A Non-parametric Approach to the Overall Estimate of Cognitive Load Using NIRS Time Series

We present a non-parametric approach to prediction of the n-back n ∈ {1, 2} task as a proxy measure of mental workload using Near Infrared Spectroscopy (NIRS) data. In particular, we focus on measuring the mental workload through hemodynamic responses in the brain induced by these tasks, thereby realizing the potential that they can offer for their detection in real world scenarios (e.g., difficulty of a conversation). Our approach takes advantage of intrinsic linearity that is inherent in the components of the NIRS time series to adopt a one-step regression strategy. We demonstrate the correctness of our approach through its mathematical analysis. Furthermore, we study the performance of our model in an inter-subject setting in contrast with state-of-the-art techniques in the literature to show a significant improvement on prediction of these tasks (82.50 and 86.40% for female and male participants, respectively). Moreover, our empirical analysis suggest a gender difference effect on the performance of the classifiers (with male data exhibiting a higher non-linearity) along with the left-lateralized activation in both genders with higher specificity in females.


COMPUTATION OF THE DIFFERENTIAL ENTROPY (DE) FEATURE
DE generalizes the concept of entropy for discrete random variables to the realm of continuous random variable. DE of a continuous random variable X is calculated as (Kumaran et al., 2016): Furthermore, when X ∼ N (µ, σ 2 ), equation (1) simplifies into: It is worth noting that the desired behaviour to follow the normal distribution is obtained via normalization of the NIRS data, thereby acquiring its standard normal distribution X ∼ N (0, 1).

SUPPLEMENTARY DEFINITIONS AND CLAIMS
Definition 4.1. Given N points, p 1 , . . . , p N ∈ R n (n ≥ 1), there exists a point x ∈ R n (n ≥ 1) that minimizes the function (Boltyanski et al., 1999): with w i , i = 1, . . . , N being the weight associated with i th data point and x is the geometric median (a.k.a generalized Fermat-Torricelli point or the Weber point) of p 1 , . . . , p N (Boltyanski et al., 1999).
CLAIM 4.2. The function f is strictly convex function if and only if the points p 1 , . . . , p N ∈ R n (n ≥ 1) are not collinear (Boltyanski et al., 1999).

OVERALL COMPLEXITY OF THE PROPOSED APPROACH
It is apparent that equations (1) and (2) are linear in the size of the input data. Furthermore, equations (4) and (5) are O(1) procedures, followed by the equation (6) that is O(M ), with M representing the size of the test set. Moreover, the latter becomes O(1) in case of realtime prediction where the system requires to estimate the state of an individual based on the NIRS time series associated with the prediction interval. On the other hand, equation (3) introduces higher degree of complexity due to the matrix-matrix multiplication in its formulation. More specifically, the first term in this equation requires the inverse of the input feature matrix that is multiplied by its transpose, indicating an O(N 3 ) computation with N being the size of the input. Whereas, the second term stays linear in the size of input i.e., O(N ) since it involves a matrix-vector multiplication. However, there are three observations to make: 1. Equation (3) reduces the overall computation of the weights associated with our model into one step, resulting in an overall O(1) complexity that is independent of the size of the input or the nature of the data involved. This is a highly desirable feature for a learning model once deployed in a real world setting. (Surname, 2002) 2. The aforementioned O(1) overall complexity becomes more attractive considering the fact that feature spaces associated with NIRS data are commonly low dimensional.
3. It is possible to speed up the computation of the first term in equation (3) via application of faster matrix multiplication technqiues such as Strassen algorithm (Strassen, 2000), achieving an O(N 2.81 ). Furthermore, the recent modification of this algorithm scales down its complexity to O(N 2.372 ) (Galli, 2014).