Approximate methods for explicit calculations of non-Gaussian moments

Abstract

We present theoretical tools from statistical physics for the calculation of non-Gaussian moments. In particular, we focus on the variational approximation and the cumulant expansion. These methods enable approximate but explicit moment calculations with non-Gaussian probability density functions (pdfs). Their use is illustrated by calculating the variance and the excess kurtosis for a univariate non-Gaussian pdf. We comment on the potential application of these methods in estimating the parameters of the recently proposed Spartan spatial random fields.

Appendices

Appendix 1

Here we derive Eqs. 52 and 53. First, note that the following is true:

$$\langle ({\mathbf{J}} \cdot {\mathbf{X}})^{m} \rangle_{0} = \langle J_{i_{1}} X_{i_{1}} J_{i_{2}} X_{i_{2}} \cdots J_{i_{m}} X_{i_{m}} \rangle_{0} = J_{i_{1}} \cdots J_{i_{m}} \langle X_{i_{1}} \cdots X_{i_{m}} \rangle_{0} $$

(82)

In Eq. 82 summation is implied over the repeated indexes i _l =1,...,m. The external field variables $J_{i_{l}},$ l=1,...,m are moved outside the angled brackets, since they do not participate in the averaging. The derivatives act only on the external field, not the quantity inside the angled brackets. Let us assume that we are interested in the m-order partial derivative $\partial^{m} ({\mathbf{J}} \cdot {\mathbf{X}})^{m}/\partial J_{i^{*}_{1}} \cdots J_{i^{*}_{m}}.$ The locations i ^*₁ ··· i ^* _m correspond to fixed locations (unlike i ₁ ··· i _m which are variable). It is not required that all i ^* _k be different, in fact they can all denote the same location. Then, we obtain

$$\begin{aligned} \frac{\partial \left({\mathbf{J}} \cdot \langle {\mathbf{X}} \rangle_{0} \right)^{m}}{\partial J_{i^{*}_{1}}} =& \delta_{i_{1},i^{*}_{1}} \left(\prod\limits_{i_{k}\neq i^{*}_{1}} J_{i_{k}} \right) \langle X_{i^{*}_{1}} \cdots X_{i_{m}} \rangle_{0} + \delta_{i_{2},i^{*}_{1}} \left(\prod\limits_{i_{k}\neq i^{*}_{1}} J_{i_{k}} \right) \langle X_{i_{1}} X_{i^{*}_{1}} \cdots X_{i_{m}} \rangle_{0} \\ & \cdots + \delta_{i_{m},i^{*}_{1}} \left(\prod\limits_{i_{k}\neq i^{*}_{1}} J_{i_{k}} \right) \langle X_{i_{1}} \cdots X_{i^{*}_{1}} \rangle_{0} = m \left(\prod\limits_{i_{k}\neq i^{*}_{1}} J_{i_{k}} \right) \left\langle X_{i^{*}_{1}} \prod _{i_{k}\neq i^{*}_{1}} X_{i_{k}} \right\rangle_{0}. \\ \end{aligned}$$

(83)

We repeat the procedure m−1 times to obtain

$$\frac{\partial^{m} \left({\mathbf{J}} \cdot \langle {\mathbf{X}} \rangle_{0}\right)^{m}}{\partial J_{i^{*}_{1}} \cdots \partial J_{i^{*}_{m}}} = m! \langle X_{i^{*}_{1}} \cdots X_{i^{*}_{m}} \rangle_{0}. $$

(84)

The same can be shown by induction, i.e., by showing (1) that Eq. 52 holds for m=1, and (2) provided that Eq. 52 is valid for m−1, then it is also valid for m. Equation 53 is also proved by this approach.

Appendix 2

The fourth order cumulant is given by Eq. 50, i.e., v ₄=〈V ⁴〉₀ − 4〈V ³〉₀〈V 〉₀ −3〈V ²〉 ²₀ +12〈V ²〉₀〈V〉 ²₀ − 6〈V〉 ⁴₀ . Since 〈V〉₀=〈H _pert〉₀, terms that are proportional to 〈V〉 ^m₀ , where m>1 are nonlinear in H _pert and do not contribute. Hence, only the terms 〈V ⁴〉₀ and −3〈V ²〉 ²₀ contribute to the active cumulant, given by Eq. 64. The latter is evaluated as follows.

Given that V=−H _pert+i J·X, the first term in the active cumulant v′₄ is given by

$$\langle V^{4} \rangle_{0} = \langle H^{4}_{\rm pert} \rangle_{0}+4 i \langle H^{3}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})\rangle_{0} - 6 \langle H^{2}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} - 4 i \langle H_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}}) \rangle_{0} + \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{4} \rangle_{0}. $$

(85)

Of the terms in Eq. 85, the first three are of order O(H ²_pert ), while the third vanishes after repeated differentiation. Hence, only the term 〈(J·X)⁴ 〉₀ contributes to Eq. 64.

The second term in v′₄ is

$$\begin{aligned} -3 \langle V^{2} \rangle_{0}^{2} =& 5 \langle H^{4}_{\rm pert}\rangle_{0} \langle H_{\rm pert}\rangle_{0} - 20 i \langle H^{3}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})\rangle_{0} \langle H_{\rm pert}\rangle_{0} - 30 \langle H^{2}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} \langle H_{\rm pert}\rangle_{0}\\ & + 20 i \langle H_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{3} \rangle_{0} \langle H_{\rm pert} \rangle_{0} +5 \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{4} \rangle_{0} \langle H_{\rm pert} \rangle_{0}.\\ \end{aligned} $$

(86)

Only the last term in Eq. 86, which is equal to 5 〈(J·X)⁴〉₀ 〈H _pert〉₀, contributes to linear order in H _pert and is included in Eq. 66.

Next, we obtain the active fifth cumulant given by Eq. 66. The contributing terms of the fifth cumulant are given by v′₅=〈V ⁵〉₀ − 5〈V ⁴〉₀ 〈V〉₀−10〈V ³〉₀ 〈V ²〉₀+30〈V ²〉 ²₀ 〈V〉₀. The first term in the active cumulant v′₅ is given by

$$\begin{aligned} \langle V^{5} \rangle_{0} =& - \langle H^{5}_{\rm pert} \rangle_{0} + 5 i \langle H^{4}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})\rangle_{0} + 10 \langle H^{3}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} - 10 i \langle H^{2}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{3} \rangle_{0} \\ & -5\langle H_{\rm pert}({\mathbf{J}} \cdot {\mathbf{X}})^{4} \rangle_{0} + i \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{5} \rangle_{0}. \\ \end{aligned}$$

(87)

Of the terms in Eq. 87, the first four are of order O(H ²_pert ), while the last one vanishes after differentiation at the limit J=0. Hence, only the term −5〈H _pert(J·X)⁴〉₀ contributes to Eq. 66.

The second term in v′₅ is

$$\begin{aligned} -5\langle V^{4}\rangle_{0} \langle V \rangle_{0} =& 5 \langle H^{4}_{\rm pert}\rangle_{0} \langle H_{\rm pert}\rangle_{0} -20 i \langle H^{3}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})\rangle_{0} \langle H_{\rm pert}\rangle_{0} - 30 \langle H^{2}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} \langle H_{\rm pert}\rangle_{0}\\ & + 20 i \langle H_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{3} \rangle_{0} \langle H_{\rm pert}\rangle_{0} + 5 \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{4} \rangle_{0} \langle H_{\rm pert}\rangle_{0}.\\ \end{aligned} $$

(88)

Only the last term in Eq. 88, which is equal to 5〈(J·X)⁴ 〉₀ 〈H _pert 〉₀, contributes to linear order in H _pert and is included in Eq. 66.

The third term in v′₅ is

$$\begin{aligned} - 10\langle V^{3} \rangle_{0} \langle V^{2} \rangle_{0} =& 10 \langle H^{3}_{\rm pert} \rangle_{0} \langle H^{2}_{\rm pert}\rangle_{0} - 30 i \langle H^{2}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0} \langle H^{2}_{\rm pert}\rangle_{0}\\ & -30 \langle H_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} \langle H^{2}_{\rm pert}\rangle_{0} + 10 i \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{3} \rangle_{0} \langle H^{2}_{\rm pert}\rangle_{0} \\ =& - 10 \langle H^{3}_{\rm pert}\rangle_{0} \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0} + 30 i \langle H^{2}_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0} \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0}\\ & + 30 \langle H_{\rm pert} ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0} - 10 i \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{3} \rangle_{0} \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0}.\\ \end{aligned} $$

(89)

Of the eight terms in Eq. 89, only the second to last term, which is equal to 30 〈H _pert (J·X)² 〉₀ 〈(J·X)²〉₀ contributes to Eq. 66.

The fourth term in v′₅ is

$$ 30\langle V^{2}\rangle_{0}^{2} \langle V \rangle_{0} = 30 \langle H^{2}_{\rm pert}\rangle_{0}^{2} \langle H_{\rm pert}\rangle_{0} + 30 \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{2}\rangle_{0}^{2} \langle H_{\rm pert}\rangle_{0} - 60 \langle H^{2}_{\rm pert} \rangle_{0} \langle ({\mathbf{J}} \cdot {\mathbf{X}})^{2} \rangle_{0} \langle H_{\rm pert}\rangle_{0}. $$

(90)

Of the three terms in Eq. 90 only the second term, which is equal to 30〈(J·X)²〉 ²₀ 〈H _pert〉₀, contributes to Eq. 66.

Appendix 3

The numerical evaluation of the moment integrals in Sect. 7 of the general form

$$I_{n} = 2 \int\limits_{0}^{\infty} \hbox{d}x\,x^{n} \hbox{e}^{-\alpha x^{2} -\beta x^{4}}, $$

(91)

involves an infinite integration domain x∈[0,∞). The integrated function decays very rapidly at large values of x, and is practically different than zero in a narrow range, which depends strongly on the value of the non-Gaussian coefficient β. To overcome numerical errors due to this dependence, we transform the integral I _n using the change of variable y=1/x as follows:

$$I_{n} = 2 \int\limits_{0}^{1} \hbox{d}x\, x^{n} \hbox{e}^{-\alpha x^{2} -\beta x^{4}} + 2 \int\limits_{\epsilon}^{1} \hbox{d}y \frac{\hbox{e}^{-\alpha/y^{2} - \beta/y^{4}}}{y^{n+2}}. $$

(92)

The integrand of the second term is zero at y=0 (because the numerator tends to zero exponentially while the denominator only as power law) and rises smoothly. As a result, it is possible to replace the lower limit of integration with a very small number ε (e.g., 10⁻¹⁶), to avoid the numerically undetermined division with zero.