-based-on-mutual-information.pdf\n", "
\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Licenses\n", "\n", "### Notebook License (CC-BY-SA 4.0)\n", "\n", "*The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).*\n", "\n", "Exercise - Mutual Information
\n", "by Diyar Oktay, Christian Herta, Klaus Strohmenger
\n", "Based on a work at https://gitlab.com/deep.TEACHING.\n", "\n", "\n", "### Code License (MIT)\n", "\n", "*The following license only applies to code cells of the notebook.*\n", "\n", "Copyright 2018 Diyar Oktay, Christian Herta, Klaus Strohmenger\n", "\n", "Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the \"Software\"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:\n", "\n", "The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.\n", "\n", "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE." ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": {  Klaus Strohmenger committed Jan 18, 2019 622  "display_name": "deep_teaching_kernel",  Klaus Strohmenger committed Nov 26, 2018 623  "language": "python",  Klaus Strohmenger committed Jan 18, 2019 624  "name": "deep_teaching_kernel"  Klaus Strohmenger committed Nov 26, 2018 625 626 627 628 629 630 631 632 633 634 635  }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3",  Klaus Strohmenger committed Jan 18, 2019 636  "version": "3.6.5"  Klaus Strohmenger committed Nov 26, 2018 637 638 639 640 641  } }, "nbformat": 4, "nbformat_minor": 2 }