Cyclic Distributions.ipynb

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    "# Cyclic Distributions\n",
    "\n",
    "Cyclic distributions can be useful to simulate pulsar signals. It would be interesting, and fun, to try to come up with analytical expressions that result in these type of distributions.\n",
    "\n",
    "## General Constraints\n",
    "\n",
    "Let $f(x)$ be some distribution on $x \\in (-\\infty, +\\infty)$ such that $f(x) = f(x + 2\\pi)$ and $f(x) \\ge 0$ then it is a cyclic distribution.\n",
    "\n",
    "I am mostly interested in symmetric single-peaked distributions for now. So, I add the following constraints (hopefully without loss of much generality) to normalize these distributions for easier comparisons:\n",
    "\n",
    "$$\\int_{-\\pi}^{\\pi} f(x) \\,\\, \\mathrm{d}x = 1$$\n",
    "\n",
    "$$f(x) = f(-x)$$\n",
    "\n",
    "$$f'(x) \\le 0 \\qquad \\mathrm{for} \\, x \\in [0, \\pi]$$\n",
    "\n",
    "This should constrain us to distributions that peak at $f(0)$, then monotonically decrease outward symmetrically. Since most distributions will have tunable shape parameters, we normalize via $0 \\le f(x) \\le 1$ and $f(0) = 1$.\n"
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    "%matplotlib inline\n",
    "\n",
    "import ipywidgets as widgets\n",
    "import numpy as np\n",
    "import scipy.integrate\n",
    "import scipy.optimize\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "np.seterr(over='ignore')\n",
    "\n",
    "plt.rc('text', usetex=True)\n",
    "plt.rc('font', family='serif')\n",
    "plt.rc('font', size=16)\n",
    "\n",
    "from IPython.display import set_matplotlib_formats\n",
    "set_matplotlib_formats('svg')"
   ]
  },
  {
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   "execution_count": 2,
   "id": "other-howard",
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   "source": [
    "def shape_parameter(fn, bracket=[0, 30]):\n",
    "    def root_func(a, fn):\n",
    "        r, *_ = scipy.integrate.quad(fn, -np.pi, np.pi, args=(a,))\n",
    "        return 1.0 - r\n",
    "\n",
    "    root = scipy.optimize.root_scalar(root_func, args=(fn,), bracket=bracket)\n",
    "    if not root.converged:\n",
    "        print(root)\n",
    "        return None\n",
    "    return root.root\n",
    "\n",
    "\n",
    "def multiple_formatter(denominator=2, number=np.pi, latex=r\"\\pi\"):\n",
    "    def gcd(a, b):\n",
    "        while b:\n",
    "            a, b = b, a % b\n",
    "        return a\n",
    "\n",
    "    def _multiple_formatter(x, pos):\n",
    "        den = denominator\n",
    "        num = np.int64(np.rint(den*x/number))\n",
    "        com = gcd(num, den)\n",
    "        num, den = int(num/com), int(den/com)\n",
    "        if den == 1:\n",
    "            if num == 0:\n",
    "                return r'$0$'\n",
    "            if num == 1:\n",
    "                return r'${}$'.format(latex)\n",
    "            elif num == -1:\n",
    "                return r'$-{}$'.format(latex)\n",
    "            else:\n",
    "                return r'${}{}$'.format(num, latex)\n",
    "        else:\n",
    "            if num == 1:\n",
    "                return r'$\\frac{{{}}}{{{}}}$'.format(latex, den)\n",
    "            elif num == -1:\n",
    "                return r'$\\frac{{-{}}}{{{}}}$'.format(latex, den)\n",
    "            else:\n",
    "                return r'$\\frac{{{}{}}}{{{}}}$'.format(num, latex, den)\n",
    "\n",
    "    return _multiple_formatter\n",
    "\n",
    "\n",
    "class Multiple:\n",
    "    def __init__(self, denominator=2, number=np.pi, latex=r\"\\pi\"):\n",
    "        self.denominator = denominator\n",
    "        self.number = number\n",
    "        self.latex = latex\n",
    "\n",
    "    def locator(self):\n",
    "        return plt.MultipleLocator(self.number / self.denominator)\n",
    "\n",
    "    def formatter(self):\n",
    "        return plt.FuncFormatter(multiple_formatter(self.denominator,\n",
    "                                                    self.number,\n",
    "                                                    self.latex))\n",
    "\n",
    "\n",
    "def plot_distribution(f, a, m=1, title=None):\n",
    "    t = np.linspace(-m*np.pi, +m*np.pi, 4097)\n",
    "    \n",
    "    plt.close(0)\n",
    "    fig = plt.figure(0, figsize=(9, 4.5), tight_layout=True)\n",
    "    ax = fig.subplots(1, 1)\n",
    "    ax.plot(t, f(t, a), 'k-')\n",
    "    ax.set_xlabel('$x$')\n",
    "    ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi))\n",
    "    ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 4))\n",
    "    ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
    "\n",
    "    if title is not None:\n",
    "        ax.set_title(title, pad=10)\n",
    "\n",
    "    plt.show()"
   ]
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   "source": [
    "### Cosine raised to a power\n",
    "\n",
    "A simple function that comes to mind is $f(x) = \\left|\\cos{x}\\right|^a$. This simplifies to\n",
    "$$f(x) = \\left(\\frac{1+\\cos{x}}{2}\\right)^a$$"
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   "id": "joint-punishment",
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   "source": [
    "def cos_power(x, a):\n",
    "    return ((1 + np.cos(x))/2)**a\n",
    "\n",
    "cos_power.name = \"Cosine Power\""
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   "source": [
    "plot_distribution(cos_power, shape_parameter(cos_power), m=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "previous-karma",
   "metadata": {
    "tags": []
   },
   "source": [
    "### von Mises distribution\n",
    "\n",
    "The [von Mises distribution](https://en.wikipedia.org/wiki/Von_Mises_distribution),\n",
    "\n",
    "$$f(x) = e^{a (\\cos{x} - 1)}$$\n",
    "\n",
    "is a maximum entropy distribution on the circle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "pacific-brazil",
   "metadata": {
    "tags": [
     "remove_cell"
    ]
   },
   "outputs": [],
   "source": [
    "def von_mises(x, a):\n",
    "    return np.exp(a * (np.cos(x) - 1))\n",
    "\n",
    "von_mises.name = \"von Mises\""
   ]
  },
  {
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   "execution_count": 6,
   "id": "every-eagle",
   "metadata": {
    "tags": [
     "remove_input"
    ]
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    "plot_distribution(von_mises, shape_parameter(von_mises), m=3)"
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    "### Any distribution with $\\tan{(x/2)}$ as argument\n",
    "\n",
    "If we take any probability distribution with infinite support from $(-\\infty, +\\infty)$, we can simply replace the argument with $\\tan(x/2)$ to get a cyclic version of the distribution!\n",
    "\n",
    "1. Normal Distribution: $$f(x) = \\exp{\\left(-a \\tan^2\\left(\\frac{x}{2}\\right)\\right)}$$\n",
    "   \n",
    "2. Cauchy Distribution: $$f(x) = \\frac{1}{1 + a \\tan^2\\left(\\frac{x}{2}\\right)}$$\n",
    "   \n",
    "3. Logistic Distribution: $$f(x) = \\operatorname{sech}^2{\\left[a\\tan^2\\left(\\frac{x}{2}\\right)\\right]}$$"
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    "def normal_tan(x, a):\n",
    "    y = np.tan(x/2)\n",
    "    return np.exp(-a*y**2)\n",
    "\n",
    "normal_tan.name = \"Normal (Tanned)\"\n",
    "\n",
    "def cauchy_tan(x, a):\n",
    "    y = np.tan(x/2)\n",
    "    return 1/(1 + a*y**2)\n",
    "\n",
    "cauchy_tan.name = \"Cauchy (Tanned)\"\n",
    "\n",
    "def logistic_tan(x, a):\n",
    "    y = np.tan(x/2)\n",
    "    return (1 / np.cosh(a*y))**2\n",
    "\n",
    "logistic_tan.name = \"Logistic (Tanned)\""
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    "t = np.linspace(-np.pi, +np.pi, 4097)\n",
    "\n",
    "plt.close(0)\n",
    "fig = plt.figure(0, figsize=(9, 4.5), tight_layout=True)\n",
    "ax = fig.subplots(1, 1)\n",
    "\n",
    "for f in [normal_tan, cauchy_tan, logistic_tan]:\n",
    "    a = shape_parameter(f)\n",
    "    ax.plot(t, f(t, a), '-', label=f.name)\n",
    "\n",
    "ax.set_xlabel('$x$')\n",
    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi/2))\n",
    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi/12))\n",
    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
    "ax.legend()\n",
    "plt.show()"
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