Mathematics & Techniques for Data Science

A personal list of topics across the data science field.

This is an opinionated list of mathematical topics and techniques essential across the data science field. Although, not an all-inclusive list, knowledgement and mastering of these provide a solid foundation useful for understanding of the diverse branches sustaining machine learning and artificial intelligence.

---

Linear Algebra

• Vectors and Matrices
  • Vector Operations: Basic operations such as addition, subtraction, and scalar multiplication.
  • Matrix Operations: Matrix multiplication, inversion, and transposition.
  • Types of Matrices: Special matrices like diagonal, symmetric, and orthogonal matrices.
• Systems of Linear Equations
  • Gaussian Elimination: Method for solving linear systems by reducing matrices to row echelon form.
  • LU Decomposition: Factorization of a matrix into lower and upper triangular matrices.
• Matrix Decompositions
  • Eigenvalues and Eigenvectors: Key concepts for understanding matrix transformations.
  • Singular Value Decomposition (SVD): Decomposition of a matrix into singular vectors and singular values.
  • Principal Component Analysis (PCA): Technique for reducing the dimensionality of data.
• Vector Spaces
  • Basis and Dimension: Fundamental properties of vector spaces.
  • Subspaces: Subsets of vector spaces that themselves are vector spaces.
  • Orthogonality and Orthogonal Projections: Concepts for projecting vectors onto subspaces.
• Linear Transformations
  • Matrix Representation: Representation of linear transformations using matrices.
  • Change of Basis: Transforming coordinates from one basis to another.

Probability and Statistics

• Probability Theory
  • Basic Probability Concepts: Definitions and rules of probability.
  • Conditional Probability and Bayes’ Theorem: Probability of events given other events.
  • Random Variables: Variables whose values are subject to randomness.
  • Probability Distributions: Descriptions of how probabilities are distributed over values.
  • Joint, Marginal, and Conditional Distributions: Relationships between multiple random variables.
  • Expectation, Variance, and Covariance: Measures of central tendency and variability.
• Statistical Inference
  • Point Estimation: Estimating population parameters from sample data.
  • Confidence Intervals: Range of values within which a parameter is expected to lie.
  • Hypothesis Testing: Procedure for testing assumptions about population parameters.
  • p-values and Significance Levels: Metrics for assessing hypothesis test results.
  • Maximum Likelihood Estimation (MLE): Method for estimating parameters by maximizing likelihood.
• Bayesian Statistics
  • Bayesian Inference: Updating probabilities based on new data.
  • Prior and Posterior Distributions: Distributions representing beliefs before and after observing data.
  • Markov Chain Monte Carlo (MCMC): Algorithms for sampling from complex distributions.
• Regression Analysis
  • Simple and Multiple Linear Regression: Modeling relationships between variables.
  • Logistic Regression: Modeling binary outcome variables.
  • Assumptions and Diagnostics: Checking the validity of regression models.
• Advanced Topics in Statistics
  • Time Series Analysis: Analyzing data points collected over time.
  • Survival Analysis: Analyzing time-to-event data.
  • Non-parametric Methods: Statistical methods not assuming a specific data distribution.

Numerical Methods

• Optimization Techniques
  • Gradient Descent: Iterative method for finding local minima of functions.
  • Stochastic Gradient Descent: Variant of gradient descent using random subsets of data.
  • Conjugate Gradient Method: Optimization algorithm for large-scale linear systems.
  • Newton’s Method: Iterative method for finding successively better approximations to roots.
• Numerical Linear Algebra
  • Matrix Factorization: Decomposing matrices into products of simpler matrices.
  • Solving Linear Systems: Methods for finding solutions to linear equations.
  • Eigenvalue Problems: Finding eigenvalues and eigenvectors of matrices.

Machine Learning

• Supervised Learning
  • Regression (Linear, Polynomial): Predicting continuous outcomes from input features.
  • Classification (k-NN, SVM, Decision Trees, Random Forests): Categorizing data into predefined classes.
• Unsupervised Learning
  • Clustering (k-Means, Hierarchical, DBSCAN): Grouping similar data points together.
  • Dimensionality Reduction (PCA, t-SNE, LDA): Reducing the number of variables in data.
• Model Evaluation
  • Cross-Validation: Technique for assessing model performance on unseen data.
  • ROC Curves and AUC: Metrics for evaluating classification model performance.
  • Precision, Recall, F1-Score: Metrics for evaluating model accuracy and relevance.
• Ensemble Methods
  • Bagging and Boosting: Techniques for improving model performance by combining multiple models.
  • Random Forests: Ensemble learning method using multiple decision trees.
  • Gradient Boosting Machines (GBM, XGBoost): Powerful ensemble methods for regression and classification.

Neural Networks and Deep Learning

• Fundamentals of Neural Networks
  • Perceptrons and Multilayer Perceptrons (MLP): Basic building blocks of neural networks.
  • Activation Functions (ReLU, Sigmoid, Tanh): Functions introducing non-linearity into neural networks.
  • Backpropagation and Gradient Descent: Algorithms for training neural networks.
• Deep Learning Architectures
  • Convolutional Neural Networks (CNNs): Networks for processing grid-like data such as images.
  • Recurrent Neural Networks (RNNs): Networks for processing sequential data.
  • Long Short-Term Memory (LSTM): RNN variant for capturing long-term dependencies.
  • Generative Adversarial Networks (GANs): Networks for generating new, synthetic data.
• Deep Learning Techniques
  • Regularization (Dropout, Batch Normalization): Techniques for preventing overfitting.
  • Transfer Learning: Leveraging pre-trained models for new tasks.
  • Hyperparameter Tuning: Optimizing model parameters for better performance.
  • Autoencoders: Networks for unsupervised learning of efficient codings.
• Advanced Topics in Deep Learning
  • Attention Mechanisms: Techniques for focusing on relevant parts of input data.
  • Transformers: Architectures for handling sequential data with attention mechanisms.
  • Reinforcement Learning: Training models to make sequences of decisions.

Dimensionality Reduction

• Principal Component Analysis (PCA)
  • Eigenvalues and Eigenvectors: Key concepts for understanding PCA.
  • Variance Explained: Measure of how much information is retained by principal components.
• Singular Value Decomposition (SVD)
  • Low-Rank Approximations: Simplifying data by reducing its dimensionality.
• Manifold Learning
  • t-SNE (t-Distributed Stochastic Neighbor Embedding): Technique for visualizing high-dimensional data.
  • UMAP (Uniform Manifold Approximation and Projection): Method for dimensionality reduction and visualization.
• Feature Selection and Extraction
  • L1 Regularization (Lasso): Technique for feature selection in regression models.
  • Recursive Feature Elimination: Method for selecting important features by recursively removing less important ones.

Additional Important Topics

• Information Theory
  • Entropy and Information Gain: Measures of uncertainty and information content.
  • Mutual Information: Measure of the mutual dependence between variables.
• Graph Theory
  • Graph Representation: Ways to represent graphs using matrices and lists.
  • Graph Algorithms (PageRank, Graph Neural Networks): Algorithms for processing graph-structured data.
• Time Series Analysis
  • Autoregressive Models (AR, MA, ARIMA): Models for analyzing and forecasting time series data.
  • Seasonal Decomposition: Breaking down time series data into seasonal components.
  • Forecasting Techniques: Methods for predicting future values in time series data.
• Natural Language Processing (NLP)
  • Text Preprocessing: Techniques for preparing text data for analysis.
  • Word Embeddings (Word2Vec, GloVe): Methods for representing words as vectors.
  • Sequence Models (RNN, LSTM, Transformer): Models for processing and understanding sequential data.