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Recognizing Sequences of Sequences

Figure 3

Recognition of a sequence of sequences.

(A): Dynamics of generated causal and hidden states at the phonemic and syllabic level, using Equation 2. At the syllabic level, there are three different syllables (1: blue, 2: green, 3: red), following the syllable sequence 1→2→3. The slowly changing state Syllable 1 causes the faster-moving phoneme sequence a→e→i→o (blue→green→red→cyan), syllable 2: o→i→e→a (cyan→red→green→blue), and syllable 3: a→i→e→o (blue→red→green→cyan). See Fig 2 for a schematic description of these sequences. At the beginning and end of the time-series (top-left plot), we introduced silence by applying a windowing function to zero time points 0 to 50 and 750 to 800. The red arrow indicates the end of the initial silent period. The phonemic states cause sound waves, resolved at 22050 Hz (see Fig. 1). These sound waves are the input to the recognition system. (B): The recognition dynamics after inverting the sound wave. At the phonemic level, the states follow the true states closely. At the syllabic level, the recognized causal state dynamics are rougher than the true states but track the true syllable sequence veridically. The high-amplitude transients of at the beginning and end of the time-series are due to the silent periods, where the syllabic recognition states experience high uncertainty (plotted in grey: confidence intervals of 95% around the mean). Note that the hidden states, at both levels, experience high uncertainty whenever a phoneme or syllable is inactive. The red arrow indicates an initial but rapidly corrected mis-recognition of the causing syllable.

doi: https://doi.org/10.1371/journal.pcbi.1000464.g003