Numpy

Creating Arrays from Python Lists

  1. Using List Create numpy array
# integer array:
np.array([1, 4, 2, 5, 3])

Output:

array([1, 4, 2, 5, 3])
  1. NumPy is constrained to arrays that all contain the same type.
  • If types do not match, NumPy will upcast if possible (here, integers are up-cast to floating point):
np.array([3.14, 4, 2, 3])

Output:

array([3.14, 4.  , 2.  , 3.  ])
  1. Explicitly set the data type of the resulting array, use the dtype keyword:
np.array([1, 2, 3, 4], dtype='float32')

Output:

array([1., 2., 3., 4.], dtype=float32)
  1. NumPy arrays can explicitly be multi-dimensional
# nested lists result in multi-dimensional arrays
a = np.array([range(i, i + 3) for i in [2, 4, 6]])
a

Output

array([[2, 3, 4], [4, 5, 6], [6, 7, 8]])

Creating Arrays from Scratch

Especially for larger arrays, it is more efficient to create arrays from scratch using routines built into NumPy.

Create array with same data

np.zeros(<createshape>, dtype=<datatype>)
np.ones(<createshape>, dtype=<datatype>)
np.full(<createshape>,<fill number>)

Create with rules

np.arange(start, end, step) #Starting at start, ending at end, stepping by step
np.linspace(start, end, step) #Starting at start, ending at end, Divide by step
np.random.random(<shape>) #Uniformly distributed
np.random.normal(a, b, <shape>) #Construct range from a to b normally distributed matrix 
np.random.randint(a, b, <shape>) #random integers in the interval [0, 10)
np.eye(n) # Construct identity matrix

NumPy Standard Data Types

  • NumPy arrays contain values of a single type, so it is important to have detailed knowledge of those types and their limitations.
  • Because NumPy is built in C, the types will be familiar to users of C language.

Note that when constructing an array, they can be specified using a string:

np.zeros(10, dtype='int16')

Or using the associated NumPy object:

np.zeros(10, dtype=np.int16)

Manipulation of NumPy Arrays

This section will present several examples of using NumPy array manipulation to access data and subarrays, and to split, reshape, and join the arrays.

Get to know them well!

We'll cover a few categories of basic array manipulations here:

  • Attributes of arrays: Determining the size, shape, memory consumption, and data types of arrays
  • Indexing of arrays: Getting and setting the value of individual array elements
  • Slicing of arrays: Getting and setting smaller subarrays within a larger array
  • Reshaping of arrays: Changing the shape of a given array
  • Joining and splitting of arrays: Combining multiple arrays into one, and splitting one array into many

NumPy Array Attributes

First let's discuss some useful array attributes.
We'll start by defining three random arrays, a one-dimensional, two-dimensional, and three-dimensional array.
We'll use NumPy's random number generator, which we will seed with a set value in order to ensure that the same random arrays are generated each time this code is run:

import numpy as np
np.random.seed(0)  # seed for reproducibility

x1 = np.random.randint(10, size=6)  # One-dimensional array
x2 = np.random.randint(10, size=(3, 4))  # Two-dimensional array
x3 = np.random.randint(10, size=(3, 4, 5))  # Three-dimensional array

Each array has attributes ndim (the number of dimensions), shape (the size of each dimension), and size (the total size of the array):

print("x3 ndim: ", x3.ndim)
print("x3 shape:", x3.shape)
print("x3 size: ", x3.size)

x3 ndim:  3
x3 shape: (3, 4, 5)
x3 size:  60

Another useful attribute is the dtype, the data type of the array:

print("dtype:", x3.dtype)

dtype: int64

Other attributes include itemsize, which lists the size (in bytes) of each array element, and nbytes, which lists the total size (in bytes) of the array:

print("itemsize:", x3.itemsize, "bytes")
print("nbytes:", x3.nbytes, "bytes")

itemsize: 8 bytes
nbytes: 480 bytes

In general, we expect that nbytes is equal to itemsize times size.

Array Indexing: Accessing Single Elements

If you are familiar with Python's standard list indexing, indexing in NumPy will feel quite familiar.
In a one-dimensional array, the value (counting from zero) can be accessed by specifying the desired index in square brackets, just as with Python lists:

x1

array([5, 0, 3, 3, 7, 9])

x1[0]

5

To index from the end of the array, you can use negative indices:

x1[-1]

9

In a multi-dimensional array, items can be accessed using a comma-separated tuple of indices:

x2

array([[3, 5, 2, 4],
       [7, 6, 8, 8],
       [1, 6, 7, 7]])

x2[2, 0]

1

x2[2, -1]

7

Q4: What is x2[-1, -2]?

Values can also be modified using any of the above index notation:

x2[0, 0] = 12
x2

array([[12,  5,  2,  4],
       [ 7,  6,  8,  8],
       [ 1,  6,  7,  7]])

Keep in mind that, unlike Python lists, NumPy arrays have a fixed type.
This means, for example, that if you attempt to insert a floating-point value to an integer array, the value will be silently truncated. Don't be caught unaware by this behavior!

x1[0] = 3.14159  # this will be truncated!
x1

array([3, 0, 3, 3, 7, 9])

Array Slicing: Accessing Subarrays

Just as we can use square brackets to access individual array elements, we can also use them to access subarrays with the slice notation, marked by the colon (:) character.
The NumPy slicing syntax follows that of the standard Python list; to access a slice of an array x, use this:

x[start:stop:step]

If any of these are unspecified, they default to the values start=0, stop=size of dimension, step=1.
We'll take a look at accessing sub-arrays in one dimension and in multiple dimensions.

One-dimensional subarrays

x = np.arange(10)
x

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

x[:5]  # first five elements

array([0, 1, 2, 3, 4])

x[5:]  # elements after index 5

array([5, 6, 7, 8, 9])

x[4:7]  # middle sub-array

array([4, 5, 6])

x[::2]  # every other element

array([0, 2, 4, 6, 8])

x[1::2]  # every other element, starting at index 1

A potentially confusing case is when the step value is negative.
In this case, the array is reversed. start is the starting index of the reversed list, while stop is the stopping index
This becomes a convenient way to reverse an array:

x[::-1]  # all elements, reversed

x[5::-1]  # reversed list starting from index 5

Q5: What is x-2-2?

Multi-dimensional subarrays

Multi-dimensional slices work in the same way, with multiple slices separated by commas.
For example:

x2

x2[:2, :3]  # first two rows, first three columns

Q5: What is x2[:4, :5]?

x2[:3, ::2]  # all rows, every other column

Finally, subarray dimensions can even be reversed together:

x2[::-1, ::-1] # reverse rows and columns together

Accessing array rows and columns

One commonly needed routine is accessing of single rows or columns of an array.
This can be done by combining indexing and slicing, using an empty slice marked by a single colon (:):

print(x2[:, 0])  # first column of x2

x2[:,0].shape

print(x2[0, :])  # first row of x2

In the case of row access, the empty slice can be omitted for a more compact syntax:

print(x2[0])  # equivalent to x2[0, :]

Subarrays as no-copy views

One important–and extremely useful–thing to know about array slices is that they return views rather than copies of the array data.
This is one area in which NumPy array slicing differs from Python list slicing: in lists, slices will be copies.
Consider our two-dimensional array from before:

print(x2)

Let's extract a subarray from this:

x2_sub = x2[:2, :2]
print(x2_sub)

Now if we modify this subarray, we'll see that the original array is changed! Observe:

x2_sub[0, 0] = 99
print(x2_sub)

print(x2)

This default behavior is actually quite useful: it means that when we work with large datasets, we can access and process pieces of these datasets without the need to copy the underlying data buffer.

Creating copies of arrays

Despite the nice features of array views, it is sometimes useful to instead explicitly copy the data within an array or a subarray. This can be most easily done with the copy() method:

x2_sub_copy = x2[:2, :2].copy()
print(x2_sub_copy)

If we now modify this subarray, the original array is not touched:

x2_sub_copy[0, 0] = 42
print(x2_sub_copy)

print(x2)

Reshaping of Arrays

Another useful type of operation is reshaping of arrays.
The most flexible way of doing this is with the reshape method.
For example, if you want to put the numbers 1 through 9 in a grid, you can do the following:

grid = np.arange(1, 10).reshape((3, 3))
print(grid)

Note that for this to work, the size of the initial array must match the size of the reshaped array.
Where possible, the reshape method will use a no-copy view of the initial array, but with non-contiguous memory buffers this is not always the case.

Another common reshaping pattern is the conversion of a one-dimensional array into a two-dimensional row or column matrix.
This can be done with the reshape method, or more easily done by making use of the newaxis keyword within a slice operation:

x = np.array([1, 2, 3])
x

# row vector via newaxis
x[np.newaxis, :]

# column vector via reshape
x.reshape((3, 1))

Q6: What has been changed after the reshape method?

# column vector via newaxis
x = np.array([1, 2, 3])
x[:, np.newaxis]

Array Concatenation and Splitting

All of the preceding routines worked on single arrays. It's also possible to combine multiple arrays into one, and to conversely split a single array into multiple arrays. We'll take a look at those operations here.

Concatenation of arrays

Concatenation, or joining of two arrays in NumPy, is primarily accomplished using the routines np.concatenate, np.vstack, and np.hstack.
np.concatenate takes a tuple or list of arrays as its first argument, as we can see here:

x = np.array([1, 2, 3])
y = np.array([3, 2, 1])
np.concatenate([x, y])

You can also concatenate more than two arrays at once:

z = [99, 99, 99]
print(np.concatenate([x, y, z]))

It can also be used for two-dimensional arrays:

grid = np.array([[1, 2, 3],
                 [4, 5, 6]])

# concatenate along the first axis
np.concatenate([grid, grid])

# concatenate along the second axis (zero-indexed)
np.concatenate([grid, grid], axis=1)

For working with arrays of mixed dimensions, it can be clearer to use the np.vstack (vertical stack) and np.hstack (horizontal stack) functions:

x = np.array([1, 2, 3])
grid = np.array([[9, 8, 7],
                 [6, 5, 4]])

# vertically stack the arrays
np.vstack([x, grid])

Q7：What may happen if we use hstack function in the previous cell?

# horizontally stack the arrays
y = np.array([[99],
              [99]])
np.hstack([grid, y])

Similary, np.dstack will stack arrays along the third axis.

Splitting of arrays

The opposite of concatenation is splitting, which is implemented by the functions np.split, np.hsplit, and np.vsplit. For each of these, we can pass a list of indices giving the split points:

x = [1, 2, 3, 99, 99, 3, 2, 1]
x1, x2, x3 = np.split(x, [3, 5]) # split at third position and fifth position.
print(x1, x2, x3)

Notice that N split-points, leads to N + 1 subarrays.
The related functions np.hsplit and np.vsplit are similar:

grid = np.arange(16).reshape((4, 4))
grid

upper, lower = np.vsplit(grid, [2])
print(upper)
print(lower)

left, right = np.hsplit(grid, [2])
print(left)
print(right)

Similarly, np.dsplit will split arrays along the third axis.

Computation on NumPy Arrays: Universal Functions ( 通用函数)

Computation on NumPy arrays can be very fast, or it can be very slow.
The key to making it fast is to use vectorized operations, generally implemented through NumPy's universal functions (ufuncs).
This section motivates the need for NumPy's ufuncs, which can be used to make repeated calculations on array elements much more efficient.
It then introduces many of the most common and useful arithmetic ufuncs available in the NumPy package.

The Slowness of Loops

The relative slowness of Python generally manifests itself in situations where many small operations are being repeated – for instance looping over arrays to operate on each element.
For example, imagine we have an array of values and we'd like to compute the reciprocal (倒数) of each.
A straightforward approach might look like this:

import numpy as np
np.random.seed(0)

def compute_reciprocals(values):
    output = np.empty(len(values))
    for i in range(len(values)):
        output[i] = 1.0 / values[i]
    return output
        
values = np.random.randint(1, 10, size=5)
compute_reciprocals(values)

This implementation probably feels fairly natural to someone from, say, a C or Java background.
But if we measure the execution time of this code for a large input, we see that this operation is very slow, perhaps surprisingly so!
We'll benchmark this with IPython's %timeit :

big_array = np.random.randint(1, 100, size=1000000)
%timeit compute_reciprocals(big_array)

It takes several seconds to compute these million operations and to store the result!
When even cell phones have processing speeds measured in Giga-FLOPS (i.e., billions of numerical operations per second), this seems almost absurdly slow.
It turns out that the bottleneck here is not the operations themselves, but the type-checking and function dispatches that CPython must do at each cycle of the loop.
Each time the reciprocal is computed, Python first examines the object's type and does a dynamic lookup of the correct function to use for that type.
If we were working in compiled code instead, this type specification would be known before the code executes and the result could be computed much more efficiently.

Introducing UFuncs

For many types of operations, NumPy provides a convenient interface into just this kind of statically typed, compiled routine. This is known as a vectorized operation.
This can be accomplished by simply performing an operation on the array, which will then be applied to each element.
This vectorized approach is designed to push the loop into the compiled layer that underlies NumPy, leading to much faster execution.

Compare the results of the following two:

print(compute_reciprocals(values))
print(1.0 / values)

Looking at the execution time for our big array, we see that it completes orders of magnitude faster than the Python loop:

%timeit (1.0 / big_array)

Vectorized operations in NumPy are implemented via ufuncs, whose main purpose is to quickly execute repeated operations on values in NumPy arrays.
Ufuncs are extremely flexible – before we saw an operation between a scalar and an array, but we can also operate between two arrays:

np.arange(5)

np.arange(5) / np.arange(1, 6)

And ufunc operations are not limited to one-dimensional arrays–they can also act on multi-dimensional arrays as well:

x = np.arange(9).reshape((3, 3))
2 ** x

Computations using vectorization through ufuncs are nearly always more efficient than their counterpart implemented using Python loops, especially as the arrays grow in size.
Any time you see such a loop in a Python script, you should consider whether it can be replaced with a vectorized expression.

Exploring NumPy's UFuncs

Ufuncs exist in two flavors: unary ufuncs, which operate on a single input, and binary ufuncs, which operate on two inputs.
We'll see examples of both these types of functions here.

Array arithmetic

NumPy's ufuncs feel very natural to use because they make use of Python's native arithmetic operators.
The standard addition, subtraction, multiplication, and division can all be used:

x = np.arange(4)
print("x     =", x)
print("x + 5 =", x + 5)
print("x - 5 =", x - 5)
print("x * 2 =", x * 2)
print("x / 2 =", x / 2)
print("x // 2 =", x // 2)  # floor division

There is also a unary ufunc for negation, and a ** operator for exponentiation, and a % operator for modulus:

print("-x     = ", -x)
print("x ** 2 = ", x ** 2)
print("x % 2  = ", x % 2)

In addition, these can be strung together however you wish, and the standard order of operations is respected:

-(0.5*x + 1) ** 2

Each of these arithmetic operations are simply convenient wrappers around specific functions built into NumPy; for example, the + operator is a wrapper for the add function:

np.add(x, 2)

The following table lists the arithmetic operators implemented in NumPy:

Operator Equivalent ufunc Description
+ np.add Addition (e.g., 1 + 1 = 2)
- np.subtract Subtraction (e.g., 3 - 2 = 1)
- np.negative Unary negation (e.g., -2)
* np.multiply Multiplication (e.g., 2 * 3 = 6)
/ np.divide Division (e.g., 3 / 2 = 1.5)
// np.floor_divide Floor division (e.g., 3 // 2 = 1)
** np.power Exponentiation (e.g., 2 ** 3 = 8)
% np.mod Modulus/remainder (e.g., 9 % 4 = 1)

Absolute value

Just as NumPy understands Python's built-in arithmetic operators, it also understands Python's built-in absolute value function:

x = np.array([-2, -1, 0, 1, 2])
abs(x)

The corresponding NumPy ufunc is np.absolute, which is also available under the alias np.abs:

np.absolute(x)

np.abs(x)

This ufunc can also handle complex data, in which the absolute value returns the magnitude:

x = np.array([3 - 4j, 4 - 3j, 2 + 0j, 0 + 1j])
np.abs(x)

Trigonometric functions

NumPy provides a large number of useful ufuncs, and some of the most useful for the data scientist are the trigonometric functions.
We'll start by defining an array of angles:

Q8: What is the Chinese of Trigonometric functions?

theta = np.linspace(0, np.pi, 3)
print(theta)

Now we can compute some trigonometric functions on these values:

print("theta      = ", theta)
print("sin(theta) = ", np.sin(theta))
print("cos(theta) = ", np.cos(theta))
print("tan(theta) = ", np.tan(theta))

The values are computed to within machine precision, which is why values that should be zero do not always hit exactly zero.
Inverse trigonometric functions are also available:

x = [-1, 0, 1]
print("x         = ", x)
print("arcsin(x) = ", np.arcsin(x))
print("arccos(x) = ", np.arccos(x))
print("arctan(x) = ", np.arctan(x))

Exponents and logarithms

Another common type of operation available in a NumPy ufunc are the exponentials:

x = [1, 2, 3]
print("x     =", x)
print("e^x   =", np.exp(x))
print("2^x   =", np.exp2(x))
print("3^x   =", np.power(3, x))

The inverse of the exponentials, the logarithms, are also available.
The basic np.log gives the natural logarithm; if you prefer to compute the base-2 logarithm or the base-10 logarithm, these are available as well:

x = [1, 2, 4, 10]
print("x        =", x)
print("ln(x)    =", np.log(x))
print("log2(x)  =", np.log2(x))
print("log10(x) =", np.log10(x))

There are also some specialized versions that are useful for maintaining precision with very small input:

x = [0, 0.001, 0.01, 0.1]
print("exp(x) - 1 =", np.expm1(x))
print("log(1 + x) =", np.log1p(x))

When x is very small, these functions give more precise values than if the raw np.log or np.exp were to be used.

There are many, many more ufuncs available in both NumPy and scipy.special.
Because the documentation of these packages is available online, a web search along the lines of "gamma function python" will generally find the relevant information.

Table Of Contents
Numpy
Creating Arrays from Python Lists
Creating Arrays from Scratch
Create array with same data
Create with rules
NumPy Standard Data Types
Manipulation of NumPy Arrays
NumPy Array Attributes
Array Indexing: Accessing Single Elements
Array Slicing: Accessing Subarrays
One-dimensional subarrays
Multi-dimensional subarrays
Accessing array rows and columns
Subarrays as no-copy views
Creating copies of arrays
Reshaping of Arrays
Array Concatenation and Splitting
Concatenation of arrays
Splitting of arrays
Computation on NumPy Arrays: Universal Functions ( 通用函数)
The Slowness of Loops
Introducing UFuncs
Exploring NumPy's UFuncs
Array arithmetic
Absolute value
Trigonometric functions
Exponents and logarithms