Blog/Python/DataScienceFromScratch/DataScienceFromScratch.ch.04.linear_algebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 밑바닥부터 시작하는 데이터 과학\n",
    "\n",
    "- 원서명 : Data Science from Scratch: First Principles with Python\n",
    "- 지은이 : Joel Grus\n",
    "- 원서 : <http://shop.oreilly.com/product/0636920033400.do>\n",
    "- 번역서 : <http://www.insightbook.co.kr/books/programming-insight>\n",
    "\n",
    "![책표지](./image/cover.png)\n",
    "\n",
    "- 출판사 예제코드 : <https://github.com/insight-book/data-science-from-scratch>\n",
    "\n",
    "위 책을 보면서 필자가 직접 코딩하면서 정리한 내용입니다.  \n",
    "책의 모든 내용을 다 포함하고 있지는 않으며, 책에 없는 부가적인 설명이 들어 갈 수 있습니다.  \n",
    "필자가 작성한 `Jupyter notebook`은 다음 Link에서 다운로드하여 실행이 가능합니다.\n",
    "\n",
    "- 본문 Jupyter notebook : <https://github.com/DevStarSJ/Study/tree/master/Blog/Python/DataScienceFromScratch>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 04 선형대수 (Linear Algebra)\n",
    "\n",
    "선형대수란 백터 공간을 다루는 수학입니다.\n",
    "\n",
    "### 1. 백터(Vector)\n",
    "\n",
    "`vector`란 유한한 공간에 존재하는 점을 표현하는 데이터 입니다.\n",
    "\n",
    "- 2차원 백터 : x,y 좌표\n",
    "- 3차원 백터 : 키, 몸무게, 나이\n",
    "- 4차원 백터 : 국어점수, 수학점수, 영어점수, 물리점수\n",
    "\n",
    "`vector`를 `python`에서 표현하기 가장 간단한 방법은 `tuple`이나 `list`로 표현하는 것입니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xy_point = ( 3,4)\n",
    "height_weight_age = [173, 68, 40]\n",
    "grades = [ 89, 100, 95, 82]\n",
    "\n",
    "a = [1, 2]\n",
    "b = [2, 1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1 vector의 덧셈\n",
    "\n",
    "각 성분끼리 더합니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vector_add(v, w):\n",
    "    return [v_i + w_i for v_i, w_i in zip(v,w)]\n",
    "\n",
    "vector_add(a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### 1.2 vector의 뺄셈\n",
    "\n",
    "각 성분끼리 빼줍니다. 각 성분에 마이너스를 한 뒤 더하는 것과 결과가 같습니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vector_subtract(v, w):\n",
    "    return [v_i - w_i for v_i, w_i in zip(v, w)]\n",
    "\n",
    "vector_subtract(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 vector의 합\n",
    "\n",
    "`vector_add`를 이용해서 여러 `vector`들의 합을 구하는 함수도 구현이 가능합니다.\n",
    "`reduce`를 이용하면 `vector_add`를 여러 번 호출하는 것을 간랸하게 구현이 가능합니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from functools import reduce\n",
    "\n",
    "def vector_sum(vectors):\n",
    "    return reduce(vector_add, vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7, 9]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = [3, 2]\n",
    "d = [1, 4]\n",
    "vectors = [a, b, c, d]\n",
    "\n",
    "vector_sum(vectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`reduce(실행할 함수, 순서형 자료)` 함수는 전달된 함수에 순서형 자료를 하나씩 대입해서 그 결과를 누적으로 적용시킵니다.\n",
    "\n",
    "<https://docs.python.org/3/library/functools.html#functools.reduce>\n",
    "\n",
    "즉 아래 코드와 완벽하게 동일한 동작을 합니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def vector_reduce(func, vectors):\n",
    "    acc = vectors[0]\n",
    "    for vector in vectors[1:]:\n",
    "        acc = func(acc, vector)\n",
    "    return acc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7, 9]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_reduce(vector_add, vectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`partial`을 이용하면 좀더 깔끔하게 정의가 가능합니다.\n",
    "\n",
    "<https://docs.python.org/3/library/functools.html#functools.partial>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from functools import partial\n",
    "\n",
    "vector_sum = partial(reduce, vector_add)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7, 9]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_sum(vectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.4 vector의 스칼라곱\n",
    "\n",
    "각 성분에 스칼라곱을 해주면 됩니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def scalar_multiply(c, v):\n",
    "    return [c * v_i for v_i in v]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 6]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scalar_multiply(3, a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.5 vector의 평균\n",
    "\n",
    "각 성분들 끼리의 평균입니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def vector_mean(vectors):\n",
    "    return scalar_multiply(1/len(vectors), vector_sum(vectors))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.75, 2.25]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_mean(vectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.6 vector의 내적\n",
    "\n",
    "각 성분별 곱한 값을 더해 준 값입니다.\n",
    "\n",
    "내적은 v가 w방향으로 얼마나 멀리 뻗어 나가는지를 나타냅니다.\n",
    "즉 v가 w방향으로 투영된 벡터의 길이를 나타냅니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def vector_dot(v, w):\n",
    "    return sum(v_i * w_i for v_i, w_i in zip(v, w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector_dot(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "내적을 그래프로 표현해 보겠습니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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1Jzr6D3bu3Enz5s2dLkd8hLWWHTt2UKdOS5599h+ULVuW0NB23HLLAKdLc4SGsopXueuu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\nIiJShMJBRESKUDiIiEgR/w8D8RofITS6+QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ddfec30940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "v = [2,1]\n",
    "w = [math.sqrt(.25), math.sqrt(.75)]\n",
    "c = vector_dot(v,w)\n",
    "vonw = scalar_multiply(c, w)\n",
    "o = [0,0]\n",
    "\n",
    "plt.arrow(0,0, v[0], v[1], width=0.002, head_width=.1, length_includes_head=True)\n",
    "plt.annotate(\"v\", v, xytext=[v[0] + 0.1, v[1]])\n",
    "\n",
    "plt.arrow(0,0, w[0],w[1], width=0.002, head_width=.1, length_includes_head=True)\n",
    "plt.annotate(\"w\", w, xytext=[w[0] - 0.1, w[1]])\n",
    "\n",
    "plt.arrow(0,0, vonw[0],vonw[1], length_includes_head=True)\n",
    "plt.annotate(\"(v·w)w\", vonw, xytext=[vonw[0]-0.1, vonw[1]+0.1])\n",
    "\n",
    "plt.arrow(v[0],v[1], vonw[0]-v[0], vonw[1]-v[1], linestyle='dotted', length_includes_head=True)\n",
    "\n",
    "plt.scatter(*zip(v,w,o), marker='.')\n",
    "\n",
    "plt.axis([0,2,0,2])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위 그림을 보면 `v·w`의 내적한 값에 `w`를 곱해주니 `w`방향으로 `v`의 크기만큼을 투영한 벡터나 나옵니다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.7 vector의 크기\n",
    "\n",
    "모든 성분들을 제곱한 값을 더한 뒤 제곱근을 구하면 됩니다.\n",
    "자기자신과의 내적값에 제곱근을 하는 것으로 쉽게 구할 수 있습니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "\n",
    "def magnitude(v):\n",
    "    return math.sqrt(vector_dot(v,v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.23606797749979"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "magnitude(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.8 vector의 거리\n",
    "\n",
    "두 벡터간의 거리를 구하는 것은 각 성분들을 뺀값을 제곱한 다음에 제곱근을 취해주면 됩니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def distance(v, w):\n",
    "    return math.sqrt(vector_dot(vector_subtract(v,w), vector_subtract(v,w)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4142135623730951"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "두 백터의 뺄셈의 크기를 구하는것과도 같습니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def distance(v, w):\n",
    "    return magnitude(vector_subtract(v,w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4142135623730951"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### 2. 행렬 (matrix)\n",
    "\n",
    "행렬은 2차원으로 구성된 숫자의 집합이며, `list`의 `list`로 구성이 가능합니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A = [[1, 2, 3],\n",
    "     [4, 5, 6]]\n",
    "\n",
    "B = [[1, 2],\n",
    "     [3, 4],\n",
    "     [5, 6]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "행렬을 만들어주는 함수를 정의해 보겠습니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_matrix(num_rows, num_cols, entry_fn):\n",
    "    return [[entry_fn(i, j)\n",
    "             for j in range(num_cols)]\n",
    "             for i in range(num_rows)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* `단위행렬(diagnoal matrix)` : 대각선의 원소는 1이고 나머지는 0인 행렬"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def is_dignoal(i, j):\n",
    "    return 1 if i == j else 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[1, 0, 0, 0, 0],\n",
       " [0, 1, 0, 0, 0],\n",
       " [0, 0, 1, 0, 0],\n",
       " [0, 0, 0, 1, 0],\n",
       " [0, 0, 0, 0, 1]]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "identity_matrix = make_matrix(5, 5, is_dignoal)\n",
    "\n",
    "identity_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1장에서 네트워크를 표현한 것을 행렬을 표현하는 것이 가능합니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "friendships = [(0, 1), (0, 2), (1, 2), (1, 3), (2, 3), (3, 4),\n",
    "               (4, 5), (5, 6), (5, 7), (6, 8), (7, 8), (8, 9)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#          user 0  1  2  3  4  5  6  7  8  9\n",
    "#\n",
    "friendships = [[0, 1, 1, 0, 0, 0, 0, 0, 0, 0], # user 0\n",
    "               [1, 0, 1, 1, 0, 0, 0, 0, 0, 0], # user 1\n",
    "               [1, 1, 0, 1, 0, 0, 0, 0, 0, 0], # user 2\n",
    "               [0, 1, 1, 0, 1, 0, 0, 0, 0, 0], # user 3\n",
    "               [0, 0, 0, 1, 0, 1, 0, 0, 0, 0], # user 4\n",
    "               [0, 0, 0, 0, 1, 0, 1, 1, 0, 0], # user 5\n",
    "               [0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 6\n",
    "               [0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 7\n",
    "               [0, 0, 0, 0, 0, 0, 1, 1, 0, 1], # user 8\n",
    "               [0, 0, 0, 0, 0, 0, 0, 0, 1, 0]] # user 9"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "이 경우 친구관계를 index로 확인이 가능합니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "friendships[0][2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "friendships[0][8]"
   ]
  }
 ],
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