220
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-23, NO. 3, MAY 1976
Latent Components in the Electrocardiogram MARTIN GRAHAM, Abstract-A new decomposition technique useful for representing a set of observed electrocardiograms is presented. This decomposition is different from past techniques in the constraints placed on the component waveforms in that they must be positive and start, stop, and overlap in a prescribed manner. The number of component waveforms is dependent on the maximum error tolerated in the reconstruction of the observed waveforms from the component representation.
A CONSIDERABLE amount of work has been published on the representation of a set of simultaneously recorded electrocardiogram waveforms as sums of component waveforms [1] -[8]. This paper describes a new technique in this class of representations, which all start with the concept that if there are K observed waveforms, Di(t), i 1, 2, - - *, K, and M component waveforms, C1(t), j = 1, 2, ,M, then the can be as represented Di(t)
D1(t) =
M
£ j=l
AijCi(t)
+ Ei(t)
where Ai are coefficients which do not vary with time, and
Es(t) is the error between the representation and the observed waveform.
A BRIEF CLASSIFICATION OF THE PUBLISHED TECHNIQUES AND THE LATENT COMPONENT TECHNIQUE 1. Specified Components Method. The most common is the Fourier series representation, where C,(t) is cos (j- l)irfot for j odd and sin j/rfot for j even, and fo is the reciprocal of the time interval that is to be represented [2]. The coefficients Aii canarebe calculated directly once the time interval and t = 0 point specified. Functions other than sines and cosines have been used [31, [4]. In general they are chosen to be an orthogonal set, which simplifies the calculations of the coefficients A ij 2. Form of Components Specified Method. An example of this representation is the use of Gaussian components, Cj(t) = [5]. The Hi, BJ and T* along with the Ai* coefficients are computed for the minimum total squared error. 3. Coefficients Specified Method. The Aii coefficients are determined from a model, not directly measured for the subject, and the C,(t) computed for the minimum total squared error [6] . The number of components M is determined by the detail of the model used. An additional constraint may be imposed on the C,(t), e.g., that they be non-negative. 4. Factor Analysis Method. The C](t) are constrained to be
Hie-Bi(Ti-t)12
Manuscript received December 7, 1973; revised October 14, 1975. This work was supported by the National Science Foundation under Grant GJ-31772. The author is with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
FELLOW, IEEE
orthogonal
to each other. The first component, Cl(t), minimizes the total squared error when only one component is used, i.e., when M = 1. Each subsequent component minimizes the total squared error with the addition of that compo-
nent
[7], [8].
5. Latent Component Representation. This new representation places the following constraints on the components. a. Each component Cj(t) is positive in the time interval from its start to its finish; i.e., from t = Si to t = F1, and is zero elsewhere. b. The components start in sequence and end in sequence; Sl W
THENW=Ei(t)
[i=1,Kandt=1,T]
andR =t,
IFR>0 GOTO Step I IF R = 0 Decomposition is complete.
N= 1,2, or 3
REFERENCES
[1] J. R. Cox, F. M. Nolle and R. M. Arthur, "Digital Analysis of the
V=O.
Calculate C)(t) [j = L, HI for minimum squared error (see ii). IF C,(t) < 0 [j = H, L] THEN V=j
THENt=t+1
IFV=0
[2]
GOTOE
IFt>Pv
THEN Fv=tl-
GOToLE
[3]
IFtO GOTO 1j
[81
V=O.
[91
Calculate C,(t) [j = L, HI for minimum squared error
178.
(see ii).
IF Ci(t)
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-23, NO. 3, MAY 1976
Latent Components in the Electrocardiogram MARTIN GRAHAM, Abstract-A new decomposition technique useful for representing a set of observed electrocardiograms is presented. This decomposition is different from past techniques in the constraints placed on the component waveforms in that they must be positive and start, stop, and overlap in a prescribed manner. The number of component waveforms is dependent on the maximum error tolerated in the reconstruction of the observed waveforms from the component representation.
A CONSIDERABLE amount of work has been published on the representation of a set of simultaneously recorded electrocardiogram waveforms as sums of component waveforms [1] -[8]. This paper describes a new technique in this class of representations, which all start with the concept that if there are K observed waveforms, Di(t), i 1, 2, - - *, K, and M component waveforms, C1(t), j = 1, 2, ,M, then the can be as represented Di(t)
D1(t) =
M
£ j=l
AijCi(t)
+ Ei(t)
where Ai are coefficients which do not vary with time, and
Es(t) is the error between the representation and the observed waveform.
A BRIEF CLASSIFICATION OF THE PUBLISHED TECHNIQUES AND THE LATENT COMPONENT TECHNIQUE 1. Specified Components Method. The most common is the Fourier series representation, where C,(t) is cos (j- l)irfot for j odd and sin j/rfot for j even, and fo is the reciprocal of the time interval that is to be represented [2]. The coefficients Aii canarebe calculated directly once the time interval and t = 0 point specified. Functions other than sines and cosines have been used [31, [4]. In general they are chosen to be an orthogonal set, which simplifies the calculations of the coefficients A ij 2. Form of Components Specified Method. An example of this representation is the use of Gaussian components, Cj(t) = [5]. The Hi, BJ and T* along with the Ai* coefficients are computed for the minimum total squared error. 3. Coefficients Specified Method. The Aii coefficients are determined from a model, not directly measured for the subject, and the C,(t) computed for the minimum total squared error [6] . The number of components M is determined by the detail of the model used. An additional constraint may be imposed on the C,(t), e.g., that they be non-negative. 4. Factor Analysis Method. The C](t) are constrained to be
Hie-Bi(Ti-t)12
Manuscript received December 7, 1973; revised October 14, 1975. This work was supported by the National Science Foundation under Grant GJ-31772. The author is with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
FELLOW, IEEE
orthogonal
to each other. The first component, Cl(t), minimizes the total squared error when only one component is used, i.e., when M = 1. Each subsequent component minimizes the total squared error with the addition of that compo-
nent
[7], [8].
5. Latent Component Representation. This new representation places the following constraints on the components. a. Each component Cj(t) is positive in the time interval from its start to its finish; i.e., from t = Si to t = F1, and is zero elsewhere. b. The components start in sequence and end in sequence; Sl W
THENW=Ei(t)
[i=1,Kandt=1,T]
andR =t,
IFR>0 GOTO Step I IF R = 0 Decomposition is complete.
N= 1,2, or 3
REFERENCES
[1] J. R. Cox, F. M. Nolle and R. M. Arthur, "Digital Analysis of the
V=O.
Calculate C)(t) [j = L, HI for minimum squared error (see ii). IF C,(t) < 0 [j = H, L] THEN V=j
THENt=t+1
IFV=0
[2]
GOTOE
IFt>Pv
THEN Fv=tl-
GOToLE
[3]
IFtO GOTO 1j
[81
V=O.
[91
Calculate C,(t) [j = L, HI for minimum squared error
178.
(see ii).
IF Ci(t)
