{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Tools - NumPy\n", "*NumPy is the fundamental library for scientific computing with Python. NumPy is centered around a powerful N-dimensional array object, and it also contains useful linear algebra, Fourier transform, and random number functions.*\n", "\n", "## Creating arrays\n", "First let's import `numpy`:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `zeros` function creates an array containing any number of zeros:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0. 0. 0. 0. 0.]\n" ] } ], "source": [ "print np.zeros(5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It's just as easy to create a 2D array (ie. a matrix) by providing a tuple with the desired number of rows and columns. For example, here's a 3x4 matrix:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]]\n" ] } ], "source": [ "print np.zeros((3,4))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Some vocabulary:\n", "\n", "* In NumPy, each dimension is called an **axis**.\n", "* The number of axes is called the **rank**.\n", " * For example, the above 3x4 matrix is an array of rank 2 (it is 2-dimensional).\n", " * The first axis has length 3, the second has length 4.\n", "* An array's list of axis lengths is called the **shape** of the array.\n", " * For example, the above matrix's shape is `(3, 4)`.\n", " * The rank is equal to the shape's length.\n", "* The **size** of an array is the total number of elements, which is the product of all axis lengths (eg. 3*4=12)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]]\n", "Shape: (3, 4)\n", "Rank: 2\n", "Size: 12\n" ] } ], "source": [ "a = np.zeros((3,4))\n", "print a\n", "print \"Shape:\", a.shape\n", "print \"Rank:\", a.ndim\n", "print \"Size:\", a.size" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can also create an N-dimensional array of arbitrary rank. For example, here's a 3D array (rank=3), with shape `(2,3,4)`:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]]\n", "\n", " [[ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0.]]]\n" ] } ], "source": [ "print np.zeros((2,3,4))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's look at the type of these arrays:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n" ] } ], "source": [ "print type(np.zeros((3,4)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Many other NumPy functions create `ndarrays`.\n", "\n", "Here's a 3x4 matrix full of ones:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 1. 1. 1. 1.]\n", " [ 1. 1. 1. 1.]\n", " [ 1. 1. 1. 1.]]\n" ] } ], "source": [ "print np.ones((3,4))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An uninitialized 2x3 array (its content is not predictable, as it is whatever is in memory at that point):" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ -1.28822975e-231 -1.28822975e-231 2.17321951e-314]\n", " [ 2.17322671e-314 -1.28822975e-231 -1.28822975e-231]]\n" ] } ], "source": [ "print np.empty((2,3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course you can initialize an `ndarray` using a regular python array (or any iterable). Just call the `array` function:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "[[ 1 2 3 4]\n", " [10 20 30 40]]\n" ] } ], "source": [ "a = np.array([[1,2,3,4], [10, 20, 30, 40]])\n", "print type(a)\n", "print a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can create an `ndarray` using NumPy's `range` function, which is similar to python's built-in `range` function:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1 2 3 4]\n" ] } ], "source": [ "print np.arange(1, 5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It also works with floats:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1. 2. 3. 4.]\n" ] } ], "source": [ "print np.arange(1.0, 5.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course you can provide a step parameter:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1. 1.5 2. 2.5 3. 3.5 4. 4.5]\n" ] } ], "source": [ "print np.arange(1, 5, 0.5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "However, when dealing with floats, the exact number of elements in the array is not always predictible. For example, consider this:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0. 0.33333333 0.66666667 1. 1.33333333 1.66666667]\n", "[ 0. 0.33333333 0.66666667 1. 1.33333333 1.66666667]\n", "[ 0. 0.33333333 0.66666667 1. 1.33333334]\n" ] } ], "source": [ "print np.arange(0, 5/3.0, 1/3.0) # depending on floating point errors, the max value is 4/3.0 or 5/3.0.\n", "print np.arange(0, 5/3.0, 0.333333333)\n", "print np.arange(0, 5/3.0, 0.333333334)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For this reason, it is generally preferable to use the `linspace` function instead of `arange` when working with floats. The `linspace` function returns an array containing a specific number of points evenly distributed between two values (note that the maximum value is *included*, contrary to `arange`):" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0. 0.33333333 0.66666667 1. 1.33333333 1.66666667]\n" ] } ], "source": [ "print np.linspace(0, 5/3.0, 6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A number of functions are available in NumPy's `random` module to create `ndarray`s initialized with random values.\n", "For example, here is a 3x4 matrix initialized with random floats between 0 and 1 (uniform distribution):" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0.31505455 0.75655077 0.15133163 0.61991085]\n", " [ 0.42809672 0.56288807 0.74871884 0.55155697]\n", " [ 0.5127511 0.65383425 0.13132709 0.79646256]]\n" ] } ], "source": [ "print np.random.rand(3,4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here's a 3x4 matrix containing random floats sampled from a univariate [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution) (Gaussian distribution) of mean 0 and variance 1:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0.10283232 1.73246206 -0.06356761 -0.07068949]\n", " [ 1.61004733 0.12150407 -1.26750323 0.97991332]\n", " [ 0.32727099 -0.86810473 -1.12277424 1.39791034]]\n" ] } ], "source": [ "print np.random.randn(3,4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To give you a feel of what these distributions look like, let's use matplotlib (see the [matplotlib tutorial](tools_matplotlib.ipynb) for more details):" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "plt.hist(np.random.rand(100000), normed=True, bins=100, histtype=\"step\", color=\"blue\", label=\"rand\")\n", "plt.hist(np.random.randn(100000), normed=True, bins=100, histtype=\"step\", color=\"red\", label=\"randn\")\n", "plt.axis([-2.5, 2.5, 0, 1.1])\n", "plt.legend(loc = \"upper left\")\n", "plt.title(\"Random distributions\")\n", "plt.xlabel(\"Value\")\n", "plt.ylabel(\"Density\")\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can also initialize an `ndarray` using a function:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 1. 2. 3. 4. 5. 6. 7. 8. 9.]]\n", "\n", " [[ 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n", " [ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.]]\n", "\n", " [[ 2. 2. 2. 2. 2. 2. 2. 2. 2. 2.]\n", " [ 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.]]]\n" ] } ], "source": [ "def my_function(z, y, x):\n", " return x * y + z\n", "\n", "print np.fromfunction(my_function, (3, 2, 10))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "NumPy first creates three `ndarrays` (one per dimension), each of shape `(3, 2, 10)`. Each array has values equal to the coordinate along a specific axis. For example, all elements in the `z` array are equal to the z-coordinate:\n", "\n", " [[[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]\n", " \n", " [[ 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n", " [ 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]]\n", " \n", " [[ 2. 2. 2. 2. 2. 2. 2. 2. 2. 2.]\n", " [ 2. 2. 2. 2. 2. 2. 2. 2. 2. 2.]]]\n", "\n", "This means that `my_function` is only called once, so the initialization is very efficient." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Array data\n", "NumPy's `ndarray`s are very efficient in part because all their elements must have the same type (usually numbers).\n", "You can check what the data type is by looking at the `dtype` attribute:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "int64 [1 2 3 4]\n", "float64 [ 1. 2. 3. 4.]\n" ] } ], "source": [ "a = np.arange(1, 5)\n", "print a.dtype, a\n", "\n", "b = np.arange(1.0, 5.0)\n", "print b.dtype, b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of letting NumPy guess what data type to use, you can set it explicitly when creating an array by setting the `dtype` parameter:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "complex64 [ 1.+0.j 2.+0.j 3.+0.j 4.+0.j]\n" ] } ], "source": [ "a = np.arange(1, 5, dtype=np.complex64)\n", "print a.dtype, a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Available data types include int8, int16, int32, int64, uint8/16/32/64, float16/32/64 and complex64/128. Check out [the documentation](http://docs.scipy.org/doc/numpy-1.10.1/user/basics.types.html) for the full list.\n", "\n", "The `itemsize` attribute returns the size (in bytes) of each item:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "8\n" ] } ], "source": [ "a = np.arange(1, 5, dtype=np.complex64)\n", "print a.itemsize" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An array's data is actually stored in memory as a flat (one dimensional) byte buffer. It is available *via* the `data` attribute (you will rarely need it, though)." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Array:\n", "[[ 1 2]\n", " [1000 2000]]\n", "Raw data:\n", "[1, 0, 0, 0, 2, 0, 0, 0, 232, 3, 0, 0, 208, 7, 0, 0]\n" ] } ], "source": [ "a = np.array([[1,2],[1000, 2000]], dtype=np.int32)\n", "print \"Array:\"\n", "print a\n", "print \"Raw data:\"\n", "print [ord(c) for c in a.data]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Several `ndarrays` can share the same data buffer, meaning that modifying one will also modify the others. We will see an example in a minute." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Reshaping an array\n", "Changing the shape of an `ndarray` is as simple as setting its `shape` attribute. However, the array's size must remain the same." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23]\n", "Rank: 1\n" ] } ], "source": [ "a = np.arange(24)\n", "print a\n", "print \"Rank:\", a.ndim" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 1 2 3]\n", " [ 4 5 6 7]\n", " [ 8 9 10 11]\n", " [12 13 14 15]\n", " [16 17 18 19]\n", " [20 21 22 23]]\n", "Rank: 2\n" ] } ], "source": [ "a.shape = (6, 4)\n", "print a\n", "print \"Rank:\", a.ndim" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0 1 2 3]\n", " [ 4 5 6 7]\n", " [ 8 9 10 11]]\n", "\n", " [[12 13 14 15]\n", " [16 17 18 19]\n", " [20 21 22 23]]]\n", "Rank: 3\n" ] } ], "source": [ "a.shape = (2, 3, 4)\n", "print a\n", "print \"Rank:\", a.ndim" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `reshape` function returns a new `ndarray` object pointing to the *same* data. This means that modifying one array will also modify the other." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 1 2 3 4 5]\n", " [ 6 7 8 9 10 11]\n", " [12 13 14 15 16 17]\n", " [18 19 20 21 22 23]]\n", "Rank: 2\n" ] } ], "source": [ "a2 = a.reshape(4,6)\n", "print a2\n", "print \"Rank:\", a2.ndim" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Set item at row 1, col 2 to 999 (more about indexing below)." ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 1 2 3 4 5]\n", " [ 6 7 999 9 10 11]\n", " [ 12 13 14 15 16 17]\n", " [ 18 19 20 21 22 23]]\n" ] } ], "source": [ "a2[1, 2] = 999\n", "print a2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The corresponding element in a has been modified." ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0 1 2 3]\n", " [ 4 5 6 7]\n", " [999 9 10 11]]\n", "\n", " [[ 12 13 14 15]\n", " [ 16 17 18 19]\n", " [ 20 21 22 23]]]\n" ] } ], "source": [ "print a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, the `ravel` function returns a new one-dimensional `ndarray` that also points to the same data:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0 1 2 3 4 5 6 7 999 9 10 11 12 13 14 15 16 17\n", " 18 19 20 21 22 23]\n" ] } ], "source": [ "a3 = a.ravel()\n", "print a3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Array indexing\n", "One-dimensional NumPy arrays can be accessed more or less like regular python arrays:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "19\n" ] } ], "source": [ "a = np.array([1, 5, 3, 19, 13, 7, 3])\n", "print a[3]" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 3 19 13]\n" ] } ], "source": [ "print a[2:5]" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 3 19 13 7]\n" ] } ], "source": [ "print a[2:-1]" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1 5]\n" ] } ], "source": [ "print a[:2]" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 3 13 3]\n" ] } ], "source": [ "print a[2::2]" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 3 7 13 19 3 5 1]\n" ] } ], "source": [ "print a[::-1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course, you can modify elements:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 5 3 999 13 7 3]\n" ] } ], "source": [ "a[3]=999\n", "print a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can also modify an `ndarray` slice:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 5 997 998 999 7 3]\n" ] } ], "source": [ "a[2:5] = [997, 998, 999]\n", "print a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And if you assign a single value, it is copied across the whole slice (this is called *broadcasting*, more on this below):" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 5 -1 -1 -1 7 3]\n" ] } ], "source": [ "a[2:5] = -1\n", "print a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Contrary to regular python arrays, you cannot grow or shrink `ndarray`s this way:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "cannot copy sequence with size 6 to array axis with dimension 3\n" ] } ], "source": [ "try:\n", " a[2:5] = [1,2,3,4,5,6] # too long\n", "except ValueError, e:\n", " print e" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You cannot delete elements either:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "cannot delete array elements\n" ] } ], "source": [ "try:\n", " del a[2:5]\n", "except ValueError, e:\n", " print e" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Multi-dimensional arrays can be accessed in a similar way by providing an index or slice for each axis, separated by commas:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 1 2 3 4 5 6 7 8 9 10 11]\n", " [12 13 14 15 16 17 18 19 20 21 22 23]\n", " [24 25 26 27 28 29 30 31 32 33 34 35]\n", " [36 37 38 39 40 41 42 43 44 45 46 47]]\n" ] } ], "source": [ "b = np.arange(48).reshape(4, 12)\n", "print b" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "14\n" ] } ], "source": [ "print b[1, 2] # row 1, col 2" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[12 13 14 15 16 17 18 19 20 21 22 23]\n" ] } ], "source": [ "print b[1, :] # row 1, all columns" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 13 25 37]\n" ] } ], "source": [ "print b[:, 1] # all rows, column 1" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 2 3 4]\n", " [26 27 28]]\n" ] } ], "source": [ "print b[(0,2), 2:5] # rows 0 and 2, columns 2 to 4 (5-1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can also provide an `ndarray` of boolean values to specify the indices that you want to access. This will come in handy later:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 1 2 3 4 5 6 7 8 9 10 11]\n", " [24 25 26 27 28 29 30 31 32 33 34 35]]\n" ] } ], "source": [ "bools = np.array([True, False, True, False])\n", "print b[bools, :] # Rows 0 and 2, all columns. Equivalent to b[(1, 3), :]" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46]\n" ] } ], "source": [ "print b[b % 3 == 1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note the subtle difference between these two expressions: " ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[12 13 14 15 16 17 18 19 20 21 22 23]\n", "[[12 13 14 15 16 17 18 19 20 21 22 23]]\n" ] } ], "source": [ "print b[1, :]\n", "print b[1:2, :]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first expression returns row 1 as a 1D array of shape `(12,)`, while the second returns that same row as a 2D array of shape `(1, 12)`." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Everything works just as well with higher dimension arrays:" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0 1 2 3 4 5]\n", " [ 6 7 8 9 10 11]]\n", "\n", " [[12 13 14 15 16 17]\n", " [18 19 20 21 22 23]]\n", "\n", " [[24 25 26 27 28 29]\n", " [30 31 32 33 34 35]]\n", "\n", " [[36 37 38 39 40 41]\n", " [42 43 44 45 46 47]]]\n" ] } ], "source": [ "c = b.reshape(4,2,6)\n", "print c" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "34\n" ] } ], "source": [ "print c[2, 1, 4] # matrix 2, row 1, col 4" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[27 33]\n" ] } ], "source": [ "print c[2, :, 3] # matrix 2, all rows, col 3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If you omit coordinates for some axes, then all elements in these axes are returned:" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[30 31 32 33 34 35]\n" ] } ], "source": [ "print c[2, 1] # Return matrix 2, row 1, all columns. This is equivalent to c[2, 1, :]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You may also write an ellipsis (`...`) to specify that all non-specified axes must be entirely included." ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[24 25 26 27 28 29]\n", " [30 31 32 33 34 35]]\n" ] } ], "source": [ "print c[2, ...] # matrix 2, all rows, all columns. This is equivalent to c[2, :, :]" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[30 31 32 33 34 35]\n" ] } ], "source": [ "print c[2, 1, ...] # matrix 2, row 1, all columns. This is equivalent to c[2, 1, :]" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[27 33]\n" ] } ], "source": [ "print c[2, ..., 3] # matrix 2, all rows, column 3. This is equivalent to c[2, :, 3]" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 3 9]\n", " [15 21]\n", " [27 33]\n", " [39 45]]\n" ] } ], "source": [ "print c[..., 3] # all matrices, all rows, column 3. This is equivalent to c[:, :, 3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Broadcasting" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we discussed above, assigning to an `ndarray` slice requires an `ndarray` of the same shape as the slice. In general, when NumPy expects arrays of the same shape but finds that this is not the case, it applies the so-called *broadcasting* rules:\n", "\n", "**First rule**: if the arrays do not have the same rank, then a 1 will be prepended to the smaller ranking arrays until their ranks match.\n", "\n" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[0 1 2 3 4 5 6 7 8 9]]]\n" ] } ], "source": [ "a = np.arange(10).reshape(1, 1, 10)\n", "print a" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Slice: [[[2 3]]]\n", "Shape: (1, 1, 2)\n" ] } ], "source": [ "print \"Slice:\", a[..., 2:4]\n", "print \"Shape:\", a[..., 2:4].shape" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's try to assign a 1D array of shape `(2,)` to this 3D array of shape `(1,1,2)`. Applying the first rule of broadcasting!" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0 1 55 66 4 5 6 7 8 9]]]\n" ] } ], "source": [ "a[..., 2:4] = [55, 66] # acts as [[[55, 56]]]\n", "print a" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Second rule**: arrays with a 1 along a particular dimension act as if they had the size of the array with the largest shape along that dimension. The value of the array element is repeated along that dimension." ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[0 1 2 3 4]\n", " [5 6 7 8 9]]\n" ] } ], "source": [ "b = np.arange(10).reshape(2, 5)\n", "print b" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Slice:\n", "[[1 2 3]\n", " [6 7 8]]\n", "Shape: (2, 3)\n" ] } ], "source": [ "print \"Slice:\"\n", "print b[..., 1:4]\n", "print \"Shape:\", b[..., 1:4].shape" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's try to assign a 2D array of shape `(2,1)` to this slice of shape `(2, 3)`. NumPy will apply the second rule of broadcasting:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 44 44 44 4]\n", " [ 5 55 55 55 9]]\n" ] } ], "source": [ "b[..., 1:4] = [[44], [55]] # acts as [[44, 44, 44], [55, 55, 55]]\n", "print b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Combining rules 1 & 2, we can do this:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 66 77 88 4]\n", " [ 5 66 77 88 9]]\n" ] } ], "source": [ "b[..., 1:4] = [66, 77, 88] # after rule 1: [[66, 77, 88]], and after rule 2: [[66, 77, 88], [66, 77, 88]]\n", "print b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And also, very simply:" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 99 99 99 4]\n", " [ 5 99 99 99 9]]\n" ] } ], "source": [ "b[..., 1:4] = 99 # after rule 1: [[99]], and after rule 2: [[99, 99, 99], [99, 99, 99]]\n", "print b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Third rule**: after rules 1 & 2, the sizes of all arrays must match." ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "could not broadcast input array from shape (2) into shape (2,3)\n" ] } ], "source": [ "try:\n", " b[..., 1:4] = [33, 44]\n", "except ValueError, e:\n", " print e" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Broadcasting rules are used in many NumPy operations, not just assignment, as we will see below.\n", "For more details about broadcasting, check out [the documentation](https://docs.scipy.org/doc/numpy-dev/user/basics.broadcasting.html)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Iterating\n", "Iterating over `ndarray`s is very similar to iterating over regular python arrays. Note that iterating over multidimensional arrays is done with respect to the first axis." ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[[ 0 1 2 3 4 5]\n", " [ 6 7 8 9 10 11]]\n", "\n", " [[12 13 14 15 16 17]\n", " [18 19 20 21 22 23]]]\n" ] } ], "source": [ "c = np.arange(24).reshape(2, 2, 6) # A 3D array (composed of two 2x6 matrices)\n", "print c" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Item:\n", "[[ 0 1 2 3 4 5]\n", " [ 6 7 8 9 10 11]]\n", "Item:\n", "[[12 13 14 15 16 17]\n", " [18 19 20 21 22 23]]\n" ] } ], "source": [ "for m in c:\n", " print \"Item:\"\n", " print m" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Item:\n", "[[ 0 1 2 3 4 5]\n", " [ 6 7 8 9 10 11]]\n", "Item:\n", "[[12 13 14 15 16 17]\n", " [18 19 20 21 22 23]]\n" ] } ], "source": [ "for i in range(len(c)): # Note that len(c) == c.shape[0]\n", " print \"Item:\"\n", " print c[i]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If you want to iterate on *all* elements in the `ndarray`, simply iterate over the `flat` attribute:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Item: 0\n", "Item: 1\n", "Item: 2\n", "Item: 3\n", "Item: 4\n", "Item: 5\n", "Item: 6\n", "Item: 7\n", "Item: 8\n", "Item: 9\n", "Item: 10\n", "Item: 11\n", "Item: 12\n", "Item: 13\n", "Item: 14\n", "Item: 15\n", "Item: 16\n", "Item: 17\n", "Item: 18\n", "Item: 19\n", "Item: 20\n", "Item: 21\n", "Item: 22\n", "Item: 23\n" ] } ], "source": [ "for i in c.flat:\n", " print \"Item:\", i" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Arithmetic operations\n", "All the usual arithmetic operators (`+`, `-`, `*`, `/`, `**`, etc.) can be used with `ndarray`s. They apply elementwise:" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a + b = [19 27 35 43]\n", "a - b = [ 9 19 29 39]\n", "a * b = [70 92 96 82]\n", "a / b = [ 2 5 10 20]\n", "a % b = [4 3 2 1]\n", "a ** b = [537824 279841 32768 1681]\n" ] } ], "source": [ "a = np.array([14, 23, 32, 41])\n", "b = np.array([5, 4, 3, 2])\n", "print \"a + b =\", a + b\n", "print \"a - b =\", a - b\n", "print \"a * b =\", a * b\n", "print \"a / b =\", a / b\n", "print \"a % b =\", a % b\n", "print \"a ** b =\", a ** b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the multiplication is *not* a matrix multiplication. We will discuss matrix operations below.\n", "\n", "The arrays must have the same shape. If they do not, NumPy will apply the broadcasting rules, as discussed above." ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 42 69 96 123]\n" ] } ], "source": [ "print a * 3 # thanks to broadcasting, this is equivalent to: a * [3, 3, 3, 3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The conditional operators also apply elementwise:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ True False True False]\n" ] } ], "source": [ "print a < [15, 16, 35, 36]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And using broadcasting:" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ True True False False]\n" ] } ], "source": [ "print a < 25 # equivalent to a < [25, 25, 25, 25]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is most useful in conjunction with boolean indexing:" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[14 23]\n" ] } ], "source": [ "print a[a < 25]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that all matching elements are returned as a 1D array, no matter the original array's shape:" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0. 0. 0. 0. 0. 0.]\n", " [ 0. 1. 2. 3. 4. 5.]\n", " [ 0. 2. 4. 6. 8. 10.]]\n" ] } ], "source": [ "p = np.fromfunction(lambda row, col: row*col, (3,6))\n", "print p" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1. 4. 4. 10.]\n" ] } ], "source": [ "print p[p%3 == 1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is possible (and quite convenient) to use boolean indexing and assignment jointly:" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0. 0. 0. 0. 0. 0.]\n", " [ 0. 99. 2. 3. 99. 5.]\n", " [ 0. 2. 99. 6. 8. 99.]]\n" ] } ], "source": [ "p[p%3 == 1] = 99\n", "print p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## To be continued..." ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }