# Recover full tick data from missing tick data

Due to some economics/regime problem, I can only have access to non full-tick data from an exchange.

To make the problem precise, a full tick data $X$ is a series of $(t_i,p_i,v_i)$ for $0 \leq i \leq N$ where $t_i$ is the timestamp, $p_i$ is the price, $v_i$ is the deal volume.

The data that I could only see is a lower resolution $\hat{X}$ of $X$, in the sense that, I can only observe the market in a sequence $j_1 < j_2 < \ldots < j_m$ and get the data like: (the sequence is not necessary deterministic or in fixed interval)

$(\hat{t_{j_k}},\hat{p_{j_k}},\hat{v_{j_k}})$ where $\hat{t_{j_k}} = t_{j_k}, \hat{p_{j_k}} = p_{j_k}$, but $$\hat{v_{j_k}} = \sum_{i=j_{k-1}+1}^{j_k} v_i$$

For instance, if the true data $X$ is:

$(0,100,1) \\ (1,102,2) \\ (2,101,1)$

I may only see the lower resolution one $\hat{X}$ as

$(0,100,1) \\ (2,101,3)$

or

$(1,102,3) \\ (2,101,1)$

The question is..

1. Suppose I only have one source of $\hat{X}$, what is the best way to recover most missing tick? I know this may be a bad question, as information has already been lost. I think I need to add some model assumption for this problem from Bayesian point of view, any reference for this?

2. Suppose I have two different source of $\hat{X}$, and because of random nature of the missing ticks, two source would be different. Any method to recover it?

P.S. I think I can think the tick data as a one-dimensional image, and lower resolution data is a pixelized version of real image data, and apply some image processing technique on it, any idea?

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If you're missing ticks, then no technique will get those ticks back.

If you have two sources, then designate one source as the primary feed and then fill-in gaps from the secondary feed. Of course, you'll have to mind the timestamps when determining whether the secondary feed can be used properly.

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