# The Statistics of EEG Unipolar References: Derivations and Properties

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## Abstract

In this brief communication, which complements the EEG reference review (Yao et al. in Brain Topogr, 2019), we provide the mathematical derivations that show: (1) any EEG reference admits the general form of a linear transformation of the ideal multichannel EEG potentials with reference to infinity; (2) the average reference (AR), the reference electrode standardization technique (REST), and its regularized version (rREST) are solving the linear inverse problems that can be derived from both the maximum likelihood estimate (MLE) and the Bayesian theory; however, REST is based on more informative prior/constraint of volume conduction than that of AR; (3) we show for the first time that REST is also a unipolar reference (UR), allowing us to define a general family of URs with unified notations; (4) some notable properties of URs are ‘no memory’, ‘rank deficient by 1’, and ‘orthogonal projector centering’; (5) we also point out here, for the first time, that rREST provides the optimal interpolating function that can be used when the reference channel is missing or the ‘bad’ channels are rejected. The derivations and properties imply that: (a) any two URs can transform to each other and referencing with URs multiple times will not accumulate artifacts; (b) whatever URs the EEG data was previously transformed with, the minimum norm solution to the reference problem will be REST and AR with and without modeling volume conduction, respectively; (c) the MLE and the Bayesian theory show the theoretical optimality of REST. The advantages and limitations of AR and REST are discussed to guide readers for their proper use.

## Keywords

The family of unipolar references Average reference REST reference Maximum likelihood estimate No memory property## Introduction

This brief communication provides the detailed mathematical demonstrations as well as some new findings that complement the EEG reference review (Yao et al. 2019). That review provides an overview of the state of the art of proposals for the EEG reference. When discussing current issues, it was evidently necessary to summarize the novel statistical approaches to the reference problem which is best described as the estimation of the potential at infinity. We thus build the family of unipolar references (URs), derive a novel maximum likelihood estimator (MLE), formulate a few notable properties and compare them to the related Bayesian estimators for this linear inverse problem first described in (Hu et al. 2018c).

### The Origin of EEG Reference Electrode Problem

### Previous Attempts and Recent Progress

Previous notable attempts to design an optimal reference have been to carry out online recording with respect to a cephalic reference electrode e.g. Cz, Fz, Oz and FCz, etc. and subsequent offline computation of a different reference (re-referencing). Examples are such as the linked mastoids (LM) (Gibbs et al. 1936; Faux et al. 1990), the average reference (AR) (Goldman 1950; Offner 1950), the reference electrode standardization technique (REST) (Yao 2001), and its regularized version (rREST) (Hu et al. 2018c). All of these are URs (Hu et al. 2018b), meaning that all the active electrodes are referenced to a unique reference signal. By contrast, other also proposed the non-URs, such as the bipolar reference recordings (Berger 1929; Niedermeyer and Da Silva 2005) and the scalp Laplacian (Hjorth 1975; Pascual-marqui et al. 1988; Perrin et al. 1989).

It is not difficult to discern that the reference signal of online recording reference electrode or the offline re-references (LM, AR) is just a linear combination to the ideal EEG potentials \( {\varvec{\upvarphi }} \) referenced at infinity. In recent years, AR and REST have been the two of the most widely adopted references. The justification of AR is that if the head is modeled as a layered spherical sphere with the neural currents spreading in a isotropic way, the discrete integral of the potentials over the head surface will be zero (Bertrand et al. 1985). REST is a method to approximately reconstruct the potentials \( {\varvec{\upvarphi }} \), by making use of the forward head model and the equivalent sources model shown in (1) (Yao 2001). A more recent development is rREST which also deals with the denoising problem via the generalized cross validation criterion (Hu et al. 2018c). It is shown as well in the same paper that the use of a population average lead field yields better results than the use of spherical lead field. With this variety of URs to pick from, it is evident that a unified model is needed to analyze interrelations between reference procedures as well as to compare their properties.

### Remaining Problems and Proposals

REST is based on the fact that EEG activities are ultimately generated by the same neural sources no matter what reference is used. The introduction of REST has stimulated an increasing number of comparative studies on how different references affect experimental data analysis (Bonfiglio et al. 2013; Tian and Yao 2013; Kugiumtzis and Kimiskidis 2015; Chella et al. 2016; Mumtaz and Malik 2018). However, we felt that purely empirical comparisons may be incomplete. Several questions remained unanswered. For instances, how can one formulate in a single model all the reference transformations? What this type of model reveals about the connections among the various references? Are all the URs dependent on each other? Is REST an UR which drags along the remaining impacts of previous use of other references? What are the common properties of the URs? Is it possible to achieve the unbiased estimator to the ideal infinity reference? What are the statistical interpretations for AR and REST? Are AR and REST valid from the view of mathematical statistics? These questions came to the forefront during drafting the accompanying review of the EEG reference problem (Yao et al. 2019).

In responding to these questions, we found that UR transformations always reduce the rank by 1 of multichannel EEG potentials referenced to infinity. Thus, estimating the ideal potentials of full rank from the singular reference transforming matrix is an underdetermined or rank deficient linear regression problem (Mardia et al. 1979; Magnus and Heinz 2007). This is therefore an inverse problem of a different nature but related to the source localization. Fortunately, the required tools to attack this problem have already been developed related to Moore–Penrose pseudoinverse of modified matrices by rank one subtraction (Meyer Jr. 1973; Trenkler 2000; Baksalary et al. 2003).

Mathematical notations

\( N_{c} \) | The number of channels | \( N_{s} \) | The number of neural sources |

\( {\mathbb{R}}^{m \times n} \) | The real matrices of \(m\) rows by \(n\) columns | \( rk \) | The matrix rank operator |

\( r \) | The unipolar references (URs) | \( {\mathbf{T}}_{r} \in {\mathbb{R}}^{{N_{c} \times N_{c} }} \) | The UR transforming operator |

\( {\varvec{\upvarphi }} \in {\mathbb{R}}^{{N_{c} \times 1}} \) | The instantaneous EEG potentials with infinity reference | \( {\mathbf{K}}_{\infty } \in {\mathbb{R}}^{{N_{c} \times N_{s} }} \) | The lead field matrix with infinity reference |

\( {\mathbf{v}}_{r} \in {\mathbb{R}}^{{N_{c} \times 1}} \) | The unipolar referenced EEG potentials over all channels | \( {\mathbf{j}} \in {\mathbb{R}}^{{N_{s} \times 1}} \) | The brain neural source currents |

\( {\mathbf{f}}_{r} \in {\mathbb{R}}^{{N_{c} \times 1}} \) | The vector of linear coefficients for URs | \( {\mathbf{1}} \in {\mathbb{R}}^{{N_{c} \times 1}} \) | A vector of ones |

\( {\mathbf{M}}^{ + } \) | The Moore–Penrose inverse of matrix \({\mathbf{M}}\) | \( {\mathbf{I}}_{{N_{c} }} \) | The identity matrix |

\( {\mathbf{M}}^{{\mathbf{T}}} \) | The transpose operator of matrix \({\mathbf{M}}\) | \( {\varvec{\Sigma}} \) | The covariance matrix |

## Demonstration

### General Form of the Reference Problem

**φ**, because the infinity reference is practically unachievable. What one observed is, instead, the referenced data \({\mathbf{x}}\), that may be either the UR recordings \({\mathbf{v}}_{r}\) or the non-UR recordings e.g. the currents by bipolar recordings and the current source density by the scalp Laplacian. Each type of referenced data is a linear transformation via pre-multiplication of the transforming matrix \({\mathbf{T}}_{\text{o}}\) with the EEG potentials

**φ**plus the sensor noise \({\varvec{\upvarepsilon}}\). Thus, the general form of the reference problem is:

**φ**is the ideal potentials with infinity reference supposed to be a deterministic fixed but unknown vector, and \({\varvec{\upvarepsilon}}\) is the non-observable random sensor noise disturbances. Apparently the estimation to

**φ**in solving the EEG reference problem (2) is an underdetermined linear regression problem.

Without loss of generality, \({\mathbf{x}}\) and \({\varvec{\upvarepsilon}}\) are considered to have the multivariate normal distribution. If the sensor noise has an independent identical distribution (IID) across channels, the covariance of sensor noise in the referenced data will be \({\varvec{\Sigma}}_{{{\varvec{\upvarepsilon}}_{\text{o}} {\varvec{\upvarepsilon}}_{\text{o}} }} = \sigma^{2} {\mathbf{T}}_{\text{o}} {\mathbf{T}}_{\text{o}}^{{\mathbf{T}}}\) because the referencing effect is taken on the noise as well during recording (Pascual-Marqui et al. 1994).

### The Family of Unipolar References (URs)

Unipolar reference (UR) is regarded if all the electrodes are referenced to a unique physical reference or a unique virtual reference. The physical reference is usually the electrode (e.g. Cz, Fz, Oz and FCz) placed on the scalp or body surface during online recording setup. The virtual reference is a linear combination of the recordings from all the electrodes, usually obtained during offline processing after the EEG data acquisition. Typical examples of virtual references are the LM, AR and REST.

The family of unipolar references

Unipolar references | \( {\mathbf{v}}_{r} = {\mathbf{T}}_{r} {\varvec{\upvarphi }} + {\varvec{\upvarepsilon}}_{r} \), \({\mathbf{T}}_{r} = {\mathbf{I}}_{{N_{c} }} - {\mathbf{1f}}_{r}^{{\mathbf{T}}}\), \({\mathbf{f}}_{r} \in \{ {\mathbf{f}}_{RR} ,{\mathbf{f}}_{LM} ,{\mathbf{f}}_{AR} ,{\mathbf{f}}_{REST} \}\) | |

Online recording references | Cz, Fz, Oz and FCz, etc. (RR) | \({\mathbf{f}}_{RR} = [0, \ldots ,0,1,0, \ldots ,0]^{{\mathbf{T}}}\) |

Offline re-references | Linked mastoids (LM) | \({\mathbf{f}}_{LM} = [0, \ldots ,0,0.5,0, \ldots ,0,0.5,0, \ldots 0]^{{\mathbf{T}}}\) |

Average reference (AR) | \({\mathbf{f}}_{AR} = {{\mathbf{1}} \mathord{\left/ {\vphantom {{\mathbf{1}} {N_{c} }}} \right. \kern-0pt} {N_{c} }}\) | |

REST | \({\mathbf{f}}_{REST} = {{{\mathbf{K}}_{\infty }^{{ + {\mathbf{T}}}} {\mathbf{K}}_{\infty }^{ + } {\mathbf{1}}} \mathord{\left/ {\vphantom {{{\mathbf{K}}_{\infty }^{{ + {\mathbf{T}}}} {\mathbf{K}}_{\infty }^{ + } {\mathbf{1}}} {[{\mathbf{1}}^{{\mathbf{T}}} {\mathbf{K}}_{\infty }^{{ + {\mathbf{T}}}} {\mathbf{K}}_{\infty }^{ + } {\mathbf{1}}}}} \right. \kern-0pt} {[{\mathbf{1}}^{{\mathbf{T}}} {\mathbf{K}}_{\infty }^{{ + {\mathbf{T}}}} {\mathbf{K}}_{\infty }^{ + } {\mathbf{1}}}}]\) |

It is justified that for a perfect layered spherical head, with neural currents spreading in an isotropic way, the integral of the potential over the head surface is zero (Bertrand et al. 1985; Yao 2017). Thus, the averaged potential over all electrodes may tend to zero and would be suitable as the reference signal.

### Demonstration of REST as an UR

Therefore, REST operator admits the same form of URs defined in (4). While the reference standardization matrix \({\mathbf{R}}_{r}\) is dependent on the prior EEG reference, the REST operator \({\mathbf{T}}_{REST}\) is independent of the specific UR in the EEG data and will be identical whichever \({\mathbf{T}}_{r}\) is adopted in (7) noting that \({\mathbf{f}}_{r}\) disappears in (11). The demonstration of REST as a UR clarifies its relation to other references.

### Properties of URs: No Memory, Rank Deficient by 1 and Orthogonal Projector Centering

We analyzed the family of URs and naturally found some valuable properties of URs summarized as ‘no memory’, ‘rank deficient by 1’ and ‘orthogonal projector centering’.

(1) No memory property

Note that \({\mathbf{f}}_{r}^{{\mathbf{T}}} {\mathbf{1}} = 1\) for \({\mathbf{f}}_{r} \in \{ {\mathbf{f}}_{RR} ,{\mathbf{f}}_{LM} ,{\mathbf{f}}_{AR} ,{\mathbf{f}}_{REST} \}\), this no memory property holds true for the family of URs including both online recording references e.g. Cz, Fz, Oz and FCz, etc. and the offline re-references such as LM, AR and REST.

(2) Rank deficient by 1 property

(3) Orthogonal projector centering property

The readers can refer to the appendix of (Hu et al. 2018c) for the proofs of ‘rank deficient by 1’ and ‘orthogonal projector centering’ properties.

From the “Demonstration of REST as an UR” section, the lead field matrices also have the ‘identity’ property \({\mathbf{K}}_{\infty } {\mathbf{K}}_{\infty }^{ + } = {\mathbf{I}}_{{N_{c} }}\), the ‘rank deficient by 1’ property \(rk({\mathbf{K}}_{r} ) = rk({\mathbf{K}}_{\infty } ) - 1\), and the ‘orthogonal projector weighted centering’ property \({\mathbf{K}}_{r}^{ + } {\mathbf{K}}_{r} = {\mathbf{I}}_{{N_{c} }} - {\mathbf{K}}_{\infty }^{ + } {\mathbf{11}}^{{\mathbf{T}}} {\mathbf{K}}_{\infty }^{{ + {\mathbf{T}}}} /{\mathbf{1}}^{{\mathbf{T}}} {\mathbf{K}}_{\infty }^{{ + {\mathbf{T}}}} {\mathbf{K}}_{\infty }^{ + } {\mathbf{1}}\). Note that the ‘rank deficient by 1’ and ‘orthogonal projector centering’ properties of the URs follow trivially if the lead field matrix \({\mathbf{K}}_{\infty }\) is an identity matrix. However, realistic lead fields \({\mathbf{K}}_{\infty }\) are far from the identity. The biophysical assumptions are the cause of the difference of REST from the other URs.

### Derivation of AR and REST from the Maximum Likelihood Estimate

It is apparent that without the constraint, estimating \({\varvec{\upvarphi }} \in R^{{N_{c} \times 1}}\) from \({\mathbf{v}}_{r - } \in R^{{(N_{c} - 1) \times 1}}\) is underdetermined.

Note that \({\mathbf{K}}_{\infty }\) is with one additional channel to \({\mathbf{K}}_{r - }\) indicating the interpolation function of REST. This function can recover the full recordings over all channels from the recordings with the reference channel missing or even be generalized to interpolate bad channels that have been rejected.

### Derivation of AR and REST from the Bayesian Theory

The REST assumes potentials generated by the Bayesian theory with the assumption of independent neural sources (Hu et al. 2018c). The influence of other covariance matrices for neural sources on REST type estimators is under study.

## Discussion

In this brief communication, the general form of the EEG reference electrodes problem is understood as a linear transformation to the potentials \({\varvec{\upvarphi }}\) referenced at infinity, that maybe either the URs or non-URs e.g. the bipolar recordings and the scalp Laplacian; the common structure of URs is recognized with unified notations; it is the first time to show the interpolation function in solving the reference problem and demonstrate REST as an UR. This allows us to study a generalized family of URs. Also, valuable properties of the URs family are summarized to establish the interrelations.

The most surprising property is ‘no memory’ indicating that URs works independently without the consequences of the URs already applied. It also means one can always safely re-reference the EEG/ERP recordings with different URs but not worry about if re-referencing multiple times will accumulate artifacts. Any two of URs can be transformed to each other and all the URs are taking effect independently. Before applying a different UR, one had better check if the present data is with UR. Transforming from non-UR to UR will damage the dataset though it is no problem to transform the data within the family of URs. This property validates that an UR can erase the effects of the other URs and it is therefore safe to apply UR multiple times. The ‘no memory’ property deserves to be kept in mind in the data preprocessing by EEG-ERP researchers.

The significant property of ‘rank deficient by 1’ implies that the URs always reduce the full rank by 1 of the ideal EEG potentials referenced at infinity. The lost one rank is because the reference signal subtracted from all the channels is a linear combination from the activities over all the channels referenced to infinity. This property tells us the impossibility of obtaining the unbiased estimator to the ideal EEG potentials \({\varvec{\upvarphi }}\) if no extra constraint is introduced. However, different URs indeed have different biases, relying on how much prior information is incorporated on estimating the ideal potentials \({\varvec{\upvarphi }}\) referenced at infinity.

The nontrivial property of ‘orthogonal projector centering’ as the AR is proved from theorem of Moore–Penrose pseudoinverse in the case of rank one modification. This property is made full use of in the derivation of AR from both the MLE and the Bayesian theory. Similarly, the lead fields engaged in REST has the properties of ‘orthogonal projector weighted centering’. Together with ‘identity’ and ‘rank deficient by 1’ properties, REST is demonstrated as an UR. It is by means of this ‘orthogonal projector (weighted) centering’ property that demonstrates that AR and REST does not depend any specific UR previously applied on the EEG data and both AR and REST can be finally derived.

Among the family of URs, AR and REST are the two main contenders. This communication shows that both the MLE and the Bayesian theory approaches allow the derivations of the AR and REST estimators. One approach is to derive AR and REST via MLE with linear and quadratic constraints respectively as general linear regression model (Magnus and Heinz 2007). AR is the best unbiased linear estimator given the linear constraint that the sum of the EEG potentials over all the channels is zero. By contrast, REST minimizes the quadratic constraint of a linear combination of equivalent sources. An alternative and more flexible approach follows from the Bayesian theory. AR was derived by assuming a priori IID covariance of multichannel EEG recordings; and REST was derived from the volume conduction and a priori IID covariance of neural sources activities.

From the view of MLE, AR is theoretically correct if the constraint of discrete integral of potentials as zero holds true. This is valid no matter which UR one starts from and the best unbiased estimator would be AR. The integral of potentials partly relies on the electrodes coverage and density (Bertrand et al. 1985; Nunez 2010). However, a recent work showed that the performance of AR is not closely related to the electrode density which is different from the common understanding to AR based on its zero integral assumption and coverage is a more important factor than the electrode density (Hu et al. 2018b). From the view of the Bayesian theory, the AR is essentially solving a generalized linear inverse problem to estimate the potentials at infinity. With the prior assumption of statistical IID across multichannel recordings, whatever the UR is, the minimum norm estimator will be the AR. The priori of IID covariance of potentials referenced to infinity is surely false, since the volume conduction effect is neglected. Also, one has to be sure the EEG data at hand is with UR before applying AR, limiting its ability in the general use (Hu et al. 2018a).

From the view of MLE, the quadratic constraint indicates that REST does not fully depend on the source configuration but rather on the effect it produces at the scalp by the equivalent source that represents infinite source configurations. When one assumes that EEG data are generated by brain sources, then in theory rREST (REST) is an optimal estimate of the potentials at infinity. REST allows the possibility that the number of channels in forward calculations can be more than those in estimating the equivalent source. With the additional channels, channels missing or rejected as ‘bad’ can be recovered with the interpolation function of REST.

It is worthwhile to emphasize that from the view of Bayesian theory, in REST the source distribution only enters the estimation as a specification of the prior; and REST is the MAP estimator for which the equivalent source approach offers the protection against sources mis-localization. The goal of REST is not to find the actual sources which are not actually necessary. rREST as the extension of REST has the ability of general application. The Eqs. (31)–(33) is rather general irrespective of which type of \({\mathbf{T}}_{\text{o}}\) applied to \({\mathbf{x}}\), meaning that the rREST can adapt to the non-UR recordings e.g. bipolar recordings, and scalp Laplacian as well.

As shown in (Hu et al. 2018c), two possible limitations of REST are the assumptions of noiseless data and the use of a spherical head model. A study with simulation and large dataset of EEG recordings was carried out, showing that unless the EEG data is with extremely high SNR, REST with spherical lead field can be used without the expense of building the realistic head models. Alternatively, an averaged lead field over a population of samples and denoising by the criterion of the generalized cross validation (GCV) should be used in the rREST practice (Hu et al. 2018c). The readers can refer to https://github.com/ShiangHu/LeadField-Pipeline for the easy runnable pipeline of the realistic head models and https://github.com/ShiangHu/Unified-EEG-reference-rREST for the use of rREST.

To conclude, this brief demonstration shows that all the common references including REST can be formulated into a family of URs with unified notations, some properties of URs e.g. ‘no memory’, ‘rank deficient by 1’ and ‘orthogonal projector centering’ may be valuable on helping the reference practice, both the MLE and the Bayesian theory can derive the AR and REST, and together with the interpolation function of REST providing the novel understanding and the statistical evidences for their use in the future EEG and ERP practice.

## Notes

### Acknowledgements

This study was funded by the National Natural Science Foundation of China (Grant No. 61871105, 81861128001), the CNS program of UESTC (No. Y03111023901014005) and the 111 project B12027.

### Author Contributions

SH developed the theories and wrote the manuscript; DY, MB and QY made valuable comments; PV posed the idea of likelihood estimate and polished the manuscript.

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