The Imaginary Sliding Window As a New Data Structure for Adaptive Algorithms

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 moment $t, t=...$ $-1, 0, 1, ...$, 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.