in Hybrid Systems: This variant of Forward algorithm is motivated by the structure of manufacturing environments that integrate process and operations control. We derive a new property of the optimal state trajectory structure which holds under a modified condition on the cost function. This allows us to develop a low-complexity, scalable algorithm for explicitly determining the optimal controls, which can be more efficient than Forward Algorithm. Continuous Forward Algorithm: A continuous forward algorithm (CFA) can be used for nonlinear modelling and identification using radial basis function (RBF) neural networks. The proposed algorithm performs the two tasks of network construction and parameter optimization within an integrated analytic framework, and offers two important advantages. First, the model performance can be significantly improved through continuous parameter optimization. Secondly, the neural representation can be built without generating and storing all candidate regressors, leading to significantly reduced memory usage and computational complexity. Complexity Complexity of Forward Algorithm is , where is the number of hidden or latent variables, like weather in the example above, and is the length of the sequence of the observed variable. This is clear reduction from the adhoc method of exploring all the possible states with a complexity of . See also Viterbi algorithm Forward-backward algorithm Baum–Welch algorithm References Further reading Russell and Norvig's Artificial Intelligence, a Modern Approach, starting on page 570 of the 2010 edition, provides a succinct exposition of this and related topics Smyth, Padhraic, David Heckerman, and Michael I. Jordan. "Probabilistic independence networks for hidden Markov probability models." Neural computation 9.2 (1997): 227-269. Read, Jonathon. "Hidden Markov Models and Dynamic Programming." University of Oslo (2011). Kohlschein, Christian, An introduction to Hidden Markov Models Manganiello, Fabio, Mirco Marchetti, and Michele Colajanni. Multistep attack detection and alert correlation in intrusion detection systems. Information Security and Assurance. Springer Berlin Heidelberg, 2011. 101-110. Zhang, Ping, and Christos G. Cassandras. "An improved forward algorithm for optimal control of a class of hybrid systems." Automatic Control, IEEE Transactions on 47.10 (2002): 1735-1739. Stratonovich, R. L. "Conditional markov processes". Theory of Probability & Its Applications 5, no. 2 (1960): 156178. Softwares Hidden Markov Model R-Package contains functionality for computing and retrieving forward procedure momentuHMM R-Package provides tools for using and inferring HMMs. GHMM Library for Python The hmm package Haskell library for HMMS, implements Forward algorithm. Library for Java contains Machine Learning and Artificial Intelligence algorithm implementations. Markov models