I have a time series data set (the Old Faithful geyser data available here: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1-2012/geyser.txt). Plotting the eruption duration on the x axis and the waiting time between eruptions on the y axis, the data looks like this:
I'm also told to plot the x values against delayed x values (for different delay lengths n):
and to plot the x values against delayed y values (for different delay lengths n):
Based on these graphs, I need to recommend one of the following models:
(i) multivariate normal (ii) Gaussian mixture model (iii) Markov chain (iv) hidden Markov model (HMM) (v) observed stochastic linear dynamical system (vi) linear Gaussian state-space model
So, I want to argue for an HMM. I think the latent variable should be able to occupy one of four states corresponding to the different cluster means we see in the delay graphs - the reason being that this would allow us to explain the long term trend in the time series. I think if each latent state corresponded to a cluster mean then the emission probability would just be a Gaussian centred on that cluster mean as this could generate the observed data.
I have two questions:
(1) What do you think of my above explanation? Have I chosen the right model? Is my reasoning correct?
(2) How can I argue against the other models? Basically I need one reason to say why each of the other models are bad. My thoughts so far are that:
(i) a multivariate normal could explain the clusters in original dataset but wouldn't be able to explain the clusters in the delay data
(ii) same argument above to get rid of Gaussian mixture model
Is that right? How to argue against the other three?