Methodology: 2D Psychometric Function

In Experiment 2, we used the GPC framework to solve a relevant multidimensional psychometric problem. In this problem, a sequence of pure tones varying in both frequency and intensity is presented to a simulated listener, who is instructed to respond whenever a tone is detected. This task is similar to the task used for traditional pure-tone audiometry (Hughson and Westlake, 1944; Carhart and Jerger, 1959), but with two key differences: for this task, sampling does not necessarily proceed one frequency at a time, and sampling and prediction resolutions in both input feature dimensions is considerably higher. The goal of this work is to construct a general multidimensional PF estimator from recorded binary responses that can be used immediately for

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pure-tone audiometry and can be readily adapted to other multidimensional psychometric estimation problems.

4.3.1 Simulation Details

To simulate the 2D psychometric field, we first defined an audiogram shape for each simulated participant. For each audiogram shape, we approximated of 1 of 4 human audiometric phenotypes (Dubno et al., 2013) using spline interpolation and linear extrapolation, forming a continuous threshold curve across frequency space. Figure 4.1 shows the 4 phenotypes, which were classified using both machine learning and physiology.

Figure 4.1: Plot of 4 human audiometric phenotypes.Reproduced from (Dubno et al., 2013).

At each frequency, we used (4.1) to generate a sigmoidal psychometric curve as a function of intensity. We selected a value for spread β between 0.2 and 10 dB/percent, and we computed the value for center point α given β and the value of audiometric threshold at that frequency, which corresponded to a detection probability of 70.7% (Levitt, 1971). The overall 2D PF is therefore a

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combination of the audiogram shape across frequency and the sigmoidal 1D PF in intensity and

provides the probability of detection ψ x

( )

_i for any input frequency/intensity pair x_i =

( )

ω ι_i, _i . As in the 1D case, the binary response y_i (success = 1; failure = 0) can be generated by

simulating one draw from a Bernoulli distribution with success probability ψ x

( )

_i . To select the set of observed frequency/intensity pairs, we use a Halton sequence (Halton, 1964), which provides a deterministic set of n well-spaced draws from the frequency/intensity domain of

interest. We use these observed samples

{(

x₁,y₁

) (

, x₁,y₁

) (

, x_n,y_n

)}

=

( )

X y, as training observations for the GPC algorithm.

4.3.2. Gaussian Process Construction

For the 2D PF inference problem, we wish to estimate a subject’s detection probability as a function of both the intensity and frequency of the presented stimulus. Our input variable x is

therefore a frequency-intensity pair, or x =

( )

ω ι, , and our output variable y is a binary response variable. We wish to infer the detection probability ψ

( )

x = p y

(

=1x

)

, and we place a GP prior on the latent function: p f

( )

= 

(

µ

( ) ( )

x K x x, , ′

)

.

As in the 1D case, the dependence of detection probability on stimulus intensity is assumed to be a monotonically increasing sigmoidal function, which is captured using the linear covariance function (Equation 12) in the intensity dimension. The dependence of the detection probability on frequency is not explicit, however, and will vary across subjects based on the shape of the audiogram. A reasonable assumption is that the overall PF is continuous along the frequency

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dimension with some smoothness (Von Békésy, 1960; Kiang et al., 1965; Green and Swets, 1966; Brant and Fozard, 1990; Leek, 2001). To reflect this behavior, we select a SE covariance function (2.9) for the frequency dimension. The full covariance function combines the linear covariance function in intensity and the SE covariance function in frequency:

( )

(( ) ( ))

2

(

)

2 2

( )

1 2 2 , , , , exp 2 K K ω ι ω ι s ω ω s ιι _{− − ′}    ′ = ′ ′ = _{ }+ ′     x x x  (4.4)

Here, s₁ and s₂ are scaling factors and  is a characteristic length scale, which regulates the smoothness of the function with respect to frequency. Again, we select a constant mean function

( )

c

µ x = for this GP.

Given a set of observed samples

( )

X y, , we again first calculate a set of best-fitting hyperparameters q =

(

c s s, , ,_{1 2} 

)

by maximizing the log marginal likelihood log p

(

y X)θ . We

)

then compute the posterior distribution p

(

f X y X* , , *

)

for a finely spaced grid of test samples

*

X across frequency-intensity space: 0.125 to 16 kHz in semitone increments for frequency and

−20 to 120 dB in 1-dB increments for intensity.

Unlike in the 1D case, we cannot readily specify a meaningful parametric form for the 2D PF across all frequencies and intensities. At any fixed frequency x_i, however, we can derive an

analytical expression for the PF by finding the inverse slope and x-intercept of the mean of the posterior latent function f at that frequency. Furthermore, the GPC method’s point estimate for

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the full PF can be computed by passing f through the likelihood function p y

(

_i =1f_i

)

and can be numerically compared to the true PF.

4.3.3. Evaluation

We evaluated overall performance of the GPC framework for a variety of psychometric and sampling parameters. Specifically, these parameters were manipulated:

• Audiogram shape: Older-normal, sensory, metabolic, and sensory + metabolic audiogram profiles (Dubno et al., 2013) were used to fix simulation values of α at each frequency. • Spread value: β values of 0.2, 0.5, 1, 2, 5, and 10 dB/percent, assumed isotropic across all

frequencies, were used to construct the PF.

• Number of observed samples: 20, 50, 100, 200, 500, and 1000 pairs

( )

X y, were used as observed data to condition the GP.

• Simulation repetition: For each unique parameter combination, we performed 40

independent repetitions of GPC inference, resulting in 5760 simulations overall.

We evaluated performance of the GPC framework by comparing the GP parameter estimates of α and β with the known values of α and β from the simulated PF. Performance was evaluated by comparing parameter values at a fine grid of frequency values (0.25 to 8 kHz in semitone increments). Edge frequencies (0.125 to 0.25 and 8 to 16 kHz) were used to train the GP but not to evaluate prediction because previous work has shown that edge effects can reduce GPC accuracy (Song et al., 2015). Accuracy was evaluated by computing the mean deviation of parameter estimates from the true value, while reliability was evaluated by computing the

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variance of GP parameter estimates across all repetitions with the same parameter values. Once again, accuracy and reliability were verified with two nonparametric numerical values: the 50% probability point and 25-75% interquartile range.

We evaluated goodness of fit of the 2D GP posterior mean to the observations using the Pearson

2

χ statistic, consistent with the 1D case. For each frequency/intensity pair x_i =

( )

ω ι_i, _i , the

statistic

( )

(

)

2 2 1 i i i i _{i i} N p P P P χ =  −  −

∑

x was computed and compared to the chi-squared distribution with J degrees of freedom, where J is the number of frequency/intensity pairs sampled. Because the GP framework with a Halton sampling scheme does not typically repeat observations at identical input values, J usually equaled the total number of observations.

In document Improving Pure-Tone Audiometry Using Probabilistic Machine Learning Classification (Page 94-99)