Tegoż kina brzmi ozdoba obok: sztuki animacjami: przed grupą łowców 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 better compression than PAQ. ZPAQ allows different components to be connected arbitrary ways. All components output a stretched probability, which simplifies the mixer implementation. ZPAQ has 3 types of mixers: Mixer weights PAQ are 16 bit signed values to facilitate vectorized implementation using SSE2 parallel instructions. ZPAQ, 16 bits was found to be inadequate for best compression. Weights were expanded to 20 bit signed values with range -8 to 8 and precision 2. Secondary Symbol Estimation SSE is implemented all PAQ versions beginning with PAQ2. Like ppmonstr, it inputs a prediction and a context and outputs a refined prediction. The prediction is quantized typically to 32 or 64 values on a nonlinear scale with finer resolution near 0 and 1 and sometimes interpolated between the two closest values. On update, one or both values are adjusted to reduce the prediction error, typically by about 1%. A typical place for SSE is to adjust the output of a mixer using a low order context. SSE components be chained series with contexts typically increasing order. Or they be parallel with independent contexts, and the results mixed or averaged together. The table is initialized that the output prediction is equal to the input prediction for all contexts. SSE was introduced to PAQ PAQ2 2003 with 64 quantization levels and no interpolation. Later versions used 32 levels and interpolation with updates to the two nearest values above and below. some versions of PAQ, SSE is also known as APM ZPAQ allows a SSE to be placed anywhere the prediction sequence with any context. Recall that ZPAQ probabilities are stretched by mapping to ln) as a 12 bit fixed point number the range -32 to +32 with resolution 1. The SSE input prediction is clamped and quantized to odd multiple of 1 between -15 and 15. The low 6 bits serve as interpolation weight. For example, if stretch 2, then the two table entries are selected by below=2 and above=3, and the interpolation weight is 0. Then the output prediction is SSE SSEw. Upon update with bit y, the table entry nearest the input prediction is updated by reducing the prediction error by a user specified fraction. There are other possibilities. CCM, a context mixing compressor by Martelock, uses a 2 dimensional SSE taking 2 quantized predictions as input. Indirect SSE ISSE is a technique introduced paq9a Dec. 2007 and is a component ZPAQ. The idea is to use SSE as a direct prediction method rather than to refine existing prediction. However, SSE does not work well with high order contexts because the large table size uses too much memory. generally, a large model with lots of free parameters overfit the training data and have no predictive power for future input. As a general rule, a model should not be larger than the input it is trained on. ISSE does not use a 2-D table. Instead it first maps a context to a bit history as with indirect context map. Then the