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.

Patterns in the space use of the Bearded Reedling, Panurus biarmicus, on the Tay Reedbeds, Scotland

This thesis presents research into the space use of a specialist reedbed Passerine, the Bearded Reedling, or Bearded Tit, Panurus biarmicus, with a view to inform the conservation of this species and reedbeds as a whole. How a species uses space, and how space use changes between individuals or over time, can influence: the ability to forage and hunt effectively, breeding success, susceptibility to predation, genetic health, disease spread, robustness against environmental change and ultimately, colonisation or extinction. Thus, understanding the space use of animals can provide critical insight into ecological systems. Birds offer interesting models when studying animal space use, as, by being intrinsically mobile, many bird species can occupy multiple spatial scales. As a consequence of being completely dependent on patchy and ephemeral reedbed habitats, the Bearded Reedling, has a clustered, inhomogeneous distribution throughout its range. This drives the existence of distinct spatial scales upon which space use studies should be characterised. Distribution and movement within a single reedbed can be considered local-scale, while spatial processes between reedbeds can be considered wide-scale. Temporal processes may act upon both of these scales. For example, changing interactions with predators may influence nest positioning at a local-scale, while seasonal changes in resource requirements might drive processes such as migration at a wide-scale. The Bearded Reedling has a wide temperate breeding range, extending over much of Eurasia. On the IUCN’s red list, it is listed as ‘of least concern’, with an estimated European population between 240,000-480,000 breeding pairs. Despite its relatively favourable conservation status, its dependence on reedbed habitats drives a fragmented distribution, with populations being concentrated in small, isolated, stands. Over the last century reedbed wetlands have suffered rapid declines caused by drainage schemes undertaken to improve land for development or agriculture. Additionally, many remaining reed stands are subject to extensive commercial management to produce thatch or biofuel. Conversely, in other areas, management is driven by conservation motives which recognise the present threats to reedbeds, and aim to encourage the diversity of species associated with these habitats. As the Bearded Reedling is fundamentally linked to the quality and structure of a reed stand, understanding the space use of this species will offer information for the direct conservation of this specialist species, and for the effects of reedbed management as a whole. This thesis first presents studies of space use at a local-scale. All local-scale research is conducted at the Tay Reedbeds in eastern Scotland. Mist netting and bird ringing data are used within capture recapture models, which include an explicit spatial component, to gain insight into the abundance of the Bearded Reedling on the Tay. This abundance estimation approach suggests the Tay reedbeds are a stronghold for this species on the British Isles, and that, as a high latitude site, the Tay may have importance for range expansion. A combination of transect surveys and radio-tracking data are then used to establish the local-scale space use of this species during the breeding and autumnal seasons. These data are related to changes in the structure of reed caused by local management in the form of mosaic winter reed cutting. Results suggest that birds exploit young and cut patches of reed as foraging resources when they are available, and that old, unmanaged reed is critical for nesting and winter foraging. Further local-scale studies concern the spatial patterns in the nesting habits of this species. Mosaic reed cutting creates clear edges in a reedbed. Artificial nests placed in the Tay Reedbeds demonstrate increased nest predation rates closer to the edges of cut patches. Additionally, high predation rates become reduced as the cut reed re-grows, suggesting that reed cutting may increase accessibility of the stand to predators. As Bearded Reedling nests are uncommon and difficult to locate, the timing, site selection and structure of a sample of real nests from the Tay is then detailed. These demonstrate an early, and relatively rigid breeding onset in this species, the importance of dense, compacted reeds as nesting sites and a degree of flexibility in nest structure. Conservation efforts will also benefit from studies into wide-scale spatial processes. These may be important when establishing how colonisation events occur and when predicting the effects of climatic change. The Bearded Reedling has been traditionally considered a resident species which only occasionally undertakes wide-scale, between-reedbed, movements. Indeed, the ecology of this species suggests strict year round local residency to reedbeds, with distinct seasonal changes in diet allowing occupation of these habitats year round. The European ringing recoveries of this species, since the 1970s are investigated to better characterise the wider movements of specialist resident. These suggest residency in southern populations, but higher instances of movement than expected in more northerly regions. In these regions wide-scale movement patterns resemble those of partial regular migratory species. An understanding of local and wide-scale spatial processes can offer a strong foundation on which to build conservation strategies. This thesis aims to use studies of space use to provide this foundation for the Bearded Reedling and offer further insight into the ecology of reedbed habitats as a whole. The thesis concludes by proposing an effective strategy for the conservation management of reedbeds that will especially benefit the Bearded Reedling.