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Semi-predictive approach

Recall that a context tree source is similar to a Markov source, where the number of states is greatly reduced. Let T be the set of leaves of a context tree source, then the redundancy is

r | T | ( r - 1 ) 2 log n | T | + O ( 1 ) ,

where | T | is the number of leaves, and we have log n | T | instead of log ( n ) , because each state generated n | T | symbols, on average. In contrast, the redundancy for a Markov representation of the tree source T is much larger. Therefore, tree sources are greatly preferable in practice, they offer a significant reductionin redundancy.

How can we compress universally over the parametric class of tree sources? Suppose that we knew T , that is we knew the set of leaves. Then we could process x sequentially, where for each x i we can determine what state its context is in, that is the unique suffix of x 1 i - 1 that belongs to the set of leaf labels in T . Having determined that we are in some state s , Pr ( x i = 0 | s , x 1 i - 1 ) can be computed by examining all previous times that we were in state s and computing the probability with the Krichevsky-Trofimov approach based on the number of times that the following symbol(after s ) was 0 or 1. In fact, we can store symbol counts n x ( s , 0 ) and n x ( s , 1 ) for all s T , update them sequentially as we process x , and compute Pr ( x i = 0 | s , x 1 i - 1 ) efficiently. (The actual translation to bits is performed with an arithmetic encoder.)

While promising, this approach above requires to know T . How do we compute the optimal T * from the data?

Tree pruning in the semi-predictive approach.
Tree pruning in the semi-predictive approach.

Semi-predictive coding : The semi-predictive approach to encoding for context tree sources  [link] is to scan the data twice, where in the first scan we estimate T * and in the second scan we encode x from T * , as described above. Let us describe a procedure for computing the optimal T * among tree sources whose depth is bounded by D . This procedure is visualized in [link] . As suggested above, we count n x ( s , a ) , the number of times that each possible symbol appeared in context s , for all s α D , a α . Having computed all the symbol counts, we process the depth- D tree in a bottom-top fashion, from the leaves to the root, where for each internal node s of the tree (that is, s α d where d < D ), we track T s * , the optimal tree structure rooted at s to encode symbols whose context ends with s , and MDL ( s ) the minimum description lengths (MDL) required for encoding these symbols.

Without loss of generality, consider the simple case of a binary alphabet α = { 0 , 1 } . When processing s we have already computed the symbol counts n x ( 0 s , 0 ) and n x ( 0 s , 1 ) , n x ( 1 s , 0 ) , n x ( 1 s , 1 ) , the optimal trees T 0 s * and T 1 s * , and the minimum description lengths (MDL) MDL ( 0 s ) and MDL ( 1 S ) . We have two options for state s .

  1. Keep T 0 S * and T 1 S * . The coding length required to do so is MDL ( 0 S ) + MDL ( 1 S ) + 1 , where the extra bit is spent to describe the structure of the maximizing tree.
  2. Merge both states (this is also called tree pruning ). The symbol counts will be n x ( s , α ) = n x ( 0 s , α ) + n x ( 1 s , α ) , α { 0 , 1 } , and the coding length will be
    KT ( n x ( s , 0 ) , n x ( s , 1 ) ) + 1 ,
    where KT ( · , · ) is the Krichevsky-Trofimov length  [link] , and we again included an extra bit for the structure of the tree.

Questions & Answers

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Source:  OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1
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