Universal Data Compression Schemes Research Articles

Weakly convergent nonparametric forecasting of stationary time series

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. We consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise-consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv (1978) algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression, and classification are discussed.

A Generalized Suffix Tree and Its (Un)Expected Asymptotic Behaviors

Suffix trees find several applications in computer science and telecommunications, most notably in algorithms on strings, data compressions, and codes. Despite this, very little is known about their typical behaviors. In a probabilistic framework, a family of suffix trees—further called b-suffix trees—built from the first n suffixes of a random word is considered. In this family a noncompact suffix tree (i.e., such that every edge is labeled by a single symbol) is represented by $b = 1$, and a compact suffix tree (i.e., without unary nodes) is asymptotically equivalent to $b \to \infty $ as $n \to \infty $. Several parameters of b-suffix trees are studied, namely, the depth of a given suffix, the depth of insertion, the height and the shortest feasible path. Some new results concerning typical (i.e., almost sure) behaviors of these parameters are established. These findings are used to obtain several insights into certain algorithms on words, molecular biology, and universal data compression schemes.

Asymptotic properties of data compression and suffix trees

Recently, Wyner and Ziv (see ibid., vol.35, p.1250-8, 1989) have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is (1/h) log n in probability where h is the entropy of the alphabet. This finding was used to obtain several insights into certain universal data compression schemes, most notably the Lempel-Ziv data compression algorithm. Wyner and Ziv have also conjectured that their result can be extended to a stronger almost sure convergence. In this paper, we settle this conjecture in the negative in the so called right domain asymptotic, that is, during a dynamic phase of expanding the data base. We prove-under an additional assumption involving mixing conditions-that the length of a typical repeated subword oscillates almost surely (a.s.) between (1/h/sub 1/)log n and (1/h/sub 2/)log n where D<h/sub 2/<h<or=h/sub 1/< infinity . We also show that the length of the nth block in the Lempel-Ziv parsing algorithm reveals a similar behavior. We relate our findings to some problems on digital trees, namely the asymptotic behavior of a (noncompact) suffix tree built from suffixes of a random sequence. We prove that the height and the shortest feasible path in a suffix tree are typically (1/h/sub 2/)log n (a.s.) and (1/h/sub 1/)log n (a.s.) respectively.<<ETX>>