Book Image

Applying Math with Python

By : Sam Morley

Book Image

Applying Math with Python

By: Sam Morley

Overview of this book

Python, one of the world's most popular programming languages, has a number of powerful packages to help you tackle complex mathematical problems in a simple and efficient way. These core capabilities help programmers pave the way for building exciting applications in various domains, such as machine learning and data science, using knowledge in the computational mathematics domain. The book teaches you how to solve problems faced in a wide variety of mathematical fields, including calculus, probability, statistics and data science, graph theory, optimization, and geometry. You'll start by developing core skills and learning about packages covered in Python’s scientific stack, including NumPy, SciPy, and Matplotlib. As you advance, you'll get to grips with more advanced topics of calculus, probability, and networks (graph theory). After you gain a solid understanding of these topics, you'll discover Python's applications in data science and statistics, forecasting, geometry, and optimization. The final chapters will take you through a collection of miscellaneous problems, including working with specific data formats and accelerating code. By the end of this book, you'll have an arsenal of practical coding solutions that can be used and modified to solve a wide range of practical problems in computational mathematics and data science.

Preface

Who this book is for

What this book covers

To get the most out of this book

Download the example code files

Conventions used

How to do it…

How it works…

There's more…

Basic Packages, Functions, and Concepts

Basic Packages, Functions, and Concepts

Technical requirements

Python numerical types

Basic mathematical functions

Further reading

Free Chapter

Mathematical Plotting with Matplotlib

Mathematical Plotting with Matplotlib

Technical requirements

Basic plotting with Matplotlib

Changing the plotting style

Adding labels and legends to plots

Adding subplots

Saving Matplotlib figures

Surface and contour plots

Customizing three-dimensional plots

Further reading

Calculus and Differential Equations

Calculus and Differential Equations

Technical requirements

Working with polynomials and calculus

Differentiating and integrating symbolically using SymPy

Solving equations

Integrating functions numerically using SciPy

Solving simple differential equations numerically

Solving systems of differential equations

Solving partial differential equations numerically

Using discrete Fourier transforms for signal processing

Further reading

Working with Randomness and Probability

Working with Randomness and Probability

Technical requirements

Selecting items at random

Generating random data

Changing the random number generator

Generating normally distributed random numbers

Working with random processes

Analyzing conversion rates with Bayesian techniques

Estimating parameters with Monte Carlo simulations

Further reading

Working with Trees and Networks

Working with Trees and Networks

Technical requirements

Creating networks in Python

Visualizing networks

Getting the basic characteristics of networks

Generating the adjacency matrix for a network

Creating directed and weighted networks

Finding the shortest paths in a network

Quantifying clustering in a network

Coloring a network

Finding minimal spanning trees and dominating sets

Further reading

Working with Data and Statistics

Working with Data and Statistics

Technical requirements

Creating Series and DataFrame objects

Loading and storing data from a DataFrame

Manipulating data in DataFrames

Plotting data from a DataFrame

Getting descriptive statistics from a DataFrame

Understanding a population using sampling

Testing hypotheses using t-tests

Testing hypotheses using ANOVA

Testing hypotheses for non-parametric data

Creating interactive plots with Bokeh

Further reading

Regression and Forecasting

Regression and Forecasting

Technical requirements

Using basic linear regression

Using multilinear regression

Classifying using logarithmic regression

Modeling time series data with ARMA

Forecasting from time series data using ARIMA

Forecasting seasonal data using ARIMA

Using Prophet to model time series data

Further reading

Geometric Problems

Geometric Problems

Technical requirements

Visualizing two-dimensional geometric shapes

Finding interior points

Finding edges in an image

Triangulating planar figures

Computing convex hulls

Constructing Bezier curves

Further reading

Finding Optimal Solutions

Finding Optimal Solutions

Technical requirements

Minimizing a simple linear function

Minimizing a non-linear function

Using gradient descent methods in optimization

Using least squares to fit a curve to data

Analyzing simple two-player games

Computing Nash equilibria

Further reading

Miscellaneous Topics

Miscellaneous Topics

Technical requirements

Keeping track of units with Pint

How to do it...

How it works...

There's more...

Accounting for uncertainty in calculations

How to do it...

How it works...

There's more...

Loading and storing data from NetCDF files

How to do it...

How it works...

There's more...

Working with geographical data

How to do it...

How it works...

Executing a Jupyter notebook as a script

How to do it...

How it works...

There's more...

Validating data

How to do it...

How it works...

Working with data streams

How to do it...

How it works...

Accelerating code with Cython

How to do it...

How it works...

There's more...

Distributing computing with Dask

How to do it...

How it works...

There's more...

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Further reading

There are many mathematical textbooks describing the basic properties of matrices and linear algebra, which is the study of vectors and matrices. A good introductory text is Blyth, T. and Robertson, E. (2013). Basic Linear Algebra. London: Springer London, Limited.

NumPy and SciPy are part of the Python mathematical and scientific computing ecosystem, and have extensive documentation that can be accessed from the official website, https://scipy.org. We will see several other packages from this ecosystem throughout this book.

More information about the BLAS and LAPACK libraries that NumPy and SciPy use behind the scenes can be found at the following links: BLAS: https://www.netlib.org/blas/ and LAPACK: https://www.netlib.org/lapack/.