{'fraction_non_alphanumeric': 0.01756198347107438, 'fraction_numerical': 0.002066115702479339, 'mean_word_length': 5.05625, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Additive and Nonadditive Fuzzy Hidden Markov Models

We present a novel approach for the development of fuzzy hidden Markov models (FHMMs) by exploiting both additive and nonadditive properties of input fuzzy sets in the fuzzy rules of generalized fuzzy model (GFM). This development utilizes 1) Gaussian mixture model (GMM) to manipulate the mixture parameters for the input fuzzy sets and 2) GFM rules for the inclusion of states in the consequent part to be able to use HMM. Taking the components of Gaussian mixture density conditioned on the past system states and making use of equivalence of GMM with GFM, parameters of the additive and nonadditive FHMMs are estimated using the forward-backward procedure of the Baum-Welch algorithm. The additive and nonadditive FHMMs are validated on three benchmark applications involving time-series prediction, and the results are compared and found to be better than or equal to those of the existing recent fuzzy models.