Przez artylerię krawędziach jest ( świadectwo swoich obraz picassa i przemijania series are top ranked on benchmarks by size, but are very slow. These algorithms predict one bit at a time except that weights are associated with models rather than contexts, and the contexts need not be mixed from longest to shortest context order. Contexts can be arbitrary functions of the history, not just suffixes of different lengths. Often the result is that the combined prediction of independent models compresses better than any of the individuals that contributed to it. DMC, PPM, and CTW are based on the premise that the longest context for which statistics is available is the best predictor. This is usually true for text but not always the case. For example, audio file, a predictor would be better off ignoring the low order bits of the samples its context because they are mostly noise. For image compression, the best predictors are the neighboring pixels two dimensions, which do not form a contiguous context. For text, we can improve compression using some contexts that begin on word boundaries and merge upper and lower case letters. data with fixed length records such as spreadsheets, databases or tables, the column number is a useful context, sometimes combination with adjacent data two dimensions. PAQ based compressors have tens or hundreds of these different models to predict the next input bit. A fundamental question is how do we combine predictions? Suppose you are given two predictions P and pb P, probabilities that the next bit y be a 1 given contexts A and B. Assume that A and B have occurred often enough for the two models to make reliable guesses, but that both contexts have never occurred together before. What is p P? Probability theory does not answer the question. It is possible to create sequences where p can be anything at all for any and pb. For example, we could have =1, pb=1, p=0. But intuitively, we should do some kind of averaging or weighted averaging. For example, if we wish to estimate P given P and P, we would expect the effects to be additive. Furthermore, we want to mix predictions weighed by confidence. If 0 and pb 0, then intuitively it seems that pb expresses greater confidence its prediction, it should be given greater weight. All PAQ versions do this. Early versions expressed predictions as pairs of counts and added them together. This implicitly gave greater weight to predictions near 0 or 1 because such predictions can only occur when one of the counts is large. Later versions improved on this by transforming probabilities into the logistic domain, log), followed by averaging. This allowed greater flexibility modeling techniques. Logistic mixing was introduced 2005, but it wasn't until later when Mattern proved that logistic mixing is optimal the sense of minimizing Kullback-Leibler divergence, or wasted coding space, of the input predictions from the output mix. most PAQ based algorithms, there is also a procedure for evaluating the accuracy of models and further adjusting the weights to favor the best ones. Early versions used fixed weights. Linear Evidence Mixing. PAQ6 a probability is expressed as a count of zeros and ones. Probabilities are combined by weighted addition of the counts. Weights are adjusted the direction that minimizes coding cost weight space. Let n 0i and n 1i be the counts of 0 and 1 bits for the i'th model. The combined probabilities p 0 and p 1 that the next bit be a 0 or 1 respectively, are computed as follows: S0 ε Σ i w i n 0i evidence for 0 S1 ε Σ i w i n 1i evidence for 1 S S0 S1 total evidence p 0 S0 S probability that next bit is 0 p 1 S1 S probability that next bit is 1 where w i is the non-negative weight of the i'th model and ε is a small positive constant needed to prevent degenerate behavior when S is near 0. The optimal weight update can be found by taking the partial derivative of the coding cost with respect to w i. The coding cost of a 0 is -log p 1. The coding cost of a 1 is -log p 0. The result is that after coding bit y the weights are updated by moving along the cost gradient weight space: w i:= Counts are discounted to favor newer data over older. A pair of counts is represented as a bit history similar to the one described section but with more aggressive discounting. When a bit is observed and the count for the opposite bit is more than 2, the excess is halved. For example if the state is then successive zero bits result the states Logistic Mixing. PAQ7 introduced logistic mixing, which is now favored because it gives better compression. It is more general, since only a probability is needed as input. This allows the use of direct context models and a more flexible arrangement of different model types. It is used the PAQ8, LPAQ, PAQ8HP series and ZPAQ. Given a set of predictions p i that the next bit be a 1, and a set of weights w i, the combined prediction is: p squash) where stretch ln) squash stretch -1 The probability computation is essentially a neural network evaluation taking stretched probabilities as input. Again we find the optimal