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15 Math Concepts Every Data Scientist Should Know

You're reading from   15 Math Concepts Every Data Scientist Should Know Understand and learn how to apply the math behind data science algorithms

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Product type Paperback
Published in Aug 2024
Publisher Packt
ISBN-13 9781837634187
Length 510 pages
Edition 1st Edition
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Author (1):
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David Hoyle David Hoyle
Author Profile Icon David Hoyle
David Hoyle
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Table of Contents (21) Chapters Close

Preface 1. Part 1: Essential Concepts
2. Chapter 1: Recap of Mathematical Notation and Terminology FREE CHAPTER 3. Chapter 2: Random Variables and Probability Distributions 4. Chapter 3: Matrices and Linear Algebra 5. Chapter 4: Loss Functions and Optimization 6. Chapter 5: Probabilistic Modeling 7. Part 2: Intermediate Concepts
8. Chapter 6: Time Series and Forecasting 9. Chapter 7: Hypothesis Testing 10. Chapter 8: Model Complexity 11. Chapter 9: Function Decomposition 12. Chapter 10: Network Analysis 13. Part 3: Selected Advanced Concepts
14. Chapter 11: Dynamical Systems 15. Chapter 12: Kernel Methods 16. Chapter 13: Information Theory 17. Chapter 14: Non-Parametric Bayesian Methods 18. Chapter 15: Random Matrices 19. Index 20. Other Books You May Enjoy

Inner and outer products of vectors

As we have said, vectors are the natural way to store data points that have many features. You probably already have some experience with manipulating vectors or arrays, for example, by performing element-wise calculations on them. But we want to do more than that. We want to combine vectors. This section introduces the two most basic but important operations we can apply to two vectors – the inner product and the outer product.

Inner product of two vectors

To calculate the inner product between two vectors, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math>, they need to be of equal length. In this and all subsequent calculations, we will assume that our vectors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> are real-valued. If those vectors are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>d</mml:mi></mml:math>-dimensional and have components <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>a</mi><mi>d</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>b</mi><mi>d</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></mrow></math>, then the inner product between them is denoted by the symbol <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo>∙</mml:mo><mml:munder underaccent="false"><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> and is defined as follows:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><munder><mi>a</mi><mo stretchy="true">_</mo></munder><mo>⋅</mo><munder><mi>b</mi><mo stretchy="true">_</mo></munder><mo>=</mo><mspace width="0.25em" /><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><mrow><msub><mi>a</mi><mi>i</mi></msub><msub><mi>b</mi><mi>i</mi></msub></mrow></mrow></mrow></mrow></math>

Eq. 1

Because of the dot between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> in the left-hand side of Eq. 1, the inner product is also called the dot-product of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:munder underaccent="false"><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math>. The result...

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