To be fair we could calculate the median and take all those below the median to be of one state and all those above in another state which you can see from the plot wold do quite well. The more distinct the distributions are the easier it ifor the model to pick up the transitions. The performance is in part due to our choise of means for the number of jelly beans removed from the jar. It is much noiser data but the HMM still does a great job. # converged at iteration 30 with logLik: -282.3745
What if the transition probabilities were much greater?ĭraws <- simulate(100, jbns = c(12, 4), switch.val = 7) This may have been a relatively easy case given we constructed the problem. \(P(X_t|X_)\) where \(X_t\) is the number of jelly beans. However because we know the data forms a sequence there is more infomration at our disposal since the probability of observing the next draw is conditional on the previous i.e. To be fair the states could be estimated by ignoring the time component and using the EM algorithm. It's impressive how well the model fits the data and filters out the noise to estimate the states. Return(grid.arrange(g0, g1, g2, g3, widths = 1, nrow = 4)) On average Bob takes 12 jelly beans and Alice takes 4. Simulationįirstly, we'll simulate the example. There is no condition saying the transition probabilities need to be the same, Bob could hand the dice over when he rolls a 2 for example meaning a probability of 1/36. Since we made up this example we can calculate the transition probability exactly i.e. The roll of the dice and the condition of passing the dice if the value is less than 4 is the transition probability. The observation is how many jelly beans were removed on that turn. In this example the state is the person who rolled the dice, Alice or Bob. How could we know who rolled the dice? HMM's. We don't know the colour, simply the final number of jelly beans that were removed from the jar on that turn. Instead we only know how many jelly beans were taken after the roll. Now assume Alice and Bob are in a different room and we can't see who is rolling the dice. If she rolls greater than 4 she takes a handful of jelly beans however she isn't a fan of any other colour than the black ones (a polarizing opinion) so puts the others back, therefore we would expect Bob to take more than Alice. If the total is equal to 2 he takes a handful jelly beans then hands the dice to Alice. Bob rolls the dice, if the total is greater than 4 he takes a handful of jelly beans and rolls again. There are 2 dice and a jar of jelly beans. The exampleīefore getting into the basic theory behind HMM's, here's a (silly) toy example which will help to understand the core concepts. The states are unknown or 'hidden' and HMM's attempt to estimate the states similar to an unsupervised clustering procedure. They are related to state space and Gaussian mixture models in the sense they aim to estimate the state which gave rise to the observation. HMM's are for modelling sequences of data whether they are derived from continuous or discrete probability distributions. What are they and why do they work so well? I can answer the first part, the second we just have to take for granted. Recently I developed a solution using a Hidden Markov Model and was quickly asked to explain myself.
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