In modern statistics, there is an increasing demand for theoretically solid methodology which can be applied beyond standard real-valued data, in the realm of more exotic data structures. In this paper, we consider a complex-valued linear mixture model based on temporally uncorrelated components. We consider the problem under discrete weakly stationary processes. We aim to find latent processes of interest when only linear mixtures of them are observable. The recovery of the latent processes is referred to as the unmixing procedure. The properties of the recovered processes themselves can be the source of interest, or alternatively, the approach can be used to reduce multivariate models into several univariate models, which can simplify the modeling burden.

In the context of linear mixture models, the assumption of uncorrelated components or stronger conditions that imply uncorrelated components is considered to be natural in several applications, for example, in finance and signal processing, see the article [14] and the book [10]. Applications where the observed time series are naturally complex-valued are frequent in, e.g., signal processing and biomedical applications, see [19, 41]. In such applications, the interest can lie in the shapes of the unobservable signals, such as the shapes of three-dimensional image valued signals, which correspond to different parts of the brain under different activities. In the signal processing literature, the problem is often referred to as the blind source separation (BSS) problem. In the BSS literature, it has been argued that discarding the special complex-valued structure can lead to loss of information, see [1]. Thus, it is often beneficial to natively work with complex-valued signals in such applications. We wish to emphasize that our C d -valued model does not directly correspond to existing R 2 d valued models, e.g., the one in [23]. In R 2 d valued BSS, it is assumed that all 2 d components are uncorrelated. In our model, the real part and the corresponding complex part are allowed to be correlated. For a collection of BSS applications, see [10].

In parallel to the signal processing community, linear mixture models, with similar model assumptions as in BSS literature, have recently received notable attention in finance, see for example [14, 22]. In financial applications, the term blind source separation is rarely used, and usually more descriptive model naming conventions are utilized. Note that our complex-valued approach is highly applicable in many real-valued financial applications. In our approach, we assume little concerning the relationship between the real and imaginary parts of a single component. Thus, from the real-valued perspective, the problem can be equivalently considered as modeling real-valued temporally uncorrelated pairs, where the elements contained in a single pair are not necessarily uncorrelated. However, transforming this complex-valued problem into a real-valued counterpart is often unfeasible, as it usually introduces additional constraints that would be otherwise embedded within the complex-valued formulation. Once the temporally uncorrelated components are recovered, one can then, for example, model volatilities bivariately (or univariately) or assess risk by modeling tail behavior with the tools of bivariate (or univariate) extreme value theory. Moreover, our approach is natural in applications where a single observation is vector valued, that is, the observations have both a magnitude and a direction, e.g., modeling the latent components of wind at an airport.

The main focus in this paper is on asymptotic behavior of a classical unmixing estimator. We consider an algorithm that is identical to the so-called Algorithm for Multiple Unknown Signals Extraction (AMUSE), [35]. In the financial side, asymptotic properties of an alternative approach, with slightly differing model assumptions compared to our approach, are given in, e.g., [14]. In the context of real-valued BSS, asymptotics have been considered in, e.g., [23–25]. Compared to the above mentioned approaches, we consider a substantially wider class of processes. In particular, we analyze the asymptotic behavior of the corresponding estimators under both long- and short-range dependent complex-valued processes. Furthermore, we take a semiparametric approach, that is, our theory is not limited to specific parametric family of distributions. As a pinnacle of the novel theory presented in this paper, we consider square-integrable processes without summable autocovariance structures, which causes the limiting distribution of the corresponding unmixing estimator to be non-Gaussian. Instead of using the classical central limit theorem, we take a modern approach and utilize general central limit theorem type results that are also applicable for processes with strong temporal dependencies. Moreover, we consider convergence rates that differ from the usual T . We wish to emphasize that modeling long-range dependent processes, that is, processes without summable autocovariance structures, are of paramount importance in many financial applications.

The paper is structured as follows. In Section 2, we recall some basic theory regarding complex-valued random variables. In Section 3, we present the temporally uncorrelated components mixing model. In Section 4, we formulate the estimation procedure and study the asymptotic properties of the estimators. In Section 5, we consider a data example involving photographs, which can be presented as complex-valued time series. The proofs for the theoretical results are presented in the supplementary Appendix.

Throughout, let ( Ω , F , P ) be a common probability space and let z • : = ( z t ) t ∈ N be a collection of random variables z t : Ω → C d , t ∈ N = { 1 , 2 , … }, where the dimension parameter d is a finite constant. Furthermore, let z t ( k ) = pr k ∘ z t , where pr k is a projection to the kth complex coordinate. We refer to the process z • ( k ) : = ( z t ( k ) ) t ∈ N as the kth component of the C d -valued stochastic process z • . Note that the components can be expressed in the form z t ( k ) = a t ( k ) + i b t ( k ) , where a t ( k ) , b t ( k ) are real-valued ∀ k ∈ 1 , 2 , … , d and i is the imaginary unit. We denote the complex conjugate of z t as z t ∗ and we denote the conjugate transpose of z t as z t H : = ( z t ∗ ) ⊤ . Furthermore, we use B ( C d ) and B ( R 2 d ) to denote the Borel-sigma algebras on C d and R 2 d , respectively.

We use the following notation for the multivariate mean and the (unsymmetrized) autocovariance matrix with lag τ ∈ 0 ∪ N: μ z t = E z t ∈ C d , S ¨ τ z t = E z t − μ z t z t + τ − μ z t + τ H ∈ C d × d . Furthermore, we use the notation S τ z t = 1 2 [ S ¨ τ [ z t ] + ( S ¨ τ [ z t ] ) H ] ∈ C d × d for the symmetrized autocovariance matrix with lag τ ∈ 0 ∪ N.

In the case of univariate stochastic processes, we use S ¨ τ [ x t , y s ] to denote S ¨ τ [ x t , y s ] = E ( x t − μ x t ) ( y s + τ − μ y s + τ ) ∗ . Note that the unsymmetrized autocovariance matrix S ¨ τ is not necessarily conjugate symmetric, i.e., Hermite symmetric or Hermitian, and it can have complex-valued diagonal entries. In contrast, the symmetrized autocovariance matrix S τ is always conjugate symmetric, that is, S τ = S τ H , and the diagonal entries of the symmetrized autocovariance matrix are by definition real-valued.

In the literature, the definition of stationarity for complex-valued stochastic processes varies. In this paper, we use the following definition.

Let Z : = Z j j ∈ T , T = 1 , 2 , … , T , be a sampled process generated by z • , where Z j is C d -valued for every j ∈ T. We use the following classical finite sample estimators for the mean vector and the autocovariance matrix with lag τ ∈ 0 , 1 , … , T − 1 : μ ˆ [ Z ] = 1 T ∑ j = 1 T Z j , S ˜ τ Z = 1 T − τ ∑ j = 1 T − τ Z j − μ ˆ [ Z ] Z j + τ − μ ˆ [ Z ] H , where for τ = 0, the scaling used in S ˜ τ is 1 / ( T − 1 ). Furthermore, the symmetrized autocovariance matrix estimator with lag τ ∈ 0 , 1 , … , T − 1 is defined as S ˆ τ Z = 1 2 [ S ˜ τ [ Z ] + ( S ˜ τ [ Z ] ) H ]. Note that the symmetrized autocovariance matrix estimator is always conjugate symmetric.

Let y 1 and y 2 be R d -valued random vectors such that the concatenated random vector, that is, y 1 ⊤ y 2 ⊤ ⊤ , follows a R 2 d -valued Gaussian distribution. Then, the C d -valued random vector y = y 1 + i y 2 follows the C d -valued Gaussian distribution with the location parameter μ y , the covariance matrix Σ y and the relation matrix P y , defined as μ y = E y = E y 1 + i E y 2 ∈ C d , Σ y = E y − μ y y − μ y H ∈ C d × d , P y = E y − μ y y − μ y ⊤ ∈ C d × d .

We distinguish between complex- and real-valued Gaussian distributions with the number of given parameters — N d ( μ y , Σ ) denotes a d-variate real-valued Gaussian distribution and N d ( μ y , Σ , P ) denotes a d-variate complex-valued Gaussian distribution.

The classical central limit theorem (CLT) for independent and identically distributed complex-valued random variables is given as follows. Let { x 1 , x 2 , … , x n } be a collection of i.i.d. C d -vectors with square-integrable components, mean μ x , covariance matrix Σ and relation matrix P. Then, 1 n ∑ j = 1 n x j − μ x → n → ∞ D x ∼ N d 0 , Σ , P , where → D denotes convergence in distribution. We can harness the rich literature on real-valued limiting theorems via the following lemma.

Lemma 1 can be, for example, applied to Gaussian vectors in a straightforward manner, see Corollary 1 in the supplementary Appendix.

In our work, we utilize Lemma 1 in order to apply real-valued results for complex-valued scenarios, but we wish to emphasize that it also works in the other way — the complex-valued asymptotic distribution automatically grants the corresponding real-valued limiting distribution. For additional details regarding complex-valued statistics, see, e.g., [13, 15, 31].

In this section, we consider a linear temporally uncorrelated components model for discrete time complex-valued processes. We define the following version of the model.

To improve the fluency of the paper, from hereon, the term mixing model is used to refer to the temporally uncorrelated components mixing model. The conditions (1)–(4) of Definition 2 imply that the latent process z • is weakly stationary in the sense of Definition 1. Note that under these model assumptions the concatenated R 2 d -process Re ( z • ⊤ ) Im ( z • ⊤ ) ⊤ is not necessarily weakly stationary in the classical real-valued sense. One could also require that the concatenated vector is stationary, which can be done by adding to the model a condition involving the complex-valued relation matrix, defined in Section 2. However, adding the extra condition would also fix the relationship, up to heterogeneous sign-changes, between the real and imaginary parts inside a single component. In this paper, we do not wish to make assumptions regarding the complex phase of a single component. Consequently, many of the following results involve a so-called phase-shift matrix J, that is, a complex-valued diagonal matrix with diagonal entries of the form exp ( i θ 1 ) , … , exp ( i θ d ). A phase-shift matrix J is by definition unitary, i.e., J J H = J H J = I d . Note that all possible phase-shift matrices can be constructed by considering phases on some suitable 2 π-length interval, e.g., the usual [ 0 , 2 π ).

Condition (3) is included in Definition 2 to ensure that we can apply limiting results that require the traditional real-valued weak stationarity. Since Condition (3) is solely used to guarantee stationarity, the only requirements for S ¨ κ z t are that it has finite elements and that it is invariant with respect to the time parameter t.

For a process x • that follows the mixing model, the corresponding unmixing problem is to uncover the latent process z • by using only the information contained in the observable process x • .

Given a process x • that follows the mixing model and a corresponding solution g : a ↦ Γ ( a − μ ), we refer to A, see Definition 2, as the mixing matrix and we refer to Γ as the unmixing matrix. The objective of the unmixing matrix is to reverse the effects of the mixing matrix. However, under our model assumptions, we cannot uniquely determine the relationship between the mixing and the unmixing matrix. The relationship between the two matrices is Γ A = J, where J ∈ C d × d is some phase-shift matrix, see Lemma 3 in the supplementary Appendix.

Under our assumptions, we cannot distinguish between solutions that contain unmixing matrices that are a phase-shift away from each other. Thus, we say that two solutions g 1 : a ↦ Γ 1 ( a − μ 1 ) and g 2 : a ↦ Γ 2 ( a − μ 2 ) are equivalent, if there exists a phase-shift matrix J such that Γ 1 = Γ 2 J.

We next provide a solution procedure for the unmixing problem. Recall that in the mixing model we assume that the mixing matrix A and the latent process z • are unobservable. Therefore, the solution procedure relies solely on statistics of the observable process x • .

One can apply Theorem 1 to find solutions by first estimating the eigenvalues and eigenvectors of S 0 x t − 1 S τ x t and then scaling the eigenvectors so that the scaling equation is satisfied. In addition, see Corollary 2 in the supplementary Appendix.

The algorithm for multiple unknown signals extraction (AMUSE), [35], is a widely applied blind source separation unmixing procedure. AMUSE and the corresponding asymptotic properties have been previously studied in the real-valued case, see, e.g, [23]. Here, we adopt the estimation part of the AMUSE algorithm. However, our underlying model assumptions are not identical to the ones in [23] and [35]. In this section, we consider the consistency and the limiting distribution of the corresponding unmixing estimator under complex-valued long-range and short-range dependent processes, significantly extending the results given in [23].

Recall that S ˆ τ is the symmetrized autocovariance matrix, and it is, by definition, conjugate symmetric. In addition, recall that the diagonal entries and the eigenvalues of a conjugate symmetric matrix are, again by definition, real-valued. Hereby, the ordering of λ ˆ τ ( j ) is always well-defined. The diagonal elements (eigenvalues) λ ˆ τ ( j ) can be seen as finite sample estimates of λ τ ( j ) . Even though the population eigenvalues λ τ ( j ) are assumed to be distinct, it is possible, due to, e.g., limited computational accuracy, that ties occur among the estimates λ ˆ τ ( j ) . Hereby, in the finite sample case, strict inequalities are not imposed on the estimates λ ˆ τ ( j ) in Definition 4.

Using Lemma 4 of the supplementary Appendix, we can straightforwardly implement the unmixing procedure. The first step is to calculate S ˆ 0 : = S ˆ 0 [ X ], from a realization X, and the corresponding conjugate symmetric inverse square root Σ ˆ 0 : = S ˆ 0 − 1 / 2 . Recall that, by assumption, the components of z • have continuous marginal distributions and the mixing matrix is nonsingular. Thus, the eigenvalues of covariance matrix estimates are always real-valued and almost surely positive. Hereby, the matrix Σ ˆ 0 can be obtained by estimating the eigendecomposition of S ˆ 0 . The next step is to choose a lag-parameter τ and estimate the eigendecomposition S ˆ τ [ X Σ ˆ 0 ⊤ ] = Σ ˆ 0 S ˆ τ [ X ] Σ ˆ 0 = V ˆ Λ ˆ τ V ˆ H . The covariance matrix and autocovariance matrix estimators are affine equivariant in the sense that S ˆ j [ X C ⊤ ] = C S ˆ j [ X ] C H , j ∈ { 0 , τ }, for all nonsingular C d × d -matrices C. An estimate for the latent process can then be found via the mapping ( X − 1 T μ ˆ ⊤ ) ( V ˆ H Σ ˆ 0 ) ⊤ , where μ ˆ is the sample mean vector of X.

In practice, one should choose the lag parameter τ ≠ 0 such that the diagonal elements of Λ ˆ τ are as distinct as possible. Strategies for choosing τ are discussed, e.g., in [9]. From a computational perspective, the estimation procedure is relatively fast. Hereby, in practice, one should try several different values of τ and study Λ ˆ τ .

We emphasize that one can apply the theory of this paper to other estimators as well. That is, under minor model assumptions, the estimators S ˆ 0 , S ˆ τ can be replaced with any matrix-valued estimators that have the so-called complex-valued affine equivariance property, see [16].

The conditions given in Definition 4 remain true if we replace Γ ˆ with J Γ ˆ, where J ∈ C d × d can be any phase-shift matrix. Thus, as in the population level, we say that the estimates Γ ˆ 1 and Γ ˆ 2 are equivalent if Γ ˆ 1 = J Γ ˆ 2 for some phase-shift matrix J.

Justified by the affine invariance property given in supplementary Appendix Lemmas 5 and 6, we can, without loss of generality, derive the rest of the theory under the assumption of trivial mixing, that is, in the case when the mixing matrix is A = I d . See also the beginning of the Proof of Lemma 2.