Extend to allow for different time models

[sempwn]

Problem statement

Currently we have only two possible models of time variation: either as a random walk or as a 2nd order random walk. For data sparse situations we may not have data to adequately infer such a complex model and instead could opt for a piecewise constant model e.g. each year has a fixed constant representing the variability in time.

Proposed solution

Add in an option to model time using a time index where for each time unit a random variable is drawn from some distribution representing time e.g. $u_t \sim N(0,\sigma_u^2)$

[sempwn]

Another proposed solution would be to include a new time index within the model data to account for how variable the time component of the model should be. For example if data are recorded weekly, but time should only vary month to month then the data table could look like the following:

region	times	orders	reported_distributed	reported_used	time_index
1	1	X	Y	Z	1
1	2	X	Y	Z	1
1	3	X	Y	Z	1
1	4	X	Y	Z	1
1	5	X	Y	Z	2
1	6	X	Y	Z	2
1	7	X	Y	Z	2
1	8	X	Y	Z	2
1	9	X	Y	Z	3
...	...	...	...	...	...