Forecasting sequence data with intermittent peaks

Question

I'm trying to forecast a sequence that looks like below: I know ARIMA, INAR, GLM, etc. but none of these works for this data. Algorithms I found for intermittent time series (ADIDA, Croston, etc.) only output a constant expected value as forecast.

After looking at Cryo's comment, I plotted the 2 graphs he mentioned.

1. heights of the peaks (I took the values >=600 as peaks):

2. time intervals between the peaks:

These graphs do seem a lot more "normal" for any model to handle. But I see no obvious trend/seasonality in either of the graphs, and their ACF and PACFs are quite small as well.

So far, I tried the GLM models (negative binomial, binomial, poisson, etc.,) which all gave me predictions of too small variance, and ARIMA (even though the data is integer) which gives me something like below (this one is the "time intervals between peaks" series):

What are some better models/ideas to try to forecast these 2 series? And what level of prediction accuracy should I expect in the end? Is it always possible to make relatively accurate forecasts for a time series? I'm a newbie in series forecasting, thanks a lot for your help...!

Answer 1

Transformed series, delay until next peak ($\tau_i$) and next peak height ($h_i$) also look better to me. If the correlation between subsequent points is low you may struggle in predicting. Are there any external variables you can bring in?

Also, can you pick up any patterns if you plot your plots in a 2d plot, with $\tau_i$ on horizontal axis as $h_i$ on vertical?

I would try to predict $\tau_i,\,h_i$ in a single model. Your height is integer valued, as I understood, and your delay would be strictly non-negative, might as well assume it is integer-valued too.

$$ \begin{align} \tau_i&\sim NB\left(r^{(\tau)}_{i},\,p^{(\tau)}_{i}\right) \\ h_i&\sim NB\left(r^{(h)}_{i},\,p^{(h)}_{i}\right) \end{align} $$

Where NB is negative binomial.This may require some more refinement, e.g. if your variance is smaller than your mean, you may need a different distribution, or may need to write it in terms thinning operators (more like INAR).

Then you will need to map your prior data points into $r$ and $p$. Might be something like (and similar for $h$):

$$ \begin{align} r^{(\tau)}_i&=\exp\left(\sum_{j=i-1-k}^{i-1}\alpha^{(r,\tau)}_j\cdot\tau_j+\sum_{j=i-1-k}^{i-1}\beta^{(r,\tau)}_j\cdot h_j\right) \\ p^{(\tau)}_i&=expit\left(\sum_{j=i-1-k}^{i-1}\alpha^{(p,\tau)}_j\cdot\tau_j+\sum_{j=i-1-k}^{i-1}\beta^{(p,\tau)}_j\cdot h_j\right) \end{align} $$

In the end you write down the likelihood and optimize to find $\alpha$-s and $\beta$-s (for some chosen look-back $k$). Clearly, a numerical solver will be necessary. I am not sure whether any package already does this, perhaps some packages aimed at modelling integer-valued sequences do. In my recent experience optax works well for these optimization tasks, but it is very much a general purpose optimizer, so some handy-work will be required