Steering law reveals a linear relationship between the movement time (MT) and the index of difficulty (ID) in trajectory-based steering tasks. However, it does not relate the variance or distribution of MT to ID. In this paper, we propose and evaluate models that predict the variance and distribution of MT based on ID for steering tasks. We first propose a quadratic variance model which reveals that the variance of MT is quadratically related to ID with the linear coefficient being 0. Empirical evaluation on a new and a previously collected dataset show that the quadratic variance model accounts for between 78% and 97% of variance of observed MT variances; it outperforms other model candidates such as linear and constant models; adding the linear coefficient leads to no improvement on the model fitness. The variance model enables predicting the distribution of MT given ID: we can use the variance model to predict the variance (or scale) parameter and Steering law to predict the mean (or location) parameter of a distribution. We have evaluated six types of distributions for predicting the distribution of MT. Our investigation also shows that positively skewed distribution such as Gamma, Lognormal, Exponentially Modified Gaussian (ExGaussian), and Extreme value distributions outperformed the symmetric distribution such as Gaussian and truncated Gaussian distribution in predicting the MT distribution, and Gamma distribution performed slightly better than other positively skewed distributions. Overall, our research advances the MT prediction of steering tasks from a point estimate to variance and distribution estimates, which provides a more complete understanding of steering behavior and quantifies the uncertainty of MT prediction.
Keywords: Steering law; probabilistic modeling.