{
  "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# GoF from binned cost functions\n",
    "\n",
    "The builtin cost functions for binned data, `BinnedNLL` and `ExtendedBinnedNLL` have a minimum value which is asymptotically chi2-distributed and thus can be used as a goodness-of-fit statistic. This example shows, that one still needs a large number of entries in each bin to reach the asymptotic regime."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from iminuit import Minuit\n",
    "from iminuit.cost import BinnedNLL, ExtendedBinnedNLL, LeastSquares\n",
    "import numpy as np\n",
    "from numba_stats import norm, expon\n",
    "import matplotlib.pyplot as plt\n",
    "import joblib\n",
    "from scipy.stats import chi2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def generate(n, seed):\n",
    "    rng = np.random.default_rng(seed)\n",
    "    s = rng.normal(1, 0.1, size=rng.poisson(n))\n",
    "    b = rng.exponential(size=rng.poisson(n))\n",
    "    x = np.append(s, b)\n",
    "    return x[(x > 0) & (x < 2)]\n",
    "\n",
    "x = generate(1000, 1)\n",
    "plt.hist(x, bins=20, range=(0, 2));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "@joblib.delayed\n",
    "def run(n, seed):\n",
    "    x = generate(n, seed)\n",
    "    xrange = (0, 2)\n",
    "    w, xe = np.histogram(x, bins=20, range=xrange)\n",
    "\n",
    "    def model1(x, z, mu, sigma, tau):\n",
    "        return z * norm.cdf(x, mu, sigma) / np.diff(norm.cdf(xrange, mu, sigma)) + (1 - z) * expon.cdf(x, 0, tau) / np.diff(expon.cdf(xrange, 0, tau))\n",
    "\n",
    "    def model2(x, s, b, mu, sigma, tau):\n",
    "        return s * n * norm.cdf(x, mu, sigma) + b * n * expon.cdf(x, 0, tau)\n",
    "\n",
    "    m = [\n",
    "        Minuit(BinnedNLL(w, xe, model1), z=0.5, mu=0.5, sigma=0.5, tau=0.5),\n",
    "        Minuit(ExtendedBinnedNLL(w, xe, model2), s=1, b=1, mu=0.5, sigma=0.5, tau=0.5),\n",
    "    ]\n",
    "    for mi in m:\n",
    "        mi.limits[\"mu\"] = (0, 2)\n",
    "        mi.limits[\"sigma\", \"tau\"] = (0.1, None)\n",
    "    m[0].limits[\"z\"] = (0, 1)\n",
    "    m[1].limits[\"s\", \"b\"] = (0, None)\n",
    "    r = []\n",
    "    for mi in m:\n",
    "        mi.migrad()\n",
    "        if mi.valid:\n",
    "            pvalue = 1 - chi2(mi.fcn._fcn.ndata - mi.nfit).cdf(mi.fval)\n",
    "            r.append(pvalue)\n",
    "        else:\n",
    "            r.append(np.nan)\n",
    "    return r\n",
    "\n",
    "pvalues = {}\n",
    "for n in (20, 100, 1000, 10000):\n",
    "    pvalues[n] = np.array(joblib.Parallel(-1)(run(n, i) for i in range(500)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x600 with 8 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(2, 4, figsize=(15, 6), constrained_layout=True)\n",
    "for i, (ni, vi) in enumerate(pvalues.items()):\n",
    "    ax[0, i].hist(vi[:, 0])\n",
    "    ax[1, i].hist(vi[:, 1])\n",
    "    ax[0, i].set_title(f\"n = {ni}, failed fits = {np.sum(np.isnan(vi[:, 0]))}\")\n",
    "    ax[1, i].set_title(f\"n = {ni}, failed fits = {np.sum(np.isnan(vi[:, 1]))}\")\n",
    "fig.supxlabel(\"pvalue\");"
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "bdbf20ff2e92a3ae3002db8b02bd1dd1b287e934c884beb29a73dced9dbd0fa3"
  },
  "kernelspec": {
   "display_name": "Python 3.8.12 ('venv': venv)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.8 (main, Oct 13 2022, 09:48:40) [Clang 14.0.0 (clang-1400.0.29.102)]"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
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