docs/notebooks/random_variates.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Random Variates\n",
    "\n",
    "When we perform a fit or load and analysis result, the parmeters of our model become distributions in the AnalysisResults object. These are actaully instantiactions of the RandomVaraiates class. \n",
    "\n",
    "While we have covered most of the functionality of RandomVariates in the AnalysisResults section, we want to highlight a few of the details here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:29.845385Z",
     "start_time": "2020-04-13T07:58:27.569112Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jburgess/coding/tml/threeml/threeML/__init__.py:12: UserWarning: No DISPLAY variable set. Using backend for graphics without display (Agg)\n",
      "  warnings.warn(\"No DISPLAY variable set. Using backend for graphics without display (Agg)\")\n",
      "/Users/jburgess/coding/tml/astromodels/astromodels/functions/function.py:139: YAMLLoadWarning: calling yaml.load() without Loader=... is deprecated, as the default Loader is unsafe. Please read https://msg.pyyaml.org/load for full details.\n",
      "  function_definition = my_yaml.load(dct['__doc__'])\n",
      "/Users/jburgess/coding/tml/astromodels/astromodels/core/parameter.py:555: UserWarning: We have set the min_value of K to 1e-99 because there was a postive transform\n",
      "  warnings.warn('We have set the min_value of %s to 1e-99 because there was a postive transform' % self.path)\n",
      "/Users/jburgess/coding/tml/astromodels/astromodels/core/parameter.py:555: UserWarning: We have set the min_value of xc to 1e-99 because there was a postive transform\n",
      "  warnings.warn('We have set the min_value of %s to 1e-99 because there was a postive transform' % self.path)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Configuration read from /Users/jburgess/.threeML/threeML_config.yml\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "WARNING RuntimeWarning: numpy.ufunc size changed, may indicate binary incompatibility. Expected 192 from C header, got 216 from PyObject\n",
      "\n",
      "WARNING: AstropyDeprecationWarning: The timefunc function is deprecated and may be removed in a future version.\n",
      "        Use astroquery.utils.timer.timefunc instead. [astropy.utils.timer]\n",
      "WARNING:astropy:AstropyDeprecationWarning: The timefunc function is deprecated and may be removed in a future version.\n",
      "        Use astroquery.utils.timer.timefunc instead.\n",
      "WARNING: AstropyDeprecationWarning: The timefunc function is deprecated and may be removed in a future version.\n",
      "        Use astroquery.utils.timer.timefunc instead. [astroquery.vo_conesearch.conesearch]\n",
      "WARNING:astropy:AstropyDeprecationWarning: The timefunc function is deprecated and may be removed in a future version.\n",
      "        Use astroquery.utils.timer.timefunc instead.\n",
      "\n",
      "WARNING YAMLLoadWarning: calling yaml.load() without Loader=... is deprecated, as the default Loader is unsafe. Please read https://msg.pyyaml.org/load for full details.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "\n",
    "from threeML import *\n",
    "\n",
    "%matplotlib inline\n",
    "jtplot.style(context=\"talk\", fscale=1, ticks=True, grid=False)\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use(\"mike\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's load back our fit of the line + gaussian from the AnalysisResults section."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:29.936536Z",
     "start_time": "2020-04-13T07:58:29.848459Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "WARNING YAMLLoadWarning: calling yaml.load() without Loader=... is deprecated, as the default Loader is unsafe. Please read https://msg.pyyaml.org/load for full details.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "ar = load_analysis_results('test_mle.fits')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we display our fit, we can see the **parameter paths** of the model. What if we want specific information on a parameter(s)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:29.992452Z",
     "start_time": "2020-04-13T07:58:29.939128Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best fit values:\n",
      "\n"
     ]
    },
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>result</th>\n",
       "      <th>unit</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>parameter</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>fake.spectrum.main.composite.a_1</th>\n",
       "      <td>(9 +/- 4) x 10^-3</td>\n",
       "      <td>1 / (cm2 keV2 s)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fake.spectrum.main.composite.b_1</th>\n",
       "      <td>1.76 +/- 0.11</td>\n",
       "      <td>1 / (cm2 keV s)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fake.spectrum.main.composite.F_2</th>\n",
       "      <td>(3.2 +/- 0.4) x 10</td>\n",
       "      <td>1 / (cm2 s)</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fake.spectrum.main.composite.mu_2</th>\n",
       "      <td>(2.521 +/- 0.012) x 10</td>\n",
       "      <td>keV</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fake.spectrum.main.composite.sigma_2</th>\n",
       "      <td>1.09 +/- 0.09</td>\n",
       "      <td>keV</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                      result              unit\n",
       "parameter                                                                     \n",
       "fake.spectrum.main.composite.a_1           (9 +/- 4) x 10^-3  1 / (cm2 keV2 s)\n",
       "fake.spectrum.main.composite.b_1               1.76 +/- 0.11   1 / (cm2 keV s)\n",
       "fake.spectrum.main.composite.F_2          (3.2 +/- 0.4) x 10       1 / (cm2 s)\n",
       "fake.spectrum.main.composite.mu_2     (2.521 +/- 0.012) x 10               keV\n",
       "fake.spectrum.main.composite.sigma_2           1.09 +/- 0.09               keV"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Correlation matrix:\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table id=\"table4383271184\">\n",
       "<tr><td>1.00</td><td>-0.85</td><td>0.00</td><td>-0.02</td><td>-0.00</td></tr>\n",
       "<tr><td>-0.85</td><td>1.00</td><td>-0.05</td><td>0.02</td><td>-0.08</td></tr>\n",
       "<tr><td>0.00</td><td>-0.05</td><td>1.00</td><td>-0.24</td><td>-0.15</td></tr>\n",
       "<tr><td>-0.02</td><td>0.02</td><td>-0.24</td><td>1.00</td><td>-0.13</td></tr>\n",
       "<tr><td>-0.00</td><td>-0.08</td><td>-0.15</td><td>-0.13</td><td>1.00</td></tr>\n",
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      "text/plain": [
       " 1.00 -0.85  0.00 -0.02 -0.00\n",
       "-0.85  1.00 -0.05  0.02 -0.08\n",
       " 0.00 -0.05  1.00 -0.24 -0.15\n",
       "-0.02  0.02 -0.24  1.00 -0.13\n",
       "-0.00 -0.08 -0.15 -0.13  1.00"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Values of -log(likelihood) at the minimum:\n",
      "\n"
     ]
    },
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>-log(likelihood)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>sim_data</th>\n",
       "      <td>24.81213</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>total</th>\n",
       "      <td>24.81213</td>\n",
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       "          -log(likelihood)\n",
       "sim_data          24.81213\n",
       "total             24.81213"
      ]
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     "output_type": "display_data"
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Values of statistical measures:\n",
      "\n"
     ]
    },
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>statistical measures</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>AIC</th>\n",
       "      <td>60.987896</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>BIC</th>\n",
       "      <td>69.184375</td>\n",
       "    </tr>\n",
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      "text/plain": [
       "     statistical measures\n",
       "AIC             60.987896\n",
       "BIC             69.184375"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ar.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at the normalization of the gaussian. To access the parameter, we take the parameter path, and we want to get the variates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:29.997305Z",
     "start_time": "2020-04-13T07:58:29.994349Z"
    }
   },
   "outputs": [],
   "source": [
    "norm = ar.get_variates('fake.spectrum.main.composite.F_2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, norm is a RandomVariate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.003076Z",
     "start_time": "2020-04-13T07:58:29.999307Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "threeML.random_variates.RandomVariates"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(norm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is essentially a wrapper around numpy NDArray with a few added properties. It is an array of samples. In the MLE case, they are samples from the covariance matrix (this is not at all a marginal distribution, but the parameter \"knows\" about the entire fit, i.e., it is *not* a profile) and in the Bayesian case, these are samples from the posterior (this is a marginal).\n",
    "\n",
    "The output representation for an RV are its 68% equal-tail and HPD uncertainties."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.009706Z",
     "start_time": "2020-04-13T07:58:30.004952Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "equal-tail: (3.2 +/- 0.4) x 10, hpd: (3.2 +/- 0.4) x 10"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can access these directly, and to any desired confidence level."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.016182Z",
     "start_time": "2020-04-13T07:58:30.012731Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(24.422018993408035, 39.46392609276965)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.equal_tail_interval(cl=0.95)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.022114Z",
     "start_time": "2020-04-13T07:58:30.018512Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(29.611419389249594, 34.79257615745331)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.highest_posterior_density_interval(cl=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As stated above, the RV is made from samples. We can histogram them to show this explicitly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.327737Z",
     "start_time": "2020-04-13T07:58:30.023779Z"
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    {
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\n",
      "text/plain": [
       "<Figure size 748.8x655.2 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.hist(norm.samples,bins=50, ec='k', fc='w', lw=1.2);\n",
    "ax.set_xlabel('norm');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can easily transform the RV through propagation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.334980Z",
     "start_time": "2020-04-13T07:58:30.329484Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "equal-tail: 1.51 -0.06 +0.05, hpd: 1.51 +/- 0.05"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_norm = np.log10(norm)\n",
    "log_norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-13T07:58:30.686272Z",
     "start_time": "2020-04-13T07:58:30.336859Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 748.8x655.2 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.hist(log_norm.samples,bins=50,ec='k', fc='w', lw=1.2);\n",
    "ax.set_xlabel('log norm');"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-01-05T05:37:36.073331Z",
     "start_time": "2018-01-05T05:37:36.067789Z"
    }
   },
   "source": [
    ".. note::\n",
    "    Some operations will destroy the RV by accessing only its NDArray substructure. For example, using an RV with astropy units will return an array of samples with the given units. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "celltoolbar": "Tags",
  "kernelspec": {
   "display_name": "3ml",
   "language": "python",
   "name": "3ml"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7"
  },
  "toc": {
   "colors": {
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