Error-Bars in Semi-Parametric Estimation
D. van Ormondt,
R. de Beer
Applied Physics,TU Delft, NL
J.W.C. van der Veen
MRS Core Facility NIMH, NIH, Bethesda, USAD.M. Sima
ESAT-SCD KU Leuven, BED. Graveron-Demilly
CREATIS-LRMN, Univ Lyon 1CNRS UMR 5220 INSERM U1044, INSA
Lyon, FR
Abstract—In in vivo metabolite-quantitation with a magnetic resonance spectroscopy (MRS) scanner, the model function of the attendant MRS signal is often only partly known. This unfavourable condition requires semi-parametric estimation. In the present study the unknown part is the form of the decay function of the MRS signal. The lack of knowledge is caused by micro-heterogeneity of the tissue hosting the metabolites. At high magnetic field, i.e., ≥ 10 Tesla, it seems reasonable to assume that the decay function, although unknown, is the same for each metabolite species. As reported in Proc. ProRISC 2012, this assumption enabled us to circumvent the often cumbersome search in function space, normally required in semi-parametric estimation. Our present paper focuses on interpreting the semi-parametric error bar provided by some leading metabolite quantitation packages. This is done by means of Monte Carlo simulations.
Index Terms — metabolite-quantitation, micro-susceptibility, signal-decay function, semi-parametric, error bar
I. INTRODUCTION
In vivo Magnetic Resonance Spectroscopy (MRS) is the sole method capable of measuring concentrations of markers of disease – metabolites – on arbitrary locations inside patients, non-invasively; see. e.g., [1]. Metabolite concentrations are estimated from an MRS-signal by fitting to it an appropriate model function.
A perennial problem is that parts of the MRS model function are unavoidably unknown. In the present study, the unknown part of the model function is the form of the decay of the in vivo MRS signal as a function of time. This condition is caused by unknown inhomogeneity on spatial micro-scale of the magnetic field within a patient due to micro-susceptibility effects [2], [3].1
Fitting a known model function to data amounts to a search in parameter space. This is relatively easy, including attendant prediction of estimation errors. Fitting a partly unknown model function to data requires not only a search in parameter space but also a search in function space; the latter is cumbersome, not in the least with respect to prediction of attendant estimation errors [4].
Combination of parameter estimation and function estimation is called semi-parametric estimation [5], [6]. It is the only way of fitting partly unknown model functions to data. Unavoidably, it results in biased estimates of the parameters
1 Note that partial ignorance about all effects that possibly contribute to any measured data is the rule rather the exception.
of the known part of the model function.
This paper focuses on errors incurred in semi-parametric estimation. The subject is especially important in clinics where a spectroscopist may have to play a key role in diagnosing a patient on the basis of concentrations of certain metabolite species. In such cases reliable error bars on estimated concentrations are indispensable. Software packages used in clinics provide the Cram´er-Rao lower bounds (CRBs) derived from the estimated parameters as substitutes for the true errors. Given the fact that the theory of Cram´er and Rao does not hold under semi-parametric conditions, we derive errors – bias and standard deviations – by Monte Carlo simulations and compare these with related CRBs provided by software packages.
II. METHODS
A. Model function
An in vivo MRS signal, s(t), is complex-valued and is acquired in the time-domain; see, e.g., [7], [8]. Apart from noise it can be modelled by
s(t) = eıϕ0
M
X
m=1
cmdm(t) sm(t) eı (2π4νmt+ϕm), (1)
in which ı = √(−1), ϕ0 is an overall phase, ϕm is a
metabolite-dependent phase, t = n4t + t0is time, with 4t is
the sampling interval, n = 0, 1, . . . , N − 1, t0 a ‘dead’ time
put to zero in this study, and m = 1, . . . , M are the indexes of the metabolites. The names of the metabolites are given in Appendix I. Furthermore:
• cm is the amount or concentration of metabolite m. It
is the most important piece of information for clinicians. Note that we refer to the concentrations also by the term amplitudes. 2
• dm(t) governs the decay of the signal of metabolite m.
In in vivo measurements, the form of dm(t) is a priori
unknown due to heterogeneity of living objects.
– In MRS practice, one often approximates the un-known decay by the surrogate model function dm(t) = eαmt with αm < 0; see, e.g., [7]. This 2In fact, the concentrations and amplitudes are proportional to each other. ’Absolute’ concentrations in vivo are hard to obtain [8], [9].
7 7 7 7 7 7 7 7 7 7 7 ' ! ' ! ' ! ' ! ' ! ' ! ' ! ' ! ' ! ' ! 7 ! 7YQCQ 'Q \ 7Q Q!
Figure 1. Real part of the FFT of the simulated MRS-signal s(t) and
signals sm(t), m = 1, . . . , 10, of ten different (fractions of) metabolite species, calculated [10] for a magnetic field strength of 11.7 T. s(t) = d(t)trueP10
m=1ctruemsm(t). The sm(t), m = 1. . . . , 10, in the model function can be given the same decay as in the simulated signal. Then we speak of fitting with ’true’ decay. Alternatively, the sm(t) in the model function can be given an approximative decay function, e.g., exponential decay. See Subsec.II-B.
approximation causes a semi-parametric condition where Cram´er-Rao theory does not hold.
– In simulations, on the other hand, one is free to put dm(t) in Eq. (1) equal to the same form as that
used for generating the signal [11], [12]. This results in a parametric condition, i.e., where Cram´er-Rao theory holds. However, note that a CRB-calculation is strictly correct only when the true parameter values are used, while in clinical practice, only estimated parameter values are available. A Monte Carlo simulation enables one to mimic this situation and to study the ramifications of computing CRBs with estimated rather than true parameter values. – Anticipating that the truth will be revealed at B0≥
11.7 T, we speculate here that dm(t) is independent
of m. (B0 is the static magnetic field of an MRS
scanner.) Hence, the subscript m is dropped from
here on: The signal of each metabolite species has the same decay.
• sm(t) = P Km
k=1am,k eı (2πνm,kt + ϕm,k) is the a priori
known, non-decaying version of the model function of metabolite m, in which am,k, νm,k, ϕm,kare the relative
amplitudes, frequencies, and phases of individual spectral components of a metabolite model function. The signals sm(t) used in this study were calculated with the
soft-ware package GAMMA [10], adapted to our needs3. The magnetic field B0 was set to 11.7 T; see Fig. 1. Note
that in this Figure each sm(t) was multiplied by the true
decay function for reasons of presentation.
• 4νm, ϕm are unknown shifts of the overall frequency
and overall phase of sm(t) due to experimental
conditions.
B. Monte Carlo procedure
The Monte Carlo procedure at hand used L = 100 different realisations ` = 1, 2, . . . , L, of white Gaussian noise, each with the same standard deviation σnoise and mean µnoise = 0.
Adding these noise signals to the noiseless MRS-signal s(t) at the bottom of Fig.1, resulted in L noisy versions of s(t), indicated by s(t)`, ` = 1, 2, . . . , L.
Using the free MRS metabolite-quantitation package jMRUI
[7], the model function of Eq. (1) was fitted to all s(t)`,
yielding L sets of estimated parameters p`, including their
CRBs. For each set p`, we estimated µp = L1PL`=1p` and
standard deviation σp = √ 1 L−1 PL `=1(p`− µp)2 . The focus was on the concentrations of the metabolite species, i.e., on p = cm. In the sequel, concentrations are also referred to
as amplitudes.
The procedure was carried out for two versions of the model function d(t) in Eq. (1), namely:
Parametric vs Semi-Parametric
Parametric Case — d(t) is the same function as that used for generating the simulated signal; referred to as ’true decay’. The Fourier transform of this true decay is an asymmetric triangle convolved with a Gaussian function. Semi-Parametric Case — d(t) = eαt, where α is a free parameter of the model-fitting algorithm of the jMRUI; referred to as ’exponential decay’. The Fourier transform of exponential decay yields a Lorentz curve which is symmet-ric. Although mathematically convenient, exponential decay is only an approximation of the truth. Therefore, it renders model-fitting semi-parametric. Bias is incurred.
The next Section gives results obtained for the two cases. III. RESULTS
We have fitted the model function of Eq. (1) to the mentioned hundred noisy versions of the noiseless simulated signal shown in the spectral domain at the bottom of Fig.1, and this for both the Parametric and Semi-parametric Cases. Fig. 5 shows the spectrum of a noisy version. The algorithm used for fitting was
QUEST [13], [14] which is contained in the free
metabolite-quantitation package jMRUI.
In the parametric case, the common true decay was incor-porated in the database signals sm(t). As a result, QUEST
3Alternatively, this could have been achieved with the
Figure 2. Estimated concentrations (amplitudes) and error bars for 100 noisy versions of a noiseless signal, using algorithmQUEST. Results for metabolite m = 1. Parametric Case. See also the text and TableI. Error bars provided by jMRUI.
estimated the common decay constant αm= α to be 0 (apart
from a small random estimation error, of course).
In the semi-parametric case, the common decay was assumed to be d(t) = eαt which is only an approximation. QUEST
estimated α such that the sum of squared residuals is mini-mal. The attendant metabolite concentrations (amplitudes) are significantly biased.
Below we present the results of the concentrations. Those for the other estimated parameters are less important for clinical practice.
A. Parametric Case
In this subsection, the correct model function is known and used. Hence, all estimations should work out according to established theory. Fig. 2 shows the estimated concentration (amplitude) of metabolite #1, c(1), for each of the 100 noisy versions of the simulated signal, using algorithm QUEST. Column 2 of Table Ilists the mean of the estimated concen-trations µcm for each metabolite m. These are to be compared
to the true values ctrue
m listed in column 3.
Column 4 lists the resulting biascm = µcm − c
true m , m =
1, 2, . . . , 10. The bias is found to be very small, in accordance with parametric estimation.
Columns 5 and 6 list the standard deviations σcm, m =
1, 2, . . . , 10 and the means of the corresponding CRB’s, µCRBcm, m = 1, 2, . . . , 10, as computed by QUEST,
respec-tively. The noise level of the signal, needed for calculating
CRB’s, was obtained from the residue of the parametric fit. Ideally, σcm ≈ µCRBcm; this appears to be the case.
B. Semi-Parametric Case
In this subsection, the correct model function is only partly known. Ruling out ’magic’, one must expect the quality of the ensuing estimations to suffer, irrespective of mathematical prowess of the estimator at hand. Fig. 3 shows the estimated concentration (amplitude) of metabolite #1, c(1), for each of
Figure 3. Estimated concentrations (amplitudes) and error bars for 100 noisy versions of a noiseless signal, using algorithmQUEST. Results for metabolite m = 1. Semi-Parametric Case. See also the text and Table II. Error bars provided by jMRUI.
the 100 noisy versions of the simulated signal, using algorithm
QUEST. The positions of the dots follow the same trend as in
Fig.2. Should one be ignorant of ctrue
1 , as is the case in clinical
practice, all seems fine. But reality is different: Possible bias is not discernible. See below.
Column 2 of TableIIlists the mean of the estimates concen-trations µcm, m = 1, 2, . . . , 10. These are to be compared to
the true values ctrue
m listed in column 3.
Column 4 reveals that biascm= µcm− c
true
m , m = 1, 2, . . . , 10,
is significant, as can be expected in semi-parametric estima-tion. However, ctruem being unknown in clinical practice, this
column is not available there. In other words, a clinician cannot be aware of the gravity of the situation w.r.t. bias. In fact, (s)he has only estimated CRB’s to rely on. ’Forewarned be forearmed’.
Table I
MONTECARLO SIMULATION OF PARAMETRIC ESTIMATION WITH ALGORITHM QUEST APPLIED TO100NOISY VERSIONS OF A NOISELESS SIGNAL. ESTIMATION RESULTS ARE LISTED FOR THE CONCENTRATIONS
(AMPLITUDES)OF METABOLITESm = 1, 2, . . . , 10. DETAILS ARE GIVEN IN THE TEXT.
m µcm † ctrue m biascm σcm ‡ µ CRBcm† 1 1.7541 1.7550 -0.0008 0.0069 0.0071 2 0.1923 0.1921 0.0002 0.0081 0.0093 3 2.0665 2.0673 -0.0008 0.0073 0.0074 4 1.3582 1.3563 0.0019 0.0067 0.0070 5 0.6330 0.6330 -0.0000 0.0055 0.0059 6 0.1472 0.1472 -0.0000 0.0072 0.0069 7 0.5523 0.5527 -0.0004 0.0036 0.0042 8 2.9743 2.9742 0.0001 0.0084 0.0083 9 0.6611 0.6609 0.0003 0.0062 0.0069 10 0.1208 0.1199 0.0009 0.0060 0.0063
m µcm c
true
m biascm σcm µCRBcm σCRBcm ratiocm ratio
∗ cm 1 2.1937 1.7550 0.4387 0.0080 0.0143 0.00009 1.2500 1.2583 2 0.2288 0.1921 0.0367 0.0111 0.0217 0.00014 1.1911 1.1718 3 2.5850 2.0673 0.5176 0.0079 0.0143 0.00010 1.2504 1.2586 4 1.6943 1.3563 0.3380 0.0078 0.0139 0.00009 1.2492 1.2549 5 0.8960 0.6330 0.2630 0.0074 0.0137 0.00009 1.4155 1.4354 6 0.1980 0.1472 0.0508 0.0097 0.0156 0.00011 1.3447 1.3611 7 0.7104 0.5527 0.1577 0.0046 0.0091 0.00006 1.2853 1.2924 8 3.7260 2.9742 0.7517 0.0088 0.0160 0.00010 1.2527 1.2605 9 0.7762 0.6609 0.1153 0.0075 0.0136 0.00009 1.1745 1.1794 10 0.1446 0.1199 0.0247 0.0076 0.0133 0.00009 1.2060 1.1942 Table II
MONTE CARLO SIMULATION OF SEMI
-PARAMETRIC ESTIMATION WITH ALGO
-RITHM QUEST APPLIED TO 100 NOISY VERSIONS OF A NOISELESS SIGNAL. ES
-TIMATION RESULTS ARE LISTED FOR THE CONCENTRATIONS(AMPLITUDES)OF METABOLITES m = 1, 2, . . . , 10, USING THE SAME SYMBOLS AS IN TABLE I. ratiocm = µcm/c
true
m , AS IS ratio ∗
m, BUT
FOR THE NOISELESS VERSION OF THE SIG
-NAL. µSTANDS FOR MEAN, σFOR STAN
-DARD DEVIATION. SEE TEXT.
Columns 5 and 6 contain the standard deviations of the concentrations σcm, m = 1, 2, . . . , 10, and means of the
corre-spondingCRB’s, µCRBcm, m = 1, 2, . . . , 10, over the estimates
for ` = 1, 2, . . . , L. Unfortunately, these informative statistical ensemble averages are not available in clinical practice either. This is because a clinician usually measures only one instance of a signal, amounting to L = 1. The Monte Carlo study revealed that in the semi-parametric case, σcmand µCRBcm may
differ significantly; an explanation will be given in Sec. IV. As in the parametric case, the noise level of the signal was estimated from the residue of the fit 4.
Column 7 brings the reassuring result that the standard deviation of the CRBcm,`, l = 1, . . . , L, is small. This implies
that CRB’s do not change much as a function of the noise realisation `. The same should hold for the parametric case, of course. Therefore, column 7 is omitted from Table I. Column 8 shows that the quantity ratiom= µcm/c
true
m depends
on the metabolite species m. For the present signal at least, it implies that bias is not proportional to concentration. Such proportionality could be expected in the case that FFT(sm(t))
has the same shape for each metabolite m and does not overlap with the spectra of other metabolites.
Column 9 lists the same quantity as does column 8, but for noise put to zero. The results indicate that the noise level had not much influence on the value of ratiom.
IV. DISCUSSION
Rationale As mentioned in Sec. I, Introduction, quoting a reliable error bars on estimated metabolite concentrations is indispensable in clinical practice. It is a perennial problem yet to be solved in semi-parametric context. An important aspect of the problem is how to be aware of shortcomings of the model function at hand in the first place. Almost all practitioners of MRS are well aware of the presence of a so-called background signal originating from macro-molecules. A myriad of methods for accommodating the macro-molecule signal is available; see, e.g., the very recent papers [15], [8]. On the other hand, ignorance of the true decay function, and its impact gets rather less attention. As an illustration of this we mention that many papers quote the width at half height
4See Sec.IV, Discussion, about the rationale of this procedure.
0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude 0 5 10 15 20 25 30 1.7 1.8 1.9 2 2.1 2.2 2.3 Occurrence Amplitude QIER ! WXHIZ ! '6&" ! QIER ! WXHIZ ! '6&" ! FMEW ! !
XVYI HIGE] I\TSRIRXMEP HIGE]
1IXEFSPMXI
1SRXI 'EVPS WMQYPEXMSR [MXL RSMWI VIEPMWEXMSRW %QTPMXYHI IWXMQEXMSR [MXL X[S JSVQW SJ HIGE] XVYI ERH I\TSRIRXMEP
8VYI EQTPMXYHI !
Figure 4. Histograms of the estimated concentration of metabolite m = 1 using true decay (green, parametric) and exponential decay (magenta, semi-parametric). The green histogram is centred on the true concentration of 1.755, as is to be expected. The magenta histogram is shifted far beyond its width, to 2.19. <CRB> stands for µCRBc1. This shift too, is to be expected.
of spectral lines, but not the shape of the spectral lines, tacitly assuming that it is Lorentzian, i.e., the Fourier transform of exponential decay. Probably this has to do with the advanced state of the art of automatic magnetic field shimming, provided by scanner manufacturers. Shimming alleviates magnetic field inhomogeneity attendant on in vivo measurements. Apparently, one expects the residual inhomogeneity to have a Lorentzian distribution. This remains to be seen for the effect of mag-netic susceptibility variations on a spatial micro-scale (micro-susceptibility).
Awareness of errors in the clinic The present study focused on errors of error bars caused by ignorance of the true decay function. Fig. 4 summarises the results given in Sec. III in the form of histograms, for metabolite m = 1. The green histogram represents parametric estimation, i.e., using the true decay. In accordance with theory, it is centred on the true con-centration. The magenta histogram represents semi-parametric estimation, i.e., using the approximative exponential decay. As is to be expected a sizeable bias is incurred, implying that
VIWMHYI
Figure 5. Bottom: Real part of the FFT of the non-exponentially decaying signal s(t)1 used in the present Monte Carlo study, based on fit algorithm
QUEST. Also shown, above the black line, are the real parts of the FFT’s of two residues of fitting an exponential decaying model function to this signal. The lower residue is for the noise level used in this paper. The effect of using an incorrect decay function (exponential) is clearly visible. The upper residue (arbitrary vertical scale) is for a ten times higher noise level. Prior to FFT, the latter residue was apodised to reduce noise. The effect of using an incorrect decay function (exponential) is hardly visible for this high noise level.
the error bars in Fig. 3 are much too small. The clinician, analysing a single, once-only measurement gets no warning to that effect. This unawareness constitutes the main message of this paper.
Can a residue reveal an incorrect decay function? An am-endment to the above is in order. The metabolite-quantitation software package jMRUIprovides graphics showing the residue of fitting a model to a signal. Fig. 5 shows the FFT of the signal related to noise realisation # 1, s(t)1, and the
residue of fitting to it the model function of Eq.(1) with exponential decay. In this study, we deliberately chose the level of the noise somewhat low to make the effect of using an incorrect model function of the decay clearly visible in a graphical presentation of the residue. By checking the residue, an experienced clinician can infer that the fit is not optimal, i.e., besides noise, distinct features indicative of a possibly incorrect model for the decay may be visible. Such features forebode biased estimates of the metabolite concentrations. Note, however, that this warning virtually disappears at 10× higher noise level, as shown in the top of Fig. 5. However, despite the obvious absence of clearly visible residual peaks, the Monte Carlo simulation still found a bias of four times the standard deviation, i.e., biasc1 = 0.320 and µCRBcm = 0.084.
Of these two numbers, a clinician gets only the latter. Hence, unknowingly, (s)he reports a much too small error. See also [16].
Noise level estimation At this point, the discrepancy between columns 5 and 6 of Table II can be explained. Column 5 lists the mean value µCRBcm of the CRBs of the L = 100
concentrations cm,` calculated by QUEST for each metabolite
m. Ideally, µCRBcm should approach the standard deviation
σcm of the 100 estimated values of cm listed in column 6.
This provides a test of estimated CRB values. According to theory, aCRBis proportional to the noise level σnoise at hand.
With jMRUI, σnoise can be obtained from three sources: i) The
residue of the fit, ii) the temporal tail of the signal assuming that it has decayed to below the noise level well before the end, iii) prior knowledge. When the model function is incorrect, or the signal has not decayed to below the noise level well before the last sample, alternatives i) and ii) cannot be used. If prior knowledge about the noise level is not available, alternative iii) cannot be used either. This constitutes a practical problem. In the present paper, alternative i) was chosen, for illustrative reasons5.
Finally, we point out that in semi-parametric estimation, µCRBcm need not be proportional to σnoise when the noise level
is estimated from the residue of the fit. This is because the relative contribution of the noise to the residue steadily drops with decreasing noise level, as explained above. A Monte Carlo simulation is the method of choice to detect this.
V. CONCLUDINGREMARKS
• This work concerns quoting error bars for parameters estimated from measurements done only once in a clinic.
• We investigated the matter by means of Monte Carlo simulations.
• The simulations were done for two cases:
1) Parametric case, i.e., the model function is correct. 2) Semi-parametric case, i.e., the model function is only
partially correct.
• In the parametric case, Cram´er-Rao bounds (CRBs) esti-mated from a single simulated noisy signal provide error bars close to those estimated from a corresponding Monte Carlo simulation. This positive result confirms theoretical expectation.
• In the semi-parametric case, CRBs estimated from a single simulated noisy signal can provide deviant error bars differing up to hundreds of per cents from those estimated by Monte Carlo simulation. This complication is unavoidable. Specifically, we mention:
– Estimation of the noise level in a signal can be sub-stantially wrong when the model function is partly unknown. EstimatedCRBs, being proportional to the standard deviation of this noise, are then equally wrong.
– Depending on the level of the noise, the ratio (bias / standard deviation) can be anywhere between infinity and zero: The lower the noise level, the higher the ratio, and vice versa.
5If the signal is sampled till well after its decay to below the noise level, alternative ii) is to be preferred. In our simulation this condition is not satisfied.
– A non-expert user of metabolite-quantitation soft-ware has little means to be asoft-ware of a specific bias, let alone being capable of estimating its size.
ACKNOWLEDGEMENT
This work was done in the context of FP7 - PEOPLE Marie Curie Initial Training Network Project PITN-GA-2012-316679-TRANSACT, http://www.transact-itn.eu/index.php, based in part on the free software package jMRUI, http://www.mrui.uab.es/mrui/.
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APPENDIXI
Table III
NAMES OF METABOLITES INVOLVED IN THIS WORK
m Abbreviation Metabolite name
1 Cho1 choline singlet
2 Cho2 choline multiplet
3 Cr1 creatine singlet 1
4 Cr2 creatine singlet 2
5 Glu glutamate
6 Gln glutamine
7 mI myo-inositol
8 NAA1 NAA singlet
9 NAA2 NAA multiplet
