{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a375bf72-967d-47ca-8a78-2f0280568fe5",
   "metadata": {},
   "source": [
    "# Linear regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fa1f6c1b-fbfa-441a-a4ba-d6d2fa73be95",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    alpha, beta, sigma = linear_regression(x, y)\n",
    "\n",
    "Linear regression y = alpha + beta * x. sigma is the standard deviation for beta.\n",
    "\"\"\"\n",
    "function linear_regression(x, y)\n",
    "    n = length(x)\n",
    "    xbar = sum(x) / n\n",
    "    ybar = sum(y) / n\n",
    "    sx2 = sum((x .- xbar) .^ 2)\n",
    "    sxy = sum((x .- xbar) .* (y .- ybar))\n",
    "    beta = sxy / sx2\n",
    "    alpha = ybar - beta * xbar\n",
    "    sigma = sqrt( sum((y .- alpha .- beta .* x) .^ 2) / ((n - 2) * sx2))\n",
    "    return alpha, beta, sigma\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b5bb2b50-99a6-448b-9c57-7f2bc807a7e7",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "xmin = 0.0\n",
    "xmax = 10.0\n",
    "np = 200\n",
    "x = range(xmin, xmax, np)\n",
    "y = 1.0 .+ 2 .* x .+ 1.5 .* randn(np);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6ff1ad04-a26b-41c0-b57d-d9fb13d5d50a",
   "metadata": {},
   "outputs": [],
   "source": [
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6a8a5b30-2fd6-4329-87a2-f8b5555ea1b7",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "plot(x, y, linestyle=\"none\", marker=\".\")\n",
    "grid(true)\n",
    "xlabel(\"x\")\n",
    "ylabel(\"y\")\n",
    "title(\"Raw data\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d548b3de-4fa8-437b-9d5b-41313e5dc6fa",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "alpha, beta, sigma = linear_regression(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2efad8ac-04ff-400e-bc0e-94d0b6d963a7",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "plot(x, y, linestyle=\"none\", marker=\".\", label=\"Raw data\")\n",
    "plot(x, alpha .+ beta .* x, linestyle=\"solid\", label=\"LSF\")\n",
    "grid(true)\n",
    "xlabel(\"x\")\n",
    "ylabel(\"y\")\n",
    "title(\"Linear regression\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "38dcfb46-f0a7-45a7-9d12-d39d737e06c9",
   "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": 5
}