Unit 742: Comprehensive Data Science Mathematics and Machine Learning Foundations

1. Introduction to Algebra

• Variables, expressions, and algebraic notation
• Linear equations and inequalities
• Quadratic equations: solving and graph interpretation
• Functions and their graphs

2. Descriptive Statistics

• Measures of central tendency: mean, median, mode
• Measures of dispersion: range, variance, standard deviation
• Graphical representations: histograms, box plots

3. Probability Theory

• Sample spaces and events
• Probability rules and axioms
• Conditional probability and independence
• Bayes' theorem and applications

4. Linear Algebra for Data Science

4.1 Introduction to Linear Algebra

• Vectors and vector operations: addition, subtraction, scalar multiplication
• Matrices: definition, operations (addition, multiplication)
• Matrix inverse and linear transformations

4.2 Eigenvalues and Eigenvectors

• Definition and significance
• Computation methods
• Applications in data analysis

4.3 Singular Value Decomposition (SVD)

• Concept and mathematical formulation
• Applications in dimensionality reduction and collaborative filtering

4.4 Principal Component Analysis (PCA)

• Dimensionality reduction technique
• Eigenvector-based data transformation
• Visualization and feature extraction

5. Calculus for Data Science

• Limits and continuity
• Derivatives and differentiation rules
• Integration and definite integrals
• Multivariable calculus: partial derivatives and gradients
• Optimization methods: maxima, minima, saddle points

6. Statistical Inference

• Hypothesis testing: null and alternative hypotheses
• Significance levels and p-values
• Confidence intervals
• Type I and Type II errors

7. Data Visualization

• Importance of visualization in data science
• Tools overview: Matplotlib, Seaborn, ggplot, Plotly
• Plot types: scatter plots, bar charts, histograms, box plots, heatmaps
• Best practices for effective communication

8. Machine Learning Fundamentals

8.1 Foundations

• Definitions: supervised, unsupervised, reinforcement learning
• Common algorithms overview: decision trees, support vector machines, clustering

8.2 Regression Analysis

• Linear regression: least squares method, model fitting
• Multiple and logistic regression
• Interpretation and evaluation of models

8.3 Clustering Algorithms

• K-means clustering
• Hierarchical clustering
• Similarity and distance metrics

8.4 Neural Networks Basics

• Feedforward networks
• Activation functions
• Backpropagation algorithm
• Training via stochastic gradient descent

9. Optimization Techniques for Data Science

• Gradient descent and variants
• Stochastic gradient descent
• Loss functions and convergence criteria

10. Time Series Analysis

• Characteristics of time series data
• Trend and seasonality analysis
• Autocorrelation and partial autocorrelation
• Forecasting methods and models

11. Bayesian Statistics

• Bayesian inference concepts
• Prior and posterior distributions
• Bayes' theorem revisited
• Applications in probabilistic modeling and decision-making

Summary and Integration

• Interrelations among topics
• Practical examples combining algebra, calculus, statistics, and machine learning
• Case studies and data science project frameworks