estimating residual noise in the auditory brain-stem response

Estimating residual noise in the auditory brain-stem response Margaret Sullivan Pepe Fred Hutchinson Cancer Research Center, Division of Public Health Sciences, Department of Biostatistics, 1124 Columbia Street, MP-665, Seattle, Washington 98104

Stephen T, Neely Boys Town National Research Hospital, 555 North 30th Street, Omaha, Nebraska 68131

(Received 25 July 1994; accepted for publication 2 May 1995)

Estimation of the residual noise in the auditory brain-stem response waveform is considered. The residual noise measures one aspect of waveform quality. Moreover, it is an important component of signal detection algorithms used for automatic termination of the test. It is shown that the most commonly used method for estimating residual noise can be severely biased. Reasons for this bias are explored and two alternative estimators are presented. ¸ 1995 Acoustical Society of America.

PACS numbers: 43.64.Ri, 43.66.Yw, 43.60.Cg

INTRODUCTION

The auditory brain-stem response (ABR) is widely used in the clinical evaluation of heating. ABRs can provide in- formation about the functional integrity of the auditory system and are especially useful for patients who are difficult to test by conventional behavioral audiometry, such as infants. Auditory-evoked potentials (such as the ABR) can be recorded from surface electrodes on the scalp in response to auditory stimulation. In addition to the ABR, the electrodes will also record unwanted potentials due to other ongoing neural activity, muscle activity, and nonphysiological envi- ronmental sources. The primary technical problem in recording the ABR is to minimize the influence of these unwanted potentials which we refer to collectively as background noise. The first step toward achieving a reduction in background noise is to use synchronous averaging of ABR recordings from many repetitions of the stimulus. The most common approach to synchronous averaging is to record a fixed number of stimulus repetitions (sweeps). A more efficient approach is to continue the averaging process until the residual noise in the averaged waveform is brought below a criterion level. Such an approach has been discussed by El- berling and Don (1984) and more recently by Sininger (1993). Accurate methods of estimating the residual background noise are essential for such measurement-based stop- ping rules and are an important component of objective detection algorithms. This paper is concerned with estimation of the residual noise.

Let us now rigorously define the term residual noise. If sk(t ) denotes the observed signal on the kth sweep at time t relative to the start of the sweep and 3(t)= 5•k/v= •s•(t)/N denotes the average value after N sweeps, then the residual noise at t is defined as RN(t) =var(3(t)). Because the noise is thought not to be synchronous with the stimulus, the residual noise in the averaged waveform is generally assumed to be constant across the poststimulus time window.

An efficient way to estimate the residual noise recom- mended by Elberling and Don (1984) is to select a single

fixed point within each sweep and to compute 1/N times the variance in the sample values observed at that point across all sweeps. This estimator can be written as RNmss=(1/N2)X;•/v__ ](s•(t)-•t)) 2, where t denotes the location of the single point chosen. The subscript mss denotes mean sum of squares. The validity of the estimator relies on the assumption that sweeps are independent. In this paper we show that the assumption of independence is violated in a set of neonatal recordings and that this violation leads to seri- ously biased estimates of residual noise. Two alternative estimators are proposed which are shown to have significantly improved performance. These estimators rely on fewer as- sumptions than the mean sum of squares estimator and are proposed for incorporation into future ABR signal detection algorithms.

I. MATERIALS AND METHODS

Data are presented for 14 babies between 33 and 52 weeks conceptual age (28 and 42 gestational age). These babies were selected as subjects for the Identification of Neo- natal Heating project, a multicenter study sponsored by the National Institute on Deafness and Communication Disor-

ders. The data described here were obtained after the main

study protocol had been completed for each baby. For ABR recording, babies were in open cribs, in a quiet but not sound-treated room, and were generally sleeping. All babies were on battery powered monitors during the testing. Elec- trodes were placed at the vertex of the head, at both mas- toids, and at the C7 region on the back of the neck.

The hardware used for the ABR recordings consisted of a PC with an Ariel DSP-16+1/O card, an Otodynamics neonatal probe, an Intelligent Hearing Systems Opti-Amp 2000 evoked potential (EP) amplifier, and a custom-designed in- terface box. The "click" stimulus was a 100-ms rectangular pulse calibrated to produce a 30 dB nHL peak (rarefaction) pressure in a reduced-volume HA-1 acoustic cavity. (The units "dB nHL" indicate decibels re: normal adult thresh-

olds. The 30 dB nHL stimulus was calibrated to have peak

2056 J. Acoust. Soc. Am. 98 (4), October 1995 0001-4966/95/98(4)/2056/6/$6.00 ¸ 1995 Acoustical Society of America 2056

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sound-pressure level of 65 dB pSPL which is equivalent to 35.6 mPa.) Using custom software (called ̂ BR•V) the I/O board was programmed to use a 50-kHz sample rate for both stimulus generation and EP recording. Although two chan- nels of EP were recorded, only the recording from vertex to mastoid was used in the analyses contained in this paper.

The responses to 4000 stimuli were recorded in blocks of 20 sweeps. Each sweep contained 2048 sample values at 20-ms intervals spanning a 41-ms time window poststimulus. The sweep repetition rate was 24.4 sweeps/s within each block of 20 sweeps. An additional 78-ms time interval elapsed between blocks to allow for uploading of data to the PC. For each block, data stored on the PC consisted of the

block average for each of the 2048 sample values, along with the 20 single point values at 17.5 ms from the start of each sweep in the block. The EP amplifier has a gain of 200 000 and the analog-to-digital converter (ADC) converts _+ 10 V to 16-bit integers. Thus each unit of the ADC values corre- sponds to about 1.5 nV between the EP electrodes.

Of primary interest to us was the stored vector of 4000 data values, one for each sweep, at the single point 17.5 ms poststimulus. From these data we estimated the residual noise in the averaged waveform for that point.

II. ESTIMATED RESIDUAL NOISE ASSUMING INDEPENDENCE

Consider a single point relative to the start of each sweep and recall that sk denotes the value at that point in the kth sweep. As an estimate of the evoked potential/x at this time point we use 3-= 5;k• •(s•/N), where N denotes the number of sweeps available. The residual noise in the average is the variance of this average:

' mXl=var(s-) = var k=l

2½ var(s0 k=l

q- • • cov(s•,sj) , (1) k= 1 jg:k

where cov(s•,sj)=E[(s•-ix)(sj-lx)]. If the noise contri- bution to {sk} is stationary and we denote var(s0=0- 2 and COV($ k ,$ k + i) = 0'2 (i), then the residual noise expression (1) can be rewritten as

var(s-)--• 1+2• 1- 0-2 i=1

=N 1+2 1- p(i) . (2) .=

The quantity p(i)=[0-2(i)]/0- 2 is called the ith term in the autocorrelation sequence (Fuller, 1976). This quantity indi- cates the strength of association between single point data values separated by i sweeps. It lies in the range [-1,1] with values of -1 and 1 indicating maximal association. If the single point values are independent of each other, then p (i) = 0 for all i.

(a)

beby=:l

0 •

-2 i • bsby==4

0 o

o

-2 [ •

bsby==2

baby==5

oVo o o baby==7

0

-2

•b2b

baby==3

baby==6

20 40

numOep of sweeps sepepal:}ng poznts (•)

bsby==8

• 0 o 4-• o

r- baLv:=:t 1

ClJ 2

.ID • o ,.-., ..• -;2 "• baby==14

o

o o .4, 0 o

o

-2

•. 1020 4b

(b)

bSby==9

bSby==:•2

o

o

' • •b2b ,•0 8c•

o

8½

bSby==10

baby==13

;•b2b 4b 8•

numben of sweeps sepenat•ng po}nts (•)

FIG. 1. Serial autocorrelations for single point values from 14 babies. Val- ues are truncated to the range [-0.2, 0.2]. Values are displayed for i= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 40, and 80.

The usual single point estimate of the residual noise is based on the assumption of independence among sweeps, so that the autocorrelations are zero and hence RN=0-2/N. An approximately unbiased estimate of 0 '2 is the mean sum of squares, %•[(s•-s-)2/N]. This is denoted by var(SP) by Elberling and Don (1984) and Sininger (1993) and the result- ant estimate of the residual noise is the mean sum of squares estimator

RNmss = • N ' k=l (3)

III. DEPENDENCE BETWEEN REPLICATIONS

We now evaluate the assumption that the autocorrelations are zero. The autocorrelation between single point data values can be estimated using the formula

N-i

/3(i)= • (Sk--S-)(Sk+i--S-) k=l N-i •=• N

(4)

For each of the 14 babies, the correlations between single point values i sweeps apart were estimated for i = 1,2, 3 .... ,8, 9, 10, 20, 40, and 80. These autocorrelations are displayed in Fig. 1. The plots suggest that for most babies, data

2057 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 M.S. Pepe and S. T. Neely: ABR residual noise estimation 2057


TABLE I. Serial autocorrelations of single point values for i= 1,2,3,20,40,80. P values [calculated using a simple linear regression of (sk-s-) on (sk_i-s-)] are shown in parentheses.

Number of sweeps separating points (i)

Baby 1 2 3 20 40 80

1 -0.05(<0.01) -0.01(0.59) 2 -0.19(<0.01) -0.01(0.46) 3 -0.25(<0.01) 4 -0.32(<0.01) 5 -0.07(<0.01) 6 -0.64(<0.01) 7 -0.01(0.73) 8 -0.04(<0.01) 9 -0.12(<0.01)

10 -0.34(<0.01) 11 -0.13(<0.01) •2 -0.52(<0.0•) 13 -0.14(<0.01) 14 -0.29(<0.01)

0.05(<0.01) -0.004(0.82) 0.011(0.49) -0.011(0.50) 0.06(<0.01) 0.029(0.07) 0.032(0.05) -0.004(0.82)

0.14(<0.01) -0.11 (<0.01) -0.067(<0.01) -0.057(<0.01) -0.021 (0.19) 0.14(<0.01) -0.14(<0.01) -0.087(<0.01) -0.043(<0.01) 0.07(<0.01) 0.01(0.70) -0.03(0.04) 0.080(<0.01) 0.44(<0.01) 0.33(<0.01) -0.192(<0.01) -0.269(<0.01)

-0.05(<0.01) -0.00(0.91) 0.046(<0.01) 0.016(0.32) 0.00(0.89) 0.01(0.46) -0.029(0.06) 0.023(0.15) 0.01 (0.46) -0.00(0.92) 0.005(0.77) -0.029(0.07) O. 16(<0.01) -0.11 (<0.01) -0.050(<0.01) -0.041 (0.01) 0.03(0.05) 0.01(0.68) 0.017(0.28) 0.010(0.52) 0.32(<0.01) -0.17(<0.01) -0.063(<0.01) -0.168(<0.01) 0.11(<0.01) -0.11(<0.01) -0.033(0.03) -0.020(0.20) 0.17(<0.01) -0.13(<0.01) 0.003(0.84) 0.020(0.21)

0.003(0.87) 0.024(0.13) 0.054(<0.01)

-0.020(0.22) 0.013(0.42)

-0.004(0.81) -0.024(0.13)

0.006(0.69) 0.038(0.02) 0.017(0.27)

-0.001(0.96)

points separated by fewer sweeps are more strongly correlated than those separated by a larger number of sweeps. The autocorrelations with the first adjacent sweep, p(1), were negative in all cases and statistically significant (p<0.01) in 13 of the 14 babies. Values are displayed in Table I. The negative correlation between consecutive sweeps suggests that values above the mean on one sweep tend to be followed by values below the mean on the next sweep. This oscillation for a single baby (baby 4) can be seen in Fig. 2 where the single point values for 100 sweeps are plotted in sequence. Figure 3 displays a scattergram of values on one sweep plotted against values on the previous sweep for this same baby. Again, the strong negative correlation is obvious.

Other terms in the autocorrelation sequence were also nonzero. Whenever the first term in the sequence had a large negative value the second term would tend to be positive, as would be expected from the pattern of oscillation seen in Fig. 2. Beyond that, however, the only consistency across babies is that the correlations tend to decrease as the number of

sweeps i separating the data points increases, with the largest correlation for i= 1. Correlations at i=40 were small, but

significantly different from zero in 7 of the 14 babies. At

BABY - 4

7354 53 -

_

_

_

-8786 47 I •ooo • 1 o

sweep

FIG. 2. Observed data at the single point plotted against sweep number for 100 consecutive sweeps. These data were derived from baby 4.

i-80 correlations were statistically significant in only two babies (babies 6 and 12).

IV. ALTERNATIVE ESTIMATES OF THE RESIDUAL NOISE

The above results showed that sweeps cannot be assumed to be independent because the autocorrelations p(i) are nonzero. Hence the estimator RNms s which relies on the p(i) being zero is a biased estimate of the residual noise. To reduce the effect of this bias, one strategy is to estimate the autocorrelation sequence from the data using the expression for •(i) in (4) and to substitute these •(i), along with an estimate of 0 '2 into Eq. (2). The resulting expression for the residual noise estimate is

RNc = RNmss' I q- 2 • 1 - IS(i) . (5) i=1

We call this the co•ected me• sum of squares esfimaton The factor in brackets in Eq. (5) co•ects the •m• estimate for possible dependence among sweeps. In many instances, it will be reasonable to choose a number I such that p(i) is

25899.5

BABY = 4

. .

_

.['

,.

.'• -26009 5

-26009.5 25899 5 signal(k-i)

FIG. 3. Scattergram of data at the single point for baby 4, with the value on one sweep plotted against the value on the previous sweep. The negative correlation is statistically significant (p < 0.01 ).



negligible for i> I. That is, data points more then I sweeps apart may reasonably be regarded as uncorrelated. The corrected mean sum of squares then can be written as.

i=1

and we call this the mean sum of squares corrected for dependence of order I. This estimate will be computationally more feasible in practice as I is likely to be much smaller than the number of sweeps N. It is also likely to be statistically more stable as fewer terms are to be estimated.

A simple alternative way to reduce the bias in RNmss due to correlations amongst sweeps is based on segmenting the single point values into consecutive blocks of equal size and computing means within each block. If blocks are of size B, the vector of N single point values is reduced to an N B- N/B

vector of block means {•1 ..... ]NB }. If block sizes are reasonably large, and serial correlations decrease as the number of sweeps separating points increases, then block means are likely to be substantially less correlated than are data points in adjacent sweeps. Since the average waveform can be writ-

ten as the average of block means, • = 5;• l(•b/NB), and now assuming that block sizes are sufficiently large to render the block means approximately uncorrelated, the residual noise can be written as

1

var() = (7) which can be estimated with

RN•u(B) = •aa N• ' (8) We call this the block average estimator of the residual noise with blocks of size B. In the next section we show that the

block average estimator and the corrected mean sum of squares estimator provide better estimates of residual noise than the usual estimator when sweeps are dependent.

V. COMPARISON OF NOISE ESTIMATES

For each baby we compared the corrected mean sum of squares estimate RNc(I) to the uncorrected value gNms s for various levels of correction. The ratio of these estimates,

RNc(I)/RNms s , is displayed in Fig. 4 for I=0, 1, 5, 20, 40, and 80. Observe that at I-0 there is no correction; hence the

ratio'RNc(0)/RNmss = 1. Our analyses of the autocorrelations suggested that they were negligible for I> 80. Therefore, we regard the residual noise estimate correcting for I= 80 as unbiased. Figure 4 shows that the correction significantly decreased the estimate of the residual noise in comparison to the value estimated by the usual method. In other words, the usual estimate tends to provide an overestimate of the amount of residual noise.

Also shown in Fig. 4 are the block average variance estimates, again displayed as a ratio relative to the usual variance estimate. For each value of I, the block size was

taken as 21, so the displayed values are RNata(2I)/RNms s . The amount of correction achieved by RNat• relative to the

baby==1

.25

• baby==4

• 5 o o

• 0 [] ['

.5

(a)

baby==2

baby==5

05 20 d•

baby==8 baby==9

• .5 .25

:::3 baby==11 baby==•2

• .75 • • 5 g o

0 ii I i i

• .75 5 • •

• o

(b)

b•3by==3

bOby==6

baby==10

..... t' • ]

baby==13

FIG. 4. Ratio of the corrected estimator to the usual mean sum of squares estimator [RNc(1)/gNms s denoted by +] and ratio of the block average estimator to the mean sum of squares estimator [RNBta(21)/RNms s denoted by ̧ ]. Values are plotted against I for != 0, 1, 5, 20, 40, and 80.

block size depends strongly on the type of autocorrelation in the data. Therefore, the interpretation for RN•t•(2I) as correcting for a given number of autocorrelation is not as straightforward as it is for RNc(I). In Fig. 4 we plotted RNc(2I)/RN. ms s against I simply because the average dis- tance between single points in adjacent blocks is then approximately I. The plot demonstrates that the block average variance estimates are also significantly smaller than the usual variance estimate for most babies. This is further evi-

dence that the usual estimate is biased in our data and pro- Vides an overestimate of the residual noise. At I--80 the

block mean and corrected estimators agree rather closely. The above results suggest that the usual estimate of the

residual noise is biased, but are not definitive since the true residual noise is unknown in each case. Only estimates of it are available and compared with each other. Computer simu- lations in which data were repeatedly generated from a statistical model allowed us to calculate the true residual noise

of the averaged waveform for the data from the model and to evaluate the various estimates of it. The model was devel-

oped to mimic data from one of the babies • (baby 4). The model, which is termed an autoregressive model (Fuller, 1976) in statistical literature and linear prediction in speech analysis literature,. is a linear regression model of the single point data s k as predicted by prior data points. The model which fit the data from baby 4 is



TABLE II. Estimates of residual noise from 1000 data sets with each data set generated by a computer from a statistical model based on data from baby 4. The true residual noise is 5994. Displayed are mean estimates and standard deviation in parentheses. The usual estimate had a mean of 9365 (s.d.-246).

1 5 20 40 80

Corrected estimate

RNC(i) 10 360(405) 5842(597) 6176(956) 6086(1268) 5914(1733)

Block average estimate RNB•t(2I) 9866(331) 7835(569) 6645(945) 6417(1284) 6249(1791)

5

s k = • ajs k_j + ca, (9) j=l

where the terms ea have mean zero. The estimated parameters were a• = -0.306, a 2 =0.017, as =0.118, a4 = -0.04, and a5=0.069 and the standard deviation of ea was estimated to be 5737.5.

In each simulation, we generated a data set of 4000 single point values from this model (9) with ea normally distributed using the parameters described above. The first five values were calculated assuming sj: 0 for j •< 0. Simu- lated data points from the model have mean 0 and have variance and autocorrelations similar to the single point values for baby 4. One-thousand simulated data sets were generated. For each data set we calculated the mean } and the residual

noise estimates RNms s , RNc(I), and RNaM(2I) for I= 1, 5, 20, 40, and 80. The true residual noise var(s-) was calculated as the variance'of the } over the 1000 data sets and was

found to be RN=5994. This is, by definition, the residual noise in • the average. Table II summarizes the estimates of this quantity obtained in the simulated data sets. The usual estimate produced values in the range 9365+_492 in 95% of the data sets. Clearly this estimate is biased too large, being substantially larger than the true residual noise value of 5994. On the other hand, the estimates which account for the dependence among the data points are much closer to the true residual noise. The corrected mean sum of squares estimate provided good estimates even with I= 5. The block average estimator required segmenting the data into blocks of 40 or more to yield valid estimates of residual noise.

Vl. DISCUSSION

In our ABR test system we found substantial autocorrelations among single point values which rendered invalid the usual method for estimating residual noise. We do not know the exact sources of these autocorrelations but clearly any noise nonuniformly distributed in frequency and which lasts for several sweeps can induce such correlation. Spectral analyses of the single point values were not entirely consis- tent across babies, However, they did indicate strong peaks at 10.7 and 11.8 Hz in several cases. We conjecture that the

ever present 60-Hz line noise may, have given rise to these double peaks. With constant sampling and exactly 41 ms between single points, 60-Hz noise would induce a spe,,ctral peak at 11.2 Hz. The sampling in our 'system was inte•rru,pted every 20 sweeps for 78 ms to upload data to the•PC. The

periodic gap in recording might have caused an 11.2-Hz peak in the spectrum to split into two peaks separated by 1.1 Hz. Other sources of noise including a and 8 waves and muscle tension could also conceivably contribute to the observed autocorrelations.

Low-frequency noise components which contribute to the observed autocorrelations could be reduced by using a high-pass filter to the data before averaging. In our experi- ments, however, use of a high-pass filter was considered to be an unacceptable signal processing strategy. Two reasons for this are, first, that significant signal components at and below 60 Hz are thought to be present and, second, that preserving the timing of waveform peaks in the final averaged evoked potential is considered important. Hence we averaged the raw data to determine the signal and focused on appropriate methods for estimating the residual noise which would accommodate low-frequency noise components.

We have shown that the corrected mean sum of squares estimate RN c and the block average estimate RNa•u both have significantly less bias than the usual estimate RNms s. We prefer the block average method because it is computationally faster (an important consideration when it must be computed repeatedly during data collection) and conceptu- ally easier. In our data from babies, the corrected estimate actually gave negative values in some cases when an insuf- ficient number of autocorrelations had been accounted for.

Block averages are also more likely to be normally distributed which has certain statistical advantages. For example, a confidence interval for RN can be obtained from RNa• using the fact that Na times RNa•/RN has a chi-squared dis- tribution with Na- 1 degrees of freedom.

Our software system stored data from a single point in the poststimulus time window. Hence our analyses pertain only to that single point. We have shown that the usual assumption that sweeps are independent was violated at that point. We suspect that the correlation between single point values at the same point in the time window but on different sweeps may well depend on the location of the single point values in the time window. Hence the usual assumption that the residual noise is constant across the time window may also be invalid. Our data did not allow us to address this

question. Our current recommendation is to estimate the residual noise at several points across the time window. Values between these points can be interpolated in order to determine the magnitude of the residual noise as a function of location in the poststimulus time window.

One issue that we have not discussed is the nonstation-



arity of the noise. Bursts of noise do occur. Elberling and Wahlgreen (1985) have suggested weighting the data within blocks by the inverse of the mean of the sum of squares within each block. This may improve the stationarity of the noise, although weighting by the square root of this quantity would seem to be more appropriate if the objective is to improve stationarity. Such weighting has been investigated and did not alter any of the conclusions presented in this paper.

Our results have important implications for objective detection of the ABR. Current detection algorithms based on a signal-to-noise ratio rely on good estimation of residual noise for the denominator component. In our system the usual mean sum of squares estimator appears to be biased too large. This makes the signal-to-noise ratio biased too small, and hence our detection algorithm is less sensitive to signal than it ought to be. Since termination of the test is dependent on the signal detection algorithm, increased test time is a consequence. Faster detection which is particularly

important for large scale screening programs may be achieved by replacing the usual estimate of residual noise with the alternatives discussed in this paper.

ACKNOWLEDGMENTS

This work was supported by NIH Grant No. R1ODC01958. We would like to thank Brenda Bergman, Jan Kaminski, Kathryn Beauchaine, and Michael Gorga for their help in collecting the data described in this paper.

Elberling, C., and Don, M. (1984). "Quality estimation of averaged auditory brainstem responses," Scand. Audiol. 13, 187-197.

Elberling, C., and Wahlgreen, O. (1985). "Estimation of auditory brainstem response, ABR, by means of Bayesian inference," Scand. Audiol. 14, 89- 96.

Fuller, W. A. (1976). Introduction to Statistical Time Series (Wiley, New York).

Sininger, Y. S. (1993). "Auditory brainstem response for objective measures of hearing," Ear Hear. 14, 23-30.



estimating residual noise in the auditory brain-stem response

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