## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 12, Number 3 (2018), 1451-1478.

### Complex-valued time series modeling for improved activation detection in fMRI studies

Daniel W. Adrian, Ranjan Maitra, and Daniel B. Rowe

#### Abstract

A complex-valued data-based model with $p$th order autoregressive errors and general real/imaginary error covariance structure is proposed as an alternative to the commonly used magnitude-only data-based autoregressive model for fMRI time series. Likelihood-ratio-test-based activation statistics are derived for both models and compared for experimental and simulated data. For a dataset from a right-hand finger-tapping experiment, the activation map obtained using complex-valued modeling more clearly identifies the primary activation region (left functional central sulcus) than the magnitude-only model. Such improved accuracy in mapping the left functional central sulcus has important implications in neurosurgical planning for tumor and epilepsy patients. Additionally, we develop magnitude and phase detrending procedures for complex-valued time series and examine the effect of spatial smoothing. These methods improve the power of complex-valued data-based activation statistics. Our results advocate for the use of the complex-valued data and the modeling of its dependence structures as a more efficient and reliable tool in fMRI experiments over the current practice of using only magnitude-valued datasets.

#### Article information

**Source**

Ann. Appl. Stat., Volume 12, Number 3 (2018), 1451-1478.

**Dates**

Received: December 2016

Revised: September 2017

First available in Project Euclid: 11 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1536652961

**Digital Object Identifier**

doi:10.1214/17-AOAS1117

**Mathematical Reviews number (MathSciNet)**

MR3852684

**Keywords**

Area under the ROC curve contrast-to-noise ratio finger-tapping motor experiment hemodynamic response function Kronecker product neurosurgical planning guide phase information signal-to-noise ratio structured covariance matrix

#### Citation

Adrian, Daniel W.; Maitra, Ranjan; Rowe, Daniel B. Complex-valued time series modeling for improved activation detection in fMRI studies. Ann. Appl. Stat. 12 (2018), no. 3, 1451--1478. doi:10.1214/17-AOAS1117. https://projecteuclid.org/euclid.aoas/1536652961

#### Supplemental materials

- Supplement to “Complex-valued time series modeling for improved activation detection in fMRI studies”. Section S-1 displays the real, imaginary, magnitude, and phase components of the data for all slices of the finger-tapping dataset used in this paper. Section S-2 provides additional details and derivations on our methodology. Further details on the analysis of the finger-tapping dataset are provided in Section S-3 while more simulation-based analyses are in Section S-4.Digital Object Identifier: doi:10.1214/17-AOAS1117SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.