Functional connectivity (FC) is one of the most common inputs to fMRI-based predictive models, due to a combination of its simplicity and robustness. However, there may be a lack of theoretical models for the generation of FC. In this work, we present a straightforward decomposition of FC into a set of basis states of sine waves with an additional jitter component. We show that the decomposition matches the predictive ability of FC after including 5–10 bases. We also find that both the decomposition and its residual have approximately equal predictive value, and when combined into an ensemble, exceed the AUC of FC-based prediction by up to 5%. Additionally, we find the residual can be used for subject fingerprinting, with 97.3% same-subject, different-scan identifiability, compared to 62.5% for FC. Unlike PCA or Factor Analysis methods, our method does not require knowledge of a population to perform its decomposition; a single subject is enough. Our decomposition of FC into two equally-predictive components may lead to a novel appreciation of group differences in patient populations. Additionally, we generate synthetic patient FC based on user-specified characteristics such as age, sex, and disease diagnosis. By creating synthetic datasets or augmentations we may reduce the high financial burden associated with fMRI data acquisition.