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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Particle Swarm Optimization Algorithm"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "by Stuart Reid (<a href=\"http://www.stuartreid.co.za\">www.stuartreid.co.za</a>)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Originally inspired by the behaviour of flocking birds, the particle swarm optimization algorithm solves optimization problems by moving a \"swarm\" of three or more particles (vectors) towards the best particle in the swarm. As such, the optimization algorithm can be classified as a direct-search, population-based, meta-heuristic, global optimization algorithm. This IPython notebook will introduce the algorithm using Numpy and mention a few of it's applications in quantitative finance."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy\n",
      "import numpy.random as nrand\n",
      "import matplotlib.pylab as plt\n",
      "import matplotlib.gridspec as gridspec"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 535
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This class contains the logic belonging to each particle. Particles are defined by three n-dimensional vectors, their current position in the search space, their best ever position in the search space, and their velocities. The velocity of a particle is an n-dimensional vector which describes in which direction it will travel. The velocity is calculated as the currency velocity plus a cognitive component and social component all multiplied by inertia. The cognitive component is the dot-product of a uniformly distributed n-dimensional random vector between zero and one and the difference "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class Particle:\n",
      "    def __init__(self, n, r):\n",
      "        self.n = n # Dimensions\n",
      "        self.position = nrand.uniform(low=r[0], high=r[1], size=n)\n",
      "        self.velocity = nrand.rand(n) / 100.0\n",
      "        self.pbest = self.position\n",
      "    \n",
      "    def update_particle(self, gbest):\n",
      "        self.update_velocity(gbest)\n",
      "        self.update_position()\n",
      "    \n",
      "    def update_velocity(self, gbest):\n",
      "        r1 = nrand.rand(self.n)\n",
      "        r2 = nrand.rand(self.n)\n",
      "        social = 1.496180 * r1 * gbest.position - self.position\n",
      "        cognitive = 1.496180 * r2 * self.pbest - self.position\n",
      "        self.velocity += social + cognitive\n",
      "        self.velocity *= 0.729844\n",
      "    \n",
      "    def update_position(self):\n",
      "        self.position += self.velocity"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 526
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This class contains a simple minimization objective function which I have created. It is simply the sum of the sin graphs for each element in the vector minus the mean of the vector. The resulting fitness lanscape is non-convex and has infinitely many local optima."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def problem_1d(x):\n",
      "    if x > 8.0 or x < -8.0:\n",
      "        return 5.0\n",
      "    return math.sin(x) - (x / 2.0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 527
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Example output for the objective function where each element in the vector is equal and increasing by 0.01"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "values = []\n",
      "solution = -9.0\n",
      "while solution < 9.0:\n",
      "    values.append(problem_1d(solution))\n",
      "    solution += 0.01\n",
      "x_axis = numpy.linspace(-9.0, 9.0, len(values))\n",
      "plt.plot(x_axis, values)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 528,
       "text": [
        "[<matplotlib.lines.Line2D at 0x330cb240>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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teoc551aa2a7ANDNbXF5tSgScc257c3aySubOuWNr8bQvgHYVvt8y4UgqqOlr\na2YPAJn84ZT/fA+2498/LUqGnHMry2//ZWbP4VtZSubZWWVmrZ1zX5lZG+Dr6h6crzZLxV7aC8Bv\nzayBme0FdAR09TsD5f+wW5yGv9gsNTcL6Ghme5pZA/wF+RcCxxRbZtbIzJqW32+MX+ZD78nsvQD0\nL7/fH3i+ugdnVZlXx8xOA/4ItAReMrPZzrnjnXMLzewpYCFQClzmNNg9U7ebWXd8u2AZcHHgeGLF\nOVdqZoOAvwJ1gQedc4sChxVnrYDnzAx8TpnsnHstbEjxYmZPAEcCLc3sc+AGYDTwlJldACwHzqz2\nGMqjIiLxp/HdIiIJoGQuIpIASuYiIgmgZC4ikgBK5iIiCaBkLiKSAErmIiIJoGQuIpIA/wcdZZu2\n0k07tAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3259a208>"
       ]
      }
     ],
     "prompt_number": 528
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This class contains the optimizer code"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class Optimizer:\n",
      "    def __init__(self, m, n, opt):\n",
      "        self.opt = opt\n",
      "        self.swarm = []\n",
      "        for i in range(m):\n",
      "            self.swarm.append(Particle(n, [-9.0, 9.0]))\n",
      "    \n",
      "    def get_best(self):\n",
      "        gbest = None\n",
      "        min_fit = float('+inf')\n",
      "        for i in range(len(self.swarm)):\n",
      "            f = problem(self.swarm[i].pbest) \n",
      "            if f < min_fit:\n",
      "                gbest = i\n",
      "                min_fit = f\n",
      "        return gbest\n",
      "    \n",
      "    def optimize(self, steps):\n",
      "        positions = [self.swarm[i].position[0] for i in range(len(self.swarm))]\n",
      "        fitnesses = [problem_1d(positions[i]) for i in range(len(positions))]\n",
      "        for i in range(steps):\n",
      "            positions, fitnesses = self.optimize_step()\n",
      "        return positions, fitnesses\n",
      "    \n",
      "    def optimize_step(self):\n",
      "        gbest = self.get_best()\n",
      "        positions = []\n",
      "        fitnesses = []\n",
      "        for i in range(len(self.swarm)):\n",
      "            if i != gbest:\n",
      "                self.swarm[i].update_particle(self.swarm[gbest])\n",
      "                pos_fit = problem(self.swarm[i].position)\n",
      "                pbest_fit = problem(self.swarm[i].pbest)\n",
      "                if pos_fit < pbest_fit:\n",
      "                    self.swarm[i].pbest = self.swarm[i].position\n",
      "                positions.append(self.swarm[i].pbest)\n",
      "                fitnesses.append(pos_fit)\n",
      "        gbest_fit = problem(self.swarm[gbest].position)\n",
      "        positions.append(self.swarm[gbest].position)\n",
      "        fitnesses.append(gbest_fit)\n",
      "        return positions, fitnesses"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 529
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "optimizer = Optimizer(10, 1, problem_1d)\n",
      "pos, fit = optimizer.optimize(0)\n",
      "plt.plot(x_axis, values)\n",
      "plt.plot(pos, fit, 'go')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 537,
       "text": [
        "[<matplotlib.lines.Line2D at 0x335b0518>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6nhoH/Bo4ofbLI4xLRERS0Ki1Wczs/wFzgA01D3UDPgT6OefWbvNaLcwiItII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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x333a2390>"
       ]
      }
     ],
     "prompt_number": 537
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pos, fit = optimizer.optimize(5)\n",
      "plt.plot(x_axis, values)\n",
      "plt.plot(pos, fit, 'go')\n",
      "plt.plot(pos[len(pos) - 1], fit[len(fit) - 1], 'ko')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 539,
       "text": [
        "[<matplotlib.lines.Line2D at 0x34364ba8>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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h1jW2ZjYLaF/Dl24Afgn0q/r0COMSEZEU1GvRkJkdil+8u6HyoX2BD4CezrmP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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x341006a0>"
       ]
      }
     ],
     "prompt_number": 539
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pos, fit = optimizer.optimize(5)\n",
      "plt.plot(x_axis, values)\n",
      "plt.plot(pos, fit, 'go')\n",
      "plt.plot(pos[len(pos) - 1], fit[len(fit) - 1], 'ko')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 540,
       "text": [
        "[<matplotlib.lines.Line2D at 0x34532da0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2p+qBr5I8vZ/B0XsZrmR75rcDVwURiNSO/ocJlt7P4Oi9DFe9k7mZDQBWO+cW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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x34372160>"
       ]
      }
     ],
     "prompt_number": 540
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}