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philcarmona
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 5,
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -26,7 +26,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -53,7 +53,7 @@
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"time $t$ : if $Z(t) = \\sum_i \\delta_{s_i}$ then $<Z(t), f>=\n",
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"\\sum_i f(s_i)$. We have the convergence in $L^1$, when we start with $X(s)=1$ one individual of phase $s$,\n",
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"\\begin{equation}\n",
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" \\lim_{n\\to +\\infty} e^{-\\alpha(nT-s)} <Z(nT),f> = h(s) \\int_S f(s)\\, d\\pi(s)\\,,\n",
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" \\lim_{n\\to +\\infty} e^{-\\alpha(nT-s)} <Z(nT),f> = h(s) \\int_S f(t)\\, d\\pi(t)\\,,\n",
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"\\end{equation}\n",
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"where the reproductive value of phase $s$ is the periodic function for $T=1$\n",
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"\\begin{equation}\n",
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@ -65,9 +65,34 @@
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"\\end{equation}\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The process is one dimensional, and the death rate is constant\n",
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"$\\mu(t)=\\mu_0$ and the birth rate is\n",
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"\\begin{equation}\n",
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" \\lambda(t) = \\lambda_0 (1 + c \\cos(2\\pi t/T))\\,.\n",
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"\\end{equation}\n",
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"The stable composition law is thus\n",
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"\\begin{equation}\n",
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"\\pi(dt) = \\frac1{e^{A(T)} -1} \\lambda(t) e^{A(t)}\\, 1_{t\\in(0,T)}\\,\n",
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"dt\\,,\n",
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"\\end{equation}\n",
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"with\n",
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"\\begin{equation}\n",
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" A(t) = \\lambda_0 (t + \\frac{ c T}{2 \\pi} \\sin(2\\pi t/T) )\n",
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"\\end{equation}\n",
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"We perform a simulation of the linear birth and death process for $N$\n",
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"periods, and we keep the phase, the birth dates modulo $T$, of the\n",
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"living individuals at time $N T$. We wait until the first non extinct\n",
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"population, and then we plot its histogram against the true density\n",
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"$\\pi$ and against the birth rate $\\lambda(t)$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 4,
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"metadata": {
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"scrolled": true
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},
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@ -75,7 +100,7 @@
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "363db6156e864761ac23dd6594f107cb",
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"model_id": "8554deb58f8549b4a98f2704c1e40090",
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"version_major": 2,
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"version_minor": 0
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},
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@ -92,38 +117,132 @@
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"<function stablecompoper.nsestimdenszchi(lzero, muzero, T, N, coeff=1.0, estimnoyau=False, image=False)>"
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]
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},
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"execution_count": 7,
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"reload(stablecompoper)\n",
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"from ipywidgets import interact, interactive, fixed, interact_manual\n",
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"import ipywidgets as widgets\n",
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"interact(stablecompoper.nsestimdenszchi,\n",
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" lzero=widgets.FloatSlider(min=0.0, max=4.0, step=0.1, value=0.8, continuous_update=False),\n",
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" muzero=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.1, continuous_update=False),\n",
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" T=widgets.IntSlider(min=1, max=10, step=1, value=2, continuous_update=False),\n",
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"N=widgets.IntSlider(min=1, max=20, step=1, value=8, continuous_update=False),\n",
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" coeff=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.5, continuous_update=False),\n",
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" )\n"
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"#reload(stablecompoper)\n",
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"#from ipywidgets import interact, interactive, fixed, interact_manual\n",
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"#import ipywidgets as widgets\n",
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"#interact(stablecompoper.nsestimdenszchi,\n",
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"# lzero=widgets.FloatSlider(min=0.0, max=4.0, step=0.1, value=0.8, continuous_update=False),\n",
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"# muzero=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.1, continuous_update=False),\n",
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"# T=widgets.IntSlider(min=1, max=10, step=1, value=2, continuous_update=False),\n",
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"#N=widgets.IntSlider(min=1, max=20, step=1, value=8, continuous_update=False),\n",
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"# coeff=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.5, continuous_update=False),\n",
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"# )\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"#reload(stablecompoper)\n",
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"#from IPython.display import display\n",
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"#w=interactive(stablecompoper.nsestimdenszchi,\n",
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"# lzero=widgets.FloatSlider(min=0.0, max=4.0, step=0.1, value=0.8, continuous_update=False),\n",
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"# muzero=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.1, continuous_update=False),\n",
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"# T=widgets.IntSlider(min=1, max=10, step=1, value=2, continuous_update=False),\n",
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"#N=widgets.IntSlider(min=1, max=20, step=1, value=8, continuous_update=False),\n",
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"# coeff=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.5, continuous_update=False),\n",
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"# )\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "c12f3784d53a4c7994ab875ce1163f87",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"interactive(children=(FloatSlider(value=0.8, continuous_update=False, description='lzero', max=4.0), FloatSlid…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"#display(w)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"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 $NT \\simeq 7.6/(\\lambda_0 -\\mu_0) \\simeq 12$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Cas constant \n",
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"#### Remarque\n",
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"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 $NT \\simeq 7.6/(\\lambda_0 -\\mu_0) \\simeq 12$\n",
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"\n",
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"Il suffit de prendre coeff=0. On trouve encore comme loi de composition stable $\\pi(dt) = C\\lambda(t) e^{A(t)} dt$"
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"Pour obtenir le cas où les taux de naissance et de mort sont constants, \n",
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"i l suffit de prendre c=0."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "83cf7c407ce24ab6bf5e6f836c282f6e",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"GridspecLayout(children=(FloatSlider(value=0.8, continuous_update=False, description='$\\\\lambda_0$', layout=La…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "dc265442f70b47cfa273d0add5e7d254",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"Output()"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"from ipywidgets import GridspecLayout,Layout,Button, AppLayout,TwoByTwoLayout,interactive_output\n",
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"import ipywidgets as widgets\n",
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"def create_expanded_button(description, button_style):\n",
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" return Button(description=description, button_style=button_style, layout=Layout(height='auto', width='auto'))\n",
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"grid = GridspecLayout(3, 3)\n",
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"blzero=widgets.FloatSlider(min=0.0, max=4.0, step=0.1, value=0.8, continuous_update=False,description=r'$\\lambda_0$')\n",
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"bmuzero=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.1, continuous_update=False,description=r'$\\mu_0$')\n",
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"bT=widgets.IntSlider(min=1, max=10, step=1, value=2, continuous_update=False,description='T')\n",
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"bN=widgets.IntSlider(min=1, max=20, step=1, value=8, continuous_update=False,description='N')\n",
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"bcoeff=widgets.FloatSlider(min=0.0, max=2.0, step=0.1, value=0.5, continuous_update=False,description='c')\n",
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"\n",
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"grid[0,0]=blzero\n",
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"grid[0,1]=bmuzero\n",
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"grid[0,2]=bcoeff\n",
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"grid[1,0]=bT\n",
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"grid[1,1]=bN\n",
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"w=interactive_output(stablecompoper.nsestimdenszchi,{'lzero':blzero,'muzero':bmuzero,'T':bT,'N':bN,'coeff':bcoeff})\n",
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"display(grid,w)\n",
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"#grid\n"
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]
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},
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{
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