{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "5f4daf7d16ce46969053a7d95bed3d2e"
},
"source": [
"## 2.2 벡터와 행렬의 연산"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "dfcec2b1bfba4b769d7bcf316d0d9949"
},
"source": [
"벡터와 행렬도 숫자처럼 덧셈, 뺄셈, 곱셈 등의 연산을 할 수 있다. 벡터와 행렬의 연산을 이용하면 대량의 데이터에 대한 계산을 간단한 수식으로 나타낼 수 있다. 물론 벡터와 행렬에 대한 연산은 숫자의 사칙 연산과는 몇 가지 다른 점이 있으므로 이러한 차이를 잘 알아야 한다."
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "7939530b71b940e2b86d687ec9e4453c"
},
"source": [
"이 절에서는 넘파이를 이용하여 벡터와 행렬의 연산을 실행하는 법도 공부한다. 다음처럼 넘파이와 맷플롯립 패키지가 임포트되어 있어야 한다. "
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"school_cell_uuid": "50bb3d967a9a4e11b884730d0684b81d"
},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pylab as plt"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "cb94846e5abe46fd92f6ff3c7a7b9a25"
},
"source": [
"### 벡터/행렬의 덧셈과 뺄셈"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "080cd979ae564a93a3c095855b315d25"
},
"source": [
"같은 크기를 가진 두 개의 벡터나 행렬은 덧셈과 뺄셈을 할 수 있다. 두 벡터와 행렬에서 같은 위치에 있는 원소끼리 덧셈과 뺄셈을 하면 된다. 이러한 연산을 **요소별(element-wise) 연산**이라고 한다. "
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "68780cec81124d158b3e8d074bbf0785"
},
"source": [
"예를 들어 벡터 $x$와 $y$가 다음과 같으면,\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x=\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"11 \\\\\n",
"12 \\\\\n",
"\\end{bmatrix}\n",
",\\;\\;\n",
"y=\n",
"\\begin{bmatrix}\n",
"0 \\\\\n",
"1 \\\\\n",
"2 \\\\\n",
"\\end{bmatrix}\n",
"\\tag{2.2.1}\n",
"\\end{align}\n",
"$$\n",
"\n",
"벡터 $x$와 $y$의 덧셈 $x+y$와 뺄셈 $x-y$는 다음처럼 계산한다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x + y =\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"11 \\\\\n",
"12 \\\\\n",
"\\end{bmatrix}\n",
"+\n",
"\\begin{bmatrix}\n",
"0 \\\\\n",
"1 \\\\\n",
"2 \\\\\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"10 + 0 \\\\\n",
"11 + 1 \\\\\n",
"12 + 2 \\\\\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"12 \\\\\n",
"14 \\\\\n",
"\\end{bmatrix}\n",
"\\tag{2.2.2}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x - y =\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"11 \\\\\n",
"12 \\\\\n",
"\\end{bmatrix}\n",
"-\n",
"\\begin{bmatrix}\n",
"0 \\\\\n",
"1 \\\\\n",
"2 \\\\\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"10 - 0 \\\\\n",
"11 - 1 \\\\\n",
"12 - 2 \\\\\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"10 \\\\\n",
"10 \\\\\n",
"\\end{bmatrix}\n",
"\\tag{2.2.3}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "1615caac7bce4f7194eb23696ca9c9f1"
},
"source": [
"벡터의 덧셈과 뺄셈을 넘파이로 계산하면 다음과 같다. 여기에서는 편의상 1차원 배열로 벡터를 표시하였다."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"school_cell_uuid": "92a4f1bd51064d95b35f88d28de98185"
},
"outputs": [],
"source": [
"x = np.array([10, 11, 12, 13, 14])\n",
"y = np.array([0, 1, 2, 3, 4])"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"school_cell_uuid": "a6e0e116c37142eabae65a0a118616eb"
},
"outputs": [
{
"data": {
"text/plain": [
"array([10, 12, 14, 16, 18])"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x + y"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"school_cell_uuid": "4ce5a014512f4b3192455936c19eb65f"
},
"outputs": [
{
"data": {
"text/plain": [
"array([10, 10, 10, 10, 10])"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x - y"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "e4ce357b185b48479f6a9fdb33a223ea"
},
"source": [
"행렬도 같은 방법으로 덧셈과 뺄셈을 할 수 있다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\begin{bmatrix}\n",
"5 & 6 \\\\\n",
"7 & 8\n",
"\\end{bmatrix}\n",
"+\n",
"\\begin{bmatrix}\n",
"10 & 20 \\\\\n",
"30 & 40 \\\\\n",
"\\end{bmatrix}\n",
"-\n",
"\\begin{bmatrix}\n",
"1 & 2 \\\\\n",
"3 & 4\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"14 & 24 \\\\\n",
"34 & 44\n",
"\\end{bmatrix}\n",
"\\tag{2.2.4}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"school_cell_uuid": "7dcf23494eac453e88f5f9528be19244"
},
"outputs": [
{
"data": {
"text/plain": [
"array([[14, 24],\n",
" [34, 44]])"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.array([[5, 6], [7, 8]]) + np.array([[10, 20], [30, 40]]) - \\\n",
" np.array([[1, 2], [3, 4]])"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "f0d566a306f6486c84dc5c75b50a505b"
},
"source": [
"### 스칼라와 벡터/행렬의 곱셈"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "7ab15da44c4b428f81ce32c3750fc858"
},
"source": [
"벡터 $x$ 또는 행렬 $A$에 스칼라값 $c$를 곱하는 것은 **벡터 $x$ 또는 행렬 $A$의 모든 원소에 스칼라값 $c$를 곱하는 것**과 같다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"c\n",
"\\begin{bmatrix}\n",
"x_1 \\\\\n",
"x_2\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"cx_1 \\\\\n",
"cx_2\n",
"\\end{bmatrix}\n",
"\\tag{2.2.5}\n",
"\\end{align}\n",
"$$\n",
"\n",
"\n",
"$$\n",
"\\begin{align}\n",
"c\n",
"\\begin{bmatrix}\n",
"a_{11} & a_{12} \\\\\n",
"a_{21} & a_{22}\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"ca_{11} & ca_{12} \\\\\n",
"ca_{21} & ca_{22}\n",
"\\end{bmatrix}\n",
"\\tag{2.2.6}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "cd72e64ecffe44e2822425e8966d75f3"
},
"source": [
"### 브로드캐스팅"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "38cea09d8db74a0091f100327cfd486e"
},
"source": [
"원래 덧셈과 뺄셈은 크기(차원)가 같은 두 벡터에 대해서만 할 수 있다. 하지만 벡터와 스칼라의 경우에는 관례적으로 다음처럼 1-벡터를 사용하여 스칼라를 벡터로 변환한 연산을 허용한다. 이를 **브로드캐스팅(broadcasting)**이라고 한다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"11 \\\\\n",
"12 \\\\\n",
"\\end{bmatrix}\n",
"- 10\n",
"=\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"11 \\\\\n",
"12 \\\\\n",
"\\end{bmatrix}\n",
"- 10\\cdot \\mathbf{1}\n",
"=\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"11 \\\\\n",
"12 \\\\\n",
"\\end{bmatrix}\n",
"-\n",
"\\begin{bmatrix}\n",
"10 \\\\\n",
"10 \\\\\n",
"10 \\\\\n",
"\\end{bmatrix}\n",
"\\tag{2.2.7}\n",
"\\end{align}\n",
"$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "f4f3364e2cf748b29051e8dbd91da376"
},
"source": [
"데이터 분석에서는 원래의 데이터 벡터 $x$가 아니라 그 데이터 벡터의 각 원소의 평균값을 뺀 **평균제거(mean removed) 벡터** 혹은 **0-평균(zero-mean) 벡터**를 사용하는 경우가 많다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x = \n",
"\\begin{bmatrix}\n",
"x_1 \\\\\n",
"x_2 \\\\\n",
"\\vdots \\\\\n",
"x_N\n",
"\\end{bmatrix}\n",
"\\;\\; \\rightarrow \\;\\;\n",
"x - m =\n",
"\\begin{bmatrix}\n",
"x_1 - m\\\\\n",
"x_2 - m \\\\\n",
"\\vdots \\\\\n",
"x_N - m\n",
"\\end{bmatrix}\n",
"\\tag{2.2.8}\n",
"\\end{align}\n",
"$$\n",
"\n",
"위 식에서 $m$은 샘플 평균이다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"m = \\dfrac{1}{N}\\sum_{i=1}^N x_i\n",
"\\tag{2.2.9}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "d85ee8dd59084a09aefdb0fe82cbaa54"
},
"source": [
"### 선형조합"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "52f7e08ae2574162b1d2d773b3378df3"
},
"source": [
"벡터/행렬에 다음처럼 스칼라값을 곱한 후 더하거나 뺀 것을 벡터/행렬의 **선형조합(linear combination)**이라고 한다. 벡터나 행렬을 선형조합해도 크기는 변하지 않는다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"c_1x_1 + c_2x_2 + c_3x_3 + \\cdots + c_Lx_L = x\n",
"\\tag{2.2.10}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"c_1A_1 + c_2A_2 + c_3A_3 + \\cdots + c_LA_L = A\n",
"\\tag{2.2.11}\n",
"\\end{align}\n",
"$$\n",
"\n",
"\n",
"$$\n",
"\\begin{align}\n",
"c_1, c_2, \\ldots, c_L \\in \\mathbf{R}\n",
"\\tag{2.2.12}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x_1, x_2, \\ldots, x_L, x \\in \\mathbf{R}^M\n",
"\\tag{2.2.13}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"A_1, A_2, \\ldots, A_L, A \\in \\mathbf{R}^{M \\times N}\n",
"\\tag{2.2.14}\n",
"\\end{align}\n",
"$$\n",
"\n",
"벡터나 행렬의 크기를 직사각형으로 표시하면 다음과 같다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\begin{matrix}\n",
"c_1\\,\\boxed{\\begin{matrix} \\phantom{\\LARGE\\mathstrut} \\\\ x_1 \\\\ \\phantom{\\LARGE\\mathstrut} \\end{matrix}} & + &\n",
"c_2\\,\\boxed{\\begin{matrix} \\phantom{\\LARGE\\mathstrut} \\\\ x_2 \\\\ \\phantom{\\LARGE\\mathstrut} \\end{matrix}} & + & \n",
"\\cdots \\!\\!\\!\\!& + & \n",
"c_L\\,\\boxed{\\begin{matrix} \\phantom{\\LARGE\\mathstrut} \\\\ x_L \\\\ \\phantom{\\LARGE\\mathstrut} \\end{matrix}}\n",
"\\end{matrix}\n",
"\\tag{2.2.15}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\begin{matrix}\n",
"c_1\\,\\boxed{\\begin{matrix} \\phantom{} & \\phantom{} & \\phantom{} \\\\ & A_1 & \\\\ \\phantom{} & \\phantom{} & \\phantom{} \\end{matrix}} \n",
"& + &\n",
"c_2\\,\\boxed{\\begin{matrix} \\phantom{} & \\phantom{} & \\phantom{} \\\\ & A_2 & \\\\ \\phantom{} & \\phantom{} & \\phantom{} \\end{matrix}} \n",
"& + &\n",
"\\cdots\n",
"& + &\n",
"c_L\\,\\boxed{\\begin{matrix} \\phantom{} & \\phantom{} & \\phantom{} \\\\ & A_L & \\\\ \\phantom{} & \\phantom{} & \\phantom{} \\end{matrix}} \n",
"\\end{matrix}\n",
"\\tag{2.2.16}\n",
"\\end{align}\n",
"$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "e0a59b5e8b814543ad79c6a660b4bb6a"
},
"source": [
"### 벡터와 벡터의 곱셈"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "f09ed6d691264628b902c4e609a30d1d"
},
"source": [
"행렬의 곱셈을 정의하기 전에 우선 두 벡터의 곱셈을 알아보자. 벡터를 곱셈하는 방법은 여러 가지가 있지만 여기서는 **내적(inner product)**에 대해서만 다룬다. 벡터 $x$와 벡터 $y$의 내적은 다음처럼 표기한다.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"x^T y \n",
"\\tag{2.2.17}\n",
"\\end{align}\n",
"$$\n",
"\n",
"내적은 다음처럼 점(dot)으로 표기하는 경우도 있어서 **닷 프로덕트(dot product)**라고도 부르고 < $x,y$ > 기호로 나타낼 수도 있다.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"x \\cdot y = \\, < x, y > \\, = x^T y \n",
"\\tag{2.2.18}\n",
"\\end{align}\n",
"$$\n",
"\n",
"\n",
"\n",
"두 벡터를 내적하려면 다음과 같은 조건이 만족되어야 한다.\n",
"\n",
"1. 우선 두 벡터의 차원(길이)이 같아야 한다. \n",
"2. 앞의 벡터가 행 벡터이고 뒤의 벡터가 열 벡터여야 한다. \n",
"\n",
"이때 내적의 결과는 스칼라값이 되며 다음처럼 계산한다. 우선 같은 위치에 있는 원소들을 요소별 곱셈처럼 곱한 다음, 그 값들을 다시 모두 더해서 하나의 스칼라값으로 만든다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x^T y = \n",
"\\begin{bmatrix}\n",
"x_{1} & x_{2} & \\cdots & x_{N} \n",
"\\end{bmatrix}\n",
"\\begin{bmatrix}\n",
"y_{1} \\\\\n",
"y_{2} \\\\\n",
"\\vdots \\\\\n",
"y_{N} \\\\\n",
"\\end{bmatrix} \n",
"= x_1 y_1 + \\cdots + x_N y_N \n",
"= \\sum_{i=1}^N x_i y_i\n",
"\\tag{2.2.19}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"x \\in \\mathbf{R}^{N \\times 1}\n",
"\\tag{2.2.20}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"y \\in \\mathbf{R}^{N \\times 1} \n",
"\\tag{2.2.21}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x^T y \\in \\mathbf{R} \n",
"\\tag{2.2.22}\n",
"\\end{align}\n",
"$$\n",
"\n",
"다음은 두 벡터의 내적의 예다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x =\n",
"\\begin{bmatrix}\n",
"1 \\\\ 2 \\\\ 3 \\\\\n",
"\\end{bmatrix}\n",
",\\;\\;\\;\n",
"y = \n",
"\\begin{bmatrix}\n",
"4 \\\\ 5 \\\\ 6 \\\\\n",
"\\end{bmatrix} \n",
"\\tag{2.2.23}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{align}\n",
"x^T y = \n",
"\\begin{bmatrix}\n",
"1 & 2 & 3\n",
"\\end{bmatrix}\n",
"\\begin{bmatrix}\n",
"4 \\\\ 5 \\\\ 6 \\\\\n",
"\\end{bmatrix} \n",
"= 1 \\cdot 4 + 2 \\cdot 5 + 3 \\cdot 6 = 32\n",
"\\tag{2.2.24}\n",
"\\end{align}\n",
"$$\n",
"\n",
"넘파이에서 벡터와 행렬의 내적은 `dot()`이라는 명령 또는 `@`(at이라고 읽는다)이라는 연산자로 계산한다. 2차원 배열로 표시한 벡터를 내적했을 때는 결과값이 스칼라가 아닌 2차원 배열이다."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"school_cell_uuid": "439ddddaf6c64f73ad9c969979ad9b0b"
},
"outputs": [
{
"data": {
"text/plain": [
"array([[32]])"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.array([[1], [2], [3]])\n",
"y = np.array([[4], [5], [6]])\n",
"\n",
"x.T @ y # 또는 np.dot(x.T, y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "bec85acb3bbe49ddb4e3345daecfced8"
},
"source": [
"넘파이에서는 1차원 배열끼리도 내적을 계산한다. 이때는 넘파이가 앞의 벡터는 행 벡터이고 뒤의 벡터는 열 벡터라고 가정한다."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"school_cell_uuid": "0cddf2f013324e719945dffbf85af20f"
},
"outputs": [
{
"data": {
"text/plain": [
"32"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.array([1, 2, 3])\n",
"y = np.array([4, 5, 6])\n",
"\n",
"x @ y # 또는 np.dot(x, y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "b6c82e49b1124ff9937d78277b64a52d"
},
"source": [
"왜 벡터의 내적은 덧셈이나 뺄셈과 달리 이렇게 복잡하게 정의된 것일까? 그 이유는 데이터 분석을 할 때 이러한 연산이 필요하기 때문이다. 벡터의 내적을 사용하여 데이터를 분석하는 몇 가지 예를 살펴보자."
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "52306dedaf4a4ef3b3b9962fe784d772"
},
"source": [
"### 가중합"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "7b2e2ce1a713483e9d6103c053dcdd78"
},
"source": [
"벡터의 내적은 가중합을 계산할 때 쓰일 수 있다. **가중합(weighted sum)**이란 복수의 데이터를 단순히 합하는 것이 아니라 각각의 수에 어떤 가중치 값을 곱한 후 이 곱셈 결과들을 다시 합한 것을 말한다. \n",
"\n",
"만약 데이터 벡터가 $x=[x_1, \\cdots, x_N]^T$이고 가중치 벡터가 $w=[w_1, \\cdots, w_N]^T$이면 데이터 벡터의 가중합은 다음과 같다.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"w_1 x_1 + \\cdots + w_N x_N = \\sum_{i=1}^N w_i x_i \n",
"\\tag{2.2.25}\n",
"\\end{align}\n",
"$$ \n",
"\n",
"이 값을 벡터 $x$와 $w$의 곱으로 나타내면 $w^Tx$ 또는 $x^Tw$ 라는 간단한 수식으로 표시할 수 있다.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"\\begin{aligned}\n",
"\\sum_{i=1}^N w_i x_i \n",
"&= \n",
"\\begin{bmatrix}\n",
"w_{1} && w_{2} && \\cdots && w_{N}\n",
"\\end{bmatrix}\n",
"\\begin{bmatrix}\n",
"x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_N\n",
"\\end{bmatrix} \n",
"&= w^Tx \n",
"\\\\\n",
"&=\n",
"\\begin{bmatrix}\n",
"x_{1} && x_{2} && \\cdots && x_{N}\n",
"\\end{bmatrix}\n",
"\\begin{bmatrix}\n",
"w_1 \\\\ w_2 \\\\ \\vdots \\\\ w_N\n",
"\\end{bmatrix}\n",
"&= x^Tw \n",
"\\end{aligned}\n",
"\\tag{2.2.26}\n",
"\\end{align}\n",
"$$\n",
"\n",
"예를 들어 쇼핑을 할 때 각 물건의 가격은 데이터 벡터, 각 물건의 수량은 가중치로 생각하여 내적을 구하면 총금액을 계산할 수 있다.\n",
"\n",
"만약 가중치가 모두 1이면 일반적인 합(sum)을 계산한다.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"w_1 = w_2 = \\cdots = w_N = 1 \n",
"\\tag{2.2.27}\n",
"\\end{align}\n",
"$$\n",
"\n",
"또는\n",
"\n",
"$$\n",
"\\begin{align}\n",
"w = \\mathbf{1}_N \n",
"\\tag{2.2.28}\n",
"\\end{align}\n",
"$$\n",
"\n",
"이면 \n",
"\n",
"$$ \n",
"\\begin{align}\n",
"\\sum_{i=1}^N x_i = \\mathbf{1}_N^T x \n",
"\\tag{2.2.29}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"bootstrap": {
"panel": {
"class": "panel-default"
}
},
"school_cell_uuid": "6c189202c6694fb2a74a8a9d64b37b94"
},
"source": [
"#### 연습 문제 2.2.1\n",
"\n",
"A, B, C 세 회사의 주식은 각각 100만원, 80만원, 50만원이다. 이 주식을 각각 3주, 4주, 5주를 매수할 때 필요한 금액을 구하고자 한다.\n",
"\n",
"(1) 주식의 가격과 수량을 각각 $p$ 벡터, $n$ 벡터로 표시하고 넘파이로 코딩한다.\n",
"\n",
"(2) 주식을 매수할 때 필요한 금액을 곱셈으로 표시하고 넘파이 연산으로 그 값을 계산한다."
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "5d686a31e45d489ab0be314fca3fd57a"
},
"source": [
"### 가중평균"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "52190e8b43a94001bb3b87357e8bce60"
},
"source": [
"가중합의 가중치값을 전체 가중치값의 합으로 나누면 **가중평균(weighted average)**이 된다. 가중평균은 대학교의 평균 성적 계산 등에 사용할 수 있다.\n",
"\n",
"예를 들어 고등학교에서는 국어, 영어, 두 과목의 평균 점수를 구할 때 단순히 두 과목의 점수(숫자)를 더한 후 2으로 나눈다. 그러나 대학교에서는 중요한 과목과 중요하지 않는 과목을 구분하는 학점(credit)이라는 숫자가 있다. 일주일에 한 시간만 수업하는 과목은 1학점짜리 과목이고 일주일에 세 시간씩 수업하는 중요한 과목은 3학점짜리 과목이다. 1학점과 3학점 과목의 점수가 각각 100점, 60점이면 학점을 고려한 가중 평균(weighted average) 성적은 다음과 같이 계산한다.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"\\dfrac{1}{1 + 3} \\times 100 + \\dfrac{3}{1 + 3} \\times 60\n",
"= 70 \n",
"\\tag{2.2.30}\n",
"\\end{align}\n",
"$$\n",
"\n",
"\n",
"벡터로 표현된 $N$개의 데이터의 단순 평균은 다음처럼 생각할 수 있다.\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\bar{x} = \\dfrac{1}{N}\\sum_{i=1}^N x_i = \\dfrac{1}{N} \\mathbf{1}_N^T x\n",
"\\tag{2.2.31}\n",
"\\end{align}\n",
"$$\n",
"\n",
"위 수식에서 보인 것처럼 $x$ 데이터의 평균은 보통 $\\bar{x}$라는 기호로 표기하고 \"엑스 바(x bar)\" 라고 읽는다.\n",
"\n",
"다음은 넘파이로 평균을 계산하는 방법이다."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"school_cell_uuid": "92725f510e5a426ba9c9f050e8d90e83"
},
"outputs": [
{
"data": {
"text/plain": [
"4.5"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.arange(10)\n",
"N = len(x)\n",
"\n",
"np.ones(N) @ x / N"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "3fd76b76bf514970b2c0bda365fd712e"
},
"source": [
"현실적으로는 ``mean()``이라는 메서드를 사용하는 것이 편하다."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"school_cell_uuid": "ed7a3c2a58294d548af6b1d38f456d64"
},
"outputs": [
{
"data": {
"text/plain": [
"4.5"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x.mean()"
]
},
{
"cell_type": "markdown",
"metadata": {
"bootstrap": {
"panel": {
"class": "panel-default"
}
},
"school_cell_uuid": "6c189202c6694fb2a74a8a9d64b37b94"
},
"source": [
"#### 연습 문제 2.2.2\n",
"\n",
"벡터 $x$의 평균 제거 벡터는 다음과 같이 계산함을 증명하라.\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"x - \\dfrac{1}{N}\\mathbf{1}_N^Tx \\mathbf{1}^{}_N\n",
"\\tag{2.2.32}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "527c7645de0d4ce195a00597bd0b006a"
},
"source": [
"### 유사도"
]
},
{
"cell_type": "markdown",
"metadata": {
"school_cell_uuid": "faa387c866e148cabbbc043be254a0a3"
},
"source": [
"벡터의 곱셈(내적)은 두 벡터 간의 유사도를 계산하는 데도 이용할 수 있다. **유사도(similarity)는 두 벡터가 닮은 정도를 정량적으로 나타낸 값**으로 두 벡터가 비슷한 경우에는 유사도가 커지고 비슷하지 앟은 경우에는 유사도가 작아진다. 내적을 이용하면 **코사인 유사도(cosine similarity)**라는 유사도를 계산할 수 있다. 추후 선형대수의 기하학적 의미를 공부할 때 코사인 유사도에 대해 살펴볼 것이다.\n",
"\n",
"예를 들어 0과 1을 나타내는 MNIST 이미지에 대해 내적을 계산해보자."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"school_cell_uuid": "93c8a93c97f34cc68e57d51e9660d347"
},
"outputs": [
{
"data": {
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