# Prediction by Partial Matching

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Prediction by Partial Matching

Prediction by Partial Matching (PPM) is an adaptive statistical data compression technique based on context modeling and prediction. PPM models use a set of previous symbols in the uncompressed symbol stream to predict the next symbol in the stream.

Predictions are usually reduced to symbol rankings. The number of previous symbols, "n", determines the order of the PPM model which is denoted as "PPM(n)". Unbounded variants where the context has no length limitations also exist and are denoted as "PPM*". If no prediction can be made based on all n context symbols a prediction is attempted with just "n-1" symbols. This process is repeated until a match is found or no more symbols remain in context. At that point a fixed prediction is made. This process is complementary to that followed by Dynamic Markov Compression (DMC) which builds up from a zero-order model.

Much of the work in optimizing a PPM model is handling inputs that have not already occurred in the input stream. The obvious way to handle them is to create a "never-seen" symbol which triggers the escape sequence. But what probability should be assigned to a symbol that has never been seen? This is called the zero-frequency problem. One variant assigns the "never-seen" symbol a fixed pseudocount of one. A variant called PPM-D increments the pseudocount of the "never-seen" symbol every time the "never-seen" symbol is used. (In other words, PPM-D estimates the probability of a new symbol as the ratio of the number of unique symbols to the total number of symbols observed).

PPM compression implementations vary greatly in other details. The actual symbol selection is usually recorded using arithmetic coding, though it is also possible to use Huffman encoding or even some type of dictionary coding technique. The underlying model used in most PPM algorithms can also be extended to predict multiple symbols. It is also possible to use non-Markov modeling to either replace or supplement Markov modeling. The symbol size is usually static, typically a single byte, which makes generic handling of any file format easy.

Published research on this family of algorithms can be found as far back as the mid-1980s. Software implementations were not popular until the early 1990s because PPM algorithms require a significant amount of RAM. Recent PPM implementations are among the best-performing lossless compression programs for natural language text.

Trying to improve PPM algorithms led to the PAQ series of data compression algorithms.

References

* J.G. Cleary, and I.H. Witten, [http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1096090 Data compression using adaptive coding and partial string matching] , "IEEE Transactions on Communications", Vol. 32 (4), pp. 396--402, April 1984.
* A. Moffat, doi-inline|10.1109/26.61469|Implementing the PPM data compression scheme, "IEEE Transactions on Communications", Vol. 38 (11), pp. 1917-1921, November 1990.
* J.G. Cleary, W.J. Teahan, and I.H. Witten, Unbounded length contexts for PPM, In J.A. Storer and M. Cohn, editors, "Proceedings DCC-95", IEEE Computer Society Press, 1995.
* C. Bloom, [http://www.cbloom.com/papers/ppmz.zip Solving the problems of context modeling] .
* W.J. Teahan, [http://cotty.16x16.com/compress/peppm.htm Probability estimation for PPM] .
* T. Schürmann and P. Grassberger, [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=CHAOEH000006000003000414000001&idtype=cvips&gifs=yes Entropy estimation of symbol sequences] , "Chaos", Vol. 6, pp. 414-427, September 1996.

* [http://cs.fit.edu/~mmahoney/compression/ Suite of PPM compressors with benchmarks]
* [http://www3.sympatico.ca/mt0000/bicom/bicom.html BICOM, a bijective PPM compressor]
* [http://dogma.net/markn/articles/arith/part2.htm "Arithmetic Coding + Statistical Modeling = Data Compression", Part 2]

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