Probability and stochastic processes. Linear algebra, Euclidian spaces.
Advanced Mathematical Tools students learn the following knowledge and develop the following skills:
1. Acquire the basic knowledge from the study of some fundamental mathematic tools useful in information and statistical signal processing applications like advanced linear algebra and theory estimation.
2. Identify, formulate and solve information and statistical signal processing problems in a multidisciplinary environment, individually or as a member of a team.
3. Analyze, design and make use of systems, procedures and algorithms in order to achieve the proposed goals in a specific information and signal processing problem, making use of simulation, analysis and application development tools in this area (MATLAB), and to analyze and understand the given results.
4. To use new e-learning techniques and tools (virtual campus, sharing documents, forums, interactive demonstration tools)
5. To acquire the following scientific skills: reading and understanding scientific papers, information searching ,verification and contrasting abilities, analysis of different state-of-the-art proposals, experimentation of a technique within the framework, proposal of different working possibilities, production of different experimentation scenarios for validation, discussion of results and proposal of future lines. It is also desirable to promote criticism and self-criticism abilities through the interaction with other people and working in a multidisciplinary team.
1. Estimation theory
1.1 Random vectors and matrix derivatives.
1.2 Minimum variance estimation (MVU) and the Cramer-Rao bound, optimal linear estimation.
1.3 Maximum likelihood estimation (ML), EM algorithm, models ocults de Markov, Gaussian Mixture Models (GMM)
1.4 Bayesian estimation, Kalman filtering
1.5 Applications
2. Linear algebra
2.1 Signal spaces
2.2 Vector-based representation and approximation
2.3 Matrix factorization
2.4 Subspace diagonalization (SVD)
2.5 Applications
3. Optimization theory
3.1 Linear optimization.
3.2 Constrained linear optimization.
3.3 Applications.
The course methodology is based on the following procedures:
- Theoretical teacher lectures covering the mathematical framework and the fundamental concepts to study.
- Laboratory lectures and practices exemplifying some practical applications of the theoretical framework, where some problems and cases of study will be proposed to be studied in groups, promoting the problem resolution through cooperation and students interactions and the final guidance of the teacher if it´s needed.
- Personal homework, as the student has to demonstrate a good understanding of the main concepts through exercises, demonstration and problem solutions that allow to better training and theory to practice knowledge acquisition.
- Presentations, as the students have to develop a specific study work (mainly related to its research framework) and they have to finally give a talk including a practical demonstration. The teacher will evaluate the possibility of doing this work in groups depending on the thematic and the proposal work complexity.
- Activity in the main subject folder (e-campus), where the teacher will facilitate complementary contents resources (scientific papers, links, etc.), discussion forums covering the main subject topics, personal or group folders management (assigned to develop the personal or group work), and teacher news to give a better guide of the course.
The student evaluation will be completed with:
A. Exams
B. Quizzes
C. Homework
D. Team reports
E. Computer assignments
F. Laboratory reports
G. Works in group
H. Works
I. Expositions
J. Laboratory participation
K. Participation in the classroom.
The subject evaluation will be structured in the following parts:
- Theoretical teacher lectures attendance, active participation and personal motivation and e-campus activity: 5% of relevance in the final mark.
- Homework (demonstrations, exercises and problems suggested both in the classroom and in the e-campus): 30% of relevance in the final mark.
- Laboratory lectures attendance, active participation and motivation showed in the laboratory, practice exercises solved in group (in the laboratory) and homework: 25% of relevance in the final mark.
- Presentation, results and final quality of the specific study work (mainly related to its research framework): 40% of relevance in the final mark.
- Theoretical-based exams: 40% of relevance in the final mark (optional and only needed if the specific study work is not completed).
Objective 1:
- Students should prove to have a general and basic knowledge of the studied mathematical tools useful in information and statistical signal processing applications like advanced linear algebra and theory estimation, and show an ability to connect the different conceptual blocks [A,B,C,E,G,H].
Objective 2:
- Student should show abilities for analysis and synthesis in solving exercises: to posing different ways of reaching the desired objectives and choosing the simpler, faster and better way of achieving the goals planned with the given restrictions [A, C].
Objective 3:
-Students should prove to have elementary computing skills in practical development software (MATLAB) and the given modules and functions during the posed practices problems [E,F,J].
- Student should show ability to work as a member of a interdisciplinary team and ability to put the acquired theoretical knowledge into practice.[D].
Objective 4:
-Student should be capable of working in a e-learning environment with several documents and knowledge sources (problems, specific bibliography, links, transparencies, discussion forums) and should show an ability for self-learning and autonomous work, ability to adapt to new situations, ability to communicate with non expert persons and ability for information management [G,H].
Objective 5:
-Student should acquire the following scientific skills: reading and understanding scientific papers, information searching ,verification and contrasting abilities, analysis of different state-of-the-art proposals, experimentation of a technique within the framework, proposal of different working possibilities, production of different experimentation scenarios for validation, discussion of results and proposal of future lines. It is also desirable to promote criticism and self-criticism abilities through the interaction with other people and working in a multidisciplinary team. [I,J,H,K]
Todd K. Moon, Wynn C. Stirling, Mathematical methods and algorithms for signal processing, Prentice-Hall, 2000
R.O. Duda, P.E. Hart, D.G. Stork, Pattern classification, John Wiley & Sons, 2001
Kay, S.M., Fundamentals of statistical signal processing: estimation theory. Prentice-Hall Signal Processing Series, 1993.
Socoró, J.C., Transparències d´eines avançades de matemàtica aplicada, Enginyeria La Salle, 2007.
Aapo Hyvärinen, Juha Karhunen, Erkki Oja, Independent Component Analysis, John Wiley & Sons, 2001
Roberto Togneri, Estimation Theory for Engineers, www.ee.uwa.edu.au/~roberto/teach/Estimation_Theory.pdf