From 01ec00cbfeae124acae247a411486f399bccec4e Mon Sep 17 00:00:00 2001
From: "J. Gergaud" <gergaud@ap240726-01.local>
Date: Sun, 17 Nov 2024 22:30:42 +0100
Subject: [PATCH] foo

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 M2/TP-PCA.pdf   |  Bin 0 -> 637744 bytes
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 create mode 100644 M2/TP-PCA.pdf

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Principal component analysis (PCA)\n",
+    "\n",
+    "## Introduction\n",
+    "\n",
+    "Principal component analysis (PCA) reduces the number of dimensions in large datasets to principal components that retain most of the original information. It does this by transforming potentially correlated variables into a smaller set of uncorrelated variables, called principal components.\n",
+    "\n",
+    "Let $X$ the matrix $(n,p)$ of data, we note $X_c$ = the centered matrix. Then the empirical variances, covariances matrix is $C = \\frac{1}{n}X_c^TX_c$. We note $\\Lambda$ the vector of the eigen value (in decrease order) of the matrix $C$ and $U$ the $(p,p)$ orthogonal matrix of the eigen vectors : \n",
+    "$$C=U\\texttt{diag}(\\Lambda) U^T.$$\n",
+    "Then We have\n",
+    "\n",
+    "* The coordinates of the $n$ observations in the new basis of the eigen vectors $(\\vec{u}_1,\\ldots,\\vec{u}_p)$ are \n",
+    "$$\\Psi = X_cU$$\n",
+    "* The The coordinates of the $p$ variables in the new basis of the eigen vectors $(\\vec{v}_1,\\ldots,\\vec{v}_p)$ are \n",
+    "$$\\Phi = \\sqrt{n}U\\texttt{diag}(\\sqrt{\\lambda_1},\\ldots,\\sqrt{\\lambda}_p)$$\n",
+    "* The total inertia (variance) is \n",
+    "$$I = \\texttt{trace}(C)=\\sum_{i=1,p}\\lambda_i$$\n",
+    "- The variance of the variable $v_i$ is $\\lambda_i$\n",
+    "\n",
+    "## Iris Data\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(150, 4)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "using Plots\n",
+    "using Statistics, LinearAlgebra\n",
+    "using RDatasets, DataFrames\n",
+    "iris = RDatasets.dataset(\"datasets\", \"iris\")  # Iris Datas\n",
+    "Names = names(iris)\n",
+    "X = Matrix(iris[:,1:4])\n",
+    "p1 = scatter(X[:,1],X[:,2], c=[:blue :red :green], group=iris.Species,xlabel = Names[1],ylabel=Names[2])\n",
+    "p2 = scatter(X[:,3],X[:,4], c=[:blue :red :green], group=iris.Species,xlabel = Names[3],ylabel=Names[4])\n",
+    "plot(p1,p2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### My PCA function\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "my_PCA (generic function with 1 method)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "using LinearAlgebra, Statistics\n",
+    "function my_PCA(X::Matrix{<:Real})::Tuple{Vector{<:Real},Matrix{<:Real},Matrix{<:Real},Matrix{<:Real},Real,Vector{<:Real}}\n",
+    "\"\"\"\n",
+    "    Compute the PCA of Data\n",
+    "    Input\n",
+    "    X : (n,p) Matrix of reals\n",
+    "         n = number of observations\n",
+    "         p = number of variables\n",
+    "    Output\n",
+    "        Λ : Vector of the p eigen value in decrease order\n",
+    "        U : (p,p) Matrix of reals\n",
+    "            eigen vectors in column\n",
+    "        Ψ : (n,p) Matrix of reals\n",
+    "            Coordinates of the observation in the new basis\n",
+    "        Φ = (p,p) Matrix of reals\n",
+    "             Coordinates of the variables in the new basis\n",
+    "        I_total : Real\n",
+    "             total inertia\n",
+    "        cum_var_ratio : p vector of reals\n",
+    "             cumulative variance ratio\n",
+    "\"\"\"\n",
+    "     n,p = size(X)\n",
+    "     Λ = zeros(p); U = zeros(p,p); Ψ = zeros(n,p); Φ = zeros(p,p); I_total=0; cum_var_ratio = zeros(p)\n",
+    "    return Λ, U, Ψ, Φ, I_total,cum_var_ratio\n",
+    "end\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "### Print results\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "4×4 Matrix{Float64}:\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "150×4 Matrix{Float64}:\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " ⋮              \n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "4×4 Matrix{Float64}:\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0\n",
+       " 0.0  0.0  0.0  0.0"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(150, 4)\n",
+      "lambda = [0.0, 0.0, 0.0, 0.0]\n",
+      "my_U = \n",
+      "my_Ψ = \n",
+      "my_Φ = \n",
+      "Total inertia = 0\n",
+      "my_cum_var_ratio = [0.0, 0.0, 0.0, 0.0]\n"
+     ]
+    }
+   ],
+   "source": [
+    "\n",
+    "my_PCA_results = my_PCA(X)\n",
+    "my_Λ, my_U, my_Ψ, my_Φ, my_I_total, my_cum_var_ratio = my_PCA_results\n",
+    "println(\"lambda = \", my_Λ)\n",
+    "println(\"my_U = \")\n",
+    "display(my_U)\n",
+    "println(\"my_Ψ = \")\n",
+    "display(my_Ψ)\n",
+    "println(\"my_Φ = \")\n",
+    "display(my_Φ)\n",
+    "println(\"Total inertia = \", my_I_total)\n",
+    "println(\"my_cum_var_ratio = \", my_cum_var_ratio)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "### Graph of the observations\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+   ],
+   "source": [
+    "p3 = scatter(my_Ψ[:,1],my_Ψ[:,2], c=[:blue :red :green], group=iris.Species,xlabel = \"PC1\", ylabel=\"PC2\")\n",
+    "p4 = scatter(my_Ψ[:,3],my_Ψ[:,4], c=[:blue :red :green], group=iris.Species,xlabel = \"PC1\", ylabel=\"PC2\")\n",
+    "plot(p3,p4)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Graph of the variables\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "\n",
+    "pvar1 = plot()\n",
+    "ech = 1.1*maximum(abs.(my_Φ))\n",
+    "for i=1:4\n",
+    "    plot!(pvar1,[0,my_Φ[i,1]], [0,my_Φ[i,2]], xlims=(-ech,ech), ylims=(-ech,ech), arrow=true, label=Names[i], legend=:bottomleft, xlabel=\"v1\", ylabel=\"v2\")\n",
+    "end\n",
+    "\n",
+    "pvar2 = plot()\n",
+    "for i=1:4\n",
+    "    plot!(pvar2,[0,my_Φ[i,3]], [0,my_Φ[i,4]], xlims=(-ech,ech), ylims=(-ech,ech), arrow=true, label=Names[i], legend=:bottomleft, xlabel=\"v3\", ylabel=\"v4\")\n",
+    "end\n",
+    "plot(pvar1,pvar2)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### With MultivariateStats package\n",
+    "\n",
+    "Use the MultivariateStats package for obtaining the same results\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Normed PCA\n",
+    "\n",
+    "### Introduction\n",
+    "\n",
+    "* $X$ the matrix $(n,p)$ of data\n",
+    "* $X_c$ is the centered matrix\n",
+    "* $Y$ is the centered and normed matrix. Each column of $Xc$ is divided by its sample standard deviation\n",
+    "* $R=Y^TY$ is the corretalion matrix of the Data $X$.\n",
+    "* $$C=U\\texttt{diag}(\\Lambda) U^T.$$\n",
+    "Then We have\n",
+    "\n",
+    "* The coordinates of the $n$ observations in the new basis of the eigen vectors $(\\vec{u}_1,\\ldots,\\vec{u}_p)$ are \n",
+    "$$\\Psi = YU$$\n",
+    "* The The coordinates of the $p$ variables in the new basis of the eigen vectors $(\\vec{v}_1,\\ldots,\\vec{v}_p)$ are \n",
+    "$$\\Phi = U\\texttt{diag}(\\sqrt{\\lambda_1},\\ldots,\\sqrt{\\lambda}_p)$$\n",
+    "* The total inertia (variance) is \n",
+    "$$I = \\texttt{trace}(C)=\\sum_{i=1,p}\\lambda_i$$\n",
+    "- The variance of the variable $v_i$ is $\\lambda_i$\n",
+    "\n",
+    "\n",
+    "### Data\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "20-element Vector{String}:\n",
+       " \"Aix-les-Bains\"\n",
+       " \"Beckerish\"\n",
+       " \"Cayranne\"\n",
+       " \"Chambon\"\n",
+       " \"Cristal-Roc\"\n",
+       " \"St Cyr\"\n",
+       " \"Evian\"\n",
+       " \"Ferita\"\n",
+       " \"St Hyppolite\"\n",
+       " \"Laurier\"\n",
+       " \"Ogeu\"\n",
+       " \"Ondine\"\n",
+       " \"Perrier\"\n",
+       " \"Ribes\"\n",
+       " \"Spa\"\n",
+       " \"Thonon\"\n",
+       " \"Veri\"\n",
+       " \"Viladreau\"\n",
+       " \"Vittel\"\n",
+       " \"Volvic\""
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "\n",
+    "# Data from Tomassone page 138 : mineral waters\n",
+    "X =[341   27   3   84   23   2  \n",
+    "263   23   9   91   5   3  \n",
+    "287   3   5   44   24   23  \n",
+    "   298   9   23   96   6   11 \n",
+    "    200   15   8   70   2   4 \n",
+    "    250   5   20   71   6   11 \n",
+    "   357   10   2   78   24   5 \n",
+    "      311   14   18   73   18   13 \n",
+    "    256   6   23   86   3   18  \n",
+    "   186   10   16   64   4   9 \n",
+    "    183   16   44   48   11   31 \n",
+    "       398   218   15   157   35   8 \n",
+    "      348   51   31   140   4   14 \n",
+    "   168   24   8   55   5   9 \n",
+    "   110   65   5   4   1   3 \n",
+    "   332   14   8   103   16   5 \n",
+    "      196   18   6   58   6   13 \n",
+    "       59   7   6   16   2   9 \n",
+    "       402   306   15   202   36   3 \n",
+    "       64   7   8   10   6   8 ]\n",
+    "\n",
+    "df = DataFrame(X,[:HCO3, :SO4, :Cl, :Ca, :Mg, :Na])\n",
+    "df[:, :Origins] = [\"Aix-les-Bains\", \"Beckerish\",\n",
+    "\"Cayranne\", \n",
+    "\"Chambon\",\n",
+    "\"Cristal-Roc\",\n",
+    "\"St Cyr\",\n",
+    "\"Evian\",\n",
+    "\"Ferita\",\n",
+    "\"St Hyppolite\",\n",
+    "\"Laurier\", \n",
+    "\"Ogeu\",\n",
+    "\"Ondine\",\n",
+    "\"Perrier\",\n",
+    "\"Ribes\", \n",
+    "\"Spa\",\n",
+    "\"Thonon\", \n",
+    "\"Veri\", \n",
+    "\"Viladreau\",\n",
+    "\"Vittel\", \n",
+    "\"Volvic\"]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.10.5",
+   "language": "julia",
+   "name": "julia-1.10"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.10.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/M2/TP-PCA.pdf b/M2/TP-PCA.pdf
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-- 
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