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Müller\n+\n+???\n+N/A\n+\n+---\n+\n+class: center, middle\n+\n+## Clustering\n+\n+???\n+\n+---\n+.center[\n+![:scale 80%](images/cluster_intro_1.png)\n+]\n+\n+???\n+\n+---\n+.center[\n+![:scale 80%](images/cluster_intro_2.png)\n+]\n+\n+???\n+\n+---\n+.center[\n+![:scale 80%](images/cluster_intro_3.png)\n+]\n+\n+???\n+\n+---\n+Group into 4 clusters\n+\n+.center[\n+![:scale 80%](images/cluster_intro_4.png)\n+]\n+\n+???\n+\n+---\n+\n+# Clustering\n+\n+- Group data into a partitioning:\n+- Identify groups by assigning “cluster labels” to\n+each data point.\n+- Points within a cluster should be “similar”.\n+- Points in different cluster should be “different”.\n+- Number of clusters might be pre-specified.\n+- Notations of “similar” and “different” depend on\n+algorithm or user specified metrics.\n+\n+???\n+\n+---\n+\n+# Goals of Clustering\n+\n+- Data Exploration\n+  - Are there coherent groups ?\n+  - How many groups are there ?\n+- Data Partitioning\n+  - Divide data by group before further processing\n+- Unsupervised feature extraction\n+  - Derive features from clusters or cluster distances\n+\n+???\n+\n+---\n+\n+class: center, middle\n+\n+## Clustering Algorithms\n+\n+???\n+\n+---\n+\n+class: center, middle\n+\n+## K-Means\n+\n+???\n+\n+---\n+\n+# K-Means Algorithm\n+\n+.narrow-right-column[\n+.smaller[\n+- Pick number of clusters k.\n+- Pick k random points as\n+“cluster center”\n+- While cluster centers change:\n+  1. Assign each data point to it’s closest cluster center\n+  2. Recompute cluster centers as the mean of the assigned points.\n+]\n+]\n+\n+.wide-left-column[\n+.center[\n+![:scale 80%](images/kmeans.png)\n+]\n+]\n+\n+???\n+\n+---\n+class: spacious\n+# Objective function for K-Means\n+\n+- Finds a local minimum of minimizing squared distances (exact solution is\n+NP hard):\n+`$$ \\min_{\\mathbf{c}_j \\in \\mathbf{R}^p, j=1,..,k} \\sum_i ||\\mathbf{x}_i - \\mathbf{c}_{x_i}||^2 $$`\n+$\\mathbf{c}_{x_i}$ is the cluster center $c_j$ closest to $x_i$\n+- New data points can be assigned cluster membership based on existing clusters.\n+\n+???\n+\n+---\n+\n+# K-Means API\n+\n+\n+.narrow-right-column[\n+.center[\n+![:scale 120%](images/kmeans_api.png)\n+]\n+]\n+\n+.wide-left-column[\n+.smaller[\n+```python\n+X, y = make_blobs(centers=4, random_state=1)\n+\n+km = KMeans(n_clusters=5, random_state=0)\n+km.fit(X)\n+print(km.cluster_centers_.shape)\n+print(km.labels_.shape)\n+# predict is the same as labels_ on training data\n+# but can be applied to new data\n+print(km.predict(X).shape)\n+plt.scatter(X[:, 0], X[:, 1], c=km.labels_)\n+plt.gca().set_aspect(\"equal\")\n+```\n+(5, 2) </br>\n+(100,) </br>\n+(100,)\n+]\n+]\n+\n+???\n+\n+---\n+\n+# Restriction of Cluster Shapes\n+\n+- Clusters are Voronoi-diagrams of centers\n+- Clusters are always convex in space\n+\n+.left-column[\n+![:scale 80%](images/cluster_shapes_1.png)\n+]\n+.right-column[\n+![:scale 90%](images/cluster_shapes_2.png)\n+]\n+\n+???\n+\n+---\n+\n+# Limitations of K-Means\n+\n+.smaller[\n+.left-column[\n+![:scale 50%](images/kmeans_limitations_1.png) </br>\n+Cluster boundaries are always in the\n+middle of the 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