ML

Mathematical Foundation

May 28, 2026
AI, ML
Machine Learning, Mathematics, Linear Algebra, Probability, Statistics, Calculus

Mathematical Foundations for Machine Learning #

Machine Learning is built on mathematical principles that allow models to:

  • represent data
  • learn patterns
  • optimise performance
flowchart LR
    DATA[Data]
    MATH[Math Models]
    OPT[Optimisation]
    MODEL[Trained Model]

    DATA --> MATH
    MATH --> OPT
    OPT --> MODEL

ML requires core mathematical tools to understand how ML algorithms work internally. Algebra deals with relationships between variables and quantities, while Calculus focuses on change and optimization.

Supervised Learning

January 3, 2026
AI, ML
AI, ML

Supervised Learning #

Trained using labelled data.
Each example in the training set includes the correct output.
The algorithm learns to generalise and make predictions on unseen data.
Generally more accurate than unsupervised methods.
Requires human intervention for labelling and setup.
Widely used due to its accuracy and efficiency.
Produces highly accurate results when trained on good-quality labelled data.

Classification #

Output is discrete (e.g. Yes/No, Spam/Not Spam).
Used for categorising data into predefined classes.
Support Vector Machine (SVM) is a common classifier (a linear classifier with margin-based separation).

Statistics

March 12, 2026
AI, ML
Statistics, Probability, Data Analysis

Statistics #

Statistical methods help you turn raw data into reliable conclusions, while understanding uncertainty, variability, and confidence.

Statistics provides the language and tools for reasoning about data, uncertainty, and inference.

ML needs understanding data behaviour, drawing conclusions, and validating machine learning models.

  • Collect Data
  • Present & Organise Data (in a systematic manner)
  • Alalyse Data
  • Infer about the Data
  • Take Decision from the Data
Statistics TopicWhat you learn (plain English)ML Connection
1. Basic Probability & StatisticsSummarise data;
understand spread;
basic probability rules		Data understanding (EDA), feature sanity checks,
detecting outliers, interpreting “average behaviour”
2. Conditional Probability & BayesUpdate probability using new information;
Bayes’ rule		Naïve Bayes, Bayesian thinking,
posterior probabilities, probabilistic classification
3. Probability DistributionsModel randomness with distributions;
expectation/variance/covariance		Likelihood models, noise assumptions (Gaussian), sampling,
probabilistic modelling foundations
4. Hypothesis TestingSampling, CLT, confidence intervals,
significance tests, ANOVA, MLE		A/B testing, evaluating model improvements,
significance vs noise, parameter estimation (MLE)
5. Prediction & ForecastingCorrelation, regression,
time series (AR/MA/ARIMA/SARIMA etc.)		Linear regression, forecasting, sequential data modelling, baseline predictive modelling
6. GMM & EMMixtures of Gaussians;
iterative estimation with EM		Unsupervised learning (soft clustering),
density estimation, latent-variable models
flowchart TD
  A["Statistical Methods<br/>AIML ZC418"] --> B["1. Basic Probability and Statistics"]
  A --> C["2. Conditional Probability and Bayes"]
  A --> D["3. Probability Distributions"]
  A --> E["4. Hypothesis Testing"]
  A --> F["5. Prediction and Forecasting"]
  A --> G["6. Gaussian Mixture Model and EM"]

  B --> B1["Central Tendency<br/>Mean - Median - Mode"]
  B --> B2["Variability<br/>Range - Variance - SD - Quartiles"]
  B --> B3["Basic Probability Concepts"]
  B3 --> B31["Axioms of Probability"]
  B3 --> B32["Definition of Probability"]
  B3 --> B33["Mutually Exclusive vs Independent"]

  C --> C1["Conditional Probability"]
  C --> C2["Independence (conditional)"]
  C --> C3["Bayes Theorem"]
  C --> C4["Naive Bayes (intro)"]

  D --> D1["Random Variables<br/>Discrete and Continuous"]
  D --> D2["Expectation - Variance - Covariance"]
  D --> D3["Transformations of RVs"]
  D --> D4["Key Distributions"]
  D4 --> D41["Bernoulli"]
  D4 --> D42["Binomial"]
  D4 --> D43["Poisson"]
  D4 --> D44["Normal (Gaussian)"]
  D4 --> D45["t - Chi-square - F (intro)"]

  E --> E1["Sampling<br/>Random and Stratified"]
  E --> E2["Sampling Distributions<br/>CLT"]
  E --> E3["Estimation<br/>Confidence Intervals"]
  E --> E4["Hypothesis Tests<br/>Means and Proportions"]
  E --> E5["ANOVA<br/>Single and Dual factor"]
  E --> E6["Maximum Likelihood"]

  F --> F1["Correlation"]
  F --> F2["Regression"]
  F --> F3["Time Series Basics<br/>Components"]
  F --> F4["Moving Averages<br/>Simple and Weighted"]
  F --> F5["Time Series Models"]
  F5 --> F51["AR"]
  F5 --> F52["ARMA / ARIMA"]
  F5 --> F53["SARIMA / SARIMAX"]
  F5 --> F54["VAR / VARMAX"]
  F --> F6["Exponential Smoothing"]

  G --> G1["GMM<br/>Mixture of Gaussians"]
  G --> G2["EM Algorithm<br/>E-step - M-step"]

  B -.-> C
  C -.-> D
  D -.-> E
  E -.-> F
  F -.-> G

Data - Types #

flowchart TD
	A[(Data)] --> B["Categorical (Qualitative)"]
    A --> C["Numerical (Quantitative)"]

    B --> B1[Nominal]
    B --> B2[Ordinal]

    C --> C1[Discrete]
    C --> C2[Continuous]

    C2 --> C21[Interval]
    C2 --> C22[Ratio]

    %% Styling
    style A fill:#E1F5FE,stroke:#333
    style B fill:#90CAF9,stroke:#333
    style B1 fill:#90CAF9,stroke:#333
    style B2 fill:#90CAF9,stroke:#333
    style C fill:#FFF9C4,stroke:#333
    style C1 fill:#FFF9C4,stroke:#333
    style C2 fill:#FFF9C4,stroke:#333
    style C21 fill:#FFF9C4,stroke:#333
    style C22 fill:#FFF9C4,stroke:#333

  1. Categorical (Qualitative) #

    express a qualitative attribute e.g. hair color, eye color

Partial Differentiation and Gradients

AI, ML
Machine Learning, Mathematics, Vector Calculus

Partial Differentiation and Gradients #

For f(x1, x2, …, xn):

[ \frac{\partial f}{\partial x_i} ]

Gradient vector:

[ \nabla f = \begin{bmatrix} \frac{\partial f}{\partial x_1} \ \vdots \ \frac{\partial f}{\partial x_n} \end{bmatrix} ]

Gradient points in direction of steepest ascent.

flowchart LR
    Input --> Function
    Function --> Gradient
    Gradient --> Optimisation

Home | Vector Calculus

Linear Independence

AI, ML
Machine Learning, Mathematics, Linear Algebra

Linear Independence #

A set of vectors is linearly independent if none of them can be written as a linear combination of the others.

\[ c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0} \;\Rightarrow\; c_1=\cdots=c_k=0 \]

Independence means each vector adds new information.

Semi-Supervised Learning

January 3, 2026
AI, ML
AI, ML

Semi-Supervised Learning #

  • A combination of labelled and unlabelled data.
  • Useful when labelling large datasets is expensive or time-consuming.
  • Works well with high-volume datasets (e.g. millions of images).
  • Only a small fraction of data is labelled (e.g. a few thousand).
  • The algorithm learns from both labelled examples and structure in unlabelled data.
  • Ideal for medical imaging where labelled data is limited.
  • For example, a radiologist can label a small set of medical scans,
    and the model uses that to learn from thousands of unlabelled scans.
  • Helps improve accuracy and generalisation with minimal manual effort.

Home | Machine Learning

Machine Learning

August 6, 2024
AI, ML
AI, ML

Machine Learning #

stateDiagram-v2

    %% ===== CLASS DEFINITIONS (Math-based colours) =====
    classDef algebra fill:#cfe8ff,stroke:#1e3a8a,stroke-width:1px
    classDef probability fill:#d1fae5,stroke:#065f46,stroke-width:1px
    classDef geometry fill:#ffedd5,stroke:#9a3412,stroke-width:1px
    classDef logic fill:#ede9fe,stroke:#5b21b6,stroke-width:1px
    classDef category font-style:italic,font-weight:bold,fill:#aaaaaa,stroke:#374151,stroke-width:3px

    %% ===== ROOT =====
    ML: Machine Learning

    %% ===== SUPERVISED =====
    ML --> SL:::category
    SL: Supervised Learning

    SL --> Regression
    Regression --> LR:::algebra
    LR: Linear Regression

    LR --> NN:::algebra
    NN: Neural Network

    NN --> DT:::logic
    DT: Decision Tree

    SL --> Classification
    Classification --> NB:::probability
    NB: Naive Bayes

    NB --> KNN:::geometry
    KNN: k-Nearest Neighbours

    KNN --> SVM:::algebra
    SVM: Support Vector Machine
    
    %% ===== UNSUPERVISED =====
    ML --> USL:::category
    USL: Unsupervised Learning

    USL --> Clustering
    Clustering --> KM:::geometry
    KM: K-Means

    KM --> GMM:::probability
    GMM: Gaussian Mixture Model

    GMM --> HMM:::probability
    HMM: Hidden Markov Model

    %% ===== REINFORCEMENT =====
    ML --> RL:::category
    RL: Reinforcement Learning

    RL --> DM:::logic
    DM: Decision Making
Mathematical Legend

Algebra / Linear Algebra (Blue) #

Used heavily when models rely on:

Gradients of Vector-Valued and Matrix Functions

AI, ML
Machine Learning, Mathematics, Vector Calculus

Gradients of Vector-Valued and Matrix Functions #

Covers gradients when outputs or parameters are vectors/matrices.

If f: R^n -> R^m, the derivative is the Jacobian.

[ J = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \dots & \frac{\partial f_1}{\partial x_n} \ \vdots & \ddots & \vdots \ \frac{\partial f_m}{\partial x_1} & \dots & \frac{\partial f_m}{\partial x_n} \end{bmatrix} ]

For scalar f(x):

[ H = \nabla^2 f ]

Hessian captures curvature.

Reinforcement Learning

AI, ML
AI, Machine Learning, Reinforcement Learning

Reinforcement Learning (RL) #

RL is learning by trial and error.

Reinforcement Learning (RL) is a type of machine learning where an autonomous agent learns to make decisions by interacting with an environment.

Instead of being told the correct answer, the agent:

  • takes actions
  • observes outcomes
  • receives rewards or penalties
  • gradually learns a strategy that maximises long-term reward

Reinforcement Learning teaches an agent how to act, not what to predict.