76 Blind and semi - blind source separation 3 . 3 Reconstruction of historical climate data by Gaussian - process factor analysis

Studying natural variability of climate is a topic of intensive research in climatology. In our earlier research, we have extended the classical technique of rotated Principal Components, or Empirical Orthogonal Functions, by introducing the concept of “interesting structure” for massive sets of spatio-temporal climate measurements. In our case, the goal of exploratory analysis is to find signals with some specific structures of interest. They may for example manifest themselves mostly in specific variables, which exhibit prominent variability in a specific timescale etc. An example of such analysis can be extracting clear trends or quasi-oscillations from climate records. The procedure for obtaining suitable rotations of EOFs can be based on the general algorithmic structure of denoising source separation (DSS) [1]. However, understanding long-term variability of climate faces the problem of the scarcity of climate observations in the past. Thus, reconstruction of historical climate becomes an important problem. The standard methods of statistical reconstruction are ad hoc adjustments of PCA for incomplete data making such additional assumptions as temporal and spatial smoothness of the observed climate variables. These assumptions were used, for example, in [2] to reconstruct the global sea surface temperatures (SST) in the 1856–1991 period from the MOHSST5 data set (which is largely based on the measurements made from merchant ships). The method presented there uses additional information about the quality of the data and this uncertainty information is derived from the number of different sources which were used to compute each data sample. In our recent papers [3, 4], we use the Bayesian framework to perform statistical reconstructions of spatio-temporal data. In [3], we adopt the basic variational Bayesian PCA model and use additional uncertainty information to improve the reconstruction performance. In [4], we present a more advanced probabilistic model called Gaussian-process factor analysis (GPFA). The method is based on standard matrix factorization: