{"id":694,"date":"2015-08-26T16:28:14","date_gmt":"2015-08-26T14:28:14","guid":{"rendered":"https:\/\/joernhees.de\/blog\/?p=694"},"modified":"2019-06-13T18:21:30","modified_gmt":"2019-06-13T16:21:30","slug":"scipy-hierarchical-clustering-and-dendrogram-tutorial","status":"publish","type":"post","link":"https:\/\/joernhees.de\/blog\/2015\/08\/26\/scipy-hierarchical-clustering-and-dendrogram-tutorial\/","title":{"rendered":"SciPy Hierarchical Clustering and Dendrogram Tutorial"},"content":{"rendered":"\r\n<!-- Note to self: the raw plugin doesn't seem to work well with:\r\n     - jetpack's markdown support\r\n     - jetpack's lazy image loading performance feature\r\n     To edit:\r\n     - first turn off jetpack's markdown feature\r\n     - then force reload current edit page!\r\n     - use the classic editor!!!\r\n     - update\r\n     - then turn markdown support back on!\r\n     - leave the lazy image loading \"feature\" off\r\n\/\/ -->\r\n\r\n<script 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*\/\r\n\r\n#notebook-container {\r\n    -webkit-box-shadow: none;\r\n    box-shadow: none;\r\n    margin: 0;\r\n    padding: 0;\r\n}\r\n\r\nspan.code_j {\r\n    font-family: Consolas,Monaco,Lucida Console,monospace;\r\n    \/*background-color: #F1F1F1;*\/\r\n}\r\n\r\ndiv.input {\r\n    display: -webkit-box;\r\n}\r\n\r\n.rendered pre {\r\n    background-color: #F1F1F1;\r\n}\r\n\r\ndiv.input_area {\r\n    border: none;\r\n    background-color: #F1F1F1;\r\n}\r\n\r\n.prompt {\r\n    min-width: 11ex;\r\n}\r\n\r\n<\/style>\r\n\r\n\r\n<!-- Loading mathjax macro -->\r\n<!-- Load mathjax -->\r\n    <script src=\"https:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\"><\/script>\r\n    <!-- MathJax configuration -->\r\n    <script type=\"text\/x-mathjax-config\">\r\n    MathJax.Hub.Config({\r\n        tex2jax: {\r\n            inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\r\n            displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ],\r\n            processEscapes: true,\r\n            processEnvironments: true\r\n        },\r\n        \/\/ Center justify equations in code and markdown cells. Elsewhere\r\n        \/\/ we use CSS to left justify single line equations in code cells.\r\n        displayAlign: 'center',\r\n        \"HTML-CSS\": {\r\n            styles: {'.MathJax_Display': {\"margin\": 0}},\r\n            linebreaks: { automatic: true }\r\n        }\r\n    });\r\n    <\/script>\r\n    <!-- End of mathjax configuration --><\/head>\r\n    <div class=\"container\" id=\"notebook-container\">\r\n\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>This is a tutorial on how to use <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/cluster.hierarchy.html\">scipy's hierarchical clustering<\/a>.<\/p>\r\n<p>One of the benefits of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hierarchical_clustering\">hierarchical clustering<\/a> is that you don't need to already know the number of clusters <span class=\"code_j\">k<\/span> in your data in advance. Sadly, there doesn't seem to be much documentation on how to actually use scipy's hierarchical clustering to make an informed decision and then retrieve the clusters.<\/p>\r\n<p>In the following I'll explain:<\/p>\r\n<ul>\r\n<li>how to <a href=\"#Perform-the-Hierarchical-Clustering\">use scipy's hierarchical clustering<\/a><\/li>\r\n<li>how to <a href=\"#Plotting-a-Dendrogram\">plot a nice dendrogram from it<\/a><\/li>\r\n<li>how to <a href=\"#Selecting-a-Distance-Cut-Off-aka-Determining-the-Number-of-Clusters\">use the dendrogram to select a distance cut-off (aka determining the number of clusters <span class=\"code_j\">k<\/span> in your data)<\/a><\/li>\r\n<li>how to <a href=\"#Retrieve-the-Clusters\">retrieve the <span class=\"code_j\">k<\/span> clusters<\/a><\/li>\r\n<li>how to <a href=\"#Visualizing-Your-Clusters\">visualize the clusters (2D case)<\/a><\/li>\r\n<\/ul>\r\n\r\n<h2 id=\"other-works:\">Other works:<a class=\"anchor-link\" href=\"#other-works:\">&#182;<\/a><\/h2>\r\n<p>Some short shameless self-advertising:<\/p>\r\n<ul>\r\n<li>I teach machines to <a href=\"https:\/\/w3id.org\/associations\/\" target=\"_blank\" rel=\"noopener noreferrer\">associate like humans<\/a>. In that project I used hierarchical clustering to group similar learned graph patterns together.<\/li>\r\n<li>I'm always searching for good students, <a href=\"\/blog\/about\/\" target=\"_blank\" rel=\"noopener noreferrer\">contact me<\/a>.<\/li>\r\n<\/ul>\r\n\r\n<h2 id=\"Naming-conventions:\">Naming conventions:<a class=\"anchor-link\" href=\"#Naming-conventions:\">&#182;<\/a><\/h2><p>Before we start, as i know that it's easy to get lost, some naming conventions:<\/p>\r\n<ul>\r\n<li><span class=\"code_j\">X<\/span> samples (<span class=\"code_j\">n<\/span> x <span class=\"code_j\">m<\/span> array), aka data points or \"singleton clusters\"<\/li>\r\n<li><span class=\"code_j\">n<\/span> number of samples<\/li>\r\n<li><span class=\"code_j\">m<\/span> number of features<\/li>\r\n<li><span class=\"code_j\">Z<\/span> cluster linkage array (contains the hierarchical clustering information)<\/li>\r\n<li><span class=\"code_j\">k<\/span> number of clusters<\/li>\r\n<\/ul>\r\n<p>So, let's go.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Imports-and-Setup\">Imports and Setup<a class=\"anchor-link\" href=\"#Imports-and-Setup\">&#182;<\/a><\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[1]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"c\"># needed imports<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">matplotlib<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">pyplot<\/span> <span class=\"k\">as<\/span> <span class=\"n\">plt<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">scipy.cluster.hierarchy<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">dendrogram<\/span><span class=\"p\">,<\/span> <span class=\"n\">linkage<\/span>\r\n<span class=\"kn\">import<\/span> <span class=\"nn\">numpy<\/span> <span class=\"kn\">as<\/span> <span class=\"nn\">np<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[2]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"c\"># some setting for this notebook to actually show the graphs inline<\/span>\r\n<span class=\"c\"># you probably won&#39;t need this<\/span>\r\n<span class=\"o\">%<\/span><span class=\"k\">matplotlib<\/span> inline\r\n<span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">set_printoptions<\/span><span class=\"p\">(<\/span><span class=\"n\">precision<\/span><span class=\"o\">=<\/span><span class=\"mi\">5<\/span><span class=\"p\">,<\/span> <span class=\"n\">suppress<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">)<\/span>  <span class=\"c\"># suppress scientific float notation<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Generating-Sample-Data\">Generating Sample Data<a class=\"anchor-link\" href=\"#Generating-Sample-Data\">&#182;<\/a><\/h2><p>You'll obviously not need this step to run the clustering if you have own data.<\/p>\r\n<p>The only thing you need to make sure is that you convert your data into a matrix <span class=\"code_j\">X<\/span> with <span class=\"code_j\">n<\/span> samples and <span class=\"code_j\">m<\/span> features, so that <span class=\"code_j\">X.shape == (n, m)<\/span>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[3]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"c\"># generate two clusters: a with 100 points, b with 50:<\/span>\r\n<span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">random<\/span><span class=\"o\">.<\/span><span class=\"n\">seed<\/span><span class=\"p\">(<\/span><span class=\"mi\">4711<\/span><span class=\"p\">)<\/span>  <span class=\"c\"># for repeatability of this tutorial<\/span>\r\n<span class=\"n\">a<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">random<\/span><span class=\"o\">.<\/span><span class=\"n\">multivariate_normal<\/span><span class=\"p\">([<\/span><span class=\"mi\">10<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"p\">[[<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">4<\/span><span class=\"p\">]],<\/span> <span class=\"n\">size<\/span><span class=\"o\">=<\/span><span class=\"p\">[<\/span><span class=\"mi\">100<\/span><span class=\"p\">,])<\/span>\r\n<span class=\"n\">b<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">random<\/span><span class=\"o\">.<\/span><span class=\"n\">multivariate_normal<\/span><span class=\"p\">([<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">20<\/span><span class=\"p\">],<\/span> <span class=\"p\">[[<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">4<\/span><span class=\"p\">]],<\/span> <span class=\"n\">size<\/span><span class=\"o\">=<\/span><span class=\"p\">[<\/span><span class=\"mi\">50<\/span><span class=\"p\">,])<\/span>\r\n<span class=\"n\">X<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">concatenate<\/span><span class=\"p\">((<\/span><span class=\"n\">a<\/span><span class=\"p\">,<\/span> <span class=\"n\">b<\/span><span class=\"p\">),)<\/span>\r\n<span class=\"k\">print<\/span> <span class=\"n\">X<\/span><span class=\"o\">.<\/span><span class=\"n\">shape<\/span>  <span class=\"c\"># 150 samples with 2 dimensions<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">1<\/span><span class=\"p\">])<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n<div class=\"output_subarea output_stream output_stdout output_text\">\r\n<pre>(150, 2)\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Perform-the-Hierarchical-Clustering\">Perform the Hierarchical Clustering<a class=\"anchor-link\" href=\"#Perform-the-Hierarchical-Clustering\">&#182;<\/a><\/h2><p>Now that we have some very simple sample data, let's do the actual clustering on it:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[4]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"c\"># generate the linkage matrix<\/span>\r\n<span class=\"n\">Z<\/span> <span class=\"o\">=<\/span> <span class=\"n\">linkage<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">,<\/span> <span class=\"s\">&#39;ward&#39;<\/span><span class=\"p\">)<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Done. That was pretty simple, wasn't it?<\/p>\r\n<p>Well, sure it was, this is python ;), but what does the weird <span class=\"code_j\">'ward'<\/span> mean there and how does this actually work?<\/p>\r\n<p>As the <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.linkage.html\">scipy <span class=\"code_j\">linkage<\/span> docs<\/a> tell us, <span class=\"code_j\">'ward'<\/span> is one of the methods that can be used to calculate the distance between newly formed clusters. <span class=\"code_j\">'ward'<\/span> causes <span class=\"code_j\">linkage()<\/span> to use the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Ward%27s_method\">Ward variance minimization algorithm<\/a>.<\/p>\r\n<p>I think it's a good default choice, but it never hurts to play around with some other common linkage methods like <a href=\"https:\/\/en.wikipedia.org\/wiki\/Single-linkage_clustering\"><span class=\"code_j\">'single'<\/span><\/a>, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Complete-linkage_clustering\"><span class=\"code_j\">'complete'<\/span><\/a>, <a href=\"https:\/\/en.wikipedia.org\/wiki\/UPGMA\"><span class=\"code_j\">'average'<\/span><\/a>, ... and the different <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.spatial.distance.pdist.html\">distance metrics<\/a> like <a href=\"https:\/\/en.wikipedia.org\/wiki\/Euclidean_distance\"><span class=\"code_j\">'euclidean'<\/span> (default)<\/a>, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Taxicab_geometry\"><span class=\"code_j\">'cityblock'<\/span> aka Manhattan<\/a>, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hamming_distance\"><span class=\"code_j\">'hamming'<\/span><\/a>, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cosine_similarity\"><span class=\"code_j\">'cosine'<\/span><\/a>... if you have the feeling that your data should not just be clustered to minimize the overall intra cluster variance in euclidean space. For example, you should have such a weird feeling with long (binary) feature vectors (e.g., word-vectors in text clustering).<\/p>\r\n<p>As you can see there's a lot of choice here and while python and scipy make it very easy to do the clustering, it's you who has to understand and make these choices. If i find the time, i might give some more practical advice about this, but for now i'd urge you to at least read up on the mentioned linked methods and metrics to make a somewhat informed choice. Another thing you can and should definitely do is check the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cophenetic_correlation\">Cophenetic Correlation Coefficient<\/a> of your clustering with help of the <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.cophenet.html\"><span class=\"code_j\">cophenet()<\/span><\/a> function. This (very very briefly) compares (correlates) the actual pairwise distances of all your samples to those implied by the hierarchical clustering. The closer the value is to 1, the better the clustering preserves the original distances, which in our case is pretty close:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[5]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"kn\">from<\/span> <span class=\"nn\">scipy.cluster.hierarchy<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">cophenet<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">scipy.spatial.distance<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">pdist<\/span>\r\n\r\n<span class=\"n\">c<\/span><span class=\"p\">,<\/span> <span class=\"n\">coph_dists<\/span> <span class=\"o\">=<\/span> <span class=\"n\">cophenet<\/span><span class=\"p\">(<\/span><span class=\"n\">Z<\/span><span class=\"p\">,<\/span> <span class=\"n\">pdist<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">c<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[5]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>0.98001483875742679<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>No matter what <span class=\"code_j\">method<\/span> and <span class=\"code_j\">metric<\/span> you pick, the <span class=\"code_j\">linkage()<\/span> function will use that method and metric to calculate the distances of the clusters (starting with your <span class=\"code_j\">n<\/span> individual samples (aka data points) as singleton clusters)) and in each iteration will merge the two clusters which have the smallest distance according the selected <span class=\"code_j\">method<\/span> and <span class=\"code_j\">metric<\/span>. It will return an array of length <span class=\"code_j\">n - 1<\/span> giving you information about the <span class=\"code_j\">n - 1<\/span> cluster merges which it needs to pairwise merge <span class=\"code_j\">n<\/span> clusters. <span class=\"code_j\">Z[i]<\/span> will tell us which clusters were merged in the <span class=\"code_j\">i<\/span>-th iteration, let's take a look at the first two points that were merged:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[6]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">Z<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[6]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([ 52.     ,  53.     ,   0.04151,   2.     ])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>We can see that each row of the resulting array has the format <span class=\"code_j\">[idx1, idx2, dist, sample_count]<\/span>.<\/p>\r\n<p>In its first iteration the <span class=\"code_j\">linkage<\/span> algorithm decided to merge the two clusters (original samples here) with indices 52 and 53, as they only had a distance of 0.04151. This created a cluster with a total of 2 samples.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[7]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">Z<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[7]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([ 14.     ,  79.     ,   0.05914,   2.     ])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>In the second iteration the algorithm decided to merge the clusters (original samples here as well) with indices 14 and 79, which had a distance of 0.04914. This again formed another cluster with a total of 2 samples.<\/p>\r\n<p>The indices of the clusters until now correspond to our samples. Remember that we had a total of 150 samples, so indices 0 to 149. Let's have a look at the first 20 iterations:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[8]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">Z<\/span><span class=\"p\">[:<\/span><span class=\"mi\">20<\/span><span class=\"p\">]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[8]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([[  52.     ,   53.     ,    0.04151,    2.     ],\r\n       [  14.     ,   79.     ,    0.05914,    2.     ],\r\n       [  33.     ,   68.     ,    0.07107,    2.     ],\r\n       [  17.     ,   73.     ,    0.07137,    2.     ],\r\n       [   1.     ,    8.     ,    0.07543,    2.     ],\r\n       [  85.     ,   95.     ,    0.10928,    2.     ],\r\n       [ 108.     ,  131.     ,    0.11007,    2.     ],\r\n       [   9.     ,   66.     ,    0.11302,    2.     ],\r\n       [  15.     ,   69.     ,    0.11429,    2.     ],\r\n       [  63.     ,   98.     ,    0.1212 ,    2.     ],\r\n       [ 107.     ,  115.     ,    0.12167,    2.     ],\r\n       [  65.     ,   74.     ,    0.1249 ,    2.     ],\r\n       [  58.     ,   61.     ,    0.14028,    2.     ],\r\n       [  62.     ,  152.     ,    0.1726 ,    3.     ],\r\n       [  41.     ,  158.     ,    0.1779 ,    3.     ],\r\n       [  10.     ,   83.     ,    0.18635,    2.     ],\r\n       [ 114.     ,  139.     ,    0.20419,    2.     ],\r\n       [  39.     ,   88.     ,    0.20628,    2.     ],\r\n       [  70.     ,   96.     ,    0.21931,    2.     ],\r\n       [  46.     ,   50.     ,    0.22049,    2.     ]])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>We can observe that until iteration 13 the algorithm only directly merged original samples. We can also observe the monotonic increase of the distance.<\/p>\r\n<p>In iteration 14 the algorithm decided to merge cluster indices 62 with 152. If you paid attention the 152 should astonish you as we only have original sample indices 0 to 149 for our 150 samples. All indices <span class=\"code_j\">idx &gt;= len(X)<\/span> actually refer to the cluster formed in <span class=\"code_j\">Z[idx - len(X)]<\/span>.<\/p>\r\n<p>This means that while <span class=\"code_j\">idx<\/span> 149 corresponds to <span class=\"code_j\">X[149]<\/span> that <span class=\"code_j\">idx<\/span> 150 corresponds to the cluster formed in <span class=\"code_j\">Z[0]<\/span>, <span class=\"code_j\">idx<\/span> 151 to <span class=\"code_j\">Z[1]<\/span>, 152 to <span class=\"code_j\">Z[2]<\/span>, ...<\/p>\r\n<p>Hence, the merge iteration 14 merged sample 62 to our samples 33 and 68 that were previously merged in iteration 3 corresponding to Z[2] (152 - 150).<\/p>\r\n<p>Let's check out the points coordinates to see if this makes sense:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[9]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">X<\/span><span class=\"p\">[[<\/span><span class=\"mi\">33<\/span><span class=\"p\">,<\/span> <span class=\"mi\">68<\/span><span class=\"p\">,<\/span> <span class=\"mi\">62<\/span><span class=\"p\">]]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[9]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([[ 9.83913, -0.4873 ],\r\n       [ 9.89349, -0.44152],\r\n       [ 9.97793, -0.56383]])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Seems pretty close, but let's plot the points again and highlight them:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[10]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">idxs<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[<\/span><span class=\"mi\">33<\/span><span class=\"p\">,<\/span> <span class=\"mi\">68<\/span><span class=\"p\">,<\/span> <span class=\"mi\">62<\/span><span class=\"p\">]<\/span>\r\n<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\">8<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">1<\/span><span class=\"p\">])<\/span>  <span class=\"c\"># plot all points<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"n\">c<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;r&#39;<\/span><span class=\"p\">)<\/span>  <span class=\"c\"># plot interesting points in red again<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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nxd\/cJFpMcrj38+Ekv3eN59Ii\/s2bHKWUkuRjSf6s1vrhNb\/q6+9v3FcX\/pt0p01rkseT\/MyatU5a\r\nqpTyE0k+nORZST5Wa71\/zENiE3o7MJzu3b0hyW\/77tqtlPKpJPuT7E23\/uM9Sf5tug2ffzDJ15K8\r\nvtb6D+MaIxvb4Lt7b5KZ+DdvIpRSXp3kj5L8aZ5ZErwn3R1sNv33pxkpAEAD2nB1IQDAtiNkAQA0\r\nQMgCAGiAkAUA0AAhCwCgAUIWAEADhCwAgAb8\/8WtrvOeQ4aWAAAAAElFTkSuQmCC\r\n\"\r\n>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>We can see that the 3 red dots are pretty close to each other, which is a good thing.<\/p>\r\n<p>The same happened in iteration 15 where the alrogithm merged indices 41 to 15 and 69:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[11]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">idxs<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[<\/span><span class=\"mi\">33<\/span><span class=\"p\">,<\/span> <span class=\"mi\">68<\/span><span class=\"p\">,<\/span> <span class=\"mi\">62<\/span><span class=\"p\">]<\/span>\r\n<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\">8<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">1<\/span><span class=\"p\">])<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"n\">c<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;r&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">idxs<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[<\/span><span class=\"mi\">15<\/span><span class=\"p\">,<\/span> <span class=\"mi\">69<\/span><span class=\"p\">,<\/span> <span class=\"mi\">41<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span><span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"n\">c<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;y&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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b6RCnl\r\nhiRfG\/eA2Jxa6\/nvyr957VdKeWa6Aesjtdbf6z3c19\/fuK8uvGHN3f1JHrnUa2mNP0nyglLKTaWU\r\nXUl+Jt0GtLRcKeVZpZTv7f383Unm4m9uEi0mOdj7+WCS37vMa2kR\/+ZNjlJKSfLBJH9Wa33\/mqf6\r\n+vsb99WF\/zrdadOa5MtJfm7NWictVUr5qSTvT\/KMJB+std475iGxCb0dGE727l6V5Hd8d+1WSvlo\r\nktuS7Em3\/uMdSf5dug2ffzDJV5K8qtb6X8c1Rja2wXf3ziQz8W\/eRCilvDzJHyf50zy9JHhXujvY\r\nbPrvTzNSAIAGtOHqQgCAbUfIAgBogJAFANAAIQsAoAFCFgBAA4QsAIAGCFkAAA34\/wHhCq7LGZO5\r\nzAAAAABJRU5ErkJggg==\r\n\"\r\n>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Showing that the 3 yellow dots are also quite close.<\/p>\r\n<p>And so on...<\/p>\r\n<p>We'll later come back to visualizing this, but now let's have a look at what's called a dendrogram of this hierarchical clustering first:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Plotting-a-Dendrogram\">Plotting a Dendrogram<a class=\"anchor-link\" href=\"#Plotting-a-Dendrogram\">&#182;<\/a><\/h2><p>A <a href=\"http:\/\/mathworks.com\/help\/stats\/dendrogram.html\">dendrogram<\/a> is a visualization in form of a tree showing the order and distances of merges during the hierarchical clustering.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[12]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"c\"># calculate full dendrogram<\/span>\r\n<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\">25<\/span><span class=\"p\">,<\/span> <span class=\"mi\">10<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">title<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;Hierarchical Clustering Dendrogram&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">xlabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;sample index&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">ylabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;distance&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># rotates the x axis labels<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">8.<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># font size for the x axis labels<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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00ksvvfTSa1pes7am9F1Ja5Xe2Hh\/jbvfUjPNpPObxjLNe37TWCbFQPmp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\r\n1G\/HdL0oAH4AAAAASUVORK5CYII=\r\n\"\r\n>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>(right click and \"View Image\" to see full resolution)\r\n<p>If this is the first time you see a dendrogram, it's probably quite confusing, so let's take this apart...<\/p>\r\n<ul>\r\n<li>On the x axis you see labels. If you don't specify anything else they are the indices of your samples in <span class=\"code_j\">X<\/span>.<\/li>\r\n<li>On the y axis you see the distances (of the <span class=\"code_j\">'ward'<\/span> method in our case).<\/li>\r\n<\/ul>\r\n<p>Starting from each label at the bottom, you can see a vertical line up to a horizontal line. The height of that horizontal line tells you about the distance at which this label was merged into another label or cluster. You can find that other cluster by following the other vertical line down again. If you don't encounter another horizontal line, it was just merged with the other label you reach, otherwise it was merged into another cluster that was formed earlier.<\/p>\r\n<p>Summarizing:<\/p>\r\n<ul>\r\n<li>horizontal lines are cluster merges<\/li>\r\n<li>vertical lines tell you which clusters\/labels were part of merge forming that new cluster<\/li>\r\n<li>heights of the horizontal lines tell you about the distance that needed to be \"bridged\" to form the new cluster<\/li>\r\n<\/ul>\r\n<p>You can also see that from distances &gt; 25 up there's a huge jump of the distance to the final merge at a distance of approx. 180. Let's have a look at the distances of the last 4 merges:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[13]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">Z<\/span><span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">4<\/span><span class=\"p\">:,<\/span><span class=\"mi\">2<\/span><span class=\"p\">]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[13]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([  15.11533,   17.11527,   23.12199,  180.27043])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Such distance jumps \/ gaps in the dendrogram are pretty interesting for us. They indicate that something is merged here, that maybe just shouldn't be merged. In other words: maybe the things that were merged here really don't belong to the same cluster, telling us that maybe there's just 2 clusters here.<\/p>\r\n<p>Looking at indices in the above dendrogram also shows us that the green cluster only has indices &gt;= 100, while the red one only has such &lt; 100. This is a good thing as it shows that the algorithm re-discovered the two classes in our toy example.<\/p>\r\n<p>In case you're wondering about where the colors come from, you might want to have a look at the <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.dendrogram.html\"><span class=\"code_j\">color_threshold<\/span><\/a> argument of <span class=\"code_j\">dendrogram()<\/span>, which as not specified automagically picked a distance cut-off value of 70 % of the final merge and then colored the first clusters below that in individual colors.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h3 id=\"Dendrogram-Truncation\">Dendrogram Truncation<a class=\"anchor-link\" href=\"#Dendrogram-Truncation\">&#182;<\/a><\/h3><p>As you might have noticed, the above is pretty big for 150 samples already and you probably have way more in real scenarios, so let me spend a few seconds on highlighting some other features of the <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.dendrogram.html\"><span class=\"code_j\">dendrogram()<\/span><\/a> function:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[14]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">title<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;Hierarchical Clustering Dendrogram (truncated)&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">xlabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;sample index&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">ylabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;distance&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">truncate_mode<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;lastp&#39;<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># show only the last p merged clusters<\/span>\r\n    <span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># show only the last p merged clusters<\/span>\r\n    <span class=\"n\">show_leaf_counts<\/span><span class=\"o\">=<\/span><span class=\"bp\">False<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># otherwise numbers in brackets are counts<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">12.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">show_contracted<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># to get a distribution impression in truncated branches<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>The above shows a truncated dendrogram, which only shows the last <span class=\"code_j\">p=12<\/span> out of our 149 merges.<\/p>\r\n<p>First thing you should notice are that most labels are missing. This is because except for <span class=\"code_j\">X[40]<\/span> all other samples were already merged into clusters before the last 12 merges.<\/p>\r\n<p>The parameter <span class=\"code_j\">show_contracted<\/span> allows us to draw black dots at the heights of those previous cluster merges, so we can still spot gaps even if we don't want to clutter the whole visualization. In our example we can see that the dots are all at pretty small distances when compared to the huge last merge at a distance of 180, telling us that we probably didn't miss much there.<\/p>\r\n<p>As it's kind of hard to keep track of the cluster sizes just by the dots, <span class=\"code_j\">dendrogram()<\/span> will by default also print the cluster sizes in brackets <span class=\"code_j\">()<\/span> if a cluster was truncated:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[15]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">title<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;Hierarchical Clustering Dendrogram (truncated)&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">xlabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;sample index or (cluster size)&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">ylabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;distance&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">truncate_mode<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;lastp&#39;<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># show only the last p merged clusters<\/span>\r\n    <span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># show only the last p merged clusters<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">12.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">show_contracted<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># to get a distribution impression in truncated branches<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>We can now see that the right most cluster already consisted of 33 samples before the last 12 merges.<\/p>\r\n<h3 id=\"Eye-Candy\">Eye Candy<a class=\"anchor-link\" href=\"#Eye-Candy\">&#182;<\/a><\/h3><p>Even though this already makes for quite a nice visualization, we can pimp it even more by also <a href=\"http:\/\/stackoverflow.com\/questions\/11917779\/how-to-plot-and-annotate-hierarchical-clustering-dendrograms-in-scipy-matplotlib\/12311618#12311618\">annotating the distances inside the dendrogram<\/a> by using some of the useful return values <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.dendrogram.html\"><span class=\"code_j\">dendrogram()<\/span><\/a>:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[16]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"k\">def<\/span> <span class=\"nf\">fancy_dendrogram<\/span><span class=\"p\">(<\/span><span class=\"o\">*<\/span><span class=\"n\">args<\/span><span class=\"p\">,<\/span> <span class=\"o\">**<\/span><span class=\"n\">kwargs<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"n\">max_d<\/span> <span class=\"o\">=<\/span> <span class=\"n\">kwargs<\/span><span class=\"o\">.<\/span><span class=\"n\">pop<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;max_d&#39;<\/span><span class=\"p\">,<\/span> <span class=\"bp\">None<\/span><span class=\"p\">)<\/span>\r\n    <span class=\"k\">if<\/span> <span class=\"n\">max_d<\/span> <span class=\"ow\">and<\/span> <span class=\"s\">&#39;color_threshold&#39;<\/span> <span class=\"ow\">not<\/span> <span class=\"ow\">in<\/span> <span class=\"n\">kwargs<\/span><span class=\"p\">:<\/span>\r\n        <span class=\"n\">kwargs<\/span><span class=\"p\">[<\/span><span class=\"s\">&#39;color_threshold&#39;<\/span><span class=\"p\">]<\/span> <span class=\"o\">=<\/span> <span class=\"n\">max_d<\/span>\r\n    <span class=\"n\">annotate_above<\/span> <span class=\"o\">=<\/span> <span class=\"n\">kwargs<\/span><span class=\"o\">.<\/span><span class=\"n\">pop<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;annotate_above&#39;<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">)<\/span>\r\n\r\n    <span class=\"n\">ddata<\/span> <span class=\"o\">=<\/span> <span class=\"n\">dendrogram<\/span><span class=\"p\">(<\/span><span class=\"o\">*<\/span><span class=\"n\">args<\/span><span class=\"p\">,<\/span> <span class=\"o\">**<\/span><span class=\"n\">kwargs<\/span><span class=\"p\">)<\/span>\r\n\r\n    <span class=\"k\">if<\/span> <span class=\"ow\">not<\/span> <span class=\"n\">kwargs<\/span><span class=\"o\">.<\/span><span class=\"n\">get<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;no_plot&#39;<\/span><span class=\"p\">,<\/span> <span class=\"bp\">False<\/span><span class=\"p\">):<\/span>\r\n        <span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">title<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;Hierarchical Clustering Dendrogram (truncated)&#39;<\/span><span class=\"p\">)<\/span>\r\n        <span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">xlabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;sample index or (cluster size)&#39;<\/span><span class=\"p\">)<\/span>\r\n        <span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">ylabel<\/span><span class=\"p\">(<\/span><span class=\"s\">&#39;distance&#39;<\/span><span class=\"p\">)<\/span>\r\n        <span class=\"k\">for<\/span> <span class=\"n\">i<\/span><span class=\"p\">,<\/span> <span class=\"n\">d<\/span><span class=\"p\">,<\/span> <span class=\"n\">c<\/span> <span class=\"ow\">in<\/span> <span class=\"nb\">zip<\/span><span class=\"p\">(<\/span><span class=\"n\">ddata<\/span><span class=\"p\">[<\/span><span class=\"s\">&#39;icoord&#39;<\/span><span class=\"p\">],<\/span> <span class=\"n\">ddata<\/span><span class=\"p\">[<\/span><span class=\"s\">&#39;dcoord&#39;<\/span><span class=\"p\">],<\/span> <span class=\"n\">ddata<\/span><span class=\"p\">[<\/span><span class=\"s\">&#39;color_list&#39;<\/span><span class=\"p\">]):<\/span>\r\n            <span class=\"n\">x<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">0.5<\/span> <span class=\"o\">*<\/span> <span class=\"nb\">sum<\/span><span class=\"p\">(<\/span><span class=\"n\">i<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">:<\/span><span class=\"mi\">3<\/span><span class=\"p\">])<\/span>\r\n            <span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">d<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n            <span class=\"k\">if<\/span> <span class=\"n\">y<\/span> <span class=\"o\">&gt;<\/span> <span class=\"n\">annotate_above<\/span><span class=\"p\">:<\/span>\r\n                <span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">plot<\/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=\"s\">&#39;o&#39;<\/span><span class=\"p\">,<\/span> <span class=\"n\">c<\/span><span class=\"o\">=<\/span><span class=\"n\">c<\/span><span class=\"p\">)<\/span>\r\n                <span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">annotate<\/span><span class=\"p\">(<\/span><span class=\"s\">&quot;<\/span><span class=\"si\">%.3g<\/span><span class=\"s\">&quot;<\/span> <span class=\"o\">%<\/span> <span class=\"n\">y<\/span><span class=\"p\">,<\/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\">xytext<\/span><span class=\"o\">=<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"o\">-<\/span><span class=\"mi\">5<\/span><span class=\"p\">),<\/span>\r\n                             <span class=\"n\">textcoords<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;offset points&#39;<\/span><span class=\"p\">,<\/span>\r\n                             <span class=\"n\">va<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;top&#39;<\/span><span class=\"p\">,<\/span> <span class=\"n\">ha<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;center&#39;<\/span><span class=\"p\">)<\/span>\r\n        <span class=\"k\">if<\/span> <span class=\"n\">max_d<\/span><span class=\"p\">:<\/span>\r\n            <span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">axhline<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"o\">=<\/span><span class=\"n\">max_d<\/span><span class=\"p\">,<\/span> <span class=\"n\">c<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;k&#39;<\/span><span class=\"p\">)<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">ddata<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[17]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">fancy_dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">truncate_mode<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;lastp&#39;<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">12.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">show_contracted<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">annotate_above<\/span><span class=\"o\">=<\/span><span class=\"mi\">10<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># useful in small plots so annotations don&#39;t overlap<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Selecting-a-Distance-Cut-Off-aka-Determining-the-Number-of-Clusters\">Selecting a Distance Cut-Off aka Determining the Number of Clusters<a class=\"anchor-link\" href=\"#Selecting-a-Distance-Cut-Off-aka-Determining-the-Number-of-Clusters\">&#182;<\/a><\/h2><p>As explained above already, a huge jump in distance is typically what we're interested in if we want to argue for a certain number of clusters. If you have the chance to do this manually, i'd always opt for that, as it allows you to gain some insights into your data and to perform some sanity checks on the edge cases. In our case i'd probably just say that our cut-off is 50, as the jump is pretty obvious:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[18]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"c\"># set cut-off to 50<\/span>\r\n<span class=\"n\">max_d<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">50<\/span>  <span class=\"c\"># max_d as in max_distance<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Let's visualize this in the dendrogram as a cut-off line:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[19]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">fancy_dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">truncate_mode<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;lastp&#39;<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">12.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">show_contracted<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">annotate_above<\/span><span class=\"o\">=<\/span><span class=\"mi\">10<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">max_d<\/span><span class=\"o\">=<\/span><span class=\"n\">max_d<\/span><span class=\"p\">,<\/span>  <span class=\"c\"># plot a horizontal cut-off line<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>As we can see, we (\"surprisingly\") have two clusters at this cut-off.<\/p>\r\n<p>In general for a chosen cut-off value <span class=\"code_j\">max_d<\/span> you can always simply count the number of intersections with vertical lines of the dendrogram to get the number of formed clusters. Say we choose a cut-off of <span class=\"code_j\">max_d = 16<\/span>, we'd get 4 final clusters:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[20]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">fancy_dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">truncate_mode<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;lastp&#39;<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">12.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">show_contracted<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">annotate_above<\/span><span class=\"o\">=<\/span><span class=\"mi\">10<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">max_d<\/span><span class=\"o\">=<\/span><span class=\"mi\">16<\/span><span class=\"p\">,<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h3 id=\"Automated-Cut-Off-Selection-(or-why-you-shouldn't-rely-on-this)\">Automated Cut-Off Selection (or why you shouldn't rely on this)<a class=\"anchor-link\" href=\"#Automated-Cut-Off-Selection-(or-why-you-shouldn't-rely-on-this)\">&#182;<\/a><\/h3><p>Now while this manual selection of a cut-off value offers a lot of benefits when it comes to checking for a meaningful clustering and cut-off, there are cases in which you want to automate this.<\/p>\r\n<p>The problem again is that there is no golden method to pick the number of clusters for all cases (which is why i think the investigative &amp; backtesting manual method is preferable). Wikipedia lists a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Determining_the_number_of_clusters_in_a_data_set\">couple of common methods<\/a>. Reading this, you should realize how different the approaches and how vague their descriptions are.<\/p>\r\n<p>I honestly think it's a really bad idea to just use any of those methods, unless you know the data you're working on really really well.<\/p>\r\n<h4 id=\"Inconsistency-Method\">Inconsistency Method<a class=\"anchor-link\" href=\"#Inconsistency-Method\">&#182;<\/a><\/h4><p>For example, let's have a look at the \"<a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.inconsistent.html\">inconsistency<\/a>\" method, which seems to be one of the defaults for the <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.fcluster.html\"><span class=\"code_j\">fcluster()<\/span><\/a> function in scipy.<\/p>\r\n<p>The question driving the inconsistency method is \"what makes a distance jump a jump?\". It answers this by comparing each cluster merge's height <span class=\"code_j\">h<\/span> to the average <span class=\"code_j\">avg<\/span> and normalizing it by the standard deviation <span class=\"code_j\">std<\/span> formed over the <span class=\"code_j\">depth<\/span> previous levels:<\/p>\r\n$$inconsistency = \\frac{h - avg}{std}$$<p>The following shows a matrix of the <span class=\"code_j\">avg, std, count, inconsistency<\/span> for each of the last 10 merges of our hierarchical clustering with <span class=\"code_j\">depth = 5<\/:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[21]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"kn\">from<\/span> <span class=\"nn\">scipy.cluster.hierarchy<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">inconsistent<\/span>\r\n\r\n<span class=\"n\">depth<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">5<\/span>\r\n<span class=\"n\">incons<\/span> <span class=\"o\">=<\/span> <span class=\"n\">inconsistent<\/span><span class=\"p\">(<\/span><span class=\"n\">Z<\/span><span class=\"p\">,<\/span> <span class=\"n\">depth<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">incons<\/span><span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">10<\/span><span class=\"p\">:]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[21]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([[  1.80875,   2.17062,  10.     ,   2.44277],\r\n       [  2.31732,   2.19649,  16.     ,   2.52742],\r\n       [  2.24512,   2.44225,   9.     ,   2.37659],\r\n       [  2.30462,   2.44191,  21.     ,   2.63875],\r\n       [  2.20673,   2.68378,  17.     ,   2.84582],\r\n       [  1.95309,   2.581  ,  29.     ,   4.05821],\r\n       [  3.46173,   3.53736,  28.     ,   3.29444],\r\n       [  3.15857,   3.54836,  28.     ,   3.93328],\r\n       [  4.9021 ,   5.10302,  28.     ,   3.57042],\r\n       [ 12.122  ,  32.15468,  30.     ,   5.22936]])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Now you might be tempted to say \"yay, let's just pick <span class=\"code_j\">5<\/span>\" as a limit in the inconsistencies, but look at what happens if we set depth to <span class=\"code_j\">3<\/span> instead:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[22]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">depth<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">3<\/span>\r\n<span class=\"n\">incons<\/span> <span class=\"o\">=<\/span> <span class=\"n\">inconsistent<\/span><span class=\"p\">(<\/span><span class=\"n\">Z<\/span><span class=\"p\">,<\/span> <span class=\"n\">depth<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">incons<\/span><span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">10<\/span><span class=\"p\">:]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[22]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([[  3.63778,   2.55561,   4.     ,   1.35908],\r\n       [  3.89767,   2.57216,   7.     ,   1.54388],\r\n       [  3.05886,   2.66707,   6.     ,   1.87115],\r\n       [  4.92746,   2.7326 ,   7.     ,   1.39822],\r\n       [  4.76943,   3.16277,   6.     ,   1.60456],\r\n       [  5.27288,   3.56605,   7.     ,   2.00627],\r\n       [  8.22057,   4.07583,   7.     ,   1.69162],\r\n       [  7.83287,   4.46681,   7.     ,   2.07808],\r\n       [ 11.38091,   6.2943 ,   7.     ,   1.86535],\r\n       [ 37.25845,  63.31539,   7.     ,   2.25872]])<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>Oups! This should make you realize that the inconsistency values heavily depend on the depth of the tree you calculate the averages over.<\/p>\r\n<p>Another problem in its calculation is that the previous <span class=\"code_j\">d<\/span> levels' heights aren't normally distributed, but expected to increase, so you can't really just treat the current level as an \"outlier\" of a normal distribution, as it's expected to be bigger.<\/p>\r\n<h4 id=\"Elbow-Method\">Elbow Method<a class=\"anchor-link\" href=\"#Elbow-Method\">&#182;<\/a><\/h4><p>Another thing you might see <a href=\"https:\/\/en.wikipedia.org\/wiki\/Elbow_method_(clustering)\">out there<\/a> is a variant of the \"elbow method\". It tries to find the clustering step where the acceleration of distance growth is the biggest (the \"strongest elbow\" of the blue line graph below, which is the highest value of the green graph below):<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[23]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">last<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Z<\/span><span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">10<\/span><span class=\"p\">:,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">last_rev<\/span> <span class=\"o\">=<\/span> <span class=\"n\">last<\/span><span class=\"p\">[::<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">idxs<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">arange<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"nb\">len<\/span><span class=\"p\">(<\/span><span class=\"n\">last<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">1<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">plot<\/span><span class=\"p\">(<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span> <span class=\"n\">last_rev<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">acceleration<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">last<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span>  <span class=\"c\"># 2nd derivative of the distances<\/span>\r\n<span class=\"n\">acceleration_rev<\/span> <span class=\"o\">=<\/span> <span class=\"n\">acceleration<\/span><span class=\"p\">[::<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">plot<\/span><span class=\"p\">(<\/span><span class=\"n\">idxs<\/span><span class=\"p\">[:<\/span><span class=\"o\">-<\/span><span class=\"mi\">2<\/span><span class=\"p\">]<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">acceleration_rev<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<span class=\"n\">k<\/span> <span class=\"o\">=<\/span> <span class=\"n\">acceleration_rev<\/span><span class=\"o\">.<\/span><span class=\"n\">argmax<\/span><span class=\"p\">()<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span>  <span class=\"c\"># if idx 0 is the max of this we want 2 clusters<\/span>\r\n<span class=\"k\">print<\/span> <span class=\"s\">&quot;clusters:&quot;<\/span><span class=\"p\">,<\/span> <span class=\"n\">k<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"output_area\"><div class=\"prompt\"><\/div>\r\n<div class=\"output_subarea output_stream output_stdout output_text\">\r\n<pre>clusters: 2\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>While this works nicely in our simplistic example (the green line takes its maximum for <span class=\"code_j\">k=2<\/span>), it's pretty flawed as well.<\/p>\r\n<p>One issue of this method has to do with the way an \"elbow\" is defined: you need at least a right and a left point, which implies that this method will never be able to tell you that all your data is in one single cluster only.<\/p>\r\n<p>Another problem with this variant lies in the <span class=\"code_j\">np.diff(Z[:, 2], 2)<\/span> though. The order of the distances in <span class=\"code_j\">Z[:, 2]<\/span> isn't properly reflecting the order of merges within one branch of the tree. In other words: there is no guarantee that the distance of <span class=\"code_j\">Z[i]<\/span> is contained in the branch of <span class=\"code_j\">Z[i+1]<\/span>. By simply computing the <span class=\"code_j\">np.diff(Z[:, 2], 2)<\/span> we assume that this doesn't matter and just compare distance jumps from different branches of our merge tree.<\/p>\r\n<p>If you still don't want to believe this, let's just construct another simplistic example but this time with very different variances in the different clusters:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[24]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">c<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">random<\/span><span class=\"o\">.<\/span><span class=\"n\">multivariate_normal<\/span><span class=\"p\">([<\/span><span class=\"mi\">40<\/span><span class=\"p\">,<\/span> <span class=\"mi\">40<\/span><span class=\"p\">],<\/span> <span class=\"p\">[[<\/span><span class=\"mi\">20<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">30<\/span><span class=\"p\">]],<\/span> <span class=\"n\">size<\/span><span class=\"o\">=<\/span><span class=\"p\">[<\/span><span class=\"mi\">200<\/span><span class=\"p\">,])<\/span>\r\n<span class=\"n\">d<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">random<\/span><span class=\"o\">.<\/span><span class=\"n\">multivariate_normal<\/span><span class=\"p\">([<\/span><span class=\"mi\">80<\/span><span class=\"p\">,<\/span> <span class=\"mi\">80<\/span><span class=\"p\">],<\/span> <span class=\"p\">[[<\/span><span class=\"mi\">30<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">30<\/span><span class=\"p\">]],<\/span> <span class=\"n\">size<\/span><span class=\"o\">=<\/span><span class=\"p\">[<\/span><span class=\"mi\">200<\/span><span class=\"p\">,])<\/span>\r\n<span class=\"n\">e<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">random<\/span><span class=\"o\">.<\/span><span class=\"n\">multivariate_normal<\/span><span class=\"p\">([<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">100<\/span><span class=\"p\">],<\/span> <span class=\"p\">[[<\/span><span class=\"mi\">100<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">100<\/span><span class=\"p\">]],<\/span> <span class=\"n\">size<\/span><span class=\"o\">=<\/span><span class=\"p\">[<\/span><span class=\"mi\">200<\/span><span class=\"p\">,])<\/span>\r\n<span class=\"n\">X2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">concatenate<\/span><span class=\"p\">((<\/span><span class=\"n\">X<\/span><span class=\"p\">,<\/span> <span class=\"n\">c<\/span><span class=\"p\">,<\/span> <span class=\"n\">d<\/span><span class=\"p\">,<\/span> <span class=\"n\">e<\/span><span class=\"p\">),)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X2<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X2<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">1<\/span><span class=\"p\">])<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>As you can see we have 5 clusters now, but they have increasing variances... let's have a look at the dendrogram again and how you can use it to spot the problem:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[25]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">Z2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">linkage<\/span><span class=\"p\">(<\/span><span class=\"n\">X2<\/span><span class=\"p\">,<\/span> <span class=\"s\">&#39;ward&#39;<\/span><span class=\"p\">)<\/span>\r\n<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\">10<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">fancy_dendrogram<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"n\">Z2<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">truncate_mode<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;lastp&#39;<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"mi\">30<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_rotation<\/span><span class=\"o\">=<\/span><span class=\"mf\">90.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">leaf_font_size<\/span><span class=\"o\">=<\/span><span class=\"mf\">12.<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">show_contracted<\/span><span class=\"o\">=<\/span><span class=\"bp\">True<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">annotate_above<\/span><span class=\"o\">=<\/span><span class=\"mi\">40<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"n\">max_d<\/span><span class=\"o\">=<\/span><span class=\"mi\">170<\/span><span class=\"p\">,<\/span>\r\n<span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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G2oYmB8vD1c5x7iIs70WxKKWexdqK40GsWElbsLYC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\r\ncWYwGAwGg8HgIYxxZjAYDAaDweAh\/j9Z194rYvSJawAAAABJRU5ErkJggg==\r\n\"\r\n>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>When looking at a dendrogram like this and trying to put a cut-off line somewhere, you should notice the very different distributions of merge distances below that cut-off line. Compare the distribution in the cyan cluster to the red, green or even two blue clusters that have even been truncated away. In the cyan cluster below the cut-off we don't really have any discontinuity of merge distances up to very close to the cut-off line. The two blue clusters on the other hand are each merged below a distance of 25, and have a gap of &gt; 155 to our cut-off line.<\/p>\r\n<p>The variant of the \"elbow\" method will incorrectly see the jump from 167 to 180 as minimal and tell us we have 4 clusters:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[26]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">last<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Z2<\/span><span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">10<\/span><span class=\"p\">:,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">last_rev<\/span> <span class=\"o\">=<\/span> <span class=\"n\">last<\/span><span class=\"p\">[::<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">idxs<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">arange<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"nb\">len<\/span><span class=\"p\">(<\/span><span class=\"n\">last<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">1<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">plot<\/span><span class=\"p\">(<\/span><span class=\"n\">idxs<\/span><span class=\"p\">,<\/span> <span class=\"n\">last_rev<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">acceleration<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">last<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span>  <span class=\"c\"># 2nd derivative of the distances<\/span>\r\n<span class=\"n\">acceleration_rev<\/span> <span class=\"o\">=<\/span> <span class=\"n\">acceleration<\/span><span class=\"p\">[::<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">plot<\/span><span class=\"p\">(<\/span><span class=\"n\">idxs<\/span><span class=\"p\">[:<\/span><span class=\"o\">-<\/span><span class=\"mi\">2<\/span><span class=\"p\">]<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">acceleration_rev<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<span class=\"n\">k<\/span> <span class=\"o\">=<\/span> <span class=\"n\">acceleration_rev<\/span><span class=\"o\">.<\/span><span class=\"n\">argmax<\/span><span class=\"p\">()<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span>  <span class=\"c\"># if idx 0 is the max of this we want 2 clusters<\/span>\r\n<span class=\"k\">print<\/span> <span class=\"s\">&quot;clusters:&quot;<\/span><span class=\"p\">,<\/span> <span class=\"n\">k<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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class=\"output_area\"><div class=\"prompt\"><\/div>\r\n<div class=\"output_subarea output_stream output_stdout output_text\">\r\n<pre>clusters: 4\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>The same happens with the inconsistency metric:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[27]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"k\">print<\/span> <span class=\"n\">inconsistent<\/span><span class=\"p\">(<\/span><span class=\"n\">Z2<\/span><span class=\"p\">,<\/span> <span class=\"mi\">5<\/span><span class=\"p\">)[<\/span><span class=\"o\">-<\/span><span class=\"mi\">10<\/span><span class=\"p\">:]<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n<div class=\"output_subarea output_stream output_stdout output_text\">\r\n<pre>[[  13.99222   15.56656   30.         3.86585]\r\n [  16.73941   18.5639    30.         3.45983]\r\n [  19.05945   20.53211   31.         3.49953]\r\n [  19.25574   20.82658   29.         3.51907]\r\n [  21.36116   26.7766    30.         4.50256]\r\n [  36.58101   37.08602   31.         3.50761]\r\n [  12.122     32.15468   30.         5.22936]\r\n [  42.6137   111.38577   31.         5.13038]\r\n [  81.75199  208.31582   31.         5.30448]\r\n [ 147.25602  307.95701   31.         3.6215 ]]\r\n<\/pre>\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>I hope you can now understand why i'm warning against blindly using any of those methods on a dataset you know nothing about. They can give you some indication, but you should always go back in and check if the results make sense, for example with a dendrogram which is a great tool for that (especially if you have higher dimensional data that you can't simply visualize anymore).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Retrieve-the-Clusters\">Retrieve the Clusters<a class=\"anchor-link\" href=\"#Retrieve-the-Clusters\">&#182;<\/a><\/h2><p>Now, let's finally have a look at how to retrieve the clusters, for different ways of determining k. We can use the  <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.fcluster.html\"><span class=\"code_j\">fcluster<\/span><\/a> function.<\/p>\r\n<h3 id=\"Knowing-max_d:\">Knowing max_d:<a class=\"anchor-link\" href=\"#Knowing-max_d:\">&#182;<\/a><\/h3><p>Let's say we determined the max distance with help of a dendrogram, then we can do the following to get the cluster id for each of our samples:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[28]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"kn\">from<\/span> <span class=\"nn\">scipy.cluster.hierarchy<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">fcluster<\/span>\r\n<span class=\"n\">max_d<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">50<\/span>\r\n<span class=\"n\">clusters<\/span> <span class=\"o\">=<\/span> <span class=\"n\">fcluster<\/span><span class=\"p\">(<\/span><span class=\"n\">Z<\/span><span class=\"p\">,<\/span> <span class=\"n\">max_d<\/span><span class=\"p\">,<\/span> <span class=\"n\">criterion<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;distance&#39;<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">clusters<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[28]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\r\n       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\r\n       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32)<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h3 id=\"Knowing-k:\">Knowing k:<a class=\"anchor-link\" href=\"#Knowing-k:\">&#182;<\/a><\/h3><p>Another way starting from the dendrogram is to say \"i can see i have k=2\" clusters. You can then use:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[29]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"n\">k<\/span><span class=\"o\">=<\/span><span class=\"mi\">2<\/span>\r\n<span class=\"n\">fcluster<\/span><span class=\"p\">(<\/span><span class=\"n\">Z<\/span><span class=\"p\">,<\/span> <span class=\"n\">k<\/span><span class=\"p\">,<\/span> <span class=\"n\">criterion<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;maxclust&#39;<\/span><span class=\"p\">)<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[29]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\r\n       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\r\n       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32)<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h3 id=\"Using-the-Inconsistency-Method-(default):\">Using the Inconsistency Method (default):<a class=\"anchor-link\" href=\"#Using-the-Inconsistency-Method-(default):\">&#182;<\/a><\/h3><p>If you're really sure you want to use the <a href=\"#Inconsistency-Method\">inconsistency method<\/a> to determine the number of clusters in your dataset, you can use the default <span class=\"code_j\">criterion<\/span> of <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.cluster.hierarchy.fcluster.html\"><span class=\"code_j\">fcluster()<\/span><\/a> and hope you picked the correct values:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[30]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><span class=\"kn\">from<\/span> <span class=\"nn\">scipy.cluster.hierarchy<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">fcluster<\/span>\r\n<span class=\"n\">fcluster<\/span><span class=\"p\">(<\/span><span class=\"n\">Z<\/span><span class=\"p\">,<\/span> <span class=\"mi\">8<\/span><span class=\"p\">,<\/span> <span class=\"n\">depth<\/span><span class=\"o\">=<\/span><span class=\"mi\">10<\/span><span class=\"p\">)<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[30]:<\/div>\r\n\r\n\r\n<div class=\"output_text output_subarea output_execute_result\">\r\n<pre>array([2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,\r\n       2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\r\n       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\r\n       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], dtype=int32)<\/pre>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h1 id=\"Visualizing-Your-Clusters\">Visualizing Your Clusters<a class=\"anchor-link\" href=\"#Visualizing-Your-Clusters\">&#182;<\/a><\/h1><p>If you're lucky enough and your data is very low dimensional, you can actually visualize the resulting clusters very easily:<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing code_cell rendered\">\r\n<div class=\"input\">\r\n<div class=\"prompt input_prompt\">In&nbsp;[31]:<\/div>\r\n<div class=\"inner_cell\">\r\n    <div class=\"input_area\">\r\n<div class=\" highlight hl-ipython2\"><pre><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\">8<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">scatter<\/span><span class=\"p\">(<\/span><span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">X<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <span class=\"n\">c<\/span><span class=\"o\">=<\/span><span class=\"n\">clusters<\/span><span class=\"p\">,<\/span> <span class=\"n\">cmap<\/span><span class=\"o\">=<\/span><span class=\"s\">&#39;prism&#39;<\/span><span class=\"p\">)<\/span>  <span class=\"c\"># plot points with cluster dependent colors<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">()<\/span>\r\n<\/pre><\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n<div class=\"output_wrapper\">\r\n<div class=\"output\">\r\n\r\n\r\n<div class=\"output_area\"><div class=\"prompt\"><\/div>\r\n\r\n\r\n<div class=\"output_png output_subarea \">\r\n<img 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5Os7LHLHVf4dxfwCVAVaAic\r\nAN51djxyw3SnRMHW2rKsJkBXYHj2kIYUUNl3buucLDj0nVfAZA8VzgFGWJaVeOm2nJx\/1yxGmh9y\r\nWtA0uzL8D04OR27c5cVmK\/DnxyiJG7Ms60T2f+ONMfNwDP+udW1UkkunjDFlLMs6aYwpC8S5OiDJ\r\nGcuyLn5W+s5zf8aYQjgSrGmWZc3PXp2r88\/VdxeWvWSxN7DjavuK2\/gNqGGMqWyM8Qbuw1GA4NLR\r\ntAAAAOFJREFUVtycMcbHGOOf\/doX6ITOuYJoIdA\/+3V\/YP419hU3ou+8gsMYY4ApwG7Lst6\/ZFOu\r\nzj9X3134NY7LphYQAwy+ZKxT3JQxpivwPuAJTLEsa7yLQ5IcyH4Cw7zsRS9ghj4792aM+QZoC5TE\r\nMf9jNLAAR8HnisAh4F7Lss67Kka5sit8dmOAMPSdVyAYY+4A1gDb+WNI8CUcT7DJ8fmnYqQiIiIi\r\nTuAOdxeKiIiI\/O0oyRIRERFxAiVZIiIiIk6gJEtERETECZRkiYiIiDiBkiwRERERJ1CSJSIiIuIE\r\n\/w9vcQbiELrWawAAAABJRU5ErkJggg==\r\n\"\r\n>\r\n<\/div>\r\n\r\n<\/div>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<p>I hope you enjoyed this tutorial. Feedback welcome ;)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"cell border-box-sizing text_cell rendered\">\r\n<div class=\"prompt input_prompt\">\r\n<\/div>\r\n<div class=\"inner_cell\">\r\n<div class=\"text_cell_render border-box-sizing rendered_html\">\r\n<h2 id=\"Further-Reading:\">Further Reading:<a class=\"anchor-link\" href=\"#Further-Reading:\">&#182;<\/a><\/h2><ul>\r\n<li>The scipy hierarchical clustering module: <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/cluster.hierarchy.html\">http:\/\/docs.scipy.org\/doc\/scipy\/reference\/cluster.hierarchy.html<\/a><\/li>\r\n<li>The scipy distance computation docs: <a href=\"http:\/\/docs.scipy.org\/doc\/scipy\/reference\/spatial.distance.html\">http:\/\/docs.scipy.org\/doc\/scipy\/reference\/spatial.distance.html<\/a><\/li>\r\n<li>The scipy hierarchical clustering module docs often refer to the MATLAB docs saying that a certain function is similar to the MATLAB one. Here's their hierarchical clustering tutorial: <a href=\"http:\/\/mathworks.com\/help\/stats\/hierarchical-clustering.html\">http:\/\/mathworks.com\/help\/stats\/hierarchical-clustering.html<\/a><\/li>\r\n<\/ul>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n    <\/div>\r\n\r\n\r\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[2],"tags":[210,25,212,211,69,132,147,171],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/pYA5n-bc","jetpack-related-posts":[{"id":526,"url":"https:\/\/joernhees.de\/blog\/2013\/06\/08\/mac-os-x-10-8-scientific-python-with-homebrew\/","url_meta":{"origin":694,"position":0},"title":"Scientific Python on Mac OS X 10.8 with homebrew","date":"2013-06-08","format":false,"excerpt":"(newer version of this guide) A step-by-step installation guide to setup a scientific python environment based on Mac OS X and homebrew. Needless to say: Make a backup (Timemachine) First install homebrew. Follow their instructions, then come back here. 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I hope you'll be able to benefit from my hours of trials and errors ;)","rel":"","context":"In &quot;Coding&quot;","img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/posts\/694"}],"collection":[{"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/comments?post=694"}],"version-history":[{"count":42,"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/posts\/694\/revisions"}],"predecessor-version":[{"id":918,"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/posts\/694\/revisions\/918"}],"wp:attachment":[{"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/media?parent=694"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/categories?post=694"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/joernhees.de\/blog\/wp-json\/wp\/v2\/tags?post=694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}