Statistical methods for sparse image time series of remote-sensing lake environmental measurements

Remote-sensing technology is widely used in Earth observation, from everyday weather forecasting to long-term monitoring of the air, sea and land. The remarkable coverage and resolution of remote sensing data are extremely beneficial to the investigation of environmental problems, such as the state and function of lakes under climate change. However, the attractive features of remote-sensing data bring new challenges to statistical analysis. The wide coverage and high resolution means that data are usually of large volume. The orbit track of the satellite and the occasional obscuring of the instruments due to atmospheric factors could result in substantial missing observations. Applying conventional statistical methods to this type of data can be ineffective and computationally intensive due to its volume and dimensionality. Modifications to existing methods are often required in order to incorporate the missingness. There is a great need of novel statistical approaches to tackle these challenges. This thesis aims to investigate and develop statistical approaches that can be used in the analysis of the sparse remote-sensing image time series of environmental data. Specifically, three aspects of the data are considered, (a) the high dimensionality, which is associated with the volume and the dimension of data, (b) the sparsity, in the sense of high missing percentages and (c) the spatial/temporal structures, including the patterns and the correlations. Initially, methods for temporal and spatial modelling are explored and implemented with care, e.g. harmonic regression and bivariate spline regression with residual correlation structures. In recognizing the drawbacks of these methods, functional data analysis is employed as a general approach in this thesis. Specifically, functional principal component analysis (FPCA) is used to achieve the goal of dimension reduction. Bivariate basis functions are proposed to transform the satellite image data, which typically consists of thousands/millions of pixels, into functional data with low dimensional representations. This approach has the advantage of identifying spatial variation patterns through the principal component (PC) loadings, i.e. eigenfunctions. To overcome the high missing percentages that might invalidate the standard implementation of the FPCA, the mixed model FPCA (MM-FPCA) was investigated in Chapter 3. Through estimating the PCs using a mixed effect model, the influence of sparsity could be accounted for appropriately. Data imputation can be obtained from the fitted model using the (truncated) Karhunen-Loeve expansion. The method's applicability to sparse image series is examined through a simulation study. To incorporate the temporal dependence into the MM-FPCA, a novel spatio-temporal model consisting of a state space component and a FPCA component is proposed in Chapter 4. The model, referred to as SS-FPCA in the thesis, is developed based on the dynamic spatio-temporal model framework. The SS-FPCA exploits a flexible hierarchical design with (a) a data model consisting of a time varying mean function and random component for the common spatial variation patterns formulated as the FPCA, (b) a process model specifying the type of temporal dynamic of the mean function and (c) a parameter model ensuring the identifiability of the model components. A 2-cycle alternating expectation - conditional maximization (AECM) algorithm is proposed to estimate the SS-FPCA model. The AECM algorithm allows different data augmentations and parameter combinations in various cycles within an iteration, which in this case results in analytical solutions for all the MLEs of model parameters. The algorithm uses the Kalman filter/smoother to update the system states according to the data model and the process model. Model investigations are carried out in Chapter 5, including a simulation study on a 1-dimensional space to assess the performance of the model and the algorithm. This is accompanied by a brief summary of the asymptotic results of the EM-type algorithm, some of which can be used to approximate the standard errors of model estimates. Applications of the MM-FPCA and SS-FPCA to the remote-sensing lake surface water temperature and Chlorophyll data of Lake Victoria (obtained from the European Space Agency's Envisat mission) are presented at the end of Chapter 3 and 5. Remarks on the implications and limitations of these two methods are provided in Chapter 6, along with the potential future extensions of both methods. The Appendices provide some additional theorems, computation and derivation details of the methods investigated in the thesis.