{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "18517e3a-e898-4a19-826b-c417fcbc5f47",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Continuous Ranked Probability Score (CRPS) for Ensembles"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54e7f1a6",
   "metadata": {},
   "source": [
    "General information about the CRPS, and how to use `crps_cdf` with a forecast expressed as a cumulative distribution function (CDF), is available in the tutorial \"Continuous Ranked Probability Score\". \n",
    "We recommend you work through that tutorial before looking at the `crps_for_ensembles` and this tutorial."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "595cd38f",
   "metadata": {},
   "source": [
    "In weather forecasting it is common to run an ensemble of deterministic models. Each ensemble member provides a deterministic forecast. The idea is that the combined ensemble forecasts can be used to estimate a probability density function (PDF), and hence a CDF. \n",
    "\n",
    "The `crps_for_ensemble` function interprets ensemble forecast input as a CDF in one of two ways and calculates the CRPS as demonstrated below.\n",
    "\n",
    "A CDF is a function of the probability of non-exceedence. It is relevant to real-valued parameters such as temperature, rainfall amount, wind speed etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "9859bbe2",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scores.probability import crps_for_ensemble\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import numpy\n",
    "import xarray"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c34ff15a",
   "metadata": {},
   "source": [
    "## Using the `crps_for_ensemble` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6e5251e4",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Uncomment the following line and run to read the documentation regarding the crps_for_ensemble function in scores.\n",
    "# crps_for_ensemble?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "149a6ba4",
   "metadata": {},
   "source": [
    "For the purpose of this tutorial, we will create a set of 10 equally likely forecasts of temperature which we will compare to a single verifying observation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "244c5cf2",
   "metadata": {},
   "outputs": [],
   "source": [
    "# In this example, each ensemble member is given a name 0 to 9\n",
    "ensemble_forecast = xarray.DataArray(\n",
    "    [1.2, 2.0, 2.7, 2.9, 3.0, 3.0, 3.1, 3.2, 3.8, 5.0],\n",
    "\tcoords=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], dims=[\"ensemble_member\"])\n",
    "\n",
    "# The observation is assumed to be 4.5\n",
    "obs_array = xarray.DataArray(4.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e2078a9f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fdf01b0d610>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 400x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# The ensemble forecast can be converted to a CDF\n",
    "# When converted in a naive way we get the empirical CDF illustrated here. The larger the ensemble the smoother and more sensible this will be.\n",
    "# The plot below also shows the CDF corresponding to the observation, and the area corrsponding to the CRPS.\n",
    "fcst_thresholds = numpy.linspace(0, 7, 700) \n",
    "empirical_cdf = xarray.DataArray(coords={'temperature': fcst_thresholds}, data=[0]*120 + [0.1]*80 + [0.2]*70 + [0.3]*20 +[0.7]*30 + [0.8]*60 + [0.9]*120 +[1]*200)\n",
    "observed_cdf = numpy.heaviside(fcst_thresholds-4.5, 1)\n",
    "plt.figure(figsize=(4, 3))\n",
    "plt.plot(fcst_thresholds, observed_cdf, label='Observation')\n",
    "plt.plot(fcst_thresholds, empirical_cdf, label='Forecast')\n",
    "plt.fill_between(fcst_thresholds, empirical_cdf, observed_cdf, color='gray', label='area for CRPS')\n",
    "plt.title(\"Empirical CDF based on Ensemble\")\n",
    "plt.xlabel(\"Temperature\")\n",
    "plt.ylabel(\"Probability\")\n",
    "plt.legend(loc=\"upper left\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c11d30f5",
   "metadata": {},
   "outputs": [
    {
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       ".xr-section-summary > span {\n",
       "  display: inline-block;\n",
       "  padding-left: 0.5em;\n",
       "}\n",
       "\n",
       ".xr-section-summary-in:disabled + label {\n",
       "  color: var(--xr-font-color2);\n",
       "}\n",
       "\n",
       ".xr-section-summary-in + label:before {\n",
       "  display: inline-block;\n",
       "  content: '►';\n",
       "  font-size: 11px;\n",
       "  width: 15px;\n",
       "  text-align: center;\n",
       "}\n",
       "\n",
       ".xr-section-summary-in:disabled + label:before {\n",
       "  color: var(--xr-disabled-color);\n",
       "}\n",
       "\n",
       ".xr-section-summary-in:checked + label:before {\n",
       "  content: '▼';\n",
       "}\n",
       "\n",
       ".xr-section-summary-in:checked + label > span {\n",
       "  display: none;\n",
       "}\n",
       "\n",
       ".xr-section-summary,\n",
       ".xr-section-inline-details {\n",
       "  padding-top: 4px;\n",
       "  padding-bottom: 4px;\n",
       "}\n",
       "\n",
       ".xr-section-inline-details {\n",
       "  grid-column: 2 / -1;\n",
       "}\n",
       "\n",
       ".xr-section-details {\n",
       "  display: none;\n",
       "  grid-column: 1 / -1;\n",
       "  margin-bottom: 5px;\n",
       "}\n",
       "\n",
       ".xr-section-summary-in:checked ~ .xr-section-details {\n",
       "  display: contents;\n",
       "}\n",
       "\n",
       ".xr-array-wrap {\n",
       "  grid-column: 1 / -1;\n",
       "  display: grid;\n",
       "  grid-template-columns: 20px auto;\n",
       "}\n",
       "\n",
       ".xr-array-wrap > label {\n",
       "  grid-column: 1;\n",
       "  vertical-align: top;\n",
       "}\n",
       "\n",
       ".xr-preview {\n",
       "  color: var(--xr-font-color3);\n",
       "}\n",
       "\n",
       ".xr-array-preview,\n",
       ".xr-array-data {\n",
       "  padding: 0 5px !important;\n",
       "  grid-column: 2;\n",
       "}\n",
       "\n",
       ".xr-array-data,\n",
       ".xr-array-in:checked ~ .xr-array-preview {\n",
       "  display: none;\n",
       "}\n",
       "\n",
       ".xr-array-in:checked ~ .xr-array-data,\n",
       ".xr-array-preview {\n",
       "  display: inline-block;\n",
       "}\n",
       "\n",
       ".xr-dim-list {\n",
       "  display: inline-block !important;\n",
       "  list-style: none;\n",
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       "\n",
       ".xr-dim-list li {\n",
       "  display: inline-block;\n",
       "  padding: 0;\n",
       "  margin: 0;\n",
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       "\n",
       ".xr-dim-list:before {\n",
       "  content: '(';\n",
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       "\n",
       ".xr-dim-list:after {\n",
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       "\n",
       ".xr-dim-list li:not(:last-child):after {\n",
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       "\n",
       ".xr-has-index {\n",
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       "\n",
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       "\n",
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       "\n",
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       "\n",
       ".xr-var-dims {\n",
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       "}\n",
       "\n",
       ".xr-var-dtype {\n",
       "  grid-column: 3;\n",
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       "\n",
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       "\n",
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       "  grid-column: 2 / 5;\n",
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       "\n",
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       ".xr-attrs dt:hover {\n",
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       "\n",
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       ".xr-var-data,\n",
       ".xr-index-data {\n",
       "  display: none;\n",
       "  background-color: var(--xr-background-color) !important;\n",
       "  padding-bottom: 5px !important;\n",
       "}\n",
       "\n",
       ".xr-var-attrs-in:checked ~ .xr-var-attrs,\n",
       ".xr-var-data-in:checked ~ .xr-var-data,\n",
       ".xr-index-data-in:checked ~ .xr-index-data {\n",
       "  display: block;\n",
       "}\n",
       "\n",
       ".xr-var-data > table {\n",
       "  float: right;\n",
       "}\n",
       "\n",
       ".xr-var-name span,\n",
       ".xr-var-data,\n",
       ".xr-index-name div,\n",
       ".xr-index-data,\n",
       ".xr-attrs {\n",
       "  padding-left: 25px !important;\n",
       "}\n",
       "\n",
       ".xr-attrs,\n",
       ".xr-var-attrs,\n",
       ".xr-var-data,\n",
       ".xr-index-data {\n",
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       "\n",
       "dl.xr-attrs {\n",
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       "\n",
       ".xr-attrs dt,\n",
       ".xr-attrs dd {\n",
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       "  margin: 0;\n",
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       "  padding-right: 10px;\n",
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       "}\n",
       "\n",
       ".xr-attrs dt {\n",
       "  font-weight: normal;\n",
       "  grid-column: 1;\n",
       "}\n",
       "\n",
       ".xr-attrs dt:hover span {\n",
       "  display: inline-block;\n",
       "  background: var(--xr-background-color);\n",
       "  padding-right: 10px;\n",
       "}\n",
       "\n",
       ".xr-attrs dd {\n",
       "  grid-column: 2;\n",
       "  white-space: pre-wrap;\n",
       "  word-break: break-all;\n",
       "}\n",
       "\n",
       ".xr-icon-database,\n",
       ".xr-icon-file-text2,\n",
       ".xr-no-icon {\n",
       "  display: inline-block;\n",
       "  vertical-align: middle;\n",
       "  width: 1em;\n",
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       "  stroke-width: 0;\n",
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       "  fill: currentColor;\n",
       "}\n",
       "</style><pre class='xr-text-repr-fallback'>&lt;xarray.DataArray ()&gt;\n",
       "array(1.111)</pre><div class='xr-wrap' style='display:none'><div class='xr-header'><div class='xr-obj-type'>xarray.DataArray</div><div class='xr-array-name'></div></div><ul class='xr-sections'><li class='xr-section-item'><div class='xr-array-wrap'><input id='section-772b0acf-9c3e-425b-b3bc-202529c48a33' class='xr-array-in' type='checkbox' checked><label for='section-772b0acf-9c3e-425b-b3bc-202529c48a33' title='Show/hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-array-preview xr-preview'><span>1.111</span></div><div class='xr-array-data'><pre>array(1.111)</pre></div></div></li><li class='xr-section-item'><input id='section-8a3ef709-110c-4052-8848-0df9f708bf8f' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-8a3ef709-110c-4052-8848-0df9f708bf8f' class='xr-section-summary'  title='Expand/collapse section'>Coordinates: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'></ul></div></li><li class='xr-section-item'><input id='section-9f08f087-24cf-4d1b-867a-c8042910be7a' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-9f08f087-24cf-4d1b-867a-c8042910be7a' class='xr-section-summary'  title='Expand/collapse section'>Indexes: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'></ul></div></li><li class='xr-section-item'><input id='section-8bdfe824-d912-42d0-ae33-d00bc0023867' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-8bdfe824-d912-42d0-ae33-d00bc0023867' class='xr-section-summary'  title='Expand/collapse section'>Attributes: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><dl class='xr-attrs'></dl></div></li></ul></div></div>"
      ],
      "text/plain": [
       "<xarray.DataArray ()>\n",
       "array(1.111)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Specifying method='ecdf' assumes the empirical CDF\n",
    "crps_for_ensemble(ensemble_forecast, obs_array, ensemble_member_dim='ensemble_member', method='ecdf').round(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "1d296010",
   "metadata": {},
   "outputs": [
    {
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       "array(1.056)</pre><div class='xr-wrap' style='display:none'><div class='xr-header'><div class='xr-obj-type'>xarray.DataArray</div><div class='xr-array-name'></div></div><ul class='xr-sections'><li class='xr-section-item'><div class='xr-array-wrap'><input id='section-1752d3f1-e890-429e-9e0a-631b8a398a15' class='xr-array-in' type='checkbox' checked><label for='section-1752d3f1-e890-429e-9e0a-631b8a398a15' title='Show/hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-array-preview xr-preview'><span>1.056</span></div><div class='xr-array-data'><pre>array(1.056)</pre></div></div></li><li class='xr-section-item'><input id='section-0ee1ad7a-c205-462c-9ad9-4a159de5f9ab' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-0ee1ad7a-c205-462c-9ad9-4a159de5f9ab' class='xr-section-summary'  title='Expand/collapse section'>Coordinates: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'></ul></div></li><li class='xr-section-item'><input id='section-7aea9360-48f7-4568-93a1-847ca9649f14' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-7aea9360-48f7-4568-93a1-847ca9649f14' class='xr-section-summary'  title='Expand/collapse section'>Indexes: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'></ul></div></li><li class='xr-section-item'><input id='section-aad01191-7a62-4338-adfc-5a4608017852' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-aad01191-7a62-4338-adfc-5a4608017852' class='xr-section-summary'  title='Expand/collapse section'>Attributes: <span>(0)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><dl class='xr-attrs'></dl></div></li></ul></div></div>"
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   "source": [
    "# Specifying method='fair' gives an approximated CRPS assuming that the ensemble values can be interpreted as a random sample from an underlying predictive distribution\n",
    "crps_for_ensemble(ensemble_forecast, obs_array, ensemble_member_dim='ensemble_member', method='fair').round(3)"
   ]
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    "### Things to try next\n",
    "\n",
    "* Use crps_for_ensemble with some real forecasts with extra dimensions in space and/or time.   \n",
    "* Read about the 'fair' method in C. Ferro (2014), \"Fair scores for ensemble forecasts\", Q J R Meteorol Soc\n",
    "        140(683):1917-1923. https://doi.org/10.1002/qj.2270"
   ]
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