{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Python implementation of the Kneedle algorithm\n",
    "Finding a “Kneedle” in a Haystack: Detecting Knee Points in System Behavior  \n",
    "Ville Satopa, Jeannie Albrecht, David Irwin, and Barath Raghavan  \n",
    "https://www1.icsi.berkeley.edu/~barath/papers/kneedle-simplex11.pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finding the knee from figure 2 from the paper"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def figure2():\n",
    "    x = np.linspace(0.0, 1, 10)\n",
    "    with np.errstate(divide='ignore'):\n",
    "        return x,np.true_divide(-1, x + 0.1) + 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 0:  Raw input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x,y = figure2()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "if not np.array_equal(np.array(x), np.sort(x)):\n",
    "    raise ValueError('x needs to be sorted')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 1: Fit a spline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.interpolate import interp1d,UnivariateSpline,InterpolatedUnivariateSpline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "N = len(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Ds = the finite set of x- and y-values that define a smooth curve, \n",
    "# one that has been fit to a smoothing spline.\n",
    "uspline = interp1d(x,y)\n",
    "\n",
    "Ds_x = np.linspace(np.min(x), np.max(x), N)\n",
    "Ds_y = uspline(Ds_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pss9S+9GvOB1JRCKEin0eTC4+/Rjrel5mV8En2fCJbzkdSUQiiIp9Hux5+i+o7fgZu3P+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\njIpdRMRlVOwiIi6jYhcRcRkVu4iIy/x/d2MzG2lbu54AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5bf7f9a6d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y);\n",
    "plt.plot(Ds_x,Ds_y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 2: Normalize the spline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def normalize(a):\n",
    "    \"\"\"return the normalized input array\"\"\"\n",
    "    return (a - min(a)) / (max(a) - min(a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# x and y normalized to unit square\n",
    "xsn = normalize(Ds_x)\n",
    "ysn = normalize(Ds_y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 3: Calculate the difference curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# the difference curve\n",
    "xd = xsn\n",
    "yd = ysn - xsn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+fDANwpvZk7QDLoNe50BkjNvVKtXkaLjXY+GmPKpMLWulVpTC2vdg4OVw6Yvu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\nCGm4K6VUEPr/YW0T1SM0TXwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5bf7f53cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title(\"Normalized spline & difference curve\");\n",
    "plt.plot(xsn,ysn);\n",
    "plt.plot(xd,yd);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 4: Identify local maxima  \n",
    "of the difference curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.signal import argrelextrema"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# local maxima for knees\n",
    "xmx_idx = argrelextrema(yd, np.greater)[0]\n",
    "xmx = xd[xmx_idx]\n",
    "ymx = yd[xmx_idx]\n",
    "#Dmx = np.stack((xmx,ymx))\n",
    "\n",
    "# local minima\n",
    "xmn_idx = argrelextrema(yd, np.less)[0]\n",
    "xmn = xd[xmn_idx]\n",
    "ymn = yd[xmn_idx]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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16mnaJKNUOGjZHq55HX70KpTsg3+cDx//BirL3K6syYRluK/ekM3w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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5bc931b390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title(\"local maxima in difference curve\");\n",
    "plt.plot(xsn,ysn);\n",
    "plt.plot(xd,yd);\n",
    "plt.hlines(ymx, plt.xlim()[0], plt.xlim()[1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 5: Calculate thresholds"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Sensitivity parameter S\n",
    "# smaller values detect knees quicker\n",
    "S = 1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def threshold(ymx_i):\n",
    "    return ymx_i - (S * np.diff(xsn).mean())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.42528736])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tmx = threshold(ymx)\n",
    "Tmx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 6: knee finding algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If any difference value (xdj, ydj), where j > i, drops below the threshold y = T|mxi\n",
    "for (x|mxi, y|mxi) before the\n",
    "next local maximum in the difference curve is reached,\n",
    "Kneedle declares a knee at the x-value of the corresponding\n",
    "local maximum x = x|xi.  \n",
    "**If the difference values reach\n",
    "a local minimum and starts to increase before y = T|mxi\n",
    "is reached, we reset the threshold value to 0 and wait for\n",
    "another local maximum to be reached.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "mxmx_iter = np.arange(xmx_idx[0], len(xsn))\n",
    "xmx_idx_iter = np.append(xmx_idx, len(xsn))\n",
    "\n",
    "knee_, norm_knee_, knee_x = 0.0, 0.0, None                                                                                                                                                                 \n",
    "for mxmx_i in range(len(xmx_idx_iter)):                                                                                                                                                                    \n",
    "\n",
    "    # stopping criteria for exhasuting array\n",
    "    if mxmx_i == len(xmx_idx_iter) - 1:                                                                                                                                                                    \n",
    "         break\n",
    "        \n",
    "    # indices between maxima/minima\n",
    "    idxs = ( mxmx_iter > xmx_idx_iter[mxmx_i] ) * ( mxmx_iter < xmx_idx_iter[mxmx_i+1] )\n",
    "    between_local_mx = mxmx_iter[np.where(idxs)]\n",
    "\n",
    "    for j in between_local_mx:\n",
    "        if j in xmn_idx:\n",
    "\n",
    "            # reached a minima, x indices are unique\n",
    "            # only need to check if j is a min\n",
    "            if yd[j + 1] > yd[j]:\n",
    "                Tmx[mxmx_i] = 0\n",
    "            elif yd[j + 1] <= yd[j]:\n",
    "                print('If this is a minima, how would you ever get here: {}'.format(j,knee))\n",
    "       \n",
    "        if yd[j] < Tmx[mxmx_i]:\n",
    "            if not knee_x:\n",
    "                knee_x = j\n",
    "            # declare a knee\n",
    "            knee = x[xmx_idx[mxmx_i]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LImPg9RthxjhdgTKMaLg3Yn6+h8gIYXSvdLvWx/5N2tuuQoMI9JkAd35uNwwp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\nV0opF/r/uvS9Z3wI/0wAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5bc8f9d160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xticks(np.arange(0,1.1,0.1))\n",
    "plt.plot(xsn,ysn);\n",
    "plt.plot(xd,yd);\n",
    "plt.hlines(Tmx[0], plt.xlim()[0], plt.xlim()[1], colors='g', linestyles='dashed');\n",
    "plt.vlines(xmx, plt.ylim()[0], plt.ylim()[1], colors='r', linestyles='dashed');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The vertical, red dashed line represents the x value of the knee point. The horizontal greeb dashed line represents the threshold value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.22222222222222221"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "knee"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.55555555555555558"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# normalized x value where the knee was determined\n",
    "xsn[knee_x]"
   ]
  }
 ],
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