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<h1>KNN</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Описание модели</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Описание данных</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Близость данных согласно метрике</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#k-nn">k-NN для классификации</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id8">Формат результата</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id9">Задание 2. Nearest Neighbors для картинок</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id10">Формат результата</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#extra-knn">Задание 3 (EXTRA). Реализация kNN</a></li>
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<section class="tex2jax_ignore mathjax_ignore" id="knn">
<h1>KNN<a class="headerlink" href="#knn" title="Permalink to this heading">#</a></h1>
<section id="id1">
<h2>Описание модели<a class="headerlink" href="#id1" title="Permalink to this heading">#</a></h2>
<p><a class="reference external" href="https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm">Метод k-ближайших соседей</a> (англ. k-nearest neighbors algorithm, k-NN) — метрический алгоритм для классификации или регрессии. В случае классификации алгоритм сводится к следующему:</p>
<ol class="arabic simple">
<li><p>Рассматриваются объекты из обучающей выборки, для которых известно, к какому классу они принадлежат.</p></li>
<li><p>Между подлежащими классификации объектами и объектами тренировочной выборки вычисляется матрица попарных расстояний согласно выбранной метрике.</p></li>
<li><p>На основе полученной матрицы расстояний для каждого из подлежащих классификации объектов определяется k ближайших объектов тренировочной выборки — k ближайших соседей.</p></li>
<li><p>Подлежащим классификации объектам приписывается тот класс, который чаще всего встречается у их k ближайших соседей.</p></li>
</ol>
<img src ="https://edunet.kea.su/repo/EduNet-content/L01/out/knn_idea.png" width="600"><p>В качестве примера работы с алгоритмом k-NN классифицируем изображение корабля из тестовой выборки CIFAR-10 с использованием <a class="reference external" href="https://scikit-learn.org/stable/modules/generated/sklearn.neighbors.KNeighborsClassifier.html">реализации алгоритма в scikit-learn</a>.</p>
<section id="id2">
<h3>Описание данных<a class="headerlink" href="#id2" title="Permalink to this heading">#</a></h3>
<p>Предположим, что мы работаем с тренировочным датасетом CIFAR-10 и хотим решить хрестоматийную задачу классификации: определить те картинки из тестового набора данных, которые относятся к классу cat. Эта задача является частным примером общей задачи классификации данных CIFAR-10, разные подходы к решению которой мы ещё неоднократно рассмотрим в ходе первых лекций.</p>
<p>Датасет CIFAR-10 содержит, как следует из названия, 10 различных классов изображений:</p>
<img src ="https://edunet.kea.su/repo/EduNet-content/L01/out/knn_on_cifar10.png" width="700" ><p>Все изображения представляют собой матрицы чисел, которые кодируют цвета отдельных пикселей. Для изображений высоты <span class="math notranslate nohighlight">\(H\)</span>, ширины <span class="math notranslate nohighlight">\(W\)</span> с <span class="math notranslate nohighlight">\(C\)</span> цветовыми каналами получаем упорядоченный набор <span class="math notranslate nohighlight">\(H \times W \times C\)</span> чисел. В данном разделе пока не будем учитывать, что значения соседних пикселей изображения могут быть значительно связаны, и будем решать задачу классификации для наивного представления изображения в виде точки в <span class="math notranslate nohighlight">\(H \times W \times C\)</span>-мерном вещественном пространстве.</p>
<img src ="https://edunet.kea.su/repo/EduNet-content/L01/out/img_to_array.png" width="700" ><p>Датасет CIFAR-10 содержит цветные (трехцветные) изображения размером <span class="math notranslate nohighlight">\(32 \times 32\)</span> пикселя. Таким образом, каждое изображение из датасета является точкой в <span class="math notranslate nohighlight">\(3072\)</span>-мерном (<span class="math notranslate nohighlight">\(32 \times 32 \times 3 = 3072\)</span>) вещественном пространстве.</p>
</section>
<section id="id3">
<h3>Близость данных согласно метрике<a class="headerlink" href="#id3" title="Permalink to this heading">#</a></h3>
<p>Пара изображений будет выглядеть практически идентично, если значения цветов соответствующих пикселей будут похожи по величине. Другими словами, практически идентичным изображениям будут соответствовать <strong>близкие</strong> точки нашего многомерного вещественного пространства. Для численной характеристики <strong>близости</strong>, можно определить функцию подсчета расстояния между парой точек — метрику.</p>
<p>Известны различные способы задания <a class="reference external" href="https://en.wikipedia.org/wiki/Metric_(mathematics)">функции расстояния между парой точек</a>. Простейшим примером является широко известная <strong>Евклидова</strong> (<span class="math notranslate nohighlight">\(L_2\)</span>) метрика:
$<span class="math notranslate nohighlight">\(L_2 (X, Y) = \sum_i (X_i - Y_i)^2,\)</span>$</p>
<p>Но, кроме неё, величина расстояния между парой точек может быть выражена рядом других функций.</p>
<p><span class="math notranslate nohighlight">\(L_1\)</span>-расстояние (манхэттенская метрика):
$<span class="math notranslate nohighlight">\(L_1 (X, Y) = \sum_i |X_i - Y_i|,\)</span>$</p>
<p>угловое расстояние:
$<span class="math notranslate nohighlight">\(ang (X, Y) = \frac{1}{\pi} \arccos \frac{\sum_i X_i Y_i}{\sqrt{\sum_i X_i^2} \sqrt{\sum_i Y_i^2}} ,\)</span>$</p>
<p>и многие другие. От выбора конкретной функции расстояния между точками будет явно зависеть представление о <strong>близости</strong> точек — объекты, близкие по одной из метрик, вовсе не обязаны оказаться близкими согласно другой.</p>
<p>Давайте попробуем вычислить <span class="math notranslate nohighlight">\(L_1\)</span>-расстояние между несколькими первыми изображениями из тестового набора данных CIFAR-10 с использованием реализованного в пакете <a class="reference external" href="https://scikit-learn.org/stable/modules/generated/sklearn.metrics.DistanceMetric.html">sklearn</a> класса <code class="docutils literal notranslate"><span class="pre">sklearn.metrics.DistanceMetric</span></code></p>
<div class="math notranslate nohighlight">
\[\large L1\;distance:\; d_1(I_1, I_2) = \sum_p|I^p_1 - I^p_2|\]</div>
<center><img src ="https://edunet.kea.su/repo/EduNet-content/L01/out/metric_to_compare_train_and_test_imgs.png" width="800"></center><p>Загрузим данные:</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">clear_output</span>
<span class="kn">from</span> <span class="nn">torchvision</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="c1"># Load dataset from torchvision.datasets</span>
<span class="n">train_set</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">CIFAR10</span><span class="p">(</span><span class="s2">"content"</span><span class="p">,</span> <span class="n">train</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">download</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">val_set</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">CIFAR10</span><span class="p">(</span><span class="s2">"content"</span><span class="p">,</span> <span class="n">train</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">download</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">labels_names</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">classes</span>
<span class="n">clear_output</span><span class="p">()</span>
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<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">ModuleNotFoundError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">2</span>
<span class="g g-Whitespace"> </span><span class="mi">1</span> <span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">clear_output</span>
<span class="ne">----> </span><span class="mi">2</span> <span class="kn">from</span> <span class="nn">torchvision</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="g g-Whitespace"> </span><span class="mi">4</span> <span class="c1"># Load dataset from torchvision.datasets</span>
<span class="g g-Whitespace"> </span><span class="mi">5</span> <span class="n">train_set</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">CIFAR10</span><span class="p">(</span><span class="s2">"content"</span><span class="p">,</span> <span class="n">train</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">download</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="ne">ModuleNotFoundError</span>: No module named 'torchvision'
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<p>Выберем три изображения из тестового набора данных:</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">img_1</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">img_2</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">img_3</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="n">fix</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"First image in CIFAR10 train data"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">img_1</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Second image in CIFAR10 train data"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">img_2</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Third image in CIFAR10 train data"</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">img_3</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sample_ship_img</span> <span class="o">=</span> <span class="n">val_set</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="mi">18</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">sample_ship_img</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<img alt="_images/eec6606383bb582cf1d2f49aeebba27e3bbfdcd7fbb9ac9e9d8357f75a1b2687.png" src="_images/eec6606383bb582cf1d2f49aeebba27e3bbfdcd7fbb9ac9e9d8357f75a1b2687.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.neighbors</span> <span class="kn">import</span> <span class="n">KNeighborsClassifier</span>
<span class="c1"># in order to limit computational time</span>
<span class="n">index_limiter</span> <span class="o">=</span> <span class="mi">5000</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">train_set</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)[:</span><span class="n">index_limiter</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">targets</span><span class="p">[:</span><span class="n">index_limiter</span><span class="p">]</span>
<span class="k">for</span> <span class="n">metric_type</span> <span class="ow">in</span> <span class="p">[</span><span class="s2">"euclidean"</span><span class="p">,</span> <span class="s2">"manhattan"</span><span class="p">,</span> <span class="s2">"chebyshev"</span><span class="p">]:</span>
<span class="nb">print</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">knn</span> <span class="o">=</span> <span class="n">KNeighborsClassifier</span><span class="p">(</span><span class="n">n_neighbors</span><span class="o">=</span><span class="n">k</span><span class="p">,</span> <span class="n">metric</span><span class="o">=</span><span class="n">metric_type</span><span class="p">)</span>
<span class="n">knn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">result_class_id</span> <span class="o">=</span> <span class="n">knn</span><span class="o">.</span><span class="n">predict</span><span class="p">([</span><span class="n">sample_ship_img</span><span class="o">.</span><span class="n">flatten</span><span class="p">()])[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">result_class</span> <span class="o">=</span> <span class="n">train_set</span><span class="o">.</span><span class="n">classes</span><span class="p">[</span><span class="n">result_class_id</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">-NN with </span><span class="si">{</span><span class="n">metric_type</span><span class="si">}</span><span class="s2"> metric</span><span class="se">\n</span><span class="s2">predicted class is: </span><span class="si">{</span><span class="n">result_class</span><span class="si">}</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>3-NN with euclidean metric
predicted class is: automobile
4-NN with euclidean metric
predicted class is: ship
5-NN with euclidean metric
predicted class is: ship
6-NN with euclidean metric
predicted class is: ship
3-NN with manhattan metric
predicted class is: automobile
4-NN with manhattan metric
predicted class is: automobile
5-NN with manhattan metric
predicted class is: truck
6-NN with manhattan metric
predicted class is: ship
3-NN with chebyshev metric
predicted class is: ship
4-NN with chebyshev metric
predicted class is: ship
5-NN with chebyshev metric
predicted class is: ship
6-NN with chebyshev metric
predicted class is: ship
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<section id="id4">
<h2>Нормализация данных<a class="headerlink" href="#id4" title="Permalink to this heading">#</a></h2>
<p>Загрузим датасет с образцами здоровой и раковой ткани. Датасет состоит из 569 примеров, где каждой строчке из 30 признаков соответствует класс <code class="docutils literal notranslate"><span class="pre">1</span></code> злокачественной (<em>malignant</em>) или <code class="docutils literal notranslate"><span class="pre">0</span></code> доброкачественной (<em>benign</em>) ткани. Задача состоит в том, чтобы по 30 признакам обучить модель определять тип ткани (злокачественная или доброкачественная).</p>
<p>Можно иметь сколь угодно хороший алгоритм для классификации, но до тех пор, пока данные на входе — мусор, на выходе из нашего чудесного классификатора мы тоже будем получать мусор <strong>(<em>garbage in - garbage out</em>)</strong>. Давайте разберемся, что конкретно надо сделать, чтобы k-NN реально заработал.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">sklearn.datasets</span>
<span class="n">cancer</span> <span class="o">=</span> <span class="n">sklearn</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">load_breast_cancer</span><span class="p">()</span> <span class="c1"># load data</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">cancer</span><span class="o">.</span><span class="n">data</span> <span class="c1"># features</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">cancer</span><span class="o">.</span><span class="n">target</span> <span class="c1"># labels(classes)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"x shape: </span><span class="si">{</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">, y shape: </span><span class="si">{</span><span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"x[0]: </span><span class="se">\n</span><span class="s2"> </span><span class="si">{</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"y[0]: </span><span class="se">\n</span><span class="s2"> </span><span class="si">{</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>x shape: (569, 30), y shape: (569,)
x[0]:
[1.799e+01 1.038e+01 1.228e+02 1.001e+03 1.184e-01 2.776e-01 3.001e-01
1.471e-01 2.419e-01 7.871e-02 1.095e+00 9.053e-01 8.589e+00 1.534e+02
6.399e-03 4.904e-02 5.373e-02 1.587e-02 3.003e-02 6.193e-03 2.538e+01
1.733e+01 1.846e+02 2.019e+03 1.622e-01 6.656e-01 7.119e-01 2.654e-01
4.601e-01 1.189e-01]
y[0]:
0
</pre></div>
</div>
</div>
</div>
<p>Посмотрим, сколько данных в классе <code class="docutils literal notranslate"><span class="pre">0</span></code> и сколько данных в классе <code class="docutils literal notranslate"><span class="pre">1</span></code></p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span> <span class="c1"># set figure size</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">target_names</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># 1 label</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">target_names</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="c1"># 0 label</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Class balance"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Num examples"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">ticks</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">labels</span><span class="o">=</span><span class="p">[</span><span class="s2">"1"</span><span class="p">,</span> <span class="s2">"0"</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper left"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/f07603ae885d13e8c8e31c0f425161d873ef9c1f45ff83168d993075b97ccbe1.png" src="_images/f07603ae885d13e8c8e31c0f425161d873ef9c1f45ff83168d993075b97ccbe1.png" />
</div>
</div>
<p>Теперь давайте посмотрим на сами данные. У нас есть 569 строк, в каждой из которых по 30 колонок. Такие колонки называют признаками или <em>features</em>. Попробуем математически описать все эти признаки (mean, std, min и тд)</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_html"><div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>0</th>
<th>1</th>
<th>2</th>
<th>3</th>
<th>4</th>
<th>5</th>
<th>6</th>
<th>7</th>
<th>8</th>
<th>9</th>
<th>...</th>
<th>20</th>
<th>21</th>
<th>22</th>
<th>23</th>
<th>24</th>
<th>25</th>
<th>26</th>
<th>27</th>
<th>28</th>
<th>29</th>
</tr>
</thead>
<tbody>
<tr>
<th>count</th>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>...</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
<td>569.000000</td>
</tr>
<tr>
<th>mean</th>
<td>14.127292</td>
<td>19.289649</td>
<td>91.969033</td>
<td>654.889104</td>
<td>0.096360</td>
<td>0.104341</td>
<td>0.088799</td>
<td>0.048919</td>
<td>0.181162</td>
<td>0.062798</td>
<td>...</td>
<td>16.269190</td>
<td>25.677223</td>
<td>107.261213</td>
<td>880.583128</td>
<td>0.132369</td>
<td>0.254265</td>
<td>0.272188</td>
<td>0.114606</td>
<td>0.290076</td>
<td>0.083946</td>
</tr>
<tr>
<th>std</th>
<td>3.524049</td>
<td>4.301036</td>
<td>24.298981</td>
<td>351.914129</td>
<td>0.014064</td>
<td>0.052813</td>
<td>0.079720</td>
<td>0.038803</td>
<td>0.027414</td>
<td>0.007060</td>
<td>...</td>
<td>4.833242</td>
<td>6.146258</td>
<td>33.602542</td>
<td>569.356993</td>
<td>0.022832</td>
<td>0.157336</td>
<td>0.208624</td>
<td>0.065732</td>
<td>0.061867</td>
<td>0.018061</td>
</tr>
<tr>
<th>min</th>
<td>6.981000</td>
<td>9.710000</td>
<td>43.790000</td>
<td>143.500000</td>
<td>0.052630</td>
<td>0.019380</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.106000</td>
<td>0.049960</td>
<td>...</td>
<td>7.930000</td>
<td>12.020000</td>
<td>50.410000</td>
<td>185.200000</td>
<td>0.071170</td>
<td>0.027290</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.156500</td>
<td>0.055040</td>
</tr>
<tr>
<th>25%</th>
<td>11.700000</td>
<td>16.170000</td>
<td>75.170000</td>
<td>420.300000</td>
<td>0.086370</td>
<td>0.064920</td>
<td>0.029560</td>
<td>0.020310</td>
<td>0.161900</td>
<td>0.057700</td>
<td>...</td>
<td>13.010000</td>
<td>21.080000</td>
<td>84.110000</td>
<td>515.300000</td>
<td>0.116600</td>
<td>0.147200</td>
<td>0.114500</td>
<td>0.064930</td>
<td>0.250400</td>
<td>0.071460</td>
</tr>
<tr>
<th>50%</th>
<td>13.370000</td>
<td>18.840000</td>
<td>86.240000</td>
<td>551.100000</td>
<td>0.095870</td>
<td>0.092630</td>
<td>0.061540</td>
<td>0.033500</td>
<td>0.179200</td>
<td>0.061540</td>
<td>...</td>
<td>14.970000</td>
<td>25.410000</td>
<td>97.660000</td>
<td>686.500000</td>
<td>0.131300</td>
<td>0.211900</td>
<td>0.226700</td>
<td>0.099930</td>
<td>0.282200</td>
<td>0.080040</td>
</tr>
<tr>
<th>75%</th>
<td>15.780000</td>
<td>21.800000</td>
<td>104.100000</td>
<td>782.700000</td>
<td>0.105300</td>
<td>0.130400</td>
<td>0.130700</td>
<td>0.074000</td>
<td>0.195700</td>
<td>0.066120</td>
<td>...</td>
<td>18.790000</td>
<td>29.720000</td>
<td>125.400000</td>
<td>1084.000000</td>
<td>0.146000</td>
<td>0.339100</td>
<td>0.382900</td>
<td>0.161400</td>
<td>0.317900</td>
<td>0.092080</td>
</tr>
<tr>
<th>max</th>
<td>28.110000</td>
<td>39.280000</td>
<td>188.500000</td>
<td>2501.000000</td>
<td>0.163400</td>
<td>0.345400</td>
<td>0.426800</td>
<td>0.201200</td>
<td>0.304000</td>
<td>0.097440</td>
<td>...</td>
<td>36.040000</td>
<td>49.540000</td>
<td>251.200000</td>
<td>4254.000000</td>
<td>0.222600</td>
<td>1.058000</td>
<td>1.252000</td>
<td>0.291000</td>
<td>0.663800</td>
<td>0.207500</td>
</tr>
</tbody>
</table>
<p>8 rows × 30 columns</p>
</div></div></div>
</div>
<p>То же самое, но в виде графика. Видно, что у признаков совершенно разные значения.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">sns</span><span class="o">.</span><span class="n">boxenplot</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">orient</span><span class="o">=</span><span class="s2">"h"</span><span class="p">,</span> <span class="n">palette</span><span class="o">=</span><span class="s2">"Set2"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xscale</span><span class="o">=</span><span class="s2">"log"</span><span class="p">,</span> <span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="mf">1e-4</span><span class="p">,</span> <span class="mf">1e4</span><span class="p">),</span> <span class="n">xlabel</span><span class="o">=</span><span class="s2">"Values"</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">"Features"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/caaf4f722a303db082c89dd3b574cbf87ae21332e2667d69ad0c1f9d58b21078.png" src="_images/caaf4f722a303db082c89dd3b574cbf87ae21332e2667d69ad0c1f9d58b21078.png" />
</div>
</div>
<p>Чтобы адекватно сравнить данные между собой, нам следует использовать нормализацию.</p>
<p><strong>Нормализация, выбор Scaler</strong></p>
<p>Нормализацией называется процедура приведения входных данных к единому масштабу (диапазону) значений. Фактически, это означает построение взаимно однозначного соответствия между некоторыми размерными величинами (которые измеряются в метрах, килограммах, годах и т. п.) и их безразмерными аналогами, принимающими значение в строго определенном числовом диапазоне (скажем, на отрезке <span class="math notranslate nohighlight">\([0,1]\)</span>). Преобразование данных к единому числовому диапазону (иногда говорят <em>домену</em>) позволяет считать их равноправными признаками и единообразно передавать их на вход модели. В некоторых источниках данная процедура явно называется <em>масштабирование</em>.</p>
<div class="math notranslate nohighlight">
\[\text{scaling map} \; : \text{some arbitrary feature domain} \rightarrow \text{definite domain} \]</div>
<p>Иногда под нормализацией данных понимают процедуру <em>стандартизации</em>, то есть приведение множеств значений всех признаков к стандартному нормальному распределению — распределению с нулевым средним значением и единичной дисперсией.</p>
<div class="math notranslate nohighlight">
\[\text{standartization map} : f_i \rightarrow (f_i - \text{mean} (\{f_i\})) \cdot \frac{1}{\text{std} (\{f_i\})}\]</div>
<p>Рассмотрим небольшой пример. Пусть у нас есть данные о некоторой группе людей, содержащие два признака: <em>возраст</em> (в годах) и <em>размер дохода</em> (в рублях). Возраст может измениться в диапазоне от 18 до 70 ( интервал 70-18 = 52). А доход от 30 000 р до 500 000 р (интервал 500 000 - 30 000 = 470 000). В таком варианте разница в возрасте имеет меньшее влияние, чем разница в доходе. Получается, что доход становится более важным признаком, изменения в котором влияют больше при сравнении схожести двух людей.</p>
<p>Должно быть так, чтобы максимальные изменения любого признака в «основной массе объектов» были одинаковы. Тогда потенциально все признаки будут равноценны.</p>
<p>Осталось определиться с выбором инструмента. Часто используют следующие варианты: <code class="docutils literal notranslate"><span class="pre">MinMaxScaler</span></code>, <code class="docutils literal notranslate"><span class="pre">StandardScaler</span></code>, <code class="docutils literal notranslate"><span class="pre">RobustScaler</span></code>.</p>
<p>Сравним <code class="docutils literal notranslate"><span class="pre">MinMaxScaler</span></code>, <code class="docutils literal notranslate"><span class="pre">StandardScaler</span></code>, <code class="docutils literal notranslate"><span class="pre">RobustScaler</span></code> для признака <code class="docutils literal notranslate"><span class="pre">data[:,0]</span></code>. <strong>Обратите внимание на ось X</strong></p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">MinMaxScaler</span><span class="p">,</span> <span class="n">RobustScaler</span>
<span class="n">test</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">24</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">141</span><span class="p">)</span> <span class="c1"># set location</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">test</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">test</span><span class="p">)),</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Num examples"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Non scaled data"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
<span class="c1"># scale data with MinMaxScaler</span>
<span class="n">test_scaled</span> <span class="o">=</span> <span class="n">MinMaxScaler</span><span class="p">()</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">142</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">test_scaled</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">test</span><span class="p">)),</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"MinMaxScaler"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
<span class="c1"># scale data with StandardScaler</span>
<span class="n">test_scaled</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">143</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">test_scaled</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">test</span><span class="p">)),</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"StandardScaler"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
<span class="c1"># scale data with RobustScaler</span>
<span class="n">test_scaled</span> <span class="o">=</span> <span class="n">RobustScaler</span><span class="p">()</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">144</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">test_scaled</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">test</span><span class="p">)),</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"RobustScaler"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Идея <strong><code class="docutils literal notranslate"><span class="pre">MinMaxScaler</span></code></strong> заключается в том, что он преобразует данные из имеющегося диапазона значений в диапазон от 0 до 1. Может быть полезно, если нужно выполнить преобразование, в котором отрицательные значения не допускаются (например, масштабирование RGB пикселей).</p>
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\[z_i=\frac{X_i-X_{min}}{X_{max}-X_{min}}\]</div>
<p><span class="math notranslate nohighlight">\(z_i\)</span> — масштабированное значение,
<span class="math notranslate nohighlight">\(X_i\)</span> — текущее значение,
<span class="math notranslate nohighlight">\(X_{min}\)</span> и <span class="math notranslate nohighlight">\(X_{max}\)</span> — минимальное и максимальное значения имеющихся данных.</p>
<p>Идея <strong><code class="docutils literal notranslate"><span class="pre">StandardScaler</span></code></strong> заключается в том, что он преобразует данные таким образом, что распределение будет иметь среднее значение 0 и стандартное отклонение 1. Большинство значений будет в диапазоне от -1 до 1. Это стандартная трансформация, и она применима во многих ситуациях.</p>
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\[z_i=\frac{X_i-u}{s}\]</div>
<p><span class="math notranslate nohighlight">\(u\)</span> — среднее значение (или 0 при <code class="docutils literal notranslate"><span class="pre">with_mean=False</span></code>) и <span class="math notranslate nohighlight">\(s\)</span> — стандартное отклонение (или 0 при <code class="docutils literal notranslate"><span class="pre">with_std=False</span></code>)</p>
<p>И <code class="docutils literal notranslate"><span class="pre">StandardScaler</span></code>, и <code class="docutils literal notranslate"><span class="pre">MinMaxScaler</span></code> очень чувствительны к наличию выбросов. <strong><code class="docutils literal notranslate"><span class="pre">RobustScaler</span></code></strong> использует медиану и основан на <em>процентилях</em>. k-й процентиль – это величина, равная или не превосходящая k процентов чисел во всем имеющемся распределении. Например, 50-й процентиль (медиана) распределения таково, что 50% чисел из распределения не меньше данного числа. Соответственно, RobustScaler не зависит от небольшого числа очень больших предельных выбросов (outliers). Следовательно, результирующий диапазон преобразованных значений признаков больше, чем для предыдущих скэйлеров и, что более важно, примерно одинаков.</p>
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\[z_i=\frac{X_i-X_{median}}{IQR}\]</div>
<p><span class="math notranslate nohighlight">\(X_{median}\)</span> — значение медианы, <span class="math notranslate nohighlight">\(IQR\)</span> — межквартильный диапазон равный разнице между 75-ым и 25-ым процентилями</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x_norm</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1"># scaled data</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">x_norm</span><span class="p">)</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span>
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