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Mastering SciPy
Mastering SciPy

Mastering SciPy: Implement state-of-the-art techniques to visualize solutions to challenging problems in scientific computing, with the use of the SciPy stack

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Profile Icon Francisco Javier B Silva Profile Icon Blanco-Silva
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Profile Icon Francisco Javier B Silva Profile Icon Blanco-Silva
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Paperback Nov 2015 404 pages 1st Edition
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Mastering SciPy

Chapter 2. Interpolation and Approximation

Approximation theory states how to find the best approximation to a given function by another function from some predetermined class and how good this approximation is. In this chapter, we are going to explore this field through two settings: interpolation and least squares approximation.

Motivation

Consider a meteorological experiment that measures the temperature of a set of buoys located on a rectangular grid at sea. We can emulate such an experiment by indicating the longitude and latitude of the buoys on a grid of 16 × 16 locations, and random temperatures on them between say 36ºF and 46ºF:

In [1]: import numpy as np, matplotlib.pyplot as plt, \
   ...: matplotlib.cm as cm; \
   ...: from mpl_toolkits.basemap import Basemap
In [2]: map1 = Basemap(projection='ortho', lat_0=20, lon_0=-60, \
   ...:                resolution='l', area_thresh=1000.0); \
   ...: map2 = Basemap(projection='merc', lat_0=20, lon_0=-60, \
   ...:                resolution='l', area_thresh=1000.0, \
   ...:                llcrnrlat=0,  urcrnrlat=45, \
   ...:                llcrnrlon=-75, urcrnrlon=-15)
In [3]: longitudes = np.linspace(-60, -30, 16); \
   ...: latitudes = np.linspace(15, 30, 16); \
   ...: lons, lats = np.meshgrid(longitudes...

Interpolation

We have three different implementation methodologies to deal with interpolation problems:

  • A procedural mode that computes a set of data points (in the form of ndarray with the required dimension) representing the actual solution.
  • In a few special cases, a functional mode that provides us with numpy functions representing the solutions.
  • An object-oriented mode that creates classes for interpolation problems. Different classes have different methods, depending on the operations that the particular kinds of interpolants enjoy. The advantage of this mode is that, through these methods, we can request more information from the solutions: not only evaluation or representation, but also relevant operations like searching for roots, computing derivatives and antiderivatives, error checking, and calculating coefficients and knots.

The choice of mode to represent our interpolants is up to us, depending mostly on how much accuracy we require, and the information/operations that we need afterwards...

Least squares approximation

Numerically, it is relatively simple to state the approximation problem for the least squares norm. This is the topic of this section.

Linear least squares approximation

In the context of linear least squares approximation, it is always possible to reduce the problem to solving a system of linear equations, as the following example shows:

Consider the sine function f(x) = sin(x) in the interval from 0 to 1. We choose as approximants the polynomials of second degree: {a0 + a1x + a2x2}. To compute the values [a0, a1, a2] that minimize this problem, we first form a 3 × 3 matrix containing the pairwise dot products (the integral of the product of two functions) of the basic functions {1, x, x2} in the given interval. Because of the nature of this problem, we obtain a Hilbert matrix of order 3:

[   < 1, 1 >    < 1, x >    < 1, x^2 > ]     [  1   1/2  1/3 ]
[   < x, 1 >    < x, x >    < x, x^2 > ]  =  [ 1/2  1/3  1/4 ]
[ < x...

Summary

In this chapter, we have explored two basic problems in the field of approximation theory: interpolation and approximation in the sense of least squares. We learned that there are three different modes to approach solutions to these problems in SciPy:

  • A procedural mode, that offers quick numerical solutions in the form of ndarrays.
  • A functional mode that offers numpy functions as the output.
  • An object-oriented mode, with great flexibility through different classes and their methods. We use this mode when we require from our solutions extra information (such as information about roots, coefficients, knots, and errors), or related objects (such as the representation of derivatives or antiderivatives).

We explored in detail all the different implementations for the interpolation coded in the scipy.interpolate module, and learned in particular that those related to splines are wrappers of several routines in the Fortran library FITPACK.

In the case of linear approximations in the least squares...

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Description

The SciPy stack is a collection of open source libraries of the powerful scripting language Python, together with its interactive shells. This environment offers a cutting-edge platform for numerical computation, programming, visualization and publishing, and is used by some of the world’s leading mathematicians, scientists, and engineers. It works on any operating system that supports Python and is very easy to install, and completely free of charge! It can effectively transform into a data-processing and system-prototyping environment, directly rivalling MATLAB and Octave. This book goes beyond a mere description of the different built-in functions coded in the libraries from the SciPy stack. It presents you with a solid mathematical and computational background to help you identify the right tools for each problem in scientific computing and visualization. You will gain an insight into the best practices with numerical methods depending on the amount or type of data, properties of the mathematical tools employed, or computer architecture, among other factors. The book kicks off with a concise exploration of the basics of numerical linear algebra and graph theory for the treatment of problems that handle large data sets or matrices. In the subsequent chapters, you will delve into the depths of algorithms in symbolic algebra and numerical analysis to address modeling/simulation of various real-world problems with functions (through interpolation, approximation, or creation of systems of differential equations), and extract their representing features (zeros, extrema, integration or differentiation). Lastly, you will move on to advanced concepts of data analysis, image/signal processing, and computational geometry.

Who is this book for?

If you are a professional with a proficiency in Python and familiarity with IPython, this book is for you. Some basic knowledge of numerical methods in scientific computing would be helpful.

What you will learn

  • Master relevant algorithms used in symbolic or numerical mathematics to address the approximation, interpolation, and optimization of scalar or multivariate functions
  • Develop different algorithms and strategies to effectively store and manipulate large matrices of data, with a view to solving various problems in numerical linear algebra
  • Understand how to model physical problems with systems of differential equations and distinguish the factors that dictate the strategies to solve them numerically
  • Perform statistical analysis, inference, data mining, and machine learning at higher level, and apply these to realworld problems
  • Adapt valuable ideas in computational geometry like Delaunay triangulations, Voronoi diagrams, geometric query problems, or Bezier curves, and apply them to various engineering problems
  • Familiarize yourself with different methods to represent and compress images, as well as techniques used in image processing, including edition, restoration, inpainting, segmentation, or feature recognition
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Table of Contents

10 Chapters
1. Numerical Linear Algebra Chevron down icon Chevron up icon
2. Interpolation and Approximation Chevron down icon Chevron up icon
3. Differentiation and Integration Chevron down icon Chevron up icon
4. Nonlinear Equations and Optimization Chevron down icon Chevron up icon
5. Initial Value Problems for Ordinary Differential Equations Chevron down icon Chevron up icon
6. Computational Geometry Chevron down icon Chevron up icon
7. Descriptive Statistics Chevron down icon Chevron up icon
8. Inference and Data Analysis Chevron down icon Chevron up icon
9. Mathematical Imaging Chevron down icon Chevron up icon
Index Chevron down icon Chevron up icon

Customer reviews

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Full star icon Full star icon Full star icon Half star icon Empty star icon 3.5
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5 star 50%
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1 star 0%
Raiyan Kamal Dec 29, 2015
Full star icon Full star icon Full star icon Full star icon Full star icon 5
I'm one of the technical reviewers of this book. The book is written for readers who are already familiar with Python and some basic knowledge on various numerical methods. This book will help you a great deal to familiarize with the various Python modules for well known scientific computing use cases.Many important topics of scientific computing are covered, including:- Differential Equations- Optimization- Computational Geometry- Image processingEach chapter takes the reader through steps working on an example problem with gradually increasing complexity. Frequent code snippets from IPython notebook and matplotlib charts make the examples easy to follow. Diagrams, textual and image output is also shown whenever necessary.I'm happy to have reviewed this book and I'm very well pleased with the result. I believe this book would prove to be useful for anyone interested in harnessing the power of SciPy stack for their scientific or engineering profession or graduate level research.
Amazon Verified review Amazon
William J. Brown Nov 23, 2021
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There is much that can be learned from the book, but it was written in 2015 and the code hasn't been updated for python3, so much of it is unusable at this point. Also, so much math is required to exercise SciPy that it's difficult write a book that shows off the library without wandering down a very large number of paths into arcane algorithms each with a limited audience.
Amazon Verified review Amazon
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