{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Common engineering numeric problems\n",
    "This notebook aims to collect a very brief summary of simple ways to solve common engineering problems in Python\n",
    "\n",
    "For a far more comprehensive discussion of chemical engineering problems solved in Python, see the [Kitchin Group pycse file](http://kitchingroup.cheme.cmu.edu/pycse/pycse.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Table lookup/interpolation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1d: numpy.interp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Problem: find P where T=130.\n",
    "\n",
    "<table>\n",
    "<tr><th>T</th><th>P</th></tr>\n",
    "<tr><td>100</td><td>200</td></tr>\n",
    "<tr><td>200</td><td>500</td></tr>\n",
    "<tr><td>300</td><td>1000</td></tr>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAASCAYAAAD2Uea5AAAABHNCSVQICAgIfAhkiAAAAoFJREFU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\ngqmr9R7v8a7xEo03o1EmuRsfAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$350.0$$"
      ],
      "text/plain": [
       "350.0"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy\n",
    "\n",
    "T = [100, 200, 300]\n",
    "P = [200, 500, 1000]\n",
    "\n",
    "numpy.interp(150, T, P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2d: scipy.interpolate.interp2d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Problem: Find the viscosity from this table for $T=230$ and $P=800$\n",
    "\n",
    "<table>\n",
    "<tr><th>T\\P</th><td>200</td><td>500</td><td>1000</td></tr>\n",
    "<tr><td>100</td><td>5</td><td>4</td><td>2</td></tr>\n",
    "<tr><td>200</td><td>3</td><td>3</td><td>2</td></tr>\n",
    "<tr><td>300</td><td>2</td><td>2</td><td>1</td></tr>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy.interpolate\n",
    "\n",
    "T = [100, 200, 300]\n",
    "P = [200, 500, 1000]\n",
    "\n",
    "visc = [[5, 4, 2],\n",
    "        [3, 3, 2],\n",
    "        [2, 2, 1]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.1])"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interpolator = scipy.interpolate.interp2d(T, P, visc)\n",
    "interpolator(230, 800)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plotting\n",
    "\n",
    "Plotting is normally handled by the matplotlib library. The pyplot interface aims for rough Matlab equivalence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The notebook allows for inline graphics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x113e10128>"
      ]
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Q0yZNhT174Gd/1kK7Zlebjwa/uar+T5IfB3YlOUAvYPoNtV61ZcuW5487nQ6dTmfYMUoj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\naWSGiSRpZIaJJGlkhokkaWT/H5D2K3soh329AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1139f26d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(T, P)\n",
    "plt.xlabel('T')\n",
    "plt.ylabel('P')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.contour.QuadContourSet at 0x11419d6d8>"
      ]
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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YAKzKDlsFLMy25wOrs+O2A9uAObV+fTMbGpfNabVy2VzPJaMZwO8l3S7pcUlflzQWmBwR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\nM7OMA8HMzAAHgpmZZRwIZmYGwP8HpnGkNtp+QFEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114175a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.contour(T, P, visc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Algebra (manipulation of equations)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sympy library supplies the ability to manipulate equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()  # This will make the expressions be printed in a more pretty way"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAAAXCAYAAABDLnAjAAAABHNCSVQICAgIfAhkiAAABIdJREFU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\n1Gn2lpSUlJSUlJSU5Oc/raY9jcYfy0EAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1$$"
      ],
      "text/plain": [
       " 4      3      2          \n",
       "x  + 4⋅x  + 6⋅x  + 4⋅x + 1"
      ]
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "equation = sympy.expand((x + 1)**4)\n",
    "equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = sympy.Symbol('y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVAAAAAaCAYAAADhXARnAAAABHNCSVQICAgIfAhkiAAABvhJREFU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\nEolEIpFIJBKJxNTwfwZOMI+VUO48AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$4 y + \\left(y + 1\\right)^{4} + 4 \\left(y + 1\\right)^{3} + 6 \\left(y + 1\\right)^{2} + 5$$"
      ],
      "text/plain": [
       "             4            3            2    \n",
       "4⋅y + (y + 1)  + 4⋅(y + 1)  + 6⋅(y + 1)  + 5"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "equation.subs(x, y + 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Systems of linear equations\n",
    "Systems of linear equations can be solved directly without iteration. In both cases we reason via matrix algebra.\n",
    "\n",
    "$$\\begin{align}\n",
    "2a + 3b + 4c &= 5 \\\\\n",
    "2b + c &= 1 \\\\\n",
    "a + b + c &= 1 \\\\\n",
    "\\end{align}$$\n",
    "\n",
    "## Analytic solution\n",
    "### sympy.solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols('a, b, c')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We state the equations in the form $f(a, b, c) = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAWCAYAAAB+OyYeAAAABHNCSVQICAgIfAhkiAAAA/JJREFU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\n",
      "text/latex": [
       "$$\\left \\{ a : -1, \\quad b : 1, \\quad c : 1\\right \\}$$"
      ],
      "text/plain": [
       "{a: -1, b: 1, c: 1}"
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve([2*a + 3*b + 4*c - 5,\n",
    "             2*b - c - 1,\n",
    "             a + b + c - 1], [a, b, c])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### scipy.linalg.solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A = numpy.matrix([[2, 3, 4],\n",
    "                  [0, 2, -1],\n",
    "                  [1, 1, 1]])\n",
    "c = numpy.matrix([5, 1, 1]).T  # .T for transpose"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Direct calculation via inverse:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[-1.],\n",
       "        [ 1.],\n",
       "        [ 1.]])"
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.I*c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For larger matrices, it can be faster to use a dedicated solver:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-1.],\n",
       "       [ 1.],\n",
       "       [ 1.]])"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.linalg.solve(A, c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Laplace transforms\n",
    "## sympy.laplace_transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t, s = sympy.symbols('t, s')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/(s**2 + 1), 0, True)"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.laplace_transform(sympy.sin(t), t, s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## sympy.inverse_laplace_transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAVCAYAAADiv3Z7AAAABHNCSVQICAgIfAhkiAAAAsNJREFU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\npv9P0mkVreTyANs6hCEMoY/4D9Aghrn3HoXtAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$e^{- t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       " -t             \n",
       "ℯ  ⋅Heaviside(t)"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.inverse_laplace_transform(1/(s + 1), s, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Systems of nonlinear equations\n",
    "## Analytic solution (sometimes):\n",
    "### sympy.solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAWCAYAAACi7pBsAAAABHNCSVQICAgIfAhkiAAAAgNJREFU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\nVgA8Ho4eAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left ( 1, \\quad -1\\right )\\right ]$$"
      ],
      "text/plain": [
       "[(1, -1)]"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve([a**2 + b,\n",
    "             a/b + 1], [a, b])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numeric solution (often)\n",
    "### scipy.optimize.fsolve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy.optimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def eqs(x):\n",
    "    a, b = x\n",
    "    return [a**2 + b,\n",
    "            a/b + 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1., -1.])"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.optimize.fsolve(eqs, [2, -1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Systems of ODEs (IVP)\n",
    "## scipy.integrate.odeint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def sys(x, t):\n",
    "    a, b = x\n",
    "    return [-2*a, -a - 2*b]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "times = numpy.linspace(0, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "scipy.integrate.odeint?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solution = scipy.integrate.odeint(sys, [1, 1], times)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x113e2b6d8>,\n",
       " <matplotlib.lines.Line2D at 0x113e2b860>]"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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grQN/VK9RZGdms2LXina/5+abvVs0k+wPHxGRuKV14JsZnx/+eZ4vi3l9OKZLLoHqali0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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113bf79e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(times, solution)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical integration (aka quadrature)\n",
    "## Given a function\n",
    "### scipy.integrate.quad"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say we're trying to integrate the surface under the curve $f(x) = \\sin(x)/x$ between 0.5 and 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return numpy.sin(x)/x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = numpy.linspace(0.01, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114550b70>]"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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iItnLtvIfBjzn7h2AScAFG2g7GHgny/MlSjqdjh1CQdD3sJa+i7X0XWQn2+Tf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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113caa828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.45297565232411635\n"
     ]
    }
   ],
   "source": [
    "integral, error = scipy.integrate.quad(f, 0.5, 1)\n",
    "print(integral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Given smooth data\n",
    "### Trapezoidal rule: scipy.signal.trapz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xdata = numpy.linspace(0.5, 1)\n",
    "ydata = f(xdata)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAAASCAYAAABmbl0zAAAABHNCSVQICAgIfAhkiAAABK1JREFU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\nKFe29QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.452974449424$$"
      ],
      "text/plain": [
       "0.452974449424"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.integrate.trapz(ydata, xdata)  # note it's _not_ x, y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simpson's rule: scipy.signal.simps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAAASCAYAAABmbl0zAAAABHNCSVQICAgIfAhkiAAABcZJREFU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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0.452975628058$$"
      ],
      "text/plain": [
       "0.452975628058"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.integrate.simps(ydata, xdata)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical derivatives\n",
    "## Given smooth data\n",
    "### central difference estimate: numpy.gradient"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = numpy.linspace(0, 2*numpy.pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = numpy.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "derivative_estimate = numpy.gradient(y, x[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114a4ee80>,\n",
       " <matplotlib.lines.Line2D at 0x114a8c8d0>]"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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f3mq5kDSp6TTClL/u/UWB8QVY02YNxbIUi/0TxBO5dOOVUqquUuqgUuqwUupfrbKUUq8o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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114147dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, x, derivative_estimate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Given noisy data\n",
    "### Savitzky-Golay scipy.signal.savgol_filter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's fine for smooth data, but what if you have some noise added?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y_noise = y + numpy.random.randn(len(y))*0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114b7a400>]"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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MbL96zutpZjPMbJaZNaL7SSS+zHwXrXRXorzpJt8uMFmyBx/xc911vmH46tUN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\nvohITPw/Ofy4TzQJSn8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114a969b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y_noise)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "derivative_estimate = numpy.gradient(y_noise, x[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114cadf98>,\n",
       " <matplotlib.lines.Line2D at 0x114cb39e8>]"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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s8XreTVWb+tXQccRIvjpw6IDT+wPxxuB/dfArjyHMb5W+xXfyvmN63iOdYWHw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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114b83748>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y_noise, x, derivative_estimate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's clearly not great."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy.signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "smooth_derivative_estimate = scipy.signal.savgol_filter(y, 5, 2, deriv=1, delta=x[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114de9a90>,\n",
       " <matplotlib.lines.Line2D at 0x114de9c50>]"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114cc6a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y_noise, x, smooth_derivative_estimate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's more like it!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimisation\n",
    "## Nonlinear\n",
    "### scipy.optimize.minimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's minimise $\\sin(x)/x$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alchemyst/anaconda3/lib/python3.5/site-packages/ipykernel/__main__.py:2: RuntimeWarning: invalid value encountered in true_divide\n",
      "  from ipykernel import kernelapp as app\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x114153860>]"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Av9eW7KFw/pDaAlWDXEWRJZxSYWbfAg4F5sWNpGEyLZGXgH8As9x9QeyYGqDY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\nCaGELyKSEP8fQItVpc+SvV4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114eff400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By inspection, we can see the minimum lies between 4 and 5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: -0.21723362821071318\n",
       " hess_inv: array([[ 4.61057157]])\n",
       "      jac: array([ -4.67523932e-07])\n",
       "  message: 'Optimization terminated successfully.'\n",
       "     nfev: 12\n",
       "      nit: 3\n",
       "     njev: 4\n",
       "   status: 0\n",
       "  success: True\n",
       "        x: array([ 4.49340729])"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scipy.optimize.minimize(f, 4.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistical analysis\n",
    "## Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simple polynomial regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = numpy.linspace(1, 4, 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y = f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1165d26d8>]"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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7zs3fesbW5blbkGTb4Nz3r/D9xuauqrb8B3g984+K3r/C928CPjN4fT5wxyTOc5PG9gZg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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1164b3860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, '.')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "quadratic = numpy.polyfit(x, y, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "smoothx = numpy.linspace(1, 4, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11670cf98>]"
      ]
     },
     "execution_count": 226,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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uFZL3JHNX9F30bt775KWm+T0jobhBoBAQEQlgX+37ilfXvcqrya9S7Zxq9G7e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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11659e128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, '.')\n",
    "plt.plot(smoothx, numpy.polyval(quadratic, smoothx))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}