# Essential Math for Data Science: New Chapters

I’m glad to announce a few updates concerning my book Essential Math for Data Science.

First, I changed the structure of the book: a first chapter on basic algebra has been removed. Part of old chapter 02 has been merged in the linear algebra part.

I restructured the table of content: I removed some content about very basic math (like what is an equation or a function) to have more space to cover slighly more advanced contents. The part

*Statistics and Probability*is now at the beginning of the book (just after a first part on Calculus). Have a look at the new TOC below to have more details.There is now

*build**hands-on projects*for each chapter. Hands-on projects are sections where you can apply the math you just learned to a practical machine learning problem (like gradient descent or regularization, for instance). The difficulty of math and code in each of these hands-on project is variable, so you should find something at the right point of your learning curve.

Here is the table of content. Click on the chapters to see what’s inside.

### Table of Content

*expand_more**expand_less*01. How to use this Book

#### PART 1. Calculus

Machine learning and data science require some experience with calculus. You’ll be introduced here to derivatives and integrals and how they are useful in machine learning and data science.

*expand_more**expand_less*02. Calculus: Derivatives and Integrals*Area under the curve**expand_more**expand_less*2.1 Derivatives2.1.1 Introduction

2.1.2 Mathematical Definition of Derivatives

2.1.3 Derivatives of Linear And Nonlinear Functions

2.1.4 Derivative Rules

2.1.5 Partial Derivatives And Gradients

*expand_more**expand_less*2.2 Integrals And Area Under The Curve2.2.1 Example

2.2.2 Riemann Sum

2.2.3 Mathematical Definition

*build*2.3 Hands-On Project: Gradient Descent2.3.1 Cost function

2.3.2 Derivative of the Cost Function

2.3.3 Implementing Gradient Descent

2.3.4 MSE Cost Function With Two Parameters

#### PART 2. Statistics and Probability

In machine learning and data science, probability and statistics are used to deal with uncertainty. This uncertainty comes from various sources, from data collection to the process you’re trying to model itself. This part will introduce you to descriptive statistics, probability distributions, bayesian statistics and information theory.

*expand_more**expand_less*03. Statistics and Probability Theory*Joint and Marginal Probability**expand_more**expand_less*3.1 Descriptive Statistics3.1.1 Variance and Standard Deviation

3.1.2 Covariance and Correlation

3.1.3 Covariance Matrix

*expand_more**expand_less*3.2 Random Variables3.2.1 Definitions and Notation

3.2.2 Discrete and Continuous Random Variables

*expand_more**expand_less*3.3 Probability Distributions3.3.1 Probability Mass Functions

3.3.2 Probability Density Functions

2.3.3 Implementing Gradient Descent

2.3.4 MSE Cost Function With Two Parameters

*expand_more**expand_less*3.4 Joint, Marginal, and Conditional Probability3.4.1 Joint Probability

3.4.2 Marginal Probability

3.4.3 Conditional Probability

*expand_more**expand_less*3.5 Cumulative Distribution Functions*expand_more**expand_less*3.6 Expectation and Variance of Random Variables3.6.1 Discrete Random Variables

3.6.2 Continuous Random Variables

3.6.3 Variance of Random Variables

*build*3.7 Hands-On Project: The Central Limit Theorem3.7.1 Continuous Distribution

3.7.2 Discrete Distribution

*expand_more**expand_less*04. Common Probability Distributions*Gaussian Distributions**expand_more**expand_less*4.1 Uniform Distribution*expand_more**expand_less*4.2 Gaussian distribution4.2.1 Formula

4.2.2 Parameters

4.2.3 Requirements

*expand_more**expand_less*4.3 Bernoulli Distribution*expand_more**expand_less*4.4 Binomial Distribution4.4.1 Description

4.4.2 Graphical Representation

*expand_more**expand_less*4.5 Poisson Distribution4.5.1 Mathematical Definition

4.5.2 Example

*expand_more**expand_less*4.6 Exponential Distribution4.6.1 Derivation from the Poisson Distribution

4.6.2 Effect of λ

*build*4.7 Hands-on Project: Waiting for the Bus

*expand_more**expand_less*05. Bayesian Statistics and Information Theory*Bayesian Inference**expand_more**expand_less*5.1 Bayes’ Theorem5.1.1 Mathematical Formulation

5.1.2 Example

5.1.3 Bayesian Interpretation

5.1.4 Bayes’ Theorem with Distributions

*expand_more**expand_less*5.2 Likelihood5.2.1 Introduction and Notation

5.2.2 Finding the Parameters of the Distribution

5.2.3 Maximum Likelihood Estimation

*expand_more**expand_less*5.3 Information Theory5.3.1 Shannon Information

5.3.2 Entropy

5.3.3 Cross Entropy

5.3.4 Kullback-Leibler Divergence (KL Divergence)

*build*5.4 Hands-On Project: Bayesian Inference5.4.1 Advantages of Bayesian Inference

5.4.2 Project

#### PART 3. Linear Algebra

Linear algebra is the core of many machine learning algorithms. The great news is that you don’t need to be able to code these algorithms yourself. It is more likely that you’ll use a great Python library instead. However, to be able to choose the right model for the right job, or to debug a broken machine learning pipeline, it is crucial to have enough understanding of what’s under the hood. The goal of this part is to give you enough understanding and intuition about the major concepts of linear algebra used in machine and data science. It is designed to be accessible, even if you never studied linear algebra.

*expand_more**expand_less*06. Scalars and Vectors*L1 Regularization. Effect of Lambda.**expand_more**expand_less*6.1 What Vectors are?6.1.1 Geometric and Coordinate Vectors

6.1.2 Vector Spaces

6.1.3 Special Vectors

*expand_more**expand_less*6.2 Operations and Manipulations on Vectors6.2.1 Scalar Multiplication

6.2.2 Vector Addition

6.2.3 Transposition

*expand_more**expand_less*6.3 Norms6.3.1 Definitions

6.3.2 Common Vector Norms

6.3.3 Norm Representations

*expand_more**expand_less*6.4 The Dot Product6.4.1 Definition

6.4.2 Geometric interpretation: Projections

6.4.3 Properties

*build*6.5 Hands-on Project: Regularization6.5.1 Introduction

6.5.2 Effect of Regularization on Polynomial Regression

6.5.3 Differences between $L^1$ and $L^2$ Regularization

*expand_more**expand_less*07. Matrices and Tensors*Scalars, vectors, matrices and tensors**expand_more**expand_less*7.1 Introduction7.1.1 Matrix Notation

7.1.2 Shapes

7.1.3 Indexing

7.1.4 Main Diagonal

7.1.5 Tensors

7.1.6 Frobenius Norm

*expand_more**expand_less*7.2 Operations and Manipulations on Matrices7.2.1 Addition and Scalar Multiplication

7.2.2 Transposition

*expand_more**expand_less*7.3 Matrix Product7.3.1 Matrices with Vectors

7.3.2 Matrices Product

7.3.3 Transpose of a Matrix Product

*expand_more**expand_less*7.4 Special Matrices7.4.1 Square Matrices

7.4.2 Diagonal Matrices

7.4.3 Identity Matrices

7.4.4 Inverse Matrices

7.4.5 Orthogonal Matrices

7.4.6 Symmetric Matrices

7.4.7 Triangular Matrices

*build*7.5 Hands-on Project: Image Classifier7.5.1 Images as Multi-dimensional Arrays

7.5.2 Data Preparation

*expand_more**expand_less*08. Span, Linear Dependency, and Space Transformation*All linear Combinations of two vectors**expand_more**expand_less*8.1 Linear Transformations8.1.1 Intuition

8.1.2 Linear Transformations as Vectors and Matrices

8.1.3 Geometric Interpretation

8.1.4 Special Cases

*expand_more**expand_less*8.2 Linear combination8.2.1 Intuition

8.2.2 All combinations of vectors

8.2.3 Span

*expand_more**expand_less*8.3 Subspaces8.3.1 Definitions

8.3.2 Subspaces of a Matrix

*expand_more**expand_less*8.4 Linear dependency8.4.1 Geometric Interpretation

8.4.2 Matrix View

*expand_more**expand_less*8.5 Basis8.5.1 Definitions

8.5.2 Linear Combination of Basis Vectors

8.5.3 Other Bases

*expand_more**expand_less*8.6 Special Characteristics8.6.1 Rank

8.6.2 Trace

8.6.3 Determinant

*build*8.7 Hands-On Project: Span

*expand_more**expand_less*09. Systems of Linear Equations*Projection of a vector onto a plane**expand_more**expand_less*9.1 System of linear equations9.1.1 Row Picture

9.1.2 Column Picture

9.1.3 Number of Solutions

9.1.4 Representation of Linear Equations With Matrices

*expand_more**expand_less*9.2 System Shape9.2.1 Overdetermined Systems of Equations

9.2.2 Underdetermined Systems of Equations

*expand_more**expand_less*9.3 Projections9.3.1 Solving Systems of Equations

9.3.2 Projections to Approximate Unsolvable Systems

9.3.3 Projections Onto a Line

9.3.4 Projections Onto a Plane

*build*9.4 Hands-on Project: Linear Regression Using Least Approximation9.4.1 Linear Regression Using the Normal Equation

9.4.2 Relationship Between Least Squares and the Normal Equation

*expand_more**expand_less*10. Eigenvectors and Eigenvalues*Principal Component Analysis on audio samples.**expand_more**expand_less*10.1 Eigenvectors and Linear Transformations*expand_more**expand_less*10.2 Change of Basis10.2.1 Linear Combinations of the Basis Vectors

10.2.2 The Change of Basis Matrix

10.2.3 Example: Changing the Basis of a Vector

*expand_more**expand_less*10.3 Linear Transformations in Different Bases10.3.1 Transformation Matrix

10.3.2 Transformation Matrix in Another Basis

10.3.3 Interpretation

*expand_more**expand_less*10.4 Eigendecomposition10.4.1 First Step: Change of Basis

10.4.2 Eigenvectors and Eigenvalues

10.4.3 Diagonalization

10.4.4 Eigendecomposition of Symmetric Matrices

*build*10.5 Hands-On Project: Principal Component Analysis10.5.1 Under the Hood

10.5.2 Making Sense of Audio

*expand_more**expand_less*11. Singular Value Decomposition*SVD Geometry**expand_more**expand_less*11.1 Nonsquare Matrices11.1.1 Different Input and Output Spaces

11.1.2 Specifying the Bases

11.1.3 Eigendecomposition is Only for Square Matrices

*expand_more**expand_less*11.2 Expression of the SVD11.2.1 From Eigendecomposition to SVD

11.2.2 Singular Vectors and Singular Values

11.2.3 Finding the Singular Vectors and the Singular Values

11.2.4 Summary

*expand_more**expand_less*11.3 Geometry of the SVD11.3.1 Two-Dimensional Example

11.3.2 Comparison with Eigendecomposition

11.3.3 Three-Dimensional Example

11.3.4 Summary

*expand_more**expand_less*11.4 Low-Rank Matrix Approximation11.4.1 Full SVD, Thin SVD and Truncated SVD

11.4.2 Decomposition into Rank One Matrices

*build**build*11.5 Hands-On Project: Image Compression

I hope that you’ll find this content useful! Feel free to contact me if you have any question, request, or feedback!