Commit 36d8290f authored by Michiel Cottaar's avatar Michiel Cottaar
Browse files

Added general stack and broadcasting rules

parent c1e5084c
%% Cell type:markdown id: tags:
# Numpy
This section introduces you to [`numpy`](http://www.numpy.org/), Python's
numerical computing library.
Numpy is not actually part of the standard Python library. But it is a
fundamental part of the Python ecosystem - it forms the basis for many
important Python libraries, and it (along with its partners
[`scipy`](https://www.scipy.org/), [`matplotlib`](https://matplotlib.org/) and
[`pandas`](https://pandas.pydata.org/)) is what makes Python an attractive
alternative to Matlab as a scientific computing platform.
## Contents
* [The Python list versus the Numpy array](#the-python-list-versus-the-numpy-array)
* [Numpy basics](#numpy-basics)
* [Creating arrays](#creating-arrays)
* [Loading text files](#loading-text-files)
* [Array properties](#array-properties)
* [Descriptive statistics](#descriptive-statistics)
* [Reshaping and rearranging arrays](#reshaping-and-rearranging-arrays)
* [Creating arrays](#creating-arrays)
* [Loading text files](#loading-text-files)
* [Array properties](#array-properties)
* [Descriptive statistics](#descriptive-statistics)
* [Reshaping and rearranging arrays](#reshaping-and-rearranging-arrays)
* [Operating on arrays](#operating-on-arrays)
* [Scalar operations](#scalar-operations)
* [Multi-variate operations](#multi-variate-operations)
* [Matrix multplication](#matrix-multiplication)
* [Broadcasting](#broadcasting)
* [Linear algebra](#linear-algebra)
* [Scalar operations](#scalar-operations)
* [Multi-variate operations](#multi-variate-operations)
* [Matrix multplication](#matrix-multiplication)
* [Broadcasting](#broadcasting)
* [Linear algebra](#linear-algebra)
* [Array indexing](#array-indexing)
* [Indexing multi-dimensional arrays](#indexing-multi-dimensional-arrays)
* [Boolean indexing](#boolean-indexing)
* [Coordinate array indexing](#coordinate-array-indexing)
* [Indexing multi-dimensional arrays](#indexing-multi-dimensional-arrays)
* [Boolean indexing](#boolean-indexing)
* [Coordinate array indexing](#coordinate-array-indexing)
* [Exercises](#exercises)
* [Load an array from a file and do stuff with it](#load-an-array-from-a-file-and-do-stuff-with-it)
* [Concatenate affine transforms](#concatenate-affine-transforms)
* [Load an array from a file and do stuff with it](#load-an-array-from-a-file-and-do-stuff-with-it)
* [Concatenate affine transforms](#concatenate-affine-transforms)
* [Appendix A: Generating random numbers](#appendix-generating-random-numbers)
* [Appendix B: Importing Numpy](#appendix-importing-numpy)
* [Appendix C: Vectors in Numpy](#appendix-vectors-in-numpy)
* [Appendix D: The Numpy `matrix`](#appendix-the-numpy-matrix)
* [Useful references](#useful-references)
<a class="anchor" id="the-python-list-versus-the-numpy-array"></a>
## The Python list versus the Numpy array
Numpy adds a new data type to the Python language - the `array` (more
specifically, the `ndarray`). A Numpy `array` is a N-dimensional array of
homogeneously-typed numerical data.
You have already been introduced to the Python `list`, which you can easily
use to store a handful of numbers (or anything else):
%% Cell type:code id: tags:
```
data = [10, 8, 12, 14, 7, 6, 11]
```
%% Cell type:markdown id: tags:
You could also emulate a 2D or ND matrix by using lists of lists, for example:
%% Cell type:code id: tags:
```
xyz_coords = [[-11.4, 1.0, 22.6],
[ 22.7, -32.8, 19.1],
[ 62.8, -18.2, -34.5]]
```
%% Cell type:markdown id: tags:
For simple tasks, you could stick with processing your data using python
lists, and the built-in
[`math`](https://docs.python.org/3/library/math.html) library. And this
might be tempting, because it does look quite a lot like what you might type
into Matlab.
But __BEWARE!__ A Python list is a terrible data structure for scientific
computing!
This is a major source of confusion for people who are learning Python, and
are trying to write efficient code. It is _crucial_ to be able to distinguish
between a Python list and a Numpy array.
___Python list == Matlab cell array:___ A list in Python is akin to a cell
array in Matlab - they can store anything, but are extremely inefficient, and
unwieldy when you have more than a couple of dimensions.
___Numpy array == Matlab matrix:___ These are in contrast to the Numpy array
and Matlab matrix, which are both thin wrappers around a contiguous chunk of
memory, and which provide blazing-fast performance (because behind the scenes
in both Numpy and Matlab, it's C, C++ and FORTRAN all the way down).
So you should strongly consider turning those lists into Numpy arrays:
%% Cell type:code id: tags:
```
import numpy as np
data = np.array([10, 8, 12, 14, 7, 6, 11])
xyz_coords = np.array([[-11.4, 1.0, 22.6],
[ 22.7, -32.8, 19.1],
[ 62.8, -18.2, -34.5]])
```
%% Cell type:markdown id: tags:
If you look carefully at the code above, you will notice that we are still
actually using Python lists. We have declared our data sets in exactly the
same way as we did earlier, by denoting them with square brackets `[` and `]`.
The key difference here is that these lists immediately get converted into
Numpy arrays, by passing them to the `np.array` function. To clarify this
point, we could rewrite this code in the following equivalent manner:
%% Cell type:code id: tags:
```
import numpy as np
# Define our data sets as python lists
data = [10, 8, 12, 14, 7, 6, 11]
xyz_coords = [[-11.4, 1.0, 22.6],
[ 22.7, -32.8, 19.1],
[ 62.8, -18.2, -34.5]]
# Convert them to numpy arrays
data = np.array(data)
xyz_coords = np.array(xyz_coords)
```
%% Cell type:markdown id: tags:
Of course, in practice, we would never create a Numpy array in this way - we
will be loading our data from text or binary files directly into a Numpy
array, thus completely bypassing the use of Python lists and the costly
list-to-array conversion. I'm emphasising this to help you understand the
difference between Python lists and Numpy arrays. Apologies if you've already
got it, [forgiveness
please](https://www.youtube.com/watch?v=ZeHflFNR4kQ&feature=youtu.be&t=128).
<a class="anchor" id="numpy-basics"></a>
## Numpy basics
Let's get started.
%% Cell type:code id: tags:
```
import numpy as np
```
%% Cell type:markdown id: tags:
<a class="anchor" id="creating-arrays"></a>
### Creating arrays
Numpy has quite a few functions which behave similarly to their equivalents in
Matlab:
%% Cell type:code id: tags:
```
print('np.zeros gives us zeros: ', np.zeros(5))
print('np.ones gives us ones: ', np.ones(5))
print('np.arange gives us a range: ', np.arange(5))
print('np.linspace gives us N linearly spaced numbers:', np.linspace(0, 1, 5))
print('np.random.random gives us random numbers [0-1]:', np.random.random(5))
print('np.random.randint gives us random integers: ', np.random.randint(1, 10, 5))
print('np.eye gives us an identity matrix:')
print(np.eye(4))
print('np.diag gives us a diagonal matrix:')
print(np.diag([1, 2, 3, 4]))
```
%% Cell type:markdown id: tags:
> See the [appendix](#appendix-generating-random-numbers) for more information
> on generating random numbers in Numpy.
The `zeros` and `ones` functions can also be used to generate N-dimensional
arrays:
%% Cell type:code id: tags:
```
z = np.zeros((3, 4))
o = np.ones((2, 10))
print(z)
print(o)
```
%% Cell type:markdown id: tags:
> Note that, in a 2D Numpy array, the first axis corresponds to rows, and the
> second to columns - just like in Matlab.
<a class="anchor" id="loading-text-files"></a>
### Loading text files
The `numpy.loadtxt` function is capable of loading numerical data from
plain-text files. By default it expects space-separated data:
%% Cell type:code id: tags:
```
data = np.loadtxt('04_numpy/space_separated.txt')
print('data in 04_numpy/space_separated.txt:')
print(data)
```
%% Cell type:markdown id: tags:
But you can also specify the delimiter to expect<sup>1</sup>:
%% Cell type:code id: tags:
```
data = np.loadtxt('04_numpy/comma_separated.txt', delimiter=',')
print('data in 04_numpy/comma_separated.txt:')
print(data)
```
%% Cell type:markdown id: tags:
> <sup>1</sup> And many other things such as file headers, footers, comments,
> and newline characters - see the
> [docs](https://docs.scipy.org/doc/numpy/reference/generated/numpy.loadtxt.html)
> for more information.
Of course you can also save data out to a text file just as easily, with
[`numpy.savetxt`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.savetxt.html):
%% Cell type:code id: tags:
```
data = np.random.randint(1, 10, (10, 10))
np.savetxt('mydata.txt', data, delimiter=',', fmt='%i')
with open('mydata.txt', 'rt') as f:
for line in f:
print(line.strip())
```
%% Cell type:markdown id: tags:
> The `fmt` argument to the `numpy.savetxt` function uses a specification
> language similar to that used in the C `printf` function - in the example
> above, `'%i`' indicates that the values of the array should be output as
> signed integers. See the [`numpy.savetxt`
> documentation](https://docs.scipy.org/doc/numpy/reference/generated/numpy.savetxt.html)
> for more details on specifying the output format.
<a class="anchor" id="array-properties"></a>
### Array properties
Numpy is a bit different than Matlab in the way that you interact with
arrays. In Matlab, you would typically pass an array to a built-in function,
e.g. `size(M)`, `ndims(M)`, etc. In contrast, a Numpy array is a Python
object which has _attributes_ that contain basic information about the array:
%% Cell type:code id: tags:
```
z = np.zeros((2, 3, 4))
print(z)
print('Shape: ', z.shape)
print('Number of dimensions: ', z.ndim)
print('Number of elements: ', z.size)
print('Data type: ', z.dtype)
print('Number of bytes: ', z.nbytes)
print('Length of first dimension: ', len(z))
```
%% Cell type:markdown id: tags:
> As depicted above, passing a Numpy array to the built-in `len` function will
> only give you the length of the first dimension, so you will typically want
> to avoid using it - instead, use the `size` attribute if you want to know
> how many elements are in an array, or the `shape` attribute if you want to
> know the array shape.
<a class="anchor" id="descriptive-statistics"></a>
### Descriptive statistics
Similarly, a Numpy array has a set of methods<sup>2</sup> which allow you to
calculate basic descriptive statistics on an array:
%% Cell type:code id: tags:
```
a = np.random.random(10)
print('a: ', a)
print('min: ', a.min())
print('max: ', a.max())
print('index of min: ', a.argmin()) # remember that in Python, list indices
print('index of max: ', a.argmax()) # start from zero - Numpy is the same!
print('mean: ', a.mean())
print('variance: ', a.var())
print('stddev: ', a.std())
print('sum: ', a.sum())
print('prod: ', a.prod())
```
%% Cell type:markdown id: tags:
These methods can also be applied to arrays with multiple dimensions:
%% Cell type:code id: tags:
```
a = np.random.randint(1, 10, (3, 3))
print('a:')
print(a)
print('min: ', a.min())
print('row mins: ', a.min(axis=1))
print('col mins: ', a.min(axis=0))
print('Min index : ', a.argmin())
print('Row min indices: ', a.argmin(axis=1))
```
%% Cell type:markdown id: tags:
Note that, for a multi-dimensional array, the `argmin` and `argmax` methods
will return the (0-based) index of the minimum/maximum values into a
[flattened](https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.ndarray.flatten.html)
view of the array.
> <sup>2</sup> Python, being an object-oriented language, distinguishes
> between _functions_ and _methods_. Hopefully we all know what a function
> is - a _method_ is simply the term used to refer to a function that is
> associated with a specific object. Similarly, the term _attribute_ is used
> to refer to some piece of information that is attached to an object, such as
> `z.shape`, or `z.dtype`.
<a class="anchor" id="reshaping-and-rearranging-arrays"></a>
### Reshaping and rearranging arrays
A numpy array can be reshaped very easily, using the `reshape` method.
%% Cell type:code id: tags:
```
a = np.arange(16).reshape(4, 4)
b = a.reshape((2, 8))
print('a:')
print(a)
print('b:')
print(b)
```
%% Cell type:markdown id: tags:
Note that this does not modify the underlying data in any way - the `reshape`
method returns a _view_ of the same array, just indexed differently:
%% Cell type:code id: tags:
```
a[3, 3] = 12345
b[0, 7] = 54321
print('a:')
print(a)
print('b:')
print(b)
```
%% Cell type:markdown id: tags:
If you need to create a reshaped copy of an array, use the `np.array`
function:
%% Cell type:code id: tags:
```
a = np.arange(16).reshape((4, 4))
b = np.array(a.reshape(2, 8))
a[3, 3] = 12345
b[0, 7] = 54321
print('a:')
print(a)
print('b:')
print(b)
```
%% Cell type:markdown id: tags:
The `T` attribute is a shortcut to obtain the transpose of an array.
%% Cell type:code id: tags:
```
a = np.arange(16).reshape((4, 4))
print(a)
print(a.T)
```
%% Cell type:markdown id: tags:
The `transpose` method allows you to obtain more complicated rearrangements
of an N-dimensional array's axes:
%% Cell type:code id: tags:
```
a = np.arange(24).reshape((2, 3, 4))
b = a.transpose((2, 0, 1))
print('a: ', a.shape)
print(a)
print('b:', b.shape)
print(b)
```
%% Cell type:markdown id: tags:
> Note again that the `T` attribute and `transpose` method return _views_ of
> your array.
Numpy has some useful functions which allow you to concatenate or stack
multiple arrays into one. The `concatenate` function does what it says on the
tin:
%% Cell type:code id: tags:
```
a = np.zeros(3)
b = np.ones(3)
print('1D concatenation:', np.concatenate((a, b)))
a = np.zeros((3, 3))
b = np.ones((3, 3))
print('2D column-wise concatenation:')
print(np.concatenate((a, b), axis=1))
print('2D row-wise concatenation:')
# The axis parameter defaults to 0,
# so it is not strictly necessary here.
print(np.concatenate((a, b), axis=0))
```
%% Cell type:markdown id: tags:
The `hstack`, `vstack` and `dstack` functions allow you to concatenate vectors
or arrays along the first, second, or third dimension respectively:
%% Cell type:code id: tags:
```
a = np.zeros(3)
b = np.ones(3)
print('a: ', a)
print('b: ', b)
hstacked = np.hstack((a, b))
vstacked = np.vstack((a, b))
dstacked = np.dstack((a, b))
print('hstacked: (shape {}):'.format(hstacked.shape))
print( hstacked)
print('vstacked: (shape {}):'.format(vstacked.shape))
print( vstacked)
print('dstacked: (shape {}):'.format(dstacked.shape))
print( dstacked)
```
%% Cell type:markdown id: tags:
Alternatively, you can use the `stack` function and give the index of the dimension along which the array
should be stacked as the `axis` keyword (so, `np.vstack((a, b))` is equivalent to `np.stack((a, b), axis=1)`).
<a class="anchor" id="operating-on-arrays"></a>
## Operating on arrays
If you are coming from Matlab, you should read this section as, while many
Numpy operations behave similarly to Matlab, there are a few important
behaviours which differ from what you might expect.
<a class="anchor" id="scalar-operations"></a>
### Scalar operations
All of the mathematical operators you know and love can be applied to Numpy
arrays:
%% Cell type:code id: tags:
```
a = np.arange(1, 10).reshape((3, 3))
print('a:')
print(a)
print('a + 2:')
print( a + 2)
print('a * 3:')
print( a * 3)
print('a % 2:')
print( a % 2)
```
%% Cell type:markdown id: tags:
<a class="anchor" id="multi-variate-operations"></a>
### Multi-variate operations
Many operations in Numpy operate on an element-wise basis. For example:
%% Cell type:code id: tags:
```
a = np.ones(5)
b = np.random.randint(1, 10, 5)
print('a: ', a)
print('b: ', b)
print('a + b: ', a + b)
print('a * b: ', a * b)
```
%% Cell type:markdown id: tags:
This also extends to higher dimensional arrays:
%% Cell type:code id: tags:
```
a = np.ones((4, 4))
b = np.arange(16).reshape((4, 4))
print('a:')
print(a)
print('b:')
print(b)
print('a + b')
print(a + b)
print('a * b')
print(a * b)
```
%% Cell type:markdown id: tags:
Wait ... what's that you say? Oh, I couldn't understand because of all the
froth coming out of your mouth. I guess you're angry that `a * b` didn't give
you the matrix product, like it would have in Matlab. Well all I can say is
that Numpy is not Matlab. Matlab operations are typically consistent with
linear algebra notation. This is not the case in Numpy. Get over it.
[Get yourself a calmative](https://youtu.be/M_w_n-8w3IQ?t=32).
<a class="anchor" id="matrix-multiplication"></a>
### Matrix multiplication
When your heart rate has returned to its normal caffeine-induced state, you
can use the `@` operator or the `dot` method, to perform matrix
multiplication:
%% Cell type:code id: tags:
```
a = np.arange(1, 5).reshape((2, 2))
b = a.T
print('a:')
print(a)
print('b:')
print(b)
print('a @ b')
print(a @ b)
print('a.dot(b)')
print(a.dot(b))
print('b.dot(a)')
print(b.dot(a))
```
%% Cell type:markdown id: tags:
> The `@` matrix multiplication operator is a relatively recent addition to
> Python and Numpy, so you might not see it all that often in existing
> code. But it's here to stay, so if you don't need to worry about
> backwards-compatibility, go ahead and use it!
One potential source of confusion for those of you who are used to Matlab's
linear algebra-based take on things is that Numpy treats row and column
vectors differently - you should take a break now and skim over the [appendix
on vectors in Numpy](#appendix-vectors-in-numpy).
For matrix-by-vector multiplications, a 1-dimensional Numpy array may be
treated as _either_ a row vector _or_ a column vector, depending on where
it is in the expression:
%% Cell type:code id: tags:
```
a = np.arange(1, 5).reshape((2, 2))
b = np.random.randint(1, 10, 2)
print('a:')
print(a)
print('b:', b)
print('a @ b - b is a column vector:')
print(a @ b)
print('b @ a - b is a row vector:')
print(b @ a)
```
%% Cell type:markdown id: tags:
If you really can't stand using `@` to denote matrix multiplication, and just
want things to be like they were back in Matlab-land, you do have the option
of using a different Numpy data type - the `matrix` - which behaves a bit more
like what you might expect from Matlab. You can find a brief overview of the
`matrix` data type in [the appendix](appendix-the-numpy-matrix).
<a class="anchor" id="broadcasting"></a>
### Broadcasting
One of the coolest features of Numpy is *broadcasting*<sup>3</sup>.
Broadcasting allows you to perform element-wise operations on arrays which
have a different shape. For each axis in the two arrays, Numpy will implicitly
expand the shape of the smaller axis to match the shape of the larger one. You
never need to use `repmat` ever again!
> <sup>3</sup>Mathworks have shamelessly stolen Numpy's broadcasting behaviour
> and included it in Matlab versions from 2016b onwards, referring to it as
> _implicit expansion_.
Broadcasting allows you to, for example, add the elements of a 1D vector to
all of the rows or columns of a 2D array:
%% Cell type:code id: tags:
```
a = np.arange(9).reshape((3, 3))
b = np.arange(1, 4)
print('a:')
print(a)
print('b: ', b)
print('a * b (row-wise broadcasting):')
print(a * b)
print('a * b.T (column-wise broadcasting):')
print(a * b.reshape(-1, 1))
```
%% Cell type:markdown id: tags:
> Here we used a handy feature of the `reshape` method - if you pass `-1` for
> the size of one dimension, it will automatically determine the size to use
> for that dimension.
Here is a more useful example, where we use broadcasting to de-mean the rows
or columns of an array:
%% Cell type:code id: tags:
```
a = np.arange(9).reshape((3, 3))
print('a:')
print(a)
print('a (cols demeaned):')
print(a - a.mean(axis=0))
print('a (rows demeaned):')
print(a - a.mean(axis=1).reshape(-1, 1))
```
%% Cell type:markdown id: tags:
> As demonstrated above, many functions in Numpy accept an `axis` parameter,
> allowing you to apply the function along a specific axis. Omitting the
> `axis` parameter will apply the function to the whole array.
Broadcasting can sometimes be confusing, but the rules which Numpy follows to
align arrays of different sizes, and hence determine how the broadcasting
should be applied, are pretty straightforward. If something is not working,
and you can't figure out why refer to the [official
documentation](https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html).
In short the broadcasting rules are:
1. If the input arrays have a different number of dimensions, the ones with fewer
dimensions will have new dimensions with length 1 prepended until all arrays
have the same number of dimensions. So adding a 2D array shaped (3, 3) with
a 1D array of length (3, ), is equivalent to adding the two 2D arrays with
shapes (3, 3) and (1, 3).
2. Once, all the arrays have the same number of dimensions, the arrays are combined
elementwise. Each dimension is compatible between the two arrays if they have
equal length or one has a length of 1. In the latter case the dimension will
be repeated using a procedure equivalent to Matlab's `repmat`).
<a class="anchor" id="linear-algebra"></a>
### Linear algebra
Numpy is first and foremost a library for general-purpose numerical computing.
But it does have a range of linear algebra functionality, hidden away in the
[`numpy.linalg`](https://docs.scipy.org/doc/numpy/reference/routines.linalg.html)
module. Here are a couple of quick examples:
%% Cell type:code id: tags:
```
import numpy.linalg as npla
a = np.array([[1, 2, 3, 4],
[0, 5, 6, 7],
[0, 0, 8, 9],
[0, 0, 0, 10]])
print('a:')
print(a)
print('inv(a)')
print(npla.inv(a))
eigvals, eigvecs = npla.eig(a)
print('eigenvalues and vectors of a:')
for val, vec in zip(eigvals, eigvecs):
print('{:2.0f} - {}'.format(val, vec))
```
%% Cell type:markdown id: tags:
<a class="anchor" id="array-indexing"></a>
## Array indexing
Just like in Matlab, slicing up your arrays is a breeze in Numpy. If you are
after some light reading, you might want to check out the [comprehensive Numpy
Indexing
reference](https://docs.scipy.org/doc/numpy/reference/arrays.indexing.html).
> As with indexing regular Python lists, array indices start from 0, and end
> indices (if specified) are exclusive.
Let's whet our appetites with some basic 1D array slicing. Numpy supports the
standard Python
[__slice__](https://www.pythoncentral.io/how-to-slice-listsarrays-and-tuples-in-python/)
notation for indexing, where you can specify the start and end indices, and
the step size, via the `start:stop:step` syntax:
%% Cell type:code id: tags:
```
a = np.arange(10)
print('a: ', a)
print('first element: ', a[0])
print('first two elements: ', a[:2])
print('last element: ', a[a.shape[0] - 1])
print('last element again: ', a[-1])
print('last two elements: ', a[-2:])
print('middle four elements: ', a[3:7])
print('Every second element: ', a[1::2])
print('Every second element, reversed: ', a[-1::-2])
```
%% Cell type:markdown id: tags:
Note that slicing an array in this way returns a _view_, not a copy, into that
array:
%% Cell type:code id: tags:
```
a = np.arange(10)
print('a:', a)
every2nd = a[::2]
print('every 2nd:', every2nd)
every2nd += 10
print('a:', a)
```
%% Cell type:markdown id: tags:
<a class="anchor" id="indexing-multi-dimensional-arrays"></a>
### Indexing multi-dimensional arrays
Multi-dimensional array indexing works in much the same way as one-dimensional
indexing but with, well, more dimensions. Use commas within the square
brackets to separate the slices for each dimension:
%% Cell type:code id: tags:
```
a = np.arange(25).reshape((5, 5))
print('a:')
print(a)
print(' First row: ', a[ 0, :])
print(' Last row: ', a[ -1, :])
print(' second column: ', a[ :, 1])
print(' Centre:')
print(a[1:4, 1:4])
```
%% Cell type:markdown id: tags:
For arrays with more than two dimensions, the ellipsis (`...`) is a handy
feature - it allows you to specify a slice comprising all elements along
more than one dimension:
%% Cell type:code id: tags:
```
a = np.arange(27).reshape((3, 3, 3))
print('a:')
print(a)
print('All elements at x=0:')
print(a[0, ...])
print('All elements at z=2:')
print(a[..., 2])
print('All elements at x=0, z=2:')
print(a[0, ..., 2])
```
%% Cell type:markdown id: tags:
<a class="anchor" id="boolean-indexing"></a>
### Boolean indexing
A numpy array can be indexed with a boolean array of the same shape. For
example:
%% Cell type:code id: tags:
```
a = np.arange(10)
print('a: ', a)
print('a > 5: ', a > 4)
print('elements in a that are > 5: ', a[a > 5])
```
%% Cell type:markdown id: tags:
In contrast to the simple indexing we have already seen, boolean indexing will
return a _copy_ of the indexed data, __not__ a view. For example:
%% Cell type:code id: tags:
```
a = np.arange(10)
b = a[a > 5]
print('a: ', a)
print('b: ', b)
print('Setting b[0] to 999')
b[0] = 999
print('a: ', a)
print('b: ', b)
```
%% Cell type:markdown id: tags:
> In general, any 'simple' indexing operation on a Numpy array, where the
> indexing object comprises integers, slices (using the standard Python
> `start:stop:step` notation), colons (`:`) and/or ellipses (`...`), will
> result in a __view__ into the indexed array. Any 'advanced' indexing
> operation, where the indexing object contains anything else (e.g. boolean or
> integer arrays, or even python lists), will result in a __copy__ of the
> data.
Logical operators `~` (not), `&` (and) and `|` (or) can be used to manipulate
and combine boolean Numpy arrays:
%% Cell type:code id: tags:
```
a = np.arange(10)
gt5 = a > 5
even = a % 2 == 0
print('a: ', a)
print('elements in a which are > 5: ', a[ gt5])
print('elements in a which are <= 5: ', a[~gt5])
print('elements in a which are even: ', a[ even])
print('elements in a which are odd: ', a[~even])
print('elements in a which are > 5 and even: ', a[gt5 & even])
print('elements in a which are > 5 or odd: ', a[gt5 | ~even])
```
%% Cell type:markdown id: tags:
Numpy also has two handy functions, `all` and `any`, which respectively allow
you to perform boolean `and` and `or` operations along the axes of an array:
%% Cell type:code id: tags:
```
a = np.arange(9).reshape((3, 3))
print('a:')
print(a)
print('rows with any element divisible by 3: ', np.any(a % 3 == 0, axis=1))
print('cols with any element divisible by 3: ', np.any(a % 3 == 0, axis=0))
print('rows with all elements divisible by 3:', np.all(a % 3 == 0, axis=1))
print('cols with all elements divisible by 3:', np.all(a % 3 == 0, axis=0))
```
%% Cell type:markdown id: tags:
<a class="anchor" id="coordinate-array-indexing"></a>
### Coordinate array indexing
You can index a numpy array using another array containing coordinates into
the first array. As with boolean indexing, this will result in a copy of the
data. Generally, you will need to have a separate array, or list, of
coordinates for each axis of the array you wish to index:
%% Cell type:code id: tags:
```
a = np.arange(16).reshape((4, 4))
print('a:')
print(a)
rows = [0, 2, 3]
cols = [1, 0, 2]
indexed = a[rows, cols]
for r, c, v in zip(rows, cols, indexed):
print('a[{}, {}] = {}'.format(r, c, v))
```
%% Cell type:markdown id: tags:
The `numpy.where` function can be combined with boolean arrays to easily
generate coordinate arrays for values which meet some condition:
%% Cell type:code id: tags:
```
a = np.arange(16).reshape((4, 4))
print('a:')
print(a)
evenrows, evencols = np.where(a % 2 == 0)
print('even row coordinates:', evenx)
print('even col coordinates:', eveny)
print(a[evenrows, evencols])
```
%% Cell type:markdown id: tags:
<a class="anchor" id="exercises"></a>
## Exercises
The challenge for each of these exercises is to complete them in as few lines
of code as possible!
> You can find example answers to the exercises in `04_numpy/.solutions`.
<a class="anchor" id="load-an-array-from-a-file-and-do-stuff-with-it"></a>
### Load an array from a file and do stuff with it
Load the file `04_numpy/2d_array.txt`, and calculate and print the mean for
each column. If your code doesn't work, you might want to __LOOK AT YOUR
DATA__, as you will have learnt if you have ever attended the FSL course.
> Bonus: Find the hidden message (hint:
> [chr](https://docs.python.org/3/library/functions.html#chr))
<a class="anchor" id="concatenate-affine-transforms"></a>
### Concatenate affine transforms
Given all of the files in `04_numpy/xfms/`, create a transformation matrix
which can transform coordinates from subject 1 functional space to subject 2
functional space<sup>4</sup>.
Write some code to transform the following coordinates from subject 1
functional space to subject 2 functional space, to test that your matrix is
correct:
| __Subject 1 coordinates__ | __Subject 2 coordinates__ |
|:-------------------------:|:-------------------------:|
| `[ 0.0, 0.0, 0.0]` | `[ 3.21, 4.15, -9.89]` |
| `[-5.0, -20.0, 10.0]` | `[-0.87, -14.36, -1.13]` |
| `[20.0, 25.0, 60.0]` | `[29.58, 27.61, 53.37]` |
> <sup>4</sup> Even though these are FLIRT transforms, this is just a toy
> example. Look
> [here](https://users.fmrib.ox.ac.uk/~paulmc/fsleyes/fslpy/latest/fsl.transform.flirt.html)
> and
> [here](https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FLIRT/FAQ#What_is_the_format_of_the_matrix_used_by_FLIRT.2C_and_how_does_it_relate_to_the_transformation_parameters.3F)
> if you actually need to work with FLIRT transforms.
<a class="anchor" id="appendix-generating-random-numbers"></a>
## Appendix A: Generating random numbers
Numpy's
[`numpy.random`](https://docs.scipy.org/doc/numpy/reference/routines.random.html)
module is where you should go if you want to introduce a little randomness
into your code. You have already seen a couple of functions for generating
uniformly distributed real or integer data:
%% Cell type:code id: tags:
```
import numpy.random as npr
print('Random floats between 0.0 (inclusive) and 1.0 (exclusive):')
print(npr.random((3, 3)))
print('Random integers in a specified range:')
print(npr.randint(1, 100, (3, 3)))
```
%% Cell type:markdown id: tags:
You can also draw random data from other distributions - here are just a few
examples:
%% Cell type:code id: tags:
```
print('Gaussian (mean: 0, stddev: 1):')
print(npr.normal(0, 1, (3, 3)))
print('Gamma (shape: 1, scale: 1):')
print(npr.normal(1, 1, (3, 3)))
print('Chi-square (dof: 10):')
print(npr.chisquare(10, (3, 3)))
```
%% Cell type:markdown id: tags:
The `numpy.random` module also has a couple of other handy functions for
random sampling of existing data:
%% Cell type:code id: tags:
```
data = np.arange(5)
print('data: ', data)
print('two random values: ', npr.choice(data, 2))
print('random permutation: ', npr.permutation(data))
# The numpy.random.shuffle function
# will shuffle an array *in-place*.
npr.shuffle(data)
print('randomly shuffled: ', data)
```
%% Cell type:markdown id: tags:
<a class="anchor" id="appendix-importing-numpy"></a>
## Appendix B: Importing Numpy
For interactive exploration/experimentation, you might want to import
Numpy like this:
%% Cell type:code id: tags:
```
from numpy import *
```
%% Cell type:markdown id: tags:
This makes your Python session very similar to Matlab - you can call all
of the Numpy functions directly:
%% Cell type:code id: tags:
```
e = array([1, 2, 3, 4, 5])
z = zeros((100, 100))
d = diag([2, 3, 4, 5])
print(e)
print(z)
print(d)
```
%% Cell type:markdown id: tags:
But if you are writing a script or application using Numpy, I implore you to
import Numpy (and its commonly used sub-modules) like this instead:
%% Cell type:code id: tags:
```
import numpy as np
import numpy.random as npr
import numpy.linalg as npla
```
%% Cell type:markdown id: tags:
The downside to this is that you will have to prefix all Numpy functions with
`np.`, like so:
%% Cell type:code id: tags:
```
e = np.array([1, 2, 3, 4, 5])
z = np.zeros((100, 100))
d = np.diag([2, 3, 4, 5])
r = npr.random(5)
print(e)
print(z)
print(d)
print(r)
```
%% Cell type:markdown id: tags:
There is a big upside, however, in that other people who have to read/use your
code will like you a lot more. This is because it will be easier for them to
figure out what the hell your code is doing. Namespaces are your friend - use
them!
<a class="anchor" id="appendix-vectors-in-numpy"></a>
## Appendix C: Vectors in Numpy
One aspect of Numpy which might trip you up, and which can be quite
frustrating at times, is that Numpy has no understanding of row or column
vectors. __An array with only one dimension is neither a row, nor a column
vector - it is just a 1D array__. If you have a 1D array, and you want to use
it as a row vector, you need to reshape it to a shape of `(1, N)`. Similarly,
to use a 1D array as a column vector, you must reshape it to have shape
`(N, 1)`.
In general, when you are mixing 1D arrays with 2- or N-dimensional arrays, you
need to make sure that your arrays have the correct shape. For example:
%% Cell type:code id: tags:
```
r = np.random.randint(1, 10, 3)
print('r is a row: ', r)
print('r.T should be a column: ', r.T, ' ... huh?')
print('Ok, make n a 2D array with one row: ', r.reshape(1, -1))
print('We could also use the np.atleast_2d function:', np.atleast_2d(r))
print('Now we can transpose r to get a column:')
print(np.atleast_2d(r).T)
```
%% Cell type:markdown id: tags:
<a class="anchor" id="appendix-the-numpy-matrix"></a>
## Appendix D: The Numpy `matrix`
By now you should be aware that a Numpy `array` does not behave in quite the
same way as a Matlab matrix. The primary difference between Numpy and Matlab
is that in Numpy, the `*` operator denotes element-wise multiplication,
whereas in Matlab, `*` denotes matrix multiplication.
Numpy does support the `@` operator for matrix multiplication, but if this is
a complete show-stopper for you - if you just can't bring yourself to write
`A @ B` to denote the matrix product of `A` and `B` - if you _must_ have your
code looking as Matlab-like as possible, then you should look into the Numpy
[`matrix`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.matrix.html)
data type.
The `matrix` is an alternative to the `array` which essentially behaves more
like a Matlab matrix:
* `matrix` objects always have exactly two dimensions.
* `a * b` denotes matrix multiplication, rather than elementwise
multiplication.
* `matrix` objects have `.H` and `.I` attributes, which are convenient ways to
access the conjugate transpose and inverse of the matrix respectively.
Note however that use of the `matrix` type is _not_ widespread, and if you use
it you will risk confusing others who are familiar with the much more commonly
used `array`, and who need to work with your code. In fact, the official Numpy
documentation [recommends against using the `matrix`
type](https://docs.scipy.org/doc/numpy-dev/user/numpy-for-matlab-users.html#array-or-matrix-which-should-i-use).
But if you are writing some very maths-heavy code, and you want your code to
be as clear and concise, and maths/Matlab-like as possible, then the `matrix`
type is there for you. Just make sure you document your code well to make it
clear to others what is going on!
<a class="anchor" id="useful-references"></a>
## Useful references
* [The Numpy manual](https://docs.scipy.org/doc/numpy/)
* [Linear algebra in `numpy.linalg`](https://docs.scipy.org/doc/numpy/reference/routines.linalg.html)
* [Broadcasting in Numpy](https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html)
* [Indexing in Numpy](https://docs.scipy.org/doc/numpy/reference/arrays.indexing.html)
* [Random sampling in `numpy.random`](https://docs.scipy.org/doc/numpy/reference/routines.random.html)
* [Python slicing](https://www.pythoncentral.io/how-to-slice-listsarrays-and-tuples-in-python/)
* [Numpy for Matlab users](https://docs.scipy.org/doc/numpy-dev/user/numpy-for-matlab-users.html)
......
......@@ -18,24 +18,24 @@ alternative to Matlab as a scientific computing platform.
* [The Python list versus the Numpy array](#the-python-list-versus-the-numpy-array)
* [Numpy basics](#numpy-basics)
* [Creating arrays](#creating-arrays)
* [Loading text files](#loading-text-files)
* [Array properties](#array-properties)
* [Descriptive statistics](#descriptive-statistics)
* [Reshaping and rearranging arrays](#reshaping-and-rearranging-arrays)
* [Creating arrays](#creating-arrays)
* [Loading text files](#loading-text-files)
* [Array properties](#array-properties)
* [Descriptive statistics](#descriptive-statistics)
* [Reshaping and rearranging arrays](#reshaping-and-rearranging-arrays)
* [Operating on arrays](#operating-on-arrays)
* [Scalar operations](#scalar-operations)
* [Multi-variate operations](#multi-variate-operations)
* [Matrix multplication](#matrix-multiplication)
* [Broadcasting](#broadcasting)
* [Linear algebra](#linear-algebra)
* [Scalar operations](#scalar-operations)
* [Multi-variate operations](#multi-variate-operations)
* [Matrix multplication](#matrix-multiplication)
* [Broadcasting](#broadcasting)
* [Linear algebra](#linear-algebra)
* [Array indexing](#array-indexing)
* [Indexing multi-dimensional arrays](#indexing-multi-dimensional-arrays)
* [Boolean indexing](#boolean-indexing)
* [Coordinate array indexing](#coordinate-array-indexing)
* [Indexing multi-dimensional arrays](#indexing-multi-dimensional-arrays)
* [Boolean indexing](#boolean-indexing)
* [Coordinate array indexing](#coordinate-array-indexing)
* [Exercises](#exercises)
* [Load an array from a file and do stuff with it](#load-an-array-from-a-file-and-do-stuff-with-it)
* [Concatenate affine transforms](#concatenate-affine-transforms)
* [Load an array from a file and do stuff with it](#load-an-array-from-a-file-and-do-stuff-with-it)
* [Concatenate affine transforms](#concatenate-affine-transforms)
* [Appendix A: Generating random numbers](#appendix-generating-random-numbers)
* [Appendix B: Importing Numpy](#appendix-importing-numpy)
......@@ -457,6 +457,8 @@ print('dstacked: (shape {}):'.format(dstacked.shape))
print( dstacked)
```
Alternatively, you can use the `stack` function and give the index of the dimension along which the array
should be stacked as the `axis` keyword (so, `np.vstack((a, b))` is equivalent to `np.stack((a, b), axis=1)`).
<a class="anchor" id="operating-on-arrays"></a>
## Operating on arrays
......@@ -655,8 +657,6 @@ print(a - a.mean(axis=0))
print('a (rows demeaned):')
print(a - a.mean(axis=1).reshape(-1, 1))
```
> As demonstrated above, many functions in Numpy accept an `axis` parameter,
> allowing you to apply the function along a specific axis. Omitting the
> `axis` parameter will apply the function to the whole array.
......@@ -668,6 +668,16 @@ should be applied, are pretty straightforward. If something is not working,
and you can't figure out why refer to the [official
documentation](https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html).
In short the broadcasting rules are:
1. If the input arrays have a different number of dimensions, the ones with fewer
dimensions will have new dimensions with length 1 prepended until all arrays
have the same number of dimensions. So adding a 2D array shaped (3, 3) with
a 1D array of length (3, ), is equivalent to adding the two 2D arrays with
shapes (3, 3) and (1, 3).
2. Once, all the arrays have the same number of dimensions, the arrays are combined
elementwise. Each dimension is compatible between the two arrays if they have
equal length or one has a length of 1. In the latter case the dimension will
be repeated using a procedure equivalent to Matlab's `repmat`).
<a class="anchor" id="linear-algebra"></a>
### Linear algebra
......
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