Dunn_Marina_Assignment1.ipynb

{
 "cells": [
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   "id": "492ea07a",
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   "source": [
    "Marina Dunn\n",
    "\n",
    "PHYS 243: Foundations of Machine Learning\n",
    "\n",
    "Assignment 1\n",
    "\n",
    "Created 10/6/21\n",
    "\n",
    "M.S. Engineering - Data Science, UCR"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "303d0253",
   "metadata": {
    "id": "303d0253"
   },
   "source": [
    "### Problem 1: Fibonacci\n",
    "\n",
    "\n",
    "Production of Fibonacci Sequence!\n",
    "Write a function that takes an integer value n, where n ∈ [0, 1, 2, 3, ...] and return the nth value in Fibonacci sequence. The function recursively and the function with for loops are both shown below. Which is faster?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e82af5be",
   "metadata": {
    "id": "e82af5be"
   },
   "source": [
    "##### Part 1: Rewrite the function with for loop.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ec3f3a16",
   "metadata": {
    "id": "ec3f3a16"
   },
   "outputs": [],
   "source": [
    "import time\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "655d95dd",
   "metadata": {
    "id": "655d95dd"
   },
   "outputs": [],
   "source": [
    "# original function\n",
    "\n",
    "def fib_rec(n=0):\n",
    "    if n==0:\n",
    "        return 1\n",
    "    elif n==1:\n",
    "        return 1\n",
    "    else:\n",
    "        return fib_rec(n - 1) + fib_rec(n - 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "902c4063",
   "metadata": {
    "id": "902c4063"
   },
   "outputs": [],
   "source": [
    "# re-written function with for loop\n",
    "\n",
    "def fib_for(n):\n",
    "    n1, n2 = 0, 1\n",
    "    if n == 0:\n",
    "        return 0\n",
    "    for i in range(n-1):\n",
    "        n1 = n2\n",
    "        n2 = n1 + n2    \n",
    "        return n2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f34313c",
   "metadata": {
    "id": "5f34313c"
   },
   "source": [
    "##### Part 2: Which function is actually faster? (Explain without running the codes)\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d5ac67e",
   "metadata": {
    "id": "7d5ac67e"
   },
   "source": [
    "In this case, iteration would be faster than recursion because with recursion, the values aren't being stored and remembered, so previous values have to be calculated once again for every future 'n'. The iterative version conveniently has both of the previous values to quickly calculate the new sum from. In order for the recursive version to be more efficient, you would need to implement some step to store the previous value so it doesn't have to be re-calculated every time. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "661c0a98",
   "metadata": {
    "id": "661c0a98"
   },
   "source": [
    "##### Part 3: Write a code to time the average time for k times function call. A function that takes three arguments (function to time (Fib rec), input of the function (n), num- ber of runs(k)) and run the Fib rec function k times for the input of n and returns the average time. Tip: You can use the time module in the python.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "28ffc885",
   "metadata": {
    "id": "28ffc885",
    "outputId": "970ac391-6920-447b-8b37-2cb3960bdf00"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average time for recursion: 2.759695053100586e-05 seconds\n",
      "Average time for iteration method: 5.960464477539062e-07 seconds\n"
     ]
    }
   ],
   "source": [
    "def timer1(n, k, f=fib_rec):\n",
    "    start = time.time()\n",
    "    for i in range(k):\n",
    "        f(n)\n",
    "    end = time.time()\n",
    "    average_time1 = (end - start) / k\n",
    "    return average_time1\n",
    "\n",
    "print(f'Average time for recursion: {timer1(10, 20, fib_rec)} seconds')\n",
    "\n",
    "def timer2(n, k, f=fib_for):\n",
    "    start = time.time()\n",
    "    for i in range(k):\n",
    "        f(n)\n",
    "    end = time.time()\n",
    "    average_time2 = (end - start) / k\n",
    "    return average_time2\n",
    "\n",
    "print(f'Average time for iteration method: {timer2(10, 20, fib_for)} seconds')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d126144",
   "metadata": {
    "id": "2d126144"
   },
   "source": [
    "#####  Part 4: Make a plot in which the x-axis is the value of the input function n and the y-axis is the average time (output of the previous function), for both recursive and non- recursive Fibonacci. (Both in the same plot; also use matplotlib package for making the plots)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b587751b",
   "metadata": {
    "id": "b587751b",
    "outputId": "e22076b8-5f7a-481c-a3e6-fb571bd9a795"
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "#plt.style.use('seaborn')\n",
    "\n",
    "n = np.arange(0,20)\n",
    "\n",
    "fib_rec_time = [timer1(i, k=50, f=fib_rec) for i in n]\n",
    "fib_for_time = [timer2(i, k=50, f=fib_for) for i in n]\n",
    "\n",
    "plt.plot(n, fib_rec_time, linewidth=2, color='red', label='Fibonacci recursion')\n",
    "plt.plot(n, fib_for_time, linewidth=2, color='blue', label='Fibonacci iteration')\n",
    "\n",
    "plt.title(r\"Average Time for Iterative v. Recursive Methods of Fibonacci Calculation\")\n",
    "\n",
    "plt.xlabel(r\"$n$\")\n",
    "plt.ylabel(r\"$Average Time$\")\n",
    "\n",
    "plt.xticks(fontsize=14)\n",
    "plt.yticks(fontsize=14)\n",
    "\n",
    "plt.legend(loc='upper left')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f78cbfff",
   "metadata": {
    "id": "f78cbfff"
   },
   "source": [
    "### Problem 2: Linear Algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1f31cce",
   "metadata": {
    "id": "a1f31cce"
   },
   "source": [
    "Given the Matrix M below answer the questions:\n",
    "\n",
    "##### Part 1: Find the determinant, transpose, inverse(if exist) for M.\n",
    "<center> M = </center>\n",
    "\n",
    "\\begin{bmatrix} \n",
    "1 & -4 & 2 \\\\\n",
    "-4 & 1 & -2\\\\\n",
    "2 & -2 & -2 \\\\\n",
    "\\end{bmatrix}\n",
    "\n",
    "How to find determinant of 3x3 matrix:\n",
    "\n",
    "<center> M = </center>\n",
    "\\begin{bmatrix} \n",
    "a & b & c \\\\\n",
    "d & e & f\\\\\n",
    "g & h & i \\\\\n",
    "\\end{bmatrix}\n",
    "<br>\n",
    "\n",
    "<center> |M| = a(ei − fh) − b(di − fg) + c(dh − eg) </center>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "81503584",
   "metadata": {
    "id": "81503584",
    "outputId": "510478c6-77e0-4b05-ef6e-0f5a417f0022"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Determinant of matrix M:\n",
      "54\n"
     ]
    }
   ],
   "source": [
    "# 1: Find determinant\n",
    "det_M = (1 * (1*-2 - (-2*-2)) ) - (-4 * ((-4*-2) - (-2*2))) + (2* ((-4*-2) - (1*2) ))\n",
    "print(\"Determinant of matrix M:\")\n",
    "print(int(det_M))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "fe6ad348",
   "metadata": {
    "id": "fe6ad348",
    "outputId": "23fee738-649e-4830-a24b-f9fd74c52eb5"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Determinant of matrix M:\n",
      "54\n"
     ]
    }
   ],
   "source": [
    "# Faster version using numpy\n",
    "M = np.array([[1, -4, 2], [-4, 1,-2],[2, -2, -2]])\n",
    "determinant = np.linalg.det(M)\n",
    "  \n",
    "print(\"Determinant of matrix M:\")\n",
    "print(int(determinant))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "206c550e",
   "metadata": {
    "id": "206c550e",
    "outputId": "6f420a8c-b4d0-4961-cd89-882e4956cf18"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transpose of M: \n",
      "[1, -4, 2]\n",
      "[-4, 1, -2]\n",
      "[2, -2, -2]\n",
      "\n",
      " M = MT. The values have switched but appear the same. \n",
      "This means the matrix is orthogonal.\n"
     ]
    }
   ],
   "source": [
    "# 2: Find transpose, where row values and column values interchange\n",
    "# using nested list comprehension\n",
    "\n",
    "M = [[1, -4, 2], [-4, 1,-2],[2, -2, -2]]\n",
    "\n",
    "# N is the dimensions of the matrix\n",
    "N = 3\n",
    "\n",
    "MT = [[M[j][i] for j in range(N)] for i in range(N)]\n",
    "\n",
    "print(\"Transpose of M: \")\n",
    "for row in MT:\n",
    "    print(row)\n",
    "    \n",
    "print(\"\\n M = MT. The values have switched but appear the same. \\nThis means the matrix is orthogonal.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "bae435c4",
   "metadata": {
    "id": "bae435c4",
    "outputId": "24c114c6-31d9-4f6e-cb4a-06bacde1dfe8"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transpose of M: \n",
      "[[ 1 -4  2]\n",
      " [-4  1 -2]\n",
      " [ 2 -2 -2]]\n"
     ]
    }
   ],
   "source": [
    "# Faster version using numpy\n",
    "M = np.array([[1, -4, 2], [-4, 1,-2],[2, -2, -2]])\n",
    "print(\"Transpose of M: \")\n",
    "print(np.transpose(M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "6026d082",
   "metadata": {
    "id": "6026d082"
   },
   "outputs": [],
   "source": [
    "#3: Find inverse of matrix M (if exist)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b47ba963",
   "metadata": {
    "id": "b47ba963"
   },
   "source": [
    "inverse of M = adj(M)/det(M)\n",
    "\n",
    "determinant must not be 0\n",
    "\n",
    "1. change signs of appropriate values\n",
    "2. transpose matrix\n",
    "3. multiply by 1/det, in this case, 54\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "01ba779a",
   "metadata": {
    "id": "01ba779a",
    "outputId": "74b3b22c-278d-47eb-b074-cfbfc0c66513"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Inverse of M: \n",
      "[[-0.11111111 -0.22222222  0.11111111]\n",
      " [-0.22222222 -0.11111111 -0.11111111]\n",
      " [ 0.11111111 -0.11111111 -0.27777778]]\n"
     ]
    }
   ],
   "source": [
    "inverse = np.linalg.inv(M)\n",
    "print(\"Inverse of M: \")\n",
    "print(inverse)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbeea255",
   "metadata": {
    "id": "bbeea255"
   },
   "source": [
    "##### Part 2: Find the eigenvalues and eigenvectors for M.\n",
    "##### find the values that satisfy det(M − λI) = 0?\n",
    "\n",
    "M − λI = \n",
    "\\begin{array}{ccc}1-\\lambda &-4&2\\\\-4&1-\\lambda &-2\\\\2&-2&-2-\\lambda \\end{array}\n",
    "<br>\n",
    "<center> M = </center>\n",
    "\\begin{bmatrix} \n",
    "a & b & c \\\\\n",
    "d & e & f\\\\\n",
    "g & h & i \\\\\n",
    "\\end{bmatrix}\n",
    "<br>\n",
    "\n",
    "<center> |M| = a(ei − fh) − b(di − fg) + c(dh − eg) </center>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "f15359ae",
   "metadata": {
    "id": "f15359ae",
    "outputId": "7cac4ca5-f84c-49d1-cd12-ed1b64a3fef4"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues: \n",
      "[-3.  6. -3.]\n",
      "\n",
      " Eigenvectors: \n",
      "[[-0.74535599 -0.66666667 -0.16169042]\n",
      " [-0.59628479  0.66666667  0.30721179]\n",
      " [ 0.2981424  -0.33333333  0.93780442]]\n"
     ]
    }
   ],
   "source": [
    "# Find eigenvalues and eigenvectors of M \n",
    "# Faster version using numpy\n",
    "from numpy import linalg as LA\n",
    "'''\n",
    "det(M - λI) = 0\n",
    "\n",
    "(1-λ)[(1-λ)(-2-λ) - (-2*-2)] + 4[(-4)*(-2-λ) - (-2*2)] + 2[(-4)(1-λ) - (1-λ)(2)] = 0\n",
    "(1-λ)[(-2-λ+2λ+λ**2) - 4] + 4[(8+λ)+4] + 2[(-4+4λ)-2+2λ] = 0\n",
    "(1-λ)(-6 + λ + λ**2) + (48 + λ) + 2(-6 + 6λ) = 0\n",
    "(-6 + λ + λ**2 + 6λ - λ**2 - λ**3) + (48 + λ) + (-12 + 12λ) = 0\n",
    "(-6 + 7λ - λ**3) + 36 + 11λ = 0\n",
    "30 + 18λ - λ**3 = 0\n",
    "\n",
    "'''\n",
    "\n",
    "M = np.array([[1, -4, 2], [-4, 1,-2],[2, -2, -2]])\n",
    "\n",
    "w, v = np.linalg.eig(M)\n",
    "print(\"Eigenvalues: \")\n",
    "print(w)\n",
    "print(\"\\n Eigenvectors: \")\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a64ae035",
   "metadata": {
    "id": "8dee4901"
   },
   "source": [
    "##### Part 3: Find the Gradient if the ∇Af(A) for the following:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cfefeebf",
   "metadata": {
    "id": "cfefeebf"
   },
   "source": [
    "<center> A = </center>\n",
    "\\begin{bmatrix} \n",
    "x11 & x12 & x13 \\\\\n",
    "x21 & x22 & x23\\\\\n",
    "x31 & x32 & x33 \\\\\n",
    "\\end{bmatrix}\n",
    "<br>\n",
    "\n",
    "<center> f(A) = (x<sub>11</sub><sup>2</sup>x<sub>22</sub>x<sub>23</sub>) + (x<sub>11</sub>x<sub>12</sub>x<sub>13</sub>x<sub>31</sub>) - (x<sub>33</sub><sup>2</sup>x<sub>32</sub> x<sub>21</sub>) </center>\n",
    "\n",
    "grad f =  < df/dx<sub>11</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>), \n",
    "\n",
    "df/dx<sub>12</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>13</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>21</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>22</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>23</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>31</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>32</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>),\n",
    "\n",
    "df/dx<sub>33</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) >\n",
    "\n",
    "<br>\n",
    "----------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "286dbc4f",
   "metadata": {
    "id": "286dbc4f"
   },
   "source": [
    "df/dx<sub>11</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "2 x<sub>11</sub>x<sub>12</sub>x<sub>13</sub> + x<sub>12</sub>x<sub>13</sub>x<sub>31</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>12</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "x<sub>11</sub>x<sub>13</sub>x<sub>31</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>13</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "x<sub>11</sub>x<sub>12</sub>x<sub>31</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>21</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "-2x<sub>33</sub>x<sub>32</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>22</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "2x<sub>11</sub>x<sub>23</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>23</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "x<sub>11</sub><sup>2</sup>x<sub>22</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>31</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "x<sub>11</sub>x<sub>12</sub>x<sub>13</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>32</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "-x<sub>33</sub><sup>2</sup>x<sub>21</sub>\n",
    "\n",
    "<br>\n",
    "-------\n",
    "\n",
    "df/dx<sub>33</sub>(x<sub>11</sub>,x<sub>12</sub>,x<sub>13</sub>,x<sub>21</sub>,x<sub>22</sub>,x<sub>23</sub>,x<sub>31</sub>,x<sub>32</sub>,x<sub>33</sub>) = \n",
    "\n",
    "-2x<sub>33</sub>x<sub>32</sub>x<sub>21</sub>\n",
    "\n",
    "<br>\n",
    "-------"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0327b2c",
   "metadata": {
    "id": "e0327b2c"
   },
   "source": [
    "\n",
    "grad f = <2 x<sub>11</sub>x<sub>12</sub>x<sub>13</sub> + x<sub>12</sub>x<sub>13</sub>x<sub>31</sub>, x<sub>11</sub>x<sub>13</sub>x<sub>31</sub>, x<sub>11</sub>x<sub>12</sub>x<sub>31</sub>, -2x<sub>33</sub>x<sub>32</sub>, 2x<sub>11</sub>x<sub>23</sub>, x<sub>11</sub><sup>2</sup>x<sub>22</sub>, x<sub>11</sub>x<sub>12</sub>x<sub>13</sub>, -x<sub>33</sub><sup>2</sup>x<sub>21</sub>, -2x<sub>33</sub>x<sub>32</sub>x<sub>21</sub> >\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d704424",
   "metadata": {
    "id": "8d704424"
   },
   "source": [
    "\\begin{bmatrix} \n",
    "2x11x12x13 + x12x13x31 & x11x13x31 & x11x12x31 \\\\\n",
    "-2x33x32 & 2x11x23 & x11^2x22\\\\\n",
    "x11x12x13 & -x33^2x21 & -2x33x32x21 \\\\\n",
    "\\end{bmatrix}\n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab657608",
   "metadata": {},
   "source": [
    "##### Part 4: Find the Hessian Matrix for g(x, y, z) = x**3y + yz sin(x) + xy**2z**5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "e244f7dc",
   "metadata": {
    "id": "e244f7dc"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\ndg/dx = 3yx**2 + yz cos(x) + y**2z**5\\nd2g/dx2 = 6yx - yz sin(x)\\nd2g/dxdy = 3x**2 + z cos(x) + 2yz**5\\nd2g/dxdz = y cos(x) + 5y**2z**4\\n\\ndg/dy = x**3 + z sin(x) + 2xyz**5\\nd2g/dy2 = 2xz**5\\nd2g/dydx = 3x**2 + z cos(x) + 2yz**5\\nd2g/dydz = sin(x) + 10xyz**4\\n\\ndg/dz = y sin(x) + 5xy**2z**4\\nd2g/dz2 = 20xy**2z**3\\nd2g/dzdx = y cos(x) + 5y**2z**4\\nd2g/dzdy = sin(x) + 10xyz**4\\n\\nHg(x,y,z) = [[d2g/dx2, d2g/dydx, d2g/dzdx], [d2g/dxdy, d2g/dy2, d2g/dzdy], [d2g/dxdz, d2g/dydz, d2g/dz2]]\\nHg = [[6xy-yzsin(x), 3x^2+zcos(x)+2yz^5, ycos(x)+5y^2z^4], [3x^2+zcos(x)+2yz^5, 2xz^5, sin(x)+10xyz^4], [ycos(x)+5y^2z^4, sin(x)+10xyz^4, 20xy^2z^3]]\\n'"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "# take second derivatives of g(x, y, z) = x**3y + yz sin(x) + xy**2z**5\n",
    "'''\n",
    "dg/dx = 3yx**2 + yz cos(x) + y**2z**5\n",
    "d2g/dx2 = 6yx - yz sin(x)\n",
    "d2g/dxdy = 3x**2 + z cos(x) + 2yz**5\n",
    "d2g/dxdz = y cos(x) + 5y**2z**4\n",
    "\n",
    "dg/dy = x**3 + z sin(x) + 2xyz**5\n",
    "d2g/dy2 = 2xz**5\n",
    "d2g/dydx = 3x**2 + z cos(x) + 2yz**5\n",
    "d2g/dydz = sin(x) + 10xyz**4\n",
    "\n",
    "dg/dz = y sin(x) + 5xy**2z**4\n",
    "d2g/dz2 = 20xy**2z**3\n",
    "d2g/dzdx = y cos(x) + 5y**2z**4\n",
    "d2g/dzdy = sin(x) + 10xyz**4\n",
    "\n",
    "Hg(x,y,z) = [[d2g/dx2, d2g/dydx, d2g/dzdx], [d2g/dxdy, d2g/dy2, d2g/dzdy], [d2g/dxdz, d2g/dydz, d2g/dz2]]\n",
    "Hg = [[6xy-yzsin(x), 3x^2+zcos(x)+2yz^5, ycos(x)+5y^2z^4], [3x^2+zcos(x)+2yz^5, 2xz^5, sin(x)+10xyz^4], [ycos(x)+5y^2z^4, sin(x)+10xyz^4, 20xy^2z^3]]\n",
    "'''"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "359415b4",
   "metadata": {
    "id": "359415b4"
   },
   "source": [
    "H(g(x,y,z)) = \n",
    "\n",
    "\\begin{bmatrix} \n",
    "6yx - yzsin(x) & 3yx^2 + zcos(x) + 2yz^5 & ycos(x) + 5xy^2z^4 \\\\\n",
    "3x^2 + zcos(x) + 2yz^5 & 2xz^5 & sin(x) + 10xyz^4 \\\\\n",
    "ycos(x) + 5y^2z^4 & sin(x) + 10xyz^4 & 20xy^2z^3 \\\\\n",
    "\\end{bmatrix}\n",
    "<br>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5693fcb4",
   "metadata": {
    "id": "5693fcb4"
   },
   "source": [
    "### Problem 3: Machine Learning"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "131d75cb",
   "metadata": {
    "id": "131d75cb"
   },
   "source": [
    "##### Part 1: Explain the difference between validation and test samples.\n",
    "\n",
    "In Machine Learning, once a computer has been given a large and diverse dataset to learn from, the training set, it must be tested on new data, the validation set. The validation set contains data the computer has not seen before, and is used to test the accuracy of the model and fine tune its parameters if necessary. The test set has the same purpose, testing unseen data, but is saved until last to assess the performance of the final full model."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32d490cc",
   "metadata": {
    "id": "32d490cc"
   },
   "source": [
    "##### Part 2: Explain the difference between supervised and unsupervised learning algorithm.\n",
    "\n",
    "Supervised Machine Learning involves a direct relationship between the input and output of data. The computer is learning from a labeled dataset and forming predictions within certain parameters, determining the outcome. (2 types of this: regression or classification)\n",
    "\n",
    "Unsupervised learning involves the computer receiving unlabeled data and interpolating within parameters in an attempt to detect patterns and group similar objects. The latter is more common with data of higher dimensions. (2 types: dimension reduction or clustering)\n",
    "\n"
   ]
  }
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