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Suppose a time series clearly has two or more “components”, e.g. a “zero” component and another one that looks like a continuous series. Example:

enter image description here

Suppose we can’t find covariates that can explain why it drops to zero, stays there, and then bounces back, so we only have this series to work with. I speculate that traditional linear models (ARIMA, Exponential Smoothing) don’t apply here. What method (that preferably has an R or Python package) would you recommend for the purpose of forecasting?

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  • $\begingroup$ Not sure what is exactly meant by "forecasting". There are for sure lots of traditional methods for changepoint detection and time-series segmentation such as the strucchange, changepoint , and ecp packages in R and the ruptures package in Python. My own R and Python package Rbeast provides a Bayesian way, though no intended for forecasting, it can be twisted a little bit for prediction. $\endgroup$
    – zhaokg
    Commented Sep 7 at 20:22
  • $\begingroup$ I might be reading too much into your diagram, but I would describe the illustration as measurements from a sensor with two modes: functioning and broken. The data is generated from two distinct processes. I would look into a latent mixture model, where the mode switches with some probability given the mode in the prior time period. $\endgroup$
    – krkeane
    Commented Sep 29 at 14:31

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A Markov switching model could probably capture this well. In a Markov switching model, parameters can shift between two or more regimes. The switching can occur in all parameters, or just in a subset. According to your drawing, it looks like the mean and variance both increase and decrease together. For example,

$$ \begin{align*} y_t &= \mu_{St} + \varepsilon_{t} \\ \mu_{St} &= \left \{ \begin{matrix} \mu_1 & \text{if} & S_t=1 \\ \mu_2 & \text{if} & S_t=2\end{matrix} \right . \\ \varepsilon_t &\sim \left \{ \begin{matrix} N(0,\sigma_1^2)& \text{if} & S_t=1 \\ N(0,\sigma_2^2) & \text{if} & S_t=2\end{matrix} \right . \\ p(S_t=1|S_{t-1}=1) &= p_{11} \\ p(S_t=2|S_{t-1}=2) &= p_{22} \\ \end{align*} $$

The transition probabilities, from one regime to the other (e.g., $p_{11}$), can be assumed constant (e.g. if the process is in regime 1, there is always a 5% probability of switching to regime 2), or they can be modeled as depending on covariates.

You could also model the mean and variance as following two separate regimes.

Some practical issues - first, MS models can struggle if your data has outliers. For example, fitting a two-regime model to Real GDP growth used to find recession and expansion periods. Fitting one now, with the presence of the massive COVID outliers, it just finds the outlier periods. So you would need to directly model the outliers or censor your data. Second, MS models tend not to forecast very well. Take again Real GDP growth as an example - a simple AR(1) or AR(2) model will usually be very competitive with or even forecast better than forecasts from a MS model.

This is a great reference text on MS models: https://direct.mit.edu/books/monograph/3265/State-Space-Models-with-Regime-SwitchingClassical

And it looks like there are a few R/Python packages that can fit these types of models. One example is here: https://cran.r-project.org/web/packages/MSwM/vignettes/examples.pdf

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I think you should have a look at the Croston method to analyse time series data. Also, a similar question was asked here.

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