python_lecture5_if_at_first.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b6771d72",
   "metadata": {},
   "source": [
    "# Python tutorial 5\n",
    "\n",
    "Let's spend another tutorial reviewing what we have learned."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ccf0f38",
   "metadata": {},
   "source": [
    "## NumPy\n",
    "\n",
    "NumPy (Numerical Python) is the most widely-used Python library. It isn't part of the default distribution, but can easily be installed. The benefit is the speed. NumPy is much faster than using ordinary lists (orders of magnitude faster). It can also handle matricies of all sizes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "7ae67355",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "fb55f61b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 3  5  7  9 11 13 15 17 19]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(3,20,2)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "7533b325",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 7  9 11 13 15 17 19 21 23]\n"
     ]
    }
   ],
   "source": [
    "a += 4\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "6c4aec65",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[14 18 22 26 30 34 38 42 46]\n"
     ]
    }
   ],
   "source": [
    "a *= 2\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8ccbfe65",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 7.  9. 11. 13. 15. 17. 19. 21. 23.]\n"
     ]
    }
   ],
   "source": [
    "a = a / 2\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a206ccb7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([11., 13., 15., 17., 19.])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[2:7]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "faf1deec",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 7.  9. 11. 13. 15. 17. 19. 21. 23.]\n",
      "[28. 36. 44. 52. 60. 68. 76. 84. 92.]\n",
      "[ 35.  45.  55.  65.  75.  85.  95. 105. 115.]\n"
     ]
    }
   ],
   "source": [
    "b = a * 4\n",
    "print(a)\n",
    "print(b)\n",
    "print(a + b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "549de95e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "eddcc83a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.163977794943222"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "bdef2de3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "26.666666666666668"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "2d34dfc2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "23.0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e17b9f1e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "135.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "d93453fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21082276215.0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.prod()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "22c48dc5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.min()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "0c224568",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.64575131, 3.        , 3.31662479, 3.60555128, 3.87298335,\n",
       "       4.12310563, 4.35889894, 4.58257569, 4.79583152])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "bce8b41d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  1.,  8., 27., 16.,  5.])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1 = np.arange(6)\n",
    "x2 = [1.0, 2.0, 3.0, 3.0, 2.0, 1.0]\n",
    "np.power(x1, x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "308d617e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[16 17 18 19]\n"
     ]
    }
   ],
   "source": [
    "c = np.array(range(20))\n",
    "print(c[c > 15])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b93dff40",
   "metadata": {},
   "source": [
    "## Special numbers\n",
    "\n",
    "$ +\\infty, \\pi, -\\infty, e, NaN $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "c5697bd3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.2                nan  4.          5.         -6.345              nan\n",
      "         inf  3.14159265  2.71828183        -inf]\n"
     ]
    }
   ],
   "source": [
    "d = np.array([1.2, np.nan, 4, 5, -6.345, np.nan, np.inf, np.pi, np.e, np.NINF])\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f1c3a615",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False,  True, False, False, False,  True, False, False, False,\n",
       "       False])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isnan(d)   # Is not a number?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "da0fd514",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False, False, False, False, False, False,  True, False, False,\n",
       "        True])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isinf(d)   # Is infinity?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "3040c479",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True, False,  True,  True,  True, False, False,  True,  True,\n",
       "       False])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isfinite(d)  # Is finite?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "a5f54a33",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1920929e-07"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.finfo(np.float32).eps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "85bb9b22",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.220446049250313e-16"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.finfo(np.float64).eps"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74b02c5a",
   "metadata": {},
   "source": [
    "## Reshape arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "38027d63",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2  3  4  5  6  7  8  9]\n",
      " [10 11 12 13 14 15 16 17 18]]\n",
      "(2, 9)\n",
      "[ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18]\n",
      "(18,)\n"
     ]
    }
   ],
   "source": [
    "arr2 = np.array([[1,2,3,4,5,6,7,8,9],\n",
    "        [10,11,12,13,14,15,16,17,18]])\n",
    "\n",
    "#reshape\n",
    "print(arr2)\n",
    "print(arr2.shape)\n",
    "print(np.reshape(arr2,-1))\n",
    "print(np.reshape(arr2,-1).shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "4fc60eaa",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "False\n"
     ]
    }
   ],
   "source": [
    "a = np.array([0.16,0.26,0.365])\n",
    "b = np.array([0.15,0.25,0.36])\n",
    "tolerance1 = 0.1 \n",
    "tolerance2 = 0.05\n",
    "print(np.allclose(a,b,tolerance1))\n",
    "print(np.allclose(a,b,tolerance2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "b1a7cfb8",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[10 13 11 15]\n",
      " [14 13 15  9]\n",
      " [13  5 13  4]]\n"
     ]
    }
   ],
   "source": [
    "# 2D array\n",
    "t = np.random.randint(1, 16, (3, 4), dtype=int)\n",
    "print(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "4e8b6acd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 3  8  4 11]\n",
      " [13 13 14  3]\n",
      " [ 7  7  7  8]]\n"
     ]
    }
   ],
   "source": [
    "# 2D array\n",
    "t = np.random.randint(1, 16, (3, 4), dtype=int)\n",
    "print(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "1f4324a6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['-1' '-1' 'C' 'C' '3.5' 'B' '2' '3.5' 'B' '2']\n"
     ]
    }
   ],
   "source": [
    "# Randomly chose 10 samples from the array with replacement\n",
    "print(np.random.choice(['A', 'B', '2', '3.5', 'C', '-1'], size=10, replace=True))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "2e200f5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.39603358, 0.67944368, 0.29023467, 0.50361549],\n",
       "       [0.86148933, 0.91616192, 0.94736819, 0.21866725],\n",
       "       [0.49469584, 0.69590266, 0.48244177, 0.67382151],\n",
       "       [0.46753565, 0.26911621, 0.30363526, 0.05223115],\n",
       "       [0.34151187, 0.89512142, 0.61937418, 0.5589788 ]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.rand(5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "d53f889a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['A' 'T' 'T' 'C' 'G' 'G' 'T' 'A' 'C' 'T' 'C' 'T' 'C' 'T' 'G' 'A' 'T' 'T'\n",
      " 'C' 'A' 'T' 'G']\n"
     ]
    }
   ],
   "source": [
    "def my_split(word):\n",
    "    return [char for char in word]\n",
    "\n",
    "dna = np.array(my_split('ATTCGGTACTCTCTGATTCATG'))\n",
    "print(dna)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "e3d2a6e7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['T' 'G' 'T' 'A' 'T' 'T' 'G' 'T' 'T' 'A' 'C' 'A' 'T' 'C' 'T' 'T' 'C' 'A'\n",
      " 'C' 'G' 'C' 'G']\n"
     ]
    }
   ],
   "source": [
    "np.random.shuffle(dna)\n",
    "print(dna)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "1cac7024",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['T' 'G' 'T' 'A' 'T' 'T' 'G' 'T' 'T' 'A' 'C' 'A' 'T' 'C' 'T' 'T' 'C' 'A'\n",
      " 'C' 'G' 'C' 'G']\n"
     ]
    }
   ],
   "source": [
    "np.random.permutation(dna)\n",
    "print(dna)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "da7b7348",
   "metadata": {},
   "outputs": [],
   "source": [
    "poiss = np.random.poisson(5, 100000) # Poisson distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "705c8c8f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt   # We can use matplotlib to plot the PDFs\n",
    "\n",
    "count, bins, ignored = plt.hist(poiss, bins=15, density=True) # Plot a histogram\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "c254c327",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "uni = np.random.uniform(low=-10.0, high=10.0, size=1000) # Uniform distribution\n",
    "\n",
    "count, bins, ignored = plt.hist(uni, bins=5, density=True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "784944e7",
   "metadata": {},
   "source": [
    "## Random can be made deterministic\n",
    "\n",
    "You can ensure you get the same sequence of random numbers by setting the **seed** first in your code. It will still be a random sequence, but whenever you restart the program it will yield the same random sequence.\n",
    "\n",
    "If you don't specify the seed, then the system chooses a random seed (e.g. the system clock) to give the appearance of a random sequence every time you re-run the program."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "49ebeab7",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(seed=816)  # Put me BEFORE any random functions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "624d4568",
   "metadata": {},
   "source": [
    "## Cool fact\n",
    "\n",
    "Numbers are represented in binary format on a computer (rather than base 10). This works great for whole numbers. However, fractions have a problem. Some fractions can be represented as a power of 2. For example, $ 0.25 = 2^{-2}$ and $0.125 = 2^{-3}$. However, ${0.1, 0.2, 0.3, 0.4}$ can not be represented exactly as a power of 2. Their binary representations are very, very close, but not exact. This is why you sometimes see strange results when adding a number many, many times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "8e15e414",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.1 is really 0.1000000000000000055511151231257827021181583404541015625 \n",
      " on the computer because 1/10 cannot be precisely represented as a power of 2.\n",
      " i.e. There is no integer power of 2 that equals 10.\n",
      "\n",
      "\n",
      "Let's add 0.1 to itself 1000000 times and then divide by 1000000\n",
      "This average should equal 0.100, but instead it is 0.100000000001332883803328854810\n"
     ]
    }
   ],
   "source": [
    "print('0.1 is really 0.1000000000000000055511151231257827021181583404541015625 \\n'\n",
    "      ' on the computer because 1/10 cannot be precisely represented as a power of 2.\\n'\n",
    "      ' i.e. There is no integer power of 2 that equals 10.')\n",
    "\n",
    "point_one = 0.1\n",
    "s = 0.0\n",
    "\n",
    "num_times = 1000000\n",
    "\n",
    "print(f\"\\n\\nLet's add {point_one} to itself {num_times} times and then divide by {num_times}\")\n",
    "for idx in range(num_times):\n",
    "    s += point_one\n",
    "    \n",
    "print(f'This average should equal {point_one:.3f}, but instead it is {(s / num_times):.30f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b0fd8a57",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.6"
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 },
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}