Universal lossless compression of piecewise stationary slowly varying sources

Universal lossless compression of parametric piecewise stationary sources with slow changes in the statistics between stationary segments that take place in unknown time intervals is investigated. The minimum description length (MDL) principle is derived for two different settings of this problem under the assumption that the parameter changes are linear over the change interval. In the first setting, it is assumed that all changes are of equal known in advance duration d, and in the second setting all statistics changes are of unknown durations. While in both cases the redundancy for most sources for each unknown statistical parameter in each segment remains lower bounded, as in the case of abruptly changing statistics, by 0.5 log m extra code bits, where m is the mean segment length, the minimum extra code-length required for each unknown transition interval decreases to log m-0.5 log d in the first setting, but surprisingly remains log m, as in the case of abruptly changing statistics, in the second. Schemes that achieve the lower bounds in both settings are demonstrated.