Graduation Year


Date of Submission


Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Arts



Reader 1

Qidi Peng

Reader 2

Asuman G. Aksoy

Reader 3

Yunied Puig de Dios


This thesis aims at introducing a new way to model time series objects in statistics using multifractional processes. It provides a detailed review of Brownian motion, fractional Brownian motion and extends the above 2 models to multifractional processes. To demonstrate a successful application to the real world, we perform pattern analysis on consumer confidence and household spending behavior. The analysis is conducted through investigating the local Holder regularity of the consumer confidence index and household expenditure. In the analysis, we first model consumer confidence index and household expenditure with a multifractional stochastic processes. We then use the index, pointwise Holder exponent (PHE), to measure the local Holder regularity of their paths. Next, several estimators of the PHE have been derived and compared using the data. Finally, we detect which household consumption factors share similar patterns of local Holder regularity to the CCI using K-means clustering.