002 - What if Covid never happened.ipynb

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     "text": [
      "WARNING (theano.tensor.blas): Using NumPy C-API based implementation for BLAS functions.\n"
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    "%load_ext nb_black\n",
    "\n",
    "import warnings\n",
    "\n",
    "warnings.simplefilter(action=\"ignore\", category=FutureWarning)\n",
    "\n",
    "import pymc3 as pm\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "import arviz as az\n",
    "import theano\n",
    "import theano.tensor as tt\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.stats.api as sms\n",
    "import altair as alt"
   ]
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Week_nr</th>\n",
       "      <th>Total</th>\n",
       "      <th>Total_scaled</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>158016</td>\n",
       "      <td>15.8016</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2</td>\n",
       "      <td>218733</td>\n",
       "      <td>21.8733</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3</td>\n",
       "      <td>268026</td>\n",
       "      <td>26.8026</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4</td>\n",
       "      <td>311895</td>\n",
       "      <td>31.1895</td>\n",
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       "      <td>358098</td>\n",
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       "    <tr>\n",
       "      <th>130</th>\n",
       "      <td>130</td>\n",
       "      <td>2308213</td>\n",
       "      <td>230.8213</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>131</th>\n",
       "      <td>131</td>\n",
       "      <td>2332833</td>\n",
       "      <td>233.2833</td>\n",
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       "    <tr>\n",
       "      <th>132</th>\n",
       "      <td>132</td>\n",
       "      <td>2346868</td>\n",
       "      <td>234.6868</td>\n",
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       "      <th>133</th>\n",
       "      <td>133</td>\n",
       "      <td>2359112</td>\n",
       "      <td>235.9112</td>\n",
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       "      <th>134</th>\n",
       "      <td>134</td>\n",
       "      <td>2368328</td>\n",
       "      <td>236.8328</td>\n",
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      "text/plain": [
       "     Week_nr    Total  Total_scaled\n",
       "1          1   158016       15.8016\n",
       "2          2   218733       21.8733\n",
       "3          3   268026       26.8026\n",
       "4          4   311895       31.1895\n",
       "5          5   358098       35.8098\n",
       "..       ...      ...           ...\n",
       "130      130  2308213      230.8213\n",
       "131      131  2332833      233.2833\n",
       "132      132  2346868      234.6868\n",
       "133      133  2359112      235.9112\n",
       "134      134  2368328      236.8328\n",
       "\n",
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    "data = pd.read_csv(\"./data/archon_arcana_weekly_20210619.csv\", thousands=\",\")\n",
    "data[\"Week_nr\"] = data.index\n",
    "data = data[data.Week_nr > 0]  # ignore first line which is day 3, start with week 1\n",
    "model_data = data[[\"Week_nr\", \"Total\"]].copy()\n",
    "model_data[\"Total_scaled\"] = (\n",
    "    model_data[\"Total\"] / 10000\n",
    ")  # PyMC3 doesn't seem to like very large numbers, so lets bring them down a few orders of magnitude\n",
    "model_data"
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   "id": "105ba65d",
   "metadata": {},
   "source": [
    "## What if COVID-19 didn't happen, how many decks would have been sold\n",
    "\n",
    "We'll take model 5 from the previous notebook and add a twist, rather than including the start of COVID-19 in the model, we'll make it a fixed value using ```pm.Data``` this will allow us to change the value later on (to something in the future) so we can create a model that would behave as if the pandemic didn't happen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "cd1b2f19",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
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     "text": [
      "Auto-assigning NUTS sampler...\n",
      "Initializing NUTS using adapt_diag...\n",
      "Multiprocess sampling (4 chains in 4 jobs)\n",
      "NUTS: [dt_interest, mm_interest, wc_interest, aoa_interest, cota_interest, decay_factor, weekly_registrations_precovid, weekly_registrations_covid, sigma]\n"
     ]
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       "      100.00% [8000/8000 18:32<00:00 Sampling 4 chains, 420 divergences]\n",
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     "text": [
      "Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 1135 seconds.\n",
      "There were 11 divergences after tuning. Increase `target_accept` or reparameterize.\n",
      "There were 21 divergences after tuning. Increase `target_accept` or reparameterize.\n",
      "There were 385 divergences after tuning. Increase `target_accept` or reparameterize.\n",
      "The acceptance probability does not match the target. It is 0.6861368264329725, but should be close to 0.8. Try to increase the number of tuning steps.\n",
      "There were 3 divergences after tuning. Increase `target_accept` or reparameterize.\n",
      "The rhat statistic is larger than 1.05 for some parameters. This indicates slight problems during sampling.\n",
      "The estimated number of effective samples is smaller than 200 for some parameters.\n"
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       "\n",
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       "                var nbb_cell_id = 4;\n",
       "                var nbb_unformatted_code = \"len_observed = len(model_data)\\nwith pm.Model() as model_5:\\n    # priors\\n    sigma = pm.Exponential(\\\"sigma\\\", lam=1.0)  # Sigma for likelihood function\\n\\n    # here we set the start of covid manually (based on previous notebook)\\n    # this way it can be changed after fitting the model\\n    covid_start = pm.Data(\\\"covid_start\\\", 68)\\n\\n    # We know from the previous that mu before and during covid should be around 0.8 and 3.0 respectively\\n    # Sigma is reduced here not to diverge to far from these values\\n    BoundNormal_0 = pm.Bound(pm.Normal, lower=0)\\n    weekly_registrations_covid = BoundNormal_0(\\n        \\\"weekly_registrations_covid\\\", mu=0.8, sigma=0.5\\n    )\\n    weekly_registrations_precovid = BoundNormal_0(\\n        \\\"weekly_registrations_precovid\\\", mu=3, sigma=0.5\\n    )\\n\\n    weekly_registrations_base = pm.math.switch(\\n        covid_start >= model_data.Week_nr,\\n        weekly_registrations_precovid,\\n        weekly_registrations_covid,\\n    )\\n\\n    # Model extra registrations due to shifting interest (like new sets being released)\\n    # The interest factor is calculated on a weekly basis\\n    decay_factor = pm.Exponential(\\\"decay_factor\\\", lam=1.0)\\n\\n    cota_interest = pm.HalfNormal(\\\"cota_interest\\\", sigma=2)\\n    aoa_interest = pm.HalfNormal(\\\"aoa_interest\\\", sigma=2)\\n    wc_interest = pm.HalfNormal(\\\"wc_interest\\\", sigma=2)\\n    mm_interest = pm.HalfNormal(\\\"mm_interest\\\", sigma=2)\\n    dt_interest = pm.HalfNormal(\\\"dt_interest\\\", sigma=2)\\n\\n    cota_surplus = pm.Deterministic(\\n        \\\"cota_surplus\\\", cota_interest * weekly_registrations_base[0]\\n    )\\n    aoa_surplus = pm.Deterministic(\\n        \\\"aoa_surplus\\\", aoa_interest * weekly_registrations_base[27]\\n    )\\n    wc_surplus = pm.Deterministic(\\n        \\\"wc_surplus\\\", wc_interest * weekly_registrations_base[50]\\n    )\\n    mm_surplus = pm.Deterministic(\\n        \\\"mm_surplus\\\", mm_interest * weekly_registrations_base[85]\\n    )\\n    dt_surplus = pm.Deterministic(\\n        \\\"dt_surplus\\\", dt_interest * weekly_registrations_base[126]\\n    )\\n\\n    interest_decayed = [cota_interest]\\n\\n    for i in range(len_observed - 1):\\n        new_element = interest_decayed[i] * decay_factor\\n        if i == 27:\\n            new_element += aoa_interest\\n        if i == 50:\\n            new_element += wc_interest\\n        if i == 85:\\n            new_element += mm_interest\\n        if i == 126:\\n            new_element += dt_interest\\n        interest_decayed.append(new_element)\\n\\n    # there were 150k decks registered the first week, that is the initial value\\n    y0 = tt.zeros(len_observed)\\n    y0 = tt.set_subtensor(y0[0], 15)\\n\\n    outputs, _ = theano.scan(\\n        fn=lambda t, y, ws, intfac: tt.set_subtensor(\\n            y[t], (ws[t] * (1 + intfac[t])) + y[t - 1]\\n        ),\\n        sequences=[tt.arange(1, len_observed)],\\n        outputs_info=y0,\\n        non_sequences=[weekly_registrations_base, interest_decayed],\\n        n_steps=len_observed - 1,\\n    )\\n\\n    total_registrations = pm.Deterministic(\\\"total_registrations\\\", outputs[-1])\\n\\n    # Likelihood\\n    likelihood = pm.Normal(\\n        \\\"y\\\", mu=total_registrations, sigma=sigma, observed=model_data.Total_scaled\\n    )\\n\\n    # posterior\\n    trace_5 = pm.sample(1000, cores=4, chains=4, init=\\\"adapt_diag\\\")\";\n",
       "                var nbb_formatted_code = \"len_observed = len(model_data)\\nwith pm.Model() as model_5:\\n    # priors\\n    sigma = pm.Exponential(\\\"sigma\\\", lam=1.0)  # Sigma for likelihood function\\n\\n    # here we set the start of covid manually (based on previous notebook)\\n    # this way it can be changed after fitting the model\\n    covid_start = pm.Data(\\\"covid_start\\\", 68)\\n\\n    # We know from the previous that mu before and during covid should be around 0.8 and 3.0 respectively\\n    # Sigma is reduced here not to diverge to far from these values\\n    BoundNormal_0 = pm.Bound(pm.Normal, lower=0)\\n    weekly_registrations_covid = BoundNormal_0(\\n        \\\"weekly_registrations_covid\\\", mu=0.8, sigma=0.5\\n    )\\n    weekly_registrations_precovid = BoundNormal_0(\\n        \\\"weekly_registrations_precovid\\\", mu=3, sigma=0.5\\n    )\\n\\n    weekly_registrations_base = pm.math.switch(\\n        covid_start >= model_data.Week_nr,\\n        weekly_registrations_precovid,\\n        weekly_registrations_covid,\\n    )\\n\\n    # Model extra registrations due to shifting interest (like new sets being released)\\n    # The interest factor is calculated on a weekly basis\\n    decay_factor = pm.Exponential(\\\"decay_factor\\\", lam=1.0)\\n\\n    cota_interest = pm.HalfNormal(\\\"cota_interest\\\", sigma=2)\\n    aoa_interest = pm.HalfNormal(\\\"aoa_interest\\\", sigma=2)\\n    wc_interest = pm.HalfNormal(\\\"wc_interest\\\", sigma=2)\\n    mm_interest = pm.HalfNormal(\\\"mm_interest\\\", sigma=2)\\n    dt_interest = pm.HalfNormal(\\\"dt_interest\\\", sigma=2)\\n\\n    cota_surplus = pm.Deterministic(\\n        \\\"cota_surplus\\\", cota_interest * weekly_registrations_base[0]\\n    )\\n    aoa_surplus = pm.Deterministic(\\n        \\\"aoa_surplus\\\", aoa_interest * weekly_registrations_base[27]\\n    )\\n    wc_surplus = pm.Deterministic(\\n        \\\"wc_surplus\\\", wc_interest * weekly_registrations_base[50]\\n    )\\n    mm_surplus = pm.Deterministic(\\n        \\\"mm_surplus\\\", mm_interest * weekly_registrations_base[85]\\n    )\\n    dt_surplus = pm.Deterministic(\\n        \\\"dt_surplus\\\", dt_interest * weekly_registrations_base[126]\\n    )\\n\\n    interest_decayed = [cota_interest]\\n\\n    for i in range(len_observed - 1):\\n        new_element = interest_decayed[i] * decay_factor\\n        if i == 27:\\n            new_element += aoa_interest\\n        if i == 50:\\n            new_element += wc_interest\\n        if i == 85:\\n            new_element += mm_interest\\n        if i == 126:\\n            new_element += dt_interest\\n        interest_decayed.append(new_element)\\n\\n    # there were 150k decks registered the first week, that is the initial value\\n    y0 = tt.zeros(len_observed)\\n    y0 = tt.set_subtensor(y0[0], 15)\\n\\n    outputs, _ = theano.scan(\\n        fn=lambda t, y, ws, intfac: tt.set_subtensor(\\n            y[t], (ws[t] * (1 + intfac[t])) + y[t - 1]\\n        ),\\n        sequences=[tt.arange(1, len_observed)],\\n        outputs_info=y0,\\n        non_sequences=[weekly_registrations_base, interest_decayed],\\n        n_steps=len_observed - 1,\\n    )\\n\\n    total_registrations = pm.Deterministic(\\\"total_registrations\\\", outputs[-1])\\n\\n    # Likelihood\\n    likelihood = pm.Normal(\\n        \\\"y\\\", mu=total_registrations, sigma=sigma, observed=model_data.Total_scaled\\n    )\\n\\n    # posterior\\n    trace_5 = pm.sample(1000, cores=4, chains=4, init=\\\"adapt_diag\\\")\";\n",
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   "source": [
    "len_observed = len(model_data)\n",
    "with pm.Model() as model_5:\n",
    "    # priors\n",
    "    sigma = pm.Exponential(\"sigma\", lam=1.0)  # Sigma for likelihood function\n",
    "\n",
    "    # here we set the start of covid manually (based on previous notebook)\n",
    "    # this way it can be changed after fitting the model\n",
    "    covid_start = pm.Data(\"covid_start\", 68)\n",
    "\n",
    "    # We know from the previous that mu before and during covid should be around 0.8 and 3.0 respectively\n",
    "    # Sigma is reduced here not to diverge to far from these values\n",
    "    BoundNormal_0 = pm.Bound(pm.Normal, lower=0)\n",
    "    weekly_registrations_covid = BoundNormal_0(\n",
    "        \"weekly_registrations_covid\", mu=0.8, sigma=0.5\n",
    "    )\n",
    "    weekly_registrations_precovid = BoundNormal_0(\n",
    "        \"weekly_registrations_precovid\", mu=3, sigma=0.5\n",
    "    )\n",
    "\n",
    "    weekly_registrations_base = pm.math.switch(\n",
    "        covid_start >= model_data.Week_nr,\n",
    "        weekly_registrations_precovid,\n",
    "        weekly_registrations_covid,\n",
    "    )\n",
    "\n",
    "    # Model extra registrations due to shifting interest (like new sets being released)\n",
    "    # The interest factor is calculated on a weekly basis\n",
    "    decay_factor = pm.Exponential(\"decay_factor\", lam=1.0)\n",
    "\n",
    "    cota_interest = pm.HalfNormal(\"cota_interest\", sigma=2)\n",
    "    aoa_interest = pm.HalfNormal(\"aoa_interest\", sigma=2)\n",
    "    wc_interest = pm.HalfNormal(\"wc_interest\", sigma=2)\n",
    "    mm_interest = pm.HalfNormal(\"mm_interest\", sigma=2)\n",
    "    dt_interest = pm.HalfNormal(\"dt_interest\", sigma=2)\n",
    "\n",
    "    cota_surplus = pm.Deterministic(\n",
    "        \"cota_surplus\", cota_interest * weekly_registrations_base[0]\n",
    "    )\n",
    "    aoa_surplus = pm.Deterministic(\n",
    "        \"aoa_surplus\", aoa_interest * weekly_registrations_base[27]\n",
    "    )\n",
    "    wc_surplus = pm.Deterministic(\n",
    "        \"wc_surplus\", wc_interest * weekly_registrations_base[50]\n",
    "    )\n",
    "    mm_surplus = pm.Deterministic(\n",
    "        \"mm_surplus\", mm_interest * weekly_registrations_base[85]\n",
    "    )\n",
    "    dt_surplus = pm.Deterministic(\n",
    "        \"dt_surplus\", dt_interest * weekly_registrations_base[126]\n",
    "    )\n",
    "\n",
    "    interest_decayed = [cota_interest]\n",
    "\n",
    "    for i in range(len_observed - 1):\n",
    "        new_element = interest_decayed[i] * decay_factor\n",
    "        if i == 27:\n",
    "            new_element += aoa_interest\n",
    "        if i == 50:\n",
    "            new_element += wc_interest\n",
    "        if i == 85:\n",
    "            new_element += mm_interest\n",
    "        if i == 126:\n",
    "            new_element += dt_interest\n",
    "        interest_decayed.append(new_element)\n",
    "\n",
    "    # there were 150k decks registered the first week, that is the initial value\n",
    "    y0 = tt.zeros(len_observed)\n",
    "    y0 = tt.set_subtensor(y0[0], 15)\n",
    "\n",
    "    outputs, _ = theano.scan(\n",
    "        fn=lambda t, y, ws, intfac: tt.set_subtensor(\n",
    "            y[t], (ws[t] * (1 + intfac[t])) + y[t - 1]\n",
    "        ),\n",
    "        sequences=[tt.arange(1, len_observed)],\n",
    "        outputs_info=y0,\n",
    "        non_sequences=[weekly_registrations_base, interest_decayed],\n",
    "        n_steps=len_observed - 1,\n",
    "    )\n",
    "\n",
    "    total_registrations = pm.Deterministic(\"total_registrations\", outputs[-1])\n",
    "\n",
    "    # Likelihood\n",
    "    likelihood = pm.Normal(\n",
    "        \"y\", mu=total_registrations, sigma=sigma, observed=model_data.Total_scaled\n",
    "    )\n",
    "\n",
    "    # posterior\n",
    "    trace_5 = pm.sample(1000, cores=4, chains=4, init=\"adapt_diag\")"
   ]
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      "  23.02791231  24.20489176  25.32756456  26.41259487  27.47153339\n",
      "  28.51238646  29.54070365  30.56033163  31.57393671  32.58336703\n",
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      "  43.60429109  44.60440709  45.60448749  48.20031234  50.30645384\n",
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      " 159.20598154 160.20598223 162.80175182 164.90785502 166.67454734\n",
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{\"min_posterior_predictor\": 2232264.2805982153, \"max_posterior_predictor\": 2311652.072288743, \"mean_posterior_predictor\": 2270936.316577412, \"Total_scaled\": 2281544.0, \"Week_nr\": 128}, {\"min_posterior_predictor\": 2251508.762632616, \"max_posterior_predictor\": 2339469.4818152147, \"mean_posterior_predictor\": 2292252.997456121, \"Total_scaled\": 2290966.0, \"Week_nr\": 129}, {\"min_posterior_predictor\": 2271169.5928965155, \"max_posterior_predictor\": 2351601.4838867527, \"mean_posterior_predictor\": 2311544.2127617397, \"Total_scaled\": 2308213.0, \"Week_nr\": 130}, {\"min_posterior_predictor\": 2275250.225349306, \"max_posterior_predictor\": 2373243.9009931944, \"mean_posterior_predictor\": 2328564.116412976, \"Total_scaled\": 2332833.0, \"Week_nr\": 131}, {\"min_posterior_predictor\": 2297468.5913359155, \"max_posterior_predictor\": 2385420.734590713, \"mean_posterior_predictor\": 2343746.8397787246, \"Total_scaled\": 2346868.0, \"Week_nr\": 132}, {\"min_posterior_predictor\": 2312453.984724053, \"max_posterior_predictor\": 2408814.210845273, \"mean_posterior_predictor\": 2357785.5381497364, \"Total_scaled\": 2359112.0, \"Week_nr\": 133}, {\"min_posterior_predictor\": 2322939.9671402415, \"max_posterior_predictor\": 2416334.133202107, \"mean_posterior_predictor\": 2369973.4355881996, \"Total_scaled\": 2368328.0, \"Week_nr\": 134}]}}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.LayerChart(...)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 5;\n",
       "                var nbb_unformatted_code = \"from pymc3_altair import plot_fit_altair\\n\\nwith model_5:\\n    pm.set_data({\\\"covid_start\\\": 68})\\n\\nchart_with_covid = plot_fit_altair(model_5, trace_5, model_data)\\nchart_with_covid\";\n",
       "                var nbb_formatted_code = \"from pymc3_altair import plot_fit_altair\\n\\nwith model_5:\\n    pm.set_data({\\\"covid_start\\\": 68})\\n\\nchart_with_covid = plot_fit_altair(model_5, trace_5, model_data)\\nchart_with_covid\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pymc3_altair import plot_fit_altair\n",
    "\n",
    "with model_5:\n",
    "    pm.set_data({\"covid_start\": 68})\n",
    "\n",
    "chart_with_covid = plot_fit_altair(model_5, trace_5, model_data)\n",
    "chart_with_covid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "e42e2524",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "                /* gets rid of default border in Firefox and Opera. */\n",
       "                border: none;\n",
       "                /* Needs to be in here for Safari polyfill so background images work as expected. */\n",
       "                background-size: auto;\n",
       "            }\n",
       "            .progress-bar-interrupted, .progress-bar-interrupted::-webkit-progress-bar {\n",
       "                background: #F44336;\n",
       "            }\n",
       "        </style>\n",
       "      <progress value='4000' class='' max='4000' style='width:300px; height:20px; vertical-align: middle;'></progress>\n",
       "      100.00% [4000/4000 01:51<00:00]\n",
       "    </div>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<div id=\"altair-viz-0a6c67915f7d462f8e3f50ea6d98cb2c\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  (function(spec, embedOpt){\n",
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{\"min_posterior_predictor\": 1933098.5670293896, \"max_posterior_predictor\": 2048012.8050694445, \"mean_posterior_predictor\": 1991913.082431347, \"Total_scaled\": 1840314.0, \"Week_nr\": 83}, {\"min_posterior_predictor\": 1963303.4333034148, \"max_posterior_predictor\": 2054147.228362601, \"mean_posterior_predictor\": 2007998.6372326706, \"Total_scaled\": 1845603.9999999998, \"Week_nr\": 84}, {\"min_posterior_predictor\": 1968922.803943805, \"max_posterior_predictor\": 2080610.2730841297, \"mean_posterior_predictor\": 2023377.4367530684, \"Total_scaled\": 1851568.0, \"Week_nr\": 85}, {\"min_posterior_predictor\": 1981717.4875066278, \"max_posterior_predictor\": 2090432.8428163326, \"mean_posterior_predictor\": 2038772.449022256, \"Total_scaled\": 1882009.0, \"Week_nr\": 86}, {\"min_posterior_predictor\": 2043024.3029107878, \"max_posterior_predictor\": 2168543.593578983, \"mean_posterior_predictor\": 2102461.233275682, \"Total_scaled\": 1918689.0, \"Week_nr\": 87}, {\"min_posterior_predictor\": 2076665.080418568, \"max_posterior_predictor\": 2237523.4496375523, \"mean_posterior_predictor\": 2158378.363373503, \"Total_scaled\": 1944323.0, \"Week_nr\": 88}, {\"min_posterior_predictor\": 2113225.36045157, \"max_posterior_predictor\": 2288764.986060241, \"mean_posterior_predictor\": 2207400.4184012506, \"Total_scaled\": 1962893.0, \"Week_nr\": 89}, {\"min_posterior_predictor\": 2158805.719877829, \"max_posterior_predictor\": 2361533.438312886, \"mean_posterior_predictor\": 2251284.0065044183, \"Total_scaled\": 1976710.0, \"Week_nr\": 90}, {\"min_posterior_predictor\": 2182980.438567222, \"max_posterior_predictor\": 2411135.034463229, \"mean_posterior_predictor\": 2290122.79382253, \"Total_scaled\": 1987082.0, \"Week_nr\": 91}, {\"min_posterior_predictor\": 2201150.4202372474, \"max_posterior_predictor\": 2455458.416800467, \"mean_posterior_predictor\": 2325811.5751309334, \"Total_scaled\": 1996842.0, \"Week_nr\": 92}, {\"min_posterior_predictor\": 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\"max_posterior_predictor\": 3032558.44245913, \"mean_posterior_predictor\": 2826207.35009162, \"Total_scaled\": 2188012.0, \"Week_nr\": 118}, {\"min_posterior_predictor\": 2696470.69225797, \"max_posterior_predictor\": 3026000.5810872167, \"mean_posterior_predictor\": 2841519.690664487, \"Total_scaled\": 2193073.0, \"Week_nr\": 119}, {\"min_posterior_predictor\": 2688174.472917463, \"max_posterior_predictor\": 3051401.073919742, \"mean_posterior_predictor\": 2857388.262691575, \"Total_scaled\": 2199799.0, \"Week_nr\": 120}, {\"min_posterior_predictor\": 2705826.2689996245, \"max_posterior_predictor\": 3074139.0565079474, \"mean_posterior_predictor\": 2872998.9295753785, \"Total_scaled\": 2206517.0, \"Week_nr\": 121}, {\"min_posterior_predictor\": 2725884.8770252857, \"max_posterior_predictor\": 3083307.70675146, \"mean_posterior_predictor\": 2888946.765936167, \"Total_scaled\": 2213017.0, \"Week_nr\": 122}, {\"min_posterior_predictor\": 2721695.418655563, \"max_posterior_predictor\": 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\"mean_posterior_predictor\": 3027594.9747730684, \"Total_scaled\": 2281544.0, \"Week_nr\": 128}, {\"min_posterior_predictor\": 2890445.385965062, \"max_posterior_predictor\": 3302060.5404640124, \"mean_posterior_predictor\": 3081182.1321915276, \"Total_scaled\": 2290966.0, \"Week_nr\": 129}, {\"min_posterior_predictor\": 2911986.3736888794, \"max_posterior_predictor\": 3358744.7867471385, \"mean_posterior_predictor\": 3128878.9658274325, \"Total_scaled\": 2308213.0, \"Week_nr\": 130}, {\"min_posterior_predictor\": 2968175.668050764, \"max_posterior_predictor\": 3403169.5789651405, \"mean_posterior_predictor\": 3171010.5933088246, \"Total_scaled\": 2332833.0, \"Week_nr\": 131}, {\"min_posterior_predictor\": 3006008.769764924, \"max_posterior_predictor\": 3467609.3009057245, \"mean_posterior_predictor\": 3208614.594683472, \"Total_scaled\": 2346868.0, \"Week_nr\": 132}, {\"min_posterior_predictor\": 3030993.8637850685, \"max_posterior_predictor\": 3484790.409740013, \"mean_posterior_predictor\": 3242797.157397348, \"Total_scaled\": 2359112.0, \"Week_nr\": 133}, {\"min_posterior_predictor\": 3062374.1834569513, \"max_posterior_predictor\": 3519579.4575792183, \"mean_posterior_predictor\": 3273677.643781491, \"Total_scaled\": 2368328.0, \"Week_nr\": 134}]}}, {\"mode\": \"vega-lite\"});\n",
       "</script>"
      ],
      "text/plain": [
       "alt.LayerChart(...)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 6;\n",
       "                var nbb_unformatted_code = \"with model_5:\\n    pm.set_data({\\\"covid_start\\\": 500})\\n\\nchart_without_covid = plot_fit_altair(model_5, trace_5, model_data)\\nchart_without_covid\";\n",
       "                var nbb_formatted_code = \"with model_5:\\n    pm.set_data({\\\"covid_start\\\": 500})\\n\\nchart_without_covid = plot_fit_altair(model_5, trace_5, model_data)\\nchart_without_covid\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with model_5:\n",
    "    pm.set_data({\"covid_start\": 500})\n",
    "\n",
    "chart_without_covid = plot_fit_altair(model_5, trace_5, model_data)\n",
    "chart_without_covid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "987db584",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div id=\"altair-viz-80c2d094e32e45f9ae31ac3975e5179e\"></div>\n",
       "<script type=\"text/javascript\">\n",
       "  (function(spec, embedOpt){\n",
       "    let outputDiv = document.currentScript.previousElementSibling;\n",
       "    if (outputDiv.id !== \"altair-viz-80c2d094e32e45f9ae31ac3975e5179e\") {\n",
       "      outputDiv = document.getElementById(\"altair-viz-80c2d094e32e45f9ae31ac3975e5179e\");\n",
       "    }\n",
       "    const paths = {\n",
       "      \"vega\": \"https://cdn.jsdelivr.net/npm//vega@5?noext\",\n",
       "      \"vega-lib\": \"https://cdn.jsdelivr.net/npm//vega-lib?noext\",\n",
       "      \"vega-lite\": \"https://cdn.jsdelivr.net/npm//vega-lite@4.8.1?noext\",\n",
       "      \"vega-embed\": \"https://cdn.jsdelivr.net/npm//vega-embed@6?noext\",\n",
       "    };\n",
       "\n",
       "    function loadScript(lib) {\n",
       "      return new Promise(function(resolve, reject) {\n",
       "        var s = document.createElement('script');\n",
       "        s.src = paths[lib];\n",
       "        s.async = true;\n",
       "        s.onload = () => resolve(paths[lib]);\n",
       "        s.onerror = () => reject(`Error loading script: ${paths[lib]}`);\n",
       "        document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "      });\n",
       "    }\n",
       "\n",
       "    function showError(err) {\n",
       "      outputDiv.innerHTML = `<div class=\"error\" style=\"color:red;\">${err}</div>`;\n",
       "      throw err;\n",
       "    }\n",
       "\n",
       "    function displayChart(vegaEmbed) {\n",
       "      vegaEmbed(outputDiv, spec, embedOpt)\n",
       "        .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));\n",
       "    }\n",
       "\n",
       "    if(typeof define === \"function\" && define.amd) {\n",
       "      requirejs.config({paths});\n",
       "      require([\"vega-embed\"], displayChart, err => showError(`Error loading script: ${err.message}`));\n",
       "    } else if (typeof vegaEmbed === \"function\") {\n",
       "      displayChart(vegaEmbed);\n",
       "    } else {\n",
       "      loadScript(\"vega\")\n",
       "        .then(() => loadScript(\"vega-lite\"))\n",
       "        .then(() => loadScript(\"vega-embed\"))\n",
       "        .catch(showError)\n",
       "        .then(() => displayChart(vegaEmbed));\n",
       "    }\n",
       "  })({\"config\": {\"view\": {\"continuousWidth\": 400, \"continuousHeight\": 300}}, \"hconcat\": [{\"layer\": [{\"mark\": {\"type\": \"area\", \"color\": \"#333333\", \"opacity\": 0.2}, \"encoding\": {\"tooltip\": [{\"type\": \"quantitative\", \"field\": \"Total_scaled\", \"format\": \",.0f\", \"title\": \"Registered Decks\"}, {\"type\": \"quantitative\", \"field\": \"max_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Max\"}, {\"type\": \"quantitative\", \"field\": \"mean_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Mean\"}, {\"type\": \"quantitative\", \"field\": \"min_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Min\"}], \"x\": {\"type\": \"quantitative\", \"field\": \"Week_nr\", \"scale\": {\"domain\": [1, 134], \"nice\": false, \"zero\": false}, \"title\": \"Week\"}, \"y\": {\"type\": \"quantitative\", \"field\": \"min_posterior_predictor\", \"title\": \"Decks Registered\"}, \"y2\": {\"field\": \"max_posterior_predictor\"}}}, {\"mark\": {\"type\": \"line\", \"color\": \"#333333\", \"opacity\": 0.7}, \"encoding\": {\"tooltip\": [{\"type\": \"quantitative\", \"field\": \"Total_scaled\", \"format\": \",.0f\", \"title\": \"Registered Decks\"}, {\"type\": \"quantitative\", \"field\": \"max_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Max\"}, {\"type\": \"quantitative\", \"field\": \"mean_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Mean\"}, {\"type\": \"quantitative\", \"field\": \"min_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Min\"}], \"x\": {\"type\": \"quantitative\", \"field\": \"Week_nr\", \"scale\": {\"domain\": [1, 134], \"nice\": false, \"zero\": false}, \"title\": \"Week\"}, \"y\": {\"type\": \"quantitative\", \"field\": \"mean_posterior_predictor\"}}}, {\"mark\": \"line\", \"encoding\": {\"tooltip\": [{\"type\": \"quantitative\", \"field\": \"Total_scaled\", \"format\": \",.0f\", \"title\": \"Registered Decks\"}, {\"type\": \"quantitative\", \"field\": \"max_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Max\"}, {\"type\": \"quantitative\", \"field\": \"mean_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Mean\"}, {\"type\": \"quantitative\", \"field\": \"min_posterior_predictor\", \"format\": \",.0f\", \"title\": \"Model Min\"}], \"x\": {\"type\": \"quantitative\", \"field\": \"Week_nr\", \"scale\": {\"domain\": [1, 134], \"nice\": false, \"zero\": false}, \"title\": \"Week\"}, \"y\": {\"type\": \"quantitative\", \"field\": \"Total_scaled\"}}}], \"data\": {\"name\": \"data-51ec81421fdf52d08a40ffd238076c89\"}, \"title\": \"Final model with COVID-19\"}, {\"layer\": [{\"mark\": {\"type\": \"area\", \"color\": \"#333333\", \"opacity\": 0.2}, \"encoding\": {\"tooltip\": [{\"type\": \"quantitative\", \"field\": \"Total_scaled\", \"format\": \",.0f\", \"title\": \"Registered Decks\"}, {\"type\": \"quantitative\", \"field\": \"max_posterior_predictor\", 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       "</script>"
      ],
      "text/plain": [
       "alt.HConcatChart(...)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "            setTimeout(function() {\n",
       "                var nbb_cell_id = 7;\n",
       "                var nbb_unformatted_code = \"combined = chart_with_covid.properties(\\n    title=\\\"Final model with COVID-19\\\"\\n) | chart_without_covid.properties(title=\\\"Final model without COVID-19\\\")\\ncombined.save(\\\"./altair_output/covid_start_data.json\\\")\\ncombined\";\n",
       "                var nbb_formatted_code = \"combined = chart_with_covid.properties(\\n    title=\\\"Final model with COVID-19\\\"\\n) | chart_without_covid.properties(title=\\\"Final model without COVID-19\\\")\\ncombined.save(\\\"./altair_output/covid_start_data.json\\\")\\ncombined\";\n",
       "                var nbb_cells = Jupyter.notebook.get_cells();\n",
       "                for (var i = 0; i < nbb_cells.length; ++i) {\n",
       "                    if (nbb_cells[i].input_prompt_number == nbb_cell_id) {\n",
       "                        if (nbb_cells[i].get_text() == nbb_unformatted_code) {\n",
       "                             nbb_cells[i].set_text(nbb_formatted_code);\n",
       "                        }\n",
       "                        break;\n",
       "                    }\n",
       "                }\n",
       "            }, 500);\n",
       "            "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "combined = chart_with_covid.properties(\n",
    "    title=\"Final model with COVID-19\"\n",
    ") | chart_without_covid.properties(title=\"Final model without COVID-19\")\n",
    "combined.save(\"./altair_output/covid_start_data.json\")\n",
    "combined"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68bac03b",
   "metadata": {},
   "source": [
    "This is by far the best way to change a model, but it doesn't offer much flexibility. In this case it is very likely that releasing a new set during COVID will have a very different effect on sales than under normal circumstances. Hence the impact of the last two sets might be exagerated. This would be an optimistic model, though it would be nice to be able to model what would happen in case Mass Mutation and Dark Tidings performed as Worlds Collide (which had the lowest impact on sales). This requires another approach (I think let me know if there is an elegant solution to this).\n",
    "\n",
    "Another method is to create a new model, with priors based on the results from the previous one, excluding the COVID-19 factor. Rather than tracing it to get posteriors, we'll use the ```prior_predictive()``` to run the model with these parameters to see what comes out at the end. This allows for some additional flexibility the latter option didn't provide\n",
    "\n",
    "We'll use two versions here, an optimistic one, where the interest in Mass Mutation (mm_interest) and Dark Tidings (dt_interest) from model_5 is used. However, as these were released during COVID, it might not be an accurate representation how they would have impacted sales during normal times. Therefore a pessimistic model, where MM and DT performed the same as the worst previous set (Worlds Collide) is build as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3c35a833",
   "metadata": {},
   "outputs": [],
   "source": [
    "with pm.Model() as model_6:\n",
    "    # priors\n",
    "    sigma = pm.Normal(\"sigma\", mu=1.17, sigma=0.07)  # Sigma for likelihood function\n",
    "\n",
    "    weekly_registrations = pm.Normal(\"weekly_registrations\", mu=1.55, sigma=0.06)\n",
    "\n",
    "    # Model extra registrations due to shifting interest (like new sets being released)\n",
    "    # The interest factor is calculated on a weekly basis\n",
    "    decay_factor = pm.Normal(\"decay_factor\", mu=0.84, sigma=0.01)\n",
    "\n",
    "    cota_interest = pm.Normal(\"cota_interest\", mu=4.11, sigma=0.19)\n",
    "    aoa_interest = pm.Normal(\"aoa_interest\", mu=1.7, sigma=0.16)\n",
    "    wc_interest = pm.Normal(\"wc_interest\", mu=0.95, sigma=0.13)\n",
    "    mm_interest = pm.Normal(\"mm_interest\", mu=3.18, sigma=0.31)\n",
    "    dt_interest = pm.Normal(\"dt_interest\", mu=3.02, sigma=0.35)\n",
    "\n",
    "    interest_decayed = [cota_interest]\n",
    "\n",
    "    for i in range(len_observed - 1):\n",
    "        new_element = interest_decayed[i] * decay_factor\n",
    "        if i == 27:\n",
    "            new_element += aoa_interest\n",
    "        if i == 50:\n",
    "            new_element += wc_interest\n",
    "        if i == 85:\n",
    "            new_element += mm_interest\n",
    "        if i == 126:\n",
    "            new_element += dt_interest\n",
    "        interest_decayed.append(new_element)\n",
    "\n",
    "    # there were 150k decks registered the first week, that is the initial value\n",
    "    y0 = tt.zeros(len_observed)\n",
    "    y0 = tt.set_subtensor(y0[0], 15)\n",
    "\n",
    "    outputs, _ = theano.scan(\n",
    "        fn=lambda t, y, intfac: tt.set_subtensor(\n",
    "            y[t], (weekly_registrations * (1 + intfac[t])) + y[t - 1]\n",
    "        ),\n",
    "        sequences=[tt.arange(1, len_observed)],\n",
    "        outputs_info=y0,\n",
    "        non_sequences=[interest_decayed],\n",
    "        n_steps=len_observed - 1,\n",
    "    )\n",
    "\n",
    "    total_registrations = pm.Deterministic(\"total_registrations\", outputs[-1])\n",
    "\n",
    "    # Likelihood\n",
    "    likelihood = pm.Normal(\n",
    "        \"y\", mu=total_registrations, sigma=sigma, observed=model_data.Total_scaled\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f3b9ee25",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_prior_predictor(m, df):\n",
    "    with m:\n",
    "        prior_predictor = pm.sample_prior_predictive()\n",
    "\n",
    "    fig, (ax1) = plt.subplots(1, 1, figsize=(7.5, 8))\n",
    "    ax1.plot(prior_predictor[\"y\"].T, color=\"0.5\", alpha=0.1)\n",
    "    ax1.plot(df.Total_scaled, color=\"red\", alpha=1)\n",
    "    ax1.set(\n",
    "        xlabel=\"Weeks (since release)\",\n",
    "        ylabel=\"Registered Decks (x10 000)\",\n",
    "        title=\"Model Fit\",\n",
    "    )\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "plot_prior_predictor(model_6, model_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ad31862c",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pymc3_altair import plot_prior_predictor_altair\n",
    "\n",
    "\n",
    "chart_prediction_1 = plot_prior_predictor_altair(model_6, model_data)\n",
    "chart_prediction_1.properties(width=\"container\").save(\n",
    "    \"./altair_output/prediction_1.json\"\n",
    ")\n",
    "chart_prediction_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "962abf93",
   "metadata": {},
   "outputs": [],
   "source": [
    "with pm.Model() as model_7:\n",
    "    # priors\n",
    "    sigma = pm.Normal(\"sigma\", mu=1.17, sigma=0.07)  # Sigma for likelihood function\n",
    "\n",
    "    weekly_registrations = pm.Normal(\"weekly_registrations\", mu=1.55, sigma=0.06)\n",
    "\n",
    "    # Model extra registrations due to shifting interest (like new sets being released)\n",
    "    # The interest factor is calculated on a weekly basis\n",
    "    decay_factor = pm.Normal(\"decay_factor\", mu=0.84, sigma=0.01)\n",
    "\n",
    "    cota_interest = pm.Normal(\"cota_interest\", mu=4.11, sigma=0.19)\n",
    "    aoa_interest = pm.Normal(\"aoa_interest\", mu=1.7, sigma=0.16)\n",
    "    wc_interest = pm.Normal(\"wc_interest\", mu=0.95, sigma=0.13)\n",
    "    mm_interest = pm.Normal(\n",
    "        \"mm_interest\", mu=0.95, sigma=0.13\n",
    "    )  # Maybe people's buying behaviour responds differently to\n",
    "    dt_interest = pm.Normal(\n",
    "        \"dt_interest\", mu=0.95, sigma=0.13\n",
    "    )  # the release of a new set, worst case scenario is setting these very low\n",
    "\n",
    "    interest_decayed = [cota_interest]\n",
    "\n",
    "    for i in range(len_observed - 1):\n",
    "        new_element = interest_decayed[i] * decay_factor\n",
    "        if i == 27:\n",
    "            new_element += aoa_interest\n",
    "        if i == 50:\n",
    "            new_element += wc_interest\n",
    "        if i == 85:\n",
    "            new_element += mm_interest\n",
    "        if i == 126:\n",
    "            new_element += dt_interest\n",
    "        interest_decayed.append(new_element)\n",
    "\n",
    "    # there were 150k decks registered the first week, that is the initial value\n",
    "    y0 = tt.zeros(len_observed)\n",
    "    y0 = tt.set_subtensor(y0[0], 15)\n",
    "\n",
    "    outputs, _ = theano.scan(\n",
    "        fn=lambda t, y, intfac: tt.set_subtensor(\n",
    "            y[t], (weekly_registrations * (1 + intfac[t])) + y[t - 1]\n",
    "        ),\n",
    "        sequences=[tt.arange(1, len_observed)],\n",
    "        outputs_info=y0,\n",
    "        non_sequences=[interest_decayed],\n",
    "        n_steps=len_observed - 1,\n",
    "    )\n",
    "\n",
    "    total_registrations = pm.Deterministic(\"total_registrations\", outputs[-1])\n",
    "\n",
    "    # Likelihood\n",
    "    likelihood = pm.Normal(\n",
    "        \"y\", mu=total_registrations, sigma=sigma, observed=model_data.Total_scaled\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eedcad89",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_prior_predictor_side_by_side(m, m2, df):\n",
    "    with m:\n",
    "        prior_predictor = pm.sample_prior_predictive()\n",
    "    with m2:\n",
    "        prior_predictor2 = pm.sample_prior_predictive()\n",
    "\n",
    "    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 8), sharey=True)\n",
    "    ax1.plot(prior_predictor[\"y\"].T, color=\"0.5\", alpha=0.1)\n",
    "    ax1.plot(df.Total_scaled, color=\"red\", alpha=1)\n",
    "    ax1.set(\n",
    "        xlabel=\"Weeks (since release)\",\n",
    "        ylabel=\"Registered Decks (x10 000)\",\n",
    "        title=\"No COVID-19 Model A (optimistic)\",\n",
    "    )\n",
    "\n",
    "    ax2.plot(prior_predictor2[\"y\"].T, color=\"0.5\", alpha=0.1)\n",
    "    ax2.plot(df.Total_scaled, color=\"red\", alpha=1)\n",
    "    ax2.set(\n",
    "        xlabel=\"Weeks (since release)\",\n",
    "        ylabel=\"Registered Decks (x10 000)\",\n",
    "        title=\"No COVID-19 Model B (pessimistic)\",\n",
    "    )\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "plot_prior_predictor_side_by_side(model_6, model_7, model_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5736df13",
   "metadata": {},
   "outputs": [],
   "source": [
    "chart_prediction_2 = plot_prior_predictor_altair(model_7, model_data)\n",
    "chart_prediction_2.properties(width=\"container\").save(\n",
    "    \"./altair_output/prediction_2.json\"\n",
    ")\n",
    "chart_prediction_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b423f98f",
   "metadata": {},
   "outputs": [],
   "source": [
    "combined = chart_prediction_1.properties(\n",
    "    title=\"No COVID-19 (Optimistic model)\"\n",
    ") | chart_prediction_2.properties(title=\"No COVID-19 (Pessimistic model)\")\n",
    "combined.save(\"./altair_output/combined_predictions.json\")\n",
    "combined"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd3a7d4b",
   "metadata": {},
   "source": [
    "So the optimistic model predicts that in a universe without COVID-19, there likely would have been over 3 million registered decks by now. Going as high up as 3.7 million, this would mean there would have been roughly 0.5 to 1.5 million more decks registered over the last 18 or so months.\n",
    "\n",
    "The pessimistic model is more modest about those numbers though the total number of registrations would still have been getting very close to 3 million, with roughly 0.5 to 1 million additional decks sold since the first lockdowns due to COVID.\n",
    "\n",
    "However, here we start the model from the very start. This isn't necessary as we know exactly how many decks were sold prior to the lockdown. So we don't have to include all the uncertainty before that point when forecasting sales in a COVID-19 free world."
   ]
  }
 ],
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