introduction-to-numpy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A Quick Introduction to Numerical Data Manipulation with Python and NumPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "tags": [],
    "ExecuteTime": {
     "end_time": "2024-05-08T09:19:53.054313400Z",
     "start_time": "2024-05-08T09:19:52.996315700Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Last updated: 2024-05-08 12:19:52.992332\n"
     ]
    }
   ],
   "source": [
    "import datetime\n",
    "print(f\"Last updated: {datetime.datetime.now()}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What is NumPy?\n",
    "\n",
    "[NumPy](https://numpy.org/doc/stable/index.html) stands for numerical Python. It's the backbone of all kinds of scientific and numerical computing in Python.\n",
    "\n",
    "And since machine learning is all about turning data into numbers and then figuring out the patterns, NumPy often comes into play.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Why NumPy?\n",
    "\n",
    "You can do numerical calculations using pure Python. In the beginning, you might think Python is fast but once your data gets large, you'll start to notice slow downs.\n",
    "\n",
    "One of the main reasons you use NumPy is because it's fast. Behind the scenes, the code has been optimized to run using C. Which is another programming language, which can do things much faster than Python.\n",
    "\n",
    "The benefit of this being behind the scenes is you don't need to know any C to take advantage of it. You can write your numerical computations in Python using NumPy and get the added speed benefits.\n",
    "\n",
    "If your curious as to what causes this speed benefit, it's a process called vectorization. [Vectorization](https://en.wikipedia.org/wiki/Vectorization) aims to do calculations by avoiding loops as loops can create potential bottlenecks.\n",
    "\n",
    "NumPy achieves vectorization through a process called [broadcasting](https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html#module-numpy.doc.broadcasting)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 0. Importing NumPy\n",
    "\n",
    "To get started using NumPy, the first step is to import it. \n",
    "\n",
    "The most common way (and method you should use) is to import NumPy as the abbreviation `np`.\n",
    "\n",
    "If you see the letters `np` used anywhere in machine learning or data science, it's probably referring to the NumPy library."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:45.060499100Z",
     "start_time": "2024-05-08T07:37:44.751482400Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.23.5\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Check the version\n",
    "print(np.__version__)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. DataTypes and attributes\n",
    "\n",
    "> **Note:** Important to remember the main type in NumPy is `ndarray`, even seemingly different kinds of arrays are still `ndarray`'s. This means an operation you do on one array, will work on another."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:35:45.710373100Z",
     "start_time": "2024-05-08T09:35:45.662352500Z"
    }
   },
   "outputs": [],
   "source": [
    "# 1-dimensonal array, also referred to as a vector\n",
    "a1 = np.array([1, 2, 3])\n",
    "\n",
    "# 2-dimensional array, also referred to as matrix\n",
    "a2 = np.array([[1, 2.0, 3.3],\n",
    "               [4, 5, 6.5]])\n",
    "\n",
    "# 3-dimensional array, also referred to as a matrix\n",
    "a3 = np.array([[[1, 2, 3],\n",
    "                [4, 5, 6],\n",
    "                [7, 8, 9]],\n",
    "                [[10, 11, 12],\n",
    "                 [13, 14, 15],\n",
    "                 [16, 17, 18]]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:45.474570500Z",
     "start_time": "2024-05-08T07:37:45.357482900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "((3,), 1, dtype('int32'), 3, numpy.ndarray)"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1.shape, a1.ndim, a1.dtype, a1.size, type(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:45.601565300Z",
     "start_time": "2024-05-08T07:37:45.554485400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "((2, 3), 2, dtype('float64'), 6, numpy.ndarray)"
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.shape, a2.ndim, a2.dtype, a2.size, type(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:45.763388800Z",
     "start_time": "2024-05-08T07:37:45.720282200Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "((2, 3, 3), 3, dtype('int32'), 18, numpy.ndarray)"
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3.shape, a3.ndim, a3.dtype, a3.size, type(a3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:35:29.564093700Z",
     "start_time": "2024-05-08T09:35:29.520926Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3]\n"
     ]
    }
   ],
   "source": [
    "print(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:36:10.933319600Z",
     "start_time": "2024-05-08T09:36:10.882212Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 229,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:35:49.362756300Z",
     "start_time": "2024-05-08T09:35:49.307657Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[ 1,  2,  3],\n        [ 4,  5,  6],\n        [ 7,  8,  9]],\n\n       [[10, 11, 12],\n        [13, 14, 15],\n        [16, 17, 18]]])"
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Anatomy of an array\n",
    "\n",
    "Key terms:\n",
    "* **Array** - A list of numbers, can be multi-dimensional.\n",
    "* **Scalar** - A single number (e.g. `7`).\n",
    "* **Vector** - A list of numbers with 1-dimension (e.g. `np.array([1, 2, 3])`).\n",
    "* **Matrix** - A (usually) multi-dimensional list of numbers (e.g. `np.array([[1, 2, 3], [4, 5, 6]])`)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pandas DataFrame out of NumPy arrays\n",
    "\n",
    "This is to examplify how NumPy is the backbone of many other libraries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:47.365186500Z",
     "start_time": "2024-05-08T07:37:47.099073400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "   a  b  c\n0  4  7  2\n1  9  8  8\n2  5  2  5\n3  4  8  2\n4  4  4  1",
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>a</th>\n      <th>b</th>\n      <th>c</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>4</td>\n      <td>7</td>\n      <td>2</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>9</td>\n      <td>8</td>\n      <td>8</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>5</td>\n      <td>2</td>\n      <td>5</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>4</td>\n      <td>8</td>\n      <td>2</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>4</td>\n      <td>4</td>\n      <td>1</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "df = pd.DataFrame(np.random.randint(10, size=(5, 3)), \n",
    "                                    columns=['a', 'b', 'c'])\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:47.429072400Z",
     "start_time": "2024-05-08T07:37:47.128073700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:47.492073Z",
     "start_time": "2024-05-08T07:37:47.174071900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "     0    1    2\n0  1.0  2.0  3.3\n1  4.0  5.0  6.5",
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>0</th>\n      <th>1</th>\n      <th>2</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>1.0</td>\n      <td>2.0</td>\n      <td>3.3</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>4.0</td>\n      <td>5.0</td>\n      <td>6.5</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = pd.DataFrame(a2)\n",
    "df2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Creating arrays\n",
    "\n",
    "* `np.array()`\n",
    "* `np.ones()`\n",
    "* `np.zeros()`\n",
    "* `np.random.rand(5, 3)`\n",
    "* `np.random.randint(10, size=5)`\n",
    "* `np.random.seed()` - pseudo random numbers\n",
    "* Searching the documentation example (finding `np.unique()` and using it)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:47.726074700Z",
     "start_time": "2024-05-08T07:37:47.443072800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 2, 3])"
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a simple array\n",
    "simple_array = np.array([1, 2, 3])\n",
    "simple_array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:47.865071700Z",
     "start_time": "2024-05-08T07:37:47.659074500Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(array([1, 2, 3]), dtype('int32'))"
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simple_array = np.array((1, 2, 3))\n",
    "simple_array, simple_array.dtype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:40:15.793788400Z",
     "start_time": "2024-05-08T09:40:15.762788900Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]]\n"
     ]
    }
   ],
   "source": [
    "# Create an array of ones\n",
    "ones = np.ones((5, 4))\n",
    "print(ones)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:48.151133600Z",
     "start_time": "2024-05-08T07:37:47.956073800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "dtype('float64')"
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The default datatype is 'float64'\n",
    "ones.dtype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:48.236875500Z",
     "start_time": "2024-05-08T07:37:48.126072600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1],\n       [1, 1]])"
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You can change the datatype with .astype()\n",
    "ones.astype(int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:48.522794400Z",
     "start_time": "2024-05-08T07:37:48.290794400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[0., 0., 0.],\n        [0., 0., 0.],\n        [0., 0., 0.]],\n\n       [[0., 0., 0.],\n        [0., 0., 0.],\n        [0., 0., 0.]],\n\n       [[0., 0., 0.],\n        [0., 0., 0.],\n        [0., 0., 0.]],\n\n       [[0., 0., 0.],\n        [0., 0., 0.],\n        [0., 0., 0.]],\n\n       [[0., 0., 0.],\n        [0., 0., 0.],\n        [0., 0., 0.]]])"
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create an array of zeros\n",
    "zeros = np.zeros((5, 3, 3))\n",
    "zeros"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:48.734600800Z",
     "start_time": "2024-05-08T07:37:48.508796700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "dtype('float64')"
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeros.dtype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:48.987474200Z",
     "start_time": "2024-05-08T07:37:48.643795100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([0, 2, 4, 6, 8])"
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create an array within a range of values\n",
    "range_array = np.arange(0, 10, 2)\n",
    "range_array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:49.203525800Z",
     "start_time": "2024-05-08T07:37:48.764585900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[7, 1, 4],\n       [2, 5, 4],\n       [4, 3, 5],\n       [3, 3, 8],\n       [5, 6, 3]])"
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Random array\n",
    "random_array = np.random.randint(10, size=(5, 3))\n",
    "random_array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:49.435527600Z",
     "start_time": "2024-05-08T07:37:48.944437300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[0.84510804, 0.51256716, 0.08500339],\n       [0.53344695, 0.31039729, 0.39002973],\n       [0.77423547, 0.41961118, 0.06986826],\n       [0.4415502 , 0.33478429, 0.08170806],\n       [0.0480138 , 0.46452611, 0.56112472]])"
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Random array of floats (between 0 & 1)\n",
    "np.random.random((5, 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:37:49.484436Z",
     "start_time": "2024-05-08T07:37:49.128436400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[0.27859392, 0.91038591, 0.92924765],\n       [0.74227083, 0.58303714, 0.49618406],\n       [0.66653583, 0.3199788 , 0.6296261 ],\n       [0.1849052 , 0.61116963, 0.59616946],\n       [0.82688689, 0.10269818, 0.60247575]])"
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Random 5x3 array of floats (between 0 & 1), similar to above\n",
    "np.random.rand(5, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NumPy uses pseudo-random numbers, which means, the numbers look random but aren't really, they're predetermined.\n",
    "\n",
    "For consistency, you might want to keep the random numbers you generate similar throughout experiments.\n",
    "\n",
    "To do this, you can use [`np.random.seed()`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.random.seed.html).\n",
    "\n",
    "What this does is it tells NumPy, \"Hey, I want you to create random numbers but keep them aligned with the seed.\"\n",
    "\n",
    "Let's see it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:48:50.594196400Z",
     "start_time": "2024-05-08T09:48:50.567180600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[6, 3, 7],\n       [4, 6, 9],\n       [2, 6, 7],\n       [4, 3, 7],\n       [7, 2, 5]])"
     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Set random seed to 0\n",
    "np.random.seed(42)\n",
    "\n",
    "# Make 'random' numbers\n",
    "np.random.randint(10, size=(5, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With `np.random.seed()` set, every time you run the cell above, the same random numbers will be generated.\n",
    "\n",
    "What if `np.random.seed()` wasn't set?\n",
    "\n",
    "Every time you run the cell below, a new set of numbers will appear."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:49:33.396818300Z",
     "start_time": "2024-05-08T09:49:33.362826400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[9, 8, 8],\n       [0, 8, 6],\n       [8, 7, 0],\n       [7, 7, 2],\n       [0, 7, 2]])"
     },
     "execution_count": 247,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Make more random numbers\n",
    "np.random.randint(10, size=(5, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see it in action again, we'll stay consistent and set the random seed to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:40:31.445822600Z",
     "start_time": "2024-05-08T07:40:31.353307400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[5, 0, 3],\n       [3, 7, 9],\n       [3, 5, 2],\n       [4, 7, 6],\n       [8, 8, 1]])"
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Set random seed to same number as above\n",
    "np.random.seed(0)\n",
    "\n",
    "# The same random numbers come out\n",
    "np.random.randint(10, size=(5, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because `np.random.seed()` is set to 0, the random numbers are the same as the cell with `np.random.seed()` set to 0 as well.\n",
    "\n",
    "Setting `np.random.seed()` is not 100% necessary but it's helpful to keep numbers the same throughout your experiments.\n",
    "\n",
    "For example, say you wanted to split your data randomly into training and test sets.\n",
    "\n",
    "Every time you randomly split, you might get different rows in each set.\n",
    "\n",
    "If you shared your work with someone else, they'd get different rows in each set too.\n",
    "\n",
    "Setting `np.random.seed()` ensures there's still randomness, it just makes the randomness repeatable. Hence the 'pseudo-random' numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:41:15.003524400Z",
     "start_time": "2024-05-08T07:41:14.934503Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "   0  1  2\n0  5  0  3\n1  3  7  9\n2  3  5  2\n3  4  7  6\n4  8  8  1",
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>0</th>\n      <th>1</th>\n      <th>2</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>5</td>\n      <td>0</td>\n      <td>3</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>3</td>\n      <td>7</td>\n      <td>9</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>3</td>\n      <td>5</td>\n      <td>2</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>4</td>\n      <td>7</td>\n      <td>6</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>8</td>\n      <td>8</td>\n      <td>1</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "df = pd.DataFrame(np.random.randint(10, size=(5, 3)))\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What unique values are in the array a3?\n",
    "\n",
    "You might want to search some like, \"how to find the unqiue values in a numpy array\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:52:16.036580700Z",
     "start_time": "2024-05-08T09:52:16.010587200Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[9 2 6]\n",
      " [9 8 3]\n",
      " [0 1 0]\n",
      " [4 4 6]\n",
      " [8 8 2]]\n"
     ]
    },
    {
     "data": {
      "text/plain": "array([0, 1, 2, 3, 4, 6, 8, 9])"
     },
     "execution_count": 257,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Your code here\n",
    "arr=np.random.randint(10, size=(5, 3))\n",
    "print(arr)\n",
    "np.unique(arr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Viewing arrays and matrices (indexing)\n",
    "\n",
    "Remember, because arrays and matrices are both `ndarray`'s, they can be viewed in similar ways.\n",
    "\n",
    "Let's check out our 3 arrays again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 2. , 3.3],\n",
       "       [4. , 5. , 6.5]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[ 1,  2,  3],\n",
       "        [ 4,  5,  6],\n",
       "        [ 7,  8,  9]],\n",
       "\n",
       "       [[10, 11, 12],\n",
       "        [13, 14, 15],\n",
       "        [16, 17, 18]]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Array shapes are always listed in the format `(row, column, n, n, n...)` where `n` is optional extra dimensions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 260,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:53:25.634832100Z",
     "start_time": "2024-05-08T09:53:25.583031800Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3]\n"
     ]
    },
    {
     "data": {
      "text/plain": "1"
     },
     "execution_count": 260,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(a1)\n",
    "a1[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:53:41.874680200Z",
     "start_time": "2024-05-08T09:53:41.830690600Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.  2.  3.3]\n"
     ]
    }
   ],
   "source": [
    "print(a2[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [4, 5, 6],\n",
       "       [7, 8, 9]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4. , 5. , 6.5])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get 2nd row (index 1) of a2\n",
    "a2[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[ 1,  2,  3],\n        [ 4,  5,  6],\n        [ 7,  8,  9]],\n\n       [[10, 11, 12],\n        [13, 14, 15],\n        [16, 17, 18]]])"
     },
     "execution_count": 263,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T09:55:26.868528600Z",
     "start_time": "2024-05-08T09:55:26.829527300Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 269,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:57:57.153106900Z",
     "start_time": "2024-05-08T09:57:57.061088700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[ 1,  2],\n        [ 4,  5]],\n\n       [[10, 11],\n        [13, 14]]])"
     },
     "execution_count": 269,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get the first 2 values of the first 2 rows of both arrays\n",
    "a3[:2, :2, :2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This takes a bit of practice, especially when the dimensions get higher. Usually, it takes me a little trial and error of trying to get certain values, viewing the output in the notebook and trying again.\n",
    "\n",
    "NumPy arrays get printed from outside to inside. This means the number at the end of the shape comes first, and the number at the start of the shape comes last."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:53:24.415131Z",
     "start_time": "2024-05-08T07:53:24.373157200Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[[3, 2, 7, 2, 0],\n         [0, 4, 5, 5, 6],\n         [8, 4, 1, 4, 9],\n         [8, 1, 1, 7, 9]],\n\n        [[9, 3, 6, 7, 2],\n         [0, 3, 5, 9, 4],\n         [4, 6, 4, 4, 3],\n         [4, 4, 8, 4, 3]],\n\n        [[7, 5, 5, 0, 1],\n         [5, 9, 3, 0, 5],\n         [0, 1, 2, 4, 2],\n         [0, 3, 2, 0, 7]]],\n\n\n       [[[5, 9, 0, 2, 7],\n         [2, 9, 2, 3, 3],\n         [2, 3, 4, 1, 2],\n         [9, 1, 4, 6, 8]],\n\n        [[2, 3, 0, 0, 6],\n         [0, 6, 3, 3, 8],\n         [8, 8, 2, 3, 2],\n         [0, 8, 8, 3, 8]],\n\n        [[2, 8, 4, 3, 0],\n         [4, 3, 6, 9, 8],\n         [0, 8, 5, 9, 0],\n         [9, 6, 5, 3, 1]]]])"
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a4 = np.random.randint(10, size=(2, 3, 4, 5))\n",
    "a4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:53:24.877971200Z",
     "start_time": "2024-05-08T07:53:24.839071Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(2, 3, 4, 5)"
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a4.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "arr=[[[[6, 7, 7, 8, 1],\n",
    "         [5, 9, 8, 9, 4],\n",
    "         [3, 0, 3, 5, 0],\n",
    "         [2, 3, 8, 1, 3]],\n",
    "\n",
    "        [[3, 3, 7, 0, 1],\n",
    "         [9, 9, 0, 4, 7],\n",
    "         [3, 2, 7, 2, 0],\n",
    "         [0, 4, 5, 5, 6]],\n",
    "\n",
    "        [[8, 4, 1, 4, 9],\n",
    "         [8, 1, 1, 7, 9],\n",
    "         [9, 3, 6, 7, 2],\n",
    "         [0, 3, 5, 9, 4]]],\n",
    "\n",
    "\n",
    "       [[[4, 6, 4, 4, 3],\n",
    "         [4, 4, 8, 4, 3],\n",
    "         [7, 5, 5, 0, 1],\n",
    "         [5, 9, 3, 0, 5]],\n",
    "\n",
    "        [[0, 1, 2, 4, 2],\n",
    "         [0, 3, 2, 0, 7],\n",
    "         [5, 9, 0, 2, 7],\n",
    "         [2, 9, 2, 3, 3]],\n",
    "\n",
    "        [[2, 3, 4, 1, 2],\n",
    "         [9, 1, 4, 6, 8],\n",
    "         [2, 3, 0, 0, 6],\n",
    "         [0, 6, 3, 3, 8]]]]"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 272,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T09:59:11.549646300Z",
     "start_time": "2024-05-08T09:59:11.508558700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[[3, 2, 7, 2],\n         [0, 4, 5, 5],\n         [8, 4, 1, 4],\n         [8, 1, 1, 7]],\n\n        [[9, 3, 6, 7],\n         [0, 3, 5, 9],\n         [4, 6, 4, 4],\n         [4, 4, 8, 4]],\n\n        [[7, 5, 5, 0],\n         [5, 9, 3, 0],\n         [0, 1, 2, 4],\n         [0, 3, 2, 0]]],\n\n\n       [[[5, 9, 0, 2],\n         [2, 9, 2, 3],\n         [2, 3, 4, 1],\n         [9, 1, 4, 6]],\n\n        [[2, 3, 0, 0],\n         [0, 6, 3, 3],\n         [8, 8, 2, 3],\n         [0, 8, 8, 3]],\n\n        [[2, 8, 4, 3],\n         [4, 3, 6, 9],\n         [0, 8, 5, 9],\n         [9, 6, 5, 3]]]])"
     },
     "execution_count": 272,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get only the first 4 numbers of each single vector\n",
    "a4[:, :, :, :4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`a4`'s shape is (2, 3, 4, 5), this means it gets displayed like so:\n",
    "* Inner most array = size 5\n",
    "* Next array = size 4\n",
    "* Next array = size 3\n",
    "* Outer most array = size 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Manipulating and comparing arrays\n",
    "* Arithmetic\n",
    "    * `+`, `-`, `*`, `/`, `//`, `**`, `%`\n",
    "    * `np.exp()`\n",
    "    * `np.log()`\n",
    "    * [Dot product](https://www.mathsisfun.com/algebra/matrix-multiplying.html) - `np.dot()`\n",
    "    * Broadcasting\n",
    "* Aggregation\n",
    "    * `np.sum()` - faster than Python's `.sum()` for NumPy arrays\n",
    "    * `np.mean()`\n",
    "    * `np.std()`\n",
    "    * `np.var()`\n",
    "    * `np.min()`\n",
    "    * `np.max()`\n",
    "    * `np.argmin()` - find index of minimum value\n",
    "    * `np.argmax()` - find index of maximum value\n",
    "    * These work on all `ndarray`'s\n",
    "        * `a4.min(axis=0)` -- you can use axis as well\n",
    "* Reshaping\n",
    "    * `np.reshape()`\n",
    "* Transposing\n",
    "    * `a3.T` \n",
    "* Comparison operators\n",
    "    * `>`\n",
    "    * `<`\n",
    "    * `<=`\n",
    "    * `>=`\n",
    "    * `x != 3`\n",
    "    * `x == 3`\n",
    "    * `np.sum(x > 3)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Arithmetic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:24:52.273198700Z",
     "start_time": "2024-05-08T08:24:52.233120300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1., 1., 1.])"
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ones = np.ones(3)\n",
    "ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2., 3., 4.])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Add two arrays\n",
    "a1 + ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 2.])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Subtract two arrays\n",
    "a1 - ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1., 2., 3.])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Multiply two arrays\n",
    "a1 * ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  4. ,  9.9],\n",
       "       [ 4. , 10. , 19.5]])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Multiply two arrays\n",
    "a1 * a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((3,), (2, 3))"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1.shape, a2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T07:56:42.289223400Z",
     "start_time": "2024-05-08T07:56:42.214221600Z"
    }
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (2,3) (2,3,3) ",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mValueError\u001B[0m                                Traceback (most recent call last)",
      "Cell \u001B[1;32mIn[83], line 2\u001B[0m\n\u001B[0;32m      1\u001B[0m \u001B[38;5;66;03m# This will error as the arrays have a different number of dimensions (2, 3) vs. (2, 3, 3) \u001B[39;00m\n\u001B[1;32m----> 2\u001B[0m \u001B[43ma2\u001B[49m\u001B[43m \u001B[49m\u001B[38;5;241;43m*\u001B[39;49m\u001B[43m \u001B[49m\u001B[43ma3\u001B[49m\n",
      "\u001B[1;31mValueError\u001B[0m: operands could not be broadcast together with shapes (2,3) (2,3,3) "
     ]
    }
   ],
   "source": [
    "# This will error as the arrays have a different number of dimensions (2, 3) vs. (2, 3, 3) \n",
    "a2 * a3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[ 1,  2,  3],\n",
       "        [ 4,  5,  6],\n",
       "        [ 7,  8,  9]],\n",
       "\n",
       "       [[10, 11, 12],\n",
       "        [13, 14, 15],\n",
       "        [16, 17, 18]]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Broadcasting\n",
    "\n",
    "- What is broadcasting?\n",
    "    - Broadcasting is a feature of NumPy which performs an operation across multiple dimensions of data without replicating the data. This saves time and space. For example, if you have a 3x3 array (A) and want to add a 1x3 array (B), NumPy will add the row of (B) to every row of (A).\n",
    "\n",
    "- Rules of Broadcasting\n",
    "    1. If the two arrays differ in their number of dimensions, the shape of the one with fewer dimensions is padded with ones on its leading (left) side.\n",
    "    2. If the shape of the two arrays does not match in any dimension, the array with shape equal to 1 in that dimension is stretched to match the other shape.\n",
    "    3. If in any dimension the sizes disagree and neither is equal to 1, an error is raised.\n",
    "    \n",
    "<img src=\"https://jakevdp.github.io/PythonDataScienceHandbook/figures/02.05-broadcasting.png\">\n",
    "\n",
    "**The broadcasting rule:**\n",
    "In order to broadcast, the size of the trailing axes for both arrays in an operation must be either the same size or one of them must be one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:04.182112300Z",
     "start_time": "2024-05-08T08:34:04.139132800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 2, 3])"
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:05.814086300Z",
     "start_time": "2024-05-08T08:34:05.734087300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(3,)"
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:07.423958300Z",
     "start_time": "2024-05-08T08:34:07.384944Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(2, 3)"
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:27.450378200Z",
     "start_time": "2024-05-08T08:34:27.405776800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:28.028479700Z",
     "start_time": "2024-05-08T08:34:27.982378900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[2. , 4. , 6.3],\n       [5. , 7. , 9.5]])"
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 + a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:29.579147700Z",
     "start_time": "2024-05-08T08:34:29.518147600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[3. , 4. , 5.3],\n       [6. , 7. , 8.5]])"
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2 + 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:34.820253900Z",
     "start_time": "2024-05-08T08:34:34.751966300Z"
    }
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (2,3) (2,3,3) ",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mValueError\u001B[0m                                Traceback (most recent call last)",
      "Cell \u001B[1;32mIn[142], line 2\u001B[0m\n\u001B[0;32m      1\u001B[0m \u001B[38;5;66;03m# Raises an error because there's a shape mismatch (2, 3) vs. (2, 3, 3)\u001B[39;00m\n\u001B[1;32m----> 2\u001B[0m \u001B[43ma2\u001B[49m\u001B[43m \u001B[49m\u001B[38;5;241;43m+\u001B[39;49m\u001B[43m \u001B[49m\u001B[43ma3\u001B[49m\n",
      "\u001B[1;31mValueError\u001B[0m: operands could not be broadcast together with shapes (2,3) (2,3,3) "
     ]
    }
   ],
   "source": [
    "# Raises an error because there's a shape mismatch (2, 3) vs. (2, 3, 3)\n",
    "a2 + a3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:36.275906200Z",
     "start_time": "2024-05-08T08:34:36.198756900Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1. 1. 1.]\n"
     ]
    },
    {
     "data": {
      "text/plain": "array([1., 2., 3.])"
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Divide two arrays\n",
    "print(ones)\n",
    "a1 / ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:37.488929900Z",
     "start_time": "2024-05-08T08:34:37.434155700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1., 1., 1.],\n       [4., 2., 2.]])"
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Divide using floor division\n",
    "a2 // a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:37.790582900Z",
     "start_time": "2024-05-08T08:34:37.733582300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 4, 9])"
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Take an array to a power\n",
    "a1 ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:38.075220100Z",
     "start_time": "2024-05-08T08:34:38.020218400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 4, 9])"
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You can also use np.square()\n",
    "np.square(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:38.319234300Z",
     "start_time": "2024-05-08T08:34:38.232220500Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 0, 1], dtype=int32)"
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Modulus divide (what's the remainder)\n",
    "a1 % 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also find the log or exponential of an array using `np.log()` and `np.exp()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:38.621220800Z",
     "start_time": "2024-05-08T08:34:38.559218600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([0.        , 0.69314718, 1.09861229])"
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the log of an array\n",
    "np.log(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:38.872220300Z",
     "start_time": "2024-05-08T08:34:38.730222900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([ 2.71828183,  7.3890561 , 20.08553692])"
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the exponential of an array\n",
    "np.exp(a1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aggregation\n",
    "\n",
    "Aggregation - bringing things together, doing a similar thing on a number of things."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:39.260222100Z",
     "start_time": "2024-05-08T08:34:39.116218900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "6"
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:39.430258400Z",
     "start_time": "2024-05-08T08:34:39.331219Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "6"
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(a1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Tip:** Use NumPy's `np.sum()` on NumPy arrays and Python's `sum()` on Python `list`s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:40.839430500Z",
     "start_time": "2024-05-08T08:34:40.791429Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(100000, numpy.ndarray)"
     },
     "execution_count": 152,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "massive_array = np.random.random(100000)\n",
    "massive_array.size, type(massive_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:50.311551100Z",
     "start_time": "2024-05-08T08:34:41.618295900Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.23 ms ± 2.04 ms per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
      "31.8 µs ± 3.92 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit sum(massive_array) # Python sum()\n",
    "%timeit np.sum(massive_array) # NumPy np.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice `np.sum()` is faster on the Numpy array (`numpy.ndarray`) than Python's `sum()`.\n",
    "\n",
    "Now let's try it out on a Python list."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:50.389571900Z",
     "start_time": "2024-05-08T08:34:50.324550100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(100000, list)"
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random \n",
    "massive_list = [random.randint(0, 10) for i in range(100000)]\n",
    "len(massive_list), type(massive_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:34:50.443628500Z",
     "start_time": "2024-05-08T08:34:50.387549300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "[4, 9, 9, 9, 0, 0, 9, 9, 1, 10]"
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "massive_list[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.685604100Z",
     "start_time": "2024-05-08T08:34:50.402550200Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "767 µs ± 128 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)\n",
      "4.75 ms ± 331 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit sum(massive_list)\n",
    "%timeit np.sum(massive_list)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NumPy's `np.sum()` is still fast but Python's `sum()` is faster on Python `list`s. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.753618200Z",
     "start_time": "2024-05-08T08:35:00.683603900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.770604100Z",
     "start_time": "2024-05-08T08:35:00.698604100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "3.6333333333333333"
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the mean\n",
    "np.mean(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.771604Z",
     "start_time": "2024-05-08T08:35:00.712605400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "6.5"
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the max\n",
    "np.max(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.771604Z",
     "start_time": "2024-05-08T08:35:00.726605500Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "1.0"
     },
     "execution_count": 160,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the min\n",
    "np.min(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.791604500Z",
     "start_time": "2024-05-08T08:35:00.742689400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "1.8226964152656422"
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the standard deviation\n",
    "np.std(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.808604600Z",
     "start_time": "2024-05-08T08:35:00.759605600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "3.3222222222222224"
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Find the variance\n",
    "np.var(a2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.933682100Z",
     "start_time": "2024-05-08T08:35:00.771604Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "1.8226964152656422"
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The standard deviation is the square root of the variance\n",
    "np.sqrt(np.var(a2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What's mean?**\n",
    "\n",
    "Mean is the same as average. You can find the average of a set of numbers by adding them up and dividing them by how many there are.\n",
    "\n",
    "**What's standard deviation?**\n",
    "\n",
    "[Standard deviation](https://www.mathsisfun.com/data/standard-deviation.html) is a measure of how spread out numbers are.\n",
    "\n",
    "**What's variance?**\n",
    "\n",
    "The [variance](https://www.mathsisfun.com/data/standard-deviation.html) is the averaged squared differences of the mean.\n",
    "\n",
    "To work it out, you:\n",
    "1. Work out the mean\n",
    "2. For each number, subtract the mean and square the result\n",
    "3. Find the average of the squared differences"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:00.990608500Z",
     "start_time": "2024-05-08T08:35:00.786604300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(4296133.472222221, 8.0)"
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Demo of variance\n",
    "high_var_array = np.array([1, 100, 200, 300, 4000, 5000])\n",
    "low_var_array = np.array([2, 4, 6, 8, 10])\n",
    "\n",
    "np.var(high_var_array), np.var(low_var_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.020605200Z",
     "start_time": "2024-05-08T08:35:00.801604300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(2072.711623024829, 2.8284271247461903)"
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(high_var_array), np.std(low_var_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.020605200Z",
     "start_time": "2024-05-08T08:35:00.817603500Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "2072.711623024829"
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The standard deviation is the square root of the variance\n",
    "np.sqrt(np.var(high_var_array))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.747434800Z",
     "start_time": "2024-05-08T08:35:00.833603800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "<Figure size 640x480 with 1 Axes>",
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "plt.hist(high_var_array)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.866434100Z",
     "start_time": "2024-05-08T08:35:01.748434400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "<Figure size 640x480 with 1 Axes>",
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(low_var_array)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reshaping"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.933433800Z",
     "start_time": "2024-05-08T08:35:01.867434100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "outputs": [],
   "source": [
    "x = np.arange(20)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.934525400Z",
     "start_time": "2024-05-08T08:35:01.883433500Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "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]\n",
      "(20,)\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "print(x)\n",
    "print(x.shape)\n",
    "        \n",
    "#prints out the dimension value of the array\n",
    "print(x.ndim)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.936448400Z",
     "start_time": "2024-05-08T08:35:01.898433900Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[[ 0  1  2  3  4]\n",
      "  [ 5  6  7  8  9]]\n",
      "\n",
      " [[10 11 12 13 14]\n",
      "  [15 16 17 18 19]]]\n",
      "(2, 2, 5)\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "x_r=np.reshape(x, [2, 2, 5])\n",
    "print(x_r)\n",
    "print(x_r.shape)\n",
    "        \n",
    "#prints out the dimension value of the array\n",
    "print(x_r.ndim)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:01.936448400Z",
     "start_time": "2024-05-08T08:35:01.913449300Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0  1  2  3  4]\n",
      " [ 5  6  7  8  9]\n",
      " [10 11 12 13 14]\n",
      " [15 16 17 18 19]]\n",
      "(4, 5)\n",
      "2\n"
     ]
    }
   ],
   "source": [
    "x_r2=np.reshape(x_r, [4, 5])\n",
    "print(x_r2)\n",
    "print(x_r2.shape)\n",
    "        \n",
    "#prints out the dimension value of the array\n",
    "print(x_r2.ndim)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:02.034119400Z",
     "start_time": "2024-05-08T08:35:01.928434700Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:35:02.105139900Z",
     "start_time": "2024-05-08T08:35:01.943435100Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1.  2.  3.3]\n",
      " [4.  5.  6.5]]\n",
      "(2, 3)\n",
      "[[[ 1  2  3]\n",
      "  [ 4  5  6]\n",
      "  [ 7  8  9]]\n",
      "\n",
      " [[10 11 12]\n",
      "  [13 14 15]\n",
      "  [16 17 18]]]\n",
      "(2, 3, 3)\n"
     ]
    }
   ],
   "source": [
    "print(a2)\n",
    "print(a2.shape)\n",
    "print(a3)\n",
    "print(a3.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 273,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T10:36:21.320205500Z",
     "start_time": "2024-05-08T10:36:21.245205Z"
    }
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (2,3) (2,3,3) ",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mValueError\u001B[0m                                Traceback (most recent call last)",
      "Cell \u001B[1;32mIn[273], line 1\u001B[0m\n\u001B[1;32m----> 1\u001B[0m \u001B[43ma2\u001B[49m\u001B[43m \u001B[49m\u001B[38;5;241;43m+\u001B[39;49m\u001B[43m \u001B[49m\u001B[43ma3\u001B[49m\n",
      "\u001B[1;31mValueError\u001B[0m: operands could not be broadcast together with shapes (2,3) (2,3,3) "
     ]
    }
   ],
   "source": [
    "a2 + a3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:40:34.740283100Z",
     "start_time": "2024-05-08T08:40:34.676219600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[1. ],\n        [2. ],\n        [3.3]],\n\n       [[4. ],\n        [5. ],\n        [6.5]]])"
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.reshape(2, 3, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:40:39.884993800Z",
     "start_time": "2024-05-08T08:40:39.794876100Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:08:37.006774200Z",
     "start_time": "2024-05-08T08:08:36.921697300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[ 2. ,  3. ,  4. ],\n        [ 6. ,  7. ,  8. ],\n        [10.3, 11.3, 12.3]],\n\n       [[14. , 15. , 16. ],\n        [18. , 19. , 20. ],\n        [22.5, 23.5, 24.5]]])"
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.reshape(2, 3, 1) + a3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transpose\n",
    "\n",
    "A tranpose reverses the order of the axes. \n",
    "\n",
    "For example, an array with shape `(2, 3)` becomes `(3, 2)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:29.062659100Z",
     "start_time": "2024-05-08T08:41:29.015660400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(2, 3)"
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:30.323237400Z",
     "start_time": "2024-05-08T08:41:30.275215200Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 4. ],\n       [2. , 5. ],\n       [3.3, 6.5]])"
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:30.833234900Z",
     "start_time": "2024-05-08T08:41:30.765214700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 4. ],\n       [2. , 5. ],\n       [3.3, 6.5]])"
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:31.547187500Z",
     "start_time": "2024-05-08T08:41:31.430204Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(3, 2)"
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.T.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For larger arrays, the default value of a tranpose is to swap the first and last axes.\n",
    "\n",
    "For example, `(5, 3, 3)` -> `(3, 3, 5)`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:34.215596900Z",
     "start_time": "2024-05-08T08:41:34.176513800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[0.71818603, 0.75389358, 0.10964897],\n        [0.63814993, 0.28945067, 0.361702  ],\n        [0.00315978, 0.04352426, 0.79419409]],\n\n       [[0.77746827, 0.56411938, 0.38736774],\n        [0.34514564, 0.41004102, 0.06705749],\n        [0.97356847, 0.06405922, 0.10844153]],\n\n       [[0.47859636, 0.87148378, 0.98030973],\n        [0.67780738, 0.87716066, 0.26757641],\n        [0.61586974, 0.44980164, 0.61761576]],\n\n       [[0.80281463, 0.03813337, 0.29641567],\n        [0.33406014, 0.50907489, 0.74504766],\n        [0.59307464, 0.34360118, 0.30266013]],\n\n       [[0.27967976, 0.92107048, 0.87532113],\n        [0.84135429, 0.87040443, 0.8213548 ],\n        [0.97830797, 0.79855969, 0.67714913]]])"
     },
     "execution_count": 182,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix = np.random.random(size=(5, 3, 3))\n",
    "matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:36.843101600Z",
     "start_time": "2024-05-08T08:41:36.800100300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(5, 3, 3)"
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:37.779902100Z",
     "start_time": "2024-05-08T08:41:37.717818500Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[0.71818603, 0.77746827, 0.47859636, 0.80281463, 0.27967976],\n        [0.63814993, 0.34514564, 0.67780738, 0.33406014, 0.84135429],\n        [0.00315978, 0.97356847, 0.61586974, 0.59307464, 0.97830797]],\n\n       [[0.75389358, 0.56411938, 0.87148378, 0.03813337, 0.92107048],\n        [0.28945067, 0.41004102, 0.87716066, 0.50907489, 0.87040443],\n        [0.04352426, 0.06405922, 0.44980164, 0.34360118, 0.79855969]],\n\n       [[0.10964897, 0.38736774, 0.98030973, 0.29641567, 0.87532113],\n        [0.361702  , 0.06705749, 0.26757641, 0.74504766, 0.8213548 ],\n        [0.79419409, 0.10844153, 0.61761576, 0.30266013, 0.67714913]]])"
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:38.307635300Z",
     "start_time": "2024-05-08T08:41:38.244449Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(3, 3, 5)"
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix.T.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:38.684450600Z",
     "start_time": "2024-05-08T08:41:38.628448700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check to see if the reverse shape is same as tranpose shape\n",
    "matrix.T.shape == matrix.shape[::-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:41:39.200452300Z",
     "start_time": "2024-05-08T08:41:39.091450800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[[ True,  True,  True,  True,  True],\n        [ True,  True,  True,  True,  True],\n        [ True,  True,  True,  True,  True]],\n\n       [[ True,  True,  True,  True,  True],\n        [ True,  True,  True,  True,  True],\n        [ True,  True,  True,  True,  True]],\n\n       [[ True,  True,  True,  True,  True],\n        [ True,  True,  True,  True,  True],\n        [ True,  True,  True,  True,  True]]])"
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check to see if the first and last axes are swapped\n",
    "matrix.T == matrix.swapaxes(0, -1) # swap first (0) and last (-1) axes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can see more advanced forms of tranposing in the NumPy documentation under [`numpy.transpose`](https://numpy.org/doc/stable/reference/generated/numpy.transpose.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot product\n",
    "\n",
    "The main two rules for dot product to remember are:\n",
    "\n",
    "1. The **inner dimensions** must match:\n",
    "  * `(3, 2) @ (3, 2)` won't work\n",
    "  * `(2, 3) @ (3, 2)` will work\n",
    "  * `(3, 2) @ (2, 3)` will work\n",
    "  \n",
    "2. The resulting matrix has the shape of the **outer dimensions**:\n",
    " * `(2, 3) @ (3, 2)` -> `(2, 2)`\n",
    " * `(3, 2) @ (2, 3)` -> `(3, 3)`\n",
    " \n",
    "**Note:** In NumPy, `np.dot()` and `@` can be used to acheive the same result for 1-2 dimension arrays. However, their behaviour begins to differ at arrays with 3+ dimensions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:38.832148900Z",
     "start_time": "2024-05-08T08:10:38.784629700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "((3, 3), (3, 2))"
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "mat1 = np.random.randint(10, size=(3, 3))\n",
    "mat2 = np.random.randint(10, size=(3, 2))\n",
    "\n",
    "mat1.shape, mat2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:39.477630100Z",
     "start_time": "2024-05-08T08:10:39.433550200Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[5, 0, 3],\n       [3, 7, 9],\n       [3, 5, 2]])"
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:39.957467Z",
     "start_time": "2024-05-08T08:10:39.875341900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[4, 7],\n       [6, 8],\n       [8, 1]])"
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:40.488756400Z",
     "start_time": "2024-05-08T08:10:40.408752700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[ 44,  38],\n       [126,  86],\n       [ 58,  63]])"
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(mat1, mat2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:41.306938200Z",
     "start_time": "2024-05-08T08:10:41.262940400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[ 44,  38],\n       [126,  86],\n       [ 58,  63]])"
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Can also achieve np.dot() with \"@\" \n",
    "# (however, they may behave differently at 3D+ arrays)\n",
    "mat1 @ mat2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:43.134654900Z",
     "start_time": "2024-05-08T08:10:43.053427800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[5, 0, 3],\n       [3, 7, 9],\n       [3, 5, 2],\n       [4, 7, 6]])"
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "mat3 = np.random.randint(10, size=(4,3))\n",
    "mat4 = np.random.randint(10, size=(4,3))\n",
    "mat3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:44.097814900Z",
     "start_time": "2024-05-08T08:10:44.053797600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[8, 8, 1],\n       [6, 7, 7],\n       [8, 1, 5],\n       [9, 8, 9]])"
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:45.795782800Z",
     "start_time": "2024-05-08T08:10:45.702673200Z"
    }
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "shapes (4,3) and (4,3) not aligned: 3 (dim 1) != 4 (dim 0)",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mValueError\u001B[0m                                Traceback (most recent call last)",
      "Cell \u001B[1;32mIn[106], line 2\u001B[0m\n\u001B[0;32m      1\u001B[0m \u001B[38;5;66;03m# This will fail as the inner dimensions of the matrices do not match\u001B[39;00m\n\u001B[1;32m----> 2\u001B[0m \u001B[43mnp\u001B[49m\u001B[38;5;241;43m.\u001B[39;49m\u001B[43mdot\u001B[49m\u001B[43m(\u001B[49m\u001B[43mmat3\u001B[49m\u001B[43m,\u001B[49m\u001B[43m \u001B[49m\u001B[43mmat4\u001B[49m\u001B[43m)\u001B[49m\n",
      "File \u001B[1;32m<__array_function__ internals>:180\u001B[0m, in \u001B[0;36mdot\u001B[1;34m(*args, **kwargs)\u001B[0m\n",
      "\u001B[1;31mValueError\u001B[0m: shapes (4,3) and (4,3) not aligned: 3 (dim 1) != 4 (dim 0)"
     ]
    }
   ],
   "source": [
    "# This will fail as the inner dimensions of the matrices do not match\n",
    "np.dot(mat3, mat4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:10:46.443870500Z",
     "start_time": "2024-05-08T08:10:46.401868400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(3, 4)"
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat3.T.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:11:24.205045900Z",
     "start_time": "2024-05-08T08:11:24.150045600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[118,  96,  77],\n       [145, 110, 137],\n       [148, 137, 130]])"
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Dot product\n",
    "np.dot(mat3.T, mat4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:11:35.112273900Z",
     "start_time": "2024-05-08T08:11:35.039259900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[40,  0,  3],\n       [18, 49, 63],\n       [24,  5, 10],\n       [36, 56, 54]])"
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Element-wise multiplication, also known as Hadamard product\n",
    "mat3 * mat4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dot product practical example, nut butter sales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:44:47.272694600Z",
     "start_time": "2024-05-08T08:44:47.219602400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[12, 15,  0],\n       [ 3,  3,  7],\n       [ 9, 19, 18],\n       [ 4,  6, 12],\n       [ 1,  6,  7]])"
     },
     "execution_count": 188,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "sales_amounts = np.random.randint(20, size=(5, 3))\n",
    "sales_amounts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:44:52.633037600Z",
     "start_time": "2024-05-08T08:44:52.585969Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "       Almond butter  Peanut butter  Cashew butter\nMon               12             15              0\nTues               3              3              7\nWed                9             19             18\nThurs              4              6             12\nFri                1              6              7",
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>Almond butter</th>\n      <th>Peanut butter</th>\n      <th>Cashew butter</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>Mon</th>\n      <td>12</td>\n      <td>15</td>\n      <td>0</td>\n    </tr>\n    <tr>\n      <th>Tues</th>\n      <td>3</td>\n      <td>3</td>\n      <td>7</td>\n    </tr>\n    <tr>\n      <th>Wed</th>\n      <td>9</td>\n      <td>19</td>\n      <td>18</td>\n    </tr>\n    <tr>\n      <th>Thurs</th>\n      <td>4</td>\n      <td>6</td>\n      <td>12</td>\n    </tr>\n    <tr>\n      <th>Fri</th>\n      <td>1</td>\n      <td>6</td>\n      <td>7</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
     },
     "execution_count": 189,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weekly_sales = pd.DataFrame(sales_amounts,\n",
    "                            index=[\"Mon\", \"Tues\", \"Wed\", \"Thurs\", \"Fri\"],\n",
    "                            columns=[\"Almond butter\", \"Peanut butter\", \"Cashew butter\"])\n",
    "weekly_sales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:14.769868900Z",
     "start_time": "2024-05-08T08:45:14.693758300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([10,  8, 12])"
     },
     "execution_count": 191,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prices = np.array([10, 8, 12])\n",
    "prices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:15.159736700Z",
     "start_time": "2024-05-08T08:45:15.058720100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(1, 3)"
     },
     "execution_count": 192,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "butter_prices = pd.DataFrame(prices.reshape(1, 3),\n",
    "                             index=[\"Price\"],\n",
    "                             columns=[\"Almond butter\", \"Peanut butter\", \"Cashew butter\"])\n",
    "butter_prices.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:16.146048900Z",
     "start_time": "2024-05-08T08:45:16.082020100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(5, 3)"
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weekly_sales.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:25.814358200Z",
     "start_time": "2024-05-08T08:45:25.716367100Z"
    }
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "shapes (3,) and (5,3) not aligned: 3 (dim 0) != 5 (dim 0)",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mValueError\u001B[0m                                Traceback (most recent call last)",
      "Cell \u001B[1;32mIn[194], line 2\u001B[0m\n\u001B[0;32m      1\u001B[0m \u001B[38;5;66;03m# Find the total amount of sales for a whole day\u001B[39;00m\n\u001B[1;32m----> 2\u001B[0m total_sales \u001B[38;5;241m=\u001B[39m \u001B[43mprices\u001B[49m\u001B[38;5;241;43m.\u001B[39;49m\u001B[43mdot\u001B[49m\u001B[43m(\u001B[49m\u001B[43msales_amounts\u001B[49m\u001B[43m)\u001B[49m\n\u001B[0;32m      3\u001B[0m total_sales\n",
      "\u001B[1;31mValueError\u001B[0m: shapes (3,) and (5,3) not aligned: 3 (dim 0) != 5 (dim 0)"
     ]
    }
   ],
   "source": [
    "# Find the total amount of sales for a whole day\n",
    "total_sales = prices.dot(sales_amounts)\n",
    "total_sales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The shapes aren't aligned, we need the middle two numbers to be the same."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:32.224280800Z",
     "start_time": "2024-05-08T08:45:32.174204200Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([10,  8, 12])"
     },
     "execution_count": 195,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:34.065363300Z",
     "start_time": "2024-05-08T08:45:34.007403Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "(3, 5)"
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sales_amounts.T.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:38.307402700Z",
     "start_time": "2024-05-08T08:45:38.257330400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([240, 138, 458, 232, 142])"
     },
     "execution_count": 197,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# To make the middle numbers the same, we can transpose\n",
    "total_sales = prices.dot(sales_amounts.T)\n",
    "total_sales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:44.154352100Z",
     "start_time": "2024-05-08T08:45:44.106357700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "((1, 3), (5, 3))"
     },
     "execution_count": 198,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "butter_prices.shape, weekly_sales.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:47.600827500Z",
     "start_time": "2024-05-08T08:45:47.559775700Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "       Mon  Tues  Wed  Thurs  Fri\nPrice  240   138  458    232  142",
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>Mon</th>\n      <th>Tues</th>\n      <th>Wed</th>\n      <th>Thurs</th>\n      <th>Fri</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>Price</th>\n      <td>240</td>\n      <td>138</td>\n      <td>458</td>\n      <td>232</td>\n      <td>142</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
     },
     "execution_count": 199,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "daily_sales = butter_prices.dot(weekly_sales.T)\n",
    "daily_sales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:50.805543700Z",
     "start_time": "2024-05-08T08:45:50.734470400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "       Almond butter  Peanut butter  Cashew butter  Total\nMon               12             15              0    240\nTues               3              3              7    138\nWed                9             19             18    458\nThurs              4              6             12    232\nFri                1              6              7    142",
      "text/html": "<div>\n<style scoped>\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n</style>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>Almond butter</th>\n      <th>Peanut butter</th>\n      <th>Cashew butter</th>\n      <th>Total</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>Mon</th>\n      <td>12</td>\n      <td>15</td>\n      <td>0</td>\n      <td>240</td>\n    </tr>\n    <tr>\n      <th>Tues</th>\n      <td>3</td>\n      <td>3</td>\n      <td>7</td>\n      <td>138</td>\n    </tr>\n    <tr>\n      <th>Wed</th>\n      <td>9</td>\n      <td>19</td>\n      <td>18</td>\n      <td>458</td>\n    </tr>\n    <tr>\n      <th>Thurs</th>\n      <td>4</td>\n      <td>6</td>\n      <td>12</td>\n      <td>232</td>\n    </tr>\n    <tr>\n      <th>Fri</th>\n      <td>1</td>\n      <td>6</td>\n      <td>7</td>\n      <td>142</td>\n    </tr>\n  </tbody>\n</table>\n</div>"
     },
     "execution_count": 200,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Need to transpose again\n",
    "weekly_sales[\"Total\"] = daily_sales.T\n",
    "weekly_sales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparison operators \n",
    "\n",
    "Finding out if one array is larger, smaller or equal to another."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:57.438695300Z",
     "start_time": "2024-05-08T08:45:57.393998600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 2, 3])"
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:45:58.956748Z",
     "start_time": "2024-05-08T08:45:58.907772900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1. , 2. , 3.3],\n       [4. , 5. , 6.5]])"
     },
     "execution_count": 202,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:46:00.140260300Z",
     "start_time": "2024-05-08T08:46:00.091263500Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[False, False, False],\n       [False, False, False]])"
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 > a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:46:00.877401500Z",
     "start_time": "2024-05-08T08:46:00.800194900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[ True,  True, False],\n       [False, False, False]])"
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 >= a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:46:01.587327400Z",
     "start_time": "2024-05-08T08:46:01.532330Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([False, False, False])"
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 > 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:46:02.265986200Z",
     "start_time": "2024-05-08T08:46:02.220971800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([ True,  True,  True])"
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 == a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:46:03.274317100Z",
     "start_time": "2024-05-08T08:46:03.176246Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[ True,  True, False],\n       [False, False, False]])"
     },
     "execution_count": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 == a2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Sorting arrays\n",
    "\n",
    "* [`np.sort()`](https://numpy.org/doc/stable/reference/generated/numpy.sort.html) - sort values in a specified dimension of an array.\n",
    "* [`np.argsort()`](https://numpy.org/doc/stable/reference/generated/numpy.argsort.html) - return the indices to sort the array on a given axis.\n",
    "* [`np.argmax()`](https://numpy.org/doc/stable/reference/generated/numpy.argmax.html) - return the index/indicies which gives the highest value(s) along an axis.\n",
    "* [`np.argmin()`](https://numpy.org/doc/stable/reference/generated/numpy.argmin.html) - return the index/indices which gives the lowest value(s) along an axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:47:37.468936700Z",
     "start_time": "2024-05-08T08:47:37.407163Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[7, 1, 4],\n       [2, 5, 4],\n       [4, 3, 5],\n       [3, 3, 8],\n       [5, 6, 3]])"
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random_array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:47:38.144387600Z",
     "start_time": "2024-05-08T08:47:38.071306200Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1, 4, 7],\n       [2, 4, 5],\n       [3, 4, 5],\n       [3, 3, 8],\n       [3, 5, 6]])"
     },
     "execution_count": 212,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sort(random_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:47:38.939804Z",
     "start_time": "2024-05-08T08:47:38.840812800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[1, 2, 0],\n       [0, 2, 1],\n       [1, 0, 2],\n       [0, 1, 2],\n       [2, 0, 1]], dtype=int64)"
     },
     "execution_count": 213,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.argsort(random_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:48:58.034562100Z",
     "start_time": "2024-05-08T08:48:57.951078600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 2, 3])"
     },
     "execution_count": 214,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:48:58.926018Z",
     "start_time": "2024-05-08T08:48:58.864021600Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([0, 1, 2], dtype=int64)"
     },
     "execution_count": 215,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Return the indices that would sort an array\n",
    "np.argsort(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:49:06.161436Z",
     "start_time": "2024-05-08T08:49:06.087365300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "0"
     },
     "execution_count": 216,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# No axis\n",
    "np.argmin(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:49:07.679788400Z",
     "start_time": "2024-05-08T08:49:07.609832400Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([[7, 1, 4],\n       [2, 5, 4],\n       [4, 3, 5],\n       [3, 3, 8],\n       [5, 6, 3]])"
     },
     "execution_count": 217,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random_array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:49:11.337341900Z",
     "start_time": "2024-05-08T08:49:11.294341300Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([0, 1, 2, 2, 1], dtype=int64)"
     },
     "execution_count": 218,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Down the vertical\n",
    "np.argmax(random_array, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-05-08T08:15:19.353429300Z",
     "start_time": "2024-05-08T08:15:19.195415Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "array([1, 0, 4], dtype=int64)"
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Across the horizontal\n",
    "np.argmin(random_array, axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:15:19.696417100Z",
     "start_time": "2024-05-08T08:15:19.560413800Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-05-08T08:15:20.613858700Z",
     "start_time": "2024-05-08T08:15:20.467857200Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false
   }
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.11.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}