Cuadrados_Minimos.ipynb

{
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  {
   "cell_type": "markdown",
   "id": "991d56ef",
   "metadata": {},
   "source": [
    "## Aproximando curvas mediante cuadrados minimos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "56e65351",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "adedcfb2",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 10\n",
    "x = range(0, 1, length=n)\n",
    "y = 2x .+ 3 + randn(n) * 0.2\n",
    "\n",
    "scatter(x, y, leg=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "dd8b47ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "E (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E(β₀, β₁) = sum((y[i] - β₀ - β₁ * x[i])^2 for i = 1:n) / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "eeee801b",
   "metadata": {},
   "outputs": [
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     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "#Estos dos los tiro a ojo modificando y viendo cuando vale la suma de los cuadrados\n",
    "β₀ = 2.9\n",
    "β₁ = 2.25\n",
    "\n",
    "scatter(x, y, leg=false)\n",
    "plot!(x -> β₀ + β₁ * x, extrema(x)..., c=:red, lw=2)\n",
    "title!(\"Suma del cuadrado de los resudiuos = $(E(β₀, β₁))\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "f6be78f5",
   "metadata": {},
   "outputs": [
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       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "β₀range = range(2.5, 3.3, length=100)\n",
    "β₁range = range(1.5, 2.5, length=100)\n",
    "p = contour(β₀range, β₁range, E, levels=50, layout=(1,2), size=(600,200))\n",
    "surface!(p[2], β₀range, β₁range, E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "4e83cfc3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " 2.9025890090905917\n",
       " 2.241007416120084"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Vemos la solucion analítica\n",
    "\n",
    "M = [n  sum(x); sum(x)  sum(x .^ 2)]\n",
    "c = [sum(y); sum(x .* y)]\n",
    "\n",
    "βsol = M \\ c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "c87e9bb6",
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Uso los valores analiticos\n",
    "\n",
    "scatter(x, y, leg=false)\n",
    "plot!(x -> βsol[1] + βsol[2] * x, extrema(x)..., c=:red, lw=2)\n",
    "title!(\"Suma del cuadrado de los resudiuos = $(E(β₀, β₁))\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5877c59b",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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