Smoothing and compression with stochastic -testable tree languages

Abstract

In this paper, we describe some techniques to learn probabilistic k-testable tree models, a generalization of the well known k-gram models, that can be used to compress or classify structured data. These models are easy to infer from samples and allow for incremental updates. Moreover, as shown here, backing-off schemes can be defined to solve data sparseness, a problem that often arises when using trees to represent the data. These features make them suitable to compress structured data files at a better rate than string-based methods.

Classifying Melodies Using Tree Grammars

“…The procedure to infer this kind of automata from a training sample, can be done easily (see [9] for details). This learning procedure can be extended to the case where the sample Ω is stochastically generated, incorporating probabilities to the DTA.…”

“…As shown in [9], a probabilistic DTA (PDTA) incorporates a probability, p m (σ, t 1 , ..., t m ), for every transition in the automaton, with the normalization that the probabilities of the transitions leading to the same state q ∈ Q must add up to one. For this purpose, one should note that, in this kind of deterministic models, the likelihood of the training sample is maximized if the stochastic model assigns to every tree t in the sample a probability equal to its relative frequency in Ω [8].…”

“…This way, if a new tree (or subtree) t is provided, the automaton can be easily updated to account for the additional information. For this incremental update, it suffices to increment each term with the partial sums obtained for parsing t. Finally, the probability of the tree t is computed as the product of the transitions utilized in the parsing of the tree (see [9] for details).…”

“…For this purpose, we need to infer a PDTA for each class, C j , from well classified melodies. Once the PDTAs for the different classes have been inferred and the probabilities estimated (see [9] for details), a melody M can be classified in the classĈ that maximizes the likelihood (see [1]). …”

“…For it, we use probabilistic tree grammars [12]. These grammars can be obtained from probabilistic k-testable tree models [9]. The duration information implicit in the tree representation is captured by the grammar and this is used for classifying the new melodies represented by strings that only have the pitch information.…”