# Estimate Hidden Markov Processes from Data

Estimate a two-state hidden Markov process with three possible emission values from the given data.

 In[1]:= Xdata = TemporalData[Automatic, {CompressedData[" 1:eJxNk9t1wzAMQwNwhg7QFTpKR8gC3f+vIi7k1ic5lig+AAj+fP98v/16vb7O /+P8zSPr/OzJY81oo7vR9PCsNrRnRE3O1s92mM05z5C1tbvbjF15GHX2e5KS tHR67SlNZ9Mn3RZM5tyahbjHG9i4xAymczhZpMckpJYrjdySTRAT1JH/ktJO YZSCkCUO5YbCL1AFViEMnAbOISJQgQWRjYx/+ClEnUcXRoZ0ssbNhHBguEKF k5CviIVw8BggyKA1bEoZ8W7fueow+d4FEAJYIo86UOKGqSLgq6y6ja5psrhO CVSnN+bDAu3sDvUVO15FJnijQm4Px7pG5B56xdKDcLj6egkV7atKd3M7U1Qz 9u15uNnFpfo708u9KgY0KdjLc0XBgyFZY/aTofTKz91W3+cjmqLmVf16l3Vf ptRFV2rfu43UJVMFT+wX64AKKw== "], {{0, 20, 1}}, 50, {"Discrete", 50}, { "Discrete", 1}, 1, {ValueDimensions -> 1, MetaInformation -> {}}}, False, 10.];
 In[2]:= Xhmm = EstimatedProcess[data, HiddenMarkovProcess[2, 3]]
 Out[2]=

Compute the loglikelihood for the data under the estimated process.

 In[3]:= XLogLikelihood[hmm, data]
 Out[3]=

Estimate a two-state process with continuous emissions.

 In[4]:= Xdata = TemporalData[Automatic, {CompressedData[" 1:eJwUm3c8le8fxo+9ixJSlBIllRIi45DZlOJLaRgNZFYiq8wklbJCZRZNaSjS dYokklKRhqIUss7e5/ye3x/+Oq9znue57899Xe/3gZ5/uMc+SRKJFEj87CN+ Hm9PzZ23WILc7KGu1dfLxtC5tOXn542hqGzJlp6BKeSxfnCK59Awa/5EVb6E EnnC8fq7jEE6Cr1t/x6QEkCUkR0wf4pEEcwcOX17Hx/cxLUT7QFTeLH778e9 a1kIfzgiqN3IxTaXDLPTMjRYXU5+dzOLhpfPsqZO/6NCXfpQ+doeOlImHipE iSQp/atGdnxcJsC1M/a8K5FMyKYN7N46zoOO8VMu6zcV1w/Zfc1L58B9L21g TzwbVynfyz9emcTNhrBX7TwOwkhRXfN+csCdSjufdIyFPbMUCy/TebjjHp8y rMRB0zOvT3qr+PA88P3V6jU0pPfa/PdyK4miX+Cu5FrDx/RwtQWeIzRIbnoy eniQA2e7s1ObzophGdQSEGUjSTbV8GVV+Ixj20/DvPZnXNifKtKe60XH8o3p /Kpfw/AU7Tjx+QmJopc2fzyinAntKrsdzsNCaGxx0Ix6xkN7UlTY619MVGYv qzl6jI2nPXKBcelc3FV59uzkcjGu++pqllSy0FtxZXcun4epqVur28p4sE0+ ZDo9gkQZMhqyW2cqxMqgWcF1P6noZa9xm7uShRg/o0G1UT52/0wt2trBx+fz ybFS/3h2NZn3XhX6j8NhS5Cda7YAui5pJ77c5iFdLjO5ro6J3Lcjcin1fFhe NN0TLivE2o3JQUFFE+BSbJcWneFiMqfOUO81G1W7xN1LxjigRY52d83loUV1 9NuJaCb2bzM4Xm0kxj+NQYMhBy5+cxNmVtyk4eMD1pI9XCpCVK/djUoXQHOn 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The overlaid histograms for each path suggest Gaussian emissions.

 In[5]:= XHistogram[data, Automatic, "PDF", PlotTheme -> "Monochrome"]
 Out[5]=

Compare the results from the default BaumWelch method and Viterbi training.

 In[6]:= XeprocBW = EstimatedProcess[data, HiddenMarkovProcess[2, "Gaussian"]]
 Out[6]=
 In[7]:= XeprocVT = EstimatedProcess[data, HiddenMarkovProcess[2, "Gaussian"], ProcessEstimator -> "ViterbiTraining"]
 Out[7]=

The data has higher loglikelihood with the BaumWelch estimated process.

 In[8]:= XLogLikelihood[#, data] & /@ {eprocBW, eprocVT}
 Out[8]=

## Mathematica

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