Comment on “Cumulant mapping as the basis of multi-dimensional spectrometry” by Leszek J. Frasinski, Phys. Chem. Chem. Phys., 2022, 24, 20776–20787

Abstract

I state a general formula for the n-variate joint cumulant of the first order and prove that it satisfies the desired properties listed in Section 3.3 of Phys. Chem. Chem. Phys., 2022, 24, 20776–20787.

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Motivation

A recent article by Frasinski¹ develops a theory of cumulant mapping, which extends covariance mapping² to any number of fragments. The central object of this theory is the n-variate joint cumulant of the first order, abbreviated nth cumulant. In Section 3.3, Frasinski lists what properties the nth cumulant should satisfy, and then gives explicit expressions for up to the 6th cumulant. How these can be found in practice is not elaborated upon.

The purpose of this comment is to show how we can find the nth cumulant—in theory and practice. I will do this by providing a general formula and describe how to evaluate it. Additionally, I will use the general formula to prove that cumulants fulfill some useful properties.

The general formula

Let X₁,…,X_n be random variables. Their multivariate cumulant-generating function is³From this function the nth cumulant is defined as³This definition is simple and useful for proving properties, but difficult to evaluate. Later in this comment I will derive the more explicit expressionwhere means all partitions of {1,…,n} into k sets.

How to evaluate it

Let us say we want to find the 7th cumulant. The main thing we should do is to find the partitions of 7 into k numbers that each are 2 or greater. For k = 1 there is the trivial 7; for k = 2 there is 5 + 2 and 4 + 3; and for k = 3 there is 3 + 2 + 2. From each nontrivial partition we then create a sum over all congruent products of covariances. Hopefully the rule becomes apparent by looking at the resultwhere I have used i as a shorthand for X_i − 〈X_i〉 inside 〈 〉. The prefactor of each sum is simply (−1)^k+1(k − 1)!. The number of products in a sum can be calculated aswhere #m is the number of parts of size m. Matlab and Python code implementing the nth cumulant is available as ESI.†

Useful properties

The four desired properties listed in Section 3.3 of the original article¹ are

• χ_n(…) ≠ 0 only if all arguments are collectively correlated;

• χ_n(…) has units of the product of all arguments;

• χ_n(…) is linear in the arguments;

• χ_n(…) is invariant under interchange of any two arguments.

I will now prove that the cumulant has these desired properties, starting with the interchange of arguments.

Property 1 (symmetric). The nth cumulant is invariant under permutation of its arguments:

χ_n(X_π(1),…,X_π(n)) = χ_n(X₁,…,X_n). (6)

Proof. Commutativity of addition, and of differentiation. □

The desired properties about linearity and units are combined into one, because the former implies the latter.

Property 2 (multilinear). The nth cumulant is linear in each of its arguments:

χ_n(aX + bY,Z₂,…) = aχ_n(X,Z₂,…) + bχ_n(Y,Z₂,…). (7)

Proof. Because of symmetry we only have to prove linearity in the first argument. By expanding the expression

K_aX+bY,…(t₁,…) − aK_X,…(t₁,…) − bK_Y,…(t₁,…) (8)

in its first argument t₁, we find that the first-order terms cancel out. Differentiating with respect to t₁ (among others) and evaluating at the origin will therefore giveNext, I phrase the desired property about correlation conversely.

Property 3 (discerning). Let (A_i)^m_i=1 and (B_j)ⁿ_j=m+1 be nonempty tuples of random variables such that A_i and B_j are independent. Then the nth cumulant of A∪B vanishes:

χ_n(A₁,…,A_m, B_m+1,…,B_n) = 0. (10)

Proof. Because and are independent, we can separate the generating function like

K_{A1,…,Am,Bm+1,…,Bn}(…) = K_A1,…,Am(…) + K_Bm+1,…,Bn(…). (11)

Differenting with respect to t_n and t₁ (among others) will annihilate both terms. □

Finally, I note an important property that follows from the last two. It tells us that independent signals simply add their contributions to a cumulant.

Property 4 (additive). Let (A_i)ⁿ_i=1 and (B_j)ⁿ_j=1 be equal-length tuples of random variables such that A_i and B_j are independent. Then the nth cumulant distributes over the addition of these tuples:

χ_n(A₁ + B₁,…,A_n + B_n) = χ_n(A₁,…,A_n) + χ_n(B₁,…,B_n). (12)

Proof. By repeatedly using linearity, we can expand the left hand side into 2ⁿ terms. The mixed terms containing both some A_i and some B_j vanish because of the discerning property. □

Explicit expression

Our strategy will be to approximate the generating function with Taylor series, starting from the inside. The exponential can be truncated by removing terms containing any t_i²The last product turns into 2ⁿ terms, one for each subset of I = {1,…,n}. The term corresponding to S contains t_i if and only if S contains i. Explicitly,Applying the expectation value is straightforward linearityIn anticipation of taking the logarithm, we extract a 1 from our expectation value taking out the term where S is the empty setNow, we plug this x into the Taylor series of the logarithmRecall that the cumulant is obtained by differentiating the above expression with respect to each variable and evaluating at the origin. This can be thought of as extracting the coefficient of the term. Hence we will focus on contributions to it.

The first term, x, contains one subterm for each nonempty subset S of I. Only the subterm corresponding to S = I will be proportional to . Its coefficient will then be .

The second term, −x²/2, contains when expanded one subterm for every ordered pair of nonempty subsets (S₁,S₂) of I. In order to get a term proportional to each index i must be an element in exactly one of S₁ and S₂. In other words, {S₁,S₂} must be a partition of I. For every partition of I there are 2! matching ordered pairs, each contributing to the coefficient.

By now the rule for the kth term is clear. It will contain subterms corresponding to each partition of I into k nonempty sets. Each subterm will be a product of the prefactor (−1)^k+1(k − 1)! and k expectation values.

We can make single-variable expectations vanish by replacing X_i with X_i − 〈X_i〉 everywhere. Using the additive property with A_i = X_i and B_i = −〈X_i〉, we see that this change preserves the cumulant.

Conflicts of interest

There are no conflicts to declare.

References

1. L. J. Frasinski, Phys. Chem. Chem. Phys., 2022, 24, 20776–20787.
2. V. Zhaunerchyk, L. Frasinski, J. H. Eland and R. Feifel, Phys. Rev. A: At., Mol., Opt. Phys., 2014, 89, 053418.
3. A. Stuart and K. Ord, Kendall's advanced theory of statistics, distribution theory, John Wiley & Sons, 2010, vol. 1.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3cp02525j

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