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 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as lin\n",
    "import matplotlib.pylab as plt\n",
    "import plotly.graph_objects as go\n",
    "np.set_printoptions(precision=3, linewidth=150, suppress=True)\n",
    "\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'retina'\n",
    "\n",
    "np.random.seed(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "## Exercise: 3D point cloud\n",
    "\n",
    "We have measurement results and we know that we must have a relationship\n",
    "quadratic between x and z:\n",
    "\n",
    "$$  z = \\alpha \\, x^2 + \\beta \\, x + \\gamma $$\n",
    "\n",
    "How to find these 3 coefficients?\n",
    "\n",
    "Of course we will do a principal component analysis but beware, this\n",
    "only works for linear relations (it gives us a vector). Therefore\n",
    "we must introduce a new variable so that our equation is linear.\n",
    "\n",
    "How to write our equation so that it is linear according to 2 variables?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### réponse :\n",
    "\n",
    "..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "### Experience Data\n",
    "\n",
    "Make a 2D point cloud with the equation\n",
    "\n",
    "$$  z = -1.3 \\, x^2 + 0.2 \\, x + 1.45 + U(-1,1)$$\n",
    "\n",
    "where U is the uniform law that simulates noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 50\n",
    "x = ...  # x varie entre -3 et 3\n",
    "z = ...\n",
    "nuage = np.array([x,z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(nuage[0], nuage[1], 'x')\n",
    "plt.title('Un nuage de points')\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "### Calculations to find the characteristics of our cloud\n",
    "\n",
    "Manufacture from our 2D point cloud a 3D point cloud by introducing the new\n",
    "variable that we have chosen.\n",
    "\n",
    "The new cloud is called `nuage3D`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = go.Figure(data=[go.Scatter3d(x=nuage3D[0], y=nuage3D[1], z=nuage3D[2], mode='markers')])\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "Calculate the covariance matrix of our cloud and its eigenvectors (we will store the eigenvectors in the `vec` variable)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = go.Figure(data=[go.Scatter3d(x=nuage3D[0], y=nuage3D[1], z=nuage3D[2], mode='markers'),\n",
    "                     go.Scatter3d(x=[0,-5*vec[0,0]], y=[0,-5*vec[1,0]], z=[0,-5*vec[2,0]]),\n",
    "                     go.Scatter3d(x=[0,vec[0,1]], y=[0,vec[1,1]], z=[0,vec[2,1]])])\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "What can we deduce from our first eigenvector?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### réponse\n",
    "\n",
    "..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "Create a 2D point cloud that no longer takes into account the impact of the coefficient\n",
    "that we just found.\n",
    "\n",
    "We call this new cloud `nuage2D` (it is not the same as our initial `cloud`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(nuage2D[0], nuage2D[1], 'x')\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "What can we deduce from this?\n",
    "\n",
    "#### response\n",
    "\n",
    "...."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"Les coefficients de z fonction polynomiale de degré 2 en x sont :\\n\")\n",
    "print(f\"alpha = {alpha}\")\n",
    "print(f\"beta  = {beta}\")\n",
    "print(f\"gamma = {gamma}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "You can rerun everything with changing the value of the randomness seed. You will see that sometimes the\n",
    "noise disturbs the results much more."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ]
}