Stochasticity in Time-Series, Dissertation part 1
Background
It’s been nearly a year since I defended my dissertation. In that time, I’ve found that people are intrested in knowing what it’s about until I try to explain it, then they feel that they don’t understand it and the discussions get lost. So, while I was playing FreeCell solitaire this morning, I thought of an experiment that could help me work with forecasting stochastic processes, including adding conformal predictions to the mixture. Part of it is because I want to keep my data science skills sharp. I also realized that this would work well as a better explanation of my dissertation also. So my plan is to split everything into three parts. This part is going to discuss stochasticity and how it plays a part of forecasting. The second part will be focused low-code and zero-shot forecasting models. The final part will be marrying these together, like I did in my dissertation, for 9-1-1 centres and how my dissertation can serve as a foundation for future research on both forecasting polystochastic processes and finding even better ways to address forecasting in centers and why it will be important.
Simple Stochasticity
This all started with a weird thought playing FreeCell. When I started playing, years ago, I was happy anytime that I had scores under 100 moves. Now, the more I’ve played and improved, I am annoyed whenever I have a count above 90. That certainly is a big difference. It made me wonder, had I thought of it earlier, could I have built a dataset that charted my progress and then forecast things such as which days are going to be more challenging?
Ultimately, this is a simple stochastic process. The “mother-ship” selects a deal at random each day. There are a finite number of choices, according to a quick search, there are \(1.75^{64}\) possibilities. The program, as part of the trivia, points out that some deals are unwinnable. So the rules that the “mother-ship” must follow also includes, for the daily challenges, the exclusion of unwinnable deals. The program also attaches the daily deal number to the day it is presented, so even if I don’t complete the daily deal, I’m guaranteed to see the same daily deal that my wife or other players saw the same day. This introduces the stochastic element through the determination of limiting factors. If the daily deal were truly random, then unwinnable deals would be possible and the deal offered would change when presented to the player depending on when it was accessed.
My analytical interest, had I started earlier would be to use my prior results to see how my play has improved over time and see if I can forecast how well I can do in the future. I also am curious to see if there is an underlying pattern of when really challenging deals are presented to players. I could correlate my wife’s results with mine to see if a second pattern emerges. This yields the possibility of two separate types of stochastic forecasts. It also allows me to create pattern recognition analyses that may shed some insight into the algorithm the company employs to select the deal for each day. This works well because my wife and I approach the puzzles in different ways, so if something is challenging to both of our styles, then it’s likely that is really difficult. If it is easy for one of us, the it might not be so difficult. Is it simply a random number generator with some values removed? Or are there other parameters that are added which restrict the lists enough to ensure a player doesn’t perceive repeated deals or perhaps you can ascertain the size of the potential pool of deals.
Double Stochasticity
This is most generally found in many commercial call centres. Most accurately, it can be described as a Cox Process. In this scenario, we have a number of calls arriving at a defined rate, but each call is randomly placed and any one call does not impact or effect the next call. The best example to describe a Cox Process in a 9-1-1 centre is that there are a random non-negative number of calls that arrive per time unit, typically an hour. Additionally, the decision of any one individual is neither influenced by or influences the decision of another individual to call the centre.
The next Stochastic level
Additionally, because of the nature of Primary Public Safety Answering Point (PSAP)s, there is an additional level of stochasticity regarding how the call for service arrives to the centre. Calls may arrive from 9-1-1 telephone lines, administrative, non-emergency, lines, Short Message Service (SMS) text messages, Telecommunications Device for the Deaf (TDD)/TeleTYpewriter (TTY) devices, radio communications from officers, or even public walk-ups for centres that have a window for the public, like the county that I live in. This additional stochasticity can magnify the challenges for both forecasting and queue development. So, when we’re trying to build models to describe this behaviour, we have to account for the additional factors that are present.
Application to the Dissertation
In my dissertation, I wanted to examine how to forecast inbound call volumes for PSAPs because it’s the first stage in a larger body of work that I hope to complete. I wanted to find a forecasting model that works well with the technical limitations that can exist in many 9-1-1 centers across the US and Canada. In many of these centers, there aren’t enough telecommunicators on duty, so there also isn’t a lot of technical support available directly to the center. When they need to create forecasts for staffing models, they simply tack on a percentage as an anecdotal guess and then hope that it’s close enough. I was interested in low-code and zero-shot models that could be deployed without incurring substantial technical debt so a center could have a hobbyist who knows just enough Python to make a useful forecast with easy-to-use tools.
The biggest problem that I faces in my research process was that I was overcomplicating it. Thankfully, two of my supervisors, Dr. Crossland and Dr. Schur, saw that and guided me away from turning into somethat I could not handle in a dissertation without it being super long and maybe not completed within the dissertation timeline that the university demanded. When I first proposed the project, I wanted to collect 5 different measurements across hourly, daily, weekly, monthly, and yearly frequencies. My original plan was to collect inbound 9-1-1 trunk line volumes, abandoned 9-1-1 trunk line volumes, administrative line volumes, abandoned administrative line volumes, and outbound call volumes. After I started thinking about that, I also wanted to collect the percentages of calls answered for 9-1-1 and admin trunk lines in 15 and 20 seconds. I had hoped for a grand correlative study where the different volumes could have been seen as exongenous variables to other variables. That would have been unwieldy. Perhaps doing one center to that depth at daily and hourly levels could be a good follow-up study, but this was not the place for it.
Had I done this study in that way, I might have also reduced the overall stochasticity of the study by discounting that additional level through the separation. the separation could have implied that these calls could be separated prior to arrival at the center. That could be a second interesting study, comparing centers that have dedicated queues solely for administrative calls against those who have singular queues.
Where Does This Go Next?
My next post discussing my dissertation will pick up with the forecasting models, specifically zero-shot and low-code models, and how they may be a better fit for for 9-1-1 centers through the country. After that, I will discuss queueing models and how 9-1-1 centers present challenges to traditional queueing models. Finally, I will discuss application of different popular queueing models, in my opinion, falls short of the promise to model 9-1-1 call centers and what alternatives could exist.