{
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   "cell_type": "markdown",
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    "# Graphical model assessment: Predictive ECDFs\n",
    "\n",
    "**[Dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/singer_transcript_counts.csv)**\n",
    "\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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       "     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
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Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(\"1001\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error() {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (var i = 0; i < css_urls.length; i++) {\n      var url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      var element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };var element = document.getElementById(\"1001\");\n  if (element == null) {\n    console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n    return false;\n  }\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  \n  var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.4.0.min.js\"];\n  var css_urls = [];\n  \n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    function(Bokeh) {\n    \n    \n    }\n  ];\n\n  function run_inline_js() {\n    \n    if (root.Bokeh !== undefined || force === true) {\n      \n    for (var i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\n    if (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import warnings\n",
    "\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import numba\n",
    "import scipy.optimize\n",
    "import scipy.stats as st\n",
    "\n",
    "import bebi103\n",
    "\n",
    "import bokeh_catplot\n",
    "\n",
    "import bokeh.io\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "We continue with graphical model assessment of repeated measurements with the Rex1 mRNA counts from the [Singer, et al. paper](https://doi.org/10.1016/j.molcel.2014.06.029). Before we begin, we need to obtain the necessary MLEs for our two models, the single Negative Binomial model, and the mixture of two Negative Binomials model. You have seen this many times before, so you can skip this large code block (which gets us MLEs for the parameters) and move on to the next section. Just recall that the Rex1 mRNA counts are stored in a Numpy array `n` and the parameters for the single Negative Binomial model are in variables `alpha` and `b`. The parameters for the mixture model are stored as `alpha1`, `b1`, `alpha2`, `b2`, and `w`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def log_like_iid_nbinom(params, n):\n",
    "    \"\"\"Log likelihood for i.i.d. NBinom measurements, parametrized\n",
    "    by alpha, b=1/beta.\"\"\"\n",
    "    alpha, b = params\n",
    "\n",
    "    if alpha <= 0 or b <= 0:\n",
    "        return -np.inf\n",
    "\n",
    "    return np.sum(st.nbinom.logpmf(n, alpha, 1/(1+b)))\n",
    "\n",
    "\n",
    "def mle_iid_nbinom(n):\n",
    "    \"\"\"Perform maximum likelihood estimates for parameters for i.i.d.\n",
    "    NBinom measurements, parametrized by alpha, b=1/beta\"\"\"\n",
    "    with warnings.catch_warnings():\n",
    "        warnings.simplefilter(\"ignore\")\n",
    "\n",
    "        res = scipy.optimize.minimize(\n",
    "            fun=lambda params, n: -log_like_iid_nbinom(params, n),\n",
    "            x0=np.array([3, 3]),\n",
    "            args=(n,),\n",
    "            method='Powell'\n",
    "        )\n",
    "\n",
    "    if res.success:\n",
    "        return res.x\n",
    "    else:\n",
    "        raise RuntimeError('Convergence failed with message', res.message)\n",
    "        \n",
    "\n",
    "def initial_guess_mix(n, w_guess):\n",
    "    \"\"\"Generate initial guess for mixture model.\"\"\"\n",
    "    n_low = n[n < np.percentile(n, 100*w_guess)]\n",
    "    n_high = n[n >= np.percentile(n, 100*w_guess)]\n",
    "    \n",
    "    alpha1, b1 = mle_iid_nbinom(n_low)\n",
    "    alpha2, b2 = mle_iid_nbinom(n_high)\n",
    "    \n",
    "    return alpha1, b1, alpha2, b2\n",
    "\n",
    "\n",
    "def log_like_mix(alpha1, b1, alpha2, b2, w, n):\n",
    "    \"\"\"Log-likeihood of binary Negative Binomial mixture model.\"\"\"\n",
    "    # Fix nonidentifieability be enforcing values of w\n",
    "    if w < 0 or w > 1:\n",
    "        return -np.inf\n",
    "    \n",
    "    # Physical bounds on parameters\n",
    "    if alpha1 < 0 or alpha2 < 0 or b1 < 0 or b2 < 0:\n",
    "        return -np.inf\n",
    "\n",
    "    logx1 = st.nbinom.logpmf(n, alpha1, 1/(1+b1))\n",
    "    logx2 = st.nbinom.logpmf(n, alpha2, 1/(1+b2))\n",
    "\n",
    "    # Multipliers for log-sum-exp\n",
    "    lse_coeffs = np.tile([w, 1-w], [len(n), 1]).transpose()\n",
    "\n",
    "    # log-likelihood for each measurement\n",
    "    log_likes = scipy.special.logsumexp(np.vstack([logx1, logx2]), axis=0, b=lse_coeffs)\n",
    "    \n",
    "    return np.sum(log_likes)\n",
    "\n",
    "\n",
    "def mle_mix(n, w_guess):\n",
    "    \"\"\"Obtain MLE estimate for parameters for binary mixture \n",
    "    of Negative Binomials.\"\"\"\n",
    "    with warnings.catch_warnings():\n",
    "        warnings.simplefilter(\"ignore\")\n",
    "\n",
    "        res = scipy.optimize.minimize(\n",
    "            fun=lambda params, n: -log_like_mix(*params, n),\n",
    "            x0=[*initial_guess_mix(n, w_guess), w_guess],\n",
    "            args=(n,),\n",
    "            method='Powell',\n",
    "            tol=1e-6,\n",
    "        )\n",
    "\n",
    "    if res.success:\n",
    "        return res.x\n",
    "    else:\n",
    "        raise RuntimeError('Convergence failed with message', res.message)    \n",
    "        \n",
    "\n",
    "# Load in data\n",
    "df = pd.read_csv(\"../data/singer_transcript_counts.csv\", comment=\"#\")\n",
    "n = df['Rex1'].values\n",
    "\n",
    "# Single Negative Binomial MLE\n",
    "alpha, b = mle_iid_nbinom(n)\n",
    "\n",
    "# Mixture model MLE\n",
    "alpha1, b1, alpha2, b2, w = mle_mix(n, 0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Predictive ECDFs\n",
    "\n",
    "If we wish to accept a model with the MLE for the parameters as a good representation of the true generative model, we would like to see how data generated from that model would look. We can do a parametric bootstrap to generate data sets and then plot confidence intervals of the resulting ECDFs. We call ECDFs that come from the generative model **predictive ECDFs**. We can try that for the single Negative Binomial model. First, we're write a function to draw samples out of a negative binomial."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "rg = np.random.default_rng()\n",
    "\n",
    "def draw_neg_binom(alpha, b, size=1):\n",
    "    return rg.negative_binomial(alpha, 1 / (1 + b), size=size)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can draw 5000 data sets from this distribution, each having 279 (the number of cells that were measured) data points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "parametric_bs_samples = draw_neg_binom(alpha, b, size=(5000, len(n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then compute the predictive ECDFs for each of these samples. We want the values of the predictive ECDF at each possible value of $n$, which requires us to compute the value of the ECDF at an arbitrary value. You wrote a function, `ecdf(x, data)`, to do this in homework 7.1(e). Referring to the homework solutions, we can use that function here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "@numba.njit\n",
    "def ecdf(x, data):\n",
    "    \"\"\"Give the value of an ECDF at arbitrary points x.\"\"\"\n",
    "    y = np.arange(len(data) + 1) / len(data)\n",
    "    return y[np.searchsorted(np.sort(data), x, side=\"right\")]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute the value of the ECDF for the respective values of $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_theor = np.arange(0, parametric_bs_samples.max() + 1)\n",
    "\n",
    "ecdfs = np.array([ecdf(n_theor, sample) for sample in parametric_bs_samples])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the ECDF values, we can compute a 95th-percentile of the value of the ECDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "ecdf_low, ecdf_high = np.percentile(ecdfs, [2.5, 97.5], axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now, we can make a plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"041d219d-2ea2-4ad0-a691-64147165df00\" data-root-id=\"1002\"></div>\n"
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Application\",\"version\":\"1.4.0\"}};\n",
       "  var render_items = [{\"docid\":\"005552a1-a49d-474c-92e1-03ede20eafc8\",\"roots\":{\"1002\":\"041d219d-2ea2-4ad0-a691-64147165df00\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bebi103.viz.fill_between(\n",
    "    x1=n_theor,\n",
    "    y1=ecdf_high,\n",
    "    x2=n_theor,\n",
    "    y2=ecdf_low,\n",
    "    patch_kwargs={\"fill_alpha\": 0.5},\n",
    "    x_axis_label=\"n\",\n",
    "    y_axis_label=\"ECDF\",\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a predictive ECDF. Is is the envelope in which we would expect 95% of the data points to lie if the generative model were true."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparing with data\n",
    "\n",
    "We can overlay the ECDF of the data to see if it falls within this envelope."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"8bacb7d1-f3b3-4869-b7f2-f2f39b01cab1\" data-root-id=\"1002\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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Application\",\"version\":\"1.4.0\"}};\n",
       "  var render_items = [{\"docid\":\"181493a1-e75f-46b0-b6f4-3d713594f256\",\"roots\":{\"1002\":\"8bacb7d1-f3b3-4869-b7f2-f2f39b01cab1\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bokeh_catplot.ecdf(data=df, val='Rex1', palette=['orange'], p=p)\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Negative Binomial model generated some counts that were quite large, which is why the plot extends past counts of 1000. If you zoom in on the region where the data lie, the deviation of the measured ECDF from the ECDF generated by the model is clear. \n",
    "\n",
    "We can also compare with the mixture model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using predictive ECDFs in the bebi103 package\n",
    "\n",
    "The `bebi103` package enables automated generation of predictive ECDFs. To demonstrate its usage, we can generate samples from the mixture model and plot a predictive ECDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw_mix(alpha1, b1, alpha2, b2, w, size):\n",
    "    \"\"\"Draw samples from Negative Binomial binary mixture model.\"\"\"\n",
    "    n = np.empty(size)\n",
    "    for i in range(size):\n",
    "        low_cell_type = rg.uniform() < w\n",
    "\n",
    "        if low_cell_type:\n",
    "            n[i] = rg.negative_binomial(alpha1, 1 / (1 + b1))\n",
    "        else:\n",
    "            n[i] = rg.negative_binomial(alpha2, 1 / (1 + b2))\n",
    "\n",
    "    return n\n",
    "\n",
    "samples = np.array([draw_mix(alpha1, b1, alpha2, b2, w, size=len(n)) for _ in range(5000)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can call `bebi103.viz.predictive_ecdf()` to make the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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Application\",\"version\":\"1.4.0\"}};\n",
       "  var render_items = [{\"docid\":\"0c5b114d-c059-4251-b3a6-d6dd9ef5aeca\",\"roots\":{\"1294\":\"6c3ac5d1-3099-448a-ba7e-442bb5a20354\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1294"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Use discrete=True for discrete data\n",
    "p = bebi103.viz.predictive_ecdf(\n",
    "    samples=samples, data=n, discrete=True, x_axis_label=\"n\"\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we see that the predictive ECDF captures the measured data much better. If you zoom into the plot, you will notice that the `bebi103.viz.predictive_ecdf()` function plots quantiles in various shades of blue. By default, the outermost region (lightest blue) contains the middle 80% of the generated ECDFs, the next one in captures the middle 60%, the next the middle 40%, and the last the middle 20%. The median ECDF from the generative model is a dark blue line in the middle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ECDF difference\n",
    "\n",
    "While the predictive ECDF plot is useful, it is perhaps more illuminating to plot the *difference* between the predictive ECDF and the measured ECDF. We can do this with the `diff=True` kwarg."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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       "\n",
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\",\"dtype\":\"float64\",\"shape\":[2864]}},\"selected\":{\"id\":\"1608\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"1609\",\"type\":\"UnionRenderers\"}},\"id\":\"1541\",\"type\":\"ColumnDataSource\"},{\"attributes\":{},\"id\":\"1608\",\"type\":\"Selection\"},{\"attributes\":{},\"id\":\"1511\",\"type\":\"LinearScale\"},{\"attributes\":{},\"id\":\"1602\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{\"line_color\":\"#084594\",\"line_width\":2,\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1557\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"1609\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1509\",\"type\":\"LinearScale\"},{\"attributes\":{\"data_source\":{\"id\":\"1556\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"1557\",\"type\":\"Line\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"1558\",\"type\":\"Line\"},\"selection_glyph\":null,\"view\":{\"id\":\"1560\",\"type\":\"CDSView\"}},\"id\":\"1559\",\"type\":\"GlyphRenderer\"}],\"root_ids\":[\"1504\"]},\"title\":\"Bokeh Application\",\"version\":\"1.4.0\"}};\n",
       "  var render_items = [{\"docid\":\"c7f59ebf-ae45-454c-8bcb-12f63ba045d2\",\"roots\":{\"1504\":\"caad4464-ac1f-44be-9c94-454318029f86\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1504"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bebi103.viz.predictive_ecdf(\n",
    "    samples=samples, data=n, diff=True, discrete=True, x_axis_label=\"n\"\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is now clear that very few of the 279 data points lie outside the 80% envelope of the predictive ECDF. We might expect more to lie outside; it is conceivable that the model is too flexible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPython 3.7.5\n",
      "IPython 7.9.0\n",
      "\n",
      "numpy 1.17.4\n",
      "pandas 0.24.2\n",
      "scipy 1.3.1\n",
      "bokeh 1.4.0\n",
      "bokeh_catplot 0.1.6\n",
      "bebi103 0.0.45\n",
      "jupyterlab 1.2.3\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,pandas,scipy,bokeh,bokeh_catplot,bebi103,jupyterlab"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}