{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Supplementary materials : Details on generating Figure 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This document is a supplementary material of the article *Detecting periodicities with Gaussian\n",
    "processes* by N. Durrande, J. Hensman, M. Rattray and N. D. Lawrence. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first step is to import the required packages. This tutorial has been written with GPy 0.8.8 which includes the kernels discussed in the article. The latter can be downloaded on the [SheffieldML github page](https://github.com/SheffieldML/GPy). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "import GPy\n",
    "\n",
    "np.random.seed(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The boundary limits for the plots are set to $[0,4 \\pi]$, and we consider a period of $2 \\pi$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class AperiodicMatern32(GPy.kern.Kern):\n",
    "    \"\"\"\n",
    "    Kernel of the aperiodic subspace (up to a given frequency) of a Matern 3/2 RKHS.\n",
    "\n",
    "    Only defined for input_dim=1.\n",
    "    \"\"\"\n",
    "\n",
    "    def __init__(self, input_dim=1, variance=1., lengthscale=1., period=2.*np.pi,\n",
    "            n_freq=10, lower=0., upper=4*np.pi,\n",
    "            active_dims=None, name='aperiodic_Matern32'):\n",
    "        self.per_kern = GPy.kern.PeriodicMatern32(input_dim, variance, lengthscale, period, n_freq, lower, upper, active_dims, name='dummy kernel')\n",
    "        self.whole_kern = GPy.kern.Matern32(input_dim, variance, lengthscale, name='dummy kernel')\n",
    "        GPy.kern.Kern.__init__(self, input_dim, active_dims, name)\n",
    "\n",
    "        self.variance = GPy.core.Param('variance', np.float64(variance), GPy.core.parameterization.transformations.Logexp())\n",
    "        self.lengthscale = GPy.core.Param('lengthscale', np.float64(lengthscale), GPy.core.parameterization.transformations.Logexp())\n",
    "        self.period = GPy.core.Param('period', np.float64(period), GPy.core.parameterization.transformations.Logexp())\n",
    "        self.link_parameters(self.variance, self.lengthscale, self.period)\n",
    "\n",
    "\n",
    "    def parameters_changed(self):\n",
    "        self.whole_kern.variance = self.variance * 1.\n",
    "        self.per_kern.variance = self.variance * 1.\n",
    "\n",
    "        self.whole_kern.lengthscale = self.lengthscale * 1.\n",
    "        self.per_kern.lengthscale = self.lengthscale * 1.\n",
    "\n",
    "        self.per_kern.period = self.period * 1.\n",
    "    \n",
    "    def K(self, X, X2=None):\n",
    "        return self.whole_kern.K(X, X2) - self.per_kern.K(X, X2)\n",
    "\n",
    "    def Kdiag(self, X):\n",
    "        return np.diag(self.K(X))\n",
    "\n",
    "    def update_gradients_full(self, dL_dK, X, X2=None):\n",
    "        self.whole_kern.update_gradients_full(dL_dK, X, X2)\n",
    "        self.per_kern.update_gradients_full(-dL_dK, X, X2)\n",
    "        self.variance.gradient = self.whole_kern.variance.gradient + self.per_kern.variance.gradient\n",
    "        self.lengthscale.gradient = self.whole_kern.lengthscale.gradient + self.per_kern.lengthscale.gradient\n",
    "        self.period.gradient = self.per_kern.period.gradient\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Domain Parameters\n",
    "a = 0.            # lower bound of the space\n",
    "b = 4*np.pi       # upper bound\n",
    "\n",
    "# kernel parameters\n",
    "per = 2*np.pi     # period\n",
    "var = 1.          # variance\n",
    "N = 20            # max frequency in the decomposition (the number of basis functions is 2N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know from the article that a Matérn kernels $k$ can be decomposed as a sum of a periodic and aperiodic kernel : $k=k_p+k_a$. In the code below, we consider 3 Matérn 3/2 kernels km1, km2, km3 with various lengthscales and we denote by kp[123] and ka[123] their associated periodic and aperiodic components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lenscl=.5\n",
    "km1 = GPy.kern.Matern32(input_dim=1,variance=var,lengthscale=lenscl)\n",
    "kp1 = GPy.kern.PeriodicMatern32(input_dim=1, variance=var, lengthscale=lenscl, period=per, n_freq=N, lower=a, upper=b)\n",
    "ka1 = AperiodicMatern32(input_dim=1, variance=var, lengthscale=lenscl, period=per, n_freq=N, lower=a, upper=b)\n",
    "\n",
    "lenscl=2.\n",
    "km2 = GPy.kern.Matern32(input_dim=1,variance=var,lengthscale=lenscl)\n",
    "kp2 = GPy.kern.PeriodicMatern32(input_dim=1, variance=var, lengthscale=lenscl, period=per, n_freq=N, lower=a, upper=b)\n",
    "ka2 = AperiodicMatern32(input_dim=1, variance=var, lengthscale=lenscl, period=per, n_freq=N, lower=a, upper=b)\n",
    "\n",
    "lenscl=5.\n",
    "km3 = GPy.kern.Matern32(input_dim=1,variance=var,lengthscale=lenscl)\n",
    "kp3 = GPy.kern.PeriodicMatern32(input_dim=1, variance=var, lengthscale=lenscl, period=per, n_freq=N, lower=a, upper=b)\n",
    "ka3 = AperiodicMatern32(input_dim=1, variance=var, lengthscale=lenscl, period=per, n_freq=N, lower=a, upper=b)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Subfigure a: Matern 3/2 kernel $k$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.1, 1.1)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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miVumD2D0gPbv5zsKo8lkCUEe3wHnIZy/w4eH2OlAbOfWokJidAjdY0Pl0GQn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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7e54b8e650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = 5.\n",
    "fig, ax = plt.subplots(figsize=(4,4))\n",
    "km1.plot(x,plot_limits=[a,b], ax=ax)#, 'all', None,'b-')\n",
    "km2.plot(x,plot_limits=[a,b], ax=ax)#, 'all', None,'b--')\n",
    "km3.plot(x,plot_limits=[a,b], ax=ax)\n",
    "plt.xlabel('')\n",
    "plt.ylabel('')\n",
    "plt.legend([\"$\\\\theta\\  =\\  0.5$\",\"$\\\\theta \\ = \\ 2$\",\"$\\\\theta \\ = \\ 5$\"],prop={'size':12},borderaxespad=0.)\n",
    "plt.ylim([-0.1,1.1])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Subfigure b: periodic sub-kernel $k_p$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.13, 0.42)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KshRbAjkZyWSkJtFtdJTGAxs+swiP28Ur79Vx7KxeHCeaRCPBGZMWxI7ufv7t\npUCu4hsPLHXU+prX48KVgMs/uUj9BDghBKVxlLcAmFyUxeeNOUM/fH5/XHhMdiHSbGEbkG/8nQeM\nS8o3btwY/LuyspLKykqeevkIbV39LJ9dxFoHbRw0GvVXDM+iUL1nATCpMJPq+lbqr3SyYHpUHUNl\nfPWeRWzbU8fhM028/v45PmVsUJToVFVVUVVVNe7XRyoWzwMVwE5gBrAdQAiRK6UMe1PKULEAOHOp\njV9VncYlBH/+hxWOnQMyEmbibZJNxGJaccCOc40dii2JHplpSXz9viX8/X/t5V+2HuTWJZMd35sT\nDcwLscmmTZvG9PqIfHsp5UEAIcQaoE1Kech4aIf5HCHEeqBCCPG1MI/JD58/gM8vuf+WcmZPyYvE\nRNtRb1Qe7JCzAJheEljo+Nzl+BELgM/eXMacKXlcae0JThPQREbEiQAp5RYp5U4p5ZaQ+ypC/v6V\nlDJfSvnzcI6348B59p1oJDs9mT/5gyWRmmcr+gd9XLzahUsIJhWqz1kATC/JBuBsHHkWAG6Xiz9/\nKPA1/M9Xj1F7qV2xRc7HVlnD1s4+/vG/3wfgG/cvJTfT2g2DY835yx34pWRyUaZtysDTDLE419gR\nd8vTLZtVxH2ryxn0+vm7X7znuD1S7IatxGLzc/tp6+qnYm5xcAXneKLOuLqV2WiPk6z0ZPKzU+kf\n9HG5Nf7apL+1bhmFuWkcqW3ihTdOqzbH0dhGLN44eIHt+8+Rmuzmr/5oRVwlNU1qGwJiMcNGYgFD\noUhdQ3yFIgCZ6cl8+4s3AvCTXx+KmxKxCmwhFm1d/Tz57D4AHntgGZNtUimINnWGWJSV2kssTPE6\ncynsApajuGXJZO68YRp9Az6+9x97dDgyTmwhFpv+Yw8tHX0sm1XI5yqdubR/OJhiMcNmYjF7cqDi\nVH2hdZRnOpe/eKiCguxUPjh9hae36Ylm48EWYvH24YtkpSez6asrla1JGWv6+r2ca+zE7RJMK85W\nbc5HmGWUp6vr49OzAMjLSmXTV1ciBPz8d0c5cOqyapMchy3EAuC7X77JsXuAhEPNxTb8UjJjYg6p\nyfZqECqflIsQAc9nYNCn2pyYsWL+RL786QX4peRv/v0d2jr7VJvkKGwhFp9fM5dbl05RbUZMOXk+\nMN9uztT8UZ5pPWkpHqYUZuHzy7jrtxjOI/cuZvHMCVxt6+Vvnn4Xr0/nL8LFFmLxZ+uWqjYh5phi\nMXeqPTtbU6gBAAANHUlEQVRSzU5Z0854xeN28fdfW0VuZgrvHWvgX351ULVJjsEWYqF6XQcrOHU+\nkDyca0PPAmC+MYksETbrKSnI4B+/fgset4vndp7k12/VqDbJEdhCLOKdvn4v1fWtuIQIJhPtxsKy\nCQAcrUuMNSCWzSriO0b/xT88u4/9JxsVW2R/tFhYwLGzzfj8kvLJuWSkJqk2Z0TmTc3H7RLU1LfR\n2+9VbY4l3LtqJl+6cx4+v+QvfrKbE+cSQyjHixYLCzh85ioAS2YWKrbk2qSmeJg1OQ+/lBxPoBWm\nHntgKWsrptLdN8if/eiNuG1MiwZaLCzgwxpDLMrtKxYAi4xQ5FDNFcWWWIfb5eJ7X13JqkWltHf3\n89g/7wouUKT5KFosYozP7+dwbSBpaGfPAqBibjEA+08mVsNSksfNE4+u5hOzi2lq7+XRH2zXU9pH\nQItFjDlxtoXOngGmFGUp34FsNJbPLkaIQNjUN5AYeQuT1GQP//TYrSwtL+RKWy+PbN6uF/wdhhaL\nGLPneAMAK+aXKLZkdHIzU5g1OY8Br5/DZ+K/hDqcjNQk/u+3bufmhYGQ5E//aQf7TugqiYkWixiz\n91hALG6aP1GxJeGxYl5A1N45clGxJWpITfHwgz+9lU/dOJ2efi/f/PEuXnjjlF4lHC0WMaW1s48j\ntU143C4+MbtYtTlhccuSyQDs/rA+YU8QjyeQ9Pwjo6y6+bn9/J9n9jHojd95M+GgxSKGVB2qxy8l\nN8wtJjM9WbU5YbFo5gRyM1Oov9oVXKwnEXG5BN9cv5zvPbySlCQ3L71VwyObtyf0tohaLGLIrg/O\nA7DGQfueuF0uVi+eBMCO/ecVW6Oeu1bM4KnH76A4L52jdc186fuv8Mqe2oT0urRYxIim9l7eP9GI\n2yW41XDtnYK5Kc8r79Ul5EkxnPnTC3j2b+9mzfKpdPd5+e7/28P/fuptmtp7VZtmKVosYsS2PbX4\n/JLViyeRm5Wq2pwxUTG3mKLcNC41dXHIaChLdHIyUnji0VX8zR/fRFqKhx0HzvO5v32ZF9+sjrtV\n0a+FFosY4PdLfvv2GSAw/8BpuF0u7v5kGQC/qtIrYpsIIbh31Uz+57v3cPPCUrp6B3ni2X185cnX\n+OB0/DeyabGIAbs/rOf8lU5K8tP55IJS1eaMi3W3zsLtEuw8cJ7LrT2qzbEVpRMy+ec/q+SJR1Yx\nISeN42ebefQHO/jzf30zrueWaLGIMlJKfmFsl/fFO+bhcTvzLS7Jz+C25VPw+SX/pbf/+xhCCNZW\nTGPr9z/LI/cuJi3Fw+4P6/n8pm385b/F5wxWoTqBJYSQqm2IJlUHL/D4v+0mNzOF3z5xH2kO3pC3\npr6VL3z/FTxuF1u//9m4XiM1Uprae3l621F+83YNA97AUn03zC1hfeUsblky2ZYXDSEEUsqwV8jW\nYhFF+ga8fOF7r3DhSid/+YUb4mJbg7/++Tu8tu8slcumsPnrt6g2x/Y0tfXy7I4TbH2zOrguSGFu\nGvetKufeVTMpybfP/CAtFgr50S8/4NntJygrzeHZv74bj8d+V5Oxcrm1hwf/9mV6+r088cgq1lZM\nU22SI+jo7mfbe3Vsrar+yA71S8sLuaNiGrd/YioTctIUWqjFQhlvHqrnL37yJi4hePo7n2KBsaZl\nPPDCG6fY/Nx+MtOS+MVf3cWUovjcMS4WSCk5cOoyW3dX89aHF+k3tlpwCcGS8kJWLixl5cJSZk3O\ntXzLTi0WCjhS28Rj/7yTnn4v37h/KV++a4Fqk6KKlJK//OlbVB28wJSiLJ56/A7lV0Un0t03yO4P\n69n+/jn2HGv4yDYEhblp3DivhCXlRSyZWcj0kuyYb7ilxcJi9hy7xHd+9hbdfV7uWjHd2PUq/nZV\n6+4b5E9+sIOT51uYXJjJDx+rtN0Gz06is2eAvccbePfoJfYca/hYN2hORjKLZhayZGYh626dRVYM\n5hZZLhZCiHVAG1AmpdwyjscdKRa9/V7+9deHeH7XKQDuvGEam7660pZZ72jR2tnHN3/8BifPt5CS\n5Oab65ex/tbZcbvlpFVIKTld38rB01f48MxVPqy5ytW2gHi4hGDXjz8Xk4WeLRULIcRyYIaUcqsQ\nYgOwX0p5MNzHjec4Siw6ewZ46a0antt5kqttvbhdgoc/s4iH716YECdNd98gm5/bz7Y9tUBgR/iv\n3LWA25dPJTkp/vd/sQIpJQ3N3RyquUpDcxcP37MoJuNYLRZPAq9LKXcJIdYAy6WUm8N93HiOrcWi\nu2+Q6gutnDzfwp5jDbx/spFBo44+d2o+f/2/VjBnij03DoolO/af40e//CDY3ZmRmsTqJZO4YU4J\nC8smMLUoKy6qQfHMWMUi0o6hXCB0v7vhJYDRHgfg1IUWkCAlSCSmdkgAKY37Cc6AvOb95uukNO4f\num0+bh5/0Ounf8BL34CPvgGv8eOjraufq+29NLf30tTeS2NLN6FaJgTcOK+EL6ydy8qFpXGZnwiH\ntRXTuHXpZLbtqeOXVac5faGVV/ee5dW9ZwFwuwQl+RlMKswkLyuV3MwUstOTyc5IJiXZQ7LHRZLH\nRbLHHfid5MYlBEIEvsQCEC7j98duB9xzBATusRarP/KZpbm28Fqj0V442n8x6n/5pe//PgpmxAaP\n28XM0hzmTstncVkhqxZPIj/bWbNIY0WSx819q8u5b3U55y938Pbhixw+08Txc800tnRzsamLi01d\nqs10PO/+5CFcLvUhXqRi0QaYPngeMLwhfrTHAeir3oapKSUzFlFStuQj6h28shhXnaH7A1eWwP3B\nez9yvylVwecZx/N4XKQmu0lN9gR/pyS5yc5IoTA3jcKcNCbkpFGUl65j8TCYWpzNF+7I5gt3BG73\nD/q41NTFpaYu2rr6ae8eoL27n87uAQa8PgYGfQHvzvg96PXjlxKQ+P1DHqAc5kEG72PIY7QSFRFz\ntLynqqoqqqqqxm9HhDmLZUCFlHKLEOJxYLuU8pAQIldK2Xatx4cdw9Y5C40mXhlrziKiDJRZ2TCS\nl20hQrBjlMc1Go3D0E1ZGk2CYqlnodFoEgctFhqNJiy0WGg0mrDQYqHRaMJCi4VGowkLLRYajSYs\ntFhoNJqw0GKh0WjCQouFRqMJCy0WGo0mLLRYaDSasNBiodFowkKLhUajCQstFhqNJiy0WGg0mrDQ\nYqHRaMJCi4VGowkLLRYajSYstFhoNJqw0GKh0WjCQouFRqMJCy0WGo0mLLRYaDSasNBiodFowkKL\nhUajCQstFhqNJiy0WGg0mrDQYqHRaMJCi4VGowkLLRYajSYstFhoNJqw0GKh0WjCImKxEEKsE0Ks\nEUJsuM5z/iHScTQajVoiEgshxHIAKeVO4/ayEZ7zCLAuknE0Go16IvUsHgRajb9rgbXDnyClfMp4\nTKPROJhIxSIXaAm5XRDh8TQajU3xROEYItIDbNy4Mfh3ZWUllZWVkR5So9EMo6qqiqqqqnG/Xkgp\nr/+EkROXLVLKrUKIJ4HtUsqdQoj1wAwp5eYRjvG6lPLOaxxfjmaDRqOJPkIIpJRhX+xH9SyklFuu\n8/DzQAWwE5gBbDeMyJVStoVrhEajsT8R5SyklAcBhBBrgDYp5SHjoR3mcwyPo0II8bVIxtJoNGoZ\nNQyJuQE6DNFolDDWMER3cGo0mrDQYqHRaMJCi4VGowkLLRYajSYsEl4sImlScfLYenw9/ljRYqHF\nQo+foOOPlYQXC41GEx5aLDQaTVjYoilLqQEaTQIzlqYs5WKh0WicgQ5DFCOEeFy1DYnE8CUew1kW\nMsbjbzB+nlQxfsj9o34PlYmF1R/SCONb+iFdw4a1wB2Kxl5ufAaq3n/LP//hSzyGsyxkjMdfA+ww\nZnaXGbctGz/k/rC+h0rEwuoPaYTxLf2QroPKGPDbUsqtQK6C938ZUGt8/rVWjT/CEo+jLgsZ4/HL\nQsasNW5bOX7woXBer8qzsPRDGgFLP6SREEIsM8VSwdjrgfcBpJSbzaUGLMZ0h8sUjQ+Kl4WUUm4J\nWS9mOcZnYiVj+R6qEouE/5CAfAVjmlQABUKIZSpyJoY41AkhWvjo90AFES8LGbEBAU/7QMh6MFYS\n9vdQZYIzYT8klV5FCE0hixdZulWDECIXqAE2AFuEEDOsHD+ENoZOljygWZEda6SU37F60LF+D1WJ\nRUJ/SATyJOuMhFO+1TkDAu93nfF3G3CDxeNvAH5m5Ew+B6y3eHyT5xkKQYPLQlqJEOIRc91aBbmz\nMX0PVYlFQn9IUsqtxokigRysT3T+iqH3PxfYZ/H4SCk7jN87CQhWzBm+xON1loW0ZHyjCvGkEKLG\nCMli+j0Y4f8f0/dQWVOWUTKrJZDgut6iwLEYey3wAoF4OR9YL6XcZaUNqjHe/xagQpEL/DiBzz/f\n6s9fMz50B6dGowkL3cGp0WjCQouFRqMJCy0WGo0mLLRYaDSasNBiodFowkKLhUajCQstFhqNJiy0\nWGg0mrD4//EuiNbbAqu2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7e54b89e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(4,4))\n",
    "km1.plot(x,plot_limits=[a,b], ax=ax)#, 'all', None,'b-')\n",
    "km2.plot(x,plot_limits=[a,b], ax=ax)#, 'all', None,'b--')\n",
    "km3.plot(x,plot_limits=[a,b], ax=ax)\n",
    "plt.xlabel('')\n",
    "plt.ylabel('')\n",
    "#plt.legend([\"$\\\\theta\\  =\\  0.5$\",\"$\\\\theta \\ = \\ 2$\",\"$\\\\theta \\ = \\ 5$\"],prop={'size':12},borderaxespad=0.)\n",
    "plt.ylim([-0.13,0.42])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Subfigure c: aperiodic sub-kernel $k_a$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.45, 1.05)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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sDnf2Bks2H+FMVS39OyfSoXWUy9otsDNRb13cIRYX9k8mPCSQvYcLyc4pcVm7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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7e51fc0650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(4,4))\n",
    "km1.plot(x,plot_limits=[a,b], ax=ax)#, 'all', None,'b-')\n",
    "km2.plot(x,plot_limits=[a,b], ax=ax)#, 'all', None,'b--')\n",
    "km3.plot(x,plot_limits=[a,b], ax=ax)\n",
    "plt.xlabel('')\n",
    "plt.ylabel('')\n",
    "plt.title('aperiodic')\n",
    "#plt.legend([\"$\\\\theta\\  =\\  0.5$\",\"$\\\\theta \\ = \\ 2$\",\"$\\\\theta \\ = \\ 5$\"],prop={'size':12},borderaxespad=0.)\n",
    "plt.ylim([-0.45,1.05])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}