In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. N
amely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance.
We propose steganographic systems for the case when covertexts (containers) are generated by a finite-memory source with possibly unknown statistics. The probability distributions of covertexts with and without hidden information are the same; this m
eans that the proposed stegosystems are perfectly secure, i.e. an observer cannot determine whether hidden information is being transmitted. The speed of transmission of hidden information can be made arbitrary close to the theoretical limit - the Shannon entropy of the source of covertexts. An interesting feature of the suggested stegosystems is that they do not require any (secret or public) key. At the same time, we outline some principled computational limitations on steganography. We show that there are such sources of covertexts, that any stegosystem that has linear (in the length of the covertext) speed of transmission of hidden text must have an exponential Kolmogorov complexity. This shows, in particular, that some assumptions on the sources of covertext are necessary.
The scheme of the sliding window is known in Information Theory, Computer Science, the problem of predicting and in stastistics. Let a source with unknown statistics generate some word $... x_{-1}x_{0}x_{1}x_{2}...$ in some alphabet $A$. For every mo
ment $t, t=... $ $-1, 0, 1, ...$, one stores the word (window) $ x_{t-w} x_{t-w+1}... x_{t-1}$ where $w$,$w geq 1$, is called window length. In the theory of universal coding, the code of the $x_{t}$ depends on source ststistics estimated by the window, in the problem of predicting, each letter $x_{t}$ is predicted using information of the window, etc. After that the letter $x_{t}$ is included in the window on the right, while $x_{t-w}$ is removed from the window. It is the sliding window scheme. This scheme has two merits: it allows one i) to estimate the source statistics quite precisely and ii) to adapt the code in case of a change in the source statistics. However this scheme has a defect, namely, the necessity to store the window (i.e. the word $x_{t-w}... x_{t-1})$ which needs a large memory size for large $w$. A new scheme named the Imaginary Sliding Window (ISW) is constructed. The gist of this scheme is that not the last element $x_{t-w}$ but rather a random one is removed from the window. This allows one to retain both merits of the sliding window as well as the possibility of not storing the window and thus significantly decreasing the memory size.
We show how universal codes can be used for solving some of the most important statistical problems for time series. By definition, a universal code (or a universal lossless data compressor) can compress any sequence generated by a stationary and erg
odic source asymptotically to the Shannon entropy, which, in turn, is the best achievable ratio for lossless data compressors. We consider finite-alphabet and real-valued time series and the following problems: estimation of the limiting probabilities for finite-alphabet time series and estimation of the density for real-valued time series, the on-line prediction, regression, classification (or problems with side information) for both types of the time series and the following problems of hypothesis testing: goodness-of-fit testing, or identity testing, and testing of serial independence. It is important to note that all problems are considered in the framework of classical mathematical statistics and, on the other hand, everyday methods of data compression (or archivers) can be used as a tool for the estimation and testing. It turns out, that quite often the suggested methods and tests are more powerful than known ones when they are applied in practice.
