{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Uczenie maszynowe 2020/2021 – laboratoria\n",
"### 3 marca 2021\n",
"# 1. Python – listy składane, indeksowanie, biblioteka _NumPy_"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Listy składane (*List comprehension*)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]\n"
]
}
],
"source": [
"lista = []\n",
"for x in range(1, 11):\n",
" lista.append(x ** 2)\n",
" \n",
"print(lista)"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]\n"
]
}
],
"source": [
"lista = [x ** 2 for x in range(1, 11)]\n",
"print(lista)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Przypuśćmy, że mamy dane zdanie i chcemy utworzyć listę, która będzie zawierać długości kolejnych wyrazów tego zdania. Możemy to zrobić w następujący sposób:"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[4, 16, 36, 64, 100]\n"
]
}
],
"source": [
"lista = []\n",
"for i in range(1, 11):\n",
" if i % 2 == 0:\n",
" lista.append(i ** 2)\n",
" \n",
"print(lista)"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[4, 16, 36, 64, 100]\n"
]
}
],
"source": [
"lista = [i ** 2 for i in range(1, 11) if i % 2 == 0]\n",
"print(lista)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"list(range(1, 11))"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tracz tarł tarcicę tak takt w takt jak takt w takt tarcicę tartak tarł\n"
]
}
],
"source": [
"zdanie = 'tracz tarł tarcicę tak takt w takt jak takt w takt tarcicę tartak tarł'\n",
"print(zdanie)"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['tracz', 'tarł', 'tarcicę', 'tak', 'takt', 'w', 'takt', 'jak', 'takt', 'w', 'takt', 'tarcicę', 'tartak', 'tarł']\n"
]
}
],
"source": [
"wyrazy = zdanie.split()\n",
"print(wyrazy)"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[5, 4, 7, 3, 4, 1, 4, 3, 4, 1, 4, 7, 6, 4]\n"
]
}
],
"source": [
"dlugosci_wyrazow = []\n",
"for wyraz in wyrazy:\n",
" dlugosci_wyrazow.append(len(wyraz))\n",
" \n",
"print(dlugosci_wyrazow)"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[5, 4, 7, 3, 4, 1, 4, 3, 4, 1, 4, 7, 6, 4]\n"
]
}
],
"source": [
"dlugosci_wyrazow = [len(wyraz) for wyraz in wyrazy]\n",
"print(dlugosci_wyrazow)"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[5, 4, 7, 3, 4, 1, 4, 3, 4, 1, 4, 7, 6, 4]\n"
]
}
],
"source": [
"zdanie = 'tracz tarł tarcicę tak takt w takt jak takt w takt tarcicę tartak tarł'\n",
"wyrazy = zdanie.split()\n",
"dlugosci_wyrazow = []\n",
"for wyraz in wyrazy:\n",
" dlugosci_wyrazow.append(len(wyraz))\n",
" \n",
"print(dlugosci_wyrazow)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Możemy to też zrobić bardziej „pythonicznie”, przy użyciu list składanych:"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[5, 4, 7, 3, 4, 1, 4, 3, 4, 1, 4, 7, 6, 4]\n"
]
}
],
"source": [
"zdanie = 'tracz tarł tarcicę tak takt w takt jak takt w takt tarcicę tartak tarł'\n",
"dlugosci_wyrazow = [len(wyraz) for wyraz in zdanie.split()]\n",
"\n",
"print(dlugosci_wyrazow)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Jeżeli chcemy, żeby był sprawdzany dodatkowy warunek, np. chcemy pomijać wyraz „takt”, to wciąż możemy użyć list składanych:"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[5, 4, 7, 3, 1, 3, 1, 7, 6, 4]\n"
]
}
],
"source": [
"zdanie = 'tracz tarł tarcicę tak takt w takt jak takt w takt tarcicę tartak tarł'\n",
"wyrazy = zdanie.split()\n",
"dlugosci_wyrazow = [len(wyraz) for wyraz in wyrazy if wyraz != 'takt']\n",
"print(dlugosci_wyrazow)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Indeksowanie"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wszystkie listy i krotki w Pythonie, w tym łańcuchy (które trakowane są jak krotki znaków), są indeksowane od 0:"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a\n",
"e\n"
]
}
],
"source": [
"napis = 'abcde'\n",
"print(napis[0]) # 'a'\n",
"print(napis[4]) # 'e'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Indeksy możemy liczyć również „od końca”:"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"e\n",
"d\n",
"a\n"
]
}
],
"source": [
"napis = 'abcde'\n",
"print(napis[-1]) # 'e' („ostatni”)\n",
"print(napis[-2]) # 'd' („drugi od końca”)\n",
"print(napis[-5]) # 'a' („piąty od końca”)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Łańcuchy możemy też „kroić na plasterki” (_slicing_):"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"bcd\n",
"b\n",
"cd\n",
"bcd\n",
"de\n",
"abc\n",
"abcde\n"
]
}
],
"source": [
"napis = 'abcde'\n",
"print(napis[1:4]) # 'bcd' („znaki od 1. włącznie do 4. wyłącznie”)\n",
"print(napis[1:2]) # 'b' (to samo co `napis[1]`)\n",
"print(napis[-3:-1]) # 'cd' (kroić można też stosując indeksowanie od końca)\n",
"print(napis[1:-1]) # 'bcd' (możemy nawet mieszać te dwa sposoby indeksowania)\n",
"print(napis[3:]) # 'de' (jeżeli koniec przedziału nie jest podany, to kroimy do samego końca łańcucha)\n",
"print(napis[:3]) # 'abc' (jeżeli początek przedziału nie jest podany, to kroimy od początku łańcucha)\n",
"print(napis[:]) # 'abcde' (kopia całego napisu)"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"abcde\n"
]
}
],
"source": [
"print(napis[:])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Biblioteka _NumPy_"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Tablice"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Głównym obiektem w NumPy jest **jednorodna**, **wielowymiarowa** tablica. Przykładem takiej tablicy jest macierz `x`.\n",
"\n",
"Macierz $x =\n",
" \\begin{pmatrix}\n",
" 1 & 2 & 3 \\\\\n",
" 4 & 5 & 6 \\\\\n",
" 7 & 8 & 9\n",
" \\end{pmatrix}$\n",
"można zapisać jako:"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2 3]\n",
" [4 5 6]\n",
" [7 8 9]]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])\n",
"\n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2 3]\n",
" [4 5 6]\n",
" [7 8 9]]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"x = np.array([[1,2,3],[4,5,6],[7,8,9]])\n",
"print(x)"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1. 2. 3. -4.5 5. ]\n",
" [ 10. 9. -8. -13. 0.39]]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"y = np.array([[1, 2, 3, -4.5, 5], [10, 9, -8, -13, 0.39]])\n",
"print(y)"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(3, 3)\n"
]
}
],
"source": [
"print(x.shape)"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(2, 5)\n"
]
}
],
"source": [
"print(y.shape)"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1. 2. ]\n",
" [ 3. -4.5 ]\n",
" [ 5. 10. ]\n",
" [ 9. -8. ]\n",
" [-13. 0.39]]\n"
]
}
],
"source": [
"z = y.reshape(5, 2)\n",
"print(z)"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(5, 2)\n"
]
}
],
"source": [
"print(z.shape)"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1. 2. 3. -4.5 5. ]\n",
" [ 10. 9. -8. -13. 0.39]]\n"
]
}
],
"source": [
"w = z.reshape(2, 5)\n",
"print(w)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Najczęsciej używane metody tablic typu `array`:"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(2, 5)\n",
"[[ 1. 2. ]\n",
" [ 3. -4.5 ]\n",
" [ 5. 10. ]\n",
" [ 9. -8. ]\n",
" [-13. 0.39]]\n",
"(5, 2)\n"
]
}
],
"source": [
"print(y.shape)\n",
"\n",
"z = y.reshape((5, 2))\n",
"print(z)\n",
"print(z.shape)"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1. 2. ]\n",
" [ 3. -4.5 ]\n",
" [ 5. 10. ]\n",
" [ 9. -8. ]\n",
" [-13. 0.39]]\n"
]
}
],
"source": [
"print(z)"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 3. -1.5 15. 1. -12.61]\n"
]
}
],
"source": [
"print(z.sum(axis=1))"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2 3]\n",
" [4 5 6]\n",
" [7 8 9]]\n",
"[2. 5. 8.]\n"
]
}
],
"source": [
"print(x)\n",
"print(x.mean(axis=1))"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1 2 3 4 5 6 7 8 9 10]\n"
]
}
],
"source": [
"lista = list(range(1, 11))\n",
"A = np.array(lista)\n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n",
"[ 1 2 3 4 5 6 7 8 9 10]\n"
]
}
],
"source": [
"lista = list(range(1, 11))\n",
"print(lista)\n",
"A = np.array(lista)\n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1 2 3 4 5 6 7 8 9 10]\n"
]
}
],
"source": [
"A = np.arange(1, 11)\n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([12, 15, 18])"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x.sum(axis=0)"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([2., 5., 8.])"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x.mean(axis=1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Do tworzenia sekwencji liczbowych jako obiekty typu `array` należy wykorzystać funkcję `arange`."
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])"
]
},
"execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.arange(10)"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14])"
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.arange(5, 15)"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 5. , 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9],\n",
" [ 6. , 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9],\n",
" [ 7. , 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9],\n",
" [ 8. , 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9],\n",
" [ 9. , 9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9],\n",
" [10. , 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, 10.8, 10.9],\n",
" [11. , 11.1, 11.2, 11.3, 11.4, 11.5, 11.6, 11.7, 11.8, 11.9],\n",
" [12. , 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.7, 12.8, 12.9],\n",
" [13. , 13.1, 13.2, 13.3, 13.4, 13.5, 13.6, 13.7, 13.8, 13.9],\n",
" [14. , 14.1, 14.2, 14.3, 14.4, 14.5, 14.6, 14.7, 14.8, 14.9]])"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.arange(5, 15, 0.1).reshape(10, 10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kształt tablicy można zmienić za pomocą metody `reshape`:"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1 2 3 4 5 6 7 8 9 10 11 12]\n",
"[[ 1 2 3 4]\n",
" [ 5 6 7 8]\n",
" [ 9 10 11 12]]\n"
]
}
],
"source": [
"x = np.arange(1, 13)\n",
"print(x)\n",
"y = x.reshape(3, 4)\n",
"print(y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Funkcją podobną do `arange` jest `linspace`, która wypełnia wektor określoną liczbą elementów z przedziału o równych automatycznie obliczonych odstępach (w `arange` należy podać rozmiar kroku):"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0. 0.22222222 0.44444444 0.66666667 0.88888889 1.11111111\n",
" 1.33333333 1.55555556 1.77777778 2. ]\n"
]
}
],
"source": [
"x = np.linspace(0, 2, 10)\n",
"print(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Dodatkowe informacje o funkcjach NumPy uzyskuje się za pomocą polecenia `help(nazwa_funkcji)`:"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Help on function shape in module numpy:\n",
"\n",
"shape(a)\n",
" Return the shape of an array.\n",
" \n",
" Parameters\n",
" ----------\n",
" a : array_like\n",
" Input array.\n",
" \n",
" Returns\n",
" -------\n",
" shape : tuple of ints\n",
" The elements of the shape tuple give the lengths of the\n",
" corresponding array dimensions.\n",
" \n",
" See Also\n",
" --------\n",
" alen\n",
" ndarray.shape : Equivalent array method.\n",
" \n",
" Examples\n",
" --------\n",
" >>> np.shape(np.eye(3))\n",
" (3, 3)\n",
" >>> np.shape([[1, 2]])\n",
" (1, 2)\n",
" >>> np.shape([0])\n",
" (1,)\n",
" >>> np.shape(0)\n",
" ()\n",
" \n",
" >>> a = np.array([(1, 2), (3, 4)], dtype=[('x', 'i4'), ('y', 'i4')])\n",
" >>> np.shape(a)\n",
" (2,)\n",
" >>> a.shape\n",
" (2,)\n",
"\n"
]
}
],
"source": [
"help(np.shape)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Tablice mogą składać się z danych różnych typów (ale tylko jednego typu danych równocześnie, stąd jednorodność)."
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"float64\n",
"[0.1 0.2 0.3]\n",
"float64\n",
"float64\n"
]
}
],
"source": [
"x = np.array([1, 2, 3, 0.5])\n",
"print(x.dtype)\n",
"x = np.array([0.1, 0.2, 0.3])\n",
"print(x)\n",
"print(x.dtype)\n",
"x = np.array([1, 2, 3], dtype='float64')\n",
"print(x.dtype)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Tworzenie tablic składających się z samych zer lub jedynek umożliwiają funkcje `zeros` oraz `ones`:"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[0. 0. 0. 0.]\n",
" [0. 0. 0. 0.]\n",
" [0. 0. 0. 0.]]\n"
]
}
],
"source": [
"x = np.zeros([3,4])\n",
"print(x)"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1. 1. 1. 1.]\n",
" [1. 1. 1. 1.]\n",
" [1. 1. 1. 1.]]\n"
]
}
],
"source": [
"y = np.ones([3,4])\n",
"print(y)"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1. 1. 1. 1.]\n",
" [1. 1. 1. 1.]\n",
" [1. 1. 1. 1.]]\n"
]
}
],
"source": [
"print(x + y)"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 0 1 2]\n",
" [ 3 4 5]\n",
" [ 6 7 8]\n",
" [ 9 10 11]]\n"
]
}
],
"source": [
"x = np.arange(12).reshape(4, 3)\n",
"print(x)"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 5 6 7]\n",
" [ 8 9 10]\n",
" [11 12 13]\n",
" [14 15 16]]\n"
]
}
],
"source": [
"y = np.arange(5, 17).reshape(4, 3)\n",
"print(y)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 0 6 14]\n",
" [ 24 36 50]\n",
" [ 66 84 104]\n",
" [126 150 176]]\n"
]
}
],
"source": [
"print(x * y)"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[0 1 2 3]\n",
" [4 5 6 7]]\n"
]
}
],
"source": [
"A = np.arange(8).reshape(2, 4)\n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1 2 3]\n",
" [ 4 5 6]\n",
" [ 7 8 9]\n",
" [10 11 12]]\n"
]
}
],
"source": [
"B = np.arange(1, 13).reshape(4, 3)\n",
"print(B)"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(2, 4) (4, 3)\n"
]
}
],
"source": [
"print(A.shape, B.shape)"
]
},
{
"cell_type": "code",
"execution_count": 101,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 48 54 60]\n",
" [136 158 180]]\n",
"(2, 3)\n"
]
}
],
"source": [
"C = np.matmul(A, B)\n",
"print(C)\n",
"print(C.shape)"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {},
"outputs": [
{
"ename": "ValueError",
"evalue": "operands could not be broadcast together with shapes (2,4) (4,3) ",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-102-a4cedde81ed0>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mA\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mB\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[1;31mValueError\u001b[0m: operands could not be broadcast together with shapes (2,4) (4,3) "
]
}
],
"source": [
"A * B"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {},
"outputs": [
{
"ename": "ValueError",
"evalue": "matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 2 is different from 4)",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-103-89ae99960b6e>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatmul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mA\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mC\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[1;31mValueError\u001b[0m: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 2 is different from 4)"
]
}
],
"source": [
"np.matmul(A, C)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Podstawowe operacje arytmetyczne"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Operatory arytmetyczne na tablicach w NumPy działają **element po elemencie**."
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2 3]\n",
" [1 2 3]]\n"
]
}
],
"source": [
"A = np.array([[1, 2, 3], [1, 2, 3]])\n",
"B = np.array([[4, 5, 6], [7, 8, 9]])\n",
" \n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[4 5 6]\n",
" [7 8 9]]\n"
]
}
],
"source": [
"print(B)"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[-3 -3 -3]\n",
" [-6 -6 -6]]\n"
]
}
],
"source": [
"print(A - B)"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[3 3 3]\n",
" [6 6 6]]\n"
]
}
],
"source": [
"print(B - A)"
]
},
{
"cell_type": "code",
"execution_count": 108,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 4 10 18]\n",
" [ 7 16 27]]\n"
]
}
],
"source": [
"print(A * B)"
]
},
{
"cell_type": "code",
"execution_count": 109,
"metadata": {},
"outputs": [
{
"ename": "ValueError",
"evalue": "matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 2 is different from 3)",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-109-04c68bb92949>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatmul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mA\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mB\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[1;31mValueError\u001b[0m: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 2 is different from 3)"
]
}
],
"source": [
"np.matmul(A, B)"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(2, 3)\n",
"(3, 4)\n",
"\n",
"[[1 2 3]\n",
" [1 2 3]]\n",
"[[1 2 3 4]\n",
" [9 8 7 6]\n",
" [2 4 6 7]]\n",
"[[25 30 35 37]\n",
" [25 30 35 37]]\n",
"\n",
"(2, 3) (3, 4)\n",
"(2, 4)\n"
]
}
],
"source": [
"print(A.shape)\n",
"B = np.array([[1, 2, 3, 4], [9, 8, 7, 6], [2, 4, 6, 7]])\n",
"print(B.shape)\n",
"\n",
"print()\n",
"\n",
"print(A)\n",
"print(B)\n",
"C = np.matmul(A, B)\n",
"print(C)\n",
"\n",
"print()\n",
"\n",
"print(A.shape, B.shape)\n",
"print(C.shape)"
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[25, 30, 35, 37],\n",
" [25, 30, 35, 37]])"
]
},
"execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.dot(A, B)"
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[2. 3. 4.]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"a = np.array([3, 4, 5])\n",
"b = np.ones(3)\n",
"print(a - b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Za mnożenie macierzy odpowiada funkcja `dot` (nie operator `*`):"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2]\n",
" [3 4]]\n"
]
}
],
"source": [
"a = np.array([[1, 2], [3, 4]])\n",
"print(a)"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2]\n",
" [3 4]]\n"
]
}
],
"source": [
"b = np.array([[1, 2], [3, 4]])\n",
"print(b)"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 1, 4],\n",
" [ 9, 16]])"
]
},
"execution_count": 115,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a * b"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 7, 10],\n",
" [15, 22]])"
]
},
"execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.dot(a,b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Przykłady innych operacji dodawania i mnożenia:"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[5., 5.],\n",
" [5., 5.]])"
]
},
"execution_count": 117,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.zeros((2, 2), dtype='float')\n",
"a += 5\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[25., 25.],\n",
" [25., 25.]])"
]
},
"execution_count": 118,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a *= 5\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": 119,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[50., 50.],\n",
" [50., 50.]])"
]
},
"execution_count": 119,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a + a"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Sklejanie tablic:"
]
},
{
"cell_type": "code",
"execution_count": 120,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1, 2, 3, 4, 5, 6, 7, 8, 9])"
]
},
"execution_count": 120,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([1, 2, 3])\n",
"b = np.array([4, 5, 6])\n",
"c = np.array([7, 8, 9])\n",
"np.hstack([a, b, c])"
]
},
{
"cell_type": "code",
"execution_count": 121,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 2, 3],\n",
" [4, 5, 6],\n",
" [7, 8, 9]])"
]
},
"execution_count": 121,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.vstack([a, b, c])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Typowe funkcje matematyczne:"
]
},
{
"cell_type": "code",
"execution_count": 122,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([3.14159265, 4.44288294, 5.44139809, 6.28318531])"
]
},
"execution_count": 122,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.arange(1, 5)\n",
"np.sqrt(x) * np.pi"
]
},
{
"cell_type": "code",
"execution_count": 123,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"16"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2**4"
]
},
{
"cell_type": "code",
"execution_count": 124,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"16"
]
},
"execution_count": 124,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.power(2, 4)"
]
},
{
"cell_type": "code",
"execution_count": 125,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0"
]
},
"execution_count": 125,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.log(np.e)"
]
},
{
"cell_type": "code",
"execution_count": 126,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4"
]
},
"execution_count": 126,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.arange(5)\n",
"x.max() - x.min()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Indeksy i zakresy"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Tablice jednowymiarowe zachowują sie podobnie do zwykłych list pythonowych."
]
},
{
"cell_type": "code",
"execution_count": 127,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([2, 3])"
]
},
"execution_count": 127,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.arange(10)\n",
"a[2:4]"
]
},
{
"cell_type": "code",
"execution_count": 128,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0, 2, 4, 6, 8])"
]
},
"execution_count": 128,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a[:10:2] # elementy do 10., co drugi element"
]
},
{
"cell_type": "code",
"execution_count": 130,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([9, 8, 7, 6, 5, 4, 3, 2, 1, 0])"
]
},
"execution_count": 130,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a[::-1] # wszytkie elementy tablicy `a`, ale w odwróconej kolejności"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Tablice wielowymiarowe mają po jednym indeksie na wymiar:"
]
},
{
"cell_type": "code",
"execution_count": 131,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 0, 1, 2, 3],\n",
" [ 4, 5, 6, 7],\n",
" [ 8, 9, 10, 11]])"
]
},
"execution_count": 131,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.arange(12).reshape(3, 4)\n",
"x"
]
},
{
"cell_type": "code",
"execution_count": 133,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"11"
]
},
"execution_count": 133,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x[2, 3] # wiersz 2, kolumna 3"
]
},
{
"cell_type": "code",
"execution_count": 134,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1, 5, 9])"
]
},
"execution_count": 134,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x[:, 1] # kolumna 1"
]
},
{
"cell_type": "code",
"execution_count": 135,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([4, 5, 6, 7])"
]
},
"execution_count": 135,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x[1, :] # wiersz 1"
]
},
{
"cell_type": "code",
"execution_count": 136,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 4, 5, 6, 7],\n",
" [ 8, 9, 10, 11]])"
]
},
"execution_count": 136,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x[1:3, :] # wiersze od 1. włącznie do 3. wyłącznie"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Warunki"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Warunki pozwalają na selekcję elementów tablicy."
]
},
{
"cell_type": "code",
"execution_count": 138,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([2, 2, 2, 3, 3, 3])"
]
},
"execution_count": 138,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([1, 1, 1, 2, 2, 2, 3, 3, 3])\n",
"a[a > 1]"
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([3, 3, 3])"
]
},
"execution_count": 139,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a[a == 3]"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(array([0, 1, 2, 3, 4, 5], dtype=int64),)"
]
},
"execution_count": 141,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.where(a < 3)"
]
},
{
"cell_type": "code",
"execution_count": 142,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0, 1, 2, 3, 4, 5], dtype=int64)"
]
},
"execution_count": 142,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.where(a < 3)[0]"
]
},
{
"cell_type": "code",
"execution_count": 143,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(array([], dtype=int64),)"
]
},
"execution_count": 143,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.where(a > 9)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Pętle i wypisywanie"
]
},
{
"cell_type": "code",
"execution_count": 145,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0 1 2 3]\n",
"[4 5 6 7]\n",
"[ 8 9 10 11]\n"
]
}
],
"source": [
"for row in x:\n",
" print(row)"
]
},
{
"cell_type": "code",
"execution_count": 146,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0\n",
"1\n",
"2\n",
"3\n",
"4\n",
"5\n",
"6\n",
"7\n",
"8\n",
"9\n",
"10\n",
"11\n"
]
}
],
"source": [
"for row in x:\n",
" for element in row:\n",
" print(element)"
]
},
{
"cell_type": "code",
"execution_count": 147,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0\n",
"1\n",
"2\n",
"3\n",
"4\n",
"5\n",
"6\n",
"7\n",
"8\n",
"9\n",
"10\n",
"11\n"
]
}
],
"source": [
"# Operacja `.flat` \"spłaszcza\" macierz\n",
"for element in x.flat:\n",
" print(element)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Liczby losowe"
]
},
{
"cell_type": "code",
"execution_count": 148,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([9, 7, 1, 6, 5])"
]
},
"execution_count": 148,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.random.randint(0, 10, 5) # Tablica złożona z 5 liczb całkowitych wylosowanych z zakresu od 0 do 10"
]
},
{
"cell_type": "code",
"execution_count": 149,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([-0.5127886 , -0.4507017 , -1.06312973, 1.22150748, 0.27790976])"
]
},
"execution_count": 149,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.random.normal(0, 1, 5) # Tablica złożona z 5 liczb wylosowanych z rozkładu normalnego (0, 1)"
]
},
{
"cell_type": "code",
"execution_count": 150,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1.34207418, 1.70197756, 1.16472435, 0.57619858, 1.22382641])"
]
},
"execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.random.uniform(0, 2, 5) # Tablica złożona z 5 liczb wylosowanych z rozkładu jednostajnego na przedziale (0, 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Macierze"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"NumPy jest pakietem wykorzystywanym do obliczeń w dziedzinie algebry liniowej, co jeszcze szczególnie przydatne w uczeniu maszynowym. \n",
"\n",
"Wektor o wymiarach $1 \\times N$ \n",
"$$\n",
" x =\n",
" \\begin{pmatrix}\n",
" x_{1} \\\\\n",
" x_{2} \\\\\n",
" \\vdots \\\\\n",
" x_{N}\n",
" \\end{pmatrix} \n",
"$$\n",
"\n",
"i jego transpozycję $x^\\top = (x_{1}, x_{2},\\ldots,x_{N})$ można wyrazić w Pythonie w następujący sposób:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Transpozycja** to operacja zamiany wierszy na kolumny i na odwrót."
]
},
{
"cell_type": "code",
"execution_count": 172,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[0 1 2]\n",
" [3 4 5]\n",
" [6 7 8]]\n"
]
}
],
"source": [
"A = np.arange(9).reshape(3, 3)\n",
"print(A)"
]
},
{
"cell_type": "code",
"execution_count": 173,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[0 3 6]\n",
" [1 4 7]\n",
" [2 5 8]]\n"
]
}
],
"source": [
"print(A.T)"
]
},
{
"cell_type": "code",
"execution_count": 151,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3, 1)"
]
},
"execution_count": 151,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import numpy as np\n",
"\n",
"x = np.array([[1, 2, 3]]).T\n",
"x.shape"
]
},
{
"cell_type": "code",
"execution_count": 152,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1, 3)"
]
},
"execution_count": 152,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xt = x.T\n",
"xt.shape"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Macierz kolumnowa** w NumPy.\n",
"$$X =\n",
" \\begin{pmatrix}\n",
" 3 \\\\\n",
" 4 \\\\\n",
" 5 \\\\\n",
" 6 \n",
" \\end{pmatrix}$$"
]
},
{
"cell_type": "code",
"execution_count": 153,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[3],\n",
" [4],\n",
" [5],\n",
" [6]])"
]
},
"execution_count": 153,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.array([[3,4,5,6]]).T\n",
"x"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Macierz wierszowa** w NumPy.\n",
"$$ X =\n",
" \\begin{pmatrix}\n",
" 3 & 4 & 5 & 6\n",
" \\end{pmatrix}$$"
]
},
{
"cell_type": "code",
"execution_count": 154,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[3, 4, 5, 6]])"
]
},
"execution_count": 154,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = np.array([[3,4,5,6]])\n",
"x"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Oprócz obiektów typu `array` istnieje wyspecjalizowany obiekt `matrix`, dla którego operacje `*` (mnożenie) oraz `**-1` (odwracanie) są określone w sposób właściwy dla macierzy (w przeciwieństwie do operacji elementowych dla obiektów `array`)."
]
},
{
"cell_type": "code",
"execution_count": 158,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1 2 3]\n",
" [4 5 6]\n",
" [7 8 9]]\n"
]
}
],
"source": [
"x = np.array([1,2,3,4,5,6,7,8,9]).reshape(3,3)\n",
"print(x)"
]
},
{
"cell_type": "code",
"execution_count": 159,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[4 6 3]\n",
" [8 7 1]\n",
" [3 0 3]]\n"
]
}
],
"source": [
"y = np.array([4,6,3,8,7,1,3,0,3]).reshape(3,3)\n",
"print(y)"
]
},
{
"cell_type": "code",
"execution_count": 160,
"metadata": {},
"outputs": [],
"source": [
"X = np.matrix(x)\n",
"Y = np.matrix(y)"
]
},
{
"cell_type": "code",
"execution_count": 161,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 4 12 9]\n",
" [32 35 6]\n",
" [21 0 27]]\n"
]
}
],
"source": [
"print(x * y) # Tablice np.array mnożone są element po elemencie"
]
},
{
"cell_type": "code",
"execution_count": 162,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 29 20 14]\n",
" [ 74 59 35]\n",
" [119 98 56]]\n"
]
}
],
"source": [
"print(X * Y) # Macierze np.matrix mnożone są macierzowo"
]
},
{
"cell_type": "code",
"execution_count": 164,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 29 20 14]\n",
" [ 74 59 35]\n",
" [119 98 56]]\n"
]
}
],
"source": [
"print(np.matmul(x, y))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Wyznacznik macierzy**"
]
},
{
"cell_type": "code",
"execution_count": 165,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"33.000000000000014"
]
},
"execution_count": 165,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.array([[3,-9],[2,5]])\n",
"np.linalg.det(a)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Macierz odwrotna**"
]
},
{
"cell_type": "code",
"execution_count": 166,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[-4, -2],\n",
" [ 5, 5]])"
]
},
"execution_count": 166,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = np.array([[-4,-2],[5,5]])\n",
"A"
]
},
{
"cell_type": "code",
"execution_count": 167,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[-0.5, -0.2],\n",
" [ 0.5, 0.4]])"
]
},
"execution_count": 167,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"invA = np.linalg.inv(A)\n",
"invA"
]
},
{
"cell_type": "code",
"execution_count": 168,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1., 0.],\n",
" [0., 1.]])"
]
},
"execution_count": 168,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.round(np.dot(A, invA))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(ponieważ $AA^{-1} = A^{-1}A = I$)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Wartości i wektory własne**"
]
},
{
"cell_type": "code",
"execution_count": 169,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1, 0, 0],\n",
" [0, 2, 0],\n",
" [0, 0, 3]])"
]
},
"execution_count": 169,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = np.diag((1, 2, 3))\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": 170,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1. 2. 3.]\n",
"[[1. 0. 0.]\n",
" [0. 1. 0.]\n",
" [0. 0. 1.]]\n"
]
}
],
"source": [
"w, v = np.linalg.eig(a)\n",
"print(w) # wartości własne\n",
"print(v) # wektory własne"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Zadania"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zadanie 1.1 (1 pkt)\n",
"\n",
"Dla danej listy `input_list` zawierającej liczby utwórz nową listę `output_list`, która będzie zawierała kwadraty liczb dodatnich z `input_list`. Użyj _list comprehension_!"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Przykładowe dane\n",
"\n",
"input_list = [34.6, -203.4, 44.9, 68.3, -12.2, 44.6, 12.7]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zadanie 1.2 (1 pkt)\n",
"\n",
"Za pomocą jednowierszowego polecenia utwórz następującą macierz jako obiekt typu `array`:\n",
"$$A = \\begin{pmatrix}\n",
"1 & 2 & \\cdots & 10 \\\\\n",
"11 & 12 & \\cdots & 20 \\\\\n",
"\\vdots & \\ddots & \\ddots & \\vdots \\\\\n",
"41 & 42 & \\cdots & 50 \n",
"\\end{pmatrix}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zadanie 1.3 (1 pkt)\n",
"\n",
"Dla macierzy $A$ z zadania 1.2:\n",
" * określ liczbę elementów, kolumn i wierszy,\n",
" * stwórz wektory średnich po wierszach oraz po kolumnach,\n",
" * wypisz jej trzecią kolumnę,\n",
" * wypisz jej czwarty wiersz.\n",
" \n",
"Użyj odpowiednich metod obiektu `array`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zadanie 1.4 (1 pkt)\n",
"\n",
"Utwórz macierze\n",
"$$ A = \\begin{pmatrix}\n",
"0 & 4 & -2 \\\\\n",
"-4 & -3 & 0\n",
"\\end{pmatrix} $$\n",
"$$ B = \\begin{pmatrix}\n",
"0 & 1 \\\\\n",
"1 & -1 \\\\\n",
"2 & 3\n",
"\\end{pmatrix} $$\n",
"oraz wektor\n",
"$$ x = \\begin{pmatrix}\n",
"2 \\\\\n",
"1 \\\\\n",
"0\n",
"\\end{pmatrix} $$\n",
"\n",
"Oblicz:\n",
" * iloczyn macierzy $A$ z wektorem $x$ \n",
" * iloczyn macierzy $A \\cdot B$\n",
" * wyznacznik $\\det(A \\cdot B)$\n",
" * wynik działania $(A \\cdot B)^\\top - B^\\top \\cdot A^\\top$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zadanie 1.5 (1 pkt)\n",
"\n",
"Czym różni się operacja `A**-1` dla obiektów typu `array` i `matrix`? Pokaż na przykładzie."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Zadanie 1.6 (1 pkt)\n",
"\n",
"Dla macierzy $X = \\left[\n",
" \\begin{array}{rrr}\n",
" 1 & 2 & 3\\\\\n",
" 1 & 3 & 6 \\\\\n",
" \\end{array}\n",
" \\right]$ oraz wektora $y = \\left[\n",
" \\begin{array}{r}\n",
" 5 \\\\\n",
" 6 \\\\\n",
" \\end{array}\n",
" \\right]$ oblicz wynikowy wektor: \n",
"$$ \\theta = (X^\\top \\, X)^{-1} \\, X^\\top \\, y \\, . $$\n",
"Wykonaj te same obliczenia raz na obiektach typu `array`, a raz na obiektach typu `matrix`.\n",
"W przypadku obiektów typu `matrix` zastosuj możliwie krótki zapis. "
]
}
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