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I am working with dataset which contains number of movie tickets sold per day. This is basically a count of total number of tickets sold, for a particular movie, for each day after its release date. I want to find out what should be the ideal duration (number of days after release) for a theater to keep the show running. In other words, I want to find out the day after which there is low probability of increase in ticket sales. So, that theaters can stop the show time for that movie and release a new movie.

I would like to know an ALGORITHM which is suited to study/analyze the graph or dataset, and provide as a result, after how many days (starting from release date of the movie), it is ideal to remove it.

Few examples:

Movie 157:

Day Tickets Sold
1   800
2   1200
3   1330
4   1300
5   1400
6   700
7   300
8   150
9   100
10  50

As we can determine after 6th and 7th day, movie sales are not increasing, so it is good to remove the movie after 7th day

However, the challenge is with non-uniform ticketing trends, like example below:

Movie 158:

Day Tickets Sold
1   800
2   900
3   600
4   900
5   1200
6   700
7   1500
8   800
9   700
10  550

Movie 159:

Day Tickets Sold
1   1300
2   900
3   600
4   400
5   300
6   500
7   800
8   1200
9   1100
10  1250

Now, I want to use an algorithm which can be used for both uniform as well as non-uniform trends.

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    $\begingroup$ I would question some of your premises if I were a theater operator. In particular, I wouldn't be interested in finding when there is a "low probability of increase in ticket sales." I would instead look at my bottom line and ask for a decision procedure to identify the point at which switching to a new film is likely to bring the greatest expected gains. When I abandon a popular film in favor of an unpopular film, simply because the audience is declining, I can lose money. Thus, analyzing these time series in isolation does not look like it will lead to good solutions. $\endgroup$ – whuber Feb 3 '16 at 19:27
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What @whuber says. Whether or not a movie should be removed must depend on the alternatives. Unfortunately, This also means that we need to forecast ticket sales for the alternatives... which will typically be new films with little ticket sales history, which poses special problems.

That said, forecasting lifecycles is an interesting problem in itself. And forecasting the tail end of lifecycles (for declining movie ticket sales) is far easier than forecasting the rest of the lifecycle based only on initial data, as for new movies.

I'd recommend you search for "lifecycle forecasting", with Bass models and similar, where you would not be so much interested in the overall saturation, but in the new sales component, which is the derivative or rate of change of the saturation. There are models that include repeat purchases, e.g., for people who watch the same movie multiple times. Consumer electronics or fashion products exhibit much the same lifecycles and similar problems. Fashion in particular may exhibit markdowns (price reductions, to clear out inventory at the end of the lifecycle) - if your movie theater reduces prices on movies after a few weeks, this may be relevant.

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