A comparison of mental arithmetic performance in time and frequency domains - Abdul-Rahman, Anmar (Frontiers in Psychology, 2022)
The Heisenberg-Gabor uncertainty principle states that information cannot be localized simultaneously in both time and frequency domains. Wavelet transformation provides a workable compromise by decomposing the signal in both time and frequency through a process of translation and scaling of a basis function followed by correlation or convolution with the original signal. Analysis of the author's performance at mental arithmetic using the Soroban was modeled in the time and frequency domains for two periods, an initial period (TI = 68 days), and a return period (TR=170 days) both separated by an interval of 370 days. The median (min,max) performance times in seconds (sec) was longer (p$<$0.001) for all tasks during the TI compared to the TR period, for addition (CT_Add 62 (45 ,127) vs 50 (38, 75) sec) and summation (CT_Sum 68 (47, 108) vs 57(43, 109) sec). Response times were longer for incorrect outcomes regardless of the study phase or task. There was an increasing phase difference for the addition and summation tasks during the TI period towards the end of the series 49.65 degrees compared to the TR period where the phase difference between the two tasks was only 2.05 degrees, indicating that both time series are likely demonstrating similar learning rates during the latter study period. A comparison between time and time/frequency domain forecasts for an additional 100 tasks demonstrated higher accuracy of the maximum overlap discrete wavelet transform (MODWT) model, where the mean absolute percentage error ranged between 5.48-8.19% and that for the time domain models (autoregressive integrated moving average (ARIMA), generalized autoregressive conditional heteroscedasticity (GARCH)) was 6.16-10.80%.