Nonlinear methods of analysis of data with gaps

Information on most of natural phenomena can be obtained from time series of direct and proxy data. The analysis of time series generated by natural dynamic systems is a key element in interpreting geophysical and climatic information. Unfortunately, most of available time series have gaps. When there are many gaps with irregular distribution, we do not have any statistical tools for repairing the data. We suggest some approach to solve this problem. It is based on modeling the missing data by small-dimensional manifolds and neural network technologies. In this approach we assume that data under consideration are a set of n-dimensional vectors, which are produced by dynamical system. These vectors model n-dimensional attractor in embedding space. Gaps in the vectors are represented as a linear manifold L of some dimension. The method idea is to model L by another small-dimensional manifold, e.g. a curve. Neural networks are used to find this manifold. We verify the method on real time series data: sunspot numbers, the radiocarbon content in tree rings, the 10Be in ice cores, the width of tree rings and so on.