{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "341ccf89",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "data_pine = np.loadtxt('pine_full.csv',skiprows=1,delimiter=\",\")\n",
    "#https://fbertran.github.io/plsRglm/reference/pine.html\n",
    "pine_y = data_pine[:,-1]\n",
    "pine_X = data_pine[:,:-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "74072c0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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HjhxRTEyMe0xhYaHHmKrXVWN+KjQ0VKGhod6cBgAAqKe8vgJTFV727dunf/7zn2rRooVHf1JSkoqLi5Wbm+tu27BhgyorK5WYmOgek5WVpfLycveYzMxMdezYUc2bN/d2yQAAwDC1DjAlJSXKy8tTXl6eJGn//v3Ky8tTQUGBysvLdfvtt2vbtm1asWKFKioq5HA45HA4VFZWJklKSEjQLbfcorFjx+rTTz/VRx99pPHjx2vEiBHuz4q55557FBISojFjxmjXrl1644039Nxzz2ny5MnemzkAADBWrd9C2rZtm2666Sb366pQMWrUKD366KN67733JElXX321x34bN25U3759JUkrVqzQ+PHj1a9fP/cH2S1cuNA91maz6YMPPlBqaqp69Oihli1bas6cOTxCDQAAJNUhwPTt21eWZZ21/+f6qkRFRbk/tO5sunbtqi1bttS2PAAAcBHgu5C+d+LECUnS9u3ba7zPyZMndeDAAbVr105hYWHnHJ+fn1/n+gAAwA8IMN/bvXu3JGns2LE+P1d4eLjPzwEAQENGgPle1RdSxsfHq0mTJjXaJz8/XyNHjtTy5cuVkJBQo33Cw8PVoUOHupYJAABEgHFr2bKl7r///jrtm5CQoO7du3u5IgAAcDY+/y4kAAAAbyPAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABgn2N8FmODEiRPavXv3Ge35+fke//yp+Ph4NWnSxKe1AQBwMSLA1MDu3bvVo0ePs/aPHDmy2vbc3Fx1797dV2UBAHDRIsDUQHx8vHJzc89oP3nypA4cOKB27dopLCys2v0AAID3BViWZfm7CF9wuVyy2WxyOp2KiIjwdzkAAKAGanr95iZeAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGCcWgeYrKwsDRkyRLGxsQoICNDq1as9+i3L0pw5c9SqVSuFhYUpOTlZ+/bt8xhz5MgRpaSkKCIiQpGRkRozZoxKSko8xnz++ee64YYb1LhxY7Vu3Vrp6em1nx0AAGiQah1gjh8/rm7dumnRokXV9qenp2vhwoVasmSJcnJy1LRpUw0YMECnTp1yj0lJSdGuXbuUmZmpjIwMZWVlady4ce5+l8ul/v37q23btsrNzdWCBQv06KOPaunSpXWYIgAAaHCs8yDJWrVqlft1ZWWlFRMTYy1YsMDdVlxcbIWGhlqvvfaaZVmW9cUXX1iSrK1bt7rHvP/++1ZAQID17bffWpZlWYsXL7aaN29ulZaWusdMnz7d6tixY41rczqdliTL6XTWdXoAAOACq+n126v3wOzfv18Oh0PJycnuNpvNpsTERGVnZ0uSsrOzFRkZqZ49e7rHJCcnKzAwUDk5Oe4xN954o0JCQtxjBgwYoD179ujo0aPVnru0tFQul8tjAwAADZNXA4zD4ZAk2e12j3a73e7uczgcio6O9ugPDg5WVFSUx5jqjvHjc/xUWlqabDabe2vduvX5TwgAANRLDeYppJkzZ8rpdLq3gwcP+rskAADgI14NMDExMZKkwsJCj/bCwkJ3X0xMjIqKijz6T58+rSNHjniMqe4YPz7HT4WGhioiIsJjAwAADZNXA0xcXJxiYmK0fv16d5vL5VJOTo6SkpIkSUlJSSouLlZubq57zIYNG1RZWanExET3mKysLJWXl7vHZGZmqmPHjmrevLk3SwYAAAaqdYApKSlRXl6e8vLyJP3vxt28vDwVFBQoICBAEydO1Ny5c/Xee+9px44d+vWvf63Y2FgNHTpUkpSQkKBbbrlFY8eO1aeffqqPPvpI48eP14gRIxQbGytJuueeexQSEqIxY8Zo165deuONN/Tcc89p8uTJXps4AAAwWG0fb9q4caMl6Yxt1KhRlmX971Hq2bNnW3a73QoNDbX69etn7dmzx+MY//3vf627777batasmRUREWHdd9991rFjxzzG/Otf/7L69OljhYaGWpdeeqk1f/78WtXJY9QAAJinptfvAMuyLD/mJ59xuVyy2WxyOp3cDwMAgCFqev1uME8hAQCAiwcBBgAAGIcAAwAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABiHAAMAAIxDgAEAAMYhwAAAAOMQYAAAgHEIMAAAwDgEGAAAYBwCDAAAMA4BBgAAGIcAAwAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHK8HmIqKCs2ePVtxcXEKCwtT+/bt9fjjj8uyLPcYy7I0Z84ctWrVSmFhYUpOTta+ffs8jnPkyBGlpKQoIiJCkZGRGjNmjEpKSrxdLgAAMJDXA8yTTz6pF198US+88ILy8/P15JNPKj09Xc8//7x7THp6uhYuXKglS5YoJydHTZs21YABA3Tq1Cn3mJSUFO3atUuZmZnKyMhQVlaWxo0b5+1yAQCAgQKsHy+NeMHgwYNlt9v18ssvu9uGDx+usLAwLV++XJZlKTY2VlOmTNHvf/97SZLT6ZTdbteyZcs0YsQI5efnq1OnTtq6dat69uwpSVq7dq1uvfVWffPNN4qNjT1nHS6XSzabTU6nUxEREd6cIgAA8JGaXr+9vgJz/fXXa/369dq7d68k6V//+pc+/PBDDRw4UJK0f/9+ORwOJScnu/ex2WxKTExUdna2JCk7O1uRkZHu8CJJycnJCgwMVE5OTrXnLS0tlcvl8tgAAEDDFOztA86YMUMul0vx8fEKCgpSRUWFnnjiCaWkpEiSHA6HJMlut3vsZ7fb3X0Oh0PR0dGehQYHKyoqyj3mp9LS0vTYY495ezoAAKAe8voKzJtvvqkVK1Zo5cqV2r59u1599VU99dRTevXVV719Kg8zZ86U0+l0bwcPHvTp+QAAgP94fQVm6tSpmjFjhkaMGCFJ6tKli77++mulpaVp1KhRiomJkSQVFhaqVatW7v0KCwt19dVXS5JiYmJUVFTkcdzTp0/ryJEj7v1/KjQ0VKGhod6eDgAAqIe8vgJz4sQJBQZ6HjYoKEiVlZWSpLi4OMXExGj9+vXufpfLpZycHCUlJUmSkpKSVFxcrNzcXPeYDRs2qLKyUomJid4uGQAAGMbrKzBDhgzRE088oTZt2qhz58767LPP9Mwzz2j06NGSpICAAE2cOFFz585Vhw4dFBcXp9mzZys2NlZDhw6VJCUkJOiWW27R2LFjtWTJEpWXl2v8+PEaMWJEjZ5AAgAADZvXA8zzzz+v2bNn66GHHlJRUZFiY2P1wAMPaM6cOe4x06ZN0/HjxzVu3DgVFxerT58+Wrt2rRo3buwes2LFCo0fP179+vVTYGCghg8froULF3q7XAAAYCCvfw5MfcHnwAAAYB6/fQ4MAACArxFgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABiHAAMAAIxDgAEAAMYhwAAAAOMQYAAAgHEIMAAAwDgEGAAAYBwCDAAAMA4BBgAAGIcAAwAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACM45MA8+2332rkyJFq0aKFwsLC1KVLF23bts3db1mW5syZo1atWiksLEzJycnat2+fxzGOHDmilJQURUREKDIyUmPGjFFJSYkvygUAAIbxeoA5evSoevfurUaNGun999/XF198oaefflrNmzd3j0lPT9fChQu1ZMkS5eTkqGnTphowYIBOnTrlHpOSkqJdu3YpMzNTGRkZysrK0rhx47xdLgAAMFCAZVmWNw84Y8YMffTRR9qyZUu1/ZZlKTY2VlOmTNHvf/97SZLT6ZTdbteyZcs0YsQI5efnq1OnTtq6dat69uwpSVq7dq1uvfVWffPNN4qNjT1nHS6XSzabTU6nUxEREd6bIAAA8JmaXr+9vgLz3nvvqWfPnrrjjjsUHR2ta665Rn/5y1/c/fv375fD4VBycrK7zWazKTExUdnZ2ZKk7OxsRUZGusOLJCUnJyswMFA5OTnVnre0tFQul8tjAwAADZPXA8y///1vvfjii+rQoYPWrVunBx98UL/73e/06quvSpIcDockyW63e+xnt9vdfQ6HQ9HR0R79wcHBioqKco/5qbS0NNlsNvfWunVrb08NAADUE14PMJWVlerevbvmzZuna665RuPGjdPYsWO1ZMkSb5/Kw8yZM+V0Ot3bwYMHfXo+AADgP14PMK1atVKnTp082hISElRQUCBJiomJkSQVFhZ6jCksLHT3xcTEqKioyKP/9OnTOnLkiHvMT4WGhioiIsJjAwAADZPXA0zv3r21Z88ej7a9e/eqbdu2kqS4uDjFxMRo/fr17n6Xy6WcnBwlJSVJkpKSklRcXKzc3Fz3mA0bNqiyslKJiYneLhkAABgm2NsHnDRpkq6//nrNmzdPd955pz799FMtXbpUS5culSQFBARo4sSJmjt3rjp06KC4uDjNnj1bsbGxGjp0qKT/rdjccsst7reeysvLNX78eI0YMaJGTyABAICGzeuPUUtSRkaGZs6cqX379ikuLk6TJ0/W2LFj3f2WZemRRx7R0qVLVVxcrD59+mjx4sW68sor3WOOHDmi8ePHa82aNQoMDNTw4cO1cOFCNWvWrEY18Bg1AADmqen12ycBpj4gwAAAYB6/fQ4MAACArxFgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABiHAAMAAIxDgAEAAMYhwAAAAOMQYAAAgHEIMAAAwDgEGAAAYBwCDAAAMA4BBgAAGIcAAwAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADG8XmAmT9/vgICAjRx4kR326lTp5SamqoWLVqoWbNmGj58uAoLCz32Kygo0KBBg9SkSRNFR0dr6tSpOn36tK/LBQAABvBpgNm6dateeuklde3a1aN90qRJWrNmjd566y1t3rxZhw4d0rBhw9z9FRUVGjRokMrKyvTxxx/r1Vdf1bJlyzRnzhxflgsAAAzhswBTUlKilJQU/eUvf1Hz5s3d7U6nUy+//LKeeeYZ3XzzzerRo4deeeUVffzxx/rkk08kSR988IG++OILLV++XFdffbUGDhyoxx9/XIsWLVJZWZmvSgYAAIbwWYBJTU3VoEGDlJyc7NGem5ur8vJyj/b4+Hi1adNG2dnZkqTs7Gx16dJFdrvdPWbAgAFyuVzatWtXtecrLS2Vy+Xy2AAAQMMU7IuDvv7669q+fbu2bt16Rp/D4VBISIgiIyM92u12uxwOh3vMj8NLVX9VX3XS0tL02GOPeaF6AABQ33l9BebgwYN6+OGHtWLFCjVu3Njbhz+rmTNnyul0ureDBw9esHMDAIALy+sBJjc3V0VFRerevbuCg4MVHByszZs3a+HChQoODpbdbldZWZmKi4s99issLFRMTIwkKSYm5oynkqpeV435qdDQUEVERHhsAACgYfJ6gOnXr5927NihvLw899azZ0+lpKS4/3ejRo20fv169z579uxRQUGBkpKSJElJSUnasWOHioqK3GMyMzMVERGhTp06ebtkAABgGK/fAxMeHq6rrrrKo61p06Zq0aKFu33MmDGaPHmyoqKiFBERoQkTJigpKUm9evWSJPXv31+dOnXSvffeq/T0dDkcDs2aNUupqakKDQ31dskAAMAwPrmJ91yeffZZBQYGavjw4SotLdWAAQO0ePFid39QUJAyMjL04IMPKikpSU2bNtWoUaP0pz/9yR/lAgCAeibAsizL30X4gsvlks1mk9Pp5H4YAAAMUdPrN9+FBAAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABiHAAMAAIxDgAEAAMYhwAAAAOMQYAAAgHEIMAAAwDgEGAAAYBwCDAAAMA4BBgAAGIcAAwAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYx+sBJi0tTddee63Cw8MVHR2toUOHas+ePR5jTp06pdTUVLVo0ULNmjXT8OHDVVhY6DGmoKBAgwYNUpMmTRQdHa2pU6fq9OnT3i4XAAAYyOsBZvPmzUpNTdUnn3yizMxMlZeXq3///jp+/Lh7zKRJk7RmzRq99dZb2rx5sw4dOqRhw4a5+ysqKjRo0CCVlZXp448/1quvvqply5Zpzpw53i4XAAAYKMCyLMuXJ/juu+8UHR2tzZs368Ybb5TT6dQll1yilStX6vbbb5ck7d69WwkJCcrOzlavXr30/vvva/DgwTp06JDsdrskacmSJZo+fbq+++47hYSEnPO8LpdLNptNTqdTERERvpwiAADwkppev31+D4zT6ZQkRUVFSZJyc3NVXl6u5ORk95j4+Hi1adNG2dnZkqTs7Gx16dLFHV4kacCAAXK5XNq1a1e15yktLZXL5fLYAABAw+TTAFNZWamJEyeqd+/euuqqqyRJDodDISEhioyM9Bhrt9vlcDjcY34cXqr6q/qqk5aWJpvN5t5at27t5dkAAID6wqcBJjU1VTt37tTrr7/uy9NIkmbOnCmn0+neDh486PNzAgAA/wj21YHHjx+vjIwMZWVl6bLLLnO3x8TEqKysTMXFxR6rMIWFhYqJiXGP+fTTTz2OV/WUUtWYnwoNDVVoaKiXZwEAAOojr6/AWJal8ePHa9WqVdqwYYPi4uI8+nv06KFGjRpp/fr17rY9e/aooKBASUlJkqSkpCTt2LFDRUVF7jGZmZmKiIhQp06dvF0yAAAwjNdXYFJTU7Vy5Uq9++67Cg8Pd9+zYrPZFBYWJpvNpjFjxmjy5MmKiopSRESEJkyYoKSkJPXq1UuS1L9/f3Xq1En33nuv0tPT5XA4NGvWLKWmprLKAgAAvP8YdUBAQLXtr7zyin7zm99I+t8H2U2ZMkWvvfaaSktLNWDAAC1evNjj7aGvv/5aDz74oDZt2qSmTZtq1KhRmj9/voKDa5a5eIwaAADz1PT67fPPgfEXAgwAAOapN58DAwAA4G0EGAAAYBwCDAAAMA4BBgAAGIcAAwAAjEOAAQAAxiHAAAAA4xBgAACAcQgwAADAOAQYAABgHAIMAAAwDgEGAAAYhwADAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABiHAAMAAIxDgAEAAMYhwAAAAOME+7sAAA1HWVmZFi9erK+++krt27fXQw89pJCQEH+XBaABIsAA8Ipp06bp2Wef1enTp91tU6dO1aRJk5Senu7HygA0RASYi8SJEye0e/fuM9pPnjypAwcOqF27dgoLC6t23/j4eDVp0sTXJcJg06ZN04IFC2S32zV37lwNHjxYGRkZmjVrlhYsWCBJhBgAXhVgWZbl7yJ8weVyyWazyel0KiIiwt/l+N327dvVo0ePOu2bm5ur7t27e7kiNBRlZWVq2rSpWrRooW+++UbBwT/8XXT69Glddtll+u9//6vjx4/zdhKAc6rp9ZsVmAZo3759OnbsmEfbyZMntXz58jPG7t+/X7Nnz9bjjz+uuLi4ao938uRJbd++3aMtPDxcHTp08F7RNVTXlSRWkXxn8eLFOn36tObOnesRXiQpODhYf/rTn/TAAw9o8eLFmjhxon+KBNDgEGAamH379unKK6+s9X6zZ8+u9T579+694CFm9+7ddVpJYhXJd7766itJ0uDBg6vtr2qvGgfURkVFhbZs2aLDhw+rVatWuuGGGxQUFOTvslAPEGAamO8Ofa1rYgI1cuTIs66o/NiJE8f11Vf/Vvv2l6tJk6Y1Osfhw4f117/+VSVHv5N0YQNMfHy8cnNzz2jPz8/XyJEjtXz5ciUkJFS7H3yjffv2kqSMjAzdf//9Z/RnZGR4jANq6p133tGUKVN04MABd1u7du309NNPa9iwYf4rDPUCAaaBKdz1obY/0EzSaqmohjvZJZV8v9VEkPTQA81UYP23LiXWWHVvhdVVdW87+ettsIai6u28Xr16KSgoSNOnT1fXrl1VXl7ufjuvUaNGmjlzpoKCgtSrVy9t376dt/MuENNXLt555x3dfvvtGjx4sF577TVdddVV2rlzp+bNm6fbb79db7/9NiHmIlevb+JdtGiRFixYIIfDoW7duun555/XddddV6N9L9abeP9z+KC2rHpZ7dq1U+PGjd3tpaWlOnTo0BnjDx06pMWLF+uhhx5SbGxstceMjY1VaGioR1vTpk3V5pp+UohvLkR1fSustvzxNphxyk6o4LP1On78uEfz/v37NWvWrFofbu7cudWuDvr6d+piYvrKRUVFha644gp16dJFq1evVmDgD5+5WllZqaFDh2rnzp3at2+fUaEMNVPT63e9DTBvvPGGfv3rX2vJkiVKTEzUn//8Z7311lvas2ePoqOjz7n/xRpgzsa0p5A++/RjjbnthrNe7H6qKqBVF7aqU3XxffndLbrmuuu9UXKDtXfzm7py49gLcq6CgcvVJnHIBTlXQ/XjlYs//OEPHisXGRkZ9XrlompVb9u2bXrggQe0bNkydenS5Yyb9D///HPdd999eumll9SzZ09W9RoY4wNMYmKirr32Wr3wwguS/pe6W7durQkTJmjGjBnn3J8A48m0z4H57B/LdM2nD/v8PPk3vqSEm0f47PhneyLsx38Z10Z1PydfvxX2ytLFev6RCT47/o+9uSFPVyR0uSDnaohMWbmoWin+qar762rr/vvvV6tWrTzaLr00VtcNHMmKnoGMfoy6rKxMubm5mjlzprstMDBQycnJys7Ornaf0tJSlZaWul+7XC6f12mSJk2anHUVpXfv3he4mnPL++aExrxU05ty6u7Nuzr77Nif5+boN4MvzOqOLy/8Q4bdqYrAkDOCbNWN07V1thutw8PDdYWPgtjZLpjSDzey11Z1N777+qJ5tnlUzeE///mPmp8q0OVh7TQ39Y4zxl3e2KVvTn6tSSNuVsuWLf02jy2rXtavip49s+P7++tq7/Uz7/krkvZfEq2464fW4Xjndq6fRV3Ux9+p2jrbAyG+mEe9DDD/+c9/VFFRIbvd7tFut9urXUWQpLS0ND322GMXojxcABfiounLC6YkfZXz/vc3VPtegeuAJN8EmJYtW1b7dNHZngirj5/Jc9YLZhX72bvOqrob33180fzZedi/3zo3k7T97GM6NZOU90ObH+Zxw6/GaNWqM9u9vgLTs39dSzync/4s6qI+/k7V1tkeCPHBPOrlW0iHDh3SpZdeqo8//lhJSUnu9mnTpmnz5s3Kyck5Y5/qVmBat27NW0gNjEkfZFfbG6prwh83VDcEF9MKzIcffqgbb7xRUVFRZ4w78t//KmvLFvXp08evKzBn8+N/vzds2KBp06bphhtu0D333KOgoCBVVFRo5cqV2rJli9LT03XzzTdL8u+/32fOgRWY812BMfoemLKyMjVp0kRvv/22hg4d6m4fNWqUiouL9e67757zGNwDA+BiY8o9MDVV3dNUcXFxeuqpp+rtjcg4fzW9fgeetcePQkJC1KNHD61fv97dVllZqfXr13usyAAAfhAUFKSnn35aGRkZGjp0qLKzs3Xs2DFlZ2dr6NChysjI0FNPPWVEeJGkYcOG6csvv9TGjRu1cuVKbdy4Ufv27SO8QFI9vQdGkiZPnqxRo0apZ8+euu666/TnP/9Zx48f13333efv0gCg3ho2bJjefvttTZkyRddf/8NN5HFxcfX6EeqzCQoKUt++ff1dBuqhehtg7rrrLn333XeaM2eOHA6Hrr76aq1du/aMG3sBAJ6GDRum2267zehP4gXOpV7eA+MN3AMDAIB5jL4HBgAA4OcQYAAAgHEIMAAAwDgEGAAAYBwCDAAAMA4BBgAAGIcAAwAAjEOAAQAAxiHAAAAA49TbrxI4X1UfMOxyufxcCQAAqKmq6/a5viigwQaYY8eOSZJat27t50oAAEBtHTt2TDab7az9Dfa7kCorK3Xo0CGFh4crICDAJ+dwuVxq3bq1Dh48aOz3LTWEOUjMoz5pCHOQGsY8GsIcJOZRn1yIOViWpWPHjik2NlaBgWe/06XBrsAEBgbqsssuuyDnioiIMPaXsUpDmIPEPOqThjAHqWHMoyHMQWIe9Ymv5/BzKy9VuIkXAAAYhwADAACMQ4A5D6GhoXrkkUcUGhrq71LqrCHMQWIe9UlDmIPUMObREOYgMY/6pD7NocHexAsAABouVmAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAaYOsrKyNGTIEMXGxiogIECrV6/2d0m1lpaWpmuvvVbh4eGKjo7W0KFDtWfPHn+XVWsvvviiunbt6v5QpaSkJL3//vv+Luu8zJ8/XwEBAZo4caK/S6mVRx99VAEBAR5bfHy8v8uqtW+//VYjR45UixYtFBYWpi5dumjbtm3+LqtW2rVrd8bPIiAgQKmpqf4urVYqKio0e/ZsxcXFKSwsTO3bt9fjjz9+zu/IqW+OHTumiRMnqm3btgoLC9P111+vrVu3+rusn3Wu65xlWZozZ45atWqlsLAwJScna9++fRe0RgJMHRw/flzdunXTokWL/F1KnW3evFmpqan65JNPlJmZqfLycvXv31/Hjx/3d2m1ctlll2n+/PnKzc3Vtm3bdPPNN+u2227Trl27/F1anWzdulUvvfSSunbt6u9S6qRz5846fPiwe/vwww/9XVKtHD16VL1791ajRo30/vvv64svvtDTTz+t5s2b+7u0Wtm6davHzyEzM1OSdMcdd/i5stp58skn9eKLL+qFF15Qfn6+nnzySaWnp+v555/3d2m1cv/99yszM1N///vftWPHDvXv31/Jycn69ttv/V3aWZ3rOpeenq6FCxdqyZIlysnJUdOmTTVgwACdOnXqwhVp4bxIslatWuXvMs5bUVGRJcnavHmzv0s5b82bN7f++te/+ruMWjt27JjVoUMHKzMz0/rFL35hPfzww/4uqVYeeeQRq1u3bv4u47xMnz7d6tOnj7/L8LqHH37Yat++vVVZWenvUmpl0KBB1ujRoz3ahg0bZqWkpPipoto7ceKEFRQUZGVkZHi0d+/e3frjH//op6pq56fXucrKSismJsZasGCBu624uNgKDQ21XnvttQtWFyswkCQ5nU5JUlRUlJ8rqbuKigq9/vrrOn78uJKSkvxdTq2lpqZq0KBBSk5O9ncpdbZv3z7Fxsbq8ssvV0pKigoKCvxdUq2899576tmzp+644w5FR0frmmuu0V/+8hd/l3VeysrKtHz5co0ePdpnX2zrK9dff73Wr1+vvXv3SpL+9a9/6cMPP9TAgQP9XFnNnT59WhUVFWrcuLFHe1hYmHErlFX2798vh8Ph8d8qm82mxMREZWdnX7A6GuyXOaLmKisrNXHiRPXu3VtXXXWVv8uptR07digpKUmnTp1Ss2bNtGrVKnXq1MnfZdXK66+/ru3bt9f798V/TmJiopYtW6aOHTvq8OHDeuyxx3TDDTdo586dCg8P93d5NfLvf/9bL774oiZPnqw//OEP2rp1q373u98pJCREo0aN8nd5dbJ69WoVFxfrN7/5jb9LqbUZM2bI5XIpPj5eQUFBqqio0BNPPKGUlBR/l1Zj4eHhSkpK0uOPP66EhATZ7Xa99tprys7O1hVXXOHv8urE4XBIkux2u0e73W53910IBBgoNTVVO3fuNPavgY4dOyovL09Op1Nvv/22Ro0apc2bNxsTYg4ePKiHH35YmZmZZ/yVZpIf/1XctWtXJSYmqm3btnrzzTc1ZswYP1ZWc5WVlerZs6fmzZsnSbrmmmu0c+dOLVmyxNgA8/LLL2vgwIGKjY31dym19uabb2rFihVauXKlOnfurLy8PE2cOFGxsbFG/Tz+/ve/a/To0br00ksVFBSk7t276+6771Zubq6/SzMabyFd5MaPH6+MjAxt3LhRl112mb/LqZOQkBBdccUV6tGjh9LS0tStWzc999xz/i6rxnJzc1VUVKTu3bsrODhYwcHB2rx5sxYuXKjg4GBVVFT4u8Q6iYyM1JVXXqkvv/zS36XUWKtWrc4IvgkJCca9FVbl66+/1j//+U/df//9/i6lTqZOnaoZM2ZoxIgR6tKli+69915NmjRJaWlp/i6tVtq3b6/NmzerpKREBw8e1Keffqry8nJdfvnl/i6tTmJiYiRJhYWFHu2FhYXuvguBAHORsixL48eP16pVq7RhwwbFxcX5uySvqaysVGlpqb/LqLF+/fppx44dysvLc289e/ZUSkqK8vLyFBQU5O8S66SkpERfffWVWrVq5e9Saqx3795nfJzA3r171bZtWz9VdH5eeeUVRUdHa9CgQf4upU5OnDihwEDPy1RQUJAqKyv9VNH5adq0qVq1aqWjR49q3bp1uu222/xdUp3ExcUpJiZG69evd7e5XC7l5ORc0PsPeQupDkpKSjz+qty/f7/y8vIUFRWlNm3a+LGymktNTdXKlSv17rvvKjw83P2+pc1mU1hYmJ+rq7mZM2dq4MCBatOmjY4dO6aVK1dq06ZNWrdunb9Lq7Hw8PAz7j1q2rSpWrRoYdQ9Sb///e81ZMgQtW3bVocOHdIjjzyioKAg3X333f4urcYmTZqk66+/XvPmzdOdd96pTz/9VEuXLtXSpUv9XVqtVVZW6pVXXtGoUaMUHGzmf+qHDBmiJ554Qm3atFHnzp312Wef6ZlnntHo0aP9XVqtrFu3TpZlqWPHjvryyy81depUxcfH67777vN3aWd1ruvcxIkTNXfuXHXo0EFxcXGaPXu2YmNjNXTo0AtX5AV73qkB2bhxoyXpjG3UqFH+Lq3GqqtfkvXKK6/4u7RaGT16tNW2bVsrJCTEuuSSS6x+/fpZH3zwgb/LOm8mPkZ91113Wa1atbJCQkKsSy+91LrrrrusL7/80t9l1dqaNWusq666ygoNDbXi4+OtpUuX+rukOlm3bp0lydqzZ4+/S6kzl8tlPfzww1abNm2sxo0bW5dffrn1xz/+0SotLfV3abXyxhtvWJdffrkVEhJixcTEWKmpqVZxcbG/y/pZ57rOVVZWWrNnz7bsdrsVGhpq9evX74L/rgVYlmEfaQgAAC563AMDAACMQ4ABAADGIcAAAADjEGAAAIBxCDAAAMA4BBgAAGAcAgwAADAOAQYAABiHAAMAAIxDgAEAAMYhwAAAAOMQYAAAgHH+H83Vt/uK9Dj1AAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.boxplot(pine_X);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c2480a11",
   "metadata": {},
   "outputs": [],
   "source": [
    "n,p = pine_X.shape\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "30f106d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_norm = (pine_X - np.mean(pine_X,axis=0))/np.std(pine_X,axis=0)\n",
    "plt.boxplot(X_norm);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "c9e429d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(58, 10)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n,p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "c846db01",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_bias = np.concatenate([np.ones((n,1)), pine_X],axis=1)\n",
    "plt.boxplot(X_bias);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "e4ed5d86",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 : 8.215225409099624\n",
      "1 : -0.00359866086722939\n",
      "2 : -0.034002709821384106\n",
      "3 : -0.02234129897657002\n",
      "4 : -0.41437365255686787\n",
      "5 : 0.12263455490896306\n",
      "6 : 1.4964268286648439\n",
      "7 : -0.4802481184136348\n",
      "8 : 0.0013816380076005514\n",
      "9 : -1.7297947641847433\n",
      "10 : 0.038454873952848814\n"
     ]
    }
   ],
   "source": [
    "alpha = np.linalg.inv(X_bias.T@X_bias)@X_bias.T@pine_y\n",
    "y_pred = X_bias @ alpha\n",
    "with np.printoptions(precision=2):\n",
    "    for i,coef in enumerate(alpha):\n",
    "        print(f\"{i} : {coef}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "d6c32739",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f81eac49960>]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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4I8x9UACYRbDII4qT70i0Akexwt4HBYBZMcuyPL2lKZVKKZFIKJlMKh6Pe/nWRYn6/zz5jROFyHUUO/NvDvUtQHTYnb8JFg4w+QKXDKYtLX5kX85TU5ltwoMbl/DfCRABdudvtkIcYP8euCQqfVAAmEWwAFCQKB/FLgYrmyh1BAsABYnyUexCRb0WC7CDBlkACpI5ip3rd/GYhibVMB7FLkQULvQDTCBYACgIrdQvyXenkDR0p9Bg2tNaecAXBAsABaOV+hAv7xQCgo4aCwBFoZU6hazASAQLAEUr9aPYFLICl7AVAgBFopAVuIRgAQBFopAVuIRgAQAGUMgKDKHGAogguj/6g0JWgGCBLJiUwo3uj/4q9UJWgGCBUZiUwi3XNeaZ7o8syQNwGzUWGOZ1S+LBtKVDJ9/TC13v6NDJ9+hKWCS6PwIIAlYsICn/pBTT0KS0rKHWyLYIKyPmcY05gCBgxQKSvG1JzGVN7qD7I4AgIFhAkneTUr6VEUvSQy++znJ9Aej+CCAIIhEs2KsvnleTUr6VEUnqTQ3oJ/veKup9ShHdHwEEQehrLNirNyMzKfUmL2ZdTYhpqNFPsZOS3RWPH/zuT/p47dX8M3Qg0/1xzfZOxaRR/xzp/gjAK6FesWCv3hyvWhI7WfHgBINzdH8E4LeYZVme/p87lUopkUgomUwqHo8X/H0G05YWP7Iv57J65jfsgxuX8BuaA26vAOX753a5Xz7waU4wFIAmZwBMszt/h3YrhKN17ri8JfE1kyqkmPTuhQEdOvle0RNUZmXkwe2dtp7nBENh6P4IwC+hDRYcrXNPZlLq6O7RN5/7g/HVi+WNdfrG0uv1g9/9Oe+znGAAgHAJbY0FR+vc5Xb9ytol16s2nvufDScYACCcQhssOFrnHi9aQ5eXxfTQFxoUk7vFogAAb4U2WHh1iqEUedWFkxMMABA9oa2xkC5NTJefYqilj0VRepP/tPWcifqVy4tFOcEAAOEW6mAhRXti8uPIYEd3j77zmzdsPWuqfoUTDAAQHaEPFlI0JyY/OopmCjbzVU6Y6sIJAIie0NZYRJkfHUXHK9jMhvoVAEA2BIuA8eJERrb3fOr3p2x1w6yedAWFlQCAnCKxFRIlXncUzbblMp5v/Y9PEioAADkRLALGy46idmsqRhqvqRUAAASLgPGqo6jTmgoKNgEAdhAsAibTUbQ3eTHrpG9qgs+35XL5e0rhKNjkVk8A8Jfj4s1XX31VK1eu1LRp0xSLxfT888+7MKzS5VVHUSdbKWHphNnR3aPFj+zTPdsOa90zXbpn22EtfmSfK6doAADZOQ4W/f39mjt3rrZu3erGeCBvWl3b3Ur51h2f0MGNS0IRKrw+ogsAGMvxVsiKFSu0YsUKN8aCEdzuKGp3y+V/LZoV+K2EfEd0Yxo6orusoTbwfy8AEHau11gMDAxoYGBg+PNUKuX2W9pW7H682/v5djqKFjqGzJbLmu2dikmjJuUw1VRI3h/RBQDk5nqwaGtr0+bNm91+G8eKbZntR8tt02OIyiVuXh7RBQCML2ZZVsEtHGOxmHbu3Km77ror5zPZVizq6+uVTCYVj8cLfeui5OrfkPndPF8dQ7F/3gSTYwj7SYpDJ9/TPdsO533ulw98mhULAChQKpVSIpHIO3+73tK7oqJC8Xh81Iefim2Z7UfLbbfHkNlyuXPetWqePSVUoUK6VC+Sa9QxDa3k0IMDANxXcneFONmPd+PPmxCEMQSJV0d0AQD5OQ4WFy5cUFdXl7q6uiRJp06dUldXl95++23TY3NFsfvxQdjPD8IYgsaLI7oAgPwcF28ePXpUn//854c/37BhgyRp1apVeuqpp4wNzC12+zdcc3VFUX++2JbbJr63m2MIIreP6AIA8nMcLG655RYVUe/pu3z9GzL+41ddeugLY2/y9Krl9niCMIagsnNEFwDgnpKrsRhvP36ks6mBrB0bg7Cf79cYBtOWDp18Ty90vaNDJ99ztUAVABBORR03LYTd4yomZTtOuedErx568YR6U7nrEDK/+R/cuGTMJN3R3aOHXnxdvalLR2lr4xVZVznc4mUvjSD07QAA+Mfu/B35203HmxC/96W5uven/zvnn83fsTHXeoE3vKopyNUzI3MPB8WRAICMSG+F5LuYat//PWvr+1x+uiLzfS9f7Tib8v7CK7d7UAShbwcAIDwiGyzsTIg7u96x9b1Gnq4otYmWnhkAACciGyzsTIjn+z9Q9aSJjjo2hn2idVqASc8MAIATka2xsDvR3TVvmn7++/+yfcOnmxOt23d2FFKASc8MAIATkQ0Wdie6ZQ21appVbfuGT7cmWrdPXRRagEnPDACAE5ENFk4mxPKymO3TFW5MtG6fushXFxLTUF3IsobaMX/PmZ4Za7Z32l7VAQCUrsjWWDhtImX3dIXp5lReFIMWWxfCPRwAALsiu2IhXZoQ7W5z+PF9nUz6hbaqNlEXwj0cAAA7Ih0sJPcmRFPf14tTF6bqQriHAwCQT+SDheTehGji+3px6uKmmZNVPWmizve/n/OZy4/VAgBQiMjWWIRFphjUSS8NJzq6e/S5774ybqiQpC/MrWNbAwBQNIKFz9y8qTRXS/NsXvxDT2S6hQIA/FPSwSIo14C7cepivNMm2QS5WygAIDxKosYim6BdA266yDTfaZNsaMsNAChWSa5Y5Lv11MvbSUcqL4upaVa1aqoqda5vaAWh0FWUQkICbbkBAMUquRWLYrpQus3kKoqTkEBbbgCAKSW3YhHU20lNr6LkO21yOdpyAwBMKLlgEcRrwN1o6z3eaZOR6mjLDQAwqOS2QoJ4Dbhbbb1ztR6fMmmi7pw3bfhmV1YqAACmlFywMHE76WDaMtoi3M1VFO74AAB4qeSCRbHXgLtxTNXtVRTu+AAAeKXkaiykwhtSuXVM1e223gAAeKXkViwynG4RuHlMtdhVFAAAgiKSKxZ2W3VntgjunHetmmdPGXfidvuYqhttvQEA8FrkVizcatXtxTFVCi0BAGEXqRULN1t1e3FM1fRpEwAAvBaZFQu3W3WbOKY6nqBdigYAQCEis2Lhdg3EeJ0siy2wDOqlaAAAOBWZYOFVDYTpAks32nkDAOCXyGyF2K1t+K93+4t6H9MFlm618wYAwA+RCRZNs6pVG69Ub2r8FYlfHnlba5dcX1RRpMlOlkG8FA0AgEJFZiukvCyme5pm5H2uNzXg+ZXo4wnipWgAABQqMsFCkq675ipbzwXpt3/aeQMAoiRSwcLub/Xv9g0EphjSzdMmAAB4LVLBIt9v/xnf+c0bWvzIvsAc46SdNwAgKmKWZXn6q3sqlVIikVAymVQ8Hjf+/TM9ISRlPcKZkQkfQZq46bwJAAgqu/N3pFYspNy//V8uiD0inFyKBgBAEEUuWEhD4eLgxiX61h2fGPe5TI+Ip35/Ku9NqAAAIL/I9LHI5m//+MDWc9/5zRvDf839HAAAFC6SKxYd3T1a/Mg+/eSVtxz/We7nAACgcJELFrku9LIriLUXAACERaSCxXgXejlR7E2oAACUqkgFi3wXejkVpA6dAACEQaSChd0gcNe8abae434OAACciVSwsBsEvnRTPfdzAADggkgFC7sXen169hTu5wAAwAWRChZOLvTifg4AAMyL3F0h0tCR0827Towq5MzV+Ir7OQAAyM/u/B3JYCERGAAAMMnu/B3Zlt6ZC73sIogAAFC8yAaLy40XHJxsnQAAgNxKIliMFxwkac32zjHdOjN3hlDICQCAfZGqsci2KrHnRG/W4JDZ5EhcdYX+nuMW1JiGTokc3LiEbREAQEkruRqLbKsStfFKXfx/g1nvDsm8litUZJ7J3BnipF7DFOo+AABhE4lgkbnRdMx2RsrMXR9+3BlC3QcAIIxC3yDL1I2m4/H6zpBcV79n6j46uns8HQ8AAHaFPliYvtF0JD/uDBkvKGVe27zrhAbTnpbGAABgS0HBYuvWrbruuutUWVmphQsX6siRI6bHZVsx2xQxSZOvumL4ry//miXp326u10t/PKNDJ9/zZDLPF5RG1n0AABA0jmssnn32WW3YsEGPPfaYFi5cqEcffVS333673nzzTdXU1LgxxnEVuk2RCRJt//NTkjSmniHx34HjB7/78/BrXtQ42A1KftR9AACQj+MVi+9///t64IEHtHr1ajU0NOixxx7TVVddpZ/97GdujC8vOzeaTr7qCtXGK0a9PvKyseWNdTq4cYl++cCn9cN/m6dvLP2Ykv/4YMyJES9qHOwGJa/rPgAAsMPRisX777+vY8eOadOmTcOvlZWVaenSpTp06FDWPzMwMKCBgYHhz1OpVIFDzS5zo+ma7Z3D2xcZI1clljXUjnt0M9MCfDBtafEj+3LWOMQ0tLqxrKHWlaOfmaDUm7yYdQyZ3hpe1n0AAGCXoxWLd999V4ODg5o6deqo16dOnare3t6sf6atrU2JRGL4o76+vvDR5mDnCvRMcLhz3rVqnj0lZyjwu8bBydXvAAAEjet9LDZt2qQNGzYMf55KpVwLF/lWJewIQo1DJiiNafhFHwsAQMA5ChbXXHONysvLdfbs2VGvnz17VrW1tVn/TEVFhSoqKrJ+zTSnN5pmE5QaB1NBCQAALznaCpk4caJuuukm7d27d/i1dDqtvXv3qrm52fjg/GCnGNSr3hZ2t28AAAgKx6dCNmzYoG3btunpp5/WG2+8oTVr1qi/v1+rV692Y3yeo8YBAIDCOa6xuPvuu/XXv/5V3/72t9Xb26t58+apo6NjTEFn0I13wRc1DgAAFCby16ZnW1mwe8EXt4sCADDE7vwdmWBhNyzkugk1Excyx1MBAMAldufv0F9CJtm/DZQLvgAAcFfog4WTsOB38ysAAKIu9MHCSVgIQvMrAACiLPTBwklYCErzKwAAoir0wcJJWAhS8ysAAKIo9MHCSVig+RUAAO4KfbBwGhbs3IQKAAAKU3J9LDJofgUAgH0l1yBLIiwAAOAWu/O347tCgszEtekAAKBwoa+xAAAAwUGwAAAAxhAsAACAMQQLAABgDMECAAAYQ7AAAADGROq4aQb9LAAA8EfkgoXTDpwAAMCcSG2FdHT3aM32zlGhQpJ6kxe1ZnunOrp7fBoZAAClITLBYjBtafOuE8rWnzzz2uZdJzSY9rSDOQAAJSUyweLIqfNjVipGsiT1JC/qyKnz3g0KAIASE5lgca4vd6go5DkAAOBcZIJFTVWlreeumVTh8kgAAChdkQkWTbOqVZeoVL5Dpf/x6z9QxAkAgEsiEyzKy2JqXdkgSeOGi7MpTogAAOCWyAQLSVreWKf2++Zrajz3dgcnRAAAcE+kgoU0FC6+9+V54z7DCREAANwRuWAhSe9eGLD1HCdEAAAwK5LBwu4JEbvPAQAAeyIZLPKdEIlp6P6QplnVXg4LAIDIi2SwGO+ESObz1pUN3HgKAIBhkQwW0qUTIrWJ0dsdtYlKtd83n5tOAQBwQeSuTR9peWOdljXU6sip8zrXd1E1VUPbH6xUAADgjkgHC2loW6R59hS/hwEAQEmI7FYIAADwHsECAAAYQ7AAAADGECwAAIAxBAsAAGAMwQIAABhDsAAAAMYQLAAAgDEECwAAYIznnTcty5IkpVIpr98aAAAUKDNvZ+bxXDwPFn19fZKk+vp6r98aAAAUqa+vT4lEIufXY1a+6GFYOp3WmTNnVFVVpVjMzGVgqVRK9fX1On36tOLxuJHvGWX8vOzjZ2UfPyv7+Fk5w8/LPjd/VpZlqa+vT9OmTVNZWe5KCs9XLMrKyjR9+nRXvnc8HudfOgf4ednHz8o+flb28bNyhp+XfW79rMZbqcigeBMAABhDsAAAAMZEIlhUVFSotbVVFRUVfg8lFPh52cfPyj5+Vvbxs3KGn5d9QfhZeV68CQAAoisSKxYAACAYCBYAAMAYggUAADCGYAEAAIyJRLDYunWrrrvuOlVWVmrhwoU6cuSI30MKpFdffVUrV67UtGnTFIvF9Pzzz/s9pMBqa2vTzTffrKqqKtXU1Oiuu+7Sm2++6fewAqm9vV1z5swZbsjT3Nysl19+2e9hhcKWLVsUi8W0fv16v4cSOA899JBisdiojxtuuMHvYQXaO++8o/vuu09TpkzRlVdeqU996lM6evSo5+MIfbB49tlntWHDBrW2tqqzs1Nz587V7bffrnPnzvk9tMDp7+/X3LlztXXrVr+HEngHDhxQS0uLDh8+rD179uiDDz7Qbbfdpv7+fr+HFjjTp0/Xli1bdOzYMR09elRLlizRnXfeqddff93voQXaa6+9pscff1xz5szxeyiB9clPflI9PT3DHwcPHvR7SIH1t7/9TYsWLdIVV1yhl19+WSdOnND3vvc9TZ482fvBWCHX1NRktbS0DH8+ODhoTZs2zWpra/NxVMEnydq5c6ffwwiNc+fOWZKsAwcO+D2UUJg8ebL15JNP+j2MwOrr67Ouv/56a8+ePdbnPvc5a926dX4PKXBaW1utuXPn+j2M0Ni4caO1ePFiv4dhWZZlhXrF4v3339exY8e0dOnS4dfKysq0dOlSHTp0yMeRIWqSyaQkqbq62ueRBNvg4KCeeeYZ9ff3q7m52e/hBFZLS4vuuOOOUf/vwlh//vOfNW3aNH3kIx/Rvffeq7ffftvvIQXWiy++qAULFuhLX/qSampqdOONN2rbtm2+jCXUweLdd9/V4OCgpk6dOur1qVOnqre316dRIWrS6bTWr1+vRYsWqbGx0e/hBNLx48d19dVXq6KiQg8++KB27typhoYGv4cVSM8884w6OzvV1tbm91ACbeHChXrqqafU0dGh9vZ2nTp1Sp/97GfV19fn99AC6S9/+Yva29t1/fXXa/fu3VqzZo2+/vWv6+mnn/Z8LJ7fbgqETUtLi7q7u9nfHcfHP/5xdXV1KZlM6rnnntOqVat04MABwsVlTp8+rXXr1mnPnj2qrKz0eziBtmLFiuG/njNnjhYuXKiZM2fqV7/6le6//34fRxZM6XRaCxYs0MMPPyxJuvHGG9Xd3a3HHntMq1at8nQsoV6xuOaaa1ReXq6zZ8+Oev3s2bOqra31aVSIkrVr1+qll17SK6+8ounTp/s9nMCaOHGiPvrRj+qmm25SW1ub5s6dqx/+8Id+Dytwjh07pnPnzmn+/PmaMGGCJkyYoAMHDuhHP/qRJkyYoMHBQb+HGFgf+tCH9LGPfUxvvfWW30MJpLq6ujFB/hOf+IQv20ehDhYTJ07UTTfdpL179w6/lk6ntXfvXvZ3URTLsrR27Vrt3LlT+/bt06xZs/weUqik02kNDAz4PYzAufXWW3X8+HF1dXUNfyxYsED33nuvurq6VF5e7vcQA+vChQs6efKk6urq/B5KIC1atGjMkfg//elPmjlzpudjCf1WyIYNG7Rq1SotWLBATU1NevTRR9Xf36/Vq1f7PbTAuXDhwqi0f+rUKXV1dam6ulozZszwcWTB09LSoh07duiFF15QVVXVcM1OIpHQlVde6fPogmXTpk1asWKFZsyYob6+Pu3YsUP79+/X7t27/R5a4FRVVY2p05k0aZKmTJlC/c5lvvnNb2rlypWaOXOmzpw5o9bWVpWXl+uee+7xe2iB9I1vfEOf+cxn9PDDD+vLX/6yjhw5oieeeEJPPPGE94Px+1iKCT/+8Y+tGTNmWBMnTrSampqsw4cP+z2kQHrllVcsSWM+Vq1a5ffQAifbz0mS9fOf/9zvoQXOV7/6VWvmzJnWxIkTrQ9/+MPWrbfeav32t7/1e1ihwXHT7O6++26rrq7OmjhxonXttddad999t/XWW2/5PaxA27Vrl9XY2GhVVFRYN9xwg/XEE0/4Mg6uTQcAAMaEusYCAAAEC8ECAAAYQ7AAAADGECwAAIAxBAsAAGAMwQIAABhDsAAAAMYQLAAAgDEECwAAYAzBAgAAGEOwAAAAxhAsAACAMf8fjCMPiXB9XyUAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(pine_y,y_pred,'o')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "dfa83922",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 8.21522541e+00 -3.59866087e-03 -3.40027098e-02 -2.23412990e-02\n",
      " -4.14373653e-01  1.22634555e-01  1.49642683e+00 -4.80248118e-01\n",
      "  1.38163801e-03 -1.72979476e+00  3.84548740e-02]\n"
     ]
    }
   ],
   "source": [
    "alpha = np.linalg.solve(X_bias.T@X_bias, X_bias.T@pine_y)\n",
    "print(alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f58bedd6",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "mlenv",
   "language": "python",
   "name": "mlenv"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}