saketkc/Knee-point.ipynb

## Knee-point.ipynb
{
  "cells": [
    {
      "metadata": {
        "run_control": {
          "frozen": false,
          "read_only": false
        }
      },
      "cell_type": "markdown",
      "source": "# Determining knee point using  [Finding a \"Kneedle\" in a Haystack: Detecting Knee Points in System Behavior](https://www1.icsi.berkeley.edu/~barath/papers/kneedle-simplex11.pdf)\n\nAnother example [here](https://github.com/saketkc/notebooks/blob/master/python/Kneedle%20Algorithm.ipynb)"
    },
    {
      "metadata": {
        "run_control": {
          "frozen": false,
          "read_only": false
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "%pylab inline\nimport scipy as sp\nimport seaborn as sns\nfrom scipy.interpolate import UnivariateSpline\nsns.set_style('white')\n\ndef draw_plot(X, Y, knee_point=None):\n    plt.plot(X, Y)\n    if knee_point:\n        plt.axvline(x=knee_point, color='k', linestyle='--')",
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": "Populating the interactive namespace from numpy and matplotlib\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "run_control": {
          "frozen": false,
          "read_only": false
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "X = [1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000]\nY = [2314, 802, 519, 417, 358, 318, 302, 284, 280]\nn = len(X)\nS = 1",
      "execution_count": 2,
      "outputs": []
    },
    {
      "metadata": {
        "collapsed": true,
        "run_control": {
          "frozen": false,
          "read_only": false
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "def _locate(Y_d, T_lm, maxima_ids):\n    n = len(Y_d)\n    for j in range(0, n):\n        for index, i in enumerate(maxima_ids):\n            if j <= i:\n                continue\n            if Y_d[j] <= T_lm[index]:\n                return index\n\n\ndef find_knee_point(X, Y, S):\n    n = len(X)\n    spl = UnivariateSpline(X, Y)\n    X_s = np.linspace(np.min(X), np.max(X), n)\n    Y_s = spl(X_s)\n    X_sn = (X_s - np.min(X_s)) / (np.max(X_s) - np.min(X_s))\n    Y_sn = (Y_s - np.min(Y_s)) / (np.max(Y_s) - np.min(Y_s))\n    X_d = X_sn\n    Y_d = Y_sn - X_sn\n    X_lm = []\n    Y_lm = []\n    maxima_ids = []\n    for i in range(1, n - 1):\n        if (Y_d[i] > Y_d[i - 1] and Y_d[i] > Y_d[i + 1]) or (np.sign(Y_d[i + 1]) != np.sign(Y_d[i - 1])):\n            X_lm.append(X_d[i])\n            Y_lm.append(Y_d[i])\n            maxima_ids.append(i)\n    print(Y_lm)\n    T_lm = Y_lm - S * np.sum(np.diff(X_sn)) / (n - 1)\n    knee_point_index = _locate(Y_d, T_lm, maxima_ids)\n    knee_point = X_lm[knee_point_index] * (np.max(X_s) - np.min(X_s)\n                                           ) + np.min(X_s)\n    return knee_point, Y_d",
      "execution_count": 3,
      "outputs": []
    },
    {
      "metadata": {
        "run_control": {
          "frozen": false,
          "read_only": false
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "knee_point, _ = find_knee_point(X, Y, S)\ndraw_plot(X, Y, knee_point)",
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": "[0.13219578562607948, -0.13339603454922289]\n",
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
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27t2ro0ePatu2bSoqKhpyKsN4y1/sb1M4caZVnUw1AwAAmLRGHHD37dsnr9er3bt36447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\nDACSCLgAEFKGYciZECtnQqwWzb/6+LN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      "source": "",
      "execution_count": null,
      "outputs": []
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