4-2_Numpy_Part2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Lecture 4-2\n",
    "\n",
    "# More NumPy\n",
    "\n",
    "## Week 4 Wednesday\n",
    "\n",
    "## Miles Chen, PhD\n",
    "\n",
    "Based on Python Data Science Handbook by Jake VanderPlas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Concatenating Arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3]\n",
      "[100 101 102 103]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(4)\n",
    "y = np.arange(100, 104)\n",
    "print(x)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0,   1,   2,   3, 100, 101, 102, 103])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.concatenate([x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "`np.concatenate` has an argument for axis. The axes are 0-indexed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0,   1,   2,   3, 100, 101, 102, 103])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.concatenate([x,y], axis = 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Let's try to concatenate in the other direction. We specify axis = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [
    {
     "ename": "AxisError",
     "evalue": "axis 1 is out of bounds for array of dimension 1",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mAxisError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[1;32mIn[5], line 1\u001b[0m\n\u001b[1;32m----> 1\u001b[0m np\u001b[38;5;241m.\u001b[39mconcatenate([x,y], axis \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m1\u001b[39m)\n",
      "\u001b[1;31mAxisError\u001b[0m: axis 1 is out of bounds for array of dimension 1"
     ]
    }
   ],
   "source": [
    "np.concatenate([x,y], axis = 1) # throws an error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4,)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.shape # you can't use axis with index 1, because axis index 1 does not exist"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0,   1,   2,   3],\n",
       "       [100, 101, 102, 103]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.vstack([x,y])   # vstack will vertically stack unidimensional arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 2, 3]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.reshape(1,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[100, 101, 102, 103]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.reshape(1,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0,   1,   2,   3],\n",
       "       [100, 101, 102, 103]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.concatenate([x.reshape(1,4), y.reshape(1,4)], axis = 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "note that when I concatenate along axis 0 for a 2-dimensional array, it concatenates by rows. In a 2D array, index 0 is for rows, and index 1 is for columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0,   1,   2,   3, 100, 101, 102, 103]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.concatenate([x.reshape(1,4), y.reshape(1,4)], axis = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 2]\n",
      " [3 4 5]]\n",
      "[[100 101 102]\n",
      " [103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "xm = np.arange(6).reshape((2,3))\n",
    "ym = np.arange(100,106,1).reshape((2,3))\n",
    "print(xm)\n",
    "print(ym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xm.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ym.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  0   1   2]\n",
      " [  3   4   5]\n",
      " [100 101 102]\n",
      " [103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "print(np.concatenate([xm,ym])) # default behavior concatenates on axis 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  0   1   2]\n",
      " [  3   4   5]\n",
      " [100 101 102]\n",
      " [103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "print(np.concatenate([xm,ym], axis = 0))  \n",
    "# axes are reported as rows, then columns.\n",
    "# concatenating along axis 0 will concatenate along rows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  0   1   2 100 101 102]\n",
      " [  3   4   5 103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "print(np.concatenate([xm,ym], axis = 1))\n",
    "# concatenating along axis 1 will concatenate along columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0,   1,   2],\n",
       "       [  3,   4,   5],\n",
       "       [100, 101, 102],\n",
       "       [103, 104, 105]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.vstack([xm, ym])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0,   1,   2, 100, 101, 102],\n",
       "       [  3,   4,   5, 103, 104, 105]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.hstack([xm, ym])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "You can always use vstack and hstack for 2D arrays."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Math Operators with numpy arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3]\n",
      "[100 101 102 103]\n"
     ]
    }
   ],
   "source": [
    "print(x)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 6, 7, 8])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([100, 102, 104, 106])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + y  # elementwise addition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0, 101, 204, 309])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x * y "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "614"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(x * y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "614"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(x,y)   # 0 * 100 + 1 * 101 + 2 * 102 + 3 * 103"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "614"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x @ y # matrix multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 2]\n",
      " [3 4 5]]\n",
      "[[100 101 102]\n",
      " [103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "print(xm)\n",
    "print(ym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5,  6,  7],\n",
       "       [ 8,  9, 10]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xm + 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[100, 102, 104],\n",
       "       [106, 108, 110]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xm + ym  # elementwise addition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 2]\n",
      " [3 4 5]]\n",
      "[[100 101 102]\n",
      " [103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "print(xm)\n",
    "print(ym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0, 101, 204],\n",
       "       [309, 416, 525]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xm * ym # element-wise multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0, 101, 204],\n",
       "       [309, 416, 525]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.multiply(xm, ym) # element-wise multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 2]\n",
      " [3 4 5]]\n",
      "[[100 101 102]\n",
      " [103 104 105]]\n"
     ]
    }
   ],
   "source": [
    "print(xm)\n",
    "print(ym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 305,  314],\n",
       "       [1214, 1250]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(xm, ym.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 305,  314],\n",
       "       [1214, 1250]])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xm.dot(ym.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 305,  314],\n",
       "       [1214, 1250]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xm @ ym.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Basic Math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(4)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[4 5 6 7]\n"
     ]
    }
   ],
   "source": [
    "print(x + 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-5 -4 -3 -2]\n"
     ]
    }
   ],
   "source": [
    "print(x - 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 2 4 6]\n"
     ]
    }
   ],
   "source": [
    "print(x * 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.  0.5 1.  1.5]\n"
     ]
    }
   ],
   "source": [
    "print(x / 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0 -1 -2 -3]\n"
     ]
    }
   ],
   "source": [
    "print(-x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 4 9]\n"
     ]
    }
   ],
   "source": [
    "print(x ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 0 1]\n"
     ]
    }
   ],
   "source": [
    "print(x % 2) # modulo division"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3]\n"
     ]
    }
   ],
   "source": [
    "print(abs(x)) # abs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Trig functions\n",
    "note that the functions are preceeded by np."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.         0.78539816 1.57079633 2.35619449 3.14159265]\n"
     ]
    }
   ],
   "source": [
    "theta = np.linspace(0, np.pi, 5)\n",
    "print(theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.00000000e+00 7.07106781e-01 1.00000000e+00 7.07106781e-01\n",
      " 1.22464680e-16]\n"
     ]
    }
   ],
   "source": [
    "print(np.sin(theta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.00000000e+00  7.07106781e-01  6.12323400e-17 -7.07106781e-01\n",
      " -1.00000000e+00]\n"
     ]
    }
   ],
   "source": [
    "print(np.cos(theta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.00000000e+00  1.00000000e+00  1.63312394e+16 -1.00000000e+00\n",
      " -1.22464680e-16]\n"
     ]
    }
   ],
   "source": [
    "print(np.tan(theta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Log and Exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.         2.30258509 4.60517019]\n",
      "[0. 1. 2.]\n"
     ]
    }
   ],
   "source": [
    "x = np.array([1, 10, 100])\n",
    "print(np.log(x))   # natural log\n",
    "print(np.log10(x)) # common log"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.         2.71828183 7.3890561 ]\n"
     ]
    }
   ],
   "source": [
    "y = np.arange(3)\n",
    "print(np.exp(y))  # e^y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1. 2. 4.]\n"
     ]
    }
   ],
   "source": [
    "print(np.exp2(y))  # 2^y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 3 9]\n"
     ]
    }
   ],
   "source": [
    "print(np.power(3, y)) # power ^ y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Aggregates\n",
    "\n",
    "you can use `sum()`\n",
    "\n",
    "or `np.sum()`\n",
    "\n",
    "`np.sum()` is faster than sum, but doesn't always behave the same way"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23\n",
      " 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47\n",
      " 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71\n",
      " 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95\n",
      " 96 97 98 99]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(100)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4950\n"
     ]
    }
   ],
   "source": [
    "print(sum(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4950\n"
     ]
    }
   ],
   "source": [
    "print(np.sum(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.47 ms ± 78.7 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
      "21.8 µs ± 1.51 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
     ]
    }
   ],
   "source": [
    "big_array = np.random.rand(10000)\n",
    "%timeit sum(big_array)\n",
    "%timeit np.sum(big_array)  # the np version is much faster"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## min and max"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0001245296552800923\n",
      "0.9997587507619023\n"
     ]
    }
   ],
   "source": [
    "print(min(big_array))\n",
    "print(max(big_array))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0001245296552800923\n",
      "0.9997587507619023\n"
     ]
    }
   ],
   "source": [
    "print(np.min(big_array))\n",
    "print(np.max(big_array))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.57 ms ± 20.3 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)\n",
      "17.3 µs ± 418 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit min(big_array)\n",
    "%timeit np.min(big_array)  # the np version is much faster"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## summaries for matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2  3  4 10]\n",
      " [ 1  6  0  7]\n",
      " [11  9  8  5]]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(1)\n",
    "# M = np.random.random((3, 4))\n",
    "M = np.arange(12)\n",
    "np.random.shuffle(M)\n",
    "M = np.reshape(M, (3,4))\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([14, 18, 12, 22])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(M) # regular sum function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "66"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(M) # np.sum function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2  3  4 10]\n",
      " [ 1  6  0  7]\n",
      " [11  9  8  5]]\n"
     ]
    }
   ],
   "source": [
    "print(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([14, 18, 12, 22])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(M, axis = 0)  # np.sum function with axis specified\n",
    "# matrices have two dimensions\n",
    "# 0 is rows, 1 is columns\n",
    "# np.sum axis = 0, will sum over rows, so you end up getting column totals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([19, 14, 33])"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(M, axis = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 3, 0, 5])"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.min(M, axis = 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2  3  4 10]\n",
      " [ 1  6  0  7]\n",
      " [11  9  8  5]]\n"
     ]
    }
   ],
   "source": [
    "print(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.452052529534663"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.49691252, 2.44948974, 3.26598632, 2.05480467])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(M, axis = 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.75, 3.5 , 8.25])"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(M, axis = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Summaries for higher dimensional arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[[1. 1. 1. 1.]\n",
      "  [1. 1. 1. 1.]\n",
      "  [1. 1. 1. 1.]]\n",
      "\n",
      " [[1. 1. 1. 1.]\n",
      "  [1. 1. 1. 1.]\n",
      "  [1. 1. 1. 1.]]]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(1)\n",
    "A = np.ones(24)\n",
    "np.random.shuffle(A)\n",
    "A = np.reshape(A, (2, 3, 4)) # two sheets, 3 rows, 4 columns\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2., 2., 2., 2.],\n",
       "       [2., 2., 2., 2.],\n",
       "       [2., 2., 2., 2.]])"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(A, axis = 0) # sum across \"sheets\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3., 3., 3., 3.],\n",
       "       [3., 3., 3., 3.]])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(A, axis = 1) # sum across rows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4., 4., 4.],\n",
       "       [4., 4., 4.]])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(A, axis = 2) # sum across columns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## dealing with nan\n",
    "nan is the float value for something that is not a number. We often use it in the place of a missing value.\n",
    "nan only exists in float type."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nan\n",
      "<class 'float'>\n"
     ]
    }
   ],
   "source": [
    "x = float(\"nan\")  # direct creation of nan\n",
    "print(x)\n",
    "print(type(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nan\n",
      "nan\n"
     ]
    }
   ],
   "source": [
    "y = float(\"inf\")  # y is the float representation of infinity\n",
    "print(y / y)  # these calculations will yield a nan result\n",
    "print(y - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum([x, 2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.nansum([x, 2])   # in R you have the option na.rm = TRUE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The following table provides a list of useful aggregation functions available in NumPy:\n",
    "\n",
    "|Function Name      |   NaN-safe Version  | Description                                   |\n",
    "|-------------------|---------------------|-----------------------------------------------|\n",
    "| ``np.sum``        | ``np.nansum``       | Compute sum of elements                       |\n",
    "| ``np.prod``       | ``np.nanprod``      | Compute product of elements                   |\n",
    "| ``np.mean``       | ``np.nanmean``      | Compute mean of elements                      |\n",
    "| ``np.std``        | ``np.nanstd``       | Compute standard deviation                    |\n",
    "| ``np.var``        | ``np.nanvar``       | Compute variance                              |\n",
    "| ``np.min``        | ``np.nanmin``       | Find minimum value                            |\n",
    "| ``np.max``        | ``np.nanmax``       | Find maximum value                            |\n",
    "| ``np.argmin``     | ``np.nanargmin``    | Find index of minimum value                   |\n",
    "| ``np.argmax``     | ``np.nanargmax``    | Find index of maximum value                   |\n",
    "| ``np.median``     | ``np.nanmedian``    | Compute median of elements                    |\n",
    "| ``np.percentile`` | ``np.nanpercentile``| Compute rank-based statistics of elements     |\n",
    "| ``np.any``        | N/A                 | Evaluate whether any elements are true        |\n",
    "| ``np.all``        | N/A                 | Evaluate whether all elements are true        |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Broadcasting\n",
    "\n",
    "This is a similar concept to recyling values in R, but only works when the dimensions are compatible"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[5 7 9]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1,2,3])\n",
    "b = np.array([4,5,6])\n",
    "print(a + b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (3,) (2,) ",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[1;32mIn[81], line 2\u001b[0m\n\u001b[0;32m      1\u001b[0m c \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39marray([\u001b[38;5;241m7\u001b[39m,\u001b[38;5;241m8\u001b[39m])\n\u001b[1;32m----> 2\u001b[0m \u001b[38;5;28mprint\u001b[39m(a \u001b[38;5;241m+\u001b[39m c)\n",
      "\u001b[1;31mValueError\u001b[0m: operands could not be broadcast together with shapes (3,) (2,) "
     ]
    }
   ],
   "source": [
    "c = np.array([7,8])\n",
    "print(a + c)  # doesn't work"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3]\n"
     ]
    }
   ],
   "source": [
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 1. 1.]\n",
      " [1. 1. 1.]\n",
      " [1. 1. 1.]]\n"
     ]
    }
   ],
   "source": [
    "e = np.ones([3,3])\n",
    "print(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2. 3. 4.]\n",
      " [2. 3. 4.]\n",
      " [2. 3. 4.]]\n"
     ]
    }
   ],
   "source": [
    "print(e + a)  # the array a gets 'broadcast' across all three rows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1]\n",
      " [2]\n",
      " [3]]\n"
     ]
    }
   ],
   "source": [
    "print(a.reshape([3,1]))  # we reshape a to be a 3x1 array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2. 2. 2.]\n",
      " [3. 3. 3.]\n",
      " [4. 4. 4.]]\n"
     ]
    }
   ],
   "source": [
    "print(e + a.reshape([3,1])) # the reshaped array is broadcast across columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 3]\n",
      " [4 5 6]]\n"
     ]
    }
   ],
   "source": [
    "d = np.vstack([a,b])  # we stack the arrays a and b vertically\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2 4 6]\n",
      " [5 7 9]]\n"
     ]
    }
   ],
   "source": [
    "print(d + a)  # a is broadcast across row"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[7 8]\n"
     ]
    }
   ],
   "source": [
    "print(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 3]\n",
      " [4 5 6]]\n"
     ]
    }
   ],
   "source": [
    "print(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (2,3) (2,) ",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[1;32mIn[92], line 1\u001b[0m\n\u001b[1;32m----> 1\u001b[0m \u001b[38;5;28mprint\u001b[39m(d \u001b[38;5;241m+\u001b[39m c)\n",
      "\u001b[1;31mValueError\u001b[0m: operands could not be broadcast together with shapes (2,3) (2,) "
     ]
    }
   ],
   "source": [
    "print(d + c)  # c does not have compatible dimensions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 8  9 10]\n",
      " [12 13 14]]\n"
     ]
    }
   ],
   "source": [
    "print(d + c.reshape([2,1]))  # after we reshape c to be a column, we can broadcast it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "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",
      "[ 0  1  2  3  4  5  6  7  8  9 10]\n"
     ]
    }
   ],
   "source": [
    "e = np.arange(10).reshape((10, 1))\n",
    "f = np.arange(11)\n",
    "print(e)\n",
    "print(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0  1  2  3  4  5  6  7  8  9 10]\n",
      " [ 1  2  3  4  5  6  7  8  9 10 11]\n",
      " [ 2  3  4  5  6  7  8  9 10 11 12]\n",
      " [ 3  4  5  6  7  8  9 10 11 12 13]\n",
      " [ 4  5  6  7  8  9 10 11 12 13 14]\n",
      " [ 5  6  7  8  9 10 11 12 13 14 15]\n",
      " [ 6  7  8  9 10 11 12 13 14 15 16]\n",
      " [ 7  8  9 10 11 12 13 14 15 16 17]\n",
      " [ 8  9 10 11 12 13 14 15 16 17 18]\n",
      " [ 9 10 11 12 13 14 15 16 17 18 19]]\n"
     ]
    }
   ],
   "source": [
    "print(e + f)  ## e and f are broadcast into compatible matrices and then added"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0  0  0  0  0  0  0  0  0  0  0]\n",
      " [ 0  1  2  3  4  5  6  7  8  9 10]\n",
      " [ 0  2  4  6  8 10 12 14 16 18 20]\n",
      " [ 0  3  6  9 12 15 18 21 24 27 30]\n",
      " [ 0  4  8 12 16 20 24 28 32 36 40]\n",
      " [ 0  5 10 15 20 25 30 35 40 45 50]\n",
      " [ 0  6 12 18 24 30 36 42 48 54 60]\n",
      " [ 0  7 14 21 28 35 42 49 56 63 70]\n",
      " [ 0  8 16 24 32 40 48 56 64 72 80]\n",
      " [ 0  9 18 27 36 45 54 63 72 81 90]]\n"
     ]
    }
   ],
   "source": [
    "print(e * f)  ## e and f are broadcast into compatible matrices and then multiplied element-wise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 3]\n",
      " [4 5 6]]\n"
     ]
    }
   ],
   "source": [
    "print(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2,  3,  4,  5,  6,  7],\n",
       "       [ 3,  4,  5,  6,  7,  8],\n",
       "       [ 4,  5,  6,  7,  8,  9],\n",
       "       [ 5,  6,  7,  8,  9, 10],\n",
       "       [ 6,  7,  8,  9, 10, 11],\n",
       "       [ 7,  8,  9, 10, 11, 12]])"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d.reshape((1,6)) + d.reshape((6,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Boolean Operators in NumPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 4 5]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(6)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ True  True  True False False False]\n"
     ]
    }
   ],
   "source": [
    "print(x < 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False False False  True  True  True]\n"
     ]
    }
   ],
   "source": [
    "print(x >= 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False False False  True False False]\n"
     ]
    }
   ],
   "source": [
    "print(x == 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3 4 5]\n"
     ]
    }
   ],
   "source": [
    "# the results can then be used to subset\n",
    "print(x[x >= 3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(x >= 3) # True = 1, False = 0, so sum counts how many are true"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x >= 3)  # finds the proportion that is True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ True  True  True False  True  True]\n"
     ]
    }
   ],
   "source": [
    "print(~(x == 3)) # use the tilde for negation of boolean values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False False False False False False]\n"
     ]
    }
   ],
   "source": [
    "print(~x == 3) # be careful if you leave off parenthesis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1, -2, -3, -4, -5, -6])"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "~x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Working with matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0  1  2  3]\n",
      " [ 4  5  6  7]\n",
      " [ 8  9 10 11]]\n"
     ]
    }
   ],
   "source": [
    "y = np.arange(12).reshape([3,4])\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[False False False False]\n",
      " [False False  True  True]\n",
      " [ True  True  True  True]]\n"
     ]
    }
   ],
   "source": [
    "print(y >= 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(y >= 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1, 2, 2])"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(y >= 6, axis = 0)  # you can perform sums and other aggregate functions axis-wise on the boolean matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 2, 4])"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(y >= 6, axis = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Bitwise (element-wise) Boolean operators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ True  True False False]\n",
      "[ True False  True False]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([True, True, False, False])\n",
    "b = np.array([True, False, True, False])\n",
    "print(a)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ True False False False]\n"
     ]
    }
   ],
   "source": [
    "print(a & b) # bitwise and"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ True  True  True False]\n"
     ]
    }
   ],
   "source": [
    "print(a | b) # bitwise or"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False  True  True False]\n"
     ]
    }
   ],
   "source": [
    "print(a ^ b) # bitwise xor (exclusive or)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False False  True  True]\n"
     ]
    }
   ],
   "source": [
    "print(~a)  # bitwise not"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.any(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.all(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# fancy indexing\n",
    "Regular lists in python do not support fancy indexing, but NumPy does!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[37 12 72  9 75  5 79 64 16  1]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(1)\n",
    "x = np.random.randint(100, size = 10)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[37 12  5]\n"
     ]
    }
   ],
   "source": [
    "index = [0, 1, 5]\n",
    "print(x[index])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 4 7]\n",
      " [2 3 8]]\n"
     ]
    }
   ],
   "source": [
    "a = [1, 4, 7]\n",
    "b = [2, 3, 8]\n",
    "ind = np.vstack([a,b])\n",
    "print(ind)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[12 75 64]\n",
      " [72  9 16]]\n"
     ]
    }
   ],
   "source": [
    "print(x[ind])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0  1  2  3]\n",
      " [ 4  5  6  7]\n",
      " [ 8  9 10 11]]\n"
     ]
    }
   ],
   "source": [
    "X = np.arange(12).reshape((3, 4))\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2,  5, 11])"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "row = np.array([0, 1, 2])\n",
    "col = np.array([2, 1, 3])\n",
    "X[row, col]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- `np.sort()`\n",
    "- `np.argsort()` gives the indexes of the values to have the proper sorting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 4 1 3 0]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(2)\n",
    "x = np.arange(5)\n",
    "np.random.shuffle(x)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 4]\n"
     ]
    }
   ],
   "source": [
    "x.sort() # sorts x in place\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[5 2 1 4]\n",
      "[2 1 3 0]\n"
     ]
    }
   ],
   "source": [
    "y = np.array([5, 2, 1, 4])\n",
    "print(y)\n",
    "print(y.argsort())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 4, 5])"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = y.argsort()\n",
    "y[d]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sorting along rows or columns\n",
    "\n",
    "A useful feature of NumPy's sorting algorithms is the ability to sort along specific rows or columns of a multidimensional array using the axis argument. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5 8 9 5 0 0]\n",
      " [1 7 6 9 2 4]\n",
      " [5 2 4 2 4 7]\n",
      " [7 9 1 7 0 6]]\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(1)\n",
    "X = np.random.randint(0, 10, (4, 6))\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 1, 2, 0, 0],\n",
       "       [5, 7, 4, 5, 0, 4],\n",
       "       [5, 8, 6, 7, 2, 6],\n",
       "       [7, 9, 9, 9, 4, 7]])"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sort each column of X\n",
    "# np.sort returns a copy of X after sorted. It does not modify X\n",
    "np.sort(X, axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 5, 5, 8, 9],\n",
       "       [1, 2, 4, 6, 7, 9],\n",
       "       [2, 2, 4, 4, 5, 7],\n",
       "       [0, 1, 6, 7, 7, 9]])"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sort each row of X\n",
    "np.sort(X, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 8, 9, 5, 0, 0])"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[0,:] # selecting a row"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5 8 9 5 0 0]\n",
      " [1 7 6 9 2 4]\n",
      " [5 2 4 2 4 7]\n",
      " [7 9 1 7 0 6]]\n"
     ]
    }
   ],
   "source": [
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 1, 0, 3], dtype=int64)"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:,1].argsort()  # the argsort for the column index 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5 2 4 2 4 7]\n",
      " [1 7 6 9 2 4]\n",
      " [5 8 9 5 0 0]\n",
      " [7 9 1 7 0 6]]\n"
     ]
    }
   ],
   "source": [
    "print(X[ X[:,1].argsort() , : ])  # 'subset' X by the argsort to arrange X by the column"
   ]
  }
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