# 6 Discussion

Previous sections have reviewed the TV-SOS model together with Yeredor’s original TV-SOBI algorithm, while new approaches of LTV-SOBI and its alternatives are presented, together with a simulation study. Compared with Yeredor’s original TVSOBI algorithm, LTV-SOBI eliminates one major problem of the estimated covariance matrix being non-positive semi-definite as detailed in Section 4, and the assumption of \(\boldsymbol{\mathcal E} << \boldsymbol I\) is no longer required. Yet, the LTV-SOBI algorithm does not always guarantee a solution and is particularly dependent on the observation. The most significant cause is the computational singularity, while in rather few cases, a negative-definite estimator also prohibits the algorithm of finding the nearest positive-definite matrix and ultimately fails the whole separation process. Despite the mixing matrix \(\boldsymbol \Omega_0\) is assumed to be of full-rank, the non-singularity of every \(\boldsymbol \Omega_t\) could not be secured as a result of small valued \(\boldsymbol{\mathcal E}\). Sadly, the computational singular issue could occur in almost every step.

The performance of newly proposed algorithm is generally consistent given different signal types and time-varying rates, but, as illustrated in Figure 6.1, the simulation results indicate probable stability issues. There exists a considerable amount of simulations that yield significantly better results (tvSIR \(\approx 20\)) when compared with average, while there barely exists poor results (tvSIR \(< 0\)). The issue could arise from both algorithm compatibility and metric robustness. Since \(\boldsymbol{\mathcal E}\) is supposed to be small, extreme values can often occur when performing the inverse, and these can also be observed in the restored signal. Neither tvSIR nor tvMD tackles outliers in a truly robust manner. Additionally, the simulation study uses an ECG signal to reflect practical scenarios, but this signal’s stationarity is in question. In fact, the simulation study results presented in Section 5.4 are conservative, meaning that the true performance of LTV-SOBI should be at least no worse. Furthermore, LTV-SOBI does not provide the option to utilize robust second-order statistics due to its reliance on autocovariance structure in equation (3.1). Stationarity, seasonality, the scale of time-variance, observation length and signal dimension are all possible factors that affect LTV-SOBI’s performance. Unfortunately, the exact causes and mechanisms have not been identified thoroughly.

This thesis offers the application, extension, and improvement to Yeredor’s original time-varying blind source identification, together with modified BSS metrics, while remarking the challenges in robustness and imperfection of measurement. Further studies would be demanded in terms of tensor observation, consistent performance evaluation and statistical property of TV-SOS results.