test-conda/StableCompositionPeriodic.ipynb

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    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.gridspec as gridspec\n",
    "import pandas as pd\n",
    "import numpy.random as rnd\n",
    "from scipy.integrate import odeint,quad\n",
    "from scipy.stats import kde,beta\n",
    "import seaborn as sns\n",
    "%matplotlib inline\n",
    "from importlib import reload\n",
    "pi=np.pi\n",
    "\n",
    "\n",
    "from numpy import linalg as LA\n",
    "from scipy.linalg import expm\n",
    "from scipy.optimize import brentq\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import stablecompoper\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The monotype linear birth and death process in a periodic environment\n",
    "\n",
    "This process $(X(t),t\\ge 0)$ with values in $\\mathbb{N}$ is described by its\n",
    "time varying generator\n",
    "\\begin{equation}\n",
    "  L_t f(x) = x\\left[\\lambda(t)(f(x+1)-f(x)) + \\mu(t) (f(x-1)-f(x))\\right]\\,,\n",
    "\\end{equation}\n",
    "where $\\lambda,\\mu$ are non negative $T$-periodic functions.\n",
    "\n",
    "Let $Z(t)$ be the  point measure on $S$ describing the set of states\n",
    "(i.e. phases) of the individuals born before $t$ and still alive at\n",
    "time $t$ : if $Z(t) = \\sum_i \\delta_{s_i}$ then $<Z(t), f>=\n",
    "\\sum_i f(s_i)$. We have the convergence in $L^1$, when we start with $X(s)=1$ one individual of phase $s$,\n",
    "\\begin{equation}\n",
    "  \\lim_{n\\to +\\infty} e^{-\\alpha(nT-s)} <Z(nT),f> = h(s) \\int_S f(s)\\, d\\pi(s)\\,,\n",
    "\\end{equation}\n",
    "where the reproductive value of phase $s$ is the periodic function for $T=1$\n",
    "\\begin{equation}\n",
    "  h(s) = e^{\\alpha s -\\varphi(s)}\\,,\n",
    "\\end{equation}\n",
    "and the measure $\\pi$ is the stable composition law\n",
    "\\begin{equation}\n",
    "  \\boxed{\\pi(dt) = \\frac1{e^{A(T)} -1} \\lambda(t) e^{A(t)}\\, 1_{t\\in(0,T)}\\, dt\\,.}\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
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   "outputs": [
    {
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      "text/plain": [
       "interactive(children=(FloatSlider(value=0.8, continuous_update=False, description='lzero', max=4.0), FloatSlid…"
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     "data": {
      "text/plain": [
       "<function stablecompoper.nsestimdenszchi(lzero, muzero, T, N, coeff=1.0, estimnoyau=False)>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reload(stablecompoper)\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "interact(stablecompoper.nsestimdenszchi,\n",
    "         lzero=widgets.FloatSlider(min=0.0, max=4.0, step=0.1, value=0.8, continuous_update=False),\n",
    "                muzero=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.1, continuous_update=False),\n",
    "                  T=widgets.IntSlider(min=1, max=10, step=1, value=2, continuous_update=False),\n",
    "N=widgets.IntSlider(min=1, max=20, step=1, value=8, continuous_update=False),\n",
    "         coeff=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.5, continuous_update=False),\n",
    "                 )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il ne faut pas partir avec un nombre de périodes $N$ trop grand, ni une période $T$ trop grande, sinon la taille de l'échantillon est beaucoup trop grande. Pour avoir une bonne estimation de la densité par l'histogramme, il suffit d'avoir une taile d'échantillon au dessus de 2000.En considerant le cas constant, on voit qu'il faut à peu près prendre $e^{N T (\\lambda_0 -\\mu_0)}\\simeq 2000$ ce qui donne $NT \\simeq 7.6/(\\lambda_0 -\\mu_0) \\simeq 12$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cas constant \n",
    "\n",
    "Il suffit de prendre coeff=0. On trouve encore comme loi de composition stable $\\pi(dt) = C\\lambda(t) e^{A(t)} dt$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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