Forecast (impute) missing discrete values in multiple time series I'm looking to forecast (impute) missing discrete values in multiple time series in order to reach a target volume in a consolidated time serie.
The context:
I have salesmen that are selling contracts in some markets over time.
For some markets, I have gaps with missing data.  
ex below: the volume of contracts over time (days) for a given market

I'm pretty confident to fill gaps on this market at the consolidated market level and obtain something like below. (Here I've just print a very correlated market, and there are plenty other)

The problem:
I'm now looking for a method able to estimate the discrete volume of contracts over days made by and for each possible salesmen on gaps period in order to reach the volume of the market I will estimate.
Another aspect to consider, is the number of salesmen on this market: roughly 5000.
As a result, many of them are "opportunist" salers with very sparse discrete activity.
Here is a tiny sample of different salers code activity for the "most productive" (yeah...).

In my context, Is there a method (several?) in order to reach my goal? 
e.g: Impute the most probable salers discrete activity according to:
- an estimated market volume over time which integrate seasonality and act as a target (the sum of all salers activity has to be equal to this target over time)
- the known activity of the salers around the gap ("past activity" before, and "future activity" after the gap)
I'm thinking and have read some things about Hidden Markov Chains, Kalman filters, ARIMA models, LSTM, Poisson regression, but I'm not an expert, and I don't really know how many methods I have to use and above all, how I have to combine them.
How would you proceed?
Thanks, Mathieu
 A: If I had your problem and my favorite piece of software , I would develop two daily models for the two regimes of observed data. The first model would predict the missing observations using approaches described here .. Forecast customer's spending and elsewhere ( see my previous posts on daily data ). Then I would take the data from the second regime and reverse the order so that the missing past would be predicted from the future ( so to speak ! ) 
  How to treat days when shops are closed in sales time series?  and Regression on default data and backward extrapolation . I would then average these two sets of forecasts to create pseudo data for the missing/omitted period.
As a coup de gras , I would then model the entire data set and use the estimated values for the missing period as my final estimates for the omitted data.
I owe this insight to a T.W Anderson https://www.google.com/search?source=hp&ei=q43sWofxF-2b5wKbqYKgDQ&q=t.+w.+anderson&oq=t.+w.+anderson&gs_l=psy-ab.3..0j0i22i30k1l9.1358.7732.0.8947.17.16.0.0.0.0.85.1110.15.16.0..2..0...1.1.64.psy-ab..1.16.1198.6..46j35i39k1j0i131k1j0i20i264k1j0i46k1.88.N5aeGxP-0n8 a guiding light in my pursuit of time series solutions/procedures/theory.
In direct response to your question about 5,000 individual time series , I can only suggest forecasting some aggregate and then pro-rationing the estimates/fitted values in order to estimate sub-count data used to form the aggregate.
