Hidden Markov Chains

Skriveni Markovljevi lanci su poznati alat za matematičko modeliranje problema koji mogu biti karakterizirani pomoću jednog nama skrivenog proces \(x_{n},n\in \mathbb{N}_{0}\) koji emitira drugi nama vidljiv opservacijski proces \(y_{n},n\in \mathbb{N}_{0}\). Ovakvi modeli su pronašli primjenu u raznovrsnim područjima znanosti poput bioinformatike, komunikacijske tehnologije te financijske matematike, a neke primjere smo naveli u samom radu. Skriveni Markovljev model je sasvim određen poznavanjem njegovih matrica prijelaznih i emitiranih vjerojatnosti Π i Φ te početne distribucije skrivenog lanca λ. Sama funkcionalnosti skrivenih Markovljevih modela se zasniva na tri osnovna problema, a pod uvjetom poznavanja parametara modela možemo riješiti problem izglađivanja pomoću Forward-Backward algoritma i problem dekodiranja pomoću Viterbi algoritma, koja su oba navedena u radu i ilustrirana na primjerima. U slučaju potrebe procjene samih parametara modela, što je poznato pod nazivom problem treniranja, koristimo se Baum-Welch algoritmom, koji je također pojašnjen u radu.Hidden Markov chains are a powerful tool for mathematical modeling of problems that can be characterized by one hidden process \(x_{n},n\in \mathbb{N}_{0}\) which emits another visible observational process \( y_{n},n\in \mathbb{N}_{0}\) . Such models have found its application in various scientific fields such as bioinformatics, communication technology and financial mathematics, and we have given some examples in the paper itself. The hidden Markov model is completely determined by knowing its matrices of transition and emission probabilities Π and Φ and the initial distribution of the hidden chain λ. The functionality of hidden Markov models is based on three basic problems, and provided we know the parameters of the model we can solve the smoothing problem using the Forward-Backward algorithm and the decoding problem by using the Viterbi algorithm, both of which are listed in the paper and illustrated on examples. In case we need to estimate the parameters of the model themselves, which is known as the training problem, we use the Baum-Welch algorithm, which is also explained in the paper