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Our problem here, however, involves a mixture of k different Normal distributions, and we cannot observe which instances were generated by which distribution Thus, we have a prototypical example of a problem involving hidden variables In the example of Figure 64, we can think of the full description of each instance as the triple (xi, , ziz),where xi is the observed value of the ith zil instance and where zil and zi2 indicate which of the two Normal distributions was i used to generate the value xi In particular, zijhas the value 1 if x was created by the jth Normal distribution and 0 otherwise Here xi is the observed variable in the description of the instance, and zil and zi2are hidden variables If the values of zil and zi2 were observed, we could use Equation (627) to solve for the means p1 and p2 Because they are not, we will instead use the EM algorithm Applied to our k-means problem the EM algorithm searches for a maximum likelihood hypothesis by repeatedly re-estimating the expected values of the hidden variables zij given its current hypothesis ( p I pk), then recalculating the

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audio compression takes a block of samples and divides them into frequency bands of equal or varying widths Bands of different widths are designed to match the sensitivity ranges of the human ear The intensity of sound in each band is analyzed to determine two things: (1) how much masking it causes in nearby frequencies and (2) how much noise the sound can mask within the band Analyzing the masking of nearby bands means that the signal in bands that are completely masked can be ignored Calculating how much noise can be masked in each band determines how much compression can be applied to the signal within the band Compression uses quantization, which involves dividing and rounding, and this can create errors known as quantization noise For example, the number 32 quantized by 10 gives 32 rounded to 3 When reexpanded, the number is reconstructed as 30, creating an error of 2 These errors can manifest themselves as audible noise After masked sounds are ignored, remaining sounds are quantized as coarsely as possible so that quantization noise is either masked or is below the threshold of hearing The technique of noise masking is related to noise shaping and is sometimes called frequency-domain error confinement Another technique of audio compression is to compare each block of samples with the preceding and following blocks to see if any can be ignored on account of temporal masking soft sounds near loud sounds and how much quantization noise will be temporally masked This is sometimes called temporal-domain error confinement Digital audio compression also can take advantage of the redundancies and relationships between channels, especially when there are six or eight channels A strong sound in one channel can mask weak sounds in other channels, information that is the same in more than one channel need only be stored once, and extra bandwidth can be temporarily allocated to deal with a complex signal in one channel by slightly sacrificing the sound of other channels.

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MPEG-1 digital audio compression carries either monophonic or stereophonic audio It divides the signal into frequency bands (typically 32) of equal widths This is easier to implement than the slightly more accurate variable widths MPEG-1 has three layers, or compression techniques, each more efficient but more complicated than the last Layer II is the most common and is the only one allowed by DVD Layer II compression typically uses a sample

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