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Multiple-generator errors are unavoidable under model misspecification
Electroencephalography and clinical Neurophysiology 95 (1995) 135-142
Multiple-generator errors are unavoidable under model misspecification Don L. Jewett *, Zhi Zhang Research Division, Abratech Corporation,
475 Gate Five Rd., Suite 255, Sausalito, CA 94965, USA
Accepted for publication: 16 February 1995
Abstract Model misspecification poses a major problem for dipole source localization (DSL) because it causes insidious multiple-generator errors (MulGenErrs) to occur in the fitted dipole parameters. This paper describes how and why this occurs, based upon simple algebraic considerations. MulGenErrs must occur, to some degree, in any DSL analysis of real data because there is model misspecification and mathematically the equations used for the simultaneously active generators must be of a different form than the equations for each generator active alone. Keywords: Fitting function; Least square fit; Dipole source localization; Dipole parameters; Model misspecification; Evoked potential map
1. Introduction Dipole source localization (DSL) methods, as means to analyze multi-channel evoked response data, have become increasingly popular and important, especially with the assumption that over the time-interval analyzed the generators do not change location and orientation, but only vary in magnitude (e.g., Scherg, 1990; Achim et al., 1991; Franssen et al., 1992). (“Rotating dipoles” are determined, when individual generators cannot be distinguished, by assuming that 3 orthogonal dipoles exist at the same location.) Fletcher et al. (19931, Zhang and Jewett (1993), Berg and Scherg (1994), and Zhang et al. (1994) have shown that, when there is model misspecification, there are insidious (undetectable) errors in the DSL fitted dipole parameters, whether applied to a single time-point or multiple time-points. These errors, so far only shown in simulations, can be as great as 3.6 cm in location, 63” in orientation, and 98% in magnitude, when a single timepoint is used (Zhang and Jewett, 1993). The errors that may occur in analysis of real data are unknown, but might be as great as, or greater, than those shown in the simulations. Unfortunately, model misspecification must always occur when a simplified model is used to analyze data from a
human recording. However, differences in the fitted dipole parameters as a function of generator wave forms may be used to detect model misspecifications, so that the fit results from a “bad” combination of dipoles and model can be rejected (Zhang and Jewett, 1994). Therefore, the issues concerning model misspecification and its manifestations with regard to errors in the fitted parameters need to be understood by users of DSL methods. The previous papers (Zhang and Jewett, 1993, 1994; Zhang et al., 1994) are heavy on mathematical detail, which means that few of the actual users of DSL methods will gain an intuitive appreciation of how these factors may influence their DSL results. This paper is intended to provide to non-mathematically oriented users sufficient background information for understanding the main issues. For this reason, this paper only requires knowledge of elementary algebra. This paper addresses errors due to model misspecification, therefore, model misspecification is always assumed, i.e., there are no errors if the model is correct. The noise effect on the DSL will also not be explicitly considered, except for one example to show the difference between the effect of model misspecification and the effect of noise.
2. Models and model misspecification
* Corresponding author. Tel.: + 1 (415) 2897455; Fax: +1 (415) 3316126. 0013-4694/95/.$09.50 0 1995 Elsevier Science Ireland Ltd. All rights reserved SD1 0013-4694(95)00047-X
The term “model” as used in this paper is equivalent to “a set of governing equations which determine the exact
EEG 94042
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D.L. Jewett, 2. Zhang/Electroencephalography
relationship between generator parameters (location, orientation, and magnitude) and the resulting potentials.” As far as the biological system is concerned, we have no way to derive the exact equations that determine the biological potentials. Instead, we set up a set of equations that we can manipulate (a model), and see if they can “fit” the results, and assume that if the fit is good, then the model is “good” (i.e., useful). Generally when recording from humans, we have no independent way to check the accuracy of our model. Instead, to check the analysis model, simulations are done using a complicated model to represent the biological system, against which a simpler analysis model can be accurately compared. We call the analysis model “FIT,” while the model meant to represent the biological system is called “GEN.” The difference between GEN and FIT is the model misspecification. To improve clarity, equations of GEN will be in upper case letters, those of FIT in lower case. Exact, perfect results in using FIT (the map is explained completely by the dipoles) can only occur when there is no model misspecification) ‘. If there is model misspecification, there will be errors in the fitted dipole parameters.
3. Single-generator
errors and multiple-generator er-
rors Under model misspecification, there are 2 kinds of errors in the DSL-fitted dipole parameters; we call them SingGenErrs (single-generator errors) and MulGenErrs errors) in Zhang et al. (1994). (multiple-generator SingGenErr is the error when only 1 generator is active. MulGenErr is the additional error when multiple generators are simultaneously active. They can be easily understood with the help of the following DSL example. Assume a spatio-temporal potential map, Vobsl, is generated in a mathematical model “GEN” by a single generator D, , and the DSL is then used to analyze this VObsl to determine the best-fit dipole parameters d, in a misspecified model “FIT.” Since FIT is misspecified from GEN, usually, d, # D,. The difference between D, and d, is the SingGenErr - the error when only 1 generator is active. This SingGenErr may be reduced by an correction (Ary et al., 1981). Similarly if “Ary-type” another map, Vobd, is generated by another single generator D, in GEN, and fitted in FIT as d,, then d, # D2, but the difference between D, and d, (the SingGenErrl may be reduced by an “Ary-type” correction. Model misspecification can be understood in 2 different ways. Physically, it means that the model on which a potential map is observed (here, the GEN model) is differ-
’A perfect fit will also occur if many dipoles are used in a misspecified model to the point that they exhaust all the degrees of freedom of the potential space.
and clinical Neurophysiology 95 (19951 135-142
ent from the model used in the DSL to analyze the observed potential map (here, FIT). For example, there is model misspecification when an evoked potential map recorded on the human scalp is analyzed using a spherical model, because a human head is different from a sphere. Mathematically, model misspecification means that 2 surface potential maps, each generated by a single dipole, are different (or not completely correlated, to be more accurate). The difference between the 2 maps can be large or small depending upon the actual dipole parameters. In this sense, model misspecification may occur even if a correct model is used, e.g., when a dipole is constrained to a wrong location, or a wrong orientation, or a wrong magnitude, or both, within the correct model. Therefore, model misspecification is equivalent to using different equations for dipoles in generating the observed surface potential maps and for dipoles in fitting the observed maps ‘. This is how model misspecification will be represented in this paper. Now, to proceed with an explanation of the MulGenErr. If we form a new map Vobs, by adding Vobs, and VobsZ together, to mimic the map when 2 generators (Dr and D2) are simultaneously active, and fit this Vobs map by 2 dipoles within FIT, we would expect the DSL to find d, and d, (i.e., to find the same solutions as are found when each generator’s map is analyzed separately, even though d, and dz are different from D, and D,, respectively). However, instead of getting d, and d,, the DSL will find d’, and d;, because of the model misspecification. The differences between d, and d’,, and between d, and d; are the MulGenErrs. MulGenErr is insidious and cannot be removed or reduced by an Ary-type correction.
4. Linear fits of d-dimensional data Because DSLs can involve problems in more than 3 dimensions, it is difficult to graphically represent them. This presentation, therefore, will proceed from the familiar linear fit of 2-dimensional data, and introduce the ideas in this analogous system, before extending the results to the DSL case. Consider the linear fit of data, as in Fig. 1. The dots are the observed datapoints and the line represents the line which minimizes the sum of the squared differences between the datapoints and the best-fit line. In this case, the underlying assumption is that the data are formed from the algebraic summation of 2 processes: (1) a linear process which is represented by the best-fit line, and (2)
z
Assume a map is generated by a dipole at location r. Notice the difference in the following two situations even when the FIT model is perfect (FIT = GEN). (a) If one dipole is placed at r to fit the map with no further constraints, then there is no model misspecification. However, (b) if the fitting dipole is constrained to a wrong orientation, even at a correct location, then there is model misspecification.
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and clinical Neurophysiology 95 (1995) 135-142
137
400 t
x
200 i
34L
~200 L, 4
Fig. 1. A set of datapoints and its least-square represent the noise-contaminated observations, sents the recovered data.
fitted line. The datapoints while the fitted line repre-
random noise added to each observation. If this assumption is correct, then the only question is how well the parameters of the best-fit line of the data analysis correspond to the parameters of the linear process that has been obscured by the noise. Thus, physically there is no model misspecification here because the model used in the analysis is the same as that used in generating the data. Now consider a situation with no noise, but with a form of model misspecification; in Fig. 2 the datapoints (solid circles) have been formed from a power function Y=Ax3
+B
(1)
where A = 1 and B = - 250. We again find the best-fit line (shown in Fig. 2, empty circles), and the equation of the line is y,=a,x+b
1
(29
where, in Fig. 2, a, = 174, and b, = - 1066. 3 In this case, the deviations of the datapoints from the line are systematic rather than random. (In statistical terminology, there is bias in this analysis.) Now if, as scientists, we wish to determine the characteristics of the process that generated the datapoints, and our analysis gives us the parameters of the linear fit (a, and b,), how should we interpret these parameters? Clearly, since there is model misspecification (i.e., the equation used in the fit is not the same equation as used to generate the data), there is no sense in which the a, and b, parameters can be correct, since it is the parameters of Eq. (1) (A and B) that are desired, not those of Eq. (2a). Indeed, the same datapoints can also be fit by other functions (other misspecified models), resulting in different fit parameters. For instance, if Y of Eq. (1) is fit by y,=a,x*
3 The fitted values are usually rounded to the nearest integer.
(2b)
Fig. 2. A function, Y = x3 - 250 (solid circles), is (linear least square) fit by (al y, = a,x+b, (empty circles), (b) yz = azx2 (empty squares), (cl y, = a3x2 +b, (empty diamonds), at x = 5, 6, 7, 8, 9, 10.
we have a2 = 5 (Fig. 2, empty squares). Or if Eq. (1) is fit by y, = a3xz + b,
(2c)
the fit results are a3 = 12, b, = -454 (Fig. 2, empty diamonds). y3 has one more term (b,) than y2, and the fit value for a3 is different from that of a2. It can be seen from Fig. 2 that the fit can be done by using different fitting functions, while none of the fits is perfect since the functions used in the fitting are each incorrect (i.e., not Eq. (1)). To reiterate: when different functions are used to fit a data set, different parameters (with different curves) will be fit. Even if the same function is used to fit the same set of datapoints, different fit values can occur if different functions are minimized (such as the absolute difference, or cubed difference) 4. Similarly, had some of the data points been missing, e.g., the one at x = 5, then the best-fit curve and parameters would have been different again. In all these cases, no matter what the criteria, since the model is wrong, so are the resulting parameters. This is equivalent to the SingGenErr of Zhang et al. (1994) and the error that may be reduced in the DSL using the Ary-type correction (Ary et al., 1981). Notice that, using the definitions of GEN and FIT, in Fig. 2, Eq. (1) is the GEN model, while Eq. (2) (a, b or c) is the FIT model. Note that in creating Fig. 2 the process was to establish GEN, choose specific values of x (5, 6, 7, 8, 9, lo), calculate the datapoints (a forward calculation to
4 The reasons that least-square fit has been widely used are: (11 It is the most useful estimation in statistics. The principle of least-square can be derived from the principle of maximum likelihood if the measurements follow a gaussian distribution. Squaring the differences means that greater importance is placed on removing the large deviations. (21 It is mathematically easy to handle. Differentiating a squared quantity leads to easy, linear terms. (3) Linear superposition holds under certain linear fits, therefore, the fitting process is simplified. (4) Adding independent effects produces additive sums of squares.
D.L. Jewett, Z. Zhang/ Electroencephalography and clinical Neurophysiology 95 (1995) 135-142
138
determine the values of y), establish FIT, and then calculate the parameters of FIT based upon the datapoints (an inverse calculation). This is directly analogous to the process in a DSL, where the biological system generates a potential map (which in a simulation would be the potentials created by GEN), and then the DSL, using the model FIT, by an iterative inverse calculation, determines which generator parameters minimize the squared differences between the observed potential map and the potential map calculated in FIT. Continuing the analogy between the example of Fig. 2 and a DSL, the values of x can be considered to represent different electrodes and the values of y represent the potentials detected at those electrodes. The parameters of Eqs. (2) (the “a”~ and “ b”s) might be considered to represent some parameters of the generator; however, note that the analogy with a DSL is not exact in all details.
900 i
450 1
x t -4501 4
5
6
7
8
9
f 10
Fig. 3. Two functions, Y, = -(x3/4> and Y2 = x4/16, are (linear least square) fit by 2 other functions, y4 = a4 x2 and ys = a5 x2, respectively, at ~=5,6,7,8,9,10.ThesumofY,andY~,Y,+~,isfitbyy~+~=~~+~x* over the same x-range. Since the same linear function (y4, ys and yi+ s) is used in different fits, there is no MulGenErr in the fitted parameters. i.e., a)4+ s = a4 + as, yi+s = y4 + ys. Also shown in the figure is the sum of y4 and ys, y4 + 5 (crosses, which is identical to y; + 5).
5. Multiple-generator error in a simple model
be the same when the best-fit solution (under FIT) is obtained to the following equation:
The preceding examples introduce the issue of models and SingGenErrs, but are not examples of MulGenErrs. MulGenErr can occur only when there are 2 (or more) generators that are simultaneously active, so we must enlarge our simplified model to include this situation. So let us have 2 generators which generate potentials at the electrodes according to the following equations:
Yi+5 = U4X2-I-a$
Y, =A,x3,
whereA,
= -l/4
(3a)
Y2=A,x4,
whereA,
= l/16
(W
These then are the 2 generators in GEN. A, and A, can be thought of as the 2 generators’ magnitudes, while x3 and x4 can be thought of as the 2 generators’ weighting functions (see later). The 2 different weighting functions (x3 and x4> mimic the 2 dipoles being at 2 different locations. In FIT we will do a linear fit against each individually:
Asking the question in another way, by rearranging the above equation: Y&S =
(a4
and then combining the a4 and a5 terms into a’4+s, Yi+5 =a;+5x
(4a)
ys = a5x2
(4b)
Solving in Eqs. (4) for the best-fit for each of the generators in Eqs. (31, yields the following values: a4 = - 2 and a5 = 5. a4 and a5 represent the 2 individually fitted magnitudes. Since the algebraic summation of the potentials of the 2 generators will occur at each electrode when the generators are simultaneously active, at each electrode on GEN, the following is true: YI+Z=yl+&
(3c)
We now ask the question: will the values of the coefficients of the best-fit equations (Eqs. (4a, 4b)), a4 and a5,
2
(4c)
we can then determine whether, when the summed data (Y1+*, Eq. (3~)) are fit by Eq. (4c), the results of the fit are just the summation of the individual fits, i.e., we are asking is the following equation true: a;+5 = a4 + a5
(44
The results from fitting Eq. (4~) to the summed data from Eq. (3~) are: ai+S = 3 a4 + US= 3
y4 = a4x2
+%)X2
(4e)
Note that MulGenErr in the fitted dipole parameters can be demonstrated either by comparing parameters (as in Eq. (4e)) or by comparing values of y, as determined by the parameters (see Fig. 3 legend). Comparing ‘‘y” values is equivalent to comparing spatio-temporal maps in a DSL. The numerical results of the above example are graphed in Fig. 3. The end result is that there is no MulGenErr in the fitted parameters. Thus, as soon as we know the fit results for Y1 and Y2 (a4 and a,> respectively, we know that the fit result for any linear combination of Y, and YZ, crYI + pY2, must be oa4 + pa,, where cwand p can be any values. The above results may at first glance seem surprising, since there is no MulGenErr in the fitted parameters even though both fits were non-perfect (wrong equations were
D.L. Jewett, Z. Zhang/Electroencephalography Table 1 A function refers to Function
y = ax y =8x2 y=ax+b y=ax2+b y=a2x
can be linear or non-linear,
depending
With respect to y Linear
Non-linear
a, x a
X
a, b, x a, b
X
X
a
on the variable(s)
and clinical Neurophysiology
it
Equation no.
5a, 5e 2b, 4a-c, 2a 2c
5b
used in the fitting); the plots of Y,, Yz and Y,+z are all curvilinear; and the most confusing part is that Eqs. (4a-4c) are all non-linear with respect to the variable “ x.” However, “x” is not the parameter that is being fit by the procedure; the parameters being fit in this example are a4, a5 and ar4+,, and these are linear in the example (Table 1). It is easy to make a mistake in trying to determine if a function is linear by looking only at the symbols used, since x, y and z are usually used as variables, while a, b and c are commonly used to denote constants (Table 1). Note especially that, when doing a ftt, the procedure fits
the equation’s constants (a, b and c), not the variable (x). The only condition under which there is no MulGenErr in the fitted parameters is when the parameters to be fit are linear (Table 2). So only if y is a linear function of the parameters to be fit, will the result be a linear fit. Clearly, there will be no MulGenErrs in the fitted parameters when the function used to fit the data is a linear function of x, as in yr of Eq. (2a), i.e., if y4, ys and yi+6 all have the form of Eq. (2a). Thus, this case illustrates the General Principle of MulGenErr, namely, there will be no
MulGenErrs in the fitted parameters when there is model misspecijication if and only if: (1) the equation used in the fitting of the summed potentials is the same as the equation used in the fitting of the potentials of the individual generators, and (2) the equation used is linear with respect to the parameter that is fit. The conditions that affect MulGenErr in the fitted parameters are shown in Table 2, which shows all possible combinations of the 2 independent requirements of the general principle. The 4 conditions of Table 2 are all assumed under model misspecification, i.e., no MulGenErr will occur in any of the conditions if there is no model misspecification. The conditions of Table 2 applicable to DSL are Ml and M3. In a DSL, the equations used in the fitting will always be linear relating to the dipole magnitude unless some constraint is imposed on the magnitude. Therefore, if only the dipole magnitude is concerned, DSL can only be within either NM or Ml of Table 1. However, as we will later show, the equations must be different, so only condition Ml applies to DSL magnitude calculations. When dipole location/orientation is concerned, DSL will involve condition M3 of Table 2, as location/orientation parame-
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95 (1995) 135-142
ters are non-linear parameters in the equations. Due to the complexity in relating condition M3 to the DSL, in the following, we will explain the Ml condition only. (The “no MulGenErr” condition NM has already been demonstrated above.) Condition Ml (Table 2) occurs when all fitting equations are linear, but all or some of them are different from each other. If the forms of the equations used to fit different individual data sets are different from each other, then it is not clear what function should be used to fit the summed data. With such differences in equations, MulGenErr will occur, and there is no reason to expect that it should not; yet this is analogous to some DSL fits. Taking the generator equations from Eqs. (31, we use the following FIT y6=a6x
(5a)
y7 = a7x2
(5b)
to fit Y, and Y2, respectively.
Another
function
Yk+7 = six + a;n2
(5c)
is fit to the sum Y, + 2 (Eq. (3~)). This can be thought of as (1) 2 individual dipoles, each at a different location (weighting functions x and x2 are different) within FIT, are used to fit 2 individual maps, respectively (Eqs. (5a, b)), and (2) 2 dipoles at the same 2 locations (weighting functions are x and x2, Eq. (5~)) are used simultaneously to fit the summed map. The fitted results are
a6 = - 18; a7 = 5 a’ = 8 a’ 6 = -43. ) 7
(5d)
MulGenErr is present since a’6 # a6 and a7 # a,. In this example, a6 and a7 represent the individually fitted dipole magnitudes, while a’, and a’, represent the simultaneously fitted dipole magnitudes. These results are plotted in Fig. 4. This is equivalent to comparing maps from GEN and FIT. It can be seen that neither y6 fits Y, nor y7 fits Y2 perfectly, because wrong equations were used in the fits. For instance, Y1 is proportional to x3 (Eq. (3a)), but the fit (using Eq. (5a)) assumes it is proportional to x (this might represent the case where
Table 2 The 4 combinations of 2 independent conditions MulGenErr will occur in the fitted parameters
that determine
whether
Function for sum compared to function for separate generators
Parameters Linear Non-linear
Same
Different
Nh4. No MulGenErr M2. MulGenErr
Ml. MulGenErr M3. MulGenErr
beingfit
Note that which equations are used in GEN does not affect the results in this table, just so long as they are different from those used in FIT, which means that the results apply to biological potentials being fit by simple models.
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D.L. Jewett, Z. Zhang/Electroencephalography
and clinical Neurophysiology 95 (1995) 135-142
ual generators. For example, take the same generator equations from Eqs. (3), but make the fitting equations for the individual generators y6 of Eq. (5a) for Y, (unchanged), and
for Yz (different weighting functions x and x + 3 represent different dipole locations), and the equation to be fit to the sum Yi+s = a:+RX
Fig. 4. The same functions, YI, Y2 and Y, + 2, as in Fig. 3, are fit by y6 = a6 x, y, = a, x2 and yi + , = a’6x + a’, x 2, respectively, at x = 5, 6, 7, 8, 9, 10. Since different functions were used in different fits, MulGenErr occurs in the fitted parameters, i.e., y;, , f y6 +y,, a$ # a6, a’, # a7. As far as the goodness of fit is concerned, yi+ 7 fits Y, + 2 much better than either y6 fits Ye or y, fits Y2. But when y: + , is plotted into 2 parts, ‘, then the fit of yi to Y, is worse than yb to Y,, yA=a’,x and y;=a’,x and the fit of y; to Y2 is worse than y, to Y2. This is analogous to the DSL under model misspecification. I’~, Y2 and YI+z are similar to the potential maps generated by 2 individual dipoles and by the 2 dipoles active together, respectively. ah and a, are similar to the fit dipole parameters under single-dipole fitting, while al, and a’T are the fit parameters for the same dipoles under simultaneously active conditions. Although the fit results, a’6 and a’,, are totally different from the single-dipole fit results, a6 and a7, the 2-dipole fitting results in a much better fit to the data map (a much smaller NLSE).
the potential maps are recorded from a human head, but the DSL assumes the head can be replaced by a sphere). The NLSEs (normalized least-square errors, see Zhang and Jewett, 1993) for the fits of y6 to Y1 and y7 to Y2 are 11% and 7%, respectively. It is important to note in Fig. 4 that although yA+7 (Eq. (5~)) does not fit Y1 + 2 perfectly, the fit is much better than either of the 2 individual fits; the y: + , to Y1+z fit has an NLSE of only 1%. Moreover, if yL+ 7 is plotted as 2 parts (based upon the parameters of Eq. (5d)), y: = abx and y: = a’7x2, then the fit of y: to Y, is much worse than y6 to Y1, and the fit of yi to Y2 is much worse than y, to Y2 (see Fig. 4), even though the yA+7 fit is better than y6 or y7 fit. Note carefully this result: with 2 generators the NLSE fit is better to the 2-generator map than the l-generator fits to l-generator maps, yet the computed generator parameters produce individual maps that are worse than the maps from the l-generator fits. A low NLSE does not guarantee accuracy of parameters when there is model misspecification. Also small differences in maps can lead to large differences in derived generator parameters. Iterative searching to reduce a small NLSE may still further incrementally increase differences in generator parameters based upon small map differences (see also the last figures in Zhang and Jewett (1993) and Zhang et al. (1994)). MulGenErr will occur even if the function used for the sum differs from just one of the equations for the individ-
then the fit parameters are a6 = - 18, aB = 27, and a’6+8 = 21. This is equivalent in the DSL to fitting both the individual maps and the summed map by 1 dipole only, so the number of generators is misspecified in the fitting. The presence of MulGenErr is shown by the fact that a6 + a, # a’,+,. Under all the 4 conditions, NM, Ml, M2, and M3 (although we only gave examples under NM and Ml), there is model misspecification, which is equivalent to finding that the potential map cannot be completely fit under single-generator cases 5. Note that whether or not there is MulGenErr in the fitted parameters does not depend upon which equation is used in GEN. Indeed, GEN can be linear or not, just so long as the equations used in GEN are different from those used in FIT there will be MulGenErrs. A review of the examples will show that this is true, by trying different equations for GEN. The skeptical reader might want to re-analyze the first example with other GEN equations, since the first example is the only case where there is no MulGenErr in the fitted parameters (NM of Table 2).
6. Discussion of the analogies between the examples and DSL In a DSL one or more spatio-temporal maps are available either from a biological recording, or from a computer simulation using a GEN. The maps are analyzed to find the best-fit dipoles within FIT. There are 6 parameters for each dipole: 3 dimensions for location, 2 angles for orientation, and a magnitude. The orientation and magnitude parameters together can also be treated as the “dipole moment,” i.e., 3 magnitudes along 3 orthogonal axes whose origin is positioned at a specific dipole location. The surface potential generated by a dipole and the dipole parameters are related by a complicated function, involving both the dipole parameters and the model parameters and the electrode locations. There is a linear relationship between the surface potential and the dipole moment, e.g., doubling the magnitude of a dipole
5 Assuming the correct summed potential map.
number
of dipoles
is specified
in fitting the
L1.L. Jewett, Z. Zhang/Electroencephalography
moment doubles the potential. Hence, 2 dipole moments at the same location can be added vectorially, which is just the algebraic summation along each axis, independently. However, the relationship is non-linear between the potential and the dipole location (or the distances from the dipole location to the electrodes). Thus, the fit in a DSL includes both linear and non-linear parameters. Since the potential is proportional to the dipole magnitude, the potential can therefore be written as v(x)
=mw(x,
I)
(6)
where v(x) is the potential at electrode “x.” 1, is equivalent to the “y’s” in the examples. m is the dipole magnitude which is proportional to v. (For simplicity, we assume the dipole orientation is fixed in a given direction at each location, thus, only dipole magnitude and location are of concern.) m is equivalent to the coefficients to be fit in the examples (the “a’s”). w(x, r), sometimes called the dipole “weighting function,” is a complicated function of the electrode “ x,” the dipole location r, and other model parameters. It is equivalent to the x of Eq. (5a) or x2 of Eq. (5b). Since w is a function of the dipole location, x and x2 of Eqs. (5) represent 2 different dipole locations in FIT. Now, to go back to the first DSL example where we introduced the concept of the SingGenErr and the MulGenErr, when Vobsl and Vobs2 are each separately fit by a dipole within FIT, the fitting functions will be (7a)
v2( x) = m2w( x, r2)
P)
respectively. Eqs. (7a) and (7b) are 2 different functions, because usually 2 dipoles at different locations will be fit, thus w(x, r,) # w(x, r2) (analogous to x # x2 in Eqs. (Sa-b)). Now, let’s examine how the summed map Vobs = Vobsl + V&Z is fit by 2 dipoles within FIT. In DSL, the fitting equation is in the following form: v;+2( x) = m’iw( x, ri) + m;w( x, r;)
(7c)
which is quite similar to the sum of Y,(X) and v,(x). Comparing the situation here with those described in Table 2, it is apparent that different functions are used in the single-fits and the summed-fit. Therefore, MulGenErr will occur in the fitted dipole magnitudes, i.e., m’, # m, and m’, f m2. This is true whether the 2 dipoles in the summed-fit are forced at the previously fitted locations (e.g., r{ = ri and ri = r2, as in the example under condition Ml) or not. When the dipole locations are allowed to vary, then there may be other locations (i.e., other weighting functions) which will make the fit better than at the locations ri and r2, and there will be MulGenErrs in the non-linear fitted dipole locations. When the DSL moves the dipoles away from the single-fitted locations, the MulGenErr in the fitted magnitudes will change. This is be-
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141
cause moving the dipoles to different locations is mathematically equivalent to using different equations in the fitting. Therefore, a realistic DSL involves both conditions Ml and M3. There is also an analogy between the condition NM of Table 2 and the DSL under model misspecification, i.e., under certain conditions, there will be no MulGenErrs even when there is model misspecification. The conditions are fitting both the individual maps and the summed map with the same number of dipoles, with either of the following conditions: (1) at the same fixed locations with no constraint on the fitting dipole orientation and magnitude, or (2) at the same fixed locations and the same fixed orientations with no constraint on the fitting dipole magnitude. Mathematically, both conditions are equivalent to fitting both the individual maps and the summed map using the same weighting function(s). This was the case in the example demonstrating condition NM, where the same weighting function x2 was used in all the fits. It is quite obvious that this is not realistic - using the same dipoles (i.e., at the same locations and/or with the same orientations) to fit both the individual and the summed maps even though different dipoles generated the different maps.
7. Forms of model misspecification GEN and FIT can differ in a variety of ways, all leading to model misspecification. The following is a partial list of the ways GEN and FIT may differ: shape parameterization (e.g., from a real head to a sphere), number of layers (e.g., using a single layer model with uniform conductivity to represent the real head, where the head has multiple layers with different conductivities: scalp, skull, and brain), conductivity and/or thickness of layers, uniformity of layer conductivity or thickness, electrode locations, reference electrodes, number of dipoles, incorrect constraints over dipole parameters. The preceding list applies to both analysis at a single time-point, and analysis of multiple timepoints.
8. Conclusion
The basic concepts of MulGenErr involved in the DSL are described. We conclude that, under model misspecification, MulGenErr must occur in the DSL method described here, because the potential map due to a single generator (e.g., a dipole) cannot be 100% fit by a single generator within a misspecified model, When such maps are added together and fit by 2 simultaneously active generators in the misspecified model, effectively, each individual map is fit by 2 generators, which will fit each individual map better than if only 1 generator were used, but the fit generator parameters will be different from the
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individual fit. Therefore, without examining the goodness of a model, the residual error between the observed and the fitted potential maps after a fit cannot and should not be used as a measure of how good the fit is. It is quite clear if a map due to 5 generators is fit by 5 generators within a misspecified model, and another map due to 2 generators is fit by 2 generators using the same misspecified model, the 5-to-5 fit will be better (smaller residual error) than the 2-to-2 fit, but probably, the MulGenErrs in the 5-to-5 fit will be larger. We have shown that MulGenErr occurs directly from simple algebraic considerations. Since in any realistic situation where evoked response data are being analyzed, model misspecification is unavoidable, MulGenErrs will occur. The issue then becomes the size of these errors which must be determined for each case, given the specific model misspecification and the generators thought to be simultaneously active.
Acknowledgement This research was supported by Grant ROlDC00328 from the National Institute on Deafness and Other Communication Diseases.
and clinical Neurophysiology 95 (1995) 135-142
References Achim, A., Richer, F. and Saint-Hilaire, J. (1991) Methodological considerations for the evaluation of spatio-temporal source models. Electroenceph. clin. Neurophysiol., 79: 227-240. Ary, J.P., Klein, S.A. and Fender, D.H. (1981) Location of sources of evoked scalp potentials: corrections for skull and scalp thicknesses. IEEE Trans. Biomed. Eng., 28: 447-452. Berg, P. and Scherg, M. (1994) A fast method for forward computation of multiple-shell spherical head models. Electroenceph. clin. Neurophysiol., 90: 58-64. Fletcher, D., Jewett, D.L., Zhang, Z. and Amir, A. (1993) The effect of skull shape on single and multiple dipole source localizations. Proc. 15th Int. Conf. IEEE Eng. Med. Biol. Sot., 15 (part 3): 1469-1470. Franssen, H., Stegeman, D.F., Moleman, J. and Schoobaar, R.P. (1992) Dipole modelling of median nerve SEPs in normal subjects and patients with small subcortical infarcts. Electroenceph. clin. Neurophysiol., 84: 401-417. Scherg, M. (1990) Fundamentals of dipole source potential analysis. In: M. Hoke, F. Grandori and G.L. Romani (Eds.), Auditory Evoked Magnetic Fields and Potentials. Karger, Basel. Zhang, Z. and Jewett, D.L. (1993) Insidious errors in dipole localization parameters at a single time-point due to model misspecification of number of shells. Electroenceph. clin. Neurophysiol., 88: l-11. Zhang, Z. and Jewett, D.L. (1994) Model misspecification detection by means of multiple-generator errors: using the observed potential map. Brain Topogr., 7: 29-39. Zhang, Z., Jewett, D.L. and Goodwill, G. (1994) Insidious errors in dipole parameters due to shell model misspecification using multiple time-points. Brain Topogr., 6: 283-298.
Multiple-generator errors are unavoidable under model misspecification Don L. Jewett *, Zhi Zhang Research Division, Abratech Corporation,
475 Gate Five Rd., Suite 255, Sausalito, CA 94965, USA
Accepted for publication: 16 February 1995
Abstract Model misspecification poses a major problem for dipole source localization (DSL) because it causes insidious multiple-generator errors (MulGenErrs) to occur in the fitted dipole parameters. This paper describes how and why this occurs, based upon simple algebraic considerations. MulGenErrs must occur, to some degree, in any DSL analysis of real data because there is model misspecification and mathematically the equations used for the simultaneously active generators must be of a different form than the equations for each generator active alone. Keywords: Fitting function; Least square fit; Dipole source localization; Dipole parameters; Model misspecification; Evoked potential map
1. Introduction Dipole source localization (DSL) methods, as means to analyze multi-channel evoked response data, have become increasingly popular and important, especially with the assumption that over the time-interval analyzed the generators do not change location and orientation, but only vary in magnitude (e.g., Scherg, 1990; Achim et al., 1991; Franssen et al., 1992). (“Rotating dipoles” are determined, when individual generators cannot be distinguished, by assuming that 3 orthogonal dipoles exist at the same location.) Fletcher et al. (19931, Zhang and Jewett (1993), Berg and Scherg (1994), and Zhang et al. (1994) have shown that, when there is model misspecification, there are insidious (undetectable) errors in the DSL fitted dipole parameters, whether applied to a single time-point or multiple time-points. These errors, so far only shown in simulations, can be as great as 3.6 cm in location, 63” in orientation, and 98% in magnitude, when a single timepoint is used (Zhang and Jewett, 1993). The errors that may occur in analysis of real data are unknown, but might be as great as, or greater, than those shown in the simulations. Unfortunately, model misspecification must always occur when a simplified model is used to analyze data from a
human recording. However, differences in the fitted dipole parameters as a function of generator wave forms may be used to detect model misspecifications, so that the fit results from a “bad” combination of dipoles and model can be rejected (Zhang and Jewett, 1994). Therefore, the issues concerning model misspecification and its manifestations with regard to errors in the fitted parameters need to be understood by users of DSL methods. The previous papers (Zhang and Jewett, 1993, 1994; Zhang et al., 1994) are heavy on mathematical detail, which means that few of the actual users of DSL methods will gain an intuitive appreciation of how these factors may influence their DSL results. This paper is intended to provide to non-mathematically oriented users sufficient background information for understanding the main issues. For this reason, this paper only requires knowledge of elementary algebra. This paper addresses errors due to model misspecification, therefore, model misspecification is always assumed, i.e., there are no errors if the model is correct. The noise effect on the DSL will also not be explicitly considered, except for one example to show the difference between the effect of model misspecification and the effect of noise.
2. Models and model misspecification
* Corresponding author. Tel.: + 1 (415) 2897455; Fax: +1 (415) 3316126. 0013-4694/95/.$09.50 0 1995 Elsevier Science Ireland Ltd. All rights reserved SD1 0013-4694(95)00047-X
The term “model” as used in this paper is equivalent to “a set of governing equations which determine the exact
EEG 94042
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D.L. Jewett, 2. Zhang/Electroencephalography
relationship between generator parameters (location, orientation, and magnitude) and the resulting potentials.” As far as the biological system is concerned, we have no way to derive the exact equations that determine the biological potentials. Instead, we set up a set of equations that we can manipulate (a model), and see if they can “fit” the results, and assume that if the fit is good, then the model is “good” (i.e., useful). Generally when recording from humans, we have no independent way to check the accuracy of our model. Instead, to check the analysis model, simulations are done using a complicated model to represent the biological system, against which a simpler analysis model can be accurately compared. We call the analysis model “FIT,” while the model meant to represent the biological system is called “GEN.” The difference between GEN and FIT is the model misspecification. To improve clarity, equations of GEN will be in upper case letters, those of FIT in lower case. Exact, perfect results in using FIT (the map is explained completely by the dipoles) can only occur when there is no model misspecification) ‘. If there is model misspecification, there will be errors in the fitted dipole parameters.
3. Single-generator
errors and multiple-generator er-
rors Under model misspecification, there are 2 kinds of errors in the DSL-fitted dipole parameters; we call them SingGenErrs (single-generator errors) and MulGenErrs errors) in Zhang et al. (1994). (multiple-generator SingGenErr is the error when only 1 generator is active. MulGenErr is the additional error when multiple generators are simultaneously active. They can be easily understood with the help of the following DSL example. Assume a spatio-temporal potential map, Vobsl, is generated in a mathematical model “GEN” by a single generator D, , and the DSL is then used to analyze this VObsl to determine the best-fit dipole parameters d, in a misspecified model “FIT.” Since FIT is misspecified from GEN, usually, d, # D,. The difference between D, and d, is the SingGenErr - the error when only 1 generator is active. This SingGenErr may be reduced by an correction (Ary et al., 1981). Similarly if “Ary-type” another map, Vobd, is generated by another single generator D, in GEN, and fitted in FIT as d,, then d, # D2, but the difference between D, and d, (the SingGenErrl may be reduced by an “Ary-type” correction. Model misspecification can be understood in 2 different ways. Physically, it means that the model on which a potential map is observed (here, the GEN model) is differ-
’A perfect fit will also occur if many dipoles are used in a misspecified model to the point that they exhaust all the degrees of freedom of the potential space.
and clinical Neurophysiology 95 (19951 135-142
ent from the model used in the DSL to analyze the observed potential map (here, FIT). For example, there is model misspecification when an evoked potential map recorded on the human scalp is analyzed using a spherical model, because a human head is different from a sphere. Mathematically, model misspecification means that 2 surface potential maps, each generated by a single dipole, are different (or not completely correlated, to be more accurate). The difference between the 2 maps can be large or small depending upon the actual dipole parameters. In this sense, model misspecification may occur even if a correct model is used, e.g., when a dipole is constrained to a wrong location, or a wrong orientation, or a wrong magnitude, or both, within the correct model. Therefore, model misspecification is equivalent to using different equations for dipoles in generating the observed surface potential maps and for dipoles in fitting the observed maps ‘. This is how model misspecification will be represented in this paper. Now, to proceed with an explanation of the MulGenErr. If we form a new map Vobs, by adding Vobs, and VobsZ together, to mimic the map when 2 generators (Dr and D2) are simultaneously active, and fit this Vobs map by 2 dipoles within FIT, we would expect the DSL to find d, and d, (i.e., to find the same solutions as are found when each generator’s map is analyzed separately, even though d, and dz are different from D, and D,, respectively). However, instead of getting d, and d,, the DSL will find d’, and d;, because of the model misspecification. The differences between d, and d’,, and between d, and d; are the MulGenErrs. MulGenErr is insidious and cannot be removed or reduced by an Ary-type correction.
4. Linear fits of d-dimensional data Because DSLs can involve problems in more than 3 dimensions, it is difficult to graphically represent them. This presentation, therefore, will proceed from the familiar linear fit of 2-dimensional data, and introduce the ideas in this analogous system, before extending the results to the DSL case. Consider the linear fit of data, as in Fig. 1. The dots are the observed datapoints and the line represents the line which minimizes the sum of the squared differences between the datapoints and the best-fit line. In this case, the underlying assumption is that the data are formed from the algebraic summation of 2 processes: (1) a linear process which is represented by the best-fit line, and (2)
z
Assume a map is generated by a dipole at location r. Notice the difference in the following two situations even when the FIT model is perfect (FIT = GEN). (a) If one dipole is placed at r to fit the map with no further constraints, then there is no model misspecification. However, (b) if the fitting dipole is constrained to a wrong orientation, even at a correct location, then there is model misspecification.
D.L. Jewett, Z. Zhang/Electroencephalography
and clinical Neurophysiology 95 (1995) 135-142
137
400 t
x
200 i
34L
~200 L, 4
Fig. 1. A set of datapoints and its least-square represent the noise-contaminated observations, sents the recovered data.
fitted line. The datapoints while the fitted line repre-
random noise added to each observation. If this assumption is correct, then the only question is how well the parameters of the best-fit line of the data analysis correspond to the parameters of the linear process that has been obscured by the noise. Thus, physically there is no model misspecification here because the model used in the analysis is the same as that used in generating the data. Now consider a situation with no noise, but with a form of model misspecification; in Fig. 2 the datapoints (solid circles) have been formed from a power function Y=Ax3
+B
(1)
where A = 1 and B = - 250. We again find the best-fit line (shown in Fig. 2, empty circles), and the equation of the line is y,=a,x+b
1
(29
where, in Fig. 2, a, = 174, and b, = - 1066. 3 In this case, the deviations of the datapoints from the line are systematic rather than random. (In statistical terminology, there is bias in this analysis.) Now if, as scientists, we wish to determine the characteristics of the process that generated the datapoints, and our analysis gives us the parameters of the linear fit (a, and b,), how should we interpret these parameters? Clearly, since there is model misspecification (i.e., the equation used in the fit is not the same equation as used to generate the data), there is no sense in which the a, and b, parameters can be correct, since it is the parameters of Eq. (1) (A and B) that are desired, not those of Eq. (2a). Indeed, the same datapoints can also be fit by other functions (other misspecified models), resulting in different fit parameters. For instance, if Y of Eq. (1) is fit by y,=a,x*
3 The fitted values are usually rounded to the nearest integer.
(2b)
Fig. 2. A function, Y = x3 - 250 (solid circles), is (linear least square) fit by (al y, = a,x+b, (empty circles), (b) yz = azx2 (empty squares), (cl y, = a3x2 +b, (empty diamonds), at x = 5, 6, 7, 8, 9, 10.
we have a2 = 5 (Fig. 2, empty squares). Or if Eq. (1) is fit by y, = a3xz + b,
(2c)
the fit results are a3 = 12, b, = -454 (Fig. 2, empty diamonds). y3 has one more term (b,) than y2, and the fit value for a3 is different from that of a2. It can be seen from Fig. 2 that the fit can be done by using different fitting functions, while none of the fits is perfect since the functions used in the fitting are each incorrect (i.e., not Eq. (1)). To reiterate: when different functions are used to fit a data set, different parameters (with different curves) will be fit. Even if the same function is used to fit the same set of datapoints, different fit values can occur if different functions are minimized (such as the absolute difference, or cubed difference) 4. Similarly, had some of the data points been missing, e.g., the one at x = 5, then the best-fit curve and parameters would have been different again. In all these cases, no matter what the criteria, since the model is wrong, so are the resulting parameters. This is equivalent to the SingGenErr of Zhang et al. (1994) and the error that may be reduced in the DSL using the Ary-type correction (Ary et al., 1981). Notice that, using the definitions of GEN and FIT, in Fig. 2, Eq. (1) is the GEN model, while Eq. (2) (a, b or c) is the FIT model. Note that in creating Fig. 2 the process was to establish GEN, choose specific values of x (5, 6, 7, 8, 9, lo), calculate the datapoints (a forward calculation to
4 The reasons that least-square fit has been widely used are: (11 It is the most useful estimation in statistics. The principle of least-square can be derived from the principle of maximum likelihood if the measurements follow a gaussian distribution. Squaring the differences means that greater importance is placed on removing the large deviations. (21 It is mathematically easy to handle. Differentiating a squared quantity leads to easy, linear terms. (3) Linear superposition holds under certain linear fits, therefore, the fitting process is simplified. (4) Adding independent effects produces additive sums of squares.
D.L. Jewett, Z. Zhang/ Electroencephalography and clinical Neurophysiology 95 (1995) 135-142
138
determine the values of y), establish FIT, and then calculate the parameters of FIT based upon the datapoints (an inverse calculation). This is directly analogous to the process in a DSL, where the biological system generates a potential map (which in a simulation would be the potentials created by GEN), and then the DSL, using the model FIT, by an iterative inverse calculation, determines which generator parameters minimize the squared differences between the observed potential map and the potential map calculated in FIT. Continuing the analogy between the example of Fig. 2 and a DSL, the values of x can be considered to represent different electrodes and the values of y represent the potentials detected at those electrodes. The parameters of Eqs. (2) (the “a”~ and “ b”s) might be considered to represent some parameters of the generator; however, note that the analogy with a DSL is not exact in all details.
900 i
450 1
x t -4501 4
5
6
7
8
9
f 10
Fig. 3. Two functions, Y, = -(x3/4> and Y2 = x4/16, are (linear least square) fit by 2 other functions, y4 = a4 x2 and ys = a5 x2, respectively, at ~=5,6,7,8,9,10.ThesumofY,andY~,Y,+~,isfitbyy~+~=~~+~x* over the same x-range. Since the same linear function (y4, ys and yi+ s) is used in different fits, there is no MulGenErr in the fitted parameters. i.e., a)4+ s = a4 + as, yi+s = y4 + ys. Also shown in the figure is the sum of y4 and ys, y4 + 5 (crosses, which is identical to y; + 5).
5. Multiple-generator error in a simple model
be the same when the best-fit solution (under FIT) is obtained to the following equation:
The preceding examples introduce the issue of models and SingGenErrs, but are not examples of MulGenErrs. MulGenErr can occur only when there are 2 (or more) generators that are simultaneously active, so we must enlarge our simplified model to include this situation. So let us have 2 generators which generate potentials at the electrodes according to the following equations:
Yi+5 = U4X2-I-a$
Y, =A,x3,
whereA,
= -l/4
(3a)
Y2=A,x4,
whereA,
= l/16
(W
These then are the 2 generators in GEN. A, and A, can be thought of as the 2 generators’ magnitudes, while x3 and x4 can be thought of as the 2 generators’ weighting functions (see later). The 2 different weighting functions (x3 and x4> mimic the 2 dipoles being at 2 different locations. In FIT we will do a linear fit against each individually:
Asking the question in another way, by rearranging the above equation: Y&S =
(a4
and then combining the a4 and a5 terms into a’4+s, Yi+5 =a;+5x
(4a)
ys = a5x2
(4b)
Solving in Eqs. (4) for the best-fit for each of the generators in Eqs. (31, yields the following values: a4 = - 2 and a5 = 5. a4 and a5 represent the 2 individually fitted magnitudes. Since the algebraic summation of the potentials of the 2 generators will occur at each electrode when the generators are simultaneously active, at each electrode on GEN, the following is true: YI+Z=yl+&
(3c)
We now ask the question: will the values of the coefficients of the best-fit equations (Eqs. (4a, 4b)), a4 and a5,
2
(4c)
we can then determine whether, when the summed data (Y1+*, Eq. (3~)) are fit by Eq. (4c), the results of the fit are just the summation of the individual fits, i.e., we are asking is the following equation true: a;+5 = a4 + a5
(44
The results from fitting Eq. (4~) to the summed data from Eq. (3~) are: ai+S = 3 a4 + US= 3
y4 = a4x2
+%)X2
(4e)
Note that MulGenErr in the fitted dipole parameters can be demonstrated either by comparing parameters (as in Eq. (4e)) or by comparing values of y, as determined by the parameters (see Fig. 3 legend). Comparing ‘‘y” values is equivalent to comparing spatio-temporal maps in a DSL. The numerical results of the above example are graphed in Fig. 3. The end result is that there is no MulGenErr in the fitted parameters. Thus, as soon as we know the fit results for Y1 and Y2 (a4 and a,> respectively, we know that the fit result for any linear combination of Y, and YZ, crYI + pY2, must be oa4 + pa,, where cwand p can be any values. The above results may at first glance seem surprising, since there is no MulGenErr in the fitted parameters even though both fits were non-perfect (wrong equations were
D.L. Jewett, Z. Zhang/Electroencephalography Table 1 A function refers to Function
y = ax y =8x2 y=ax+b y=ax2+b y=a2x
can be linear or non-linear,
depending
With respect to y Linear
Non-linear
a, x a
X
a, b, x a, b
X
X
a
on the variable(s)
and clinical Neurophysiology
it
Equation no.
5a, 5e 2b, 4a-c, 2a 2c
5b
used in the fitting); the plots of Y,, Yz and Y,+z are all curvilinear; and the most confusing part is that Eqs. (4a-4c) are all non-linear with respect to the variable “ x.” However, “x” is not the parameter that is being fit by the procedure; the parameters being fit in this example are a4, a5 and ar4+,, and these are linear in the example (Table 1). It is easy to make a mistake in trying to determine if a function is linear by looking only at the symbols used, since x, y and z are usually used as variables, while a, b and c are commonly used to denote constants (Table 1). Note especially that, when doing a ftt, the procedure fits
the equation’s constants (a, b and c), not the variable (x). The only condition under which there is no MulGenErr in the fitted parameters is when the parameters to be fit are linear (Table 2). So only if y is a linear function of the parameters to be fit, will the result be a linear fit. Clearly, there will be no MulGenErrs in the fitted parameters when the function used to fit the data is a linear function of x, as in yr of Eq. (2a), i.e., if y4, ys and yi+6 all have the form of Eq. (2a). Thus, this case illustrates the General Principle of MulGenErr, namely, there will be no
MulGenErrs in the fitted parameters when there is model misspecijication if and only if: (1) the equation used in the fitting of the summed potentials is the same as the equation used in the fitting of the potentials of the individual generators, and (2) the equation used is linear with respect to the parameter that is fit. The conditions that affect MulGenErr in the fitted parameters are shown in Table 2, which shows all possible combinations of the 2 independent requirements of the general principle. The 4 conditions of Table 2 are all assumed under model misspecification, i.e., no MulGenErr will occur in any of the conditions if there is no model misspecification. The conditions of Table 2 applicable to DSL are Ml and M3. In a DSL, the equations used in the fitting will always be linear relating to the dipole magnitude unless some constraint is imposed on the magnitude. Therefore, if only the dipole magnitude is concerned, DSL can only be within either NM or Ml of Table 1. However, as we will later show, the equations must be different, so only condition Ml applies to DSL magnitude calculations. When dipole location/orientation is concerned, DSL will involve condition M3 of Table 2, as location/orientation parame-
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95 (1995) 135-142
ters are non-linear parameters in the equations. Due to the complexity in relating condition M3 to the DSL, in the following, we will explain the Ml condition only. (The “no MulGenErr” condition NM has already been demonstrated above.) Condition Ml (Table 2) occurs when all fitting equations are linear, but all or some of them are different from each other. If the forms of the equations used to fit different individual data sets are different from each other, then it is not clear what function should be used to fit the summed data. With such differences in equations, MulGenErr will occur, and there is no reason to expect that it should not; yet this is analogous to some DSL fits. Taking the generator equations from Eqs. (31, we use the following FIT y6=a6x
(5a)
y7 = a7x2
(5b)
to fit Y, and Y2, respectively.
Another
function
Yk+7 = six + a;n2
(5c)
is fit to the sum Y, + 2 (Eq. (3~)). This can be thought of as (1) 2 individual dipoles, each at a different location (weighting functions x and x2 are different) within FIT, are used to fit 2 individual maps, respectively (Eqs. (5a, b)), and (2) 2 dipoles at the same 2 locations (weighting functions are x and x2, Eq. (5~)) are used simultaneously to fit the summed map. The fitted results are
a6 = - 18; a7 = 5 a’ = 8 a’ 6 = -43. ) 7
(5d)
MulGenErr is present since a’6 # a6 and a7 # a,. In this example, a6 and a7 represent the individually fitted dipole magnitudes, while a’, and a’, represent the simultaneously fitted dipole magnitudes. These results are plotted in Fig. 4. This is equivalent to comparing maps from GEN and FIT. It can be seen that neither y6 fits Y, nor y7 fits Y2 perfectly, because wrong equations were used in the fits. For instance, Y1 is proportional to x3 (Eq. (3a)), but the fit (using Eq. (5a)) assumes it is proportional to x (this might represent the case where
Table 2 The 4 combinations of 2 independent conditions MulGenErr will occur in the fitted parameters
that determine
whether
Function for sum compared to function for separate generators
Parameters Linear Non-linear
Same
Different
Nh4. No MulGenErr M2. MulGenErr
Ml. MulGenErr M3. MulGenErr
beingfit
Note that which equations are used in GEN does not affect the results in this table, just so long as they are different from those used in FIT, which means that the results apply to biological potentials being fit by simple models.
140
D.L. Jewett, Z. Zhang/Electroencephalography
and clinical Neurophysiology 95 (1995) 135-142
ual generators. For example, take the same generator equations from Eqs. (3), but make the fitting equations for the individual generators y6 of Eq. (5a) for Y, (unchanged), and
for Yz (different weighting functions x and x + 3 represent different dipole locations), and the equation to be fit to the sum Yi+s = a:+RX
Fig. 4. The same functions, YI, Y2 and Y, + 2, as in Fig. 3, are fit by y6 = a6 x, y, = a, x2 and yi + , = a’6x + a’, x 2, respectively, at x = 5, 6, 7, 8, 9, 10. Since different functions were used in different fits, MulGenErr occurs in the fitted parameters, i.e., y;, , f y6 +y,, a$ # a6, a’, # a7. As far as the goodness of fit is concerned, yi+ 7 fits Y, + 2 much better than either y6 fits Ye or y, fits Y2. But when y: + , is plotted into 2 parts, ‘, then the fit of yi to Y, is worse than yb to Y,, yA=a’,x and y;=a’,x and the fit of y; to Y2 is worse than y, to Y2. This is analogous to the DSL under model misspecification. I’~, Y2 and YI+z are similar to the potential maps generated by 2 individual dipoles and by the 2 dipoles active together, respectively. ah and a, are similar to the fit dipole parameters under single-dipole fitting, while al, and a’T are the fit parameters for the same dipoles under simultaneously active conditions. Although the fit results, a’6 and a’,, are totally different from the single-dipole fit results, a6 and a7, the 2-dipole fitting results in a much better fit to the data map (a much smaller NLSE).
the potential maps are recorded from a human head, but the DSL assumes the head can be replaced by a sphere). The NLSEs (normalized least-square errors, see Zhang and Jewett, 1993) for the fits of y6 to Y1 and y7 to Y2 are 11% and 7%, respectively. It is important to note in Fig. 4 that although yA+7 (Eq. (5~)) does not fit Y1 + 2 perfectly, the fit is much better than either of the 2 individual fits; the y: + , to Y1+z fit has an NLSE of only 1%. Moreover, if yL+ 7 is plotted as 2 parts (based upon the parameters of Eq. (5d)), y: = abx and y: = a’7x2, then the fit of y: to Y, is much worse than y6 to Y1, and the fit of yi to Y2 is much worse than y, to Y2 (see Fig. 4), even though the yA+7 fit is better than y6 or y7 fit. Note carefully this result: with 2 generators the NLSE fit is better to the 2-generator map than the l-generator fits to l-generator maps, yet the computed generator parameters produce individual maps that are worse than the maps from the l-generator fits. A low NLSE does not guarantee accuracy of parameters when there is model misspecification. Also small differences in maps can lead to large differences in derived generator parameters. Iterative searching to reduce a small NLSE may still further incrementally increase differences in generator parameters based upon small map differences (see also the last figures in Zhang and Jewett (1993) and Zhang et al. (1994)). MulGenErr will occur even if the function used for the sum differs from just one of the equations for the individ-
then the fit parameters are a6 = - 18, aB = 27, and a’6+8 = 21. This is equivalent in the DSL to fitting both the individual maps and the summed map by 1 dipole only, so the number of generators is misspecified in the fitting. The presence of MulGenErr is shown by the fact that a6 + a, # a’,+,. Under all the 4 conditions, NM, Ml, M2, and M3 (although we only gave examples under NM and Ml), there is model misspecification, which is equivalent to finding that the potential map cannot be completely fit under single-generator cases 5. Note that whether or not there is MulGenErr in the fitted parameters does not depend upon which equation is used in GEN. Indeed, GEN can be linear or not, just so long as the equations used in GEN are different from those used in FIT there will be MulGenErrs. A review of the examples will show that this is true, by trying different equations for GEN. The skeptical reader might want to re-analyze the first example with other GEN equations, since the first example is the only case where there is no MulGenErr in the fitted parameters (NM of Table 2).
6. Discussion of the analogies between the examples and DSL In a DSL one or more spatio-temporal maps are available either from a biological recording, or from a computer simulation using a GEN. The maps are analyzed to find the best-fit dipoles within FIT. There are 6 parameters for each dipole: 3 dimensions for location, 2 angles for orientation, and a magnitude. The orientation and magnitude parameters together can also be treated as the “dipole moment,” i.e., 3 magnitudes along 3 orthogonal axes whose origin is positioned at a specific dipole location. The surface potential generated by a dipole and the dipole parameters are related by a complicated function, involving both the dipole parameters and the model parameters and the electrode locations. There is a linear relationship between the surface potential and the dipole moment, e.g., doubling the magnitude of a dipole
5 Assuming the correct summed potential map.
number
of dipoles
is specified
in fitting the
L1.L. Jewett, Z. Zhang/Electroencephalography
moment doubles the potential. Hence, 2 dipole moments at the same location can be added vectorially, which is just the algebraic summation along each axis, independently. However, the relationship is non-linear between the potential and the dipole location (or the distances from the dipole location to the electrodes). Thus, the fit in a DSL includes both linear and non-linear parameters. Since the potential is proportional to the dipole magnitude, the potential can therefore be written as v(x)
=mw(x,
I)
(6)
where v(x) is the potential at electrode “x.” 1, is equivalent to the “y’s” in the examples. m is the dipole magnitude which is proportional to v. (For simplicity, we assume the dipole orientation is fixed in a given direction at each location, thus, only dipole magnitude and location are of concern.) m is equivalent to the coefficients to be fit in the examples (the “a’s”). w(x, r), sometimes called the dipole “weighting function,” is a complicated function of the electrode “ x,” the dipole location r, and other model parameters. It is equivalent to the x of Eq. (5a) or x2 of Eq. (5b). Since w is a function of the dipole location, x and x2 of Eqs. (5) represent 2 different dipole locations in FIT. Now, to go back to the first DSL example where we introduced the concept of the SingGenErr and the MulGenErr, when Vobsl and Vobs2 are each separately fit by a dipole within FIT, the fitting functions will be (7a)
v2( x) = m2w( x, r2)
P)
respectively. Eqs. (7a) and (7b) are 2 different functions, because usually 2 dipoles at different locations will be fit, thus w(x, r,) # w(x, r2) (analogous to x # x2 in Eqs. (Sa-b)). Now, let’s examine how the summed map Vobs = Vobsl + V&Z is fit by 2 dipoles within FIT. In DSL, the fitting equation is in the following form: v;+2( x) = m’iw( x, ri) + m;w( x, r;)
(7c)
which is quite similar to the sum of Y,(X) and v,(x). Comparing the situation here with those described in Table 2, it is apparent that different functions are used in the single-fits and the summed-fit. Therefore, MulGenErr will occur in the fitted dipole magnitudes, i.e., m’, # m, and m’, f m2. This is true whether the 2 dipoles in the summed-fit are forced at the previously fitted locations (e.g., r{ = ri and ri = r2, as in the example under condition Ml) or not. When the dipole locations are allowed to vary, then there may be other locations (i.e., other weighting functions) which will make the fit better than at the locations ri and r2, and there will be MulGenErrs in the non-linear fitted dipole locations. When the DSL moves the dipoles away from the single-fitted locations, the MulGenErr in the fitted magnitudes will change. This is be-
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cause moving the dipoles to different locations is mathematically equivalent to using different equations in the fitting. Therefore, a realistic DSL involves both conditions Ml and M3. There is also an analogy between the condition NM of Table 2 and the DSL under model misspecification, i.e., under certain conditions, there will be no MulGenErrs even when there is model misspecification. The conditions are fitting both the individual maps and the summed map with the same number of dipoles, with either of the following conditions: (1) at the same fixed locations with no constraint on the fitting dipole orientation and magnitude, or (2) at the same fixed locations and the same fixed orientations with no constraint on the fitting dipole magnitude. Mathematically, both conditions are equivalent to fitting both the individual maps and the summed map using the same weighting function(s). This was the case in the example demonstrating condition NM, where the same weighting function x2 was used in all the fits. It is quite obvious that this is not realistic - using the same dipoles (i.e., at the same locations and/or with the same orientations) to fit both the individual and the summed maps even though different dipoles generated the different maps.
7. Forms of model misspecification GEN and FIT can differ in a variety of ways, all leading to model misspecification. The following is a partial list of the ways GEN and FIT may differ: shape parameterization (e.g., from a real head to a sphere), number of layers (e.g., using a single layer model with uniform conductivity to represent the real head, where the head has multiple layers with different conductivities: scalp, skull, and brain), conductivity and/or thickness of layers, uniformity of layer conductivity or thickness, electrode locations, reference electrodes, number of dipoles, incorrect constraints over dipole parameters. The preceding list applies to both analysis at a single time-point, and analysis of multiple timepoints.
8. Conclusion
The basic concepts of MulGenErr involved in the DSL are described. We conclude that, under model misspecification, MulGenErr must occur in the DSL method described here, because the potential map due to a single generator (e.g., a dipole) cannot be 100% fit by a single generator within a misspecified model, When such maps are added together and fit by 2 simultaneously active generators in the misspecified model, effectively, each individual map is fit by 2 generators, which will fit each individual map better than if only 1 generator were used, but the fit generator parameters will be different from the
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individual fit. Therefore, without examining the goodness of a model, the residual error between the observed and the fitted potential maps after a fit cannot and should not be used as a measure of how good the fit is. It is quite clear if a map due to 5 generators is fit by 5 generators within a misspecified model, and another map due to 2 generators is fit by 2 generators using the same misspecified model, the 5-to-5 fit will be better (smaller residual error) than the 2-to-2 fit, but probably, the MulGenErrs in the 5-to-5 fit will be larger. We have shown that MulGenErr occurs directly from simple algebraic considerations. Since in any realistic situation where evoked response data are being analyzed, model misspecification is unavoidable, MulGenErrs will occur. The issue then becomes the size of these errors which must be determined for each case, given the specific model misspecification and the generators thought to be simultaneously active.
Acknowledgement This research was supported by Grant ROlDC00328 from the National Institute on Deafness and Other Communication Diseases.
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