{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d735264a-f7aa-491b-b804-aa7adf93ea53",
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra, Plots, DelimitedFiles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c66f5d41-623c-4e7c-a56f-db00dea5dcb7",
   "metadata": {},
   "outputs": [],
   "source": [
    "M = convert(Matrix{Int}, readdlm(\"../09-29/salaries.csv\", ',', skipstart=1)[:, 2:end])\n",
    "A = M[1:end, 1:3];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e41b86f2-4c79-4d8e-8b18-2939d3e94fe8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "398×4 Matrix{Int64}:\n",
       " 101  132   491   460\n",
       " 507  153   719  1073\n",
       "  15   10    18    23\n",
       " 111   77   132   166\n",
       " 596  163  1249  1682\n",
       " 301  117   562   746\n",
       " 499  171   839  1167\n",
       "  27   25    65    67\n",
       "  38   15    29    52\n",
       " 109   71   412   450\n",
       " 594  230   703  1067\n",
       " 269  134   352   487\n",
       " 216  192   832   856\n",
       "   ⋮             \n",
       "  75   25    59   109\n",
       " 345  174  1153  1324\n",
       " 146  101   395   440\n",
       "  21    8    25    38\n",
       " 174   82   586   678\n",
       " 248  173   549   624\n",
       " 238  179   977  1036\n",
       " 476  147  1178  1507\n",
       " 352  146   463   669\n",
       " 122   47   182   257\n",
       " 228  193  1150  1185\n",
       " 529  146  1040  1423"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = hcat(A, A[:, 1] .- A[:, 2] .+ A[:, 3]) # new column!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "474db68f-18a1-4465-86d2-8858b5e673f0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 26031.994273985612\n",
       "  3136.8196381998196\n",
       "  1113.9150221554164\n",
       "     1.2532368455541195e-12"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, S, V = svd(A)\n",
    "S # last eigen very close to 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "996fde91-0545-4c13-b698-f98695f4f9e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       "     0.0002891389119208513\n",
       "  1113.9150221554394\n",
       "  3136.8196381998237\n",
       " 26031.994273985627"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt.(eigvals(A' * A)) # no help because numerical error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "bd0558d1-a325-453e-b11c-33fb8fa08179",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " -4.70433529065995e17\n",
       "  4.7043352906598285e17\n",
       " -4.704335290659996e17\n",
       "  4.704335290660089e17"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = M[1:end, 4]\n",
    "\n",
    "x = V * inv(diagm(S)) * U' * y # large values :("
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "8e337ee5-c9d9-4c36-a186-d6f1f249fc99",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       "  -6312.126276909091\n",
       "   1760.8043960629386\n",
       " -17914.38753206213\n",
       "     -9.408670581319933e17"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diagm(S) \\ (U' * y) # same problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "174145da-2232-4b8c-a41f-fdbfb6d1d983",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       "  -7381.269979698599\n",
       "  -5911.720495010535\n",
       " -11150.23174887978\n",
       "  20666.47182920933"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A' * A) \\ (A' * y) # same problem but not apparent here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "07671bb0-0d01-4a9f-a075-522fbd0f691e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Real}:\n",
       " 3.841426782270476e-5\n",
       " 0.0003187942296146447\n",
       " 0.0008977345489649724\n",
       " 0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# truncate the svd?\n",
    "Ss = (x -> if x > 1e-10 1/x else 0 end).(S)\n",
    "# very pragmatic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "294319d8-71ce-412d-8c09-ff2a032e2b52",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Any}:\n",
       "    940.2932859956918\n",
       " -14233.28376070477\n",
       "  -2828.6684831854627\n",
       "  12344.908563515011"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       "    940.2932859956919\n",
       " -14233.28376070477\n",
       "  -2828.6684831854627\n",
       "  12344.908563515011"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_reg = V * diagm(Ss) * (U' * y)\n",
    "x_reg |> display\n",
    "\n",
    "# the same as\n",
    "x_reg = V[:, 1:3] * (diagm(S[1:3]) \\ (U[:, 1:3]' * y))\n",
    "x_reg |> display\n",
    "\n",
    "# much better but a bit arbitrary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "96678996-a2b6-4f96-be9b-b5107015c237",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 26031.99442670313\n",
       "  3136.8195355612684\n",
       "  1113.914857400499\n",
       "     0.001975927891687143"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "E = 1e-4 * randn(size(A))\n",
    "AA = A .+ E\n",
    "U, S, V = svd(AA)\n",
    "\n",
    "display(S) # perturbed by noise, the small value became much larger"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Julia 1.9.3",
   "language": "julia",
   "name": "julia-1.9"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.9.3"
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 "nbformat": 4,
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}