Numerical Computing with Python and Numpy

This tutorial covers the following topics:

• Working with numerical data in Python

• Going from Python lists to Numpy arrays

• Multi-dimensional Numpy arrays and their benefits

• Array operations, broadcasting, indexing, and slicing

• Working with CSV data files using Numpy

Working with numerical data

The "data" in Data Analysis typically refers to numerical data, e.g., stock prices, sales figures, sensor measurements, sports scores, database tables, etc. The Numpy library provides specialized data structures, functions, and other tools for numerical computing in Python. Let's work through an example to see why & how to use Numpy for working with numerical data.

Suppose we want to use climate data like the temperature, rainfall, and humidity to determine if a region is well suited for growing apples. A simple approach for doing this would be to formulate the relationship between the annual yield of apples (tons per hectare) and the climatic conditions like the average temperature (in degrees Fahrenheit), rainfall (in millimeters) & average relative humidity (in percentage) as a linear equation.

yield_of_apples = w1 * temperature + w2 * rainfall + w3 * humidity

We're expressing the yield of apples as a weighted sum of the temperature, rainfall, and humidity. This equation is an approximation since the actual relationship may not necessarily be linear, and there may be other factors involved. But a simple linear model like this often works well in practice.

Based on some statical analysis of historical data, we might come up with reasonable values for the weights w1, w2, and w3. Here's an example set of values:

w1, w2, w3 = 0.3, 0.2, 0.5

Given some climate data for a region, we can now predict the yield of apples. Here's some sample data:

To begin, we can define some variables to record climate data for a region.

kanto_temp = 73
kanto_rainfall = 67
kanto_humidity = 43

We can now substitute these variables into the linear equation to predict the yield of apples.

In [3]:

kanto_yield_apples = kanto_temp * w1 + kanto_rainfall * w2 + kanto_humidity * w3
kanto_yield_apples

Out[3]:

56.8

In [4]:

print("The expected yield of apples in Kanto region is {} tons per hectare.".format(kanto_yield_apples))

The expected yield of apples in Kanto region is 56.8 tons per hectare.

To make it slightly easier to perform the above computation for multiple regions, we can represent the climate data for each region as a vector, i.e., a list of numbers.

In [5]:

kanto = [73, 67, 43]
johto = [91, 88, 64]
hoenn = [87, 134, 58]
sinnoh = [102, 43, 37]
unova = [69, 96, 70]

The three numbers in each vector represent the temperature, rainfall, and humidity data, respectively.

We can also represent the set of weights used in the formula as a vector.

In [6]:

weights = [w1, w2, w3]

We can now write a function crop_yield to calcuate the yield of apples (or any other crop) given the climate data and the respective weights.

In [7]:

def crop_yield(region, weights):
    result = 0
    for x, w in zip(region, weights):
        result += x * w
    return result

In [8]:

crop_yield(kanto, weights)

Out[8]:

56.8

In [9]:

crop_yield(johto, weights)

Out[9]:

76.9

In [10]:

crop_yield(unova, weights)

Out[10]:

74.9

Going from Python lists to Numpy arrays

The calculation performed by the crop_yield (element-wise multiplication of two vectors and taking a sum of the results) is also called the dot product. Learn more about dot product here: https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-dot-product-and-vector-length .

The Numpy library provides a built-in function to compute the dot product of two vectors. However, we must first convert the lists into Numpy arrays.

Let's install the Numpy library using the pip package manager.

In [11]:

!pip install numpy --upgrade --quiet

Next, let's import the numpy module. It's common practice to import numpy with the alias np.

In [12]:

import numpy as np

We can now use the np.array function to create Numpy arrays.

In [13]:

kanto = np.array([73, 67, 43])

In [14]:

kanto

Out[14]:

array([73, 67, 43])

In [15]:

weights = np.array([w1, w2, w3])

In [16]:

weights

Out[16]:

array([0.3, 0.2, 0.5])

Numpy arrays have the type ndarray.

In [17]:

type(kanto)

Out[17]:

numpy.ndarray

In [18]:

type(weights)

Out[18]:

numpy.ndarray

Just like lists, Numpy arrays support the indexing notation [].

In [19]:

weights[0]

Out[19]:

0.3

In [20]:

kanto[2]

Out[20]:

43

Operating on Numpy arrays

We can now compute the dot product of the two vectors using the np.dot function.

In [21]:

np.dot(kanto, weights)

Out[21]:

56.8

We can achieve the same result with low-level operations supported by Numpy arrays: performing an element-wise multiplication and calculating the resulting numbers' sum.

In [22]:

(kanto * weights).sum()

Out[22]:

56.8

The * operator performs an element-wise multiplication of two arrays if they have the same size. The sum method calculates the sum of numbers in an array.

In [23]:

arr1 = np.array([1, 2, 3])
arr2 = np.array([4, 5, 6])

In [24]:

arr1 * arr2

Out[24]:

array([ 4, 10, 18])

In [25]:

arr2.sum()

Out[25]:

15

Benefits of using Numpy arrays

Numpy arrays offer the following benefits over Python lists for operating on numerical data:

• Ease of use: You can write small, concise, and intuitive mathematical expressions like (kanto * weights).sum() rather than using loops & custom functions like crop_yield.

• Performance: Numpy operations and functions are implemented internally in C++, which makes them much faster than using Python statements & loops that are interpreted at runtime

Here's a comparison of dot products performed using Python loops vs. Numpy arrays on two vectors with a million elements each.

In [26]:

# Python lists
arr1 = list(range(1000000))
arr2 = list(range(1000000, 2000000))

# Numpy arrays
arr1_np = np.array(arr1)
arr2_np = np.array(arr2)

In [27]:

%%time
result = 0
for x1, x2 in zip(arr1, arr2):
    result += x1*x2
result

CPU times: user 151 ms, sys: 1.35 ms, total: 153 ms
Wall time: 152 ms

Out[27]:

833332333333500000

In [28]:

%%time
np.dot(arr1_np, arr2_np)

CPU times: user 1.96 ms, sys: 751 µs, total: 2.71 ms
Wall time: 1.6 ms

Out[28]:

833332333333500000

As you can see, using np.dot is 100 times faster than using a for loop. This makes Numpy especially useful while working with really large datasets with tens of thousands or millions of data points.

Multi-dimensional Numpy arrays

We can now go one step further and represent the climate data for all the regions using a single 2-dimensional Numpy array.

In [31]:

climate_data = np.array([[73, 67, 43],
                         [91, 88, 64],
                         [87, 134, 58],
                         [102, 43, 37],
                         [69, 96, 70]])

In [32]:

climate_data

Out[32]:

array([[ 73,  67,  43],
       [ 91,  88,  64],
       [ 87, 134,  58],
       [102,  43,  37],
       [ 69,  96,  70]])

If you've taken a linear algebra class in high school, you may recognize the above 2-d array as a matrix with five rows and three columns. Each row represents one region, and the columns represent temperature, rainfall, and humidity, respectively.

Numpy arrays can have any number of dimensions and different lengths along each dimension. We can inspect the length along each dimension using the .shape property of an array.

# 2D array (matrix)
climate_data.shape

Out[33]:

(5, 3)

In [34]:

weights

Out[34]:

array([0.3, 0.2, 0.5])

In [35]:

# 1D array (vector)
weights.shape

Out[35]:

(3,)

In [36]:

# 3D array 
arr3 = np.array([
    [[11, 12, 13], 
     [13, 14, 15]], 
    [[15, 16, 17], 
     [17, 18, 19.5]]])

In [37]:

arr3.shape

Out[37]:

(2, 2, 3)

All the elements in a numpy array have the same data type. You can check the data type of an array using the .dtype property.

In [38]:

weights.dtype

Out[38]:

dtype('float64')

In [39]:

climate_data.dtype

Out[39]:

dtype('int64')

If an array contains even a single floating point number, all the other elements are also converted to floats.

In [40]:

arr3.dtype

Out[40]:

dtype('float64')

We can now compute the predicted yields of apples in all the regions, using a single matrix multiplication between climate_data (a 5x3 matrix) and weights (a vector of length 3). Here's what it looks like visually:

You can learn about matrices and matrix multiplication by watching the first 3-4 videos of this playlist: https://www.youtube.com/watch?v=xyAuNHPsq-g&list=PLFD0EB975BA0CC1E0&index=1 .

We can use the np.matmul function or the @ operator to perform matrix multiplication.

In [41]:

np.matmul(climate_data, weights)

Out[41]:

array([56.8, 76.9, 81.9, 57.7, 74.9])

In [42]:

climate_data @ weights

Out[42]:

array([56.8, 76.9, 81.9, 57.7, 74.9])

Working with CSV data files

Numpy also provides helper functions reading from & writing to files. Let's download a file climate.txt, which contains 10,000 climate measurements (temperature, rainfall & humidity) in the following format:

temperature,rainfall,humidity
25.00,76.00,99.00
39.00,65.00,70.00
59.00,45.00,77.00
84.00,63.00,38.00
66.00,50.00,52.00
41.00,94.00,77.00
91.00,57.00,96.00
49.00,96.00,99.00
67.00,20.00,28.00
...

This format of storing data is known as comma-separated values or CSV.

CSVs: A comma-separated values (CSV) file is a delimited text file that uses a comma to separate values. Each line of the file is a data record. Each record consists of one or more fields, separated by commas. A CSV file typically stores tabular data (numbers and text) in plain text, in which case each line will have the same number of fields. (Wikipedia)

To read this file into a numpy array, we can use the genfromtxt function.

In [44]:

climate_data = np.genfromtxt('climate.txt', delimiter=',', skip_header=1)

In [45]:

climate_data

Out[45]:

array([[25., 76., 99.],
       [39., 65., 70.],
       [59., 45., 77.],
       ...,
       [99., 62., 58.],
       [70., 71., 91.],
       [92., 39., 76.]])

climate_data.shape

Out[46]:

(10000, 3)

We can now perform a matrix multiplication using the @ operator to predict the yield of apples for the entire dataset using a given set of weights.

In [47]:

weights = np.array([0.3, 0.2, 0.5])

In [48]:

yields = climate_data @ weights

In [49]:

yields

Out[49]:

array([72.2, 59.7, 65.2, ..., 71.1, 80.7, 73.4])

In [50]:

yields.shape

Out[50]:

(10000,)

Let's add the yields to climate_data as a fourth column using the np.concatenate function.

In [51]:

climate_results = np.concatenate((climate_data, yields.reshape(10000, 1)), axis=1)

In [52]:

climate_results

Out[52]:

array([[25. , 76. , 99. , 72.2],
       [39. , 65. , 70. , 59.7],
       [59. , 45. , 77. , 65.2],
       ...,
       [99. , 62. , 58. , 71.1],
       [70. , 71. , 91. , 80.7],
       [92. , 39. , 76. , 73.4]])

There are a couple of subtleties here:

• Since we wish to add new columns, we pass the argument axis=1 to np.concatenate. The axis argument specifies the dimension for concatenation.

• The arrays should have the same number of dimensions, and the same length along each except the dimension used for concatenation. We use the np.reshape function to change the shape of yields from (10000,) to (10000,1).

Here's a visual explanation of np.concatenate along axis=1 (can you guess what axis=0 results in?):

The best way to understand what a Numpy function does is to experiment with it and read the documentation to learn about its arguments & return values. Use the cells below to experiment with np.concatenate and np.reshape.

Let's write the final results from our computation above back to a file using the np.savetxt function.

In [53]:

climate_results

Out[53]:

array([[25. , 76. , 99. , 72.2],
       [39. , 65. , 70. , 59.7],
       [59. , 45. , 77. , 65.2],
       ...,
       [99. , 62. , 58. , 71.1],
       [70. , 71. , 91. , 80.7],
       [92. , 39. , 76. , 73.4]])

In [54]:

np.savetxt('climate_results.txt', 
           climate_results, 
           fmt='%.2f', 
           delimiter=',',
           header='temperature,rainfall,humidity,yeild_apples', 
           comments='')

The results are written back in the CSV format to the file climate_results.txt.

temperature,rainfall,humidity,yeild_apples
25.00,76.00,99.00,72.20
39.00,65.00,70.00,59.70
59.00,45.00,77.00,65.20
84.00,63.00,38.00,56.80
...

Numpy provides hundreds of functions for performing operations on arrays. Here are some commonly used functions:

• Mathematics: np.sum, np.exp, np.round, arithemtic operators

• Array manipulation: np.reshape, np.stack, np.concatenate, np.split

• Linear Algebra: np.matmul, np.dot, np.transpose, np.eigvals

• Statistics: np.mean, np.median, np.std, np.max

How to find the function you need? The easiest way to find the right function for a specific operation or use-case is to do a web search. For instance, searching for "How to join numpy arrays" leads to this tutorial on array concatenation.

You can find a full list of array functions here: https://numpy.org/doc/stable/reference/routines.html

Arithmetic operations, broadcasting and comparison

Numpy arrays support arithmetic operators like +, -, *, etc. You can perform an arithmetic operation with a single number (also called scalar) or with another array of the same shape. Operators make it easy to write mathematical expressions with multi-dimensional arrays.

In [58]:

arr2 = np.array([[1, 2, 3, 4], 
                 [5, 6, 7, 8], 
                 [9, 1, 2, 3]])

In [59]:

arr3 = np.array([[11, 12, 13, 14], 
                 [15, 16, 17, 18], 
                 [19, 11, 12, 13]])

In [60]:

# Adding a scalar
arr2 + 3

Out[60]:

array([[ 4,  5,  6,  7],
       [ 8,  9, 10, 11],
       [12,  4,  5,  6]])

In [61]:

# Element-wise subtraction
arr3 - arr2

Out[61]:

array([[10, 10, 10, 10],
       [10, 10, 10, 10],
       [10, 10, 10, 10]])

In [62]:

# Division by scalar
arr2 / 2

Out[62]:

array([[0.5, 1. , 1.5, 2. ],
       [2.5, 3. , 3.5, 4. ],
       [4.5, 0.5, 1. , 1.5]])

In [63]:

# Element-wise multiplication
arr2 * arr3

Out[63]:

array([[ 11,  24,  39,  56],
       [ 75,  96, 119, 144],
       [171,  11,  24,  39]])

In [64]:

# Modulus with scalar
arr2 % 4

Out[64]:

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

Array Broadcasting

Numpy arrays also support broadcasting, allowing arithmetic operations between two arrays with different numbers of dimensions but compatible shapes. Let's look at an example to see how it works.

In [65]:

arr2 = np.array([[1, 2, 3, 4], 
                 [5, 6, 7, 8], 
                 [9, 1, 2, 3]])

In [66]:

arr2.shape

Out[66]:

(3, 4)

In [67]:

arr4 = np.array([4, 5, 6, 7])

In [68]:

arr4.shape

Out[68]:

(4,)

In [69]:

arr2 + arr4

Out[69]:

array([[ 5,  7,  9, 11],
       [ 9, 11, 13, 15],
       [13,  6,  8, 10]])

When the expression arr2 + arr4 is evaluated, arr4 (which has the shape (4,)) is replicated three times to match the shape (3, 4) of arr2. Numpy performs the replication without actually creating three copies of the smaller dimension array, thus improving performance and using lower memory.

Broadcasting only works if one of the arrays can be replicated to match the other array's shape.

In [70]:

arr5 = np.array([7, 8])

In [71]:

arr5.shape

Out[71]:

(2,)

In [72]:

arr2 + arr5

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-72-c22e92053c39> in <module>
----> 1 arr2 + arr5

ValueError: operands could not be broadcast together with shapes (3,4) (2,) 

In the above example, even if arr5 is replicated three times, it will not match the shape of arr2. Hence arr2 + arr5 cannot be evaluated successfully. Learn more about broadcasting here: https://numpy.org/doc/stable/user/basics.broadcasting.html .

Array Comparison

Numpy arrays also support comparison operations like ==, !=, > etc. The result is an array of booleans.

In [73]:

arr1 = np.array([[1, 2, 3], [3, 4, 5]])
arr2 = np.array([[2, 2, 3], [1, 2, 5]])

In [74]:

arr1 == arr2

Out[74]:

array([[False,  True,  True],
       [False, False,  True]])

In [75]:

arr1 != arr2

Out[75]:

array([[ True, False, False],
       [ True,  True, False]])

In [76]:

arr1 >= arr2

Out[76]:

array([[False,  True,  True],
       [ True,  True,  True]])

In [77]:

arr1 < arr2

Out[77]:

array([[ True, False, False],
       [False, False, False]])

Array comparison is frequently used to count the number of equal elements in two arrays using the sum method. Remember that True evaluates to 1 and False evaluates to 0 when booleans are used in arithmetic operations.

(arr1 == arr2).sum()

Out[78]:

3

Array indexing and slicing

Numpy extends Python's list indexing notation using [] to multiple dimensions in an intuitive fashion. You can provide a comma-separated list of indices or ranges to select a specific element or a subarray (also called a slice) from a Numpy array.

In [79]:

arr3 = np.array([
    [[11, 12, 13, 14], 
     [13, 14, 15, 19]], 
    
    [[15, 16, 17, 21], 
     [63, 92, 36, 18]], 
    
    [[98, 32, 81, 23],      
     [17, 18, 19.5, 43]]])

In [80]:

arr3.shape

Out[80]:

(3, 2, 4)

In [81]:

# Single element
arr3[1, 1, 2]

Out[81]:

36.0

In [82]:

# Subarray using ranges
arr3[1:, 0:1, :2]

Out[82]:

array([[[15., 16.]],

       [[98., 32.]]])

In [83]:

# Mixing indices and ranges
arr3[1:, 1, 3]

Out[83]:

array([18., 43.])

In [84]:

# Mixing indices and ranges
arr3[1:, 1, :3]

Out[84]:

array([[63. , 92. , 36. ],
       [17. , 18. , 19.5]])

In [85]:

# Using fewer indices
arr3[1]

Out[85]:

array([[15., 16., 17., 21.],
       [63., 92., 36., 18.]])

In [86]:

# Using fewer indices
arr3[:2, 1]

Out[86]:

array([[13., 14., 15., 19.],
       [63., 92., 36., 18.]])

In [87]:

# Using too many indices
arr3[1,3,2,1]

---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
<ipython-input-87-fbde713646b3> in <module>
      1 # Using too many indices
----> 2 arr3[1,3,2,1]

IndexError: too many indices for array: array is 3-dimensional, but 4 were indexed

The notation and its results can seem confusing at first, so take your time to experiment and become comfortable with it. Use the cells below to try out some examples of array indexing and slicing, with different combinations of indices and ranges. Here are some more examples demonstrated visually:

Other ways of creating Numpy arrays

Numpy also provides some handy functions to create arrays of desired shapes with fixed or random values. Check out the official documentation or use the help function to learn more.

In [88]:

# All zeros
np.zeros((3, 2))

Out[88]:

array([[0., 0.],
       [0., 0.],
       [0., 0.]])

In [89]:

# All ones
np.ones([2, 2, 3])

Out[89]:

array([[[1., 1., 1.],
        [1., 1., 1.]],

       [[1., 1., 1.],
        [1., 1., 1.]]])

In [90]:

# Identity matrix
np.eye(3)

Out[90]:

array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

In [91]:

# Random vector
np.random.rand(5)

Out[91]:

array([0.8208279 , 0.93801216, 0.44954753, 0.17816933, 0.32049174])

In [92]:

# Random matrix
np.random.randn(2, 3) # rand vs. randn - what's the difference?

Out[92]:

array([[ 0.83775   ,  1.13851471, -0.79147694],
       [ 0.56320765,  1.00386056, -0.42502339]])

In [93]:

# Fixed value
np.full([2, 3], 42)

Out[93]:

array([[42, 42, 42],
       [42, 42, 42]])

In [94]:

# Range with start, end and step
np.arange(10, 90, 3)

Out[94]:

array([10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58,
       61, 64, 67, 70, 73, 76, 79, 82, 85, 88])

In [95]:

# Equally spaced numbers in a range
np.linspace(3, 27, 9)

Out[95]:

array([ 3.,  6.,  9., 12., 15., 18., 21., 24., 27.])