04_numpy.ipynb

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    "# Numpy\n",
    "\n",
    "\n",
    "This section introduces you to [`numpy`](http://www.numpy.org/), Python's\n",
    "numerical computing library.\n",
    "\n",
    "\n",
    "Numpy is not actually part of the standard Python library. But it is a\n",
    "fundamental part of the Python ecosystem - it forms the basis for many\n",
    "important Python libraries, and it (along with its partners\n",
    "[`scipy`](https://www.scipy.org/), [`matplotlib`](https://matplotlib.org/) and\n",
    "[`pandas`](https://pandas.pydata.org/)) is what makes Python an attractive\n",
    "alternative to Matlab as a scientific computing platform.\n",
    "\n",
    "\n",
    "## Contents\n",
    "\n",
    "\n",
    "* [The Python list versus the Numpy array](#the-python-list-versus-the-numpy-array)\n",
    "* [Numpy basics](#numpy-basics)\n",
    " * [Creating arrays](#creating-arrays)\n",
    " * [Operating on arrays](#operating-on-arrays)\n",
    " * [Array properties](#array-properties)\n",
    " * [Descriptive statistics](#descriptive-statistics)\n",
    " * [Reshaping and rearranging arrays](#reshaping-and-rearranging-arrays)\n",
    "* [Multi-variate operations](#multi-variate-operations)\n",
    " * [Matrix multplication](#matrix-multiplication)\n",
    " * [Broadcasting](#broadcasting)\n",
    "* [Array indexing](#array-indexing)\n",
    " * [Indexing multi-dimensional arrays](#indexing-multi-dimensional-arrays)\n",
    " * [Boolean indexing](#boolean-indexing)\n",
    " * [Coordinate array indexing](#coordinate-array-indexing)\n",
    "* [Generating random numbers](#generating-random-numbers)\n",
    "\n",
    "* [Appendix: Importing Numpy](#appendix-importing-numpy)\n",
    "* [Appendix: Vectors in Numpy](#appendix-vectors-in-numpy)\n",
    "\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"the-python-list-versus-the-numpy-array\"></a>\n",
    "## The Python list versus the Numpy array\n",
    "\n",
    "\n",
    "Numpy adds a new data type to the Python language - the `array` (more\n",
    "specifically, the `ndarray`). A Numpy `array` is a N-dimensional array of\n",
    "homogeneously-typed numerical data.\n",
    "\n",
    "\n",
    "You have already been introduced to the Python `list`, which you can easily\n",
    "use to store a handful of numbers (or anything else):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = [10, 8, 12, 14, 7, 6, 11]"
   ]
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   "metadata": {},
   "source": [
    "You could also emulate a 2D or ND matrix by using lists of lists, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xyz_coords = [[-11.4,   1.0,  22.6],\n",
    "              [ 22.7, -32.8,  19.1],\n",
    "              [ 62.8, -18.2, -34.5]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For simple tasks, you could stick with processing your data using python\n",
    "lists, and the built-in\n",
    "[`math`](https://docs.python.org/3.5/library/math.html) library. And this\n",
    "might be tempting, because it does look quite a lot like what you might type\n",
    "into Matlab.\n",
    "\n",
    "\n",
    "But __BEWARE!__ A Python list is a terrible data structure for scientific\n",
    "computing!\n",
    "\n",
    "\n",
    "This is a major source of confusion for those poor souls who have spent their\n",
    "lives working in Matlab, but have finally seen the light and switched to\n",
    "Python. It is _crucial_ to be able to distinguish between a Python list and a\n",
    "Numpy array.\n",
    "\n",
    "\n",
    "___Python list == Matlab cell array:___ A list in python is akin to a cell\n",
    "array in Matlab - they can store anything, but are extremely inefficient, and\n",
    "unwieldy when you have more than a couple of dimensions.\n",
    "\n",
    "\n",
    "___Numy array == Matlab matrix:___ These are in contrast to the Numpy array\n",
    "and Matlab matrix, which are both thin wrappers around a contiguous chunk of\n",
    "memory, and which provide blazing-fast performance (because behind the scenes\n",
    "in both Numpy and Matlab, it's C, C++ and FORTRAN all the way down).\n",
    "\n",
    "\n",
    "So you should strongly consider turning those lists into Numpy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "data = np.array([10, 8, 12, 14, 7, 6, 11])\n",
    "\n",
    "xyz_coords = np.array([[-11.4,   1.0,  22.6],\n",
    "                       [ 22.7, -32.8,  19.1],\n",
    "                       [ 62.8, -18.2, -34.5]])"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you look carefully at the code above, you will notice that we are still\n",
    "actually using Python lists. We have declared our data sets in exactly the\n",
    "same way as we did earlier, by denoting them with square brackets `[` and `]`.\n",
    "\n",
    "\n",
    "The key difference here is that these lists immediately get converted into\n",
    "Numpy arrays, by passing them to the `np.array` function.  To clarify this\n",
    "point, we could rewrite this code in the following equivalent manner:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Define our data sets as python lists\n",
    "data       = [10, 8, 12, 14, 7, 6, 11]\n",
    "xyz_coords = [[-11.4,   1.0,  22.6],\n",
    "              [ 22.7, -32.8,  19.1],\n",
    "              [ 62.8, -18.2, -34.5]]\n",
    "\n",
    "# Convert them to numpy arrays\n",
    "data       = np.array(data)\n",
    "xyz_coords = np.array(xyz_coords)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, in practice, we would never create a Numpy array in this way - we\n",
    "will be loading our data from text or binary files directly into a Numpy\n",
    "array, thus completely bypassing the use of Python lists and the costly\n",
    "list-to-array conversion.  I'm emphasising this to help you understand the\n",
    "difference between Python lists and Numpy arrays. Apologies if you've already\n",
    "got it, forgiveness please.\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"numpy-basics\"></a>\n",
    "## Numpy basics\n",
    "\n",
    "\n",
    "Let's get started."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"creating-arrays\"></a>\n",
    "### Creating arrays\n",
    "\n",
    "\n",
    "Numpy has quite a few functions which behave similarly to their equivalents in\n",
    "Matlab:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print('np.zeros gives us zeros:                       ', np.zeros(5))\n",
    "print('np.ones gives us ones:                         ', np.ones(5))\n",
    "print('np.arange gives us a range:                    ', np.arange(5))\n",
    "print('np.linspace gives us N linearly spaced numbers:', np.linspace(0, 1, 5))\n",
    "print('np.random.random gives us random numbers:      ', np.random.random(5))\n",
    "print('np.random.randint gives us random integers:    ', np.random.randint(1, 10, 5))\n",
    "print('np.eye gives us an identity matrix:')\n",
    "print(np.eye(4))\n",
    "print('np.diag gives us a diagonal matrix:')\n",
    "print(np.diag([1, 2, 3, 4]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> There will be more on random numbers [below](#generating-random-numbers).\n",
    "\n",
    "\n",
    "The `zeros` and `ones` functions can also be used to generate N-dimensional\n",
    "arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = np.zeros((3, 4))\n",
    "o = np.ones((2, 10))\n",
    "print(z)\n",
    "print(o)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Note that, in a 2D Numpy array, the first axis corresponds to rows, and the\n",
    "> second to columns - just like in Matlab.\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"operating-on-arrays\"></a>\n",
    "### Operating on arrays\n",
    "\n",
    "\n",
    "All of the mathematical operators you know and love can be applied to Numpy\n",
    "arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (3, 3))\n",
    "print('a:')\n",
    "print(a)\n",
    "print('a + 2:')\n",
    "print( a + 2)\n",
    "print('a * 3:')\n",
    "print( a * 3)\n",
    "print('a % 2:')\n",
    "print( a % 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"array-properties\"></a>\n",
    "### Array properties\n",
    "\n",
    "\n",
    "Numpy is a bit different than Matlab in the way that you interact with\n",
    "arrays. In Matlab, you would typically pass an array to a built-in function,\n",
    "e.g. `size(M)`, `ndims(M)`, etc. In contrast, a Numpy array is a Python\n",
    "object which has _attributes_ that contain basic information about the array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = np.zeros((2, 3, 4))\n",
    "print(z)\n",
    "print('Shape:                     ', z.shape)\n",
    "print('Number of dimensions:      ', z.ndim)\n",
    "print('Number of elements:        ', z.size)\n",
    "print('Data type:                 ', z.dtype)\n",
    "print('Number of bytes:           ', z.nbytes)\n",
    "print('Length of first dimension: ', len(z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> As depicted above, passing a Numpy array to the built-in `len` function will\n",
    "> only give you the length of the first dimension, so you will typically want\n",
    "> to avoid using it - use the `size` attribute instead.\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"descriptive-statistics\"></a>\n",
    "### Descriptive statistics\n",
    "\n",
    "\n",
    "Similarly, a Numpy array has a set of methods<sup>1</sup> which allow you to\n",
    "calculate basic descriptive statisics on an array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.random(10)\n",
    "print('a: ', a)\n",
    "print('min:          ', a.min())\n",
    "print('max:          ', a.max())\n",
    "print('index of min: ', a.argmin())  # remember that in Python, list indices\n",
    "print('index of max: ', a.argmax())  # start from zero - Numpy is the same!\n",
    "print('mean:         ', a.mean())\n",
    "print('variance:     ', a.var())\n",
    "print('stddev:       ', a.std())\n",
    "print('sum:          ', a.sum())\n",
    "print('prod:         ', a.prod())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> <sup>1</sup> Python, being an object-oriented language, distinguishes\n",
    "> between _functions_ and _methods_. _Method_ is simply the term used to refer\n",
    "> to a function that is associated with a specific object. Similarly, the term\n",
    "> _attribute_ is used to refer to some piece of information that is attached\n",
    "> to an object, such as `z.shape`, or `z.dtype`.\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"reshaping-and-rearranging-arrays\"></a>\n",
    "### Reshaping and rearranging arrays\n",
    "\n",
    "\n",
    "A numpy array can be reshaped very easily, using the `reshape` method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (4, 4))\n",
    "b = a.reshape((2, 8))\n",
    "print('a:')\n",
    "print(a)\n",
    "print('b:')\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this does not modify the underlying data in any way - the `reshape`\n",
    "method returns a _view_ of the same array, just indexed differently:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a[3, 3] = 12345\n",
    "b[0, 7] = 54321\n",
    "print('a:')\n",
    "print(a)\n",
    "print('b:')\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you need to create a reshaped copy of an array, use the `np.array`\n",
    "function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (4, 4))\n",
    "b = np.array(a.reshape(2, 8))\n",
    "a[3, 3] = 12345\n",
    "b[0, 7] = 54321\n",
    "print('a:')\n",
    "print(a)\n",
    "print('b:')\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `T` attribute is a shortcut to obtain the transpose of an array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (4, 4))\n",
    "print(a)\n",
    "print(a.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `transpose` method allows you to obtain more complicated rearrangements\n",
    "of an array's axes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (2, 3, 4))\n",
    "b = a.transpose((2, 0, 1))\n",
    "print('a: ', a.shape)\n",
    "print(a)\n",
    "print('b:', b.shape)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Note again that the `T` attribute and `transpose` method return _views_ of\n",
    "> your array.\n",
    "\n",
    "\n",
    "Numpy has some useful functions which allow you to concatenate or stack\n",
    "multiple arrays into one. The `concatenate` function does what it says on the\n",
    "tin:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.zeros(3)\n",
    "b = np.ones(3)\n",
    "\n",
    "print('1D concatenation:', np.concatenate((a, b)))\n",
    "\n",
    "a = np.zeros((3, 3))\n",
    "b = np.ones((3, 3))\n",
    "\n",
    "print('2D column-wise concatenation:')\n",
    "print(np.concatenate((a, b), axis=1))\n",
    "\n",
    "print('2D row-wise concatenation:')\n",
    "\n",
    "# The axis parameter defaults to 0,\n",
    "# so it is not strictly necessary here.\n",
    "print(np.concatenate((a, b), axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `hstack`, `vstack` and `dstack` functions allow you to concatenate vectors\n",
    "or arrays along the first, second, or third dimension respectively:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.zeros(3)\n",
    "b = np.ones(3)\n",
    "\n",
    "print('a: ', a)\n",
    "print('b: ', b)\n",
    "\n",
    "hstacked = np.hstack((a, b))\n",
    "vstacked = np.vstack((a, b))\n",
    "dstacked = np.dstack((a, b))\n",
    "\n",
    "print('hstacked: (shape {}):'.format(hstacked.shape))\n",
    "print( hstacked)\n",
    "print('vstacked: (shape {}):'.format(vstacked.shape))\n",
    "print( vstacked)\n",
    "print('dstacked: (shape {}):'.format(dstacked.shape))\n",
    "print( dstacked)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"multi-variate-operations\"></a>\n",
    "## Multi-variate operations\n",
    "\n",
    "\n",
    "Many operations in Numpy operate on an elementwise basis. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (5))\n",
    "b = np.random.randint(1, 10, (5))\n",
    "\n",
    "print('a:     ', a)\n",
    "print('b:     ', b)\n",
    "print('a + b: ', a + b)\n",
    "print('a * b: ', a * b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This also extends to higher dimensional arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (4, 4))\n",
    "b = np.random.randint(1, 10, (4, 4))\n",
    "\n",
    "print('a:')\n",
    "print(a)\n",
    "print('b:')\n",
    "print(b)\n",
    "\n",
    "print('a + b')\n",
    "print(a + b)\n",
    "print('a * b')\n",
    "print(a * b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wait ... what's that you say? Oh, I couldn't understand because of all the\n",
    "froth coming out of your mouth. I guess you're angry that `a * b` didn't give\n",
    "you the matrix product, like it would have in Matlab.  Well all I can say is\n",
    "that Python is not Matlab. Get over it. Take a calmative.\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"matrix-multiplication\"></a>\n",
    "*## Matrix multiplication\n",
    "\n",
    "\n",
    "When your heart rate has returned to its normal caffeine-induced state, you\n",
    "can use the `dot` method, or the `@` operator, to perform matrix\n",
    "multiplication:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (4, 4))\n",
    "b = np.random.randint(1, 10, (4, 4))\n",
    "\n",
    "print('a:')\n",
    "print(a)\n",
    "print('b:')\n",
    "print(b)\n",
    "\n",
    "print('a @ b')\n",
    "print(a @ b)\n",
    "\n",
    "print('a.dot(b)')\n",
    "print(a.dot(b))\n",
    "\n",
    "print('b.dot(a)')\n",
    "print(b.dot(a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> The `@` matrix multiplication operator is a relatively recent addition\n",
    "> to Python and Numpy, so you might not see it all that often in existing\n",
    "> code. But it's here to stay, so go ahead and use it!\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"broadcasting\"></a>\n",
    "### Broadcasting\n",
    "\n",
    "\n",
    "One of the coolest (and possibly confusing) features of Numpy is its\n",
    "[_broadcasting_\n",
    "rules](https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html).\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"array-indexing\"></a>\n",
    "## Array indexing\n",
    "\n",
    "\n",
    "Just like in Matlab, slicing up your arrays is a breeze in Numpy.  If you are\n",
    "after some light reading, you might want to check out the [comprehensive Numpy\n",
    "Indexing\n",
    "reference](https://docs.scipy.org/doc/numpy/reference/arrays.indexing.html).\n",
    "\n",
    "\n",
    "> As with indexing regular Python lists, array indices start from 0, and end\n",
    "> indices (if specified) are exclusive.\n",
    "\n",
    "\n",
    "Let's whet our appetites with some basic 1D array slicing. Numpy supports the\n",
    "standard Python __slice__ notation for indexing, where you can specify the\n",
    "start and end indices, and the step size, via the `start:stop:step` syntax:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, 10)\n",
    "\n",
    "print('a:                              ', a)\n",
    "print('first element:                  ', a[0])\n",
    "print('first two elements:             ', a[:2])\n",
    "print('last element:                   ', a[a.shape[0] - 1])\n",
    "print('last element again:             ', a[-1])\n",
    "print('last two elements:              ', a[-2:])\n",
    "print('middle four elements:           ', a[3:7])\n",
    "print('Every second element:           ', a[::2])\n",
    "print('Every second element, reversed: ', a[1::-2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that slicing an array in this way returns a _view_, not a copy, into that\n",
    "array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, 10)\n",
    "print('a:', a)\n",
    "every2nd = a[::2]\n",
    "print('every 2nd:', every2nd)\n",
    "every2nd += 10\n",
    "print('a:', a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"indexing-multi-dimensional-arrays\"></a>\n",
    "### Indexing multi-dimensional arrays\n",
    "\n",
    "\n",
    "Multi-dimensional array indexing works in much the same way as one-dimensional\n",
    "indexing but with, well, more dimensions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (5, 5))\n",
    "print('a:')\n",
    "print(a)\n",
    "print(' First row:     ', a[  0, :])\n",
    "print(' Last row:      ', a[ -1, :])\n",
    "print(' second column: ', a[  :, 1])\n",
    "print(' Centre:')\n",
    "print(a[1:4, 1:4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For arrays with more than two dimensions, the ellipsis (`...`) is a handy\n",
    "feature - it allows you to specify a slice comprising all elements along\n",
    "more than one dimension:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (3, 3, 3))\n",
    "print('a:')\n",
    "print(a)\n",
    "print('All elements at x=0:')\n",
    "print(a[0, ...])\n",
    "print('All elements at z=2:')\n",
    "print(a[..., 2])\n",
    "print('All elements at x=0, z=2:')\n",
    "print(a[0, ..., 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"boolean-indexing\"></a>\n",
    "### Boolean indexing\n",
    "\n",
    "\n",
    "A numpy array can be indexed with a boolean array of the same shape. For\n",
    "example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, 10)\n",
    "\n",
    "print('a:                          ', a)\n",
    "print('a > 5:                      ', a > 4)\n",
    "print('elements in a that are > 5: ', a[a > 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In contrast to the simple indexing we have already seen, boolean indexing will\n",
    "return a _copy_ of the indexed data, __not__ a view. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, 10)\n",
    "b = a[a > 5]\n",
    "print('a: ', a)\n",
    "print('b: ', b)\n",
    "print('Setting b[0] to 999')\n",
    "b[0] = 999\n",
    "print('a: ', a)\n",
    "print('b: ', b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> In general, any 'simple' indexing operation on a Numpy array, where the\n",
    "> indexing object comprises integers, slices (using the standard Python\n",
    "> `start:stop:step` notation), colons (`:`) and/or ellipses (`...`), will\n",
    "> result in a __view__ into the indexed array. Any 'advanced' indexing\n",
    "> operation, where the indexing object contains anything else (e.g. boolean or\n",
    "> integer arrays, or even python lists), will result in a __copy__ of the\n",
    "> data.\n",
    "\n",
    "\n",
    "Logical operators `~` (not), `&` (and) and `|` (or) can be used to manipulate\n",
    "and combine boolean Numpy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a    = np.random.randint(1, 10, 10)\n",
    "gt5  = a > 5\n",
    "even = a % 2 == 0\n",
    "\n",
    "print('a:                                    ', a)\n",
    "print('elements in a which are > 5:          ', a[gt5])\n",
    "print('elements in a which are <= 5:         ', a[~gt5])\n",
    "print('elements in a which are even:         ', a[even])\n",
    "print('elements in a which are odd:          ', a[~even])\n",
    "print('elements in a which are > 5 and even: ', a[gt5 &  even])\n",
    "print('elements in a which are > 5 or odd:   ', a[gt5 | ~even])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"coordinate-array-indexing\"></a>\n",
    "### Coordinate array indexing\n",
    "\n",
    "\n",
    "You can index a numpy array using another array containing coordinates into\n",
    "the first array.  As with boolean indexing, this will result in a copy of the\n",
    "data.  Generally, you will need to have a separate array, or list, of\n",
    "coordinates into each data axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.randint(1, 10, (4, 4))\n",
    "print(a)\n",
    "\n",
    "rows    = [0, 2, 3]\n",
    "cols    = [1, 0, 2]\n",
    "indexed = a[rows, cols]\n",
    "\n",
    "for r, c, v in zip(rows, cols, indexed):\n",
    "    print('a[{}, {}] = {}'.format(r, c, v))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a class=\"anchor\" id=\"generating-random-numbers\"></a>\n",
    "## Generating random numbers\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"appendix-importing-numpy\"></a>\n",
    "## Appendix: Importing Numpy\n",
    "\n",
    "\n",
    "For interactive exploration/experimentation, you might want to import\n",
    "Numpy like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This makes your Python session very similar to Matlab - you can call all\n",
    "of the Numpy functions directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e = array([1, 2, 3, 4, 5])\n",
    "z = zeros((100, 100))\n",
    "d = diag([2, 3, 4, 5])\n",
    "\n",
    "print(e)\n",
    "print(z)\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But if you are writing a script or application using Numpy, I implore you to\n",
    "Numpy like this instead:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The downside to this is that you will have to prefix all Numpy functions with\n",
    "`np.`, like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e = np.array([1, 2, 3, 4, 5])\n",
    "z = np.zeros((100, 100))\n",
    "d = np.diag([2, 3, 4, 5])\n",
    "\n",
    "print(e)\n",
    "print(z)\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a big upside, however, in that other people who have to read/use your\n",
    "code will like you a lot more. This is because it will be easier for them to\n",
    "figure out what the hell your code is doing. Namespaces are your friend - use\n",
    "them!\n",
    "\n",
    "\n",
    "<a class=\"anchor\" id=\"appendix-vectors-in-numpy\"></a>\n",
    "## Appendix: Vectors in Numpy\n",
    "\n",
    "\n",
    "One aspect of Numpy which might trip you up, and which can be quite\n",
    "frustrating at times, is that Numpy has no understanding of row or column\n",
    "vectors.  __An array with only one dimension is neither a row, nor a column\n",
    "vector - it is just a 1D array__.  If you have a 1D array, and you want to use\n",
    "it as a row vector, you need to reshape it to a shape of `(1, N)`. Similarly,\n",
    "to use a 1D array as a column vector, you must reshape it to have shape\n",
    "`(N, 1)`.\n",
    "\n",
    "\n",
    "In general, when you are mixing 1D arrays with 2- or N-dimensional arrays, you\n",
    "need to make sure that your arrays have the correct shape. For example:"
   ]
  },
  {
   "cell_type": "code",