Kmeans.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# K-means clustering\n",
    "\n",
    "\n",
    "The basic method for lossy data compression and summarization. Unsupervised method\n",
    "\n",
    "- Initialize $k$ cluster centers\n",
    "- Associations: Find the points closest to each cluster center and form groups\n",
    "- Recalculate means: Set the cluster center to the mean of each group"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation\n",
    "import numpy as np\n",
    "plt.rcParams[\"animation.html\"] = \"jshtml\"\n",
    "\n",
    "from IPython import display\n",
    "\n",
    "N = 30\n",
    "D = 2\n",
    "\n",
    "# Generate a data set\n",
    "X1 = 0.3*np.random.randn(D,N//2) \n",
    "X2 = 0.8*np.random.randn(D,N//2) + 3*np.ones((D, N//2))\n",
    "X = np.hstack((np.mat(X1), np.mat(X2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    document.getElementById(this.slider_id).max = this.frames.length - 1;\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<div class=\"animation\" align=\"center\">\n",
       "    <img id=\"_anim_img0d2828ede3db4b27800386b3175b4875\">\n",
       "    <br>\n",
       "    <input id=\"_anim_slider0d2828ede3db4b27800386b3175b4875\" type=\"range\" style=\"width:350px\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           onchange=\"anim0d2828ede3db4b27800386b3175b4875.set_frame(parseInt(this.value));\"></input>\n",
       "    <br>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.previous_frame()\">\n",
       "        <i class=\"fa fa-step-backward\"></i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.reverse_animation()\">\n",
       "        <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "        </button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0d2828ede3db4b27800386b3175b4875.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "  <form action=\"#n\" name=\"_anim_loop_select0d2828ede3db4b27800386b3175b4875\" class=\"anim_control\">\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"once\" > Once </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"loop\" checked> Loop </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"reflect\" > Reflect </input>\n",
       "  </form>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img0d2828ede3db4b27800386b3175b4875\";\n",
       "    var slider_id = \"_anim_slider0d2828ede3db4b27800386b3175b4875\";\n",
       "    var loop_select_id = \"_anim_loop_select0d2828ede3db4b27800386b3175b4875\";\n",
       "    var frames = new Array(0);\n",
       "    \n",
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       "UrVyQgAAAABJRU5ErkJggg==\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim0d2828ede3db4b27800386b3175b4875 = new Animation(frames, img_id, slider_id, 200.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = 3;\n",
    "\n",
    "idx = np.random.choice(N, replace=False, size=K)\n",
    "mu = 0*X[:,idx] + 3*np.random.randn(D, K)\n",
    "#plt.plot(mu[0,:],mu[1,:],'ro')\n",
    "\n",
    "# Number of epochs\n",
    "EP = 20\n",
    "\n",
    "fig = plt.figure(figsize=(6,6))\n",
    "plt.plot(X[0,:], X[1,:],'kx')\n",
    "\n",
    "ax = fig.gca()\n",
    "ln = plt.Line2D(xdata=mu[0,:], ydata=mu[1,:], marker='o', color='r',linestyle=None,linewidth=0)\n",
    "ax.add_line(ln)\n",
    "\n",
    "\n",
    "def animate(e):\n",
    "    global mu\n",
    "    ln.set_xdata(mu[0,:])\n",
    "    ln.set_ydata(mu[1,:])\n",
    "\n",
    "    dist = np.zeros((N,K))\n",
    "    for i in range(N):\n",
    "        for c in range(K):\n",
    "            err = X[:,i]-mu[:,c]\n",
    "            dist[i,c] = float(err.T*err)\n",
    "    \n",
    "    # Assignments\n",
    "    a = np.argmin(dist, axis=1)\n",
    "    \n",
    "    mu = np.mat(np.zeros((D,K)))\n",
    "    count = np.zeros((K))\n",
    "    for i,c in enumerate(a):\n",
    "        count[c] += 1\n",
    "        mu[:,c] = (count[c]-1)/count[c]*mu[:,c] + 1./count[c]*X[:,i]\n",
    "    \n",
    "    \n",
    "\n",
    "\n",
    "\n",
    "animation = matplotlib.animation.FuncAnimation(fig, animate, frames=EP)\n",
    "\n",
    "from IPython.display import HTML\n",
    "HTML(animation.to_jshtml())\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "$\\newcommand{\\trace}{\\mathop{\\text{Tr}}}$\n",
    "\n",
    "Suppose we are given a data matrix $X$ of size $D \\times N$ ($N$ samples each having dimension $D$). We wish\n",
    "to represent this dataset using $K$ cluster centers. The cluster centers are stored in a $D \\times K$ matrix. \n",
    "A binary assignment matrix $C$ encodes at each column the nearest cluster center with the convention that $C(k, i) = 1$ when the nearest \n",
    "cluster center to $X(:,i)$ is $M(:,k)$ and $C(k, i) = 0$ otherwise.\n",
    "\n",
    "Using this encoding, K-means clustering can be viewed as an approximate matrix factorization problem where we \n",
    "wish to minimize \n",
    "$$\n",
    "\\| E\\|_F^2\n",
    "$$\n",
    "where \n",
    "\\begin{eqnarray}\n",
    "E &= & X - M C \\\\\n",
    "\\end{eqnarray}\n",
    "with the additional constrained that the assignment matrix is binary, with the constraint\n",
    "$\\sum_k C(k,j) = 1$. \n",
    "\n",
    "Given the assignment matrix, the cluster center matrix $M$ can be found \n",
    "by minimizing the error with respect to $M$ (see Matrix Calculus.ipynb) We find the gradient and set \n",
    "it to zero.\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\|E\\|_F^2 & = & \\trace \\left( X^\\top X + C^\\top M^\\top M C - 2 X^\\top M C \\right) \\\\\n",
    "\\frac{d\\|E\\|_F^2}{d M} & = & 2 M C C^\\top - 2 X C^\\top \\\\\n",
    "M & = & X C^\\top (C C^\\top)^{-1}\n",
    "\\end{eqnarray}\n",
    "This is the means of the assignments. To see this fix $C$\n",
    "As the assignment matrix $C$ is binary $X C^\\top$ is \n",
    "$$\n",
    "\\sum_k X(d,i) (C^\\top)(i,k)\n",
    "$$\n",
    "This $k$'th column of this matrix is the sums of all data points asigned to $k$'th cluster center. The $K \\times K$ matrix\n",
    "$$\n",
    "C C^\\top\n",
    "$$\n",
    "is diagonal, with the number of points assigned to cluster $k$ at $(C C^\\top)(k,k)$. The inverse of a diagonal matrix is just the reciprocal of each entry at the diagonal -- postmultiplying $(C C^\\top)^{-1}$ with $X C^\\top$ results in dividing each column with the number of assignments so we obtain the arithmetic mean of each cluster. \n",
    "\n",
    "Given the cluster center matrix $M$, we can now find the best $C$ by finding the nearest cluster center for each $X(:,i)$ and setting $C(k,i) = 1$ when $M(:,k)$ is the nearest.\n",
    "\n",
    "This is an alternating minimization schema where we greedly minimize the error for $C$ with fixed $M$, than for $M$ with fixed $C$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Traceback (most recent call last):\n",
      "  File \"/Users/cemgil/anaconda/envs/py36/lib/python3.6/site-packages/matplotlib/cbook/__init__.py\", line 388, in process\n",
      "    proxy(*args, **kwargs)\n",
      "  File \"/Users/cemgil/anaconda/envs/py36/lib/python3.6/site-packages/matplotlib/cbook/__init__.py\", line 228, in __call__\n",
      "    return mtd(*args, **kwargs)\n",
      "  File \"/Users/cemgil/anaconda/envs/py36/lib/python3.6/site-packages/matplotlib/animation.py\", line 1081, in _start\n",
      "    self._init_draw()\n",
      "  File \"/Users/cemgil/anaconda/envs/py36/lib/python3.6/site-packages/matplotlib/animation.py\", line 1792, in _init_draw\n",
      "    self._draw_frame(next(self.new_frame_seq()))\n",
      "  File \"/Users/cemgil/anaconda/envs/py36/lib/python3.6/site-packages/matplotlib/animation.py\", line 1814, in _draw_frame\n",
      "    self._drawn_artists = self._func(framedata, *self._args)\n",
      "  File \"<ipython-input-2-a015364e2c87>\", line 27, in animate\n",
      "    dist[i,c] = float(err.T*err)\n",
      "TypeError: only length-1 arrays can be converted to Python scalars\n"
     ]
    },
    {
     "data": {
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M7Fdm9paZfa3sQVXZ5o8sW9doNDQ3N7dxn5UnAMo0MNxmdpuk70j6jKR7JT1hZveWPbDI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\nAMEQbgAI5v8Bem8BRLnQqbIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ed97eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ed97ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11a55c978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11aaae208>"
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     "metadata": {},
     "output_type": "display_data"
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      "text/plain": [
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ac2a4a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "D = 2\n",
    "N = 20\n",
    "K = 3\n",
    "sig = 4\n",
    "\n",
    "M = 10*np.random.randn(D,K)\n",
    "C = np.random.multinomial(1, np.ones(K)/K, N).T\n",
    "C\n",
    "\n",
    "X_hat = M.dot(C) \n",
    "X = X_hat + sig*np.random.randn(D, N)\n",
    "M = X.dot(C.T).dot(np.linalg.inv(C.dot(C.T)))\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(X[0,:],X[1,:],'x')\n",
    "plt.plot(M[0,:],M[1,:],'or')\n",
    "\n",
    "plt.show()\n",
    "\n",
    "plt.figure(figsize=(N/2, D/2))\n",
    "plt.imshow(X, cmap='gray_r')\n",
    "plt.show()\n",
    "\n",
    "plt.figure(figsize=(N/2, D/2))\n",
    "plt.imshow(X_hat, cmap='gray_r')\n",
    "plt.show()\n",
    "\n",
    "plt.figure(figsize=(K/2, D/2))\n",
    "plt.imshow(M, cmap='gray_r')\n",
    "plt.show()\n",
    "\n",
    "plt.figure(figsize=(N/2, K/2))\n",
    "plt.imshow(C, cmap='gray_r')\n",
    "plt.show()\n",
    "\n",
    "\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:py36]",
   "language": "python",
   "name": "conda-env-py36-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
  },
  "toc": {
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
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   "toc_position": {},
   "toc_section_display": "block",
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  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}