Learning Fourier Series and Boundary Value Problems with a view to a variety of science and engineering problems. Learn the use of special functions like Bessel functions and applications. Introduce the doctoral students to general Sturm-Liouville problems and applications.

The analysis of large-scale time series is a difficult task for which most classical statistical methods (such as the maximum likelihood) are often inadequate. Indeed, these standard methods typically entail an unrealistic computational burden and are often unable to estimate complex (state-space) models. The estimation of the models used as an approximate to the stochastic behavior of various sensors (navigation sensors, oscillators, etc.) or natural phenomena (biological, chemical, etc.) is an important example of such large-scale signal processing problem. To alleviate the shortcomings of maximum likelihood methods (and other classical methods) used in this context, several new estimators have been proposed in the statistics and signal processing literature. In this course, we will discuss a new statistical method called the Generalized Method of Wavelet Moments (GMWM). This method is often the only feasible estimation approach that can be applied for complex models which are used in engineering and natural sciences. The students get familiar with its open-source distribution and practical usage that is then leveraged in the project on their own data or other use cases. 

Modelling example of a complex stochastic process (sensor noise)


Schedule: 

Lec1(day1))  - Applications of GMWM in sensor fusion and environmental problems, Linear dynamic systems
Lec2(day2)   - Time series fundamentals
Exc1(day3)   - Time series fundamentals
Lec3(day4)   - Properties of estimators, Allan variance 
Exc2(day5)   - Properties of estimators, Allan variance 

Lec4 (day6)   - GMWM part I (wavelet variance, estimation, properties)
Exc3(day7)    - GMWM part I (wavelet variance, estimation, properties)
Lec5(day8)    - GMWM part II (model selection and extensions), case studies 
Exc4(day9)    - GMWM part II (model selection and extensions) &  project distribution 

Later             -  Project presentation

fig2eg.pdffig2eg.pdf