{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Session 4.1 Preclass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import minimize\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# objective function\n",
    "# x = [mu, lambda, alpha, beta]\n",
    "mu_e = 2.3\n",
    "mu_var = 0.5\n",
    "sig_e = 2.75\n",
    "sig_var = 1\n",
    "def f(x):\n",
    "    mu = (mu_e - x[0])**2\n",
    "    sig = (sig_e - (x[3]/(x[2]-1)))**2\n",
    "    mu_vr = (mu_var - (x[3]/((x[2]-1)*x[1])))**2\n",
    "    sig_vr = (sig_var - (x[3]**2/((x[2]-1)**2*(x[2]-2))))**2\n",
    "    return mu+sig+mu_vr+sig_vr\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Started at [1 1 3 3]\n",
      "Ended at [ 2.29999808  5.50002766  9.562539   23.5469938 ]\n",
      "f([ 2.29999808  5.50002766  9.562539   23.5469938 ]) = 0.000000\n"
     ]
    }
   ],
   "source": [
    "initial = np.array([1, 1,3,3])\n",
    "result = minimize(f, initial)\n",
    "final = result.x\n",
    "print('Started at', initial)\n",
    "print('Ended at', final)\n",
    "print('f(%s) = %.6f' % (final, f(final)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.3       5.5       9.5625   23.546875]\n"
     ]
    }
   ],
   "source": [
    "# solution 2: solve for system of equation\n",
    "from scipy.optimize import fsolve\n",
    "import math\n",
    "\n",
    "def equations(x):\n",
    "    mu = x[0] - mu_e\n",
    "    sig = (x[3]/(x[2]-1)) -sig_e\n",
    "    mu_vr = (x[3]/((x[2]-1)*x[1])) - mu_var\n",
    "    sig_vr =(x[3]**2/((x[2]-1)**2*(x[2]-2))) -  sig_var\n",
    "    return (mu, sig, mu_vr, sig_vr)\n",
    "\n",
    "result =  fsolve(equations, (1, 1,3,3))\n",
    "\n",
    "print(result)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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