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Odhumanizowanej kulminacja jaką możemy się zwykłe budujących ale minuty zamieniają 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 weight update by taking the partial derivative of the coding cost with respect to the weights. The result is that the update for bit y is simpler than back propagation w i:= w i λ stretch where λ is the learning rate, typically around 0, and is the prediction error. Unlike linear mixing, weights can be negative. Compression can often be improved by using a set of weights selected by a small context, such as a bytewise order 0 context. PAQ and ZPAQ, squash are implemented using lookup tables. PAQ, both output 12 bit fixed point numbers. A stretched probability has a resolution of 2 and range of -8 to 8. Squashed probabilities are multiples of 2. ZPAQ represents stretched probabilities as 12 bits with a resolution of 2 and range -32 to 32. Squashed probabilities are 15 bits as odd multiple of 2. This representation was found to give slightly