notebooks/initial_design.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## **Initial Design Space**\n",
    "\n",
    "To get recommendations for novel data to sample in order to enhance an emulator, a general assumption is that a surrogate model exists. This need not be the case. There are applications where it is necessary to start with some particular scheme of sampling to effectively span the entire design space at hand. `twinLab` can do this with a simple call to the function `Emulator.design` where `Emulator` is the class that contains the attributes and methods of the surrogate model we wish to build.\n",
    "\n",
    "In this notebook, we will see:\n",
    "\n",
    "- How to get recommendations of points to be sampled in the design space before training an emulator\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start by importing all the necessary packages and dependencies\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "          ====== TwinLab Client Initialisation ======\n",
      "          Version     : 2.10.0\n",
      "          User        : alexander@digilab.co.uk\n",
      "          Server      : https://twinlab.digilab.co.uk/v3\n",
      "          Environment : /Users/mead/digiLab/twinLab/tutorials/.env\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Package imports\n",
    "import matplotlib.pyplot as plt\n",
    "import plotly.graph_objects as go\n",
    "\n",
    "# twinLab imports\n",
    "import twinlab as tl"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`twinLab` allows the user to specify the prior distribution for each of the input dimensions. Here, these priors are in the form of uniform distributions that are defined with a lower and upper bound. Let's go ahead and create priors for a two-dimensional input.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the upper and lower bounds for the uniform distribution\n",
    "xmin1, xmax1 = 4.0, 6.0\n",
    "xmin2, xmax2 = -1.0, 1.0\n",
    "\n",
    "# Define the number of points to be sampled\n",
    "num_samples = 10\n",
    "\n",
    "# Create a list of Prior objects that contains the distribution for each dimension\n",
    "priors = [\n",
    "    tl.Prior(\"x1\", tl.distributions.Uniform(xmin1, xmax1)),\n",
    "    tl.Prior(\"x2\", tl.distributions.Uniform(xmin2, xmax2)),\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can call the `design` method of the `Emulator` class. The function takes in a list of priors (one prior for each input variable), a sampling method which specifies the strategy, and a number of points to be sampled. This function returns a `pandas` `DataFrame` that contains the recommended points.\n",
    "\n",
    "Let's first generate some samples with the Latin Hypercube Sampling method.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>5.838095</td>\n",
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       "    <tr>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
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       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "         x1        x2\n",
       "0  5.152440  0.391614\n",
       "1  4.774644  0.816693\n",
       "2  5.313341 -0.350914\n",
       "3  5.777005 -0.748730\n",
       "4  4.014842 -0.116069\n",
       "5  4.537036 -0.821577\n",
       "6  4.896734 -0.419376\n",
       "7  5.838095  0.102245\n",
       "8  5.537168  0.799658\n",
       "9  4.285963  0.533917"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Initialise an emulator\n",
    "emulator = tl.Emulator(\"test_emulator\")\n",
    "\n",
    "# Call the design method to sample some data points with Latin Hypercube sampling\n",
    "initial_points_lhc = emulator.design(priors, num_samples, tl.DesignParams())\n",
    "display(initial_points_lhc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also generate some samples with a Uniform Random method by using `sampling_method = tl.sampling.UniformRandom()`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
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       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
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       "      <th>1</th>\n",
       "      <td>5.775828</td>\n",
       "      <td>0.567853</td>\n",
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       "         x1        x2\n",
       "0  4.782405  0.282683\n",
       "1  5.775828  0.567853\n",
       "2  5.929219 -0.267390\n",
       "3  5.038211  0.889014\n",
       "4  4.046887 -0.065461\n",
       "5  5.308098  0.094007\n",
       "6  5.492113 -0.890557\n",
       "7  4.263958  0.743582\n",
       "8  4.414892 -0.645300\n",
       "9  4.879744 -0.545616"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Define the InitialDesignParams object with Uniform Random Sampling Method\n",
    "params = tl.DesignParams(sampling_method=tl.sampling.UniformRandom())\n",
    "\n",
    "# Call the design method to sample some data points with Uniform Random sampling\n",
    "initial_points_random = emulator.design(\n",
    "    priors=priors, num_points=num_samples, params=params\n",
    ")\n",
    "display(initial_points_random)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot the generated samples on a two-dimensional plot and visualise them.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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EZLmwsPatR+SgmCSJyHLx8YanWBUK88sVCkCtNtQjcmJMkkRkOaXS8JoHYJoom+dzc/m+JDk9JkkiujppacCWLUC3btLyiAhDOd+TJBfAzgSI6OqlpQEpKexxh1wWkyQRXRulEkhIsHcURFbB5lYiIiIZTJJEREQymCSJiIhkMEkSERHJYJIkIiKSwSRJREQkg0mSiIhIBpMkERGRDHYmQGQhvZ4dzBC5CyZJIgvk5wNz50rHG46IMPT1za5KiVwPm1uJ2ig/Hxg3TpogAaC62lCen2+fuIjIepgkidpArzfcQQphuqy5LDPTUI+IXAeTJFEbFBeb3kFeTgigqspQj4hcB5MkURvU1LRvPSJyDkySRG0QFta+9YjIOTBJErVBfLzhKVaFwvxyhQJQqw31iMh12CRJrly5ElFRUfDx8UFsbCxKS0tl665evRrx8fEICAhAQEAAEhMTTepPmTIFCoVCMiUnJ1v7MMiNKZWG1zwA00TZPJ+by/cliVyN1ZPkpk2bkJWVhSVLlmDXrl0YNGgQkpKScOLECbP1i4qKMGnSJGzbtg0lJSVQq9W4/fbbUV1dLamXnJyMmpoa47RhwwZrHwq5ubQ0YMsWoFs3aXlEhKGc70kSuR6FEOYeam8/sbGxuPnmm/Haa68BAJqamqBWq/Hwww9j4cKFra6v1+sREBCA1157DRkZGQAMd5J1dXX48MMP2xRDQ0MDGhoajPM6nQ5qtRparRYqlcrygyK3xh53iOxPp9PBz8/P6t/jVr2TbGxsRFlZGRITE//coYcHEhMTUVJS0qZtnDt3DhcvXkSXLl0k5UVFRQgODkafPn0wc+ZMnDp1SnYbOTk58PPzM05qtfrqDogIhoSYkABMmmT4ZIIkcl1WTZK//fYb9Ho9QkJCJOUhISHQaDRt2saCBQsQHh4uSbTJycl4++23UVhYiOeeew5fffUV7rjjDuhl3uTOzs6GVqs1TlVVVVd/UERE5DYcuu/WZ599Fhs3bkRRURF8fHyM5RMnTjT+ecCAARg4cCCuv/56FBUVYfTo0Sbb8fb2hre3t01iJiIi12HVO8nAwEAolUrU1tZKymtraxEaGtriusuXL8ezzz6LL774AgMHDmyxbs+ePREYGIhDhw5dc8xERETNrJokvby8EBMTg8LCQmNZU1MTCgsLERcXJ7ve888/j2XLlqGgoABDhw5tdT/Hjh3DqVOnEMY3uYmIqB1Z/RWQrKwsrF69GuvWrUNFRQVmzpyJs2fPYurUqQCAjIwMZGdnG+s/99xzWLRoEd566y1ERUVBo9FAo9Ggvr4eAFBfX4/58+fju+++w5EjR1BYWIiUlBT06tULSUlJ1j4cIiJyI1b/TXLChAk4efIkFi9eDI1Gg8GDB6OgoMD4MM/Ro0fh4fFnrn799dfR2NiIcePGSbazZMkSPPHEE1Aqldi9ezfWrVuHuro6hIeH4/bbb8eyZcv4uyMREbUrq78n6Yhs9X4NERFZh0u8J0lEROTMmCSJiIhkMEkSERHJYJIkIiKSwSRJREQkg0mSiIhIBpMkERGRDCZJIiIiGUySREREMpgkiYiIZDBJEhERyWCSJCIiksEkSUREJINJkoiISAaTJBERkQyrD7pMREQuSq8HiouBmhogLAyIjweUSntH1a6YJInI6bjBd7Pjy88H5s4Fjh37sywiAlixAkhLs19c7YzNrUTkVPLzgagoYNQo4N57DZ9RUYZyspH8fGDcOGmCBIDqakO5C/1lMEkSkdNwo+9mx6XXG+4ghTBd1lyWmWmo5wKYJInIKbjZd7PjKi42/V/K5YQAqqoM9VwAkyQROQU3+252XDU17VvPwTFJEpFTcLPvZscVFta+9RwckyQROQU3+252XPHxhqdYFQrzyxUKQK021HMBTJJE5BTc7LvZcSmVhtc8ANO/jOb53FyXeSeHSZKInIKbfTc7trQ0YMsWoFs3aXlEhKHchd6TVAhh7lkx16bT6eDn5wetVguVSmXvcIjIAubeYVerDQnShb6bnYMde3Ww1fc4kySTJJHTYY87ZKvvcXZLR0ROR6kEEhLsHQW5A/4mSUREJINJkoiISAaTJBERkQwmSSIiIhlMkkRERDKYJImIiGQwSRIREcmwSZJcuXIloqKi4OPjg9jYWJSWlrZYf/Pmzejbty98fHwwYMAA/Oc//5EsF0Jg8eLFCAsLQ8eOHZGYmIiDBw9a8xCIiMgNWT1Jbtq0CVlZWViyZAl27dqFQYMGISkpCSdOnDBb/9tvv8WkSZMwbdo0/PDDD0hNTUVqair27t1rrPP888/jlVdewapVq7Bjxw506tQJSUlJuHDhgrUPh4iI3IjVu6WLjY3FzTffjNdeew0A0NTUBLVajYcffhgLFy40qT9hwgScPXsW//73v41lt9xyCwYPHoxVq1ZBCIHw8HDMmzcPjz76KABAq9UiJCQEeXl5mDhxosk2Gxoa0NDQYJzX6XRQq9Xslo6IyEnZqls6q95JNjY2oqysDImJiX/u0MMDiYmJKCkpMbtOSUmJpD4AJCUlGetXVlZCo9FI6vj5+SE2NlZ2mzk5OfDz8zNOarX6Wg+NiIjcgFWT5G+//Qa9Xo+QkBBJeUhICDQajdl1NBpNi/WbPy3ZZnZ2NrRarXGqqqq6quMhIiL34hYdnHt7e8Pb29veYRARkZOx6p1kYGAglEolamtrJeW1tbUIDQ01u05oaGiL9Zs/LdkmERHR1bBqkvTy8kJMTAwKCwuNZU1NTSgsLERcXJzZdeLi4iT1AWDr1q3G+j169EBoaKikjk6nw44dO2S3SUREdDWs3tyalZWFyZMnY+jQoRg2bBhyc3Nx9uxZTJ06FQCQkZGBbt26IScnBwAwd+5cjBw5Ei+++CLGjh2LjRs34vvvv8c///lPAIBCoUBmZiaeeuop9O7dGz169MCiRYsQHh6O1NRUax8OERG5EasnyQkTJuDkyZNYvHgxNBoNBg8ejIKCAuODN0ePHoWHx583tMOHD8f69evx+OOP429/+xt69+6NDz/8EP379zfWeeyxx3D27FnMmDEDdXV1uPXWW1FQUAAfHx9rHw4REbkRq78n6Yhs9X4NERFZh62+x93i6VZL6fVAcTFQUwOEhQHx8YBSae+oiIjI1pgkr5CfD8ydCxw79mdZRASwYgWQlma/uIiIyPY4Cshl8vOBceOkCRIAqqsN5fn59omLiIjsg0nyD3q94Q7S3C+0zWWZmYZ6RETkHpgk/1BcbHoHeTkhgKoqQz0iInIPTJJ/qKlp33pEROT8mCT/EBbWvvWIiMj5MUn+IT7e8BSrQmF+uUIBqNWGekRE5B6YJP+gVBpe8wBME2XzfG4u35ckInIn7p0k9XqgqAjYsAEoKkJaih5btgDdukmrRUQAW7bwPUkiInfj3p0J9O8PHD/+53xEBNJWrEDKkTT2uENERG7edysASY9/ze2qvG0kInJotuq71b2bW6/EXgOIiOgyTJJXYq8BRET0ByZJOew1gIjI7TFJymGvAUREbs+9n241R6EwvPPBXgOIiNwe7yQvx14DiIjoMu6dJMPDpfPsNYCIiC7j3s2te/cCP/7IXgOIiMgs906SSiWQkGDvKIiIyEG5d3MrERFRC5gkiYiIZDBJEhERyWCSJCIiksEkSUREJINJkoiISAaTJBERkQwmSSIiIhlMkkRERDKYJImIiGQwSRIREclgkiQiIpLBJElERCSDSZKIiEiGVZPk6dOnkZ6eDpVKBX9/f0ybNg319fUt1n/44YfRp08fdOzYEd27d8ecOXOg1Wol9RQKhcm0ceNGax4KERG5IauOJ5meno6amhps3boVFy9exNSpUzFjxgysX7/ebP3jx4/j+PHjWL58OaKjo/Hrr7/iwQcfxPHjx7FlyxZJ3bVr1yI5Odk47+/vb81DISIiN6QQQghrbLiiogLR0dHYuXMnhg4dCgAoKCjAmDFjcOzYMYSHh7dpO5s3b8Z9992Hs2fPokMHQ05XKBT44IMPkJqaelWx6XQ6+Pn5QavVQqVSXdU2iIjo2un1QHExUFMDhIUB8fGAUtn6erb6Hrdac2tJSQn8/f2NCRIAEhMT4eHhgR07drR5O80noDlBNps1axYCAwMxbNgwvPXWW2gp1zc0NECn00kmIiKyr/x8ICoKGDUKuPdew2dUlKHcUVgtSWo0GgQHB0vKOnTogC5dukCj0bRpG7/99huWLVuGGTNmSMqffPJJvPfee9i6dSvuvvtuPPTQQ3j11Vdlt5OTkwM/Pz/jpFarLT8gIiJqN/n5wLhxwLFj0vLqakO5oyRKi5PkwoULzT44c/m0f//+aw5Mp9Nh7NixiI6OxhNPPCFZtmjRIowYMQJDhgzBggUL8Nhjj+GFF16Q3VZ2dja0Wq1xqqqquub4iIjo6uj1wNy5gLkGwOayzExDPXuz+MGdefPmYcqUKS3W6dmzJ0JDQ3HixAlJ+aVLl3D69GmEhoa2uP7vv/+O5ORkXHfddfjggw/g6enZYv3Y2FgsW7YMDQ0N8Pb2Nlnu7e1ttpyIiGyvuNj0DvJyQgBVVYZ6CQk2C8ssi5NkUFAQgoKCWq0XFxeHuro6lJWVISYmBgDw5ZdfoqmpCbGxsbLr6XQ6JCUlwdvbGx9//DF8fHxa3Vd5eTkCAgKYCImInEBNTfvWsyarvQLSr18/JCcnY/r06Vi1ahUuXryI2bNnY+LEicYnW6urqzF69Gi8/fbbGDZsGHQ6HW6//XacO3cO//rXvyQP2QQFBUGpVOKTTz5BbW0tbrnlFvj4+GDr1q145pln8Oijj1rrUIiIqB2FhbVvPWuy6nuS7777LmbPno3Ro0fDw8MDd999N1555RXj8osXL+LAgQM4d+4cAGDXrl3GJ1979eol2VZlZSWioqLg6emJlStX4pFHHoEQAr169cJLL72E6dOnW/NQiIioncTHAxERhod0zP0uqVAYlsfH2z42k1is9Z6kI+N7kkRE9tX8dCsgTZQKheFzyxYgLU1+fad/T5KIiEhOWpohEXbrJi2PiGg9QdqSVZtbiYiI5KSlASkpV9fjjq0wSRIRkd0olfZ/zaMlbG4lIiKSwSRJREQkg0mSiIhIBn+TJCLndLVjLBFZgEmSiJxPfr6hh+zLOwCNiABWrHCcdwfIJbC5lYici7OMsUQugUmSiJyHM42xRC6BSZKInIclYywRtQMmSSJyHs40xhK5BCZJInIezjTGErkEJkkich7NYyw1DxVxJYUCUKsdY4wlcglMkkTkPJRKw2segGmibJ7PzeX7ktRumCSJyLk4yxhL5BLYmQAROR9nGGOJXAKTJBE5J0cfY4lcAptbiYiIZPBOksgRsLNuIofEJElkb+ysm8hhsbmVyJ7YWTeRQ2OSJLIXdtZN5PCYJInshZ11Ezk8Jkkie2Fn3UQOj0mSyF7YWTeRw2OSJLIXdtZN5PCYJInshZ11Ezk8Jkkie2Jn3UQOjZ0JENkbO+smclhMkkSOgJ11EzkkNrcSERHJYJIkIiKSwSRJREQkg0mSiIhIhlWT5OnTp5Geng6VSgV/f39MmzYN9fX1La6TkJAAhUIhmR588EFJnaNHj2Ls2LHw9fVFcHAw5s+fj0uXLlnzUIiIyA1Z9enW9PR01NTUYOvWrbh48SKmTp2KGTNmYP369S2uN336dDz55JPGeV9fX+Of9Xo9xo4di9DQUHz77beoqalBRkYGPD098cwzz1jtWIiIyP0ohDA3Ts+1q6ioQHR0NHbu3ImhQ4cCAAoKCjBmzBgcO3YM4eHhZtdLSEjA4MGDkZuba3b5Z599hjvvvBPHjx9HSEgIAGDVqlVYsGABTp48CS8vr1Zj0+l08PPzg1arhUqluroDJCIiu7HV97jVmltLSkrg7+9vTJAAkJiYCA8PD+zYsaPFdd99910EBgaif//+yM7Oxrlz5yTbHTBggDFBAkBSUhJ0Oh327dtndnsNDQ3Q6XSSiYiIqDVWa27VaDQIDg6W7qxDB3Tp0gUajUZ2vXvvvReRkZEIDw/H7t27sWDBAhw4cAD5f4zQrtFoJAkSgHFebrs5OTlYunTptRwOERG5IYuT5MKFC/Hcc8+1WKeiouKqA5oxY4bxzwMGDEBYWBhGjx6Nw4cP4/rrr7+qbWZnZyMrK8s4r9PpoFarrzpGIiJyDxYnyXnz5mHKlCkt1unZsydCQ0Nx4sQJSfmlS5dw+vRphIaGtnl/sbGxAIBDhw7h+uuvR2hoKEpLSyV1amtrAUB2u97e3vD29m7zPomIiICrSJJBQUEICgpqtV5cXBzq6upQVlaGmJgYAMCXX36JpqYmY+Jri/LycgBA2B8Dz8bFxeHpp5/GiRMnjM25W7duhUqlQnR0tIVHQ2Q/ej37NCdydFZ7cKdfv35ITk7G9OnTUVpaiu3bt2P27NmYOHGi8cnW6upq9O3b13hnePjwYSxbtgxlZWU4cuQIPv74Y2RkZOC2227DwIEDAQC33347oqOj8de//hU//vgjPv/8czz++OOYNWsW7xbJaeTnA1FRwKhRwL33Gj6jogzlRORAhBWdOnVKTJo0SXTu3FmoVCoxdepU8fvvvxuXV1ZWCgBi27ZtQgghjh49Km677TbRpUsX4e3tLXr16iXmz58vtFqtZLtHjhwRd9xxh+jYsaMIDAwU8+bNExcvXmxzXFqtVgAw2S6RLbz/vhAKhRCAdFIoDNP779s7QiLHZ6vvcau9J+nI+J4k2Yteb7hjPHbM/HKFwjDecmUlm16JWuL070kSkaniYvkECRjuKauqDPWIyP6YJIlsqKamfesRkXUxSRLZ0B8PabdbPSKyLiZJIhuKjzf85qhQmF+uUABqtaEeEdkfkySRDSmVwIoVhj9fmSib53Nz+dAOkaNgkiSysbQ0YMsWoFs3aXlEhKE8Lc0+cRGRKauOJ0lE5qWlASkp7HGHyNExSRLZiVIJJCTYOwoiagmbW4mIiGQwSRIREclgkiQiIpLBJElERCSDSZKIiEgGkyQREZEMJkkiIiIZfE+SWqXX86V3InJPTJLUovx8YO5c6RiIERGG/kfZfRoRuTo2t5Ks/Hxg3DjTQYKrqw3l+fn2iYuIyFaYJMksvd5wBymE6bLmssxMQz0iIlfFJElmFReb3kFeTgigqspQj4jIVTFJklk1Ne1bj4jIGTFJkllhYe1bj4jIGTFJklnx8YanWBUK88sVCkCtNtQjInJVTJJkllJpeM0DME2UzfO5uXxfkohcG5MkyUpLA7ZsAbp1k5ZHRBjK+Z4kEbk6diZALUpLA1JS2OMOEbknJklqlVIJJCTYOwoiIttjcysREZEMJkkiIiIZTJJEREQymCSJiIhk8MEdcjh6vR4XL160dxhEAABPT08o+Ti322KSJIchhIBGo0FdXZ29QyGS8Pf3R2hoKBRyXVCRy2KSJIfRnCCDg4Ph6+vLLySyOyEEzp07hxMnTgAAwthZsdthkiSHoNfrjQmya9eu9g6HyKhjx44AgBMnTiA4ONi1m171evYccgWrPrhz+vRppKenQ6VSwd/fH9OmTUN9fb1s/SNHjkChUJidNm/ebKxnbvnGjRuteShkZc2/Qfr6+to5EiJTzdelS/9Wnp8PREUBo0YB995r+IyKMpS7MasmyfT0dOzbtw9bt27Fv//9b3z99deYMWOGbH21Wo2amhrJtHTpUnTu3Bl33HGHpO7atWsl9VJTU615KGQjbGIlR+Ty12V+PjBunOlI69XVhnI3TpRWa26tqKhAQUEBdu7ciaFDhwIAXn31VYwZMwbLly9HeHi4yTpKpRKhoaGSsg8++AD33HMPOnfuLClv/iGdiIiugV4PzJ0LCGG6TAjDsD+ZmYZOnN2w6dVqd5IlJSXw9/c3JkgASExMhIeHB3bs2NGmbZSVlaG8vBzTpk0zWTZr1iwEBgZi2LBheOuttyDM/QX/oaGhATqdTjIREREMv0FeeQd5OSGAqipDPTdktSSp0WgQHBwsKevQoQO6dOkCjUbTpm2sWbMG/fr1w/DhwyXlTz75JN577z1s3boVd999Nx566CG8+uqrstvJycmBn5+fcVKr1ZYfEJGNRUVFITc3195hOAWFQoEPP/zQ3mE4p5qa9q3nYixOkgsXLpR9uKZ52r9//zUHdv78eaxfv97sXeSiRYswYsQIDBkyBAsWLMBjjz2GF154QXZb2dnZ0Gq1xqmqquqa4yMHptcDRUXAhg2GT73eqrubMmXKNf0mnpeXB39/f5PynTt3tvgbflskJCQgMzOzzfskN9TW11rc9PUXi3+TnDdvHqZMmdJinZ49eyI0NNT4blGzS5cu4fTp0236LXHLli04d+4cMjIyWq0bGxuLZcuWoaGhAd7e3ibLvb29zZaTC8rPN/y+cnnzUUQEsGKF040SHRQUZO8QrK6xsRFeXl72DsO9xccb/o1UV5v/XVKhMCyPj7d9bA7A4jvJoKAg9O3bt8XJy8sLcXFxqKurQ1lZmXHdL7/8Ek1NTYiNjW11P2vWrMH//d//temLory8HAEBAUyE7s5Bn9B76aWXMGDAAHTq1AlqtRoPPfSQ8VWooqIiTJ06FVqt1tgS88QTTwAwbW5VKBR488038Ze//AW+vr7o3bs3Pv7442uO78iRI/Dw8MD3338vKc/NzUVkZCSamppQVFQEhUKBTz/9FAMHDoSPjw9uueUW7N27V7LON998g/j4eHTs2BFqtRpz5szB2bNnjcujoqKwbNkyZGRkQKVSGe+Ut2/fjoSEBPj6+iIgIABJSUk4c+aM2fMAAIMHDzaep2Y1NTW444470LFjR/Ts2RNbtmyRLK+qqsI999wDf39/dOnSBSkpKThy5Mg1nDkXoVQa/hMJGBLi5Zrnc3Pd8qEdwIq/Sfbr1w/JycmYPn06SktLsX37dsyePRsTJ040PtlaXV2Nvn37orS0VLLuoUOH8PXXX+P+++832e4nn3yCN998E3v37sWhQ4fw+uuv45lnnsHDDz9srUMhZ9DaE3qA4Qk9Kze9muPh4YFXXnkF+/btw7p16/Dll1/iscceAwAMHz4cubm5UKlUxteZHn30UdltLV26FPfccw92796NMWPGID09HadPn76m+KKiopCYmIi1a9dKyteuXYspU6bAw+PPr4n58+fjxRdfxM6dOxEUFIS77rrL+O7g4cOHkZycjLvvvhu7d+/Gpk2b8M0332D27NmS7S5fvhyDBg3CDz/8gEWLFqG8vByjR49GdHQ0SkpK8M033+Cuu+6C3sK/q0WLFuHuu+/Gjz/+iPT0dEycOBEVFRUADO83JiUl4brrrkNxcTG2b9+Ozp07Izk5GY2NjVdz2lxLWhqwZQvQrZu0PCLCUO5krTDtSljRqVOnxKRJk0Tnzp2FSqUSU6dOFb///rtxeWVlpQAgtm3bJlkvOztbqNVqodfrTbb52WeficGDB4vOnTuLTp06iUGDBolVq1aZrStHq9UKAEKr1V71sVH7On/+vPjpp5/E+fPnr24D27YJYUiHLU9XXGvtYfLkySIlJaXN9Tdv3iy6du1qnF+7dq3w8/MzqRcZGSlefvll4zwA8fjjjxvn6+vrBQDx2Wefye5r5MiRwtPTU3Tq1EkyeXt7S/a5adMmERAQIC5cuCCEEKKsrEwoFApRWVkphBBi27ZtAoDYuHGjcZ1Tp06Jjh07ik2bNgkhhJg2bZqYMWOGZP/FxcXCw8PD+PcaGRkpUlNTJXUmTZokRowYIXsMV54HIYQYNGiQWLJkieTcPPjgg5I6sbGxYubMmUIIId555x3Rp08f0dTUZFze0NAgOnbsKD7//HPZfTe75uvTWVy6ZPg3sn694fPSJXtHJMtW3+NW7ZauS5cuWL9+vezyqKgos69uPPPMM3jmmWfMrpOcnIzk5OR2i5FchAM/offf//4XOTk52L9/P3Q6HS5duoQLFy7g3LlzFvcwNHDgQOOfO3XqBJVKZfLb/5XS09Px97//XVKWn58v+TeWmpqKWbNm4YMPPsDEiRORl5eHUaNGISoqSrJeXFyc8c9dunRBnz59jHdrP/74I3bv3o13333XWEcIgaamJlRWVqJfv34AIHktDDD8XDJ+/Pg2HH3LLo+teb68vNwY26FDh3DddddJ6ly4cAGHDx++5n27DKUSSEiwdxQOhX23kmtw0Cf0jhw5gjvvvBMzZ87E008/jS5duuCbb77BtGnT0NjYaHGS9PT0lMwrFAo0NTW1uI6fnx969eolKbvy9SwvLy9kZGRg7dq1SEtLw/r167Gi+XeqNqqvr8cDDzyAOXPmmCzr3r278c+dOnWSLGvuG1WOh4eHyX+mLe0err6+HjExMZIE3swdHpCiq8ckSfZzeWfK3boBgYFXvy0HfUKvrKwMTU1NePHFF42/7b333nuSOl5eXhb//mYN999/P/r3749//OMfuHTpEtLM/A713XffGRPemTNn8PPPPxvvEG+66Sb89NNPJgm5NQMHDkRhYSGWLl1qdnlQUBBqLmsB0Ol0qKysNBvb5U/Df/fddxgyZIgxtk2bNiE4OBgqlcqi+Mi9WbXvViJZV3amnJFhSHBa7dVtz85P6Gm1WpSXl0umqqoq9OrVCxcvXsSrr76KX375Be+88w5WrVolWTcqKgr19fUoLCzEb7/9hnPnzlklxtb069cPt9xyCxYsWIBJkyaZvcN78sknUVhYiL1792LKlCkIDAw0viO6YMECfPvtt5g9ezbKy8tx8OBBfPTRRyYP7lwpOzsbO3fuxEMPPYTdu3dj//79eP311/Hbb78BAP7nf/4H77zzDoqLi7Fnzx5MnjzZ7EgcmzdvxltvvYWff/4ZS5YsQWlpqXHf6enpCAwMREpKCoqLi1FZWYmioiLMmTMHx1rqbYbcHpMk2Z7cqxp6vaH7qz8e/beYHZ/QKyoqwpAhQyTT0qVLMWjQILz00kt47rnn0L9/f7z77rvIycmRrDt8+HA8+OCDmDBhAoKCgvD8889bLc7WNDcD/7//9//MLn/22Wcxd+5cxMTEQKPR4JNPPjG+5zhw4EB89dVX+PnnnxEfH48hQ4Zg8eLFZvtpvtwNN9yAL774Aj/++COGDRuGuLg4fPTRR+jQwdDQlZ2djZEjR+LOO+/E2LFjkZqaiuuvv95kO0uXLsXGjRsxcOBAvP3229iwYQOio6MBGEbx+Prrr9G9e3ekpaWhX79+mDZtGi5cuMA7S2qRQph7csbF6XQ6+Pn5QavV8h+Iren1hjvIKxLkhchIVK5ahR6BgfDx8gIGDDC9I7RkHxwT76osW7YMmzdvxu7duyXlRUVFGDVqFM6cOeOWPfVcuHABlZWV6NGjB3x8fOwdDsF23+P8TZJsq7XOlAGgsRGorweueBKxzfiEnsXq6+tx5MgRvPbaa3jqqafsHQ6Rw2BzK9lWW1/B4AveNjV79mzExMQgISFBtqmVyB3xTpJsq62vYLA/T5vKy8tDXl6e7PKEhIQWh6MjclW8kyTban5Vo6XfG728gCsG2SYisgcmSbKtll7VaKZWX/1DO0RE7YhJkmxP7lUNpdKQIAMC7BMXEdEV+Jsk2UdaGpCSYtrjjp+fvSMjIjJikiT7ufxVjQsXADNdjRER2RObW4mIiGQwSRI5kKioKOTm5hrnNRoN/vd//xedOnVyy55urpSXl+dQ50GvB4qKgA0bDJ8O0E89tTMmSXI5tv7iSkhIQGZmpkn51Xyh79y5EzNmzDDOv/zyy6ipqUF5eTl+/vnna4y0fSgUCuOkUqlw880346OPPrJ3WDZ3ZR/9o0YZ5vPz7R0ZtScmSXIpzv7FFRQUJBlj8vDhw4iJiUHv3r1NxoBsq0Yr9F60du1a1NTU4Pvvv8eIESMwbtw47Nmzp93346jk+uivrjaUO8v1Rq1jkiSX4ehfXFOmTEFqaiqWL1+OsLAwdO3aFbNmzZIMIHx5c2tUVBTef/99vP3221AoFJgyZQoA4OjRo0hJSUHnzp2hUqlwzz33oLa21riNJ554AoMHD8abb74p6ZBboVDgjTfewJ133glfX1/069cPJSUlOHToEBISEtCpUycMHz4chw8fbvVY/P39ERoaihtuuAHLli3DpUuXsG3bNuPygoIC3HrrrfD390fXrl1x5513SrZ75MgRKBQK5OfnY9SoUfD19cWgQYNQUlIi2U9eXh66d+8OX19f/OUvf8GpU6dMYnn99ddx/fXXw8vLC3369ME777wjWd6exw0YWibmzjU/bGlzWWYmm15dBZMkuQRn+eLatm0bDh8+jG3btmHdunUtdge3c+dOJCcn45577kFNTQ1WrFiBpqYmpKSk4PTp0/jqq6+wdetW/PLLL5gwYYJk3UOHDuH9999Hfn4+ysvLjeXLli1DRkYGysvL0bdvX9x777144IEHkJ2dje+//x5CiFbHf7zcpUuXsGbNGgAwDpkFAGfPnkVWVha+//57FBYWwsPDA3/5y1/Q1NQkWf/vf/87Hn30UZSXl+OGG27ApEmTcOnSJQDAjh07MG3aNOP4lKNGjTLpfP2DDz7A3LlzMW/ePOzduxcPPPAApk6dKknY7X3crfXRL4RhxLfi4jZtjhydcENarVYAEFqt1t6h0B/Onz8vfvrpJ3H+/PmrWn/bNiEMX08tT9u2tWvYQgghRo4cKebOnWtSvnbtWuHn52ecnzx5soiMjBSXLl0ylo0fP15MmDDBOB8ZGSlefvll43xKSoqYPHmycf6LL74QSqVSHD161Fi2b98+AUCUlpYKIYRYsmSJ8PT0FCdOnJDEA0A8/vjjxvmSkhIBQKxZs8ZYtmHDBuHj49Pi8QIQPj4+olOnTsLDw0MAEFFRUeLUqVOy65w8eVIAEHv27BFCCFFZWSkAiDfffNPkOCoqKoQQQkyaNEmMGTNGsp0JEyZIzunw4cPF9OnTJXXGjx8vWa89jvvy63P9+rZda+vXy26O2oGtvsd5J0kuoa2Di7S1nrXceOONUF42tmVYWBhOnDjR5vUrKiqgVquhVquNZdHR0fD390dFRYWxLDIyEkFBQSbrDxw40PjnkJAQAMCAAQMkZRcuXIBOp2sxjpdffhnl5eX47LPPEB0djTfffBNdunQxLj948CAmTZqEnj17QqVSISoqCoChqVgunrA/Or9vPh8VFRWIjY2V1I+Li5PMV1RUYMSIEZKyESNGSM5Fex63Ic5Wq1hUjxwbOxMgl2DPLy6VSgWtVmtSXldXB78rehDy9PSUzCsUCpMmyPbQqVMns+WX71/xR/+45spaiyk0NBS9evVCr169sHbtWowZMwY//fST8eGiu+66C5GRkVi9ejXCw8PR1NSE/v37mzxEdDX7vhrtddzAn330V1ebb95XKAzL4+OvMWhyCLyTJJfQ2uAiCoWhW1hrfHH16dMHu3btMinftWsXbrjhhnbdV79+/VBVVYWqqipj2U8//YS6ujpER0e3677aatiwYYiJicHTTz8NADh16hQOHDiAxx9/HKNHj0a/fv1w5swZi7fbr18/7NixQ1L23XffmdTZvn27pGz79u1WPRct9dHfPJ+ba6hHzo9JklyCPb+4Zs6ciZ9//hlz5szB7t27ceDAAbz00kvYsGED5s2b1677SkxMxIABA5Ceno5du3ahtLQUGRkZGDlyJIYOHdqu+7JEZmYm3njjDVRXVyMgIABdu3bFP//5Txw6dAhffvklsrKyLN7mnDlzUFBQgOXLl+PgwYN47bXXUFBQIKkzf/585OXl4fXXX8fBgwfx0ksvIT8/H48++mh7HZpZcn30R0QYytPSrLp7siEmSXIZ9vri6tmzJ77++mvs378fiYmJiI2NxXvvvYfNmzcjOTm5XfelUCjw0UcfISAgALfddhsSExPRs2dPbNq0qV33Y6nk5GT06NEDTz/9NDw8PLBx40aUlZWhf//+eOSRR/DCCy9YvM1bbrkFq1evxooVKzBo0CB88cUXePzxxyV1UlNTsWLFCixfvhw33ngj3njjDaxduxYJzX0CW1FaGnDkCLBtG7B+veGzspIJ0tUohHC/4cZ1Oh38/Pyg1WqhUqnsHQ4BuHDhAiorKyXv9V0tvf7PwUXCwgxNrGz6omvRntcntQ9bfY/zwR1yOZcPLkJEdC3Y3EpERCSDSZKIiEgGkyQREZEMJklyKNZ4kZzoWvG6dF98cIccgpeXFzw8PHD8+HEEBQXBy8vL2AsKkb0IIdDY2IiTJ0/Cw8ND0ok7uQcmSXIIHh4e6NGjB2pqanD8+HF7h0Mk4evri+7du8PDg41v7oZJkhyGl5cXunfvjkuXLkFv7zGtiP6gVCrRoUMHtmy4KSZJcigKhQKenp4mHYETEdkD2w6IiIhkWC1JPv300xg+fDh8fX3h7+/fpnWEEFi8eDHCwsLQsWNHJCYm4uDBg5I6p0+fRnp6OlQqFfz9/TFt2jTU19db4QiIiMjdWS1JNjY2Yvz48Zg5c2ab13n++efxyiuvYNWqVdixYwc6deqEpKQkXLhwwVgnPT0d+/btw9atW/Hvf/8bX3/9NWbMmGGNQyAiIjdn9Q7O8/LykJmZibq6uhbrCSEQHh6OefPmGYe50Wq1CAkJQV5eHiZOnIiKigpER0dj586dxmGBCgoKMGbMGBw7dgzh4eFmt93Q0ICGhgbjvFarRffu3VFVVcUOzomInJBOp4NarTY7uHm7Ela2du1a4efn12q9w4cPCwDihx9+kJTfdtttYs6cOUIIIdasWSP8/f0lyy9evCiUSqXIz8+X3faSJUsEAE6cOHHi5GLT4cOHLc5LlnCYp1s1Gg0AICQkRFIeEhJiXKbRaBAcHCxZ3qFDB3Tp0sVYx5zs7GzJoK91dXWIjIzE0aNHrfs/kHbW/D8nZ7sDdta4AeeNnXHbFuO2veYWwS5dulh1PxYlyYULF+K5555rsU5FRQX69u17TUG1N29vb3h7e5uU+/n5Od2FAQAqlYpx25izxs64bYtx2561O3iwKEnOmzcPU6ZMabFOz549ryqQ0NBQAEBtbS3CwsKM5bW1tRg8eLCxzokTJyTrXbp0CadPnzauT0RE1F4sSpJBQUEICgqySiA9evRAaGgoCgsLjUlRp9Nhx44dxidk4+LiUFdXh7KyMsTExAAAvvzySzQ1NSE2NtYqcRERkfuy2n3q0aNHUV5ejqNHj0Kv16O8vBzl5eWSdxr79u2LDz74AIChp5XMzEw89dRT+Pjjj7Fnzx5kZGQgPDwcqampAIB+/fohOTkZ06dPR2lpKbZv347Zs2dj4sSJsk+2muPt7Y0lS5aYbYJ1ZIzb9pw1dsZtW4zb9mwWu7WeCJo8ebLZJ5G2bdtmrANArF271jjf1NQkFi1aJEJCQoS3t7cYPXq0OHDggGS7p06dEpMmTRKdO3cWKpVKTJ06Vfz+++/WOgwiInJjVn9PkoiIyFmx71YiIiIZTJJEREQymCSJiIhkMEkSERHJcKkk+eyzzxpfJWnJ5s2b0bdvX/j4+GDAgAH4z3/+I1ku2jBkV3trS+yrV69GfHw8AgICEBAQgMTERJSWlkrqTJkyBQqFQjIlJyfbNe68vDyTmHx8fCR1bH3O2xJ3QkKCSdwKhQJjx4411rHF+X7iiSdM9tFar1aOcI1bGrejXN+Wxu0o17elcTvK9Q0A1dXVuO+++9C1a1d07NgRAwYMwPfff9/iOkVFRbjpppvg7e2NXr16IS8vz6TOypUrERUVBR8fH8TGxppcT23hMkly586deOONNzBw4MAW63377beYNGkSpk2bhh9++AGpqalITU3F3r17jXXaMmSXPWIvKirCpEmTsG3bNpSUlECtVuP2229HdXW1pF5ycjJqamqM04YNG+waN2Do9urymH799VfJclue87bGnZ+fL4l57969UCqVGD9+vKSeLc73jTfeKNnHN998I1vXka5xS+J2pOvbkrgBx7m+LYnbUa7vM2fOYMSIEfD09MRnn32Gn376CS+++CICAgJk16msrMTYsWMxatQolJeXIzMzE/fffz8+//xzY51NmzYhKysLS5Yswa5duzBo0CAkJSWZ9NrWKnu+f9Jefv/9d9G7d2+xdetWMXLkSDF37lzZuvfcc48YO3aspCw2NlY88MADQgjDu5qhoaHihRdeMC6vq6sT3t7eYsOGDXaN/UqXLl0S1113nVi3bp2xbPLkySIlJaXd47ySJXG3NhKMLc/5tZzvl19+WVx33XWivr7eWGaL871kyRIxaNCgNtd3lGvc0rivZK/r29K4HeX6vtbzba/re8GCBeLWW2+1aJ3HHntM3HjjjZKyCRMmiKSkJOP8sGHDxKxZs4zzer1ehIeHi5ycHIv25RJ3krNmzcLYsWORmJjYat2SkhKTeklJSSgpKQFg+B+KRqOR1PHz80NsbKyxTnuyJPYrnTt3DhcvXjTpBb+oqAjBwcHo06cPZs6ciVOnTrVXuEaWxl1fX4/IyEio1WqkpKRg3759xmW2POfXcr7XrFmDiRMnolOnTpJyW5zvgwcPIjw8HD179kR6ejqOHj0qW9eRrnFL4r6SPa9vS+N2lOv7Ws63va7vjz/+GEOHDsX48eMRHByMIUOGYPXq1S2u09o13tjYiLKyMkkdDw8PJCYmWnzOHWaorKu1ceNG7Nq1Czt37mxTfY1G0+pwXM1lcnXai6WxX2nBggUIDw+XXAjJyclIS0tDjx49cPjwYfztb3/DHXfcgZKSEiiVSrvE3adPH7z11lsYOHAgtFotli9fjuHDh2Pfvn2IiIiw2Tm/lvNdWlqKvXv3Ys2aNZJyW5zv2NhY5OXloU+fPqipqcHSpUsRHx+PvXv34rrrrjOp7yjXuKVxX8le17elcTvK9X0t59ue1/cvv/yC119/HVlZWfjb3/6GnTt3Ys6cOfDy8sLkyZPNriN3jet0Opw/fx5nzpyBXq83W2f//v2WBWjRfaeDOXr0qAgODhY//vijsay1JjRPT0+xfv16SdnKlStFcHCwEEKI7du3CwDi+PHjkjrjx48X99xzj11jv1xOTo4ICAiQrG9O82DW//3vf68lXKNrjVsIIRobG8X1118vHn/8cSGEbc75tcY9Y8YMMWDAgFbrtff5NufMmTNCpVKJN9980+xyR7nGr9Ra3Jez1/VtjiVxC2Gf69scS+K25/Xt6ekp4uLiJGUPP/ywuOWWW2TX6d27t3jmmWckZZ9++qkAIM6dOyeqq6sFAPHtt99K6syfP18MGzbMovicurm1rKwMJ06cwE033YQOHTqgQ4cO+Oqrr/DKK6+gQ4cO0Ov1JuuEhoaitrZWUlZbW2scauvyIbvk6tgr9mbLly/Hs88+iy+++KLVh0969uyJwMBAHDp0yO5xN/P09MSQIUOMMdninF9L3GfPnsXGjRsxbdq0VvfT3ufbHH9/f9xwww2y+3CUa/xKrcXdzJ7XtzltjbuZPa5vc9oat72v77CwMERHR0vK+vXr12JTsdw1rlKp0LFjRwQGBkKpVLbLOXfqJDl69Gjs2bPHOMJIeXk5hg4divT0dJSXl5ttDoiLi0NhYaGkbOvWrYiLiwMgHbKrWfOQXc117BU7YHhKbtmyZSgoKMDQoUNb3c+xY8dw6tQpyRid9oj7cnq9Hnv27DHGZItzfi1xb968GQ0NDbjvvvta3U97n29z6uvrcfjwYdl9OMo1bmncgP2vb3PaEvfl7HF9m9PWuO19fY8YMQIHDhyQlP3888+IjIyUXae1a9zLywsxMTGSOk1NTSgsLLT8nFt03+kErmxC++tf/yoWLlxonN++fbvo0KGDWL58uaioqBBLliwRnp6eYs+ePcY6zz77rPD39xcfffSR2L17t0hJSRE9evQQ58+ft2vszz77rPDy8hJbtmwRNTU1xql5FJTff/9dPProo6KkpERUVlaK//73v+Kmm24SvXv3FhcuXLBb3EuXLhWff/65OHz4sCgrKxMTJ04UPj4+Yt++fZJjs/U5by3uZrfeequYMGGCSbmtzve8efNEUVGRqKysFNu3bxeJiYkiMDBQnDhxwmzcjnKNWxq3o1zflsbtKNe3pXE3s/f1XVpaKjp06CCefvppcfDgQfHuu+8KX19f8a9//ctYZ+HCheKvf/2rcf6XX34Rvr6+Yv78+aKiokKsXLlSKJVKUVBQYKyzceNG4e3tLfLy8sRPP/0kZsyYIfz9/YVGo7EoPpdPkiNHjhSTJ0+W1HnvvffEDTfcILy8vMSNN94oPv30U8nytgzZZY/YIyMjzQ4/tmTJEiGEEOfOnRO33367CAoKEp6eniIyMlJMnz7d4ouivePOzMwU3bt3F15eXiIkJESMGTNG7Nq1S7INe5zztlwr+/fvFwDEF198YbK+rc73hAkTRFhYmPDy8hLdunUTEyZMEIcOHWoxbke4xi2N21Gub0vjdpTr+2quE0e4voUQ4pNPPhH9+/cX3t7eom/fvuKf//ynZPnkyZPFyJEjJWXbtm0TgwcPFl5eXqJnz56SYRebvfrqq8a/m2HDhonvvvvO4tg4VBYREZEMp/5NkoiIyJqYJImIiGQwSRIREclgkiQiIpLBJElERCSDSZKIiEgGkyQREZEMJkkiIiIZTJJEREQymCSJiIhkMEkSERHJ+P+BDRy5IOQmrAAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create 2-D scatter plot to visualise the generated data points\n",
    "plt.scatter(\n",
    "    initial_points_lhc[\"x1\"],\n",
    "    initial_points_lhc[\"x2\"],\n",
    "    color=\"red\",\n",
    "    label=\"Latin Hypercube\",\n",
    ")\n",
    "plt.scatter(\n",
    "    initial_points_random[\"x1\"],\n",
    "    initial_points_random[\"x2\"],\n",
    "    color=\"blue\",\n",
    "    label=\"Uniform Random\",\n",
    ")\n",
    "plt.xlim(xmin1, xmax1)\n",
    "plt.ylim(xmin2, xmax2)\n",
    "plt.legend()\n",
    "plt.gca().set_aspect(\"equal\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the recommended points are from different regions of the space and this is a good way to start with when there is no initial data at hand. Moreover, we can see that the Latin Hypercube design fills the space more evenly than Uniform Random sampling.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## **Three-dimensional plots**\n",
    "\n",
    "To understand this better, let's also try visualising some data in three dimensions. As in two dimensions, we start with defining the priors for each dimension.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the priors for the 3 dimensions\n",
    "xmin1, xmax1 = 0, 1\n",
    "xmin2, xmax2 = -1, 1\n",
    "xmin3, xmax3 = 1, 5\n",
    "num_samples = 20\n",
    "priors = [\n",
    "    tl.Prior(\"x1\", tl.distributions.Uniform(xmin1, xmax1)),\n",
    "    tl.Prior(\"x3\", tl.distributions.Uniform(xmin3, xmax3)),\n",
    "    tl.Prior(\"x2\", tl.distributions.Uniform(xmin2, xmax2)),\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate some points in the three-dimensional space using the Latin HyperCube sampling method.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x3</th>\n",
       "      <th>x2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.663275</td>\n",
       "      <td>2.516673</td>\n",
       "      <td>-0.448634</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.414869</td>\n",
       "      <td>3.873020</td>\n",
       "      <td>-0.808536</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.396221</td>\n",
       "      <td>2.362963</td>\n",
       "      <td>-0.073499</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.506995</td>\n",
       "      <td>1.098995</td>\n",
       "      <td>0.587711</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.085142</td>\n",
       "      <td>1.955937</td>\n",
       "      <td>-0.741418</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.843611</td>\n",
       "      <td>1.414040</td>\n",
       "      <td>-0.104625</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.939660</td>\n",
       "      <td>4.404250</td>\n",
       "      <td>-0.601657</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.234513</td>\n",
       "      <td>1.244551</td>\n",
       "      <td>-0.570116</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.723374</td>\n",
       "      <td>4.798774</td>\n",
       "      <td>0.408920</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.028380</td>\n",
       "      <td>3.350137</td>\n",
       "      <td>0.096712</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.457147</td>\n",
       "      <td>4.125053</td>\n",
       "      <td>0.115411</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.261148</td>\n",
       "      <td>4.927352</td>\n",
       "      <td>-0.347988</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.781703</td>\n",
       "      <td>2.947065</td>\n",
       "      <td>-0.972371</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.577224</td>\n",
       "      <td>3.437354</td>\n",
       "      <td>-0.267792</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.326597</td>\n",
       "      <td>3.111354</td>\n",
       "      <td>0.353709</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.968010</td>\n",
       "      <td>2.150680</td>\n",
       "      <td>0.231679</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.183975</td>\n",
       "      <td>2.648610</td>\n",
       "      <td>0.816878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.878907</td>\n",
       "      <td>3.653508</td>\n",
       "      <td>0.794339</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.609374</td>\n",
       "      <td>1.799206</td>\n",
       "      <td>0.990851</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.102506</td>\n",
       "      <td>4.301224</td>\n",
       "      <td>0.659168</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          x1        x3        x2\n",
       "0   0.663275  2.516673 -0.448634\n",
       "1   0.414869  3.873020 -0.808536\n",
       "2   0.396221  2.362963 -0.073499\n",
       "3   0.506995  1.098995  0.587711\n",
       "4   0.085142  1.955937 -0.741418\n",
       "5   0.843611  1.414040 -0.104625\n",
       "6   0.939660  4.404250 -0.601657\n",
       "7   0.234513  1.244551 -0.570116\n",
       "8   0.723374  4.798774  0.408920\n",
       "9   0.028380  3.350137  0.096712\n",
       "10  0.457147  4.125053  0.115411\n",
       "11  0.261148  4.927352 -0.347988\n",
       "12  0.781703  2.947065 -0.972371\n",
       "13  0.577224  3.437354 -0.267792\n",
       "14  0.326597  3.111354  0.353709\n",
       "15  0.968010  2.150680  0.231679\n",
       "16  0.183975  2.648610  0.816878\n",
       "17  0.878907  3.653508  0.794339\n",
       "18  0.609374  1.799206  0.990851\n",
       "19  0.102506  4.301224  0.659168"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Initialise an emulator\n",
    "new_emulator = tl.Emulator(\"3D_emulator\")\n",
    "\n",
    "# Define the InitialDesignParams object\n",
    "params = tl.DesignParams()\n",
    "\n",
    "# Call the design method to sample some data points with Latin Hypercube sampling\n",
    "initial_points_lhc = emulator.design(priors, num_samples, params=params)\n",
    "display(initial_points_lhc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also do the same with Uniform Random sampling for comparison.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x1</th>\n",
       "      <th>x3</th>\n",
       "      <th>x2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.733925</td>\n",
       "      <td>3.866178</td>\n",
       "      <td>0.161589</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.384208</td>\n",
       "      <td>3.486594</td>\n",
       "      <td>-0.360965</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.665548</td>\n",
       "      <td>2.773607</td>\n",
       "      <td>-0.953394</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.616252</td>\n",
       "      <td>2.417605</td>\n",
       "      <td>0.982409</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.850639</td>\n",
       "      <td>2.018563</td>\n",
       "      <td>0.416636</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.197053</td>\n",
       "      <td>1.254959</td>\n",
       "      <td>0.672306</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.995332</td>\n",
       "      <td>3.084010</td>\n",
       "      <td>-0.021980</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.245574</td>\n",
       "      <td>2.930389</td>\n",
       "      <td>-0.610964</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.504518</td>\n",
       "      <td>4.945792</td>\n",
       "      <td>-0.120311</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.843837</td>\n",
       "      <td>3.775162</td>\n",
       "      <td>0.714899</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.911225</td>\n",
       "      <td>4.502642</td>\n",
       "      <td>-0.564896</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.090088</td>\n",
       "      <td>3.385987</td>\n",
       "      <td>0.877961</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.302272</td>\n",
       "      <td>2.378045</td>\n",
       "      <td>0.099993</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.463364</td>\n",
       "      <td>4.366634</td>\n",
       "      <td>0.575504</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.418122</td>\n",
       "      <td>1.640041</td>\n",
       "      <td>-0.706834</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.583209</td>\n",
       "      <td>1.553834</td>\n",
       "      <td>0.218399</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.040863</td>\n",
       "      <td>1.814247</td>\n",
       "      <td>-0.213006</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.773586</td>\n",
       "      <td>1.079272</td>\n",
       "      <td>-0.442174</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.281515</td>\n",
       "      <td>4.697074</td>\n",
       "      <td>0.388213</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.117242</td>\n",
       "      <td>4.174713</td>\n",
       "      <td>-0.894143</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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       "          x1        x3        x2\n",
       "0   0.733925  3.866178  0.161589\n",
       "1   0.384208  3.486594 -0.360965\n",
       "2   0.665548  2.773607 -0.953394\n",
       "3   0.616252  2.417605  0.982409\n",
       "4   0.850639  2.018563  0.416636\n",
       "5   0.197053  1.254959  0.672306\n",
       "6   0.995332  3.084010 -0.021980\n",
       "7   0.245574  2.930389 -0.610964\n",
       "8   0.504518  4.945792 -0.120311\n",
       "9   0.843837  3.775162  0.714899\n",
       "10  0.911225  4.502642 -0.564896\n",
       "11  0.090088  3.385987  0.877961\n",
       "12  0.302272  2.378045  0.099993\n",
       "13  0.463364  4.366634  0.575504\n",
       "14  0.418122  1.640041 -0.706834\n",
       "15  0.583209  1.553834  0.218399\n",
       "16  0.040863  1.814247 -0.213006\n",
       "17  0.773586  1.079272 -0.442174\n",
       "18  0.281515  4.697074  0.388213\n",
       "19  0.117242  4.174713 -0.894143"
      ]
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     "metadata": {},
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    }
   ],
   "source": [
    "# Define the DesignParams object with Uniform Random Sampling Method\n",
    "params = tl.DesignParams(sampling_method=tl.sampling.UniformRandom())\n",
    "\n",
    "# Generate the points with Uniform Random sampling and display\n",
    "initial_points_random = emulator.design(priors, num_samples, params=params)\n",
    "display(initial_points_random)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot a nice three-dimensional visualisation of the data points generated using the `design` method with both the Latin Hypercube and Uniform Random sampling schemes. Feel free to move the plot around an visualise the data across the various dimensions.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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    "# Create 3D scatter plot\n",
    "fig = go.Figure()\n",
    "\n",
    "# Add the first set of points\n",
    "fig.add_trace(\n",
    "    go.Scatter3d(\n",
    "        x=initial_points_random[\"x1\"],\n",
    "        y=initial_points_random[\"x2\"],\n",
    "        z=initial_points_random[\"x3\"],\n",
    "        mode=\"markers\",\n",
    "        name=\"Uniform Random\",\n",
    "    )\n",
    ")\n",
    "\n",
    "# Add the second set of points\n",
    "fig.add_trace(\n",
    "    go.Scatter3d(\n",
    "        x=initial_points_lhc[\"x1\"],\n",
    "        y=initial_points_lhc[\"x2\"],\n",
    "        z=initial_points_lhc[\"x3\"],\n",
    "        mode=\"markers\",\n",
    "        name=\"Latin Hypercube\",\n",
    "    )\n",
    ")\n",
    "\n",
    "# Set labels and title\n",
    "fig.update_layout(\n",
    "    scene=dict(xaxis_title=\"x1\", yaxis_title=\"x2\", zaxis_title=\"x3\"),\n",
    "    title=\"3D Scatter Plot\",\n",
    "    width=900,\n",
    "    height=600,\n",
    ")\n",
    "\n",
    "# Show plot\n",
    "fig.show()"
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  {
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   "source": [
    "We can also visualise the two-dimensional projections of the sampled points in each of the three possible two-dimensional projections from three dimensions. This can help us understand how the different sampling strategies are covering the design space across various dimensions. You can see that the Latin Hypercube design fills the space well in the projected dimensions\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
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            "type": "scattergl"
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          "font": {
           "color": "#2a3f5f"
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          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
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          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
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          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
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           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
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          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "3D Scatter Plot"
        },
        "width": 600
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set layout options\n",
    "fig.update_layout(scene=dict(aspectmode=\"auto\"))\n",
    "\n",
    "# Create 2D projections on each plane\n",
    "fig.update_layout(scene_camera=dict(eye=dict(x=2, y=0, z=0)), width=600, height=550)\n",
    "fig.show()\n",
    "\n",
    "fig.update_layout(scene_camera=dict(eye=dict(x=0, y=0, z=2)), width=600, height=550)\n",
    "fig.show()\n",
    "\n",
    "fig.update_layout(scene_camera=dict(eye=dict(x=0, y=2, z=0)), width=600, height=550)\n",
    "fig.show()"
   ]
  }
 ],
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