QUANTITATIVE MODELS FOR DATA SCIENCE
Instructional goals
The course aims at providing students with adequate mathematical foundations of linear algebra and multivariable calculus and at introducing them to mathematical methods used in data science. Students will also make use of the MATLAB programming language and with the aid of this tool examples will be discussed.
Intended learning outcomes
Knowledge and understanding: the course will give students a solid mathematical background in linear algebra and calculus and will provide them with mathematical tools and methods to analyze data. Students will also acquire a basic knowledge of the MATLAB programming language.
Applying knowledge and understanding: thanks to a wide range of examples discussed during the course (using also innovative teaching methods, like "flipped classroom") and thanks to project work, students will be able to:
- Comprehend the mathematical foundations of data science problems.
- Use mathematical methods to analyze data and propose quantitative solutions to problems in business and economics.
- Use the MATLAB programming language to provide numerical analyses and visualize results.
Making judgments: students will be able to assess the validity of the models used in data analysis and critically interpret the results obtained to provide innovative solutions for business processes.
Communications skills: students will be able to efficiently illustrate mathematical models and methods used in data analysis, thanks to the proper understanding of these tools and the ability to critically interpret their results. Students will also be able to visualize and explain the results of data analyses, thanks to the MATLAB programming language. These skills will be verified during the course thanks to innovative teaching methods, like "flipped classroom", and project work.
Learning skills: students will be able to apply mathematical models and methods commonly used in data science, determine which method is best suited to the problem under study, and interpret and illustrate the results obtained.
Course Contents
Multivariable calculus: domain, level sets, limits and continuity, differentiability, gradient vector, second-order differentials, Hessian matrix. Unconstrained optimization of real-valued functions. An introduction to constrained optimization problems. Applications to the least-squares method and data fitting.
Linear algebra: orthogonality, the Gram-Schmidt algorithm, the eigenvalue problem, diagonalization. A brief introduction to symmetric matrices and quadratic forms. Applications to the least-squares method and data fitting.
An introduction to the MATLAB programming language.
Reference Books
- Stewart, James. Calculus: Early Transcendentals. Eighth ed. Boston: Cengage learning, 2016. https://tinyurl.com/ydvcla3z
- Lay, David C., Steven R. Lay, Judith McDonald, Judi J. McDonald, Judith Joanne McDonald, and J. J. McDonald. Linear Algebra and its Applications. Fifth, Global ed. Boston: Pearson, 2016. https://tinyurl.com/yz8gqugq
Further reading material will be provided on the official course's web page, via the learn.luiss.it platform.
Teaching Methods
Lectures and exercise sessions. Parts of lectures will be devoted to interactive sessions on MATLAB and to the discussion of examples and case studies, possibly adopting the flipped classroom teaching method.
Students’ participation during lectures is strongly encouraged.
Assessment Method
Tests, final written exam.
Thesis assignment criteria
Not applicable.
Week 1 Contenuto sessioni on line e on campus
On-campus: Introduction to the course. Recap on Cartesian curves.
Online: Recap on linear algebra.
On-campus: Recap on linear algebra. Multivariable calculus: motivations, examples. Domain, range, graph.
Online: Level sets.
Week 2 Contenuto sessioni on line e on campus
On-campus: Exercise session on domains of functions and level sets.
Online: Curves in explicit, implicit, and parametric form. Elements of Euclidean topology.
On-campus: Elements of Euclidean topology. Multivariable calculus: Limits, and continuity of functions of several variables. Directional derivatives.
Online: Partial derivatives; gradient vector and its significance.
Week 3 Contenuto sessioni on line e on campus
On-campus: Exercise session on directional and partial derivatives and gradient.
Online: Gradient as direction of maximum increase. Differentiability.
On-campus: Differentiability, first-order differential, and tangent plane.
Online: Linearization and approximation of functions. Second-order directional and partial derivatives, Schwarz’s Theorem..
Week 4 Contenuto sessioni on line e on campus
On-campus: exercise session on differentials, tangent planes, linearization.
Online: Hessian matrix. Second-order differential.
On-campus: Intermediate test. Taylor's theorem for twice-differentiable functions.
Online: Introduction to optimization. Local and global extrema of functions. First-order necessary condition and critical points. Convex functions. Second-order conditions for convexity.
Week 5 Contenuto sessioni on line e on campus
On-campus: exercise session on second-order differentials and convex functions. Recap on MATLAB and data import.
On-campus: First-order sufficient condition for extreme points of convex functions. Definition of saddle point. Classification of critical points. Ad hoc methods to find global extreme points. Least-squares method.
Online: Least-squares method.
Week 6 Contenuto sessioni on line e on campus
On-campus: exercise session on unconstrained optimization and least-squares problems.
MATLAB session on data visualization and least-squares problems.
On-campus: Constrained optimization problems for real-valued functions of two variables with an equality constraint. First-order necessary condition; Lagrange function.
Online: Exercises on contrained optimization.
Week 7 Contenuto sessioni on line e on campus
On-campus: Again on Constrained optimization. First-order necessary condition.
Constrained optimization. Methods to find solutions. Exercises.
Online: Recap on linear algebra.
Week 8 Contenuto sessioni on line e on campus
On-campus: Exercises on linear systems and inverse of matrices.
On-campus: Introduction to orthogonal projections; Recap on linear algebra.
Online: Orthogonal sets. Orthogonal projections..
Week 9 Contenuto sessioni on line e on campus
On-campus: Orthogonal representation and decomposition theorems. Gram-Schmidt algorithm. Exercise session on orthogonal projections and the Gram-Schmidt algorithm.
Online: QR decomposition.
On-campus: Exercise session on orthogonal projections and the Gram-Schmidt algorithm.
Online: The least-squares method and applications to data fitting.
Week 10 Contenuto sessioni on line e on campus
On-campus: Exercise session on QR factorization and least-squares problems.
Online: least-squares problems and QR decomposition.
On-campus: MATLAB session: least-squares problems and data fitting.
Online: Introduction to the Principal Components Analysis. Eigenvalues and eigenvectors.
Week 11 Contenuto sessioni on line e on campus
On-campus: Eigenvectors, eigenspaces, and geometric multiplicity. Exercise session on the eigenvalue problem.
Online: Eigenvalues, characteristic equation, characteristic polynomial, algebraic multiplicity and its relationship with the geometric multiplicity.
On-campus: Exercise session on the eigenvalue problem. Diagonalization of matrices.
Online: Diagonalization of matrices.
Week 12 Contenuto sessioni on line e on campus
On-campus: Exercise session on diagonalization.
Online: Symmetric matrices: orthogonal diagonalization. spectral theorem, principal axes.
On-campus: Principal Components Analysis.
Online: mock final.