Linear multivariate models for physiological signal analysis: theory

ELSEVIER Computer Methods and Programs in Biomedicine 51 (1996) 85-94

Linear multivariate models for physiological signal analysis: theory

Ilkka Korhonen*“, Luca Mainardib, Pekka Loula”, Guy Carrault”, Giuseppe Basellib, Anna Bianchib

a VTT Information Technology, Multimedia Systems, Tampere, Finland bPolytechnic University of Milan, Department of Biomedical Engineering Milan, Italy

‘Universite de Rennes, Laboratoire Traitement du Signal et de L ‘Image, Rennes, France

Abstract

The general linear parametric multivariate modelling concept is presented. This model combines a variety of different kinds of multivariate linear models. The concept of partial spectral analysis is derived from the general model. Some emphasis is laid on the causality demands of the model, and it is shown that the classic strictly-causal structure must be abandoned in order to utilise the modelling in many practical situations. Two special sub-class models are described in detail: the multivariate autoregressive model and the multivariate dynamic adjustment model. Furthermore, time-varying modelling is considered. The modelling of the real system is presented on a general level as a system identification cycle. The application of the methods to real physiological data is presented in the companion paper.

Keywords: Multivariate; Linear; Modelling; Parametric; Spectral analysis; Dynamic

1. Introduction

Multivariate processes are perceived when instead of observing just a single process y(k), we observe simultaneously several processes y,(k),..., y,(k), which are more or less related to each other via certain joint probabilities. From the methodological point of view, the real system producing these separately observed processes

*Corresponding author.

may be heterogeneous (e.g. heart rate (HR), blood pressure (BP) and respiration CR)), or, more or less homogeneous (e.g. multichannel electroen- cephalogram (EEG)). So considered, the multivariate processes are commonly observed, for example in patient monitoring during intensive care or other daily clinical practices in medicine.

Conventionally, all the observed processes are studied separately as univariate processes. Since univariate methods do not provide any information on the joint probability distributions of different processes, the observed data are not fully

0169-2607,/96/$15.00 0 1996 Elsevier Science Ireland Ltd. All rights reserved. PII SO1 69-2607(96) 01764-6

8f) 1. Korhonen et al. / Comptraer Methods nrtd Prosyrams in Biomedicine 51 (1996) 85-94

utilised. To achieve a more complete description of the system behaviour, we need methods to estimate these joint probability distributions. This brings us to multivariate modelling, the most im- portant feature of which is to provide information on the interrelationships between the multiple variables observed.

The concept of multivariate modelling is a very broad one, comprising all the models which utilise the joint information of two or more variables. Hence, complete management of the topic within one paper is not possible. In the present paper, we limit our consideration to discrete-time linear multivariate parametric models which describe the second-order stochastic properties of the process variables. The restriction is based on the great practical importance of discrete-time, linear methods in digital computer applications, and on the observation that these linear models are able to fruitfully describe the complex interactions among different biological signals in many experimental conditions, for example in fields concern- ing the cardiovascular system [l-8] and neuro- sciences [9- 111.

The aim of the present paper is to present state-of-the-art multivariate modelling techniques with special focus on physiological signal analysis. The methodology and principles for linear multivariate parametric modelling are presented first. A general linear time series model, which brings together most of the commonly used linear multivariate models, is introduced. The model properties, the principles for the model utilisation, and the model identification and validation are then addressed, with particular attention to the problems involved with the analysis of signals originating from physiological systems. In the companion paper [12], three application examples of the given modelling technique are presented,

2. Linear multivariate models

2.1. The general structure

Usually the analysis of the system is restricted to the second-order stochastic properties, that is, mean, variance, auto- and cross-correlation (or, in frequency domain, auto- and cross-spectral) func-

tion, in which case the model only needs to deal with these properties of the system. This is characteristic of the parametric models presented in this paper.

In the literature, there are various general multivariate model structures, for example state-space models and difference equation models. However, all these presentations can be thought of as special cases of the time-series presentation of a linear system [13]. It can be shown that any discrete-time, linear, time-independent, strictly causal system can be modelled by a general model [131

B(q) A(q)y(k) = F(q) ---au(k) + - c(q) e(k)

D(q) (1)

where

MA ‘4(q) =z+ c a(k)qm”

x= I

R-l,, B(q) = c b(k)q-k

k=U

MF F(q) = c .f(k>q-k

k=U

(2)

(3)

(4)

MC C(q) = z + c C(k)q-k (5)

k= I

D(q) =I+ b; d(k)@ /i= I

(6)

Above, q represents the unit delay operator, k is the sample number, and Z is the identity matrix. y(k) is the observed output vector, u(k) is the observed input vector, and e(k) is the non-measurable white noise disturbance vector with the same dimension as y(k). MA,B,C,F,D are the orders of the corresponding model terms. y(k) and u(k) represent the observations of the system variables, and e(k) represents the stochastic part of the process (Fig. 1). e(k) is thus considered as the part of the system that cannot be predicted by the model: the residual, or prediction, error [13]. In Eq. 1, the time-independence of the second

I. Korhonerl et al. /Computer Methods and Programs in Biomedicine 51 (1996) M-94 87

Fig. 1. A general time-series model of a linear system. y and r~ represent the measured output and input signals, respectively. e represents the non-me~urable disturbance, for example noise or the effect of other, non-measurable variables A, B. C, D and F represent the model coefficient matrices (Eq. 1 ).

order properties means that, besides the system properties, u(k) and e(k) are also jointly stationary in second-order sense (i.e. their cross-spectral properties are time-inva~ant).

In the model parameter identification, t.he parameters of the transfer functions are calcu- lated on the basis of the observed data, and the probability density function of e(k), ~,CX>, is de- termined in terms of a few numerical characteristics, typically mean and variance 2131. When the parameters of the model are optimally identified on the basis of the informat~e data of length IV, it converges to the real system in the sense [13]

uniformly in LCJ Co = 2rrf/f?, f, being the sampling frequency), as N and model order M (N > > M) tend to infinity. In Eq. 7, C represents the true transfer function from the input to the output, and II the true transfer function from the disturbance to the output. Thus, the model is capable of describing the second order properties of the real system well, within the limits set by data length and quality, regardless of the structure of the real system.

2.2. The analysis of dynamic interactions of the system

The analysis of the dynamic ~teractions in the system by using the model in Eq. 1 (or its sub- model) is based on the model properties according to Eq. 7: the model is capable of describing the system total transfer functions. These transfer functions are usually studied in the spectral domain, which gives a more direct physical interpretation of the system characteristics.

Assuming the input u(k) uncorrelated with the disturbance e(k), the effect of u(k) on the spectrum of the output y(k), L!$,:~,(w), is

S,,,,,(w) = T,(e’“)S,,(w)T,(e’“)* (8)

where * denotes matrix conjugate transpose, Sv:r,(~l is the spectrum of the ~(~~, and

?;;(e’“) = B(e’“)

A(e’“)F(e’“) (9)

is the total transfer function matrix from the input to the output (G refers to Eq. 7). If the inputs z&k1 are mutually unco~elated, S,,(o) be- comes diagonal, and the effects of each u;(k) may be studied separately by partial spectra

S!,:,,,(w) = T,(e”“)S,,,(w)T,(e’“)* (101

where S,,,(w) is the input spectral matrix where other elements are zero except when t2 = m = i (m and n being the column and row indexes, respectively). This presentation aflows one to study the effect of each input to the spectrum of the output. To study the (squared) transfer functions, the input spectrum in Eq. 10 may be re- placed by the spectrum of the white signal with power equal to the power of u,(k).

Similarly, the effect of the disturbance e(k) on the output spectrum is

S (w) = T y .c M (e’“‘)rT (I?“‘)” - fi (11)

88 I. Korhonen et al. /Computer Methods and Progmms in Biomedicine 51 (1996) 85-94

where 2.3. Causality of the model

q&+“) := C(eiw) A(e’“)D(e’“)

(12)

C is the variance-covariance matrix of the white noise sources e(k), defining the spectra of the white noises. Again, if e(k) are mutually uncorrelated (C is diagonal), their effect may be studied individually by noise conditioned spectra, or, partial spectra [141:

S,:,$w) == TH(e’w)C,TH(e’w>*

where the elements of Zi are zero, except for n = m = i. Correspondingly, SyZei(ti) is that part of the output spectrum that gets its origin purely from the noise source e,(k). Because of the diagonal C (uncorrelated e(k)) there exists a one-to- one correspondence between the uncorrelated variability of each y,(k) and ei(k). Hence, the effect of ej(k) computed in Eq. 13 may be inter- preted as the effect of y,(k) on the other variables. This allows the use of noise conditioned spectrum in the analysis of system interactions.

To analyse the relative importance of each input or disturbance for the variability in each signal, the normalisation of the partial spectrum may be utilised. This yields the concept of noise source contribution (NSC) 111

Sjj:iC”)

bcw) = sjj( W)

I I

(14)

where Sj,i,.j(w> is the partial spectrum of Yj(k), originating from e,(k), or ui(k), and Sjj(W> is the spectrum of yj(k). NSC states how great a share of the total variability of the signal originates from other signals, as a function of frequency. To study the extent of NSC in a frequency range from wi to 02, we may utilise NSC ratio (NSCR) [14] which states how much of the signal power in the frequency range from o, to wZ originates from the other variable.

The strict sense causality of the model in Eq. 1 means that: (i) the system does not exhibit imme- diate (zero-delay) or negative delay transfer functions between the output variables y(k); and (ii) there cannot be negative delay transmission from the input u(k) to the output y(k). In physical terms this means that the response cannot occur before the cause. The causality in strict sense yields unambiguousness of the model - there cannot be two models which have the same input and output but different transfer functions, in the sense of the second order properties. Hence, no presumptions of the system transfer function are needed in the modelling, but all the model properties derive from the observed data implic- itly. The linear strictly causal model presented can describe all the correlations of the y(k) to the u(k- i), i = O..maxW,,M,), and to the y(k -i>, i = l...M,, as causal relationships. If the outputs y(k) exhibit some zero-delay correlation not arising from the u(k) or y(k -j), j > 0, the causal relationships may not be judged from the data, and this correlation is modelled by the joint properties of e(k), i.e., in correlation between e(k).

In practice, for example in cardiovascular dynamics, simultaneous components of variability in the signals involved are common [ 1,2], which yields the correlation of e(k) in the model. On the other hand, most of the tools for the analysis of the system dynamics are based on the assumption of uncorrelated e(k) (Section 2.2.) [l-3]. To make these tools accessible, the correlation between ei(k) and ej(k), i + i, must be shifted to the model transfer functions. As a consequence, the definition of strict sense causality must be abandoned. This is done by adding the zero-delay transfer paths to the model which yields

A(q) = F a(k)q-k (15) k=O

where a(O) is not necessarily diagonal. Corre- spondingly, the same changes may be made for C(q) and D(q). As a consequence, the model is

I. Korhonen et al. /Computer Methods and FWgrams in Biomedicine 51 (1996) 85-94 89

no longer unambiguous. To make the model iden- tifiable, the direction of the instantaneous transfer paths must be pre-defined, keeping in mind that no closed loops at zero-delay are allowed UJ- While defining the direction of these instantaneous (or faster than sampling period) transfer mechanisms, a priori knowledge of the system may be utilised. After these definitions and model identification, the strength of the instantaneous transfer as well as transfer in other delays can be studied by utilising the standard methods.

Similarly, if EC(~) exhibit mutual correlation, the correlation may be shifted to the model by modelling the u(k) with special structures, for example by shifting some u,(k) to the y(k) and utilising the method presented above.

2.4. Some multivariate models

Equation 1 gives a general model structure which as it stands cannot reasonably be applied. This is due to its high computational complexity. Depending on what terms are included in the model, this generalised model has several sub- types which are more commonly applied in modelling (Table 1). Furthermore, the individual terms may be further partitioned and these partitions may have different orders, which leads to cus- tomised models [3]. In the following pa~~ap~, some special cases of the general parametric models are introduced.

2.4.1. Multivariate autoregressiue model In the case when the system input u(k) is not

known, the multivariate autoregressive (MAR)

Table 1 Some common linear model types

model is a convenient choice for the model structure, since it considers every variable as both an input and an output variable, enabling the analysis of system dynamics (Fig. 2) [1,2,4,5f. MAR model structure provides a simple structure with wide flexibility, and it can be identified with the standard and numerically effective algorithms, for example the Levinson algorithm [IS]. MAR model can be expressed as

Afq)y(k) = e(k) (16)

As in Eq. 12, MAR model may be completely presented in a filter form

y(k) =A(q)-‘e(k) *

y(k) = T,(qMk) (17)

Thus, MAR model identifies a mult~ariable filter, which produces the output of the system, y(k), from the white noise sources, e(k). To have a different view of the model structure, Eq. 16 can be written in the form

y(k) = - C a(i)y(k - i) -t e(k) i=O

It can be seen that MAR model can be considered as a one-step-ahead prediction model, where the present values of y(k) are predicted by the past values of all the signals involved, and by the stochastic residual. The present value of each

Polynomials used in Eq. 1 -___ A I3 c AB AC AD ABC BF BFDC -~

Abbreviation

AR FIR MA ARX ARMA ARXAR,‘DA ARMAX OE l3J

Name of the model structure

Autoregressive Finite impulse response Moving average Autoregressive with exogeneous input Autoregressive moving average Dynamic adjustment mode1 Autoregressive moving average with exogeneous input Output error Box-Jenkins

90 1. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1996) 85-94

(a)

(b) el e2

eL

Fig. 2. (a, MAR model of a linear system of three variables. A’s represent the linear transfer functions estimated by the model, and e’s the unknown disturbance to each variable.

MAR model enables transfer paths in all directions between all the variables involved.(b) A multivariate filter presentation

of the I.-variable MAR model.

disturbance, e,(k), affects without delay only the present value of the corresponding y,(k). Hence, each driving noise source e;(k) is unique to each output y,(k). Without a loss in generality, e,(k) may be assumed to be uncorrelated (if they are not, they can be made uncorrelated, see Section 2.3.). From this, it follows that the unique effect of the variable y,(k) on the system can be assessed by studying the effect of e,(k) in the model (Eq. 11).

2.4.2. Multivariate dynamic adjustment models Multivariate dynamic adjustment (MDA) mod-

els [3,6] have the general structure of an MAR network of interactions fed by monovariate AR noises at the level of each signal in place of the white noises that feed an MAR process. MDA models may be expressed as

A(q)y(k) =D(q)F’e(kl (19)

where the D(q) is diagonal, thus the e,(k) feeds the MAR network only at the y,(k). The denomi- nation of MDA models is given by analogy with the well-known single-output dynamic adjustment (DA) models, or ARXAR models, in which an AR noise and an exogenous (X1 input feeds an AR loop [16]. Indeed, by opening the loops of the MAR network, an MDA model is divided into L (L = number of signals) DA models in which one of the signals is taken as single-output while the other L-l ones act as exogenous inputs.

The main feature of an MDA model is its capability to detect and classify different rhyth- mogenic sites by associating to each oscillation different types of poles. In fact, the MAR network is formed by finite impulse response (FIR) filters which represent the causal interactions among the signals; so the closed-loops of the network generate poles which are the zeroes of the MAR network determinant. In an MAR model, these are the only type of poles recognised, which, therefore, are able to enhance but not classify rhythms. In an MDA model, in contrast, these poles are distinguished from those of each AR input.

Thus, L + 1 classes of poles are recognised: the closed-loop poles (cl-poles), relevant to the closed-loop interactions of the MAR network; and the poles relevant to the L AR inputs. The former type represents resonances of feedbacks and regulation mechanisms represented by the model, while the latter reveal mechanisms exter- nal to the model and indicate which signal is affected first [7].

The identification of the MDA model transfer functions is obtained through a direct method [3] which minimises a scalar function of the prediction error matrix C, i.e., its determinant. ‘, is required to have some properties: in particular, it has to be diagonal in order to preserve the uncorrelation between input noises also at zero-lag (Section 2.3.). Such a property allows one to minimise C by minimising each element on its diagonal (i.e., each single-output prediction error variances). In other words, the model prediction of each output can be optimised separately, by virtually opening the MAR network and mod-

I. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1996) 85-94 91

elling each output individually. In such a way, we identify L single-output linear systems which have the structure of a single-output DA model (ARXAR model):

MA y,(k) - - c a,,(i)y,,(k -i)

i= I

- c ya,,,(i)y,Jk- i>

mil i=l (20)

+ u,(k)

MI) u,(k) = - c d,,(i)u,(k - i) + e,(k)

i= I

where MAi, MA,, MD are the orders of the corresponding model parameters. The correctness of the prediction has to be assessed a posteriori in order to verify the assumptions made on the characteristic of C; the whiteness of each noise, uncorrelation between each pair of input noises, and diagonal@ of Z are therefore tested.

After MDA identification it is possible to obtain a quantitative evaluation of the different rhythm sources by means of a multivariate spectral decomposition procedure. The spectrum of one of the signals is first divided into partial spectra each due to one of the L inputs. Next, each partial spectrum is decomposed by means of a residual method [6,171 into components relevant to its poles, which are both the MAR network poles and the poles of the input.

2.4.3. Time-varying models Depending on the second order stationarity of

the system, the model may or may not be time-dependent. The most usually employed methods are the time-invariant ones which are based on the assumption of the system to be jointly (second- order) stationary during the observation period. However, in the case that the stationarity condi- tion is not met, the methods may be implemented as time-variant ones. The dynamic, or, time- dependent, multivariate models are usually mod- ifications of the traditional time-invariant ones [5, 111.

When the considered multivariate model has MAR form (Eq. 16) [5], several well-established monovariate AR algorithms of recursive identification can be extended to the identification of multivariate input/output systems. The recursive identification makes A(q) time-dependent adjust- ing its values in response to dynamic changes in the signals or in their mutual relationships. At each time instant k, a new estimate of A(q) may be obtained according to

A(q,k) =A(q,k - 1) +K(k)e(k) (21)

where K(k) is the gain of the algorithm and it is analogous to the Kalman gain vector for Kalman adaptive filter [18]. The updating procedure for K(k) depends on the selected algorithm. For the recursive least square algorithm, K(k) is obtained by minimising the following figure of merit

J= t &“Ze(kMkf (22) k= --XI

where w (0 < w < 1) is exponentially weighting the prediction errors and gives more emphasis to the current errors and progressively forgets the past terms, making the model able to adapt itself to changing conditions. Thus, the model may be utilised in monitoring the changes in system dynamics over time.

2.5. ZdentiJcation and validation of the model

Multivariate modelling may be considered as a system identification task. The purpose of the model determines the level which has to be reached in the identification procedure. In general, the identification procedure may be depicted as in Fig. 3. This procedure may be roughly divided into four main steps, which are all guided, on the one hand, by the prior knowledge of the system to be analysed and, on other, the final goal of the identification.

2.51. Eqeriment design and data collection The basis of the modelling is the informative

experimental data, which contains the system dynamics. The quality of the data is usually the main restraint of the identification procedure,

1. Korhonen et al. /Computer Meth0d.v and Programs in Biomedicine SI (1996) 85-94

Prior Knowledge

Choose Model 4

structure

v +.

Substitute Variables to Model

t OK:UseIt!

Fig. 3. A schematic presentation of the modelling cycle [13].

and good models cannot be obtained with poor data. Proper experiment design, based on strong physiological knowledge, establishes the base of the system identification procedure.

2.5.2. Model structure selection This step involves the selection of the model

family (i.e., MAR, MDA, etc.), the classification of the variables (input, output) and, finally, the selection of the model order. The optimal model order allows a good description of the data, seen as representative of the process, without too good fitting to the actual samples which characterise the particular (noisy) realisation in addition to the generating stochastic process. To guide the model order selection, several objective criteria have

been proposed [18]. They include Akaike’s Infor- mation Criterion, the Rissanen criterion for minimum description length, and Parzen’s criterion [19,20]. However, these criteria do not necessarily behave optimally with real data [15,18]. Hence, in the model structure selection, some subjective decisions and utilisation of a priori knowledge (e..g., physiological knowledge, heuristic/analyti- cal models, etc.> must always been involved.

2.5.3. Identification of the model parameters Identification of the model parameters involves

the calculation of the model parameters to fit the data by minimising some figure of merit, usually the power of the prediction (residual) error e(k). A variety of numerical methods for different model structures are presented in the literature [3,13,15,18].

2.54. Model validation Model validation involves two main levels, the

numerical validation and the final validation. The numerical validation includes the inspection of whether the presumptions made were justified, i.e., stationarity, whiteness and independence of the residuals, etc. To do this, a variety of statistical tests exist [13,21]. Furthermore, one may com- pute the prediction ratio, which tells how much the model can explain the variability of the signals [13]. These assessments aim to provide objective criteria on whether the model has the capability to resolve the system dynamics. At this level, nothing is stated about the physiological significance of the model.

The final validation aims to resolve if the model is good enough for its purpose. The real implementation of this step depends on the purpose of the modelling, and in general no objective criteria exist apart from ‘common sense’. For example, in the modelling of the cardiovascular dynamics, the aim is to resolve some physiologically relevant information. Thus, the proposed model may be validated against the physiological knowledge: if certain known phenomena are depicted by the detailed analysis of the model, it may be considered a good model, otherwise it may be considered a bad model. In contrast, for example in

1. Korhonen et al. /Computer Methods and Programs in Biomedicine 51 (1996185-94 93

multichannel data segmentation, the performance of the model in rupture detection is a convenient measure for the model goodness.

The steps presented are usually repeated in an iterative manner, and the real implementation of the system identification scheme varies from application to application. The main ideas to note from above are that (i) the real identification procedure is a complex task involving many related sub-tasks; (ii) physiological knowledge must be strongly involved in order to draw relevant results.

3. Discussion

In this paper, multivariate linear parametric signal models, capable of describing the second- order properties of a causal linear system, were presented. The bases of the presentation were the concepts of the general linear model and the partial spectrum, which were introduced in order to bond together the variety of existing linear multivariate parametric modelling methods. Two special model structures were highlighted: MAR and MDA models. The MAR model structure provides a simple basic model which enables analysis of dynamic interactions between the different variables. This model may be efficient~ utilised, for example in the analysis of cardiovascular dynamics both in static [1,2] and dynamic [51 conditions. The MDA model structure offers a more detailed model which provides access to single transfer paths between the variables included. This structure has been used in the analysis of the origin of different cardiovascular rhythms [3,6-a]. These examples show that the methodoIo~ presented makes it possible to obtain extra information about cardiovascular control system that is not obtained with traditional methods. The appli- cability of the methods to the analysis of different physiological signals is presented in more detail in the companion paper [21].

Characteristic of the presentation was the signal analysis point of view. Signal models are not to be considered models of the physiological systems but models of the signals - the output real~ations of the system under study. Thus, the inter-relationships described by the models are to

be considered features of the signals, not necessarily the system. As these signals are closely related to the behaviour of the real system, the models built for the signals may be used in the analysis of the real physiological system, for example in the tasks of decision, detection, classification, or interpretation, Thus, they may have the power to detect and describe the pathologic phenomenon of the system, but lack the power to explain the origin of the abnormality.

The difference of principle between the signal model and the real system is of great importance. In practice great care should be taken to keep the difference to a minimum. This is achieved by acquiring maximally representative and informative signals, and, by building optimum models to depict those signals. In the data collection phase, phase errors, delays in signal transmission, and erroneous data samples (noise), may cause the signal model achieved to significantly differ from the real system and its behaviour f13f. In the model identification phase, the model structure selected must be capable of depicting the system dynamic sufficiently well, while being numerically robust. In the validation phase, good sense should be involved in selecting proper measures for the goodness of the model. It can be con- cluded that the importance of the whole system identification cycle has to be emphasised in order to efficiently utilise the methods presented.

Due to the advanced state of linear methods and their analysis tools (e.g., concept of spectrum, or, transfer function), including efficient digital computer applications, they are able to provide clinically valuable, relevant, and interpretable information on the physiological systems (e.g., sym- patho-vagal balance, baroreceptor sensitivity, etc.). This has been shown in innumerable stud- ies. Thus, their use is clinically strongly justified. However, physiological systems may exhibit non- linear behaviour [22-241. Due to this difference in the nature of the linear model and non-linear reality, linear models may be deficient in extract- ing all the available information from the observation data. Novel methods, like non-linear models and artificial neural networks, do not implic- itly suffer from this rest~~tion, and thus provide some potential to enhance the utilisation of the

94 1. Korhonen et al. /Computer Methods and Prqrams in 3~o~dici~~ 51 (1996185-94

information content in the data. However, due to the variety of probIems faced in the use of non- linear methods (e.g., interpretation of the results, numerical problems, definition of the type of non-lineari~, etc.>, they do not seem to be capable of superseding the linear methods. Instead of this, non-linear methods may be utihsed in addition to the linear ones, as parallel methods to optimally utiiise the information content of the data. Still, the linear methods, for example those presented in this paper, will retain their position as the primary tool in physiological signal analysis.

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Linear multivariate models for physiological signal analysis: theory