A Bayesian Lifetime Model for the “Hot 100” Billboard Songs

Abstract

People have long been enamored by ranked lists of celebrities (e.g., “best-dressed” lists), places (e.g., best cities to live in), things (e.g., most popular songs, books, and movies), and countless other entities. Likewise, people are equally interested in watching these rankings evolve over time and speculating about possible future changes (e.g., who will be ##1 next week?) We focus on a popular, but fairly typical ranked list (the Billboard “Hot 100” songs) to explain and model the simultaneous movement of multiple items (songs) up and down the chart over time. Although our interest in Billboard data partly reflects the glamour of the music industry, these charts provide a very rich and general data structure. Surprisingly little research has been done on time-series models for ranked objects. We further enrich the dataset by adding covariates (e.g., artist history) to capture additional sources of variation across songs and over time. We posit a model for the time series of charts based on a latent lifetime (worth) process. Specifically, the latent popularity of each song is assumed to follow a generalized gamma (GG) lifetime curve with double exponentially (DE) distributed errors. The immense flexibility in the GG family allows the mean of a song's latent worth process to follow an exponential, Weibull, lognormal, or gamma curve (among others), reflecting the many possible paths that it might take through the chart. The DE error structure is used for convenience, as it leads to a well-established “exploding” multinomial-logit likelihood. This framework is embedded in a Bayesian structure in which parameters of the GG curve are song specific, with means related to observed covariates, and assumed to come from a multivariate lognormal prior. Inferences from the model are obtained from posterior samples using Markov chain Monte Carlo techniques.