A comparison of mental arithmetic performance in time and frequency domains

The Heisenberg-Gabor uncertainty principle defines the limits of information resolution in both time and frequency domains. The limit of resolution discloses unique properties of a time series by frequency decomposition. However, classical methods such as Fourier analysis are limited by spectral leakage, particularly in longitudinal data with shifting periodicity or unequal intervals. Wavelet transformation provides a workable compromise by decomposing the signal in both time and frequency through translation and scaling of a basis function followed by correlation or convolution with the original signal. This study aimed to compare the accuracy of predictive models in mental arithmetic in time and frequency domains. Analysis of the author's response time at mental arithmetic using a soroban was modeled for two periods, an initial period (T-I = 68 days), and a return period (T-R = 170 days) both separated by an interval of 370 days. The median (min,max) response times in seconds (s) was longer for all tasks during the T-I compared to the T-R period (p < 0.001), for addition [CTAdd 62 (45, 127) vs 50 (38, 75) s] and summation [CTSum 68 (47, 108) vs 57(43, 109) s]. Response times were longer for errors regardless of the study period or task. There was an increasing phase difference for the addition and summation tasks during the T-I period toward the end of the series 49.65(o) compared to the T-R period where the phase difference between the two tasks was only 2.05(o), indicating that both tasks 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 and 8.19% and that for the time domain models [autoregressive integrated moving average (ARIMA), generalized autoregressive conditional heteroscedasticity (GARCH)] was 6.16-10.80%.

Automated summary

The Heisenberg-Gabor uncertainty principle defines the limits of information resolution in both time and frequency domains, while wavelet transformation provides a workable compromise by decomposing the signal in both time and frequency. This study aimed to compare the accuracy of predictive models in mental arithmetic in time and frequency domains.