Some of my recent work has used “interior” data to compute metrics which are then analysed by regular time-series methods. I've recently been thinking about paths to generalize this methodology for the analysis of high-frequency data.
The above diagram, I hope, illustrates what I mean by interior data. With regular time-series analysis we typically look at price changes over a homogeneous sequence of intervals and construct linear functions of lagged price changes. This is represented by the lower half of the data.
With higher frequency data, it is easy (i.e. cheap) to obtain data that summarizes trading activity down to resolutions of around one minute via the standard structure of “price bars.” The problem with this data is that, as the data frequency is increased, the data sequence becomes sparse, i.e. there are many intervals that do not contain trading activity, and quantized, i.e. we start to see the fundamental pricing interval of $0.01 strongly. Both of these factors disrupt the utility of classic time-series analysis.
These factors push one to analyze larger time intervals in order that a decent quantity of acceptably continuous data exists. With that constraint, it is easy to then only look in the direction of the data sampled at these larger time intervals. However, there is a set of data that exists within the interior of the larger time intervals, and my thoughts are that we should be able to assemble some kind of non-linear analytical machine to process the set of interior state vectors and forecast the evolution of the exterior process from that.