Learn Mathematics for Data Science

Complete Course Syllabus

Course Overview

This course provides a comprehensive foundation in the mathematical concepts essential for data science, machine learning, and statistical analysis. Students will develop proficiency in linear algebra, calculus, statistics, probability, and optimization techniques commonly used in data science applications.

Prerequisites: High school algebra and basic familiarity with programming concepts

Duration: 16 weeks (can be adapted to 12 or 20 weeks)

Level: Intermediate to Advanced

Module 1: Foundations and Review (Weeks 1-2)

Week 1: Mathematical Foundations

  • Set Theory and Logic
    • Sets, subsets, unions, intersections
    • Logical operations and proof techniques
    • Mathematical induction
  • Functions and Relations
    • Domain, codomain, range
    • One-to-one, onto, bijective functions
    • Inverse functions
  • Number Systems and Properties
    • Real numbers, complex numbers
    • Properties of real numbers
    • Absolute value and inequalities

Week 2: Essential Algebra and Precalculus

  • Polynomial and Rational Functions
  • Exponential and Logarithmic Functions
    • Properties of logs and exponentials
    • Natural logarithm and e
  • Trigonometric Functions
  • Sequences and Series
    • Arithmetic and geometric sequences
    • Convergence and divergence

Assessment: Problem sets covering foundational concepts

Module 2: Linear Algebra (Weeks 3-6)

Week 3: Vectors and Vector Spaces

  • Vector Basics
    • Vector notation and operations
    • Dot product and cross product
    • Vector norms (L1, L2, L∞)
  • Vector Spaces
    • Definition and properties
    • Linear independence and dependence
    • Basis and dimension
  • Applications in Data Science
    • Feature vectors and data representation using Julia arrays
    • Distance metrics with LinearAlgebra.jl functions
    • Implementing custom similarity measures

Week 4: Matrices and Matrix Operations

  • Matrix Fundamentals
    • Matrix notation and types
    • Matrix addition, multiplication, and transpose
    • Special matrices (identity, diagonal, symmetric)
  • Matrix Properties
    • Determinants and their properties
    • Matrix inverse and pseudo-inverse
    • Rank and nullity
  • Systems of Linear Equations
    • Gaussian elimination
    • Row echelon form
    • Applications to data fitting

Week 5: Eigenvalues and Eigenvectors

  • Eigenvalue Problems
    • Definition and computation
    • Characteristic polynomial
    • Geometric and algebraic multiplicity
  • Diagonalization
    • Diagonalizable matrices
    • Spectral decomposition
  • Applications
    • PCA implementation using LinearAlgebra.jl
    • Markov chains with sparse matrices
    • Graph analysis using LightGraphs.jl

Week 6: Advanced Linear Algebra Topics

  • Matrix Decompositions
    • LU decomposition
    • QR decomposition
    • Singular Value Decomposition (SVD)
  • Positive Definite Matrices
  • Matrix Calculus Basics
    • Derivatives with respect to vectors and matrices
  • Applications
    • Dimensionality reduction with MultivariateStats.jl
    • Collaborative filtering algorithms
    • Image compression using FFTW.jl and Images.jl

Assessment: Linear algebra project implementing PCA or SVD using Julia’s efficient linear algebra routines

Module 3: Calculus (Weeks 7-9)

Week 7: Single Variable Calculus

  • Limits and Continuity
    • Limit definition and properties
    • Continuity and discontinuities
  • Derivatives
    • Definition and interpretation
    • Differentiation rules
    • Chain rule and implicit differentiation
  • Applications of Derivatives
    • Optimization problems
    • Related rates
    • L’Hôpital’s rule

Week 8: Integration and Series

  • Integration
    • Definite and indefinite integrals
    • Fundamental theorem of calculus
    • Integration techniques
  • Infinite Series
    • Taylor and Maclaurin series
    • Convergence tests
    • Applications to approximation

Week 9: Multivariable Calculus

  • Partial Derivatives
    • Definition and computation
    • Gradient vector and directional derivatives
    • Chain rule for multivariable functions
  • Multiple Integration
    • Double and triple integrals
    • Applications to probability
  • Vector Calculus
    • Divergence and curl
    • Line and surface integrals
  • Optimization
    • Critical points and second derivative test
    • Constrained optimization and Lagrange multipliers

Assessment: Calculus applications in machine learning optimization using Optim.jl and automatic differentiation

Module 4: Probability Theory (Weeks 10-12)

Week 10: Foundations of Probability

  • Sample Spaces and Events
    • Probability axioms
    • Conditional probability and independence
    • Bayes’ theorem and applications
  • Combinatorics
    • Permutations and combinations
    • Counting principles
  • Discrete Probability Distributions
    • Bernoulli, binomial, geometric using Distributions.jl
    • Poisson distribution and applications
    • Expected value and variance calculations

Week 11: Continuous Probability

  • Continuous Random Variables
    • Probability density functions
    • Cumulative distribution functions
  • Important Continuous Distributions
    • Uniform, normal (Gaussian) with Distributions.jl
    • Exponential, gamma, beta distributions
    • Chi-square, t-distribution, F-distribution
    • Random sampling and Monte Carlo methods
  • Functions of Random Variables
    • Transformation techniques
    • Moment generating functions

Week 12: Multivariate Probability

  • Joint Distributions
    • Joint PMF and PDF
    • Marginal and conditional distributions
    • Independence of random variables
  • Covariance and Correlation
    • Definition and properties
    • Correlation coefficient
  • Multivariate Normal Distribution
    • Properties and applications
  • Central Limit Theorem
    • Statement and applications
    • Law of large numbers

Assessment: Probability modeling project with real data using StatsBase.jl and HypothesisTests.jl

Module 5: Statistics (Weeks 13-14)

Week 13: Descriptive Statistics and Estimation

  • Descriptive Statistics
    • Measures of central tendency
    • Measures of variability and dispersion
    • Quantiles and percentiles
  • Parameter Estimation
    • Point estimation and estimator properties
    • Method of moments
    • Maximum likelihood estimation
    • Bias, consistency, and efficiency
  • Confidence Intervals
    • Construction and interpretation
    • Bootstrap methods using Bootstrap.jl

Week 14: Hypothesis Testing and Regression

  • Hypothesis Testing
    • Null and alternative hypotheses
    • Type I and Type II errors
    • P-values and significance levels
    • Common tests (t-test, chi-square, ANOVA)
  • Linear Regression
    • Simple and multiple regression with GLM.jl
    • Least squares estimation
    • Model assumptions and diagnostics
    • R-squared and model evaluation using StatsModels.jl
  • Non-parametric Methods
    • Bootstrap and permutation tests
    • Rank-based methods

Assessment: Statistical analysis of a real dataset using Julia’s statistical ecosystem

Module 6: Optimization (Weeks 15-16)

Week 15: Optimization Fundamentals

  • Unconstrained Optimization
    • Necessary and sufficient conditions
    • Gradient descent implementation with Optim.jl
    • Newton’s method and BFGS using LineSearches.jl
  • Constrained Optimization
    • Equality constraints and Lagrange multipliers
    • Inequality constraints and KKT conditions
    • Linear programming with JuMP.jl and GLPK
  • Convex Optimization
    • Convex sets and functions
    • Convex optimization problems
    • Applications in machine learning

Week 16: Advanced Topics and Applications

  • Stochastic Optimization
    • Stochastic gradient descent implementation
    • Mini-batch methods with efficient Julia arrays
    • Adaptive learning rates using Flux.jl optimizers
  • Regularization Techniques
    • Ridge regression (L2) with GLMNet.jl
    • Lasso regression (L1) implementation
    • Elastic net using MLJ.jl framework
  • Applications Integration
    • Support vector machines with LIBSVM.jl
    • Neural network optimization using Flux.jl
    • Cross-validation with MLJ.jl ecosystem

Final Assessment: Comprehensive project integrating multiple mathematical concepts

Learning Objectives

By the end of this course, students will be able to:

  1. Linear Algebra Proficiency
    • Perform matrix operations and solve linear systems
    • Apply eigenvalue decomposition and SVD to real problems
    • Understand geometric interpretations of linear transformations
  2. Calculus Mastery
    • Compute derivatives and integrals for optimization problems
    • Apply multivariable calculus to machine learning algorithms
    • Use Taylor series for function approximation
  3. Probability and Statistics
    • Model uncertainty using probability distributions
    • Perform statistical inference and hypothesis testing
    • Apply Bayes’ theorem to real-world problems
  4. Optimization Skills
    • Solve constrained and unconstrained optimization problems
    • Implement gradient-based optimization algorithms
    • Apply regularization techniques appropriately
  5. Integration and Application
    • Connect mathematical concepts to data science applications
    • Implement mathematical algorithms efficiently in Julia
    • Leverage Julia’s performance for large-scale computations
    • Critically evaluate mathematical assumptions in data science models

Assessment Structure

  • Problem Sets (40%): Weekly assignments reinforcing key concepts
  • Projects (35%): Four major projects applying mathematics to data science problems
  • Midterm Exam (10%): Covering Modules 1-3
  • Final Exam (15%): Comprehensive exam covering all modules
  • Primary: “Mathematics for Machine Learning” by Deisenroth, Faisal, and Ong
  • Linear Algebra: “Linear Algebra and Its Applications” by David C. Lay
  • Calculus: “Calculus: Early Transcendentals” by James Stewart
  • Probability: “Introduction to Probability” by Blitzstein and Hwang
  • Statistics: “An Introduction to Statistical Learning” by James, Witten, Hastie, and Tibshirani

Software Tools

  • Programming: Julia with LinearAlgebra.jl, Statistics.jl, Distributions.jl
  • Mathematical Computing: Pluto.jl notebooks, SymPy.jl for symbolic math
  • Numerical Computing: DifferentialEquations.jl, Optim.jl, JuMP.jl
  • Data Manipulation: DataFrames.jl, CSV.jl, Query.jl
  • Visualization: Plots.jl, PlotlyJS.jl, StatsPlots.jl, Makie.jl
  • Machine Learning Integration: MLJ.jl, Flux.jl
  • Optional: R (via RCall.jl), Python (via PyCall.jl) for cross-language integration

Prerequisites for Success

  • Solid foundation in high school algebra and trigonometry
  • Basic programming experience (preferably Julia, but Python/MATLAB background acceptable)
  • Comfort with abstract mathematical thinking
  • Willingness to engage with both theoretical concepts and practical applications
  • Familiarity with package management and development environments

Course Policies

  • Attendance: Regular participation in lectures and discussion sections
  • Collaboration: Encouraged for understanding concepts, but individual work required for assessments
  • Late Work: Penalty structure for late submissions
  • Academic Integrity: Clear guidelines on acceptable collaboration and resource usage

Pricing

  • This course costs Rs 60,000/-
  • To be paid in 4 monthly installments

This syllabus serves as a comprehensive guide but may be adapted based on student background, course duration, and specific program requirements.