Regressão Logística.ipynb

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      "text/plain": [
       "Plots.GRBackend()"
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   ],
   "source": [
    "using Plots\n",
    "gr()"
   ]
  },
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       "0.0"
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   ],
   "source": [
    "L(p) = p^402 * (1 - p)^598\n",
    "L(0.1)"
   ]
  },
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   ],
   "source": [
    "ℓ(p) = 402 * log(p) + 598 * log(1 - p)\n",
    "ℓ(0.1)"
   ]
  },
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   "source": [
    "ℓ(0.2)"
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   "source": [
    "ℓ(0.4)"
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   "source": [
    "ℓ(0.5)"
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   "source": [
    "ps = range(0.401, 0.403, length=100)\n",
    "plot(ps, ℓ.(ps), lw=2)"
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     "execution_count": 13,
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   "source": [
    "σ(t) = 1 / (1 + exp(-t))\n",
    "plot(σ, -5, 5, leg=false)"
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  {
   "cell_type": "code",
   "execution_count": 83,
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     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Random\n",
    "\n",
    "σ(t) = 1 / (1 + exp(-t))\n",
    "\n",
    "Random.seed!(0)\n",
    "n = 100\n",
    "x = sort(rand(n))\n",
    "y = [xi + randn() * 0.0 < 0.5 ? 0 : 1 for xi in x]\n",
    "#x = [0.0; 0.4; 0.6; 1.0]\n",
    "#y = [0; 1; 0; 1]\n",
    "scatter(x, y, leg=false)\n",
    "\n",
    "β = [-2.0; 4.0]\n",
    "p(β, x) = σ(β[1] + β[2] * x)\n",
    "px = [p(β, xi) for xi in x]\n",
    "\n",
    "plot!(x, px, c=:red, lw=2)\n",
    "\n",
    "threshold = 0.5\n",
    "y_pred = [p(β, xi) > threshold ? 1 : 0 for xi in x]\n",
    "\n",
    "scatter!(x, y_pred, c=:magenta, ms=2)\n",
    "plot!([minimum(x), maximum(x)], [threshold, threshold], c=:red)\n",
    "\n",
    "x_thr = (log(threshold / (1 - threshold)) - β[1]) / β[2]\n",
    "\n",
    "plot!([x_thr, x_thr], [0, 1], c=:red)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info:   iter      f(x)         π         Δ     ratio   inner      bk         cgstatus  \n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:64\n",
      "┌ Info:      0   5.9e+01   1.2e+01   1.2e+00   9.9e-01       2       0  on trust-region boundary\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      1   5.2e+01   1.5e+01   1.8e+00   1.0e+00       3       0  on trust-region boundary\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      2   3.8e+01   5.7e+00   2.6e+00   1.1e+00       3       0  on trust-region boundary\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      3   2.9e+01   6.1e+00   4.0e+00   1.2e+00       3       0  solution good enough given atol and rtol\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      4   2.3e+01   1.6e+00   4.9e+00   1.1e+00       3       0  solution good enough given atol and rtol\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      5   2.2e+01   2.2e-01   4.9e+00   1.0e+00       3       0  solution good enough given atol and rtol\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      6   2.2e+01   6.2e-03   4.9e+00   1.0e+00       3       0  solution good enough given atol and rtol\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      7   2.2e+01   5.5e-06   4.9e+00   1.0e+00       3       0  solution good enough given atol and rtol\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:154\n",
      "┌ Info:      8   2.2e+01   4.4e-12   4.9e+00\n",
      "└ @ JSOSolvers /home/abel/.julia/packages/JSOSolvers/9g3cJ/src/trunk.jl:197\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\"Execution stats: first-order stationary\""
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# JuliaSmoothOptimizers\n",
    "using LinearAlgebra, NLPModels, JSOSolvers\n",
    "\n",
    "# ℓ(β) = ∑ yᵢ ln pᵢ + (1 - yᵢ) ln (1 - pᵢ)\n",
    "# pᵢ = p(β, xᵢ)\n",
    "λ = 1e-1\n",
    "ℓ(β) = -sum(y[i] * log(p(β, x[i])) + (1 - y[i]) * log(1 - p(β, x[i])) for i = 1:length(x)) + λ * dot(β, β) / 2\n",
    "nlp = ADNLPModel(ℓ, [0.0; 1.0])\n",
    "output = trunk(nlp, max_time=3.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Generic Execution stats\n",
      "  status: \"first-order stationary\"\n",
      "  objective value: 21.742219294254316\n",
      "  primal feasibility: 0.0\n",
      "  dual feasibility: 4.404995668373065e-12\n",
      "  solution: [-5.17449141661845  10.397267182576813]\n",
      "  iterations: 8\n",
      "  elapsed time: 0.39601802825927734\n"
     ]
    }
   ],
   "source": [
    "print(output)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
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     "output_type": "execute_result"
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   ],
   "source": [
    "scatter(x, y, leg=false)\n",
    "\n",
    "β = output.solution\n",
    "p(β, x) = σ(β[1] + β[2] * x)\n",
    "xg = range(0, 1, length=100)\n",
    "px = [p(β, xi) for xi in xg]\n",
    "\n",
    "plot!(xg, px, c=:red, lw=2)\n",
    "\n",
    "threshold = 0.5\n",
    "y_pred = [p(β, xi) > threshold ? 1 : 0 for xi in x]\n",
    "\n",
    "scatter!(x, y_pred, c=:magenta, ms=2)\n",
    "plot!([minimum(x), maximum(x)], [threshold, threshold], c=:red)\n",
    "\n",
    "x_thr = (log(threshold / (1 - threshold)) - β[1]) / β[2]\n",
    "\n",
    "plot!([x_thr, x_thr], [0, 1], c=:red)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(a,b) = σ(1.0 + 0.6a + 0.6b)\n",
    "ag = range(-7, 7, length=20)\n",
    "bg = range(-7, 7, length=20)\n",
    "#surface(ag, bg, f)\n",
    "wireframe(ag, bg, f)"
   ]
  }
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