Data Science

2.

Mathematical and Statistical Foundations

2.1.

2.1.1.

Vectors and Vector Operations

2.1.1.1.

Vector Definition

2.1.1.2.

Vector Addition and Subtraction

2.1.1.3.

Scalar Multiplication

2.1.1.4.

2.1.1.5.

2.1.1.6.

2.1.2.

2.1.2.1.

Vector Space Definition

2.1.2.2.

2.1.2.3.

Linear Independence

2.1.2.4.

Basis and Dimension

2.1.2.5.

2.1.3.

Matrices and Matrix Operations

2.1.3.1.

Matrix Definition

2.1.3.2.

Matrix Addition and Subtraction

2.1.3.3.

Matrix Multiplication

2.1.3.4.

2.1.3.5.

2.1.3.6.

2.1.3.7.

2.1.4.

Systems of Linear Equations

2.1.4.1.

Gaussian Elimination

2.1.4.2.

Matrix Form of Linear Systems

2.1.4.3.

Solution Methods

2.1.5.

Eigenvalues and Eigenvectors

2.1.5.1.

2.1.5.2.

Characteristic Equation

2.1.5.3.

Eigendecomposition

2.1.5.4.

Diagonalization

2.1.5.5.

Applications in Data Science

2.1.6.

Singular Value Decomposition

2.1.6.1.

2.1.6.2.

2.1.6.3.

Applications in Dimensionality Reduction

2.1.6.4.

Applications in Recommendation Systems

2.2.

2.2.1.

Limits and Continuity

2.2.1.1.

Limit Definition

2.2.1.2.

2.2.1.3.

Applications to Optimization

2.2.2.

2.2.2.1.

Derivative Definition

2.2.2.2.

Differentiation Rules

2.2.2.3.

2.2.2.4.

Applications in Machine Learning

2.2.3.

Partial Derivatives

2.2.3.1.

Multivariable Functions

2.2.3.2.

Partial Derivative Definition

2.2.3.3.

Gradient Vectors

2.2.3.4.

Directional Derivatives

2.2.4.

2.2.4.1.

Critical Points

2.2.4.2.

Local and Global Extrema

2.2.4.3.

Lagrange Multipliers

2.2.4.4.

Gradient Descent

2.2.4.5.

Stochastic Gradient Descent

2.2.4.6.

Learning Rate and Convergence

2.2.5.

Multivariable Calculus

2.2.5.1.

Jacobian Matrices

2.2.5.2.

Hessian Matrices

2.2.5.3.

Taylor Series Expansion

2.3.

Probability Theory

2.3.1.

Basic Probability Concepts

2.3.1.1.

Sample Spaces and Events

2.3.1.2.

Probability Axioms

2.3.1.3.

Probability Rules

2.3.1.4.

2.3.2.

Conditional Probability

2.3.2.1.

2.3.2.2.

Multiplication Rule

2.3.2.3.

Law of Total Probability

2.3.3.

2.3.3.1.

Statement and Proof

2.3.3.2.

Applications in Machine Learning

2.3.3.3.

Naive Bayes Classifier Foundation

2.3.4.

Random Variables

2.3.4.1.

Discrete Random Variables

2.3.4.2.

Continuous Random Variables

2.3.4.3.

Probability Mass Functions

2.3.4.4.

Probability Density Functions

2.3.4.5.

Cumulative Distribution Functions

2.3.5.

Expected Value and Variance

2.3.5.1.

Expected Value Definition

2.3.5.2.

Properties of Expected Value

2.3.5.3.

Variance Definition

2.3.5.4.

Properties of Variance

2.3.5.5.

2.3.5.6.

2.3.6.

Common Probability Distributions

2.3.6.1.

Discrete Distributions

2.3.6.1.1.

Bernoulli Distribution

2.3.6.1.2.

Binomial Distribution

2.3.6.1.3.

Poisson Distribution

2.3.6.1.4.

Geometric Distribution

2.3.6.1.5.

Negative Binomial Distribution

2.3.6.2.

Continuous Distributions

2.3.6.2.1.

Uniform Distribution

2.3.6.2.2.

Normal Distribution

2.3.6.2.3.

Exponential Distribution

2.3.6.2.4.

Gamma Distribution

2.3.6.2.5.

Beta Distribution

2.3.6.2.6.

Chi-Square Distribution

2.3.6.2.7.

2.3.6.2.8.

2.3.7.

Joint Distributions

2.3.7.1.

Joint Probability Mass Functions

2.3.7.2.

Joint Probability Density Functions

2.3.7.3.

Marginal Distributions

2.3.7.4.

Independence of Random Variables

2.3.8.

Central Limit Theorem

2.3.8.1.

2.3.8.2.

Sampling Distributions

2.4.

Descriptive Statistics

2.4.1.

Measures of Central Tendency

2.4.1.1.

2.4.1.2.

2.4.1.3.

2.4.1.4.

2.4.1.5.

2.4.2.

Measures of Dispersion

2.4.2.1.

2.4.2.2.

2.4.2.3.

Standard Deviation

2.4.2.4.

Interquartile Range

2.4.2.5.

Mean Absolute Deviation

2.4.2.6.

Coefficient of Variation

2.4.3.

Measures of Shape

2.4.3.1.

2.4.3.2.

2.4.4.

Measures of Position

2.4.4.1.

2.4.4.2.

2.4.4.3.

2.4.5.

Correlation and Association

2.4.5.1.

Pearson Correlation Coefficient

2.4.5.2.

Spearman Rank Correlation

2.4.5.3.

2.4.5.4.

2.4.5.5.

Correlation vs Causation

2.5.

Inferential Statistics

2.5.1.

Sampling Theory

2.5.1.1.

Population vs Sample

2.5.1.2.

Sampling Methods

2.5.1.2.1.

Simple Random Sampling

2.5.1.2.2.

Stratified Sampling

2.5.1.2.3.

Cluster Sampling

2.5.1.2.4.

Systematic Sampling

2.5.1.3.

2.5.1.4.

Sampling Distributions

2.5.2.

2.5.2.1.

Point Estimation

2.5.2.2.

Properties of Estimators

2.5.2.3.

Method of Moments

2.5.2.4.

Maximum Likelihood Estimation

2.5.2.5.

Interval Estimation

2.5.3.

Confidence Intervals

2.5.3.1.

2.5.3.2.

2.5.3.3.

Confidence Intervals for Means

2.5.3.4.

Confidence Intervals for Proportions

2.5.3.5.

Confidence Intervals for Differences

2.5.4.

Hypothesis Testing

2.5.4.1.

Null and Alternative Hypotheses

2.5.4.2.

Test Statistics

2.5.4.3.

2.5.4.4.

Significance Levels

2.5.4.5.

Type I and Type II Errors

2.5.4.6.

Statistical Power

2.5.4.7.

One-tailed vs Two-tailed Tests

2.5.5.

Common Statistical Tests

2.5.5.1.

2.5.5.1.1.

One-sample t-test

2.5.5.1.2.

Two-sample t-test

2.5.5.1.3.

2.5.5.2.

Chi-square Tests

2.5.5.2.1.

Goodness of Fit

2.5.5.2.2.

Test of Independence

2.5.5.3.

2.5.5.3.1.

2.5.5.3.2.

2.5.5.4.

Non-parametric Tests

2.5.5.4.1.

Mann-Whitney U Test

2.5.5.4.2.

Wilcoxon Signed-rank Test

2.5.5.4.3.

Kruskal-Wallis Test

2.5.6.

Experimental Design

2.5.6.1.

Principles of Experimental Design

2.5.6.2.

2.5.6.3.

2.5.6.4.

Factorial Designs

2.5.7.

2.5.7.1.

Design Principles

2.5.7.2.

Sample Size Calculation

2.5.7.3.

Statistical Analysis

2.5.7.4.

Interpreting Results

2.5.7.5.

Multiple Testing Corrections

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