Formulae

JumpDiff.formulae.M_formula(power, tau=True)

Generate the formula for the conditional moments with second-order corrections based on the relation with the ordinary Bell polynomials

\[M_n(x^{\prime},\tau) \sim (n!)\tau D_n(x^{\prime}) + \frac{(n!)\tau^2}{2} \sum_{m=1}^{n-1} D_m(x^{\prime}) D_{n-m}(x^{\prime})\]

Parameters:power (int) – Desired order of the formula. Returns:term – Expression up to given power. Return type:sympy.symbols

JumpDiff.formulae.F_formula(power)

Generate the formula for the conditional moments with second-order corrections based on the relation with the ordinary Bell polynomials

\[\begin{split}D_n(x) &= \frac{1}{\tau (n!)} \bigg[ \hat{B}_{n,1} \left(M_1(x,\tau),M_2(x,\tau),\ldots,M_{n}(x,\tau)\right) \\ &\qquad \left.-\frac{\tau}{2} \hat{B}_{n,2} \left(M_1(x,\tau),M_2(x,\tau),\ldots,M_{n-1}(x,\tau)\right)\right].\end{split}\]

Parameters:power (int) – Desired order of the formula. Returns:term – Expression up to given power. Return type:sympy.symbols

JumpDiff.formulae.F_formula_solver(power)

Generate the reciprocal relation of the moments to the Kramers─Moyal coefficients by sequential iteration.

\[\begin{split}D_n(x) &= \frac{1}{\tau (n!)} \bigg[ \hat{B}_{n,1} \left(M_1(x,\tau),M_2(x,\tau),\ldots,M_{n}(x,\tau)\right) \\ &\qquad \left.-\frac{\tau}{2} \hat{B}_{n,2} \left(M_1(x,\tau),M_2(x,\tau),\ldots,M_{n-1}(x,\tau)\right)\right].\end{split}\]

Parameters:power (int) – Desired order of the formula. Returns:term – Expression up to given power. Return type:sympy.symbols