A nonparametric efficient evaluation of partial directed coherence

Abstract

Studying the flow of information between different areas of the brain can be performed using the so-called partial directed coherence (PDC). This measure is usually evaluated by first identifying a multivariate autoregressive model and then using Fourier transforms of the impulse responses identified and applying appropriate normalizations. Here, we present another way to evaluate PDCs in multivariate time series. The method proposed is nonparametric and utilizes a strong spectral factorization of the inverse of the spectral density matrix of a multivariate process. To perform the factorization, we have recourse to an algorithm developed by Davis and his collaborators. We present simulations as well as an application on a real data set (local field potentials in a sleeping mouse) to illustrate the methodology. A detailed comparison with the common approach in terms of complexity is made. For long autoregressive models, the proposed approach is of interest.

Appendices

Appendix 1: Spectral factorization

The aim here is to present the main steps for the derivation of the Davis and Dickinson algorithm. The code is presented in the next appendix. The algorithm relies on the equivalence between Kalman filtering and Wiener filtering. Note that the complete proof is lengthy and requires a considerable amount of algebraic manipulation. The complete proof is given in some detail in Dickinson’s master’s thesis but does not appear in other publications. This is the main reason to include here the main steps of the proof. The only difference with Dickinson’s proof is the faster method we use to obtain Eq. (9) in what follows.

The proof consists in expressing the spectral factors used in Wiener filtering in terms of the elements of the solution of Kalman filtering. Then the spectral factor essentially depends on the covariance of the error, which is given by the solution of a Riccati equation. This equation has no closed-form solution (except in very rare cases). Using a Newton–Raphson recursion to solve the Riccati equation makes it possible to obtain as a byproduct the Davis algorithm.

Suppose we have the following state and observation equations:

$$\begin{aligned} {\varvec{x}}_k&= {\varvec{A}}{\varvec{x}}_{k-1} + {\varvec{B}}{\varvec{u}}_k, \\ {\varvec{y}}_k&= {\varvec{C}}{\varvec{x}}_k + {\varvec{v}}_k, \end{aligned}$$

where ${\varvec{u}}$ and ${\varvec{v}}$ are independent white sequences with zero mean and corresponding covariance matrices ${\varvec{Q}}$ and ${\varvec{R}}$. The aim in filtering is to estimate ${\varvec{x}}_k$ from the observation ${\varvec{y}}$ up to time $k$.

The covariance matrix is ${\varvec{S}}_{yy}(z) = {\varvec{C}}{\varvec{S}}_{xx}(z) {\varvec{C}}^\top + {\varvec{R}}$ and can be written as

$$\begin{aligned} {\varvec{S}}_{yy}(z) = {\varvec{R}}+ {\varvec{C}}({\varvec{I}}z - {\varvec{A}})^{-1} {\varvec{B}}{\varvec{Q}}{\varvec{B}}^\top ({\varvec{I}}z^{-\star } - {\varvec{A}})^{-\dagger }{\varvec{C}}^\top . \end{aligned}$$

The Wiener filter necessitates having a strong spectral factorization of this spectral matrix in the form

$$\begin{aligned} {\varvec{S}}_{yy}(z) = {\varvec{F}}(z) {\varvec{W}}{\varvec{F}}^{\dagger }(z^{-\star }). \end{aligned}$$

Note that in this sketch of the proof we adopt for convenience a convention other than that in Eq. (5), the derivation here being done in accordance with the common definition of correlation. The difference is simply a transposition. Then the spectral factor is given by ${\varvec{F}}(z) = {\varvec{I}}+ {\varvec{C}}({\varvec{I}}z-{\varvec{A}})^{-1} {\varvec{K}}$, where the Kalman gain (steady state) reads ${\varvec{K}}= {\varvec{A}}{\varvec{P}}{\varvec{C}}^\top {\varvec{W}}^{-1}$ and ${\varvec{W}}= {\varvec{R}}+ {\varvec{C}}{\varvec{P}}{\varvec{C}}^\top$. ${\varvec{P}}$ is the solution of the Riccati equation ${\varvec{P}}= {\varvec{A}}{\varvec{P}}{\varvec{A}}^\top - {\varvec{A}}{\varvec{P}}{\varvec{C}}^\top {\varvec{W}}^{-1} {\varvec{C}}{\varvec{P}}{\varvec{A}}^\top + {\varvec{B}}{\varvec{Q}}{\varvec{B}}^\top$.

The problem reduces to one of obtaining the solution of the Riccati equation, which is far from being obvious. For this, Davis proposed using a Newton–Raphson algorithm to solve

$$\begin{aligned} 0&= f({\varvec{P}}) \\&= -{\varvec{P}}+ {\varvec{A}}{\varvec{P}}{\varvec{A}}^\top - {\varvec{A}}{\varvec{P}}{\varvec{C}}^\top {\varvec{W}}^{-1} {\varvec{C}}{\varvec{P}}{\varvec{A}}^\top + {\varvec{B}}{\varvec{Q}}{\varvec{B}}^\top . \end{aligned}$$

The Newton–Raphson iteration for solving this is

$$\begin{aligned}&{\varvec{P}}_{n+1} - ({\varvec{A}}- {\varvec{K}}_n {\varvec{C}}) {\varvec{P}}_{n+1} ({\varvec{A}}- {\varvec{K}}_n {\varvec{C}})^\top \\&\quad ={\varvec{K}}_n {\varvec{R}}{\varvec{K}}_n^\top + {\varvec{B}}{\varvec{Q}}{\varvec{B}}^\top . \end{aligned}$$

A first trick is to use the representation in series ${\varvec{X}}=\sum _{n\le 0} {\varvec{A}}^{n} {\varvec{\Gamma }}{\varvec{A}}^{\top n }$ for the solution of ${\varvec{X}}+ {\varvec{A}}{\varvec{X}}{\varvec{A}}^\top ={\varvec{\Gamma }}$. If ${\varvec{\Gamma }}$ is positive-definite, then the series is an inner product, and we can use Parseval’s equality to obtain the equivalent form in the $z$-domain. Apply this to ${\varvec{P}}_{n+1}$ to obtain

$$\begin{aligned} {\varvec{P}}_{n+1}&= \frac{1}{2i\pi } \oint (z{\varvec{I}}- {\varvec{A}}+ {\varvec{K}}_n {\varvec{C}})^{-1} \big ( {\varvec{K}}_n {\varvec{R}}{\varvec{K}}_n^\top + {\varvec{B}}{\varvec{Q}}{\varvec{B}}^\top \big ) \\&(z^{-\star } {\varvec{I}}- {\varvec{A}}+ {\varvec{K}}_n {\varvec{C}})^{-\dagger } \frac{\hbox {d}z}{z}. \end{aligned}$$

Then pre- and postmultiplying of ${\varvec{P}}_{n+1}$ by ${\varvec{C}}$, using some algebra and recalling the definitions of ${\varvec{S}}_{yy}$ and ${\varvec{F}}$ and the fact that $(2i\pi )^{-1}\oint {\varvec{F}}_n(z)^{-1} \hbox {d}z/z={\varvec{I}}$, allows us to obtain

$$\begin{aligned} {\varvec{R}}+{\varvec{C}}{\varvec{P}}_{n+1} {\varvec{C}}^\top&= \frac{1}{2i\pi } \oint {\varvec{F}}_n(z)^{-1} {\varvec{S}}_{yy}(z) {\varvec{F}}_n(z^{-\star })^{-\dagger }\frac{\hbox {d}z}{z}\\&:= {\varvec{W}}_n, \end{aligned}$$

which constitutes the first part of the algorithm.

To obtain the iteration on the spectral factor, Dickinson proposes to examine $\Delta {\varvec{P}}_n := {\varvec{P}}_{n+1}-{\varvec{P}}_n$. Substracting two successive iterations of the Newton–Raphson iteration leads to

$$\begin{aligned}&\Delta {\varvec{P}}_{n} - ({\varvec{A}}- {\varvec{K}}_n {\varvec{C}}) \Delta {\varvec{P}}_n ({\varvec{A}}- {\varvec{K}}_n {\varvec{C}})^\top \\&\quad = -({\varvec{K}}_n- {\varvec{K}}_{n-1}){\varvec{W}}_{n-1}({\varvec{K}}_n- {\varvec{K}}_{n-1})^\top :=- {\varvec{T}}_n. \end{aligned}$$

This last matrix ${\varvec{T}}_n$ is positive-definite since ${\varvec{W}}_{n-1}$ is positive-definite.

Since $\Delta {\varvec{P}}_{n}$ satisfies an equation of the type ${\varvec{X}}+ {\varvec{A}}{\varvec{X}}{\varvec{A}}^\top ={\varvec{\Gamma }}$, we use the series representation for $\Delta {\varvec{P}}_{n}$,

$$\begin{aligned} \Delta {\varvec{P}}_{n} = -\sum _{k\ge 0}({\varvec{A}}- {\varvec{K}}_n {\varvec{C}})^k {\varvec{T}}_n({\varvec{A}}- {\varvec{K}}_n {\varvec{C}})^{(k)\top }, \end{aligned}$$

and since ${\varvec{T}}_n$ is positive-definite, we can have an equivalent form in the $z$-domain,

$$\begin{aligned} \Delta {\varvec{P}}_{n}&= \frac{-1}{2i\pi } \oint \big ({\varvec{I}}- z^{-1}({\varvec{A}}- {\varvec{K}}_n {\varvec{C}})\big )^{-1}{\varvec{T}}_n \nonumber \\&\times \Big (z^{-\star } ( {\varvec{I}}- z^{\star }({\varvec{A}}- {\varvec{K}}_n {\varvec{C}})\Big )^{-\dagger } \frac{\hbox {d}z}{z}. \end{aligned}$$

(9)

Then we must solve the equation, which can be verified by direct evaluation:

$$\begin{aligned}&{\varvec{C}}(z {\varvec{I}}- {\varvec{A}})^{-1}({\varvec{A}}-{\varvec{K}}_n {\varvec{C}})^\top \Delta {\varvec{P}}_n {\varvec{C}}^\top \nonumber \\&\quad = ({\varvec{F}}_{n+1}(z) -{\varvec{F}}_n(z)) {\varvec{W}}_{n}. \end{aligned}$$

Inserting (9), a lengthy calculation leads to

$$\begin{aligned}&-2i\pi ({\varvec{F}}_{n+1}(z) -{\varvec{F}}_n(z)) {\varvec{W}}_{n}\\&\quad = ({\varvec{F}}_n(z)- {\varvec{F}}_{n-1}(z)) {\varvec{W}}_{n-1} \nonumber \\&\qquad \times \oint \frac{\hbox {d}v}{(v-z)} ( {\varvec{F}}_n(v^{-\star })- {\varvec{F}}_{n-1}(v^{-\star }))^{\dagger } {\varvec{F}}_n(v^{-\star })^{-\dagger }\nonumber \\&\qquad -\, {\varvec{F}}_n(z) \nonumber \\&\qquad \times \oint \frac{dv}{(v-z)} {\varvec{F}}_n(v)^{-1} \Big ( {\varvec{F}}_n(v) {\varvec{W}}_{n-1} {\varvec{F}}_n(v^{-\star }) \!-\!{\varvec{S}}_{yy}(v) \Big )\nonumber \\&\qquad \times \,\,{\varvec{F}}_n(v^{-\star })^{-\dagger }, \nonumber \end{aligned}$$

(10)

an expression that can be linked with the causal projection.

If $H(z)$ is the $z$-transform of a sequence $h_k, k\in \mathbb {Z}$, recall that

$$\begin{aligned} (P_+H)(z)&= \sum _{k\ge 0} h_k z^{-k}\\&= \frac{1}{2i\pi }\oint \frac{\hbox {d}v}{v} \frac{z}{z-v} H(v). \end{aligned}$$

Thus, note that the integrals appearing in expression (11) are of the form

$$\begin{aligned} \oint \frac{\hbox {d}v}{v} \frac{v}{v-z} H(v)&= \oint \frac{\hbox {d}v}{v} \frac{v+z-z}{v-z}H(v) \\&= \oint \frac{\hbox {d}v}{v} H(v) -2i\pi (P_+H)(z). \end{aligned}$$

The first integral in (11) concerns an anticausal quantity with no constant term and is therefore equal to zero. Noting that ${\varvec{W}}_{n-1}$ does not depend on $v$, and therefore its causal part is equal to itself, we finally get the beautiful result

$$\begin{aligned}&{\varvec{W}}_{n} = \frac{1}{2i\pi }\oint \frac{\hbox {d}v}{v}{\varvec{F}}_n(v)^{-1}{\varvec{S}}_{yy}(v){\varvec{F}}_n(v^{-\star })^{-\dagger }\\&{\varvec{F}}_{n+1}(z) = {\varvec{F}}_n(z) \Big (P_+\big [ {\varvec{F}}_n^{-1}{\varvec{S}}_{yy}{\varvec{F}}_n^{-\dagger }\big ]\Big )(z) {\varvec{W}}_{n}^{-1}. \end{aligned}$$

Appendix 2: Code for spectral factorization

The following is MATLAB code for the spectral factorization. It uses three-dimensional arrays. No test for positive definiteness is included, and if the assumptions on matrix $S$ are inadequate, then the algorithm should not converge.