# Piecewise fittings of the saccadic velocity profiles using the rational power functions

Hsuan Hung Lin, Yung Fu Chen, Tainsong Chen, Tze Tung Tsai

## 摘要

The relation for the product of peak velocity and duration against saccadic amplitude was tightly correlated with correlation coefficient according to the previous study. Previous study described the velocity profile as triangular and referred to the saccadic amplitude as an integration of the profile that the amplitude is proportional to V mD. From our observation and derivation, in addition to the triangular profile, the rational power function could also be applied to explain the linear relationship between the saccadic amplitude and the product of peak velocity and duration. The saccadic amplitude is proportional to the product of peak velocity and duration (V mD) only when the velocity profile is symmetry. For greater amplitudes, the profiles become asymmetrical and it is more suitable for describing the saccadic amplitude as the sum of the acceleration and the deceleration amplitudes. Although the correlation coefficients, after fitting two rational power functions to the acceleration phase and the deceleration phase, were satisfactory, thist however cannot resolve the inflection points which divide two phases into four segments: two are located in the acceleration phase and the other two in the deceleration phase. In this study, the inflection points were obtained by differentiating the velocity profile. Rational power functions were then applied individually to fit these 4 separate segments. The results show that the rational power functions were fitted very well to velocity profiles for amplitudes 5°, 10°, 20°, and 30° with correlation coefficients greater than 0.999. A comparison for the shape parameters indicates that the curvature for segment 1 (n 1) and 2 (n 2) is significantly larger than segment 3 (n 3) and 4 (n 4). Also n 3 is very close to 1, which mimics that it approximates to a line. In conclusion, rational power functions are efficient in piecewise fitting for the saccadic velocity profiles. The shape parameters are also illuminated for describing the velocity profile characteristics of saccades.

原文 English Proceedings of the International Conference on Mathematics and Engineering Techniques in Medicine and Biological Sciences, METMBS'04 F. Valafar, H. Valafar 486-489 4 Published - 2004 Proceedings of the International Conference on Mathematics and Engineering Techniques in medicine and Biological Sciences, METMBS'04 - Las Vegas, NV, United States持續時間: 2004 六月 21 → 2004 六月 24

### 出版系列

名字 Proceedings of the International Conference on Mathematics and Engineering Techniques in Medicine and Biological Sciences, METMBS'04

### Other

Other Proceedings of the International Conference on Mathematics and Engineering Techniques in medicine and Biological Sciences, METMBS'04 United States Las Vegas, NV 04-06-21 → 04-06-24

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