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+{
+ "cells": [
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": [
+    "# Show how to extract prediction contributions for each distribution parameter\n",
+    "\n",
+    "This example shows how to get the contribution of every feature for each distributional parameter for a given data set. This allows similar inferences as one might get from SHAP but comes from lightGBM's internal workings. We can use output for example to get for a given prediction which features are causing the most impact to a given distributional parameter.\n",
+    "\n",
+    "These contributions are created before the response function is applied. As such in the case of the identity function, for a given row of data the sum of the contributions should equal the parameter value.\n"
+   ],
+   "id": "bf95ab4267d5a34"
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": [
+    "# Imports\n",
+    "\n",
+    "First, we import necessary functions. "
+   ],
+   "id": "bbea43740b87eb"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:33:07.019505Z",
+     "start_time": "2024-09-26T09:33:01.235342Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "from lightgbmlss.model import *\n",
+    "from lightgbmlss.distributions.Gaussian import *\n",
+    "from lightgbmlss.datasets.data_loader import load_simulated_gaussian_data\n",
+    "from scipy.stats import norm\n",
+    "\n",
+    "import plotnine\n",
+    "from plotnine import *\n",
+    "\n",
+    "plotnine.options.figure_size = (12, 8)"
+   ],
+   "id": "b5f2d07ce70bb24b",
+   "outputs": [],
+   "execution_count": 1
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "# Data",
+   "id": "bd7bba77a5e0fa2f"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:33:07.067189Z",
+     "start_time": "2024-09-26T09:33:07.019505Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "train, test = load_simulated_gaussian_data()\n",
+    "\n",
+    "X_train, y_train = train.filter(regex=\"x\"), train[\"y\"].values\n",
+    "X_test, y_test = test.filter(regex=\"x\"), test[\"y\"].values\n",
+    "\n",
+    "dtrain = lgb.Dataset(X_train, label=y_train)"
+   ],
+   "id": "1062b4b851a12bc9",
+   "outputs": [],
+   "execution_count": 2
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": [
+    "# Get a Trained Model\n",
+    "\n",
+    "As this example is about th uses of a trained model, we wont do any hyper-parameter searching. We will also use a Gaussian distribution as the response function of the loc parameter is the identity function, this will allow us to more easily compare the outputs of a standard parameter prediction to a contributions prediction."
+   ],
+   "id": "170feafe1dccf85c"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:58:18.921694Z",
+     "start_time": "2024-09-26T09:57:36.453028Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "lgblss = LightGBMLSS(\n",
+    "    Gaussian()\n",
+    ")\n",
+    "lgblss.train(\n",
+    "    params=dict(),\n",
+    "    train_set=dtrain\n",
+    ")\n",
+    "\n",
+    "param_dict = {\n",
+    "    \"eta\":                      [\"float\", {\"low\": 1e-5,   \"high\": 1,     \"log\": True}],\n",
+    "    \"max_depth\":                [\"int\",   {\"low\": 1,      \"high\": 10,    \"log\": False}],\n",
+    "    \"num_leaves\":               [\"int\",   {\"low\": 255,    \"high\": 255,   \"log\": False}],  # set to constant for this example\n",
+    "    \"min_data_in_leaf\":         [\"int\",   {\"low\": 20,     \"high\": 20,    \"log\": False}],  # set to constant for this example\n",
+    "    \"min_gain_to_split\":        [\"float\", {\"low\": 1e-8,   \"high\": 40,    \"log\": False}],\n",
+    "    \"min_sum_hessian_in_leaf\":  [\"float\", {\"low\": 1e-8,   \"high\": 500,   \"log\": True}],\n",
+    "    \"subsample\":                [\"float\", {\"low\": 0.2,    \"high\": 1.0,   \"log\": False}],\n",
+    "    \"feature_fraction\":         [\"float\", {\"low\": 0.2,    \"high\": 1.0,   \"log\": False}],\n",
+    "    \"boosting\":                 [\"categorical\", [\"gbdt\"]],\n",
+    "}\n",
+    "\n",
+    "np.random.seed(123)\n",
+    "opt_param = lgblss.hyper_opt(param_dict,\n",
+    "                             dtrain,\n",
+    "                             num_boost_round=100,        # Number of boosting iterations.\n",
+    "                             nfold=5,                    # Number of cv-folds.\n",
+    "                             early_stopping_rounds=20,   # Number of early-stopping rounds\n",
+    "                             max_minutes=10,             # Time budget in minutes, i.e., stop study after the given number of minutes.\n",
+    "                             n_trials=30 ,               # The number of trials. If this argument is set to None, there is no limitation on the number of trials.\n",
+    "                             silence=True,               # Controls the verbosity of the trail, i.e., user can silence the outputs of the trail.\n",
+    "                             seed=123,                   # Seed used to generate cv-folds.\n",
+    "                             hp_seed=123                 # Seed for random number generator used in the Bayesian hyperparameter search.\n",
+    "                            )\n",
+    "\n",
+    "np.random.seed(123)\n",
+    "\n",
+    "opt_params = opt_param.copy()\n",
+    "n_rounds = opt_params[\"opt_rounds\"]\n",
+    "del opt_params[\"opt_rounds\"]\n",
+    "\n",
+    "# Train Model with optimized hyperparameters\n",
+    "lgblss.train(opt_params,\n",
+    "             dtrain,\n",
+    "             num_boost_round=n_rounds\n",
+    "             )\n"
+   ],
+   "id": "f45c868160f1f08b",
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "  0%|          | 0/30 [00:00<?, ?it/s]"
+      ],
+      "application/vnd.jupyter.widget-view+json": {
+       "version_major": 2,
+       "version_minor": 0,
+       "model_id": "fe4d614b202c4516931ba7cd69d2c733"
+      }
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "Hyper-Parameter Optimization successfully finished.\n",
+      "  Number of finished trials:  30\n",
+      "  Best trial:\n",
+      "    Value: 2.0839056900194977\n",
+      "    Params: \n",
+      "    eta: 0.042322345196562056\n",
+      "    max_depth: 3\n",
+      "    num_leaves: 255\n",
+      "    min_data_in_leaf: 20\n",
+      "    min_gain_to_split: 10.495083287505906\n",
+      "    min_sum_hessian_in_leaf: 4.025662198099785e-06\n",
+      "    subsample: 0.41879883505881144\n",
+      "    feature_fraction: 0.7628021535153005\n",
+      "    boosting: gbdt\n",
+      "    opt_rounds: 72\n"
+     ]
+    }
+   ],
+   "execution_count": 25
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "# Get parameter predictions and parameter contribution predictions",
+   "id": "3c8358d79ec85438"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:58:28.354001Z",
+     "start_time": "2024-09-26T09:58:28.322325Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "pred_params = lgblss.predict(X_test, pred_type=\"parameters\")\n",
+    "pred_param_contributions = lgblss.predict(X_test, pred_type=\"contributions\")\n"
+   ],
+   "id": "c0bab6ad5807cd8d",
+   "outputs": [],
+   "execution_count": 26
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "As location parameter uses identity function the sum of these predictions should equal the value in pred_params. However this is not true for the scale params, as response functions have not been applied when contributions are created.",
+   "id": "c39c97f68b1ba929"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:58:30.937621Z",
+     "start_time": "2024-09-26T09:58:30.930606Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "sum_of_contributions = pred_param_contributions.groupby(level=\"distribution_arg\", axis=1).sum()\n",
+    "location_values_are_all_close =  np.allclose(pred_params[\"loc\"], sum_of_contributions[\"loc\"])\n",
+    "scale_values_are_all_close =  np.allclose(pred_params[\"scale\"], sum_of_contributions[\"scale\"])\n",
+    "\n",
+    "\n",
+    "print(f\"{location_values_are_all_close=}\")\n",
+    "print(f\"{scale_values_are_all_close=}\")\n"
+   ],
+   "id": "87e216c88a4ff947",
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "location_values_are_all_close=True\n",
+      "scale_values_are_all_close=False\n"
+     ]
+    },
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "C:\\Users\\SimonRobertPike\\AppData\\Local\\Temp\\ipykernel_47316\\1135199838.py:1: FutureWarning: DataFrame.groupby with axis=1 is deprecated. Do `frame.T.groupby(...)` without axis instead.\n"
+     ]
+    }
+   ],
+   "execution_count": 27
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "### Show contributions for each feature for location parameter",
+   "id": "90e4b7d2544afd58"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:58:40.765411Z",
+     "start_time": "2024-09-26T09:58:40.738780Z"
+    }
+   },
+   "cell_type": "code",
+   "source": "pred_param_contributions.xs(\"loc\", axis=1, level=\"distribution_arg\")",
+   "id": "1b6b7013a1f7e957",
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
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+       "0                       0.0       0.0       0.0       0.0       0.0       0.0   \n",
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+       "2999                    0.0       0.0       0.0       0.0       0.0       0.0   \n",
+       "\n",
+       "FeatureContribution  x_noise6  x_noise7  x_noise8  x_noise9  x_noise10  \\\n",
+       "0                         0.0       0.0       0.0       0.0        0.0   \n",
+       "1                         0.0       0.0       0.0       0.0        0.0   \n",
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+       "2999                      0.0       0.0       0.0       0.0        0.0   \n",
+       "\n",
+       "FeatureContribution     Const  \n",
+       "0                    9.979979  \n",
+       "1                    9.979979  \n",
+       "2                    9.979979  \n",
+       "3                    9.979979  \n",
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+       "  </tbody>\n",
+       "</table>\n",
+       "<p>3000 rows × 12 columns</p>\n",
+       "</div>"
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "execution_count": 28
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "### Show contributions for each feature for scale parameter",
+   "id": "eaf2ad3ecc736152"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T09:58:55.637299Z",
+     "start_time": "2024-09-26T09:58:55.621654Z"
+    }
+   },
+   "cell_type": "code",
+   "source": "pred_param_contributions.xs(\"scale\", axis=1, level=\"distribution_arg\")\n",
+   "id": "c5453f7e5a378096",
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "FeatureContribution    x_true  x_noise1  x_noise2  x_noise3  x_noise4  \\\n",
+       "0                    0.410550  0.002106       0.0       0.0  0.000034   \n",
+       "1                    0.411261  0.000684       0.0       0.0 -0.000340   \n",
+       "2                   -0.597674  0.002106       0.0       0.0 -0.000340   \n",
+       "3                    0.848748  0.002832       0.0       0.0  0.000034   \n",
+       "4                    0.414522  0.001565       0.0       0.0  0.000866   \n",
+       "...                       ...       ...       ...       ...       ...   \n",
+       "2995                 0.411230  0.002832       0.0       0.0 -0.000340   \n",
+       "2996                 0.380649  0.002106       0.0       0.0 -0.000340   \n",
+       "2997                -0.597582  0.001647       0.0       0.0  0.000034   \n",
+       "2998                -0.607346 -0.001425       0.0       0.0 -0.001143   \n",
+       "2999                 0.410550  0.002106       0.0       0.0  0.000034   \n",
+       "\n",
+       "FeatureContribution  x_noise5  x_noise6  x_noise7  x_noise8  x_noise9  \\\n",
+       "0                    0.000197  0.004102 -0.000127       0.0 -0.000608   \n",
+       "1                    0.000197  0.004813 -0.000127       0.0 -0.000608   \n",
+       "2                    0.000197  0.004102 -0.000127       0.0 -0.000608   \n",
+       "3                    0.000197  0.001399 -0.000127       0.0 -0.000608   \n",
+       "4                    0.000123  0.002716 -0.004167       0.0  0.053916   \n",
+       "...                       ...       ...       ...       ...       ...   \n",
+       "2995                 0.000197  0.002135 -0.000127       0.0 -0.000608   \n",
+       "2996                 0.000197  0.004400 -0.000127       0.0 -0.000608   \n",
+       "2997                 0.000197 -0.004547 -0.000127       0.0 -0.000700   \n",
+       "2998                 0.000887  0.002013  0.003194       0.0 -0.000029   \n",
+       "2999                 0.000197  0.004102 -0.000127       0.0 -0.000608   \n",
+       "\n",
+       "FeatureContribution  x_noise10     Const  \n",
+       "0                    -0.000503  0.653589  \n",
+       "1                    -0.000129  0.653589  \n",
+       "2                    -0.000129  0.653589  \n",
+       "3                     0.001529  0.653589  \n",
+       "4                     0.001894  0.653589  \n",
+       "...                        ...       ...  \n",
+       "2995                  0.000432  0.653589  \n",
+       "2996                  0.002376  0.653589  \n",
+       "2997                 -0.000893  0.653589  \n",
+       "2998                 -0.004395  0.653589  \n",
+       "2999                 -0.000503  0.653589  \n",
+       "\n",
+       "[3000 rows x 12 columns]"
+      ],
+      "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>FeatureContribution</th>\n",
+       "      <th>x_true</th>\n",
+       "      <th>x_noise1</th>\n",
+       "      <th>x_noise2</th>\n",
+       "      <th>x_noise3</th>\n",
+       "      <th>x_noise4</th>\n",
+       "      <th>x_noise5</th>\n",
+       "      <th>x_noise6</th>\n",
+       "      <th>x_noise7</th>\n",
+       "      <th>x_noise8</th>\n",
+       "      <th>x_noise9</th>\n",
+       "      <th>x_noise10</th>\n",
+       "      <th>Const</th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>0</th>\n",
+       "      <td>0.410550</td>\n",
+       "      <td>0.002106</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.000034</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.004102</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>-0.000503</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>1</th>\n",
+       "      <td>0.411261</td>\n",
+       "      <td>0.000684</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000340</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.004813</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>-0.000129</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2</th>\n",
+       "      <td>-0.597674</td>\n",
+       "      <td>0.002106</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000340</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.004102</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>-0.000129</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>3</th>\n",
+       "      <td>0.848748</td>\n",
+       "      <td>0.002832</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.000034</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.001399</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>0.001529</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>4</th>\n",
+       "      <td>0.414522</td>\n",
+       "      <td>0.001565</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.000866</td>\n",
+       "      <td>0.000123</td>\n",
+       "      <td>0.002716</td>\n",
+       "      <td>-0.004167</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.053916</td>\n",
+       "      <td>0.001894</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>...</th>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "      <td>...</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2995</th>\n",
+       "      <td>0.411230</td>\n",
+       "      <td>0.002832</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000340</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.002135</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>0.000432</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2996</th>\n",
+       "      <td>0.380649</td>\n",
+       "      <td>0.002106</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000340</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.004400</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>0.002376</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2997</th>\n",
+       "      <td>-0.597582</td>\n",
+       "      <td>0.001647</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.000034</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>-0.004547</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000700</td>\n",
+       "      <td>-0.000893</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2998</th>\n",
+       "      <td>-0.607346</td>\n",
+       "      <td>-0.001425</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.001143</td>\n",
+       "      <td>0.000887</td>\n",
+       "      <td>0.002013</td>\n",
+       "      <td>0.003194</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000029</td>\n",
+       "      <td>-0.004395</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2999</th>\n",
+       "      <td>0.410550</td>\n",
+       "      <td>0.002106</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>0.000034</td>\n",
+       "      <td>0.000197</td>\n",
+       "      <td>0.004102</td>\n",
+       "      <td>-0.000127</td>\n",
+       "      <td>0.0</td>\n",
+       "      <td>-0.000608</td>\n",
+       "      <td>-0.000503</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "<p>3000 rows × 12 columns</p>\n",
+       "</div>"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "execution_count": 29
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "# Show Mean Feature Impact for Data set",
+   "id": "394e64d247168fa0"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T10:02:54.858851Z",
+     "start_time": "2024-09-26T10:02:54.838744Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "sum_of_contributions_column = \"SumOfContributions\"\n",
+    "mean_parameter_contribution = pred_param_contributions.abs().mean().unstack(\"distribution_arg\")\n",
+    "mean_parameter_contribution[sum_of_contributions_column] = mean_parameter_contribution.sum(1)\n",
+    "\n",
+    "mean_parameter_contribution.sort_values(sum_of_contributions_column, ascending=False).drop(columns=sum_of_contributions_column)\n"
+   ],
+   "id": "54d4970cf1957735",
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "distribution_arg         loc     scale\n",
+       "FeatureContribution                   \n",
+       "Const                9.97998  0.653589\n",
+       "x_true               0.00000  0.591846\n",
+       "x_noise6             0.00000  0.004865\n",
+       "x_noise7             0.00000  0.004410\n",
+       "x_noise1             0.00000  0.003991\n",
+       "x_noise10            0.00000  0.002688\n",
+       "x_noise9             0.00000  0.002582\n",
+       "x_noise4             0.00000  0.001666\n",
+       "x_noise5             0.00000  0.000585\n",
+       "x_noise2             0.00000  0.000000\n",
+       "x_noise3             0.00000  0.000000\n",
+       "x_noise8             0.00000  0.000000"
+      ],
+      "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>distribution_arg</th>\n",
+       "      <th>loc</th>\n",
+       "      <th>scale</th>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>FeatureContribution</th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>Const</th>\n",
+       "      <td>9.97998</td>\n",
+       "      <td>0.653589</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_true</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.591846</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise6</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.004865</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise7</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.004410</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise1</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.003991</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise10</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.002688</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise9</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.002582</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise4</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.001666</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise5</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.000585</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise2</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.000000</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise3</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.000000</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise8</th>\n",
+       "      <td>0.00000</td>\n",
+       "      <td>0.000000</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ]
+     },
+     "execution_count": 36,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "execution_count": 36
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "# Get correlation between contributions for the scale parameter ",
+   "id": "f7c73f303f04d4ff"
+  },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": "",
+   "id": "8d5dc9e448d5c322"
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-09-26T10:07:21.265976Z",
+     "start_time": "2024-09-26T10:07:21.249801Z"
+    }
+   },
+   "cell_type": "code",
+   "source": "pred_param_contributions.xs(\"scale\", axis=1, level=\"distribution_arg\").corr().dropna(how=\"all\").dropna(axis=1,how=\"all\")\n",
+   "id": "f331d8603042908",
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "FeatureContribution    x_true  x_noise1  x_noise4  x_noise5  x_noise6  \\\n",
+       "FeatureContribution                                                     \n",
+       "x_true               1.000000  0.007743 -0.001227 -0.047812 -0.021568   \n",
+       "x_noise1             0.007743  1.000000 -0.006627 -0.022206  0.136683   \n",
+       "x_noise4            -0.001227 -0.006627  1.000000 -0.015965 -0.030661   \n",
+       "x_noise5            -0.047812 -0.022206 -0.015965  1.000000  0.006217   \n",
+       "x_noise6            -0.021568  0.136683 -0.030661  0.006217  1.000000   \n",
+       "x_noise7             0.015344  0.002144  0.474505  0.021826  0.029863   \n",
+       "x_noise9             0.024361 -0.006972  0.013089  0.016001  0.009558   \n",
+       "x_noise10            0.035479  0.012114 -0.035713 -0.001433  0.028450   \n",
+       "\n",
+       "FeatureContribution  x_noise7  x_noise9  x_noise10  \n",
+       "FeatureContribution                                 \n",
+       "x_true               0.015344  0.024361   0.035479  \n",
+       "x_noise1             0.002144 -0.006972   0.012114  \n",
+       "x_noise4             0.474505  0.013089  -0.035713  \n",
+       "x_noise5             0.021826  0.016001  -0.001433  \n",
+       "x_noise6             0.029863  0.009558   0.028450  \n",
+       "x_noise7             1.000000  0.023556  -0.015318  \n",
+       "x_noise9             0.023556  1.000000  -0.030408  \n",
+       "x_noise10           -0.015318 -0.030408   1.000000  "
+      ],
+      "text/html": [
+       "<div>\n",
+       "<style scoped>\n",
+       "    .dataframe tbody tr th:only-of-type {\n",
+       "        vertical-align: middle;\n",
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+       "\n",
+       "    .dataframe tbody tr th {\n",
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+       "    }\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>FeatureContribution</th>\n",
+       "      <th>x_true</th>\n",
+       "      <th>x_noise1</th>\n",
+       "      <th>x_noise4</th>\n",
+       "      <th>x_noise5</th>\n",
+       "      <th>x_noise6</th>\n",
+       "      <th>x_noise7</th>\n",
+       "      <th>x_noise9</th>\n",
+       "      <th>x_noise10</th>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>FeatureContribution</th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "      <th></th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>x_true</th>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>0.007743</td>\n",
+       "      <td>-0.001227</td>\n",
+       "      <td>-0.047812</td>\n",
+       "      <td>-0.021568</td>\n",
+       "      <td>0.015344</td>\n",
+       "      <td>0.024361</td>\n",
+       "      <td>0.035479</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise1</th>\n",
+       "      <td>0.007743</td>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>-0.006627</td>\n",
+       "      <td>-0.022206</td>\n",
+       "      <td>0.136683</td>\n",
+       "      <td>0.002144</td>\n",
+       "      <td>-0.006972</td>\n",
+       "      <td>0.012114</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise4</th>\n",
+       "      <td>-0.001227</td>\n",
+       "      <td>-0.006627</td>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>-0.015965</td>\n",
+       "      <td>-0.030661</td>\n",
+       "      <td>0.474505</td>\n",
+       "      <td>0.013089</td>\n",
+       "      <td>-0.035713</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise5</th>\n",
+       "      <td>-0.047812</td>\n",
+       "      <td>-0.022206</td>\n",
+       "      <td>-0.015965</td>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>0.006217</td>\n",
+       "      <td>0.021826</td>\n",
+       "      <td>0.016001</td>\n",
+       "      <td>-0.001433</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise6</th>\n",
+       "      <td>-0.021568</td>\n",
+       "      <td>0.136683</td>\n",
+       "      <td>-0.030661</td>\n",
+       "      <td>0.006217</td>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>0.029863</td>\n",
+       "      <td>0.009558</td>\n",
+       "      <td>0.028450</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise7</th>\n",
+       "      <td>0.015344</td>\n",
+       "      <td>0.002144</td>\n",
+       "      <td>0.474505</td>\n",
+       "      <td>0.021826</td>\n",
+       "      <td>0.029863</td>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>0.023556</td>\n",
+       "      <td>-0.015318</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise9</th>\n",
+       "      <td>0.024361</td>\n",
+       "      <td>-0.006972</td>\n",
+       "      <td>0.013089</td>\n",
+       "      <td>0.016001</td>\n",
+       "      <td>0.009558</td>\n",
+       "      <td>0.023556</td>\n",
+       "      <td>1.000000</td>\n",
+       "      <td>-0.030408</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>x_noise10</th>\n",
+       "      <td>0.035479</td>\n",
+       "      <td>0.012114</td>\n",
+       "      <td>-0.035713</td>\n",
+       "      <td>-0.001433</td>\n",
+       "      <td>0.028450</td>\n",
+       "      <td>-0.015318</td>\n",
+       "      <td>-0.030408</td>\n",
+       "      <td>1.000000</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ]
+     },
+     "execution_count": 44,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "execution_count": 44
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "outputs": [],
+   "execution_count": null,
+   "source": "",
+   "id": "ae0a0247ad688b42"
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}