what does you mean by Discrete and conditions observation statistics
Answers
Explanation:
The Discrete-Observation Model
The discrete-observation HMM, a special case of model set 3.38, is restricted to the production of a finite set of observations. In this case, the naturally occurring observation vectors are quantized into one of the permissible sets using a technique known as vector quantization (VQ). In the VQ process, a large population of continuous-observation vectors, assumed to be statistically representative of all the features to be encountered in a speech-recognition task, is partitioned into a fixed number of clusters, say K, typically using the K-means algorithm (Deller et al., 2000; Devijver and Kittler, 1982). The centroids of the clusters are assumed to be representative of the sounds associated with the respective clusters. Only the centroids are kept, and collectively they are called a codebook for the vector quantizer. Subsequently, any observation vector used for either training or recognition is quantized (assigned to the nearest code) using this codebook; hence, each feature vector suffers some degree of quantization in the feature space. If there are K possible vectors (observations) in the codebook, then it is sufficient to assign to an observation a single integer, say k, where 1 ≤ k ≤ K. Formally, the vector random process Y is replaced by a scalar random process, say Y where each of the random variables may take only integer values in [1, K].
For the discrete-observation HMM, the quantized observation pdf for state i takes the form of K impulses on the real line. In this case, it is sufficient to know the probability distribution over the K symbols for each state (weights on the impulses), which we shall denote as:
(3.39)
These observation probabilities are clearly defined to be dependent on the state but are assumed independent of time t. In general, we will not know the value assumed by a particular observation, , and we will write:
(3.40)
Similar to the definition of the transition probability matrix, A, we define the observation probability matrix, B, with (k, i) element b(k|i). The general mathematical specification of the HMM of equation 3.38 can be modified to reflect the discrete observations:
(3.41)
where , the set of K vectors in the VQ codebook.