Math for Data Science

Data Science is a large field that requires vast knowledge and being at a beginner's level, that's a fair question to ask "How much maths is required to become a Data Scientist?"  or "How much do you need to know in Data Science?". The point is when you'll be working on solving real-life problems, you'll be required to work on a wide scale and that would certainly need to have clear concepts of Mathematics.

The very first skill that you need to learn in Mathematics is Linear Algebra, following which Statistics, Calculus, etc. We will be providing you with a structure of Mathematics that you need to learn to become a Data Scientist.

Section 1: Linear Algebra

Linear Algebra is the foundation for understanding many data science algorithms.

• Scalars, Vectors, and Matrices : Scalars are single values, vectors are arrays of values representing features, and matrices are 2D structures used to represent datasets.
• Linear Combinations : Used in regression models and PCA.
• Vector Operations and Dot Product for gradient descent
• Types of matrices and Matrix operations : Essential for solving equations and optimizing machine learning models
• Linear Transformation of Matrix : Operations for reshaping data, often used in PCA and feature scaling.
• Solving systems of linear equations : Essential for finding model parameters, such as in linear regression.
• Eigenvalues and Eigenvectors for understanding variance and principal components.
• Singular Value Decomposition (SVD): Decomposes a matrix into three smaller matrices, widely used in tasks like data compression, noise reduction, and dimensionality reduction.
• Norms and Distance Measures
  • Cosine similarity
  • Vector norms for regularization techniques like Lasso and Ridge
• Linear Mapping to transform input data

Refer to master article : Linear Algebra Operations For Machine Learning

Section 2. Probability and Statistics

Both are essential pillars of Data Science, providing the mathematical framework to analyze, interpret, and predict patterns within data. In predictive modeling, these concepts help in building reliable models that quantify uncertainty and make data-driven decisions.

Probability for data science

• Sample space , and types of events : helps in understanding possible outcomes and patterns in data, essential for anomaly detection and risk assessment.
• Probability Rules : enables accurate forecasting and prediction of events, helping in model evaluation.
• Conditional Probability : Used in machine learning for tasks like classification and recommendation systems where past data impacts future outcomes.
• Bayes' Theorem : Key for updating predictions with new data, in models like Naive Bayes.
• Random Variables and probability distributions : Helps model uncertainty in data, select appropriate algorithms, and perform hypothesis testing, forming the basis for statistical analysis in machine learning.

Statistics for data science

• Central Limit Theorem : Ensures that sample means approximate a normal distribution, important for making inferences from samples.
• Descriptive Statistics: Summarizes dataset characteristics (mean, median, variance), helping understand and visualize data patterns.
• Inferential Statistics : Draws conclusions about a population from a sample, essential for predicting and testing hypotheses in data science.
  • Point estimates and confidence intervals
  • Hypothesis testing, p-value , Type I and II errors
  • T-test
  • Paired T-test
  • F-Test
  • z-test
  • Chi-square Test for Feature Selection : Assesses the independence of categorical features, useful for selecting relevant features in machine learning.
• Correlation: Help quantify the similarity between datasets - Pearson for linear, Cosine for similarity, Spearman for ranked data.
• Differentiating correlation from causation : Correlation shows a relationship, but causation proves one variable influences another, crucial for avoiding misleading conclusions
• Types of Sampling techniques

Section 3: Calculus

Calculus is crucial for optimizing models. Master article "Mastering Calculus for Machine Learning" provides a comprehensive overview of the foundational role of calculus in machine learning. For a deeper dive into specific areas and their relevance to machine learning, explore the individual articles outlined below:

• Differentiation: Learn how derivatives are used to measure changes in model parameters and optimize loss functions in machine learning.
• Partial Derivatives: Understand how to compute gradients for multivariable functions, crucial for training models with multiple parameters.
• Gradient Descent Algorithm : Relies on gradients to iteratively adjust parameters and minimize loss functions, forming the backbone of most optimization techniques in machine learning.
• Backpropagation in neural networks
• Chain Rule: Discover how this rule enables backpropagation in neural networks by calculating gradients for composite functions.
• Jacobian and Hessian Matrices: Provide higher-order information about functions. Jacobians are used for mapping gradients in vector-valued functions, while Hessians are critical for second-order optimization techniques like Newton’s method.
• Taylor’s series : Approximates functions near a specific point, simplifying complex functions into polynomial representations, which facilitates gradient computation and optimization processes.
• Higher-Order Derivatives : Capture curvature and sensitivity of a function, which is important for understanding convergence properties in optimization.
• Fourier Transformations : Useful for understanding and optimizing functions in the frequency domain, especially in signal processing and feature extraction tasks.
• Area under the curve : Involves integration (inverse of differentiation) and is vital for evaluating performance metrics like AUC-ROC, commonly used in classification problems.

Section 4: Geometry and Graph Knowledge

Graph Theory is a branch of mathematics which consist of vertices (nodes) connected by edges a crucial field for analyzing relationships and structures in data for network analysis. Let's cover the foundational concepts and essential principles of graph theory in 2 parts:

• Graph Theory Basics – Set 1
• Graph Theory Basics – Set 2

Remember: Data science is not about memorizing formulas; it’s about developing a mindset that leverages mathematical principles to extract meaningful patterns and predictions from data. Invest time in understanding these sections deeply, and you'll be well-equipped to navigate the exciting challenges of the field.

As you advance in your data science journey, revisit these mathematical concepts often. They form the backbone of data science and will empower you to tackle diverse problems with confidence and precision.