Maths for Data Science Course

Lesson 1 • Logarithms and Exponentials

Explores exponential growth and logarithmic scaling critical for loss functions and data normalisation. Links log rules to entropy and information theory.

Lesson 2 • Number Systems and Arithmetic

Covers integers, rationals, reals, and floating-point representation. Establishes numerical literacy needed for all subsequent quantitative work.

Lesson 3 • Algebraic Expressions and Equations

Teaches simplification, factorising, and solving linear and quadratic equations. Provides the symbolic manipulation skills used in model formulation.

Lesson 4 • Summation and Product Notation

Introduces sigma and pi notation used throughout statistics and machine learning formulas. Builds comfort reading and writing compact mathematical expressions.

Lesson 1 • Vectors and Vector Spaces

Defines vectors, norms, and inner products as the language of feature representation. Grounds geometric intuition for distance and similarity metrics.

Lesson 2 • Eigenvalues and Eigenvectors

Derives eigendecomposition and its geometric meaning. Directly enables PCA and spectral methods used in dimensionality reduction.

Lesson 3 • Systems of Linear Equations

Solves linear systems using Gaussian elimination and matrix methods. Connects solution existence to rank and data consistency.

Lesson 4 • Singular Value Decomposition

Presents SVD as a generalisation of eigendecomposition for rectangular matrices. Applies SVD to data compression and latent factor models.

Lesson 1 • Limits and Continuity

Establishes the limit concept and conditions for continuity. Provides the theoretical basis for derivatives and convergence analysis.

Lesson 2 • Partial Derivatives and Gradients

Extends differentiation to multivariate functions and introduces the gradient vector. Gradient computation is the core operation in backpropagation.

Lesson 3 • Optimisation Using Calculus

Identifies critical points via first and second derivative tests. Connects these techniques to loss minimisation in supervised learning.

Lesson 4 • Differentiation Rules and Techniques

Covers power, product, quotient, and chain rules for computing derivatives. These rules are applied directly when differentiating loss functions.

Lesson 1 • Continuous Probability Distributions

Examines Uniform, Normal, Exponential, and Beta distributions with their PDFs. Applies these to feature modelling and likelihood functions.

Lesson 2 • Conditional Probability and Independence

Introduces conditional probability, Bayes' theorem, and statistical independence. These concepts underpin Naive Bayes classifiers and causal reasoning.

Lesson 3 • Joint Distributions and Covariance

Defines joint, marginal, and conditional distributions for multiple variables. Covariance and correlation link directly to feature relationships in datasets.

Lesson 4 • Probability Fundamentals

Defines sample spaces, events, and axioms of probability. Establishes the formal language used throughout statistical modelling.

Lesson 1 • Descriptive Statistics and Visualisation

Computes mean, median, variance, skewness, and kurtosis to summarise distributions. Pairs numerical summaries with appropriate chart types.

Lesson 2 • Hypothesis Testing Framework

Formalises null and alternative hypotheses, p-values, and decision rules. Connects Type I and Type II errors to model evaluation thresholds.

Lesson 3 • Sampling and Estimation

Covers random sampling methods and point estimation via MLE and MOM. Introduces bias-variance tradeoff in estimator quality.

Lesson 4 • Common Statistical Tests

Applies t-tests, chi-square tests, and ANOVA to compare groups and test associations. Guides test selection based on data type and research question.

Lesson 1 • KL Divergence and Cross-Entropy

Defines KL divergence as an asymmetric distance between distributions. Cross-entropy loss in classification models is derived directly from this concept.

Lesson 2 • Joint and Conditional Entropy

Extends entropy to joint and conditional settings for multiple variables. Provides the foundation for mutual information and feature relevance scoring.

Lesson 3 • Shannon Entropy and Information

Defines self-information and Shannon entropy as measures of uncertainty. Connects entropy to optimal encoding and decision tree splitting criteria.

Lesson 4 • Information Theory in Model Evaluation

Applies information-theoretic metrics to compare model predictions with true distributions. Links perplexity, log-loss, and AIC to entropy-based reasoning.

Lesson 1 • Gradient Descent Fundamentals

Derives the gradient descent update rule from first principles. Establishes the connection between loss surface geometry and parameter updates.

Lesson 2 • Convexity and Convergence Analysis

Defines convex functions and sets, and proves convergence guarantees for convex objectives. Identifies non-convex challenges in deep learning optimisation.

Lesson 3 • Advanced Optimisation Algorithms

Covers momentum, RMSProp, and Adam optimisers that accelerate convergence. Explains adaptive learning rates and their practical advantages.

Lesson 4 • Variants of Gradient Descent

Compares batch, stochastic, and mini-batch gradient descent in terms of speed and stability. Guides algorithm selection for different dataset sizes.

Lesson 1 • Logistic Regression and MLE

Derives logistic regression from the Bernoulli likelihood using MLE. Links cross-entropy loss to the probabilistic model formulation.

Lesson 2 • Mathematical Evaluation of Models

Derives evaluation metrics including MSE, R-squared, precision, recall, and AUC from first principles. Connects metric choice to the mathematical properties of the task.

Lesson 3 • Principal Component Analysis

Derives PCA using covariance matrix eigendecomposition and variance maximisation. Applies PCA to high-dimensional datasets for visualisation and noise reduction.

Lesson 4 • Mathematical Derivation of Linear Regression

Derives the ordinary least-squares solution using calculus and linear algebra. Connects the normal equations to gradient descent convergence.