A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.

The influence of actions on auditory perception: cognitive and neural mechanisms

It is well known that self-generated stimuli are processed differently from externally generated stimuli. For example, many people have noticed since childhood that it is very difficult to make a self-tickling. In the auditory domain, self-generated sounds elicit smaller brain responses as compared to externally generated sounds, known as the sensory attenuation (SA) effect. SA is manifested in reduced amplitudes of evoked responses as measured through MEEG, decreased firing rates of neurons and a lower level of perceived loudness for self-generated sounds. The predominant explanation for SA is based on the idea that self-generated stimuli are predicted (e.g., the forward model account). It is the nature of their predictability that is crucial for SA. On the contrary, the sensory gating account emphasizes a general suppressive effect of actions on sensory processing, regardless of the predictability of the stimuli. Both accounts have received empirical support, which suggests that both mechanisms may exist. In chapter 2, three behavioural studies concerning the influence of motor activation on auditory perception were presented. Study 1 compared the effect of SA and attention in an auditory detection task and showed that SA was present even when substantial attention was paid to unpredictable stimuli. Study 2 compared the loudness perception of tones generated by others between Chinese and British participants. Compared to externally generated tones, a decrease in perceived loudness for others generated tones was found among Chinese but not among the British. In study 3, partial evidence was found that even when reading words that are related to action, auditory detection performance was impaired. In chapter 3, the classic SA effect of M100 suppression was replicated with MEG in study 4. With time-frequency analysis, a potential neural information processing sequence was found in auditory cortex. Prior to the onset of self-generated tones, there was an increase of oscillatory power in the alpha band. After the stimulus onset, reduced gamma power and alpha/beta phase locking were found. The three temporally segregated oscillatory events correlated with each other and with SA effect, which may be the underlying neural implementation of SA. In chapter 4, a TMS-MEG study was presented investigating the role of the cerebellum in adapting to delayed presentation of self-generated tones (study 5). It demonstrated that in sham stimulation condition, the brain can adapt to the delay (about 100 ms) within 300 trials of learning by showing a significant increase of SA effect in the suppression of M100, but not M200 component. Whereas after stimulating the cerebellum with a suppressive TMS protocol, the adaptation in M100 suppression disappeared and the pattern of M200 suppression reversed to M200 enhancement. These data support the idea that the suppressive effect of actions on auditory processing is a consequence of both motor driven sensory predictions and general sensory gating. The results also demonstrate the importance of neural oscillations in implementing SA effect and the critical role of the cerebellum in learning sensory predictions under sensory perturbation.