Suppose I have a five state Markov chain. The states are observable (in fact I defined them, they are outcomes of an classification algorithm). So I have a long time series, see the first picture.
The states correspond to a real stochastic situation, it represents the environment for a car driver, e.g., the highway, motorway, city etc. The data comes from several independent drivers and I think this should be “purely” stochastic, so I can’t use any regression method, correct? So I chose the (time discrete) Markov chains. My goal is to create a model-switching Markov chain. In each state there is another “model” to predict the speed of the car. The prediction is done via sampling from the speed distribution which has different parameters for each Markov-chain state.
I’ve read about hidden Markov chains with autoregressive models, but here I don’t have hidden states.
Now I try to estimate the transfer-probability matrices via a maximum-likelihood estimation. The TPM for example is
0.85393 0.033708 0.078652 0.033708 0 0.11111 0.66667 0.11111 0.055556 0.055556 0.2 0.1 0.66667 0.033333 0 0.11905 0 0.02381 0.83333 0.02381 0 0 0 0.5 0.5
I calculate the realization, according to Handbook of MCM, chapter 3 and two possible timeseries are:
Fine, does not really look like the original time series. Sure, it’s stochastic. So, how do I prove, that these series represent the real time series? I think that looking at the time series is not sufficient. If I had many states, I would go for a histogram. I tried the autocorrelation function to see, if there are dependencies, but I'm not sure, if there are ways to validate my Markov Chain with the auto- or crosscorrelation function.
Or is the only possibility to use the states and predict the vehicle speed and validate it via MSE or similar? But again there is the problem, that it is not a “real” time series, since it is stochastic.