diff --git a/notebooks/diagnostics.ipynb b/notebooks/diagnostics.ipynb
index 457dedf..9490c69 100644
--- a/notebooks/diagnostics.ipynb
+++ b/notebooks/diagnostics.ipynb
@@ -6,7 +6,18 @@
    "metadata": {
     "block_hidden": true
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/Users/ryannazareth/anaconda3/envs/fbprophet/lib/python3.7/site-packages/rpy2/robjects/pandas2ri.py:17: FutureWarning: pandas.core.index is deprecated and will be removed in a future version.  The public classes are available in the top-level namespace.\n",
+      "  from pandas.core.index import Index as PandasIndex\n",
+      "/Users/ryannazareth/Documents/Python_sprints/prophet/python/fbprophet/diagnostics.py:10: TqdmExperimentalWarning: Using `tqdm.autonotebook.tqdm` in notebook mode. Use `tqdm.tqdm` instead to force console mode (e.g. in jupyter console)\n",
+      "  from tqdm.autonotebook import tqdm\n"
+     ]
+    }
+   ],
    "source": [
     "%load_ext rpy2.ipython\n",
     "%matplotlib inline\n",
@@ -468,17 +479,343 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The size of the rolling window in the figure can be changed with the optional argument `rolling_window`, which specifies the proportion of forecasts to use in each rolling window. The default is 0.1, corresponding to 10% of rows from `df_cv` included in each window; increasing this will lead to a smoother average curve in the figure.\n",
+    "The size of the rolling window in the figure can be changed with the optional argument `rolling_window`, which specifies the proportion of forecasts to use in each rolling window. The default is 0.1, corresponding to 10% of rows from `df_cv` included in each window; increasing this will lead to a smoother average curve in the figure. The `initial` period should be long enough to capture all of the components of the model, in particular seasonalities and extra regressors: at least a year for yearly seasonality, at least a week for weekly seasonality, etc.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Hyperparameter Optimisation\n",
     "\n",
-    "The `initial` period should be long enough to capture all of the components of the model, in particular seasonalities and extra regressors: at least a year for yearly seasonality, at least a week for weekly seasonality, etc."
+    "Auto parameter tuning can also be carried out by evaluating the parameter combinations in serial and using the in-built parallelization over cutoffs. An example implementation with multi-processing in Python is shown below, with a grid of six combinations of `changepoint_prior_scale` and `changepoint_range` parameters. The function `create_param_combinaitons` creates a dataframe of parameter combinations, which can be evaluated serially to call `single_cv_run` with the `parallel` keyword to parallelze over cutoffs. The best parameter combination is selected based on the best `rmse` score but can be switched to another performance metric depending on the use case.\n",
+    "\n",
+    "As an alternative for creating parameter combinations, one could also use the `ParameterGrid` class in `sklearn.model_selection`. This would need to be installed and imported separately if required, as it is not included with Prophet."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      " The best param combination is {'changepoint_prior_scale': 0.05, 'changepoint_range': 0.8}\n"
+     ]
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div>\n",
+       "<style scoped>\n",
+       "    .dataframe tbody tr th:only-of-type {\n",
+       "        vertical-align: middle;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe tbody tr th {\n",
+       "        vertical-align: top;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe thead th {\n",
+       "        text-align: right;\n",
+       "    }\n",
+       "</style>\n",
+       "<table border=\"1\" class=\"dataframe\">\n",
+       "  <thead>\n",
+       "    <tr style=\"text-align: right;\">\n",
+       "      <th></th>\n",
+       "      <th>horizon</th>\n",
+       "      <th>rmse</th>\n",
+       "      <th>mape</th>\n",
+       "      <th>params</th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>0</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.450030</td>\n",
+       "      <td>0.034958</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.05, 'changepoint_range': 0.8}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>1</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.453755</td>\n",
+       "      <td>0.035471</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.05, 'changepoint_range': 0.9}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.456887</td>\n",
+       "      <td>0.035469</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.5, 'changepoint_range': 0.8}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>3</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.490453</td>\n",
+       "      <td>0.039134</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.5, 'changepoint_range': 0.9}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>4</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.463969</td>\n",
+       "      <td>0.036428</td>\n",
+       "      <td>{'changepoint_prior_scale': 5.0, 'changepoint_range': 0.8}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>5</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.512077</td>\n",
+       "      <td>0.040488</td>\n",
+       "      <td>{'changepoint_prior_scale': 5.0, 'changepoint_range': 0.9}</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "   horizon      rmse      mape  \\\n",
+       "0 200 days  0.450030  0.034958   \n",
+       "1 200 days  0.453755  0.035471   \n",
+       "2 200 days  0.456887  0.035469   \n",
+       "3 200 days  0.490453  0.039134   \n",
+       "4 200 days  0.463969  0.036428   \n",
+       "5 200 days  0.512077  0.040488   \n",
+       "\n",
+       "                                                        params  \n",
+       "0  {'changepoint_prior_scale': 0.05, 'changepoint_range': 0.8}  \n",
+       "1  {'changepoint_prior_scale': 0.05, 'changepoint_range': 0.9}  \n",
+       "2   {'changepoint_prior_scale': 0.5, 'changepoint_range': 0.8}  \n",
+       "3   {'changepoint_prior_scale': 0.5, 'changepoint_range': 0.9}  \n",
+       "4   {'changepoint_prior_scale': 5.0, 'changepoint_range': 0.8}  \n",
+       "5   {'changepoint_prior_scale': 5.0, 'changepoint_range': 0.9}  "
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from fbprophet.diagnostics import cross_validation, performance_metrics\n",
+    "import itertools\n",
+    "\n",
+    "def create_param_combinations(**param_dict):\n",
+    "    param_iter = itertools.product(*param_dict.values())\n",
+    "    params =[]\n",
+    "    for param in param_iter:\n",
+    "        params.append(param) \n",
+    "    params_df = pd.DataFrame(params, columns=list(param_dict.keys()))\n",
+    "    return params_df\n",
+    "\n",
+    "def single_cv_run(history_df, metrics, param_dict, parallel):\n",
+    "    m = Prophet(**param_dict)\n",
+    "    m.fit(history_df)\n",
+    "    df_cv = cross_validation(m, initial='2600 days', period='100 days', horizon = '200 days', parallel=parallel)\n",
+    "    df_p = performance_metrics(df_cv, rolling_window=1)\n",
+    "    df_p['params'] = str(param_dict)\n",
+    "    df_p = df_p.loc[:, metrics]\n",
+    "    return df_p\n",
+    "\n",
+    "\n",
+    "pd.set_option('display.max_colwidth', None)\n",
+    "param_grid = {  \n",
+    "                'changepoint_prior_scale': [0.05, 0.5, 5],\n",
+    "                'changepoint_range': [0.8, 0.9],\n",
+    "              }\n",
+    "metrics = ['horizon', 'rmse', 'mape', 'params'] \n",
+    "results = []\n",
+    "\n",
+    "\n",
+    "params_df = create_param_combinations(**param_grid)\n",
+    "for param in params_df.values:\n",
+    "    param_dict = dict(zip(params_df.keys(), param))\n",
+    "    cv_df = single_cv_run(df,  metrics, param_dict, parallel=\"processes\")\n",
+    "    results.append(cv_df)\n",
+    "results_df = pd.concat(results).reset_index(drop=True)\n",
+    "best_param = results_df.loc[results_df['rmse'] == min(results_df['rmse']), ['params']]\n",
+    "print(f'\\n The best param combination is {best_param.values[0][0]}')\n",
+    "results_df"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Alternatively, in some cases one could benefit from parallelizing over parameter values instead, when the number of parameter combinations are large and the user has access to a large number of cores or a cluster. In the example below,  parameter combinations are evaluated in parallel using `dask.distributed.Client`. The helper function `parallelize_param_combinations` parallelizes the calls to `single_cv_run` for each parameter combination. The cutoffs in `cross_validation` are then evaluated serially. To switch to other parallel modes in this example, import the `concurrent.futures` module and set `pool=concurrent.futures.ThreadPoolExecutor()`for execution in threads or `pool=concurrent.futures.ProcessPoolExecutor()` for multi-processing."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      " The best param combination is {'changepoint_prior_scale': 0.05, 'changepoint_range': 0.8}\n"
+     ]
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div>\n",
+       "<style scoped>\n",
+       "    .dataframe tbody tr th:only-of-type {\n",
+       "        vertical-align: middle;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe tbody tr th {\n",
+       "        vertical-align: top;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe thead th {\n",
+       "        text-align: right;\n",
+       "    }\n",
+       "</style>\n",
+       "<table border=\"1\" class=\"dataframe\">\n",
+       "  <thead>\n",
+       "    <tr style=\"text-align: right;\">\n",
+       "      <th></th>\n",
+       "      <th>horizon</th>\n",
+       "      <th>rmse</th>\n",
+       "      <th>mape</th>\n",
+       "      <th>params</th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>0</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.450030</td>\n",
+       "      <td>0.034958</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.05, 'changepoint_range': 0.8}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>1</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.453755</td>\n",
+       "      <td>0.035471</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.05, 'changepoint_range': 0.9}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.456887</td>\n",
+       "      <td>0.035469</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.5, 'changepoint_range': 0.8}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>3</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.490453</td>\n",
+       "      <td>0.039134</td>\n",
+       "      <td>{'changepoint_prior_scale': 0.5, 'changepoint_range': 0.9}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>4</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.463969</td>\n",
+       "      <td>0.036428</td>\n",
+       "      <td>{'changepoint_prior_scale': 5.0, 'changepoint_range': 0.8}</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>5</th>\n",
+       "      <td>200 days</td>\n",
+       "      <td>0.512077</td>\n",
+       "      <td>0.040488</td>\n",
+       "      <td>{'changepoint_prior_scale': 5.0, 'changepoint_range': 0.9}</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "   horizon      rmse      mape  \\\n",
+       "0 200 days  0.450030  0.034958   \n",
+       "1 200 days  0.453755  0.035471   \n",
+       "2 200 days  0.456887  0.035469   \n",
+       "3 200 days  0.490453  0.039134   \n",
+       "4 200 days  0.463969  0.036428   \n",
+       "5 200 days  0.512077  0.040488   \n",
+       "\n",
+       "                                                        params  \n",
+       "0  {'changepoint_prior_scale': 0.05, 'changepoint_range': 0.8}  \n",
+       "1  {'changepoint_prior_scale': 0.05, 'changepoint_range': 0.9}  \n",
+       "2   {'changepoint_prior_scale': 0.5, 'changepoint_range': 0.8}  \n",
+       "3   {'changepoint_prior_scale': 0.5, 'changepoint_range': 0.9}  \n",
+       "4   {'changepoint_prior_scale': 5.0, 'changepoint_range': 0.8}  \n",
+       "5   {'changepoint_prior_scale': 5.0, 'changepoint_range': 0.9}  "
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from dask.distributed import Client\n",
+    "import functools\n",
+    "from concurrent.futures import ThreadPoolExecutor, ProcessPoolExecutor\n",
+    "\n",
+    "\n",
+    "def parallelize_param_combinations(history_df, params_df, single_cv_callable, pool):\n",
+    "    results = []\n",
+    "    for param in params_df.values:\n",
+    "        param_dict = dict(zip(params_df.keys(), param))\n",
+    "        if isinstance(pool,(ThreadPoolExecutor, ProcessPoolExecutor)):\n",
+    "            future = pool.submit(single_cv_callable, history_df, param_dict=param_dict)\n",
+    "            results.append(future.result())\n",
+    "        elif isinstance(pool, Client):\n",
+    "            remote_df = pool.scatter(history_df)\n",
+    "            future = pool.submit(single_cv_callable, remote_df, param_dict=param_dict)\n",
+    "            results.append(future)\n",
+    "    if isinstance(pool, Client):\n",
+    "        results = pool.gather(results)\n",
+    "    results_df = pd.concat(results).reset_index(drop=True)\n",
+    "    \n",
+    "    return results_df\n",
+    "\n",
+    "\n",
+    "single_cv_callable = functools.partial(single_cv_run, metrics=metrics, parallel=None)\n",
+    "\n",
+    "pool = Client()\n",
+    "results_df  = parallelize_param_combinations(df, params_df, single_cv_callable, pool=pool)\n",
+    "best_param = results_df.loc[results_df['rmse'] == min(results_df['rmse']), ['params']]\n",
+    "print(f'\\n The best param combination is {best_param.values[0][0]}')\n",
+    "results_df"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Recommended Hyperparameter Ranges"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In the examples above, we have used recommended initial settings for `changepoint_prior_scale:[0.05, 0.5, 5]` and \n",
+    "`changepoint_range: [0.8, 0.9]`. We could alternatively also use a random search to carry out a sweep between a range of values e.g. `np.random.uniform(0.05, 5, 3)`. Other parameters like the `seasonality_prior_scale`, `holidays_prior_scale` and `seasonality_mode` could also be optimised for. For the seasonality and holiday prior scales, recommended values to start with are `[0.1,1,10]`, and it is better to set these  values on a log scale in the grid e.g. `np.random.uniform(-1, 1, 5)`."
    ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python (fbprophet)",
    "language": "python",
-   "name": "python3"
+   "name": "fbprophet"
   },
   "language_info": {
    "codemirror_mode": {
@@ -490,7 +827,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.6"
+   "version": "3.7.0"
   }
  },
  "nbformat": 4,