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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 FREE CHAPTER
2. Chapter 1: Recap of Mathematical Notation and Terminology 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

The kernel trick

To learn how the kernel trick allows us to do feature construction implicitly and efficiently, we will first have to learn what a kernel is.

What is a kernel?

The simplest way to think about a kernel is to consider it as a mapping that takes two vectors as input and returns a scalar. It is a mapping that maps <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></mrow></math>. This means that a kernel is a function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>f</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:munder underaccent="false"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:munder underaccent="false"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:mrow></mml:mfenced></mml:math>, with the input vectors being <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>x</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>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo>.</mml:mo></mml:math> The value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>f</mml:mi></mml:math> is a real number. This means that the inner product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:msup><mml:mrow><mml:munder underaccent="false"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:mrow><mml:mrow><mml:mi mathvariant="normal">⊤</mml:mi></mml:mrow></mml:msup><mml:munder underaccent="false"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> is an example of a kernel function.

That is a high-level mathematical definition of what a kernel is, but what is the intuition behind this? An <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>f</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:munder underaccent="false"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:munder underaccent="false"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:mrow></mml:mfenced></mml:math> kernel function applied to the 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>x</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>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> is typically used to measure the similarity between those vectors. Consequently, we usually want our kernel function to have its largest values when <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>x</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>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> are most similar and its lowest values when <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>x</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>y</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math> are least similar. We want the function to decrease smoothly and monotonically in between those two scenarios.

Commonly...

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