7.1040-GaussNewton.ipynb

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   "source": [
    "using Plots\n",
    "using Statistics\n",
    "using LinearAlgebra\n",
    "using JSON\n",
    "\n",
    "include(\"readclassjson.jl\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# ; wget http://ee263.stanford.edu/hw/gauss_fit_data.json -P data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.1040 Gauss Newton Method\n",
    "Get the data $t, y($ and $N)$ from the file gauss_fit_data. $\\mathrm{m},$ available on the class website. Implement the Gauss-Newton (as outlined in part\n",
    "(a) above). You'll need an initial guess for the parameters. You can visually estimate them (giving a short justification), or estimate them by any other method (but you must explain your method). Plot the RMS error $E$ as a function of the iteration number. (You should plot enough iterations to convince yourself that the algorithm has nearly converged.) Plot the final Gaussian function obtained along with the data on the same plot. Repeat for another reasonable, but different initial guess for the parameters. Repeat for another set of parameters that is not reasonable, i.e., not a good guess for the parameters. (It's possible, of course, that the Gauss-Newton algorithm doesn't converge, or fails at some step; if this occurs, say\n",
    "so.) Briefly comment on the results you obtain in the three cases."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any,Any} with 3 entries:\n",
       "  \"N\" => 200\n",
       "  \"t\" => [0.0, 0.502513, 1.00503, 1.50754, 2.01005, 2.51256, 3.01508, 3.51759, …\n",
       "  \"y\" => [3.5784, 0.7447, 3.557, 0.1211, 4.1369, -1.8362, 3.0537, 1.0934, 1.090…"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gaussian = readclassjson(\"data/gauss_fit_data.json\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = gaussian[\"t\"]\n",
    "y = gaussian[\"y\"];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define initial parameters: initial guesses for $mu$ and $\\sigma$ use the rough empirical mean and standard deviation for t since this aligns with the definition of a Gaussian distribution. The normalizing constant was initialized by plugging those parameters into $f(t)$ with $a_0 = 1$ and taking $y / f(t)$ to see how different the scales were. This turned out to be roughly 20."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       " 20.0\n",
       " 50.0\n",
       " 29.085017343694034"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu0 = 50\n",
    "sigma0 = std(t)\n",
    "a0 = 20\n",
    "\n",
    "p0 = [a0 ; mu0; sigma0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "jacobian (generic function with 1 method)"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function normal(t, p)\n",
    "    p[1] .* exp.(-(t .- p[2]).^2 ./ (p[3]^2))\n",
    "end\n",
    "\n",
    "function rmse(p)\n",
    "    sqrt(1/gaussian[\"N\"]) * norm(normal(t, p) .- y)\n",
    "end\n",
    "\n",
    "function jacobian(p)\n",
    "    f = normal(t, p)\n",
    "    dr = 2 * (f .- y)\n",
    "    exponent = exp.(-(t .- p[2]).^2 ./ (p[3]^2))\n",
    "    da =  dr .* exponent\n",
    "    dm =  dr .* ((2*p[1] .*(t .- p[2]))./ (p[3]^2)).* exponent\n",
    "    ds =  dr .* ((2*p[1] .*((t .- p[2]).^2))./ (p[3]^3)).* exponent\n",
    "    \n",
    "    j = [da dm ds]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fitter (generic function with 2 methods)"
      ]
     },
     "execution_count": 218,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fitter(p0, iters=100)\n",
    "    p = p0\n",
    "    rmse_vec = zeros(iters)\n",
    "    for i = 1:iters\n",
    "        Jk = jacobian(p)\n",
    "        Ak = Jk\n",
    "        bk = Jk * p .- rmse(p)\n",
    "\n",
    "        F = qr(Ak)\n",
    "        p_k1 = inv(F.R) * transpose(F.Q[:,1:3]) * bk\n",
    "\n",
    "        rmse_vec[i] = rmse(p)\n",
    "\n",
    "        p = p_k1\n",
    "        \n",
    "    end\n",
    "    return(p, rmse_vec)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [],
   "source": [
    "p1, rmse_vec1 = fitter(p0);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Output parameter values for converged solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       " 12.098632354668037\n",
       " 54.80866643570745\n",
       " 42.021670289800674"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The G-N method converges within 25 iterations when we use the reasonable initialized parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(rmse_vec1, label=false)\n",
    "title!(\"RMSE by Iteration\")\n",
    "xlabel!(\"Iteration\")\n",
    "ylabel!(\"RMSE\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The estimated parameters fit the data very well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {},
   "outputs": [
    {
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       "  302.519,1145.41 313.035,939.355 323.551,1097.68 334.066,919.943 344.582,1140.8 355.098,1050.9 365.613,993.68 376.129,877.047 386.645,945.719 397.16,1077.27 \n",
       "  407.676,1235.6 418.192,1123.37 428.707,960.356 439.223,1155.83 449.739,951.104 460.254,818.619 470.77,942.372 481.285,946.528 491.801,901.493 502.317,769.505 \n",
       "  512.832,725.631 523.348,1095.11 533.864,898.544 544.379,831.324 554.895,952.923 565.411,522.526 575.926,764.127 586.442,1015.98 596.958,752.889 607.473,760.865 \n",
       "  617.989,742.591 628.505,1111.21 639.02,605.385 649.536,1076.88 660.052,652.024 670.567,981.693 681.083,743.805 691.599,645.744 702.114,658.693 712.63,768.535 \n",
       "  723.145,906.138 733.661,772.194 744.177,868.231 754.692,990.364 765.208,864.128 775.724,626.959 786.239,622.642 796.755,761.231 807.271,678.869 817.786,608.96 \n",
       "  828.302,639.022 838.818,620.702 849.333,332.814 859.849,989.401 870.365,580.457 880.88,578.822 891.396,512.87 901.912,693.828 912.427,460.539 922.943,561.121 \n",
       "  933.458,448.988 943.974,422.83 954.49,359.628 965.005,563.077 975.521,343.769 986.037,502.06 996.552,747.358 1007.07,543.97 1017.58,590.572 1028.1,770.086 \n",
       "  1038.61,562.313 1049.13,585.293 1059.65,374.174 1070.16,425.596 1080.68,335.892 1091.19,420.875 1101.71,480.257 1112.22,465.627 1122.74,401.867 1133.26,262.392 \n",
       "  1143.77,311.797 1154.29,636.783 1164.8,406.161 1175.32,453.778 1185.83,327.901 1196.35,456.78 1206.87,233.507 1217.38,430.363 1227.9,623.49 1238.41,182.773 \n",
       "  1248.93,325.487 1259.44,303.592 1269.96,493.343 1280.47,257.595 1290.99,724.752 1301.51,213.354 1312.02,379.736 1322.54,624.865 1333.05,564.865 1343.57,344.647 \n",
       "  1354.08,342.065 1364.6,529.287 1375.12,444.809 1385.63,162.047 1396.15,447.002 1406.66,372.234 1417.18,256.273 1427.69,242.636 1438.21,360.201 1448.73,630.534 \n",
       "  1459.24,636.86 1469.76,322.156 1480.27,502.946 1490.79,306.893 1501.3,607.845 1511.82,329.437 1522.33,307.244 1532.85,349.261 1543.37,566.255 1553.88,468.851 \n",
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     },
     "execution_count": 210,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(t, y, label = \"y\")\n",
    "plot!(t, normal(t, p1), label=\"Fit\")\n",
    "title!(\"Fitted Gaussian with P0 = [20, 50, 29]\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With a new guess within a reasonable range, we still get convergence and a good fit. Note from the RMSE plot that convergence takes somewhat longer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [
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     "execution_count": 214,
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     "output_type": "execute_result"
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   ],
   "source": [
    "p02 = [10, 70, 15]\n",
    "p2, rmse_vec2 = fitter(p02);\n",
    "plot(t, y, label = \"y\")\n",
    "plot!(t, normal(t, p2), label=\"Fit\")\n",
    "title!(\"Fitted Gaussian with P0 = [10, 70, 15]\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(rmse_vec2, label=false)\n",
    "title!(\"RMSE by Iteration\")\n",
    "xlabel!(\"Iteration\")\n",
    "ylabel!(\"RMSE\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With a very unreasonable guess, the algorithm did not converge and we are not left with a solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {},
   "outputs": [
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     "execution_count": 231,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "p03 = [-100, 0.001, 100]\n",
    "p3, rmse_vec3 = fitter(p03, 500);\n",
    "plot(t, y, label = \"y\")\n",
    "plot!(t, normal(t, p3), label=\"Fit\")\n",
    "title!(\"Fitted Gaussian with P0 = [-100, 0.001, 100]\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(rmse_vec3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore we see that Gauss-Newton works quite well with good initialization. Even with guesses that are off from the truth, it still converges in reasonable time. However some domain knowledge is warranted to avoid divergence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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