{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SciPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The SciPy library is one of the core packages that make up the SciPy stack. It provides many user-friendly and efficient numerical routines such as routines for numerical integration and optimization.\n",
    "\n",
    "Library documentation: <a>http://www.scipy.org/scipylib/index.html</a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# needed to display the graphs\n",
    "%matplotlib inline\n",
    "from pylab import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from numpy import *\n",
    "from scipy.integrate import quad, dblquad, tplquad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 0.0)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# integration\n",
    "val, abserr = quad(lambda x: exp(-x ** 2),  Inf, Inf)\n",
    "val, abserr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import odeint, ode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "duUAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaa65fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# differential equation\n",
    "def dy(y, t, zeta, w0):\n",
    "    x, p = y[0], y[1]\n",
    "    \n",
    "    dx = p\n",
    "    dp = -2 * zeta * w0 * p - w0**2 * x\n",
    "\n",
    "    return [dx, dp]\n",
    "\n",
    "# initial state\n",
    "y0 = [1.0, 0.0]\n",
    "\n",
    "# time coodinate to solve the ODE for\n",
    "t = linspace(0, 10, 1000)\n",
    "w0 = 2*pi*1.0\n",
    "\n",
    "# solve the ODE problem for three different values of the damping ratio\n",
    "y1 = odeint(dy, y0, t, args=(0.0, w0)) # undamped\n",
    "y2 = odeint(dy, y0, t, args=(0.2, w0)) # under damped\n",
    "y3 = odeint(dy, y0, t, args=(1.0, w0)) # critial damping\n",
    "y4 = odeint(dy, y0, t, args=(5.0, w0)) # over damped\n",
    "\n",
    "fig, ax = subplots()\n",
    "ax.plot(t, y1[:,0], 'k', label=\"undamped\", linewidth=0.25)\n",
    "ax.plot(t, y2[:,0], 'r', label=\"under damped\")\n",
    "ax.plot(t, y3[:,0], 'b', label=r\"critical damping\")\n",
    "ax.plot(t, y4[:,0], 'g', label=\"over damped\")\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.fftpack import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "U/x/8QPGQJ9lUrYAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xad52978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# fourier transform\n",
    "N = len(t)\n",
    "dt = t[1]-t[0]\n",
    "\n",
    "# calculate the fast fourier transform\n",
    "# y2 is the solution to the under-damped oscillator from the previous section\n",
    "F = fft(y2[:,0]) \n",
    "\n",
    "# calculate the frequencies for the components in F\n",
    "w = fftfreq(N, dt)\n",
    "\n",
    "fig, ax = subplots(figsize=(9,3))\n",
    "ax.plot(w, abs(F));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear Algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = array([[1,2,3], [4,5,6], [7,8,9]])\n",
    "b = array([1,2,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.33333333,  0.66666667,  0.        ])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve a system of linear equations\n",
    "x = solve(A, b)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.34211023,  0.22314378, -0.11368553])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# eigenvalues and eigenvectors\n",
    "A = rand(3,3)\n",
    "B = rand(3,3)\n",
    "\n",
    "evals, evecs = eig(A)\n",
    "\n",
    "evals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.24560175, -0.93054452,  0.46758953],\n",
       "       [-0.84521949, -0.00332278, -0.56599975],\n",
       "       [-0.4746407 ,  0.36616371,  0.67897299]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evecs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[-0.23411483,  0.96016594,  0.15255034],\n",
       "        [-0.84815445, -0.12501447, -0.51478677],\n",
       "        [-0.47520972, -0.24990547,  0.84363676]]),\n",
       " array([ 1.3457633 ,  0.23442523,  0.1079208 ]),\n",
       " array([[-0.21657397, -0.81553512, -0.53665463],\n",
       "        [ 0.7658064 ,  0.19902251, -0.61149865],\n",
       "        [-0.60550497,  0.54340824, -0.58143892]]))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "svd(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "QcnZWQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb0e82b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return 4*x**3 + (x-2)**2 + x**4\n",
    "\n",
    "fig, ax  = subplots()\n",
    "x = linspace(-5, 3, 100)\n",
    "ax.plot(x, f(x));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 2.804988\n",
      "         Iterations: 4\n",
      "         Function evaluations: 24\n",
      "         Gradient evaluations: 8\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 0.46961745])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_min = optimize.fmin_bfgs(f, -0.5)\n",
    "x_min"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "kYxSgBcRySgFeBGRjPr/SYBJHv8M/N8AAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15241390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# create a (continous) random variable with normal distribution\n",
    "Y = stats.norm()\n",
    "\n",
    "x = linspace(-5,5,100)\n",
    "\n",
    "fig, axes = subplots(3,1, sharex=True)\n",
    "\n",
    "# plot the probability distribution function (PDF)\n",
    "axes[0].plot(x, Y.pdf(x))\n",
    "\n",
    "# plot the commulative distributin function (CDF)\n",
    "axes[1].plot(x, Y.cdf(x));\n",
    "\n",
    "# plot histogram of 1000 random realizations of the stochastic variable Y\n",
    "axes[2].hist(Y.rvs(size=1000), bins=50);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 1.0, 1.0)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.mean(), Y.std(), Y.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.193733722708372, 0.23272385793970249)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# t-test example\n",
    "t_statistic, p_value = stats.ttest_ind(Y.rvs(size=1000), Y.rvs(size=1000))\n",
    "t_statistic, p_value"
   ]
  }
 ],
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