Poisson Regression
Josh Iden
Introduction
In this blog, we’ll look at how Poisson regression can be used to predict ticket sales over time during an event sales cycle, from the beginning of sales to the conclusion of the event.
About Poisson Regression
Poisson regression is used to model count data, specifically the relationship between the mean of the count data and a set of predictor variables. It is based on the Poisson distribution, a discrete probability distribution which gives the probability of an event happening a certain number of times within a given interval, represented by the formula
\(f(x) = P(X=x) = \frac{(e^λλ^χ)}{x!}\)
Where
- x is a non-negative discrete variable that follows the Poisson distribution,
- λ is the average rate of value, and the variance, and > 0
- e is the Euler’s number (e = 2.718)
In Poisson Regression, the response variable is assumed to have a Poisson distribution, and the model is fitted using maximum likelihood estimation, which involves finding the parameters that maximize the likelihood of observing the given data.
Poisson Regression To Predict Ticket Sales
Concert ticket sales frequently follow a Poisson distribution; the majority of ticket sales occur during the first 5-10 days after concert sales begin, and 5-10 days before the concert takes place. This is due to a number of factors that are beyond the scope of this blog, but some of which are addressed here. Here is an example graphic visualizing this distribution,
(Graphic Courtesy of Patchboard)
Ticket sales per day are in fact counts of data, and are monitored by concert organizers to project if or when the concert will “sell out”, if ticket prices need to be adjusted in order to maximize revenue, or if more advertising or promotions are necessary. By combining the use of historical data and the variables available to concert organizers covered in previous blogs along with bootstrapping techniques, we can build a Poisson regression model that will predict the probability distribution of ticket counts and can be used to determine if any actions are necessary to maximize revenue.
Conclusion
Poisson Regression is a useful method to model count data that can be used to predict non-linear concert ticket sales over time and arm concert organizers with models that maximize sales revenue.