Fig. 3

Derivative of the state. Period A:Period B = 1:n, where n is a positive integer. \( \frac{{{\text{d}}S}}{{{\text{d}}t}} \) is the derivative of S with respect to time; \( \frac{{{\text{d}}S}}{{{\text{d}}t}} \) = k + δ, where k is constant and δ (δ _i ; i = 1, 2,…, n) is noise which follows the normal distribution. We observe the state (n + 1)-times at the interval of the period A during the period B. The counterfactual derivative of the state is postulated as (k + δz), where δz follows the same distribution as δ _is