From 493455e0f80c190c5e01ec63554b81c5f0801d55 Mon Sep 17 00:00:00 2001
From: statmlben <bdai.hk@protonmail.com>
Date: Tue, 12 Nov 2024 11:56:41 +0800
Subject: [PATCH] add img RR ipynb

---
 .gitignore                                    |   6 +-
 .../RankRegression-checkpoint.ipynb           | 103 +++++++++++++-----
 doc/source/examples/RankRegression.ipynb      | 103 +++++++++++++-----
 3 files changed, 155 insertions(+), 57 deletions(-)

diff --git a/.gitignore b/.gitignore
index faa2713..0d29f81 100644
--- a/.gitignore
+++ b/.gitignore
@@ -33,4 +33,8 @@ eigen3.zip
 .history/
 
 # Built Visual Studio Code Extensions
-*.vsix
\ No newline at end of file
+*.vsix
+
+# python checkpoint
+*-checkpoint.ipynb
+*-checkpoint.py
diff --git a/doc/source/examples/.ipynb_checkpoints/RankRegression-checkpoint.ipynb b/doc/source/examples/.ipynb_checkpoints/RankRegression-checkpoint.ipynb
index 781baa2..eb93e0c 100644
--- a/doc/source/examples/.ipynb_checkpoints/RankRegression-checkpoint.ipynb
+++ b/doc/source/examples/.ipynb_checkpoints/RankRegression-checkpoint.ipynb
@@ -69,7 +69,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 200,
+   "execution_count": 1,
    "metadata": {
     "id": "r3VoUcR4FceK"
    },
@@ -100,7 +100,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 201,
+   "execution_count": 2,
    "metadata": {
     "id": "OcdNUtvrn_Hy"
    },
@@ -127,7 +127,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 202,
+   "execution_count": 3,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -144,15 +144,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 203,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 811 ms, sys: 105 ms, total: 916 ms\n",
-      "Wall time: 915 ms\n"
+      "CPU times: user 793 ms, sys: 132 ms, total: 925 ms\n",
+      "Wall time: 925 ms\n"
      ]
     }
    ],
@@ -175,7 +175,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 204,
+   "execution_count": 5,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -197,15 +197,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 205,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 744 ms, sys: 77.2 ms, total: 821 ms\n",
-      "Wall time: 819 ms\n"
+      "CPU times: user 737 ms, sys: 95 ms, total: 832 ms\n",
+      "Wall time: 830 ms\n"
      ]
     }
    ],
@@ -227,7 +227,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 206,
+   "execution_count": 7,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -239,13 +239,13 @@
     {
      "data": {
       "text/html": [
-       "<style>#sk-container-id-23 {color: black;}#sk-container-id-23 pre{padding: 0;}#sk-container-id-23 div.sk-toggleable {background-color: white;}#sk-container-id-23 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-23 label.sk-toggleable__label-arrow:before {content: \"▸\";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-23 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-23 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-23 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-23 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-23 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-23 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: \"▾\";}#sk-container-id-23 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-23 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-23 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-23 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-23 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-23 div.sk-parallel-item::after {content: \"\";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-23 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-23 div.sk-serial::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-23 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-23 div.sk-item {position: relative;z-index: 1;}#sk-container-id-23 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-23 div.sk-item::before, #sk-container-id-23 div.sk-parallel-item::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-23 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-23 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-23 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-23 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-23 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-23 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-23 div.sk-label-container {text-align: center;}#sk-container-id-23 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-23 div.sk-text-repr-fallback {display: none;}</style><div id=\"sk-container-id-23\" class=\"sk-top-container\"><div class=\"sk-text-repr-fallback\"><pre>LinearRegression(fit_intercept=False)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class=\"sk-container\" hidden><div class=\"sk-item\"><div class=\"sk-estimator sk-toggleable\"><input class=\"sk-toggleable__control sk-hidden--visually\" id=\"sk-estimator-id-23\" type=\"checkbox\" checked><label for=\"sk-estimator-id-23\" class=\"sk-toggleable__label sk-toggleable__label-arrow\">LinearRegression</label><div class=\"sk-toggleable__content\"><pre>LinearRegression(fit_intercept=False)</pre></div></div></div></div></div>"
+       "<style>#sk-container-id-1 {color: black;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: \"▸\";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: \"▾\";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: \"\";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id=\"sk-container-id-1\" class=\"sk-top-container\"><div class=\"sk-text-repr-fallback\"><pre>LinearRegression(fit_intercept=False)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class=\"sk-container\" hidden><div class=\"sk-item\"><div class=\"sk-estimator sk-toggleable\"><input class=\"sk-toggleable__control sk-hidden--visually\" id=\"sk-estimator-id-1\" type=\"checkbox\" checked><label for=\"sk-estimator-id-1\" class=\"sk-toggleable__label sk-toggleable__label-arrow\">LinearRegression</label><div class=\"sk-toggleable__content\"><pre>LinearRegression(fit_intercept=False)</pre></div></div></div></div></div>"
       ],
       "text/plain": [
        "LinearRegression(fit_intercept=False)"
       ]
      },
-     "execution_count": 206,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -257,8 +257,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 207,
-   "metadata": {},
+   "execution_count": 8,
+   "metadata": {
+    "scrolled": true
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -291,6 +293,60 @@
     "    print(f\"{str(key).ljust(20)} {str(value).ljust(20)}\")"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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zzT/77DPl5+dX+Dr++c9/6g9/+INb77rIyEgNGTKkwuc4nclkci1e6nQ6lZWV5bql6tTb4E5nNpsVGBioL7/8UseOHav06wMAap8TJ05Ikho3bnzG40r322y2Cp87KipK33//vcaMGaNjx45p0aJFGjx4sJo0aaLnnnuuzK3WniQlJSkyMlLx8fG64447FBwcrNWrVysuLk7SyXZmGzZs0F133aUTJ064cnZWVpZ69uypvXv36tChQ5KksLAw7d69W3v37q3wdfzzn/+UJD388MNu46Uz7Crr1FmHx48f12+//aauXbtq//79bi3hThcWFqZff/21Qq0CAADuzpZTJKlBgwaudTtKSkqUnZ0th8Ohjh07lvu96k9/+pPbbOTSO7v2799f5tiNGzeqZ8+e6tGjh1auXHlOa2ddeumlioyMVIsWLTRy5EhdcsklWrNmjWuNsV27dmnv3r0aPHiwsrKyXPkwLy9PPXr00KZNm1RSUiKn06kvvvhCAwYMcLtb+ZJLLlHv3r3Lfe2uXbvq8ssvd20bhqGPPvpI/fr1k2EYbt+Ze/bsqePHj7veq4rkrbCwMG3dulWHDx+u8Pvxz3/+Uw0aNCiTnydNmiTDMLRmzRq38aSkJCUmJrq2r7zySoWEhJT791Sep59+WpGRkYqOjtYNN9ygPXv2aO7cubrjjjtcx1S0XhEWFqa8vDy3dnJns337dh09elRjxoxxW1dm+PDhHu+Ar4hTP4/k5eXpt99+0/XXXy/DMPTdd995fF7pXfx///vfvbrgK6oWRXTAhzRt2vS8Vpfeu3ev1q5d6yqClz6SkpIkqUxP8/K0aNFC69at02effaaFCxeqadOmslqtbr1Z9+7dK8Mw1LJlyzKvtWfPHtfrJCQkaOLEiXrzzTd10UUXqWfPnnrttdfO+OVXkiwWi1q2bFlm/NJLLz2Xt8NNSUmJXnnlFbVs2VImk0kXXXSRIiMj9cMPP5wxHpPJpNmzZ2vNmjWKiorSjTfeqBdffFEZGRmVjgUAUDuUFsdLi+meVLTYfrqYmBilpKToyJEj+vnnnzV//nxFRkZq2rRpeuuttyp0jtdee03r1q3Thx9+qD59+ui3335zKzrs27dPhmFo6tSpZXL2008/Lel/nw+mT5+unJwctWrVSm3bttXkyZP1ww8/nPH1LRaL/P393b54S+eXsyXpm2++UVJSkoKDgxUWFqbIyEhXH9Ez5e3HH39cjRo10rXXXquWLVvqoYce4tZpAKigs+WUUu+8846uvPJK1/oZkZGR+sc//lHu/5+bNWvmtl1aUD99glJhYaFuueUWtW/fXu+///45fy/+6KOPtG7dOq1YsUJ/+MMfdPToUbcCaOkPxMOGDSuTD998803Z7XYdP35cR48eVUFBgS655JIyr1HemHTye++prFarcnJyXOupnPoonTRXmnsrkrdefPFF/ec//1F8fLyuvfZaPfPMM2ctblssFsXGxpb5bHLZZZe59p/q9L8n6eTfVUUnko0ePVrr1q3TJ5984uoZ7nQ63Y6paL3iwQcfVKtWrdS7d2/FxcVp5MiRZ+3PXno9p9cRGjZseE4Lv5/u4MGDGj58uMLDw9WoUSNFRkaqa9euks78eeRPf/qTOnfurPvuu09RUVEaNGiQ3n//fQrqtRw90QEfcq693k5PSiUlJfrjH/+oxx57rNzjW7VqddZzBgcHu5KYJHXu3FlXX321/vznP2v+/Pmu1/Hz89OaNWvUoEGDMudo1KiR689z587V8OHD9fe//12ff/65Hn74Yc2aNUtbtmxxm9FQ1U5/b55//nlNnTpVI0eO1HPPPafw8HD5+/tr/PjxZ01k48ePV79+/bRq1Sp99tlnmjp1qmbNmqUNGzaoffv21XYNAADvCg0NVUxMzFkLyT/88IOaNm2qkJCQSr2On5+fWrVqpVatWumWW25Ry5YttXz5ct13331nfe61116rjh07SpIGDBigLl26aPDgwfr555/VqFEjV4579NFH1bNnz3LPUVoUuPHGG5WamurK2W+++aZeeeUVLVq0qEKxVOQ6y3N6zk5NTVWPHj3UunVrvfzyy4qPj1dgYKD++c9/6pVXXjlj3r7sssv0888/69NPP9XatWv10UcfaeHChZo2bZqeffbZ874GAKjLzpZTJOmvf/2rhg8frgEDBmjy5Mlq0qSJGjRooFmzZpVZRFJSud8XJZW548pkMqlPnz76+9//rrVr16pv377nFPuNN96oiy66SJLUr18/tW3bVkOGDNGOHTvk7+/vyh0vvfRSmb7tpRo1aqTCwsJzel2p7Pf40te65557PK57cuWVV0qqWN666667dMMNN+jjjz/W559/rpdeekmzZ8/WypUrPc6OP1cV/XvypGXLlq46Qt++fdWgQQM98cQT6t69u+vfVEXrFU2aNNGuXbv02Wefac2aNVqzZo2WLFmie++9V++88865XloZFf084nQ69cc//lHZ2dl6/PHH1bp1awUHB+vQoUMaPnz4GT+PmM1mbdq0SRs3btQ//vEPrV27Vv/3f/+nm266SZ9//rnH9xu+jSI6UAtceOGFysnJcRsrKirSkSNH3MYSExOVm5vrVgQ/X1deeaXuuecevf7663r00UfVrFkzJSYmyjAMJSQkVKgw37ZtW7Vt21ZPPfWU/v3vf6tz585atGiRZsyYUe7xzZs3L/dW8p9//rnMWEXfmw8//FDdu3cvM7MvJyfH9WHrTBITEzVp0iRNmjRJe/fu1VVXXaW5c+fqr3/961mfCwCovfr27as33nhDX3/9tbp06VJm/1dffaUDBw7ogQceqJLXu/jii3XhhReWyWMVUVrE6N69u1599VU98cQTrtlXDRs2rNDng/DwcI0YMUIjRoxQbm6ubrzxRj3zzDMei+jNmzdXSUmJUlNT3WafVzRnS2Vnw33yySey2+1avXq128y40nZxZxMcHKw//elP+tOf/qSioiLdfvvtmjlzpqZMmeJ2Zx0AwLPycop08nvVxRdfrJUrV7oVI0vvbqosPz8/LV++XP3799edd96pNWvWqFu3bpU6V6NGjfT0009rxIgRev/99zVo0CDXHVMhISFnzIdNmjRRUFCQ9u3bV2ZfeWPliYyMVOPGjeV0OiuUeyuSt2JiYvTggw/qwQcf1NGjR3X11Vdr5syZHovozZs31xdffKETJ064zUb/73//69pfnZ588km98cYbeuqpp1yzyM+lXhEYGKh+/fqpX79+Kikp0YMPPqjXX39dU6dOLfeOgNLr2bt3r2666SbXeHFxsdLS0tSuXTvXWOndEKd/Jjn988iPP/6oX375Re+8847uvfde13hF28z4+/urR48e6tGjh15++WU9//zzevLJJ7Vx48Yqrdmg5tDOBagFEhMTy/QzX7x4cZlfSu+66y5t3rxZn332WZlz5OTkyOFwVOr1H3vsMRUXF+vll1+WJN1+++1q0KCBnn322TK/TBuGoaysLEkne8Oe/ppt27aVv7+/7Ha7x9fr06ePtmzZom+//dY1ZrVatXz58jLHVvS9adCgQZlYP/jgA1cfWE/y8/PLzEZITExU48aNz3gNAIC6YfLkyTKbzXrggQdc+a1Udna2xowZowsuuECTJ08+p/Nu3bpVeXl5Zca//fZbZWVlVbodSrdu3XTttddq3rx5KiwsVJMmTdStWze9/vrr5RbmrVar68+nX1+jRo10ySWXnDHflX55L71brdS8efPKHJuYmKjjx4+7zew/cuSIPv74Y7fjSmdnnZq3jx8/riVLlniMw9M1BAYG6vLLL5dhGCouLj7r8wEA/3N6TpHK/3/01q1btXnz5vN+vcDAQK1cuVLXXHON+vXr5/Z98FwNGTJEcXFxmj17tiSpQ4cOSkxM1Jw5c5Sbm1vm+NJ82KBBAyUlJWnVqlVuPcj37dtXpo+4Jw0aNNDAgQP10Ucf6T//+Y/H15LOnrecTmeZtiFNmjRRbGzsWb9TO51Ovfrqq27jr7zyivz8/KpsBrsnYWFheuCBB/TZZ59p165dkiperzj9PfH393fN3Pd0zR07dlRkZKQWLVqkoqIi1/jSpUvLFMtLf1A5tY7gdDq1ePFit+PK+7duGIb+8pe/eLzuUtnZ2WXGSu+AoI5QezETHagF7rvvPo0ZM0YDBw7UH//4R33//ff67LPPysygnjx5slavXq2+fftq+PDh6tChg/Ly8vTjjz/qww8/1IEDByo06/p0l19+ufr06aM333xTU6dOVWJiombMmKEpU6bowIEDGjBggBo3bqy0tDR9/PHHGj16tB599FFt2LBBY8eO1Z133qlWrVrJ4XDo3XffdX2o8OSxxx7Tu+++q169eumRRx5RcHCwFi9erObNm5e5pb6i703fvn01ffp0jRgxQtdff71+/PFHLV++/Kz90X755Rf16NFDd911ly6//HIFBATo448/VmZmpgYNGnTO7yUAoHZp2bKl3nnnHQ0ZMkRt27bVqFGjlJCQoAMHDuitt97Sb7/9pr/97W9leoJL0ttvv11uD89HHnlE7777rpYvX67bbrtNHTp0UGBgoPbs2aO3335bQUFBrv7flTF58mTdeeedWrp0qcaMGaPXXntNXbp0Udu2bXX//ffr4osvVmZmpjZv3qxff/1V33//vaST+b5bt27q0KGDwsPDtX37dn344YcaO3asx9e66qqrdPfdd2vhwoU6fvy4rr/+eq1fv77c2XqDBg3S448/rttuu00PP/yw8vPzlZKSolatWrktRnfzzTe7ZqA98MADys3N1RtvvKEmTZqcdYb+zTffrOjoaHXu3FlRUVHas2ePXn31Vd1yyy3n3LMeAFA2p/Tt21crV67UbbfdpltuuUVpaWlatGiRLr/88nKL0+fKbDbr008/1U033aTevXvrX//6l6644opzPk/Dhg31yCOPaPLkyVq7dq169eqlN998U71791abNm00YsQINW3aVIcOHdLGjRsVEhKiTz7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nm80me1aWJplMijebteXYMc10OORkNjpw3sjdACrr1NnmJU6HDmdkqri4yO2YiOhYWuIAVYzcDQCA7/CvqhO1bNlSL7zwQplfywHgXMWbzUoMDlZMUFCZffbiYlksFqWmpio1NVVWq9ULEQJ1A7kbQEWcOtu804AhKikpkdPp9HZYQL1E7gYAwDsqvbBouScLCNDhw4er8pQA4JJVVKT9FoteGDfO1XfVFBGhlBUruGUcqCRyN4CKioiOVWBg4Dk/79T1TiQWHgXOF7kb9d255hXW3QJQFSpVRF+9erXbtmEYOnLkiF599VV17ty5SgID6iKSd1mlM8uls78nuU6nAh0OTQgMVKuwMKUXFGhuVhZ9V4EKIHcD8Aar1aqRo8foREGha4yFR4GKIXcDZVmtViXfN1j23CzXmKlRhFLeLH9i1enH5+XblZmRLnsx624BODeVKqIPGDDAbdvPz0+RkZG66aabNHfu3KqIC6hzWDSzrNNnllf0PYkLClJicPDJDbu9BiIFaj9yNwBvsNlsOlFQqG5DkxURE8fCo8A5IHcDZdlsNtlzszSpn0nxkWalWws09xPPE6tOP37LnmOaucwhp5MJbQDOTaWK6CUlJVUdB1DnsWhmWafPLOc9AaoPuRuAN0XExLHwKHCOyN2AZ/GRZiU2/X1ildwnVpV3B3h8ZJgSmwbLkllQw5HWDvaiit8hDtRXVdoTHajvKtKbrXTRTEsBybtU6cxy3hMAAAAAQGXRvuXcZdmKtD/Nohem/X6HOO8ZUK5KFdEnTpxY4WNffvnlyrwEUOuc3q5FYtFLAL6D3A0AQO1C7gbOHe1bzl1ugVOB/g5N6BuoVvFhvGeAB5Uqon/33Xf67rvvVFxcrEsvvVSS9Msvv6hBgwa6+uqrXcf5+flVTZRALXB6uxYWvQTgS8jdAADULuRuoPJK273QvqXi4iKDeM+AM6hUEb1fv35q3Lix3nnnHV144YWSpGPHjmnEiBG64YYbNGnSpCoNEqhNStu1SGLRy9/Zi+mvBngbuRsAgNqF3A0AgO+oVBF97ty5+vzzz12JXJIuvPBCzZgxQzfffDPJHIBLVlGR9lssemHc7/3V7HZlpqfLHkp/NaAmkbsBAKhdyN0AAPgO/8o8yWazyWq1lhm3Wq06ceLEeQcFoO7IdToV6HBoQmCg5oWFaVRgoAyHQ05mowM1itwNAEDtQu4GAMB3VKqIftttt2nEiBFauXKlfv31V/3666/66KOPNGrUKN1+++1VHSOAOiAuKEiJwcGKCQrydihAvUTuBgCgdiF3AwDgOyrVzmXRokV69NFHNXjwYBUXF588UUCARo0apZdeeqlKAwQAAOeP3A0AQO1C7gYAwHdUqoh+wQUXaOHChXrppZeUmpoqSUpMTFRw6WKKAADAp5C7AQCoXcjdAAD4jkq1cyl15MgRHTlyRC1btlRwcLAMw6iquIB6yV5cLIvFotTUVFksFjnoGw6gipG7AQCoXcjdAAB4X6VmomdlZemuu+7Sxo0b5efnp7179+riiy/WqFGjdOGFF2ru3LlVHSdQ52UVFWm/xaIXxo2TyWRSnt2uzPR02UNDvR0agDqA3A2gphQV2WWxWCSJSQHAeSB3AwDgOyo1E33ChAlq2LChDh48qAsuuMA1/qc//Ulr166tsuCA+iTX6VSgw6EJgYGaFxamUYGBMhwOOfniCaAKkLsB1IQTOdlKS92vJ597XqMeelhTnpkuy8GDKi4u8nZoQK1D7gYAwHdUaib6559/rs8++0xxcXFu4y1btnTNOgFQOXFBQUoMDpaloMDboQCoQ8jdQNWwWq2y2WySmGVdnsL8PPk3bKiuQ5PVtEWi9u7apo8WzpHT6fR2aECtQ+4GAMB3VKqInpeX5/ZLeKns7GyZTKbzDgoAAFQtcjdw/qxWq0aOHqMTBYWSpIL8PB3OyGSWdTkiomMV3TxB1sPp3g4FqLXI3QAA+I5KtXO54YYbtGzZMte2n5+fSkpK9OKLL6p79+5VFhwAAKga5G7g/NlsNp0oKFS3ocka+NhMdRowRCUlJcyyBlAtyN0AAPiOSs1Ef/HFF9WjRw9t375dRUVFeuyxx7R7925lZ2frm2++qeoYAQDAeSJ3A1UnIiaOWdYAql1N5u5Dhw7p8ccf15o1a5Sfn69LLrlES5YsUceOHSVJhmHo6aef1htvvKGcnBx17txZKSkpatmyZZXGAQCAr6rUTPQrrrhCv/zyi7p06aL+/fsrLy9Pt99+u7777jslJiZWdYwAAOA8kbsBAKhdaip3Hzt2TJ07d1bDhg21Zs0a/fTTT5o7d64uvPBC1zEvvvii5s+fr0WLFmnr1q0KDg5Wz549VVhYWGVxAADgy855JnpxcbF69eqlRYsW6cknn6yOmAAAQBUidwMAULvUZO6ePXu24uPjtWTJEtdYQkKC68+GYWjevHl66qmn1L9/f0nSsmXLFBUVpVWrVmnQoEHVGh8AAL7gnGeiN2zYUD/88EN1xAIAAKoBuRuoPKvVqtTUVKWmpspiscjhcHg7JAD1QE3m7tWrV6tjx46688471aRJE7Vv315vvPGGa39aWpoyMjKUlJTkGgsNDVWnTp20efPmGokRAABvq1Q7l3vuuUdvvfVWVccCAACqCbkbOHdWq1UjR4/RqIce1qiHHtaUZ6bLcvCgiouLvB0agHqgpnL3/v37Xf3NP/vsMyUnJ+vhhx/WO++8I0nKyMiQJEVFRbk9LyoqyrWvPHa7XTabze0BVAd7UbEsFgs/eAOoVpVaWNThcOjtt9/WF198oQ4dOig4ONht/8svv1wlwQEAgKpB7gbOnc1m04mCQnUbmqyImDjt3bVNHy2cI6fT6e3QANQDNZW7S0pK1LFjRz3//POSpPbt2+s///mPFi1apGHDhlX6vLNmzdKzzz5bJTECnmTZirQ/zaIXpo2TyWRSXr5dmRnpsheHejs0AHXMORXR9+/frxYtWug///mPrr76aknSL7/84naMn59f1UUHAADOC7kbOH8RMXGKbp4g6+F0b4cCoB6o6dwdExOjyy+/3G3ssssu00cffSRJio6OliRlZmYqJibGdUxmZqauuuoqj+edMmWKJk6c6Nq22WyKj4+vsrgBScotcCrQ36EJfQPVKj5MW/Yc08xlDjmdzEYHULXOqYjesmVLHTlyRBs3bpQk/elPf9L8+fPL3NZVlQ4dOqTHH39ca9asUX5+vi655BItWbJEHTt2lHRykZOnn35ab7zxhnJyctS5c2fXrWgAANR33sjdAACg8mo6d3fu3Fk///yz29gvv/yi5s2bSzq5yGh0dLTWr1/vKprbbDZt3bpVycnJHs9rMplkMpmqJWbgdHGRQUpsGixLZoG3QwFQR51TT3TDMNy216xZo7y8vCoN6FTHjh1T586d1bBhQ61Zs0Y//fST5s6dqwsvvNB1zIsvvqj58+dr0aJF2rp1q4KDg9WzZ08VFhZWW1wAANQWNZ27AQDA+anp3D1hwgRt2bJFzz//vPbt26cVK1Zo8eLFeuihhySdnPU+fvx4zZgxQ6tXr9aPP/6oe++9V7GxsRowYEC1xQUAgC+pVE/0Uqcn96o2e/ZsxcfHa8mSJa6xhIQEt9efN2+ennrqKfXv31+StGzZMkVFRWnVqlUaNGhQtcYHAEBtU925GwAAVK3qzt3XXHONPv74Y02ZMkXTp09XQkKC5s2bpyFDhriOeeyxx5SXl6fRo0crJydHXbp00dq1axUUFFStsQEA4CvOaSa6n59fmd5r1dlHdfXq1erYsaPuvPNONWnSRO3bt9cbb7zh2p+WlqaMjAwlJSW5xkJDQ9WpUydt3rzZ43lZJRwAUF/UdO6WTrZiu+eeexQRESGz2ay2bdtq+/btrv2GYWjatGmKiYmR2WxWUlKS9u7dW60xAQBQW3gjd/ft21c//vijCgsLtWfPHt1///1lXn/69OnKyMhQYWGhvvjiC7Vq1apaYwIAwJec00x0wzA0fPhwV1+zwsJCjRkzpswq4StXrqyS4Pbv36+UlBRNnDhRf/7zn7Vt2zY9/PDDCgwM1LBhw5SRkSFJZXrDRUVFufaVh1XCAQD1RU3n7tJWbN27d9eaNWsUGRmpvXv3ltuK7Z133lFCQoKmTp2qnj176qeffmJGGwCg3qvp3A0AAM7unIrow4YNc9u+5557qjSY05WUlKhjx456/vnnJUnt27fXf/7zHy1atKhMLOeCVcIBAPVFTeduWrEBAHB+ajp3AwCAszunIvqpX4hrQkxMjC6//HK3scsuu0wfffSRJCk6OlqSlJmZqZiYGNcxmZmZrlXDy8Mq4agp9uJiWSwWSZLFYpHD4fByRADqm5rO3atXr1bPnj1155136l//+peaNm2qBx980HVb+NlasVFEBwDUdzWduwEAwNmd18Ki1a1z5876+eef3cZ++eUXNW/eXNLJmW3R0dFav369q2hus9m0detWJScn13S4gJusoiLtt1j0wrhxMplMyrPblZmeLntoqLdDA4BqU12t2Ox2u+x2u2ub9UwAAAAAADXFp4voEyZM0PXXX6/nn39ed911l7799lstXrxYixcvlnRycZPx48drxowZatmypauvamxsrAYMGODd4FHv5TqdCnQ4NCEwUK3CwrTl2DHNdDjkZDY6gDqsulqxsZ4JAAAAAMBb/L0dwJlcc801+vjjj/W3v/1NV1xxhZ577jnNmzdPQ4YMcR3z2GOPady4cRo9erSuueYa5ebmau3atSxMBp8RFxSkxOBgxfBvEkA94KkV28GDByW5t2I7VWZmpmtfeaZMmaLjx4+7Hunp6VUcOQAAAAAA5fPpmeiS1LdvX/Xt29fjfj8/P02fPl3Tp0+vwagAAEB5qqsVG+uZAAAAAAC8xeeL6AAAoPagFRsAAAAAoK6hiA4AAKpMaSu2KVOmaPr06UpISCi3FVteXp5Gjx6tnJwcdenShVZsAAAAAACfRREdAABUKVqxAQA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Sy+XSP//5T1111VW+dZmZmbrqqqv0t7/9LYE1S22nT5+WJN53UZo3b56+//3vB7wfgVRCDI8v4nfniN/xR+zuHmI3UhUxO76I2Z0jZscfMbv7iNuxy050BXCG1+vV7NmzNWfOHI0fP15Hjx5NdJVS3sGDB/XQQw9p1apVia5KUvr888/l8Xh07rnnBqw/99xztX///gTVKrW1tbXp9ttv17e+9S199atfTXR1UsZzzz2nPXv2aPfu3YmuCtAtxPD4In6HR/yOL2J39xC7kaqI2fFFzA6PmB1fxOzuI27HB0+i97C7775bGRkZYV/79+/XQw89pIaGBi1evDjRVU46Xb2G/o4fP64pU6bohz/8oW699dYE1Rx9zbx58/TRRx/pueeeS3RVUobVatXPf/5zPfvss8rNzU10dYAAxPDYEL+RCojd0SN2IxkRs2NDzEYqIGZ3D3E7fjK8Xq830ZVIZzabTXa7Pew+559/vm644Qa9+uqrysjI8K33eDzKysrSTTfdpKeeeqqnq5q0unoNTSaTJKmmpkbl5eX65je/qSeffFKZmfxWFIrL5VK/fv304osvatq0ab71s2bN0qlTp7R169bEVS4FzZ8/X1u3btXbb7+t0tLSRFcnZbzyyiv6wQ9+oKysLN86j8ejjIwMZWZmyjCMgG1AbyKGx4b43TOI3/FD7O4eYjeSETE7NsTsnkHMjh9idvcRt+OHTvQkcezYMTkcDt9yTU2NJk+erBdffFETJ05UcXFxAmuXOo4fP64rrrhC48aN0zPPPENDEMHEiRM1YcIEPfTQQ5LODI8677zzNH/+fN19990Jrl1q8Hq9qqys1Msvv6wdO3Zo1KhRia5SSmloaFB1dXXAuptvvlkXXnihfvnLXzJMDymBGB474nd0iN+xIXbHhtiNVEbMjh0xOzrE7NgQs2NH3I4fcqInifPOOy9gecCAAZKksrIyAnkXHT9+XOXl5RoxYoRWrVolm83m21ZUVJTAmiWvRYsWadasWRo/frwmTJigtWvXqqmpSTfffHOiq5Yy5s2bp82bN2vr1q0aOHCgamtrJUl5eXmyWCwJrl3yGzhwYIeg3b9/fxUUFBDMkTKI4bEhfkeP+B0bYndsiN1IZcTs2BCzo0fMjg0xO3bE7fihEx1pY/v27Tp48KAOHjzY4QMQAy5Cu/HGG2Wz2bRkyRLV1tbq4osv1ptvvtlh4hN0rqqqSpJUXl4esH7jxo2aPXt271cIAFIM8Tt6xO/YELsBoHuI2dEjZseGmI1kQjoXAAAAAAAAAAA6wewPAAAAAAAAAAB0gk50AAAAAAAAAAA6QSc6AAAAAAAAAACdoBMdAAAAAAAAAIBO0IkOAAAAAAAAAEAn6EQHAAAAAAAAAKATdKIDAAAAAAAAANAJOtEBAAAAAAAAAOgEnehAH3L06FFlZGRo7969ne6zY8cOZWRk6NSpU3E9d0ZGhl555ZW4lhmsvLxct99+e4+eAwCA3kTsBgAgtRC7gfREJzqQRGbPnq2MjAxlZGQoJydHpaWluuuuu9TS0hKX8ktKSnTixAl99atfjUt58XLttddqypQpIbe98847ysjI0H/+859erhUAAJERuzsidgMAkhmxuyNiNxAZnehAkpkyZYpOnDihw4cPa82aNXr00Ue1dOnSuJSdlZWloqIiZWdnx6W8eKmoqND27dv16aefdti2ceNGjR8/XhdddFECagYAQGTE7kDEbgBAsiN2ByJ2A5HRiQ4kGbPZrKKiIpWUlGjatGm66qqrtH37dt/2trY2rVixQqWlpbJYLBo7dqxefPFF3/b6+nrddNNNKiwslMVi0ahRo7Rx40ZJoYeVvf766xo9erQsFouuuOIKHT16NKA+v/71r3XxxRcHrFu7dq1GjhzpW969e7e++93vavDgwcrLy9OkSZO0Z8+eLv/NU6dOVWFhoZ588smA9Y2NjXrhhRdUUVEhu92uH/3oRxo+fLj69eunr33ta/rjH/8YttxQQ9ny8/MDzmO1WnXDDTcoPz9f55xzjq677roO1wAAgHCI3V8gdgMAUgGx+wvEbqBr6EQHkthHH32k999/XyaTybduxYoV2rRpk9avX6+PP/5YCxcu1I9//GPt3LlTknTffffpv//9r9544w3t27dPVVVVGjx4cMjyrVarrr/+el177bXau3evfvrTn+ruu++Oup4NDQ2aNWuW3n33Xe3atUujRo3SNddco4aGhi4dn52drZ/85Cd68skn5fV6fetfeOEFeTwe/ehHP1JLS4vGjRunbdu26aOPPtJtt92mmTNn6oMPPoi6vu1aW1s1efJkDRw4UO+8847ee+89DRgwQFOmTJHL5ep2uQCAvovYTewGAKQWYjexG+iK5BpbAkCvvfaaBgwYILfbLcMwlJmZqYcffliSZBiGli9frr/85S+69NJLJUnnn3++3n33XT366KOaNGmSjh07pq9//esaP368JAX8ch2sqqpKZWVlWr16tSTpS1/6kj788EP97ne/i6rOV155ZcDyY489pvz8fO3cuVNTp07tUhm33HKLfv/732vnzp0qLy+XdGZI2fTp05WXl6e8vDzdeeedvv0rKyv15z//Wc8//7wmTJgQVX3bbdmyRW1tbXr88ceVkZHhO2d+fr527Nihq6++ulvlAgD6FmI3sRsAkFqI3cRuIFp0ogNJ5oorrlBVVZWampq0Zs0aZWdna/r06ZKkgwcPqrm5Wd/97ncDjnG5XPr6178uSZo7d66mT5+uPXv26Oqrr9a0adN02WWXhTzXvn37NHHixIB17R8SolFXV6d7771XO3bs0GeffSaPx6Pm5mYdO3asy2VceOGFuuyyy7RhwwaVl5fr4MGDeuedd/Tb3/5WkuTxeLR8+XI9//zzOn78uFwulwzDUL9+/aKub7t///vfOnjwoAYOHBiwvqWlRYcOHep2uQCAvoXYTewGAKQWYjexG4gWnehAkunfv78uuOACSdKGDRs0duxYPfHEE6qoqFBjY6Mkadu2bRo+fHjAcWazWZL0ve99T9XV1Xr99de1fft2fec739G8efO0atWqbtUnMzMzYKiXdGY4lr9Zs2bJbrfrwQcf1IgRI2Q2m3XppZdGPTSroqJClZWVWrdunTZu3KiysjJNmjRJkvT73/9eDz74oNauXauvfe1r6t+/v26//faw58jIyAhb98bGRo0bN07PPvtsh2MLCwujqjsAoO8idhO7AQCphdhN7AaiRU50IIllZmbqnnvu0b333iun06kvf/nLMpvNOnbsmC644IKAV0lJie+4wsJCzZo1S88884zWrl2rxx57LGT5Y8aM6ZDbbNeuXQHLhYWFqq2tDQiK/hOkSNJ7772nBQsW6JprrtFXvvIVmc1mff7551H/vTfccIMyMzO1efNmbdq0SbfccotvuNd7772n6667Tj/+8Y81duxYnX/++frf//4XtrzCwkKdOHHCt3zgwAE1Nzf7li+55BIdOHBAQ4YM6XA98/Lyoq4/AADEbmI3ACC1ELuJ3UBX0IkOJLkf/vCHysrK0rp16zRw4EDdeeedWrhwoZ566ikdOnRIe/bs0UMPPaSnnnpKkrRkyRJt3bpVBw8e1Mcff6zXXntNY8aMCVn2nDlzdODAAf3iF7/QJ598os2bN3eYqbu8vFw2m00PPPCADh06pHXr1umNN94I2GfUqFF6+umntW/fPv3973/XTTfdJIvFEvXfOmDAAN14441avHixTpw4odmzZwecY/v27Xr//fe1b98+/exnP1NdXV3Y8q688ko9/PDD+te//qV//OMfmjNnjnJycnzbb7rpJg0ePFjXXXed3nnnHR05ckQ7duzQggUL9Omnn0ZdfwAAJGK3/zmI3QCAVEDs/uIcxG4gNDrRgSSXnZ2t+fPn64EHHlBTU5Puv/9+3XfffVqxYoXGjBmjKVOmaNu2bSotLZUkmUwmLV68WBdddJEuv/xyZWVl6bnnngtZ9nnnnaeXXnpJr7zyisaOHav169dr+fLlAfuMGTNGjzzyiNatW6exY8fqgw8+CJhoRJKeeOIJ1dfX65JLLtHMmTO1YMECDRkypFt/b0VFherr6zV58mQNGzbMt/7ee+/VJZdcosmTJ6u8vFxFRUWaNm1a2LJWr16tkpISffvb39aMGTN05513BuRy69evn95++22dd955uv766zVmzBhVVFSopaVFgwYN6lb9AQAgdp9B7AYApApi9xnEbqBzGd7gxEUAAAAAAAAAAEAST6IDAAAAAAAAANApOtEBAAAAAAAAAOgEnegAAAAAAAAAAHSCTnQAAAAAAAAAADpBJzoAAAAAAAAAAJ2gEx0AAAAAAAAAgE7QiQ4AAAAAAAAAQCfoRAcAAAAAAAAAoBN0ogMAAAAAAAAA0Ak60QEAAAAAAAAA6ASd6AAAAAAAAAAAdIJOdAAAAAAAAAAAOvH/rEklmCK3TugAAAAASUVORK5CYII=",
+      "text/plain": [
+       "<Figure size 1500x500 with 3 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "## plot residual results\n",
+    "import pandas as pd\n",
+    "import seaborn as sns\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "res_true = y - X.dot(beta_true)\n",
+    "res_ols = y - ols.predict(X)\n",
+    "res_RR = y - np.mean(y) - X.dot(RankReg.coef_)\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "# Create a figure with three subplots\n",
+    "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(15, 5))\n",
+    "\n",
+    "# Histogram for true residuals\n",
+    "ax1.hist(res_true, bins=100, alpha=0.7, color='red', edgecolor='black')\n",
+    "ax1.set_title('True Residuals')\n",
+    "ax1.set_xlabel('Residual Value')\n",
+    "ax1.set_ylabel('Frequency')\n",
+    "\n",
+    "# Histogram for OLS residuals\n",
+    "ax2.hist(res_ols, bins=100, alpha=0.7, color='skyblue', edgecolor='black')\n",
+    "ax2.set_title('OLS Residuals')\n",
+    "ax2.set_xlabel('Residual Value')\n",
+    "ax2.set_ylabel('Frequency')\n",
+    "\n",
+    "# Histogram for RR residuals\n",
+    "ax3.hist(res_RR, bins=100, alpha=0.7, color='orange', edgecolor='black')\n",
+    "ax3.set_title('Rank Regression Residuals')\n",
+    "ax3.set_xlabel('Residual Value')\n",
+    "ax3.set_ylabel('Frequency')\n",
+    "\n",
+    "# Set layout and show plot\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {
@@ -308,22 +364,13 @@
    "provenance": []
   },
   "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
+   "display_name": "Python 3",
    "name": "python3"
   },
   "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.10.12"
-  }
+   "name": "python"
+  },
+  "nbformat_minor": 0
  },
  "nbformat": 4,
  "nbformat_minor": 4
diff --git a/doc/source/examples/RankRegression.ipynb b/doc/source/examples/RankRegression.ipynb
index 781baa2..eb93e0c 100644
--- a/doc/source/examples/RankRegression.ipynb
+++ b/doc/source/examples/RankRegression.ipynb
@@ -69,7 +69,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 200,
+   "execution_count": 1,
    "metadata": {
     "id": "r3VoUcR4FceK"
    },
@@ -100,7 +100,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 201,
+   "execution_count": 2,
    "metadata": {
     "id": "OcdNUtvrn_Hy"
    },
@@ -127,7 +127,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 202,
+   "execution_count": 3,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -144,15 +144,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 203,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 811 ms, sys: 105 ms, total: 916 ms\n",
-      "Wall time: 915 ms\n"
+      "CPU times: user 793 ms, sys: 132 ms, total: 925 ms\n",
+      "Wall time: 925 ms\n"
      ]
     }
    ],
@@ -175,7 +175,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 204,
+   "execution_count": 5,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -197,15 +197,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 205,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 744 ms, sys: 77.2 ms, total: 821 ms\n",
-      "Wall time: 819 ms\n"
+      "CPU times: user 737 ms, sys: 95 ms, total: 832 ms\n",
+      "Wall time: 830 ms\n"
      ]
     }
    ],
@@ -227,7 +227,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 206,
+   "execution_count": 7,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -239,13 +239,13 @@
     {
      "data": {
       "text/html": [
-       "<style>#sk-container-id-23 {color: black;}#sk-container-id-23 pre{padding: 0;}#sk-container-id-23 div.sk-toggleable {background-color: white;}#sk-container-id-23 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-23 label.sk-toggleable__label-arrow:before {content: \"▸\";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-23 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-23 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-23 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-23 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-23 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-23 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: \"▾\";}#sk-container-id-23 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-23 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-23 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-23 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-23 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-23 div.sk-parallel-item::after {content: \"\";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-23 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-23 div.sk-serial::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-23 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-23 div.sk-item {position: relative;z-index: 1;}#sk-container-id-23 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-23 div.sk-item::before, #sk-container-id-23 div.sk-parallel-item::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-23 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-23 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-23 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-23 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-23 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-23 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-23 div.sk-label-container {text-align: center;}#sk-container-id-23 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-23 div.sk-text-repr-fallback {display: none;}</style><div id=\"sk-container-id-23\" class=\"sk-top-container\"><div class=\"sk-text-repr-fallback\"><pre>LinearRegression(fit_intercept=False)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class=\"sk-container\" hidden><div class=\"sk-item\"><div class=\"sk-estimator sk-toggleable\"><input class=\"sk-toggleable__control sk-hidden--visually\" id=\"sk-estimator-id-23\" type=\"checkbox\" checked><label for=\"sk-estimator-id-23\" class=\"sk-toggleable__label sk-toggleable__label-arrow\">LinearRegression</label><div class=\"sk-toggleable__content\"><pre>LinearRegression(fit_intercept=False)</pre></div></div></div></div></div>"
+       "<style>#sk-container-id-1 {color: black;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: \"▸\";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: \"▾\";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: \"\";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: \"\";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id=\"sk-container-id-1\" class=\"sk-top-container\"><div class=\"sk-text-repr-fallback\"><pre>LinearRegression(fit_intercept=False)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class=\"sk-container\" hidden><div class=\"sk-item\"><div class=\"sk-estimator sk-toggleable\"><input class=\"sk-toggleable__control sk-hidden--visually\" id=\"sk-estimator-id-1\" type=\"checkbox\" checked><label for=\"sk-estimator-id-1\" class=\"sk-toggleable__label sk-toggleable__label-arrow\">LinearRegression</label><div class=\"sk-toggleable__content\"><pre>LinearRegression(fit_intercept=False)</pre></div></div></div></div></div>"
       ],
       "text/plain": [
        "LinearRegression(fit_intercept=False)"
       ]
      },
-     "execution_count": 206,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -257,8 +257,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 207,
-   "metadata": {},
+   "execution_count": 8,
+   "metadata": {
+    "scrolled": true
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -291,6 +293,60 @@
     "    print(f\"{str(key).ljust(20)} {str(value).ljust(20)}\")"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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zzT/77DPl5+dX+Dr++c9/6g9/+INb77rIyEgNGTKkwuc4nclkci1e6nQ6lZWV5bql6tTb4E5nNpsVGBioL7/8UseOHav06wMAap8TJ05Ikho3bnzG40r322y2Cp87KipK33//vcaMGaNjx45p0aJFGjx4sJo0aaLnnnuuzK3WniQlJSkyMlLx8fG64447FBwcrNWrVysuLk7SyXZmGzZs0F133aUTJ064cnZWVpZ69uypvXv36tChQ5KksLAw7d69W3v37q3wdfzzn/+UJD388MNu46Uz7Crr1FmHx48f12+//aauXbtq//79bi3hThcWFqZff/21Qq0CAADuzpZTJKlBgwaudTtKSkqUnZ0th8Ohjh07lvu96k9/+pPbbOTSO7v2799f5tiNGzeqZ8+e6tGjh1auXHlOa2ddeumlioyMVIsWLTRy5EhdcsklWrNmjWuNsV27dmnv3r0aPHiwsrKyXPkwLy9PPXr00KZNm1RSUiKn06kvvvhCAwYMcLtb+ZJLLlHv3r3Lfe2uXbvq8ssvd20bhqGPPvpI/fr1k2EYbt+Ze/bsqePHj7veq4rkrbCwMG3dulWHDx+u8Pvxz3/+Uw0aNCiTnydNmiTDMLRmzRq38aSkJCUmJrq2r7zySoWEhJT791Sep59+WpGRkYqOjtYNN9ygPXv2aO7cubrjjjtcx1S0XhEWFqa8vDy3dnJns337dh09elRjxoxxW1dm+PDhHu+Ar4hTP4/k5eXpt99+0/XXXy/DMPTdd995fF7pXfx///vfvbrgK6oWRXTAhzRt2vS8Vpfeu3ev1q5d6yqClz6SkpIkqUxP8/K0aNFC69at02effaaFCxeqadOmslqtbr1Z9+7dK8Mw1LJlyzKvtWfPHtfrJCQkaOLEiXrzzTd10UUXqWfPnnrttdfO+OVXkiwWi1q2bFlm/NJLLz2Xt8NNSUmJXnnlFbVs2VImk0kXXXSRIiMj9cMPP5wxHpPJpNmzZ2vNmjWKiorSjTfeqBdffFEZGRmVjgUAUDuUFsdLi+meVLTYfrqYmBilpKToyJEj+vnnnzV//nxFRkZq2rRpeuuttyp0jtdee03r1q3Thx9+qD59+ui3335zKzrs27dPhmFo6tSpZXL2008/Lel/nw+mT5+unJwctWrVSm3bttXkyZP1ww8/nPH1LRaL/P393b54S+eXsyXpm2++UVJSkoKDgxUWFqbIyEhXH9Ez5e3HH39cjRo10rXXXquWLVvqoYce4tZpAKigs+WUUu+8846uvPJK1/oZkZGR+sc//lHu/5+bNWvmtl1aUD99glJhYaFuueUWtW/fXu+///45fy/+6KOPtG7dOq1YsUJ/+MMfdPToUbcCaOkPxMOGDSuTD998803Z7XYdP35cR48eVUFBgS655JIyr1HemHTye++prFarcnJyXOupnPoonTRXmnsrkrdefPFF/ec//1F8fLyuvfZaPfPMM2ctblssFsXGxpb5bHLZZZe59p/q9L8n6eTfVUUnko0ePVrr1q3TJ5984uoZ7nQ63Y6paL3iwQcfVKtWrdS7d2/FxcVp5MiRZ+3PXno9p9cRGjZseE4Lv5/u4MGDGj58uMLDw9WoUSNFRkaqa9euks78eeRPf/qTOnfurPvuu09RUVEaNGiQ3n//fQrqtRw90QEfcq693k5PSiUlJfrjH/+oxx57rNzjW7VqddZzBgcHu5KYJHXu3FlXX321/vznP2v+/Pmu1/Hz89OaNWvUoEGDMudo1KiR689z587V8OHD9fe//12ff/65Hn74Yc2aNUtbtmxxm9FQ1U5/b55//nlNnTpVI0eO1HPPPafw8HD5+/tr/PjxZ01k48ePV79+/bRq1Sp99tlnmjp1qmbNmqUNGzaoffv21XYNAADvCg0NVUxMzFkLyT/88IOaNm2qkJCQSr2On5+fWrVqpVatWumWW25Ry5YttXz5ct13331nfe61116rjh07SpIGDBigLl26aPDgwfr555/VqFEjV4579NFH1bNnz3LPUVoUuPHGG5WamurK2W+++aZeeeUVLVq0qEKxVOQ6y3N6zk5NTVWPHj3UunVrvfzyy4qPj1dgYKD++c9/6pVXXjlj3r7sssv0888/69NPP9XatWv10UcfaeHChZo2bZqeffbZ874GAKjLzpZTJOmvf/2rhg8frgEDBmjy5Mlq0qSJGjRooFmzZpVZRFJSud8XJZW548pkMqlPnz76+9//rrVr16pv377nFPuNN96oiy66SJLUr18/tW3bVkOGDNGOHTvk7+/vyh0vvfRSmb7tpRo1aqTCwsJzel2p7Pf40te65557PK57cuWVV0qqWN666667dMMNN+jjjz/W559/rpdeekmzZ8/WypUrPc6OP1cV/XvypGXLlq46Qt++fdWgQQM98cQT6t69u+vfVEXrFU2aNNGuXbv02Wefac2aNVqzZo2WLFmie++9V++88865XloZFf084nQ69cc//lHZ2dl6/PHH1bp1awUHB+vQoUMaPnz4GT+PmM1mbdq0SRs3btQ//vEPrV27Vv/3f/+nm266SZ9//rnH9xu+jSI6UAtceOGFysnJcRsrKirSkSNH3MYSExOVm5vrVgQ/X1deeaXuuecevf7663r00UfVrFkzJSYmyjAMJSQkVKgw37ZtW7Vt21ZPPfWU/v3vf6tz585atGiRZsyYUe7xzZs3L/dW8p9//rnMWEXfmw8//FDdu3cvM7MvJyfH9WHrTBITEzVp0iRNmjRJe/fu1VVXXaW5c+fqr3/961mfCwCovfr27as33nhDX3/9tbp06VJm/1dffaUDBw7ogQceqJLXu/jii3XhhReWyWMVUVrE6N69u1599VU98cQTrtlXDRs2rNDng/DwcI0YMUIjRoxQbm6ubrzxRj3zzDMei+jNmzdXSUmJUlNT3WafVzRnS2Vnw33yySey2+1avXq128y40nZxZxMcHKw//elP+tOf/qSioiLdfvvtmjlzpqZMmeJ2Zx0AwLPycop08nvVxRdfrJUrV7oVI0vvbqosPz8/LV++XP3799edd96pNWvWqFu3bpU6V6NGjfT0009rxIgRev/99zVo0CDXHVMhISFnzIdNmjRRUFCQ9u3bV2ZfeWPliYyMVOPGjeV0OiuUeyuSt2JiYvTggw/qwQcf1NGjR3X11Vdr5syZHovozZs31xdffKETJ064zUb/73//69pfnZ588km98cYbeuqpp1yzyM+lXhEYGKh+/fqpX79+Kikp0YMPPqjXX39dU6dOLfeOgNLr2bt3r2666SbXeHFxsdLS0tSuXTvXWOndEKd/Jjn988iPP/6oX375Re+8847uvfde13hF28z4+/urR48e6tGjh15++WU9//zzevLJJ7Vx48Yqrdmg5tDOBagFEhMTy/QzX7x4cZlfSu+66y5t3rxZn332WZlz5OTkyOFwVOr1H3vsMRUXF+vll1+WJN1+++1q0KCBnn322TK/TBuGoaysLEkne8Oe/ppt27aVv7+/7Ha7x9fr06ePtmzZom+//dY1ZrVatXz58jLHVvS9adCgQZlYP/jgA1cfWE/y8/PLzEZITExU48aNz3gNAIC6YfLkyTKbzXrggQdc+a1Udna2xowZowsuuECTJ08+p/Nu3bpVeXl5Zca//fZbZWVlVbodSrdu3XTttddq3rx5KiwsVJMmTdStWze9/vrr5RbmrVar68+nX1+jRo10ySWXnDHflX55L71brdS8efPKHJuYmKjjx4+7zew/cuSIPv74Y7fjSmdnnZq3jx8/riVLlniMw9M1BAYG6vLLL5dhGCouLj7r8wEA/3N6TpHK/3/01q1btXnz5vN+vcDAQK1cuVLXXHON+vXr5/Z98FwNGTJEcXFxmj17tiSpQ4cOSkxM1Jw5c5Sbm1vm+NJ82KBBAyUlJWnVqlVuPcj37dtXpo+4Jw0aNNDAgQP10Ucf6T//+Y/H15LOnrecTmeZtiFNmjRRbGzsWb9TO51Ovfrqq27jr7zyivz8/KpsBrsnYWFheuCBB/TZZ59p165dkiperzj9PfH393fN3Pd0zR07dlRkZKQWLVqkoqIi1/jSpUvLFMtLf1A5tY7gdDq1ePFit+PK+7duGIb+8pe/eLzuUtnZ2WXGSu+AoI5QezETHagF7rvvPo0ZM0YDBw7UH//4R33//ff67LPPysygnjx5slavXq2+fftq+PDh6tChg/Ly8vTjjz/qww8/1IEDByo06/p0l19+ufr06aM333xTU6dOVWJiombMmKEpU6bowIEDGjBggBo3bqy0tDR9/PHHGj16tB599FFt2LBBY8eO1Z133qlWrVrJ4XDo3XffdX2o8OSxxx7Tu+++q169eumRRx5RcHCwFi9erObNm5e5pb6i703fvn01ffp0jRgxQtdff71+/PFHLV++/Kz90X755Rf16NFDd911ly6//HIFBATo448/VmZmpgYNGnTO7yUAoHZp2bKl3nnnHQ0ZMkRt27bVqFGjlJCQoAMHDuitt97Sb7/9pr/97W9leoJL0ttvv11uD89HHnlE7777rpYvX67bbrtNHTp0UGBgoPbs2aO3335bQUFBrv7flTF58mTdeeedWrp0qcaMGaPXXntNXbp0Udu2bXX//ffr4osvVmZmpjZv3qxff/1V33//vaST+b5bt27q0KGDwsPDtX37dn344YcaO3asx9e66qqrdPfdd2vhwoU6fvy4rr/+eq1fv77c2XqDBg3S448/rttuu00PP/yw8vPzlZKSolatWrktRnfzzTe7ZqA98MADys3N1RtvvKEmTZqcdYb+zTffrOjoaHXu3FlRUVHas2ePXn31Vd1yyy3n3LMeAFA2p/Tt21crV67UbbfdpltuuUVpaWlatGiRLr/88nKL0+fKbDbr008/1U033aTevXvrX//6l6644opzPk/Dhg31yCOPaPLkyVq7dq169eqlN998U71791abNm00YsQINW3aVIcOHdLGjRsVEhKiTz7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+      "text/plain": [
+       "<Figure size 1500x500 with 3 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "## plot residual results\n",
+    "import pandas as pd\n",
+    "import seaborn as sns\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "res_true = y - X.dot(beta_true)\n",
+    "res_ols = y - ols.predict(X)\n",
+    "res_RR = y - np.mean(y) - X.dot(RankReg.coef_)\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "# Create a figure with three subplots\n",
+    "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(15, 5))\n",
+    "\n",
+    "# Histogram for true residuals\n",
+    "ax1.hist(res_true, bins=100, alpha=0.7, color='red', edgecolor='black')\n",
+    "ax1.set_title('True Residuals')\n",
+    "ax1.set_xlabel('Residual Value')\n",
+    "ax1.set_ylabel('Frequency')\n",
+    "\n",
+    "# Histogram for OLS residuals\n",
+    "ax2.hist(res_ols, bins=100, alpha=0.7, color='skyblue', edgecolor='black')\n",
+    "ax2.set_title('OLS Residuals')\n",
+    "ax2.set_xlabel('Residual Value')\n",
+    "ax2.set_ylabel('Frequency')\n",
+    "\n",
+    "# Histogram for RR residuals\n",
+    "ax3.hist(res_RR, bins=100, alpha=0.7, color='orange', edgecolor='black')\n",
+    "ax3.set_title('Rank Regression Residuals')\n",
+    "ax3.set_xlabel('Residual Value')\n",
+    "ax3.set_ylabel('Frequency')\n",
+    "\n",
+    "# Set layout and show plot\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {
@@ -308,22 +364,13 @@
    "provenance": []
   },
   "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
+   "display_name": "Python 3",
    "name": "python3"
   },
   "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.10.12"
-  }
+   "name": "python"
+  },
+  "nbformat_minor": 0
  },
  "nbformat": 4,
  "nbformat_minor": 4