Time Series Analysis and Classifying Sequences or Identifying Sequence Motifs Imagine if I have a game that dynamically increases or decreases the difficulty between levels each day and players can choose when they want to play/quit. Additionally, the players always get to see the changes in the difficulty level for the last 365 days so they might be able to guess that there is a local minima/maxima/trend/seasonality.

In this example, there are 4 independent players (A, B, C, and D) that play the game on different days. Based on this visualization alone, we see that Player A (red circles) likes to play easy levels and stops playing when the difficulty level begins to increase. Player B (orange squares) only likes to play when the difficulty level is approximately increasing but doesn't like to play when it gets too hard. Player C (yellow triangles) is a master and only likes to play the hardest levels. Player D (green diamonds) only plays when the levels are decreasing in difficulty.
I have several thousand other time series data for these players (and other players) and the first thing I want to do is identify/classify the multiple different playing preferences (i.e., what are the different playing motifs?). 
Then, I want to confirm whether the above trends/preferences also hold across multiple time series for each player. For instance, I want to know whether Player A exclusively plays when the game is easiest or if Player B only plays when the difficulty is increasing. A player can even spend a part of their time playing one way and the rest of their time playing a different way (i.e., 25% of the time spent playing the easiest difficulty levels and 75% of the time playing the hardest levels) 
 A: I've been giving your query some thought. In my opinion, the biggest challenge you face is in appropriately accounting for a very messy, complex dataset. In other words, as I understand it you've got an underlying black box algorithm dynamically, sequentially and uncontrollably spewing out level changes and rewards to players who can drop in and out at will. Given this, what would be ideal for your understanding would be to set up A/B type experiments to test and tease out the drivers of participation. Regrettably, it doesn't sound like that would be possible. On the other hand, you do have 365 days of historic information. 
Would it be possible to decompose or classify this evolving, 365 day background history into a comprehensive set of binary (y/n) background metrics? Since I know so little about the data, I'm struggling with the specifics of what these background metrics might be. These metrics could then form a set of rules against which participant responses can be evaluated. E.g., for some unit of time -- a minute, an hour, a day, a week, whatever -- across and against this set of metrics, this is that period's profile. It sounds like lags in the metrics could also be useful. Next, as each metric is manipulated, e.g., switched on or off, evaluate participant response. For me, the biggest problem with this approach is that there are so many things going on at the same time, drilling down to very tiny, minute changes to gain insight could be challenging or even impossible. 
Key target variables could be player participation (opting in or dropping out), duration (survival) of play, intensity of play, extent of rewards, and so on. One way to describe this approach is as a type of regression discontinuity design where everything is known and controlled up to the point at which a change is introduced. Needless to say, there would be a lot of discontinuities to evaluate. Alternatively, this approach could also amount to a kind of agent based model or historic simulation in the context of a massive contingency table matrix of data. Based on their responses to the changes, player profiles could be formed and their drivers (the metrics) evaluated.
It's always best to start with the simplest analyses, considering cross-sectional swaths of data and ignoring the sequential component, e.g., participation (opting in or dropping out) can be evaluated at this preliminary level of analysis. This should help clarify the information. Motifs or "states of play" capturing participant clusters or segments at an aggregate or global level can be developed. Then, additional levels of complication can be successively introduced as needed.
A few functional forms suggest themselves: logistic regression for the regression discontinuities, massive contingency tables, agent-based models, and so on. Once the sequential, time series component is reintroduced, there are lots of additional models that might work. Aggarwal and Reddy's book Data Clustering has an excellent overview of time series clustering methodologies. Given that "states of play" have been created, hidden markov models based on the transitions in those states might be helpful. Steve Scott's papers on his Google website (https://research.google.com/pubs/author57989.html) would be helpful here...or Oded Netzer's work on hierarchical, hidden markov models (here ... https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/2618/HMM%20of%20Customer%20Relationship%20Dynamics.pdf) Another approach that I particularly like that is not driven by the moments of any distribution is the information theoretic, complexity-based, permutation distribution method of Andreas Brandmaier (here ... https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/2618/HMM%20of%20Customer%20Relationship%20Dynamics.pdf) Finally, recurrent, deep learning neural networks could also be brought to bear as outlined in this excellent review by Lin and Tegmark, Critical Behavior from Deep Dynamics (here ... https://ai2-s2-pdfs.s3.amazonaws.com/5ba0/3a03d844f10d7b4861d3b116818afe2b75f2.pdf).
Anyway, those are a few thoughts. Hope they help.
* A few possible global metrics, later edit *
A few global metrics that could be used to create the motifs, segments or states of play could include:


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*Various measures based on the classic analysis of time series such as the presence (absence) of a unit root, drift and/or stationarity of the series, deterministic (linear) trend, polynomials in trend, and so on. By creating aggregate, global metrics for each user, patterns in those profiles could be created. The shapes in the graph you provided suggest that some users have an upward trend in utilization with a sharp changepoint leading to a downward trend. Identifying subjects with or without such sharp changepoints, a count of the number of changepoints (0,1,2,3...), and so on, would help in classification. Trends can be analyzed using a Spearman correlation or the Pettit-Mann-Whitney nonparametric test, etc.

*Presumably, some sort of learning is taking place within subjects (participants). Metrics that tease this out would be useful. These metrics could be based on cumulative measures of "success," however defined. Learning could be defined based on an increasing slope of that cumulant. Next, given that a stimulus or change in the "black box" occurs, how long is the time to participant response to this change? If learning is taking place, then one could hypothesize that the response time should be getting shorter, correct?

*Simple percentage changes in the rate of success (however defined) across various time frames might be useful, e.g., 1 day, 1 week, 1 month % change, and so on...
