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Home > Data > Scientific Computing > Mastering Numerical Computing with NumPy
Mastering Numerical Computing with NumPy
Mastering Numerical Computing with NumPy

Mastering Numerical Computing with NumPy: Master scientific computing and perform complex operations with ease

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Mastering Numerical Computing with NumPy

Linear Algebra with NumPy

One of the major divisions of mathematics is algebra, and linear algebra in particular, focuses on linear equations and mapping linear spaces, namely vector spaces. When we create a linear map between vector spaces, we are actually creating a data structure called a matrix. The main usage of linear algebra is to solve simultaneous linear equations, but it can also be used for approximations for non-linear systems. Imagine a complex model or system that you are trying to understand, think of it as a non-linear model. In such cases, you can reduce the complex, non-linear characteristics of the problem into simultaneous linear equations, and you can solve them with the help of linear algebra.

In computer science, linear algebra is heavily used in machine learning (ML) applications. In ML applications, you deal with high-dimensional arrays, which can easily...

Vector and matrix mathematics

In the previous chapter, you practiced introductory operations with vectors and matrices. In this section, you will practice more advanced vector and matrix operations that are heavily used in linear algebra. Let's remember the dot product perspective on matrix manipulation and how it can be done with different methods when you have 2-D arrays. The following code block shows alternative ways of performing dot product calculation:

In [1]: import numpy as np 
        a = np.arange(12).reshape(3,2) 
        b = np.arange(15).reshape(2,5) 
        print(a) 
        print(b)
Out[1]: 
[[ 0 1] 
 [ 2 3] 
 [ 4 5]] 
[[ 0 1 2 3 4] 
 [ 5 6 7 8 9] 
In [2]: np.dot(a,b)
Out[2]: array([[ 5,  6,  7,  8,  9], 
               [15, 20, 25, 30, 35], 
               [25, 34, 43, 52, 61]]) 
In [3]: np.matmul(a,b)
Out[3]: array([[ 5,  6,  7,  8,  9], 
               [15, 20, 25, 30...

What's an eigenvalue and how do we compute it?

An eigenvalue is a coefficient of an eigenvector. By definition, an eigenvector is a non zero vector that only changes by a scalar factor when linear transformation is applied. In general, when linear transformation is applied to a vector, its span (the line passing through its origin) is shifted, but some special vectors are not affected by these linear transformations and remain on their own span. These are what we call eigenvectors. The linear transformation affects them only by stretching or squishing them as you are multiplying this vector with a scalar. The value of this scalar is called the eigenvalue. Let's say we have a matrix A, which will be used in linear transformation. We can represent the eigenvalue and eigenvector in a mathematical statements as follows:

Here, is the eigenvector and denotes the eigenvalue...

Computing the norm and determinant

This subsection will introduce two important values in linear algebra, namely the norm and determinant. Briefly, the norm gives length of a vector. The most commonly used norm is the L2-norm, which is also known as the Euclidean norm. Formally, the Lp-norm of x is calculated as follows:

The L0-norm is actually the cardinality of a vector. You can calculate it by just counting the total number of non-zero elements. For example, the vector A =[2,5,9,0] contains three non-zero elements, therefore ||A||0 = 3. The following code block shows the same norm calculation with numpy:

In [24]: import numpy as np 
         x = np.array([2,5,9,0]) 
         np.linalg.norm(x,ord=0)
Out[24]: 3.0

In NumPy, you can calculate the norm of the vector with the use of the linalg.norm() method. The first parameter is the input array and the ord parameter is for order...

Solving linear equations

In this section, you will learn how to solve linear equations by using the linalg.solve() method. When you have a linear equation to solve, as in the form , in simple cases you can just calculate the inverse of A and then multiply it by B to get the solution, but when A has a high dimensionality, that makes it very hard computationally to calculate the inverse of A. Let's start with an example of three linear equations with three unknowns, as follows:

So, these equations can be formalized as follows with matrices:

Then, our problem is to solve . We can calculate the solution with a plain vanilla NumPy without using linalg.solve(). After inverting the A matrix, you will multiply with B in order to get results for x. In the following code block, we calculate the dot product for the inverse matrix of A and B in order to calculate :

In [44]: A =...

Computing gradient

When you have a linear line, you take the derivative so the derivative shows the slope of this line. Gradient is a generalization of the derivative when you have a multiple variable in your function, therefore the result of gradient is actually a vector function rather than a scalar value in derivative. The main goal of ML is actually finding the best model that fits your data. You can evaluate the meaning of the best as minimizing your loss function or objective function. Gradient is used for finding the value of the coefficients or a function that will minimize your loss or cost function. A well-known way of finding optimum points is taking the derivative of the objective function then setting it to zero to find your model coefficients. If you have more than one coefficient then it becomes a gradient rather than a derivative, and it becomes a vector equation...

Summary

In this chapter, we covered vector and matrix operations for linear algebra. We looked at advanced matrix operations, especially featuring dot operations. You also learned about eigenvalues and eigenvectors and then practiced their use in principal component analysis (PCA). Moreover, we covered the norm and determinant calculation and mentioned their importance and usage in ML. In the last two subsections, you learned how to convert linear equations into matrices and solve them, and looked at the computation and importance of gradients.

In the next chapter, we will use NumPy statistics to do explanatory data analysis to explore the 2015 United States Housing data.

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Key benefits

  • • Grasp all aspects of numerical computing and understand NumPy
  • • Explore examples to learn exploratory data analysis (EDA), regression, and clustering
  • • Access NumPy libraries and use performance benchmarking to select the right tool

Description

NumPy is one of the most important scientific computing libraries available for Python. Mastering Numerical Computing with NumPy teaches you how to achieve expert level competency to perform complex operations, with in-depth coverage of advanced concepts. Beginning with NumPy's arrays and functions, you will familiarize yourself with linear algebra concepts to perform vector and matrix math operations. You will thoroughly understand and practice data processing, exploratory data analysis (EDA), and predictive modeling. You will then move on to working on practical examples which will teach you how to use NumPy statistics in order to explore US housing data and develop a predictive model using simple and multiple linear regression techniques. Once you have got to grips with the basics, you will explore unsupervised learning and clustering algorithms, followed by understanding how to write better NumPy code while keeping advanced considerations in mind. The book also demonstrates the use of different high-performance numerical computing libraries and their relationship with NumPy. You will study how to benchmark the performance of different configurations and choose the best for your system. By the end of this book, you will have become an expert in handling and performing complex data manipulations.

Who is this book for?

Mastering Numerical Computing with NumPy is for you if you are a Python programmer, data analyst, data engineer, or a data science enthusiast, who wants to master the intricacies of NumPy and build solutions for your numeric and scientific computational problems. You are expected to have familiarity with mathematics to get the most out of this book.

What you will learn

  • • Perform vector and matrix operations using NumPy
  • • Perform exploratory data analysis (EDA) on US housing data
  • • Develop a predictive model using simple and multiple linear regression
  • • Understand unsupervised learning and clustering algorithms with practical use cases
  • • Write better NumPy code and implement the algorithms from scratch
  • • Perform benchmark tests to choose the best configuration for your system
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Publication date, Length, Edition, Language, ISBN-13
Publication date : Jun 28, 2018
Length: 248 pages
Edition : 1st
Language : English
ISBN-13 : 9781788993357
Category :
Data
Languages :
Python
Concepts :
Scientific Computing
Tools :
NumPy

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Length: 248 pages
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Language : English
ISBN-13 : 9781788993357
Category :
Data
Languages :
Python
Concepts :
Scientific Computing
Tools :
NumPy

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Table of Contents

10 Chapters
Working with NumPy Arrays Chevron down icon Chevron up icon
Working with NumPy Arrays
Technical requirements
Why do we need NumPy?
Who uses NumPy?
Introduction to vectors and matrices
Basics of NumPy array objects
NumPy array operations
Working with multidimensional arrays
Indexing, slicing, reshaping, resizing, and broadcasting
Summary
Linear Algebra with NumPy Chevron down icon Chevron up icon
Linear Algebra with NumPy
Vector and matrix mathematics
What's an eigenvalue and how do we compute it?
Computing the norm and determinant
Solving linear equations
Computing gradient
Summary
Exploratory Data Analysis of Boston Housing Data with NumPy Statistics Chevron down icon Chevron up icon
Exploratory Data Analysis of Boston Housing Data with NumPy Statistics
Loading and saving files
Exploring our dataset
Looking at basic statistics
Computing histograms
Explaining skewness and kurtosis
Trimmed statistics
Box plots
Computing correlations
Summary
Predicting Housing Prices Using Linear Regression Chevron down icon Chevron up icon
Predicting Housing Prices Using Linear Regression
Supervised learning and linear regression
Independent and dependent variables
Hyperparameters
Loss and error functions
Univariate linear regression with gradient descent
Using linear regression to model housing prices
Summary
Clustering Clients of a Wholesale Distributor Using NumPy Chevron down icon Chevron up icon
Clustering Clients of a Wholesale Distributor Using NumPy
Unsupervised learning and clustering
Hyperparameters
The loss function
Implementing our algorithm for a single variable
Modifying our algorithm
Summary
NumPy, SciPy, Pandas, and Scikit-Learn Chevron down icon Chevron up icon
NumPy, SciPy, Pandas, and Scikit-Learn
NumPy and SciPy
NumPy and pandas
SciPy and scikit-learn
Summary
Advanced Numpy Chevron down icon Chevron up icon
Advanced Numpy
NumPy internals
Summary
Overview of High-Performance Numerical Computing Libraries Chevron down icon Chevron up icon
Overview of High-Performance Numerical Computing Libraries
BLAS and LAPACK
ATLAS
Intel Math Kernel Library
OpenBLAS
Configuring NumPy with low-level libraries using AWS EC2
Compute-intensive tasks for benchmarking
Summary
Performance Benchmarks Chevron down icon Chevron up icon
Performance Benchmarks
Why do we need a benchmark?
Preparing for a performance benchmark
Results
Summary
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Mert Cakmak
Umit Mert Cakmak is a data scientist at IBM, where he excels at helping clients solve complex data science problems, from inception to delivery of deployable assets. His research spans multiple disciplines beyond his industry and he likes sharing his insights at conferences, universities, and meet-ups.
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Tiago Antao
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Tiago Antao is a bioinformatician currently working in the field of genomics. A former computer scientist, Tiago moved into computational biology with an MSc in Bioinformatics from the Faculty of Sciences at the University of Porto (Portugal) and a PhD on the spread of drug-resistant malaria from the Liverpool School of Tropical Medicine (UK). Postdoctoral, Tiago has worked with human datasets at the University of Cambridge (UK) and with mosquito whole genome sequencing data at the University of Oxford (UK), before helping to set up the bioinformatics infrastructure at the University of Montana. He currently works as a data engineer in the biotechnology field in Boston, MA. He is one of the co-authors of Biopython, a major bioinformatics package written in Python.
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Mert Cuhadaroglu is a BI Developer in EPAM, developing E2E analytics solutions for complex business problems in various industries, mostly investment banking, FMCG, media, communication, and pharma. He consistently uses advanced statistical models and ML algorithms to provide actionable insights. Throughout his career, he has worked in several other industries, such as banking and asset management. He continues his academic research in AI for trading algorithms.
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  2. Sadly, if your eBook or Video file is faulty or a fault occurs during the eBook or Video being made available to you, i.e. during download then you should contact Customer Relations Team within 14 days of purchase on customercare@packt.com who will be able to resolve this issue for you.
  3. You will have a choice of replacement or refund of the problem items.(damaged, defective or incorrect)
  4. Once Customer Care Team confirms that you will be refunded, you should receive the refund within 10 to 12 working days.
  5. If you are only requesting a refund of one book from a multiple order, then we will refund you the appropriate single item.
  6. Where the items were shipped under a free shipping offer, there will be no shipping costs to refund.

On the off chance your printed book arrives damaged, with book material defect, contact our Customer Relation Team on customercare@packt.com within 14 days of receipt of the book with appropriate evidence of damage and we will work with you to secure a replacement copy, if necessary. Please note that each printed book you order from us is individually made by Packt's professional book-printing partner which is on a print-on-demand basis.

What tax is charged? Chevron down icon Chevron up icon

Currently, no tax is charged on the purchase of any print book (subject to change based on the laws and regulations). A localized VAT fee is charged only to our European and UK customers on eBooks, Video and subscriptions that they buy. GST is charged to Indian customers for eBooks and video purchases.

What payment methods can I use? Chevron down icon Chevron up icon

You can pay with the following card types:

  1. Visa Debit
  2. Visa Credit
  3. MasterCard
  4. PayPal
What is the delivery time and cost of print books? Chevron down icon Chevron up icon

Shipping Details

USA:

'

Economy: Delivery to most addresses in the US within 10-15 business days

Premium: Trackable Delivery to most addresses in the US within 3-8 business days

UK:

Economy: Delivery to most addresses in the U.K. within 7-9 business days.
Shipments are not trackable

Premium: Trackable delivery to most addresses in the U.K. within 3-4 business days!
Add one extra business day for deliveries to Northern Ireland and Scottish Highlands and islands

EU:

Premium: Trackable delivery to most EU destinations within 4-9 business days.

Australia:

Economy: Can deliver to P. O. Boxes and private residences.
Trackable service with delivery to addresses in Australia only.
Delivery time ranges from 7-9 business days for VIC and 8-10 business days for Interstate metro
Delivery time is up to 15 business days for remote areas of WA, NT & QLD.

Premium: Delivery to addresses in Australia only
Trackable delivery to most P. O. Boxes and private residences in Australia within 4-5 days based on the distance to a destination following dispatch.

India:

Premium: Delivery to most Indian addresses within 5-6 business days

Rest of the World:

Premium: Countries in the American continent: Trackable delivery to most countries within 4-7 business days

Asia:

Premium: Delivery to most Asian addresses within 5-9 business days

Disclaimer:
All orders received before 5 PM U.K time would start printing from the next business day. So the estimated delivery times start from the next day as well. Orders received after 5 PM U.K time (in our internal systems) on a business day or anytime on the weekend will begin printing the second to next business day. For example, an order placed at 11 AM today will begin printing tomorrow, whereas an order placed at 9 PM tonight will begin printing the day after tomorrow.


Unfortunately, due to several restrictions, we are unable to ship to the following countries:

  1. Afghanistan
  2. American Samoa
  3. Belarus
  4. Brunei Darussalam
  5. Central African Republic
  6. The Democratic Republic of Congo
  7. Eritrea
  8. Guinea-bissau
  9. Iran
  10. Lebanon
  11. Libiya Arab Jamahriya
  12. Somalia
  13. Sudan
  14. Russian Federation
  15. Syrian Arab Republic
  16. Ukraine
  17. Venezuela