L05_SciPy.ipynb

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    "<img align=\"left\" src=\"./img/lu.png\" hspace=\"20\"/> <img align=\"right\" src=\"./img/midlands+.png\"/> \n",
    "<br/><br/><br/><br/><br/><br/><br/>\n",
    "\n",
    "------"
   ]
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    "## SciPy\n",
    "[SciPy](https://docs.scipy.org/doc/scipy/reference/) that is a Python standard scientific-computing library built on top of NumPy contains various toolboxes (similar to Matlab toolboxes) and offers a large number of high-level routines for mathematical algorithms. SciPy is organized into subpackages covering different scientific computing domains. Each subpackage provides many routines that are optimized and tested, and therefore should be used when possible. In this lecture we will go through the following subpackages, each of which will be provided with examples to have a general idea of how to use SciPy for scientific computing.\n",
    "* [Special functions](https://docs.scipy.org/doc/scipy/reference/special.html)\n",
    "* [Interpolation](https://docs.scipy.org/doc/scipy/reference/tutorial/interpolate.html)\n",
    "* [Integration](https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html)\n",
    "* [Optimisation](https://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html)\n",
    "* [Signal processing](https://docs.scipy.org/doc/scipy/reference/tutorial/signal.html)\n",
    "* [Fourier Transforms](https://docs.scipy.org/doc/scipy/reference/tutorial/fftpack.html)\n",
    "* [Linear Algebra](https://docs.scipy.org/doc/scipy/reference/tutorial/linalg.html)\n",
    "* [Sparse Eigenvalue Problems](https://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html)\n",
    "* [Statistics](https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html)\n",
    "* [Multi-dimensional image processing](https://docs.scipy.org/doc/scipy/reference/tutorial/ndimage.html)\n",
    "* [File I/O](https://docs.scipy.org/doc/scipy/reference/tutorial/io.html)\n",
    "\n",
    "Before we go into some of the subpackages to get an idea of how to use these toolboxes, make sure SciPy package is installed (`pip install scipy`). "
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    "------\n",
    "\n",
    "### Special functions\n",
    "SciPy provides a large collection of mathematical special functions that are widely used in computational physics problems. Available functions that can be found [here](https://docs.scipy.org/doc/scipy/reference/special.html) include airy, elliptic, bessel, gamma, beta, hypergeometric, parabolic cylinder, mathieu, spheroidal wave, struve, and kelvin. The docstring of the `scipy.special` module is very well written in terms of mathematical description, Python implementation and how to use it. Here is an example ($Y_n(x)$: Bessel function of the second kind of integer order and real argument) for demonstrating how to use the special function module:"
   ]
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      "0.021073628036873716\n"
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       "<matplotlib.legend.Legend at 0x2b4a0864ac8>"
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from __future__ import print_function\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "from scipy.special import yn\n",
    "\n",
    "print(yn(1, 15.))  # 1 is the order\n",
    "\n",
    "x = np.linspace(0., 10., 1000)\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, yn(0, x), '-', linewidth=0.6, label='$Y_0(x)$')   # 0 order\n",
    "ax.plot(x, yn(1, x), '-.', linewidth=0.6, label='$Y_1(x)$')  # 1st order\n",
    "ax.plot(x, yn(2, x), '--', linewidth=0.6, label='$Y_2(x)$')  # 2nd order\n",
    "ax.set_xlabel('x', fontsize=11)\n",
    "ax.set_ylabel('Y(x)', fontsize=11)\n",
    "ax.set(ylim=(-3, 1))\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### Interpolation\n",
    "`scipy.interpolate` module offers routines for one-dimensional (e.g. `interp1d`) and multi-dimensional (e.g. `interp2d` and `griddata`) interpolation functionalities. For each interpolation function, it supports different interpolation methods (e.g. linear and cubic) and the default is always linear interpolation.\n",
    "* Univariate interpolation\n",
    "\n",
    "`interp1d` is a commonly used function to perform 1D interpolation. It starts with creating an interpolator object based on the raw `x` and `y` data to be interpolated. Then pass the new `x` data to the interpolator and the interpolator will perform the interpolation and return the new interpolated `y` values. See the example below: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2b49ddb3a20>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy import interpolate\n",
    "\n",
    "x_raw = np.arange(11)\n",
    "y_raw = np.sin(x_raw)\n",
    "\n",
    "linear = interpolate.interp1d(x_raw, y_raw)  # linear interpolation\n",
    "cubic = interpolate.interp1d(x_raw, y_raw, kind='cubic')  # cubic interpolation\n",
    "x = np.linspace(0, 10, 100)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x_raw, y_raw, '*', label='Raw')\n",
    "ax.plot(x, linear(x), '-', linewidth=0.6, label='Linear')\n",
    "ax.plot(x, cubic(x), '--', linewidth=0.6, label='Cubic')\n",
    "ax.set_xlabel('x', fontsize=11)\n",
    "ax.set_ylabel('Y(x)', fontsize=11)\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Multivariate interpolation\n",
    "\n",
    "There are different Python functions that can be used to perform multivariate interpolation for both unstructured data and data on a regular grid. `interp2d` and `griddata` are commonly used functions for interpolating unstructured data. The former is used to carry out 2D interpolation over a 2-D grid, while the later is for interpolating unstructured N-dimensional data. Note that many interpolation functions for unstructured data can also be used to interpolate data on regular data as a special case. Let's take $Z(x, y) = \\frac{xy}{x^2 + y^2}$ as an example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.QuadMesh at 0x2b49de55b70>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return x * y / (x**2 + y**2)\n",
    "\n",
    "\n",
    "# old data\n",
    "x1 = y1 = np.linspace(-1., 1., 10)  # old coordinates\n",
    "X1, Y1 = np.meshgrid(x1, y1)\n",
    "z1 = f(X1, Y1)  # old data\n",
    "\n",
    "# interpolate to a finer grid\n",
    "x2 = y2 = np.linspace(-1., 1., 100)  # new coordinates\n",
    "cubic_2d = interpolate.interp2d(x1, y1, z1, kind='cubic')\n",
    "z2 = cubic_2d(x2, y2)  # new data\n",
    "X2, Y2 = np.meshgrid(x2, y2)  # for plot\n",
    "\n",
    "fig, ax = plt.subplots(nrows=1, ncols=2)\n",
    "ax[0].pcolormesh(X1, Y1, z1)  # raw data\n",
    "ax[1].pcolormesh(X2, Y2, z2)  # interpolated data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the data is on a regular grid, we just need to pass the coordinates of one column and one row to the interpolation function (the present case above for example). Otherwise, a full coordinates for all data must be provided.\n",
    "\n",
    "The `interpolation` module also contains many spline functions and classes for spline interpolation on both unstructured and regular grids. Read more [here](https://docs.scipy.org/doc/scipy/reference/interpolate.html#module-scipy.interpolate). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### Integration\n",
    "\n",
    "The subpackage `scipy.integrate` provides several integration techniques including single, double and triple integrals, as well as an ordinary differential equation integrator.\n",
    "\n",
    "* Quadrature\n",
    "\n",
    "The functions for single, double and triple integrations are `quad`, `dblquad` and `tplquad` respectively. Let's take `quad` as an example. `quad` is used to integrate a function of one variable over two points that define the domain of the function. The abstract form of the function to be integrated can be expressed as: $I = \\int_{a}^{b} f(x) dx$. As an example, let $a = 0$, $b = \\pi$ and $f(x) = \\sin(x)$ and its integration can be computed numerically as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Integral value of sin(x) is 2.0 and the abs error is 2.220446049250313e-14.\n"
     ]
    }
   ],
   "source": [
    "from scipy import integrate\n",
    "\n",
    "def f(x):\n",
    "    return np.sin(x)\n",
    "\n",
    "I, err = integrate.quad(f, 0, np.pi)\n",
    "print('Integral value of sin(x) is %s and the abs error is %s.' % (I, err))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Ordinary differential equations (ODEs)\n",
    "\n",
    "SciPy provides two different ways to solve ODEs: an API based on the function `odeint`, and object-oriented API based on the class `ode`. To demonstrate how to solve ODEs numerically by using SciPy, we choose `odeint` as an example to solve two different systems: first-order ODE and second-order ODE.\n",
    "\n",
    "##### First-order ODEs\n",
    "\n",
    "You might have noticed that we have implemented our own method (class `ForwardEuler`) for solving the first-order ODE in lecture L03. Yes, I was re-inventing the wheel. Don’t do that. Let’s take the same simple decaying system that has been described in lecture L03 and solve it using SciPy `odeint`. The system is governed by a first-order ODE: $$u'(t) = -10u(t),\\quad t\\in(0, 2],\\quad u(0)=1$$ and the SciPy `odeint` solution is shown below:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2b4a1130a20>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(u, t):\n",
    "    return -10. * u\n",
    "\n",
    "def f_ext(t, U0):\n",
    "    return U0 * np.exp(-10. * t)\n",
    "\n",
    "# Exact solution\n",
    "t_ext = np.linspace(0, 1, 1000)\n",
    "u_ext = f_ext(t_ext, 1)\n",
    "\n",
    "# odeint solution\n",
    "t = np.linspace(0, 1, 50)\n",
    "u = integrate.odeint(f, 1., t)\n",
    "\n",
    "# plot\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(t_ext, u_ext, 'k-', label='Exact')\n",
    "ax.plot(t, u, linestyle='None', marker='*', markersize=3, label='odeint')\n",
    "ax.set_xlabel('t', fontsize=11)\n",
    "ax.set_ylabel('u', fontsize=11)\n",
    "ax.set(xlim=(0, 1))\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You probably have seen that there is a discrepancy between the solutions of class `ForwardEuler` and SciPy `odeint`, and the later gave a better solution. Why?\n",
    "\n",
    "##### Second-order ODEs\n",
    "\n",
    "For `odeint` to compute a second-order ODEs, the ODEs need to be transformed to two first-order equations $Y = (y, y')$, where $Y$ is a vector to store the two reduced first-order ODEs. This can be done by introducing intermittent variables.  Let’s try to solve a second-order Homogeneous equation that has a form of: $$y''(x) + 2y'(x) + 5y(x) = 0 \\quad x\\in(0, 10], \\quad y(0) = 1; y'(0) = 0$$ \n",
    "\n",
    "To reduce the equation into two first-order equations, let $y_1 = y$ and $y_2 = y'$. Substituting to the raw ODE and we have: $$y_1' = y_2, \\quad y_1(0) = 1$$ $$y_2' = -2y_2 - 5y_1, \\quad y_2(0) = 0$$\n",
    "\n",
    "Now we have two first-order ODEs, each of which has a known initial condition. The exact solution is $y = e^{-x} \\cos(2x) + \\frac{1}{2} e^{-x} \\sin(2x)$ which will be used to verify the numerical solution. The numerical solution using `odeint` is shown below: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2b4a11998d0>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(Y, x):\n",
    "    # Here Y is a vector containing two elements: Y[0] = y1, Y[1] = y2\n",
    "    y1, y2 = Y[0], Y[1]\n",
    "    dy1 = y2\n",
    "    dy2 = -2 * y2 - 5 * y1\n",
    "    return [dy1, dy2]\n",
    "\n",
    "def f_ext(x):\n",
    "    return np.exp(-x) * np.cos(2*x) + 1/2 * np.exp(-x) * np.sin(2*x)\n",
    "\n",
    "# Exact solution\n",
    "x_ext = np.linspace(0, 10, 500)\n",
    "y_ext = f_ext(x_ext)\n",
    "\n",
    "# odeint solution\n",
    "x = np.linspace(0, 10, 100)\n",
    "y = integrate.odeint(f, [1, 0], x)\n",
    "\n",
    "# plot\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x_ext, y_ext, 'k-', label='Exact')\n",
    "ax.plot(x, y[:, 0], linestyle='None', marker='*', markersize=3, label='odeint')\n",
    "ax.set_xlabel('x', fontsize=11)\n",
    "ax.set_ylabel('y', fontsize=11)\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### Fourier Transforms\n",
    "Fourier transform that is a method for expressing a function as a sum of periodic components are widely used in computational physics. SciPy `fftpack` module provides functions for accessing the classic [FFTPACK](http://www.netlib.org/fftpack/) FORTRAN library to compute fast Fourier transforms (FFTs) for both complex and real periodic sequences.\n",
    "\n",
    "To demonstrate how to use the `fftpack` module, we take the damped harmonic oscillator function as the example: $\\frac{d^2y}{dx^2} + 2 \\zeta \\omega_0 \\frac {dy}{dx} + {\\omega_0}^2 y = 0$, where $\\zeta$ is the damping ratio and $\\omega_0$ is the frequency.\n",
    "\n",
    "Let's define $\\zeta = \\frac {\\sqrt 5}{5}$ and $\\omega_0 = \\sqrt 5$, then the function becomes to the one used in the previous section. We have already had its solution and we use it here as the signal to compute the FFT: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy import fftpack\n",
    "\n",
    "# Number of sample points\n",
    "N = len(x_ext)\n",
    "\n",
    "# sample spacing\n",
    "T = x_ext[1] - x_ext[0]\n",
    "\n",
    "# calculate FFT of y_ext\n",
    "fy = fftpack.fft(y_ext)\n",
    "\n",
    "# fftfreq calculates the frequencies for the components in fy and returns the FFT sample frequency points\n",
    "omega = fftpack.fftfreq(N, T)\n",
    "\n",
    "# plot\n",
    "fig, ax = plt.subplots(figsize=(12, 3))\n",
    "ax.plot(omega[np.where(omega > 0)], np.abs(fy[np.where(omega > 0)]))\n",
    "ax.set_xlabel('Frequency (Hz)', fontsize=11)\n",
    "ax.set_xlim(0, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, a peak in the spectrum occurs at around 0.35Hz that is corresponding to our definition $\\omega_0 = \\sqrt 5$ that gives the ordinary frequency $f = \\frac {\\omega_0}{2\\pi} = \\frac {\\sqrt5}{2 \\times 3.14} = 0.35588Hz$.\n",
    "\n",
    "Note that there is an alternative FFT Python package ([PyFFTW](https://github.com/pyFFTW/pyFFTW)) that is dependent on the [FFTW](http://www.fftw.org/) library. It is not included in SciPy due to licence issue. PyFFTW has a re-implementation of several functions in `scipy.fftpack` and is usually significant faster. For users who are concerning the speed of FFT computation PyFFTW should be used."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### Optimisation\n",
    "\n",
    "Mathematical optimisation is a large area of applied mathematics, and it deals with problems of finding best values (minima, maxima or zeros) of so-called *cost functions* in a given domain. `scipy.optimize` subpackage provides several commonly-used optimisation algorithms for problems such as unconstrained and constrained minimization of multivariate scalar functions, global optimisation, least-squares minimization, scalar univariate functions minimisation and root finding. \n",
    "Below are a few example for demonstrating the basic usage of `scipy.optimize` routines.\n",
    "\n",
    "* Finding minima\n",
    "\n",
    "Below are a few example for demonstrating the basic usage of `scipy.optimize` routines.\n",
    "Consider the problem of minimising the following function that has a single variable: $$f(x) = \\cos(3 \\pi x)/x, \\quad x \\in [0.1, 1.2]$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-4, 6)]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return np.cos(3 * np.pi * x) / x\n",
    "    \n",
    "x = np.linspace(0.1, 1.2, 100)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x), 'k-', linewidth=0.6)\n",
    "ax.set_xlabel('x', fontsize=11)\n",
    "ax.set_ylabel('f(x)', fontsize=11)\n",
    "ax.set(ylim=(-4, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As seen in the figure above, this function has a global minimum and local minimum. We can use the `scipy.optimize.minimize_scalar` function to find its minimum at a defined domain of $x$. See the code below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The minimum of f(x) with x between 0.1 and 1.2 is -3.171517111340773 at x = 0.29691854717096766.\n",
      "The minimum of f(x) with x between 0.7 and 1.1 is -1.005698713865335 at x = 0.98865627812584.\n"
     ]
    }
   ],
   "source": [
    "from scipy import optimize\n",
    "\n",
    "# minimum between 0.1 < x < 1.2\n",
    "res = optimize.minimize_scalar(f, bounds=(0.1, 1.2), method='bounded')\n",
    "print('The minimum of f(x) with x between 0.1 and 1.2 is %s at x = %s.' % (res.fun, res.x))\n",
    "\n",
    "# minimum between 0.7 < x < 1.1\n",
    "res = optimize.minimize_scalar(f, bounds=(0.7, 1.1), method='bounded')\n",
    "print('The minimum of f(x) with x between 0.7 and 1.1 is %s at x = %s.' % (res.fun, res.x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Finding roots\n",
    "\n",
    "To find the roots of the (non-linear) equations defined by $f(x) = 0$ we can use the SciPy `fsolve` function by giving an initial guess. Consider a problem of finding the roots of $f(x) = x^3 - 2$: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Root is at x = 1.2599210498948732\n"
     ]
    }
   ],
   "source": [
    "def f(x):\n",
    "    return x**3 - 2\n",
    "    \n",
    "x = np.linspace(-2, 2, 50)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x), 'k-', linewidth=0.6)\n",
    "ax.set_xlabel('x', fontsize=11)\n",
    "ax.set_ylabel('f(x)', fontsize=11)\n",
    "plt.show()\n",
    "\n",
    "root = optimize.newton(f, -1.5)\n",
    "print('Root is at x = %s' % root)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### Linear Algebra\n",
    "\n",
    "`scipy.linalg` module contains a number of matrix related functions for standard linear algebra operations, such as linear equation solving, eigenvalue problems, matrix functions, decomposition and etc. A few of examples are provided below to demonstrate how to use the `scipy.linalg` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1., -1.,  6.])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import linalg\n",
    "\n",
    "# solve Ax = b\n",
    "A = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])  # 3 x 3 matrix\n",
    "b = np.array([1., 2., 1.])  # vector\n",
    "x = linalg.solve(A, b)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00,  0.00000000e+00,  0.00000000e+00],\n",
       "       [-3.33066907e-17,  1.00000000e+00,  0.00000000e+00],\n",
       "       [ 1.66533454e-16,  1.66533454e-16,  1.00000000e+00]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve AX = B\n",
    "A = B = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])  # 3 x 3 matrix\n",
    "X = linalg.solve(A, B)\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Eigenvalue problems\n",
    "\n",
    "Computation of eigenvalues and eigenvectors of a matrix is straightforward by using the `scipy.linalg.eig` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "val, vec = linalg.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.        +0.j, -1.44948974+0.j,  3.44948974+0.j])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# eigenvalues\n",
    "val"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        ,  0.19399248, -0.89052821],\n",
       "       [ 0.        , -0.43158379, -0.20014165],\n",
       "       [ 1.        ,  0.88096671, -0.40853743]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# eigenvectors\n",
    "vec"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenvector corresponds to the $n$th eigenvalue is the $n$th column vector of the eigenvector matrix (`vec` for this case), e.g. the corresponding eigenvector for the first eigenvalue (`val[0]`) is `vec[:,0]`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Matrix operations\n",
    "\n",
    "SciPy provides functions for computing Determinant, Finding inverse and computing norms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-5.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# determinant\n",
    "linalg.det(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.00000000e-01,  4.00000000e-01,  1.85037171e-18],\n",
       "       [ 2.00000000e-01, -6.00000000e-01, -7.40148683e-18],\n",
       "       [-1.00000000e+00,  3.00000000e+00,  1.00000000e+00]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# inverse\n",
    "linalg.inv(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.4031242374328485"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# norms\n",
    "linalg.norm(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### Statistics\n",
    "\n",
    "The `scipy.stats` module contains a large number of statistical functions, probability distributions including continuous and discrete distributions as well as statistical tests. A full set of functions are available [here](https://docs.scipy.org/doc/scipy/reference/stats.html).\n",
    "\n",
    "* continuous distributions\n",
    "\n",
    "Let's take the normal distribution as an example. First we create a (continous) random variable with normal distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['__class__',\n",
       " '__delattr__',\n",
       " '__dict__',\n",
       " '__dir__',\n",
       " '__doc__',\n",
       " '__eq__',\n",
       " '__format__',\n",
       " '__ge__',\n",
       " '__getattribute__',\n",
       " '__gt__',\n",
       " '__hash__',\n",
       " '__init__',\n",
       " '__init_subclass__',\n",
       " '__le__',\n",
       " '__lt__',\n",
       " '__module__',\n",
       " '__ne__',\n",
       " '__new__',\n",
       " '__reduce__',\n",
       " '__reduce_ex__',\n",
       " '__repr__',\n",
       " '__setattr__',\n",
       " '__sizeof__',\n",
       " '__str__',\n",
       " '__subclasshook__',\n",
       " '__weakref__',\n",
       " 'a',\n",
       " 'args',\n",
       " 'b',\n",
       " 'cdf',\n",
       " 'dist',\n",
       " 'entropy',\n",
       " 'expect',\n",
       " 'interval',\n",
       " 'isf',\n",
       " 'kwds',\n",
       " 'logcdf',\n",
       " 'logpdf',\n",
       " 'logpmf',\n",
       " 'logsf',\n",
       " 'mean',\n",
       " 'median',\n",
       " 'moment',\n",
       " 'pdf',\n",
       " 'pmf',\n",
       " 'ppf',\n",
       " 'random_state',\n",
       " 'rvs',\n",
       " 'sf',\n",
       " 'stats',\n",
       " 'std',\n",
       " 'var']"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "\n",
    "# X is a Python object representing a normal distribution \n",
    "X = stats.norm(loc=0., scale=2.5)  # loc specifies the mean and scale is the std  \n",
    "\n",
    "# Let's see what methods are available within X\n",
    "dir(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To generate a sequence of random variates, use the `rvs` method and `size` keyword argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.57310512, -1.15621127, -4.57290586, -0.89383331,  1.78447276,\n",
       "       -1.54766297, -0.64798752,  0.13107752, -3.38200813,  2.61890261,\n",
       "       -1.15547588, -2.54636235,  1.78197183,  2.02878081,  1.78289908,\n",
       "        2.75706243,  3.13965917,  1.84384369,  2.54850305, -1.64235895])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.rvs(size=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 2.5)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate mean and std\n",
    "X.mean(), X.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# calculate the probability distribution function (PDF) at a number of points\n",
    "x = np.linspace(-8, 8, 100)\n",
    "pdf = X.pdf(x)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, pdf, 'k-', linewidth=0.6)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# calculate the cumulative distribution function (CDF)\n",
    "x = np.linspace(-8, 8, 100)\n",
    "cdf = X.cdf(x)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, cdf, 'k-', linewidth=0.6)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* discrete distributions\n",
    "\n",
    "The functions for discreate distributions are similar to that of continuous distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Y is a Python object representing a Poisson distribution \n",
    "Y = stats.poisson(4.)\n",
    "\n",
    "# calculate the probability mass function (PMF) at a number of points\n",
    "x = np.arange(15)\n",
    "pmf = Y.pmf(x)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, pmf, '*-', linewidth=0.6)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Statistical tests\n",
    "\n",
    "Assume that we have two set of data that are subject to the Gaussian distribution. We can then test if the mean values of the two sets of data the means are significantly different. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_indResult(statistic=-22.56554002785575, pvalue=1.2628884892286962e-100)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1 = stats.norm(0, 1)\n",
    "X2 = stats.norm(1, 1)\n",
    "\n",
    "stats.ttest_ind(X1.rvs(size=1000), X2.rvs(size=1000)) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the p value that is the probability of both data sets being identical is very close to zero, it is very likely that the two data sets have different mean values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------\n",
    "\n",
    "### More reading\n",
    "* [SciPy : high-level scientific computing](https://www.scipy-lectures.org/intro/scipy.html#statistics-and-random-numbers-scipy-stats)\n",
    "* [SciPy documentation](https://docs.scipy.org/doc/scipy/reference/)"
   ]
  }
 ],
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