A complex-valued linear mixture model is considered for discrete weakly stationary processes. Latent components of interest are recovered, which underwent a linear mixing. Asymptotic properties are studied of a classical unmixing estimator which is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag

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 [

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 [

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), [

The paper is structured as follows. In Section

Throughout, let

We use the following notation for the multivariate mean and the (unsymmetrized) autocovariance matrix with lag

In the case of univariate stochastic processes, we use

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

The

Let

Let

We distinguish between complex- and real-valued Gaussian distributions with the number of given parameters —

The classical central limit theorem (CLT) for independent and identically distributed complex-valued random variables is given as follows. Let

Lemma

In our work, we utilize Lemma

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.

The

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

Condition (3) is included in Definition

For a process

Let

Given a process

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

We next provide a solution procedure for the unmixing problem. Recall that in the mixing model we assume that the mixing matrix

One can apply Theorem

The algorithm for multiple unknown signals extraction (AMUSE), [

Let

Recall that

Using Lemma

In practice, one should choose the lag parameter

We emphasize that one can apply the theory of this paper to other estimators as well. That is, under minor model assumptions, the estimators

The conditions given in Definition

Justified by the affine invariance property given in supplementary Appendix Lemmas

We next consider limiting properties of the finite sample solutions. Note that the finite sample statistics and the sampled stochastic process

Hereby, under the assumptions of Lemma

By Theorem

Theorem

In this section, we consider a class of stochastic processes that satisfy the Breuer–Major theorem for weakly stationary processes. The Breuer–Major theorem is considered in the context of Gaussian subordinated processes, see, e.g., [

We emphasize that Gaussian subordinated processes form a very rich model class. For example, recently in [

We next give the definitions of univariate real-valued Hermite polynomials and Hermite ranks. We define the

The Hermite rank for a function

Let

Note that in the one-dimensional setting, the Hermite rank

The proof of Theorem

We next present the following assumption which enables us to find the asymptotic behavior of the unmixing matrix estimator using the Breuer–Major theorem.

Note that the finite-dimensional distributions of the

We again want to emphasize that a wide class of stochastic processes satisfy Assumption

We are now ready to consider the asymptotic distribution for the unmixing matrix estimator under Assumption

Note that it is possible to present Theorem

In this section, we provide examples where the convergence rate of the unmixing estimator differs from the standard

In comparison to Assumption

If the autocovariance functions

We say that a

Note that the definition of long-range dependence varies in the literature. For details on long-range dependent processes and their different definitions, we refer to [

There is a large literature on limit theorems under long-range dependence. See, e.g., [

In view of Proposition

We stress that Assumption

In view of Proposition

In the presence of long-range dependency, the mean estimator

In order to obtain the limiting distribution for

We remark that it might be that most of the elements vanish. Indeed, the coefficient

The restoration of images, which have been mixed together by some transformation, is a classical example in blind source separation (BSS), see, e.g., [

In our example, we consider colored photographs and conduct the example using the statistical software R ([

Original images

We apply a bijective cube to unit sphere transformation for every color triplet. Then, we use the well-known stereographic projection, which is an almost bijective transformation between the unit sphere and the complex plane. The stereographic projection is bijective everywhere except for the north pole of the unit sphere. For almost all photographs, we can choose the color coordinate system such that no pixel has a color triplet located on the north pole. This holds for our example and we can apply the inverse mappings in order to get back to the color cube surface.

We then have a single complex-number corresponding to every pixel of the color corrected images and we can present the images in a

Mixed images

We then apply the unmixing procedure presented in Section

Unmixed images using the AMUSE procedure with

We tried several different lag parameters and the best performance, in the light of the MD index and visual inspection, was attained with

Unmixed images produced by equivalent solutions

Comparing the original color corrected images in Figure

Both directions of the claim follow directly from the multivariate version of the continuous mapping theorem. Note that the mapping

The corollary follows directly by applying Lemma

Let

For the second part of the proof, let

In order to simplify the notation, we denote

Next, let Equations (

Under the assumptions of Definition

The second part of the proof follows directly from Theorem

Note that Lemma

By Lemma

By Lemma

The assumption

Using part (i) and Slutsky’s lemma, we get that the inverse is uniformly tight, since

Denote

Under trivial mixing, we have that

We next denote

Next, let

After the diagonal entries have been rotated to the positive real axis, the rotated diagonal elements are equal to the corresponding moduli, that is,

The claim of the lemma can then be written as

Let

The left part of Equation (

Recall that

In order to incorporate the shift

The unsymmetrized autocovariance matrix estimator with lag

We can reformulate the estimator as

The symmetrized autocovariance estimator can then be expressed as

Let

The off-diagonal element

Under Assumption

Recall that the real part of the

With

Using independence of the processes

Similarly as in the proof of Theorem

Following the proof of Theorem

Hence, due to independence, it suffices to prove that, for every

The authors would like to thank Katariina Kilpinen for providing the photographs to Section