{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to mathematical statistics \n",
    "\n",
    "Welcome to the lecture 5 in 02403\n",
    "\n",
    "During the lectures we will present both slides and notebooks. \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: Probability region"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Sigma = np.array([[1,1],[1,2]])\n",
    "print(stats.chi2.ppf(0.95,2))\n",
    "print(\"Sigma = \",Sigma)\n",
    "print(\"Sigma^-1= \",np.linalg.inv(Sigma))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Probability region"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y =  np.arange(-np.sqrt(6),np.sqrt(6),0.001)\n",
    "plt.plot(y,y+np.sqrt(6-y**2))\n",
    "plt.plot(y,y-np.sqrt(6-y**2))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add eigen vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Lambda, V = np.linalg.eig(Sigma)\n",
    "\n",
    "print(V)\n",
    "plt.plot(y,y+np.sqrt(6-y**2),scalex=[-3,3])\n",
    "plt.plot(y,y-np.sqrt(6-y**2))\n",
    "plt.plot([0,V[0,0]*np.sqrt(Lambda[0]*6)],[0,V[1,0]*np.sqrt(Lambda[0]*6)])\n",
    "plt.plot([0,-V[0,1]*np.sqrt(Lambda[1]*6)],[0,-V[1,1]*np.sqrt(Lambda[1]*6)])\n",
    "plt.axis([-3.5,3.5,-3.5,3.5])\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: projecton mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[1/2,-1/2],[-1/2,1/2]])\n",
    "I = np.array([[1,0],[0,1]])\n",
    "print(I-A)\n",
    "print(A)\n",
    "\n",
    "print(A@A)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Items on a scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X1 = np.array([[1,0],[0,1],[1,1]])\n",
    "print(np.linalg.inv(X1.T@X1)@X1.T)\n",
    "\n",
    "X2 = np.array([[1,0],[1,1],[2,1]])\n",
    "print(np.linalg.inv(X2.T@X2)@X2.T)\n",
    "\n",
    "X3 = np.array([[1,1],[1,-1],[2,0]])\n",
    "print(np.linalg.inv(X3.T@X3)@X3.T)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Items on a scale orthogonal?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(X1.T@X1)\n",
    "print(X2.T@X2)\n",
    "print(X3.T@X3)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Items on a scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "H1 = X1 @ np.linalg.inv(X1.T@X1)@X1.T\n",
    "H2 = X2 @ np.linalg.inv(X2.T@X2)@X2.T\n",
    "H3 = X3 @ np.linalg.inv(X3.T@X3)@X3.T\n",
    "\n",
    "print(np.max(np.abs(H1-H2)))\n",
    "print(np.max(np.abs(H1-H3)))\n",
    "H1"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "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": "3.10.16"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}