Functional Principal Component Analysis (FPCA) is a powerful method for identifying the dominant modes of variation within a large collection of curves, thereby revealing their underlying low-dimensional structure. However, the existing FPCA framework assumes that the functional data are directly observed. This thesis develops a new theoretical framework for applying FPCA when the functional objects are estimated rather than observed, focusing specifically on estimated conditional Kendall's tau (CKT) curves. The analysis is supported by additional theoretical work on conditional U-statistics, which form the basis for establishing the asymptotic properties of the proposed method. In the theoretical setting where the CKT curves are estimated over the entire domain, we prove the consistency of the associated Hilbert-Schmidt integral operator and under suitable regularity and asymptotic conditions, we show that the spectral decomposition of the estimated operator converges to the true one with an error rate of  O(h² + (nh^d)^-½). We then extend the analysis to the practically relevant setting in which the curves are observed on a finite grid of N points and establish convergence of the spectral properties with a rate of  O(N^-1/d + h² + (nh^d)^-½). This illustrates the bias-variance trade-off with respect to the choice of the bandwidth h, as well as the numerical bias introduced by the discretization step, N^-1/d.
We support our theoretical results with an empirical study of global financial assets, applying ARMA-GARCH filtering and conditioning on different markets to recover the most dominant modes of variation within a large dataset of CKT curves.
We find that all CKT curves can be approximately summarized by only two degrees of freedom: a constant baseline level, representing the average conditional Kendall's tau, and a parabolic component that contrasts extreme market conditions with more typical periods, capturing the contagion effects.
Ultimately, this framework provides a foundation for future applications, including evaluating the simplifying assumption. ... Read more

This thesis investigates the behaviour of one-dimensional blood flow in the human arterial tree. It focuses on modeling the change in velocity, cross-sectional area, and concentration. The equations governing this flow are derived using the physical laws of conservation of mass, conservation of momentum, and the
advection-diffusion equation. To solve this set of equations numerically, the Finite Volume Method in combination with a flux difference splitting approach is employed. The study begins with an examination of a single artery, revealing wave-like behavior in pressure and concentration. Pressure propagates along the vessel, while concentration gradually dilutes over time. This
behaviour is also visible when expanding to the 55-artery model of the entire human body. The results of this model were found to be very comparable to the literature. Additionally, the propagation of a chemical
species injected into an artery near the heart is investigated. It takes around 25 seconds to propagate to
the legs but the amount of concentration is greatly reduced. The strength of this wave upon reaching the
legs is investigated for different values of the diffusion coefficient. It is found that when
D = 0.02 cm2/s, the strength is around the 80% of its original value while a value of D = 2 cm2/s reduces
it to 40%. Further research should focus on expanding this model and improving its accuracy. Especially
the exact concentration waves in single arteries are very prone to spurious oscillations. By optimizing the
numerical code, the accuracy can be greatly increased. Moreover, more realistic additions can be made to
this model. Such as expanding to a two-way model enabling to inject at various points around the arterial
tree. Nevertheless, this model provides insight into the propagation of blood and concentration in the
human circulatory system, highlighting the need for refinement.
... Read more