The Phonetic Analysis of Speech Corpora

Various studies in the last fifty years have been concerned with phonetic vowel reduction that is with the changes in vowel quality brought about by segmental, prosodic, and situational contexts. In Lindblom's (1990) hyper- and hypoarticulation theory, speech production varies along a continuum from clear to less clear speech. According to this theory, speakers make as much effort to speak clearly as is required by the listener for understanding what is being said. Thus, the first time that a person's name is mentioned, the production is likely to be clear because this is largely unpredictable information for the listener; but subsequent productions of the same name in an ongoing dialogue are likely to be less clear, because the listener can more easily predict its occurrence from context (Fowler & Housum, 1987).

A vowel that is spoken less clearly tends to be reduced which means that there is a deviation from its position in an acoustic space relative either to a clear, or citation-form production. The deviation is often manifested as centralisation, in which the vowel is produced nearer to the centre of the speaker's vowel space than in clear speech. Equivalently, in clear speech there is an expansion of the vowel space. There is articulatory evidence for this type of vowel space expansion when vowels occur in prosodically accented words, often because these tend to be points of information focus, that is points of the utterance that are especially important for understanding what is being said (de Jong, 1995; Harrington, Fletcher & Beckman, 2000).

One of the ways of quantifying vowel space expansion is to measure the Euclidean or straight line distance between a vowel and the centre of the vowel space. Wright (2003) used just such a measure to compare so called easy and hard words on their distances to the centre of the vowel space. Easy words are those that have high lexical frequency (i.e., occur often) and low neighborhood density (there are few words that are phonemically similar). Since such words tend to be easier for the listener to understand, then, applying Lindblom’s (1990) model, the vowels should be more centralised compared with hard words which are both infrequent and high in neighborhood density.

sqrt( (0 - 3)^2 + (0 - 4)^2 )

5
Because of the nice way that vectors work in R, the same result is given by:
a = c(0, 0)

b = c(3, 4)

sqrt( sum( (a - b)^2 ))

5
So a function to calculate the Euclidean distance between any two points a and b is:

{

# Function to calculate Euclidean distance between a and b;

sqrt(sum((a - b)^2))

}
In fact, this function works not just in a two-dimensional space, but in an n-dimensional space. So if there are two vowels, a and b, in a three-dimensional F1, F2, F3 space with coordinates for vowel a F1 = 500 Hz, F2 = 1500 Hz, F3 = 2500 Hz and for vowel b F1 = 220 Hz, F2 = 2400 Hz, F3 = 3000 Hz, then the straight line, Euclidean distance between a and b is just over 1066 Hz as follows:
a = c(500, 1500, 2500)

b = c(220, 2400, 3000)

euclid(a, b)

1066.958
Exactly the same principle (and hence the same function) works in 4, 5, …n dimensional spaces even though any space higher than three dimensions cannot be seen or drawn. The only obligation on the function is that the vectors should be of the same length. The function can be made to break giving an error message, if the user should try to do otherwise:

{

# Function to calculate Euclidean distance between a and b;

if(length(a) != length(b))

stop("a and b must be of the same length")

sqrt(sum((a - b)^2))

}
a = c(3, 4)

b = c(10, 1, 2)

euclid(a, b)

Error in euclid(a, b) : a and b must be of the same length

m.av = apply(vowlax.fdat.5[temp,1:2], 2, mean)

m.av

T1 T2

f.av

533.8439 1965.8293

a E I O

f = apply(vowlax.fdat.5[!temp,1:2], 2, tapply, vowlax.l[!temp], mean)

f

T1 T2

E 515.9268 2202.268

I 358.0941 2318.812

O 520.0000 1160.813

f.av

545.041 1805.51

eplot(vowlax.fdat.5[temp,1:2], vowlax.l[temp], dopoints=T, form=T, xlab="F2 (Hz)", ylab="F1 (Hz)", doellipse=F)

text(-f.av[2], -f.av[1], "X", cex=3)
The Euclidean distances of each data point to X in Fig. 6.14 can be obtained by applying euclid() to the rows of the matrix using apply()with a second argument of 1 (meaning apply to rows):
temp = vowlax.spkr=="68"

e.f = apply(vowlax.fdat.5[temp,1:2], 1, euclid, f.av)

edistances = rep(0, nrow(vowlax.fdat.5))

# Logical vector to identify speaker 67

temp = vowlax.spkr == "67"

# The next two commands give the male speaker's centroid analogous to f.av

m = apply(vowlax.fdat.5[temp,1:2], 2, tapply, vowlax.l[temp], mean)

m.av = apply(m, 2, mean)

# Distances to the centroid for the male speaker

edistances[temp] = apply(vowlax.fdat.5[temp,1:2], 1, euclid, m.av)

# Distances to the centroid for the female speaker

edistances[!temp] = apply(vowlax.fdat.5[!temp,1:2], 1, euclid, f.av)
Since all the objects are parallel to each other, it only takes one line to produce a boxplot of the results comparing the Euclidean distances for the male and female speakers separately by vowel category (Fig 6.15):
boxplot(edistances ~ factor(vowlax.spkr) * factor(vowlax.l), ylab= "Distance (Hz)")
Fig. 6.15 about here
Fig. 6.15 confirms what was suspected: the Euclidean distances are greater on every vowel category for the female speaker.
6.5.2 Relative distance between vowel categories

In the study of dialect and sound change, there is often a need to compare the relative position of two vowel categories in a formant space. The sound change can sometimes be linked to age and social class, as the various pioneering studies by Labov (1994, 2001) have shown. It might be hypothesised that a vowel is in the process of fronting or raising: for example, the vowel in who'd in the standard accent of English has fronted in the last fifty years (Harrington et al, 2008; Hawkins & Midgley, 2005), there has been a substantial rearrangement of the front lax vowels in New Zealand English (Maclagen & Hay, 2007), and there is extensive evidence in Labov (1994, 2001) of numerous diachronic changes to North American vowels.

Vowels are often compared across two different age groups so that if there is a vowel change in progress, the position of the vowel in the older and younger groups might be different (this type of study is known as an apparent time study: see e.g., Bailey et al, 1991). Of course independently of sound change, studies comparing different dialects might seek to provide quantitative evidence for the relative differences in vowel positions: whether, for example, the vowel in Australian English head is higher and/or fronter than that of Standard Southern British English.

There are a number of ways of providing quantitative data of this kind. The one to be illustrated here is concerned with determining whether the position of a vowel in relation to other vowels is different in one set of data compared with another. I used just this technique (Harrington, 2006) to assess whether the long, final lax vowel in words like city, plenty, ready, was relatively closer to the tense vowel in [i] (heed) than in the lax vowel in [ɪ] (hid) in the more recent Christmas messages broadcast by Queen Elizabeth II over a fifty year period.

For illustrative purposes, the analysis will again make use of the lax vowel data. Fig. 6.10 suggests that [ɛ] is closer to [ɪ] than it is to [a] in the female than in the male speaker. Perhaps this is a sound change in progress, perhaps the female subject does not speak exactly the same variety as the male speaker; or perhaps it has something to do with differences between the speakers along the hyper- and hypoarticulation continuum, or perhaps it is an artefact of anatomical differences in the vocal tract between the male and female speaker. Whatever the reasons, it is just this sort of problem that can arise in sociophonetics in dealing with gradual and incremental sound change.

The way of addressing this issue based on Harrington (2006) is to work out two Euclidean distances: d₁, the distance of all of the [ɛ] tokens to the centroid of [ɪ]; and d₂, the distance of all of the same [ɛ] tokens to the centroid of [a]. The ratio of these two distances, d₁/d₂ is indicative of how close (in terms of Euclidean distances) the [ɛ] tokens are to [ɪ] in relation to [a].The logarithm of this ratio, which will be termed E_RATIO, gives the same information but in a more convenient form. More specifically, since

= log(d₁) – log(d₂)

temp = vowlax.spkr == "68" & vowlax.l=="I"

mean.I = apply(vowlax.fdat.5[temp,1:2], 2, mean)
# Next two lines calculate the centroid of female [a]

temp = vowlax.spkr == "68" & vowlax.l=="a"

mean.a = apply(vowlax.fdat.5[temp,1:2], 2, mean)
# Logical vector to identify all the female speaker's [ɛ] vowels

temp = vowlax.spkr == "68" & vowlax.l=="E"

etoI = apply(vowlax.fdat.5[temp,1:2], 1, euclid, mean.I)

etoa = apply(vowlax.fdat.5[temp,1:2], 1, euclid, mean.a)

ratio.log.f = log(etoI/etoa)

col = "steelblue"; xlab=expression(E[RATIO])

hist(ratio.log.f, xlim=xlim, col=col, xlab=xlab, main="Speaker 67")

hist(ratio.log.m, xlim=xlim, col=col, xlab=xlab, main="Speaker 68")

The calculation of the Euclidean distance to the centre of the vowel space discussed in 6.5.1 is one of the possible methods for measuring vowel undershoot, a term first used by Lindblom (1963) to refer to the way in which vowels failed to reach their targets due to contextual influences such as the flanking consonants and stress. But in such calculations, the extent of vowel undershoot (or expansion) is being measured only at a single time point. The technique to be discussed in this section is based on a parameterisation of the entire formant trajectory. These parameterisations involve reducing an entire formant trajectory to a set of coefficients – this can be thought of as the analysis mode. A by-product of this reduction is that if the formant trajectories are reconstructed from the coefficients that were obtained in the analysis mode, then a smoothed formant contour can be derived – this is the synthesis mode and it is discussed more fully at the end of this section.

The type of coefficients to be considered are due to van Bergem (1993) and involve fitting a parabola, that is an equation of the form F = c₀ + c₁t + c₂t², where F is a formant from the start to the end of a vowel that changes as a function of time t. As the equation shows, there are three coefficients c₀, c₁, and c₂ that have to be calculated for each vowel separately from the formant's trajectory. The shape of a parabola is necessarily curved in an arc, either U-shaped if c₂ is positive, or ∩-shaped if c₂ is negative. The principle that lies behind fitting such an equation is as follows. The shape of a formant trajectory, and in particular that of F2, over the extent of a vowel, is influenced mostly both by the vowel and by the immediately preceding and following sounds: that is by the left and right contexts. At the vowel target, which for most monophthongs is nearest the vowel's temporal midpoint, the shape is predominantly determined by the phonetic quality of the vowel, but it is reasonable to assume that the influence from the context increases progressively nearer the vowel onset and offset (e.g. Broad & Fertig, 1970). Consider for example the case in which the vowel has no target at all. Just this hypothesis has been suggested for schwa vowels by Browman & Goldstein (1992b) in an articulatory analysis and by van Bergem (1994) using acoustic data. In such a situation, a formant trajectory might approximately follow a straight line between its values at the vowel onset and vowel offset: this would happen if the vowel target has no influence so that the trajectory's shape is entirely determined by the left and right contexts. On the other hand, if a vowel has a prominent target, as if often the case if it is emphasised or prosodically accented (Pierrehumbert & Talkin, 1990), then, it is likely to deviate considerably from a straight line joining its onset and offset. Since a formant trajectory often follows reasonably well a parabolic trajectory (Lindblom, 1963), one way to measure the extent of deviation from the straight line and hence to estimate how much it is undershot is to fit a parabola to the formant and then measure the parabola's curvature. If the formant is heavily undershot and follows more or less a straight line path between its endpoints, then the curvature will be almost zero; on the other hand, the more prominent the target, the greater the deviation from the straight line, and the greater the magnitude of the curvature, in either a positive or a negative direction.

The way that the parabola is fitted in van Bergem (1993) is essentially to rescale the time axis of a trajectory linearly between t = -1 and t = 1. This rescaled time-axis can be obtained using the seq() function, if the length of the trajectory in data points is known. As an example, the length of the F2-trajectory for the first segment in the lax vowel data is given by:

N

18

times = seq(-1, 1, length=N)

F2int = approx(frames(vowlax.fdat[1,2]), n=N)

times = seq(-1, 1, length=N)

par(mfrow=c(1,2));

plot(times, F2int$y, type="b", xlab="Normalized time", ylab="F2 (Hz)")

# The original F2 for this segment

plot(vowlax.fdat[1,2], type="b", xlab="Time (ms)", ylab="")
Fig. 6.17 about here
There are three unknown coefficients to be found in the parabola F = c₀ + c₁t + c₂t² that is to be fitted to the data of the left panel in Fig. 6.17 and this requires inserting three sets of data points into this equation. It can be shown (van Bergem, 1993) that when the data is extended on the time axis between t = ± 1, the coefficients have the following values:
# c₀ is the value at t = 0.

c0 = F2int$y[times==0]

# c₁ is half of the difference between the first and last data points.

c1 <- 0.5 * (F2int$y[N] - F2int$y[1])

# c₂ is half of the sum of the first and last data points minus c₀

c2 <- 0.5 * (F2int$y[N] + F2int$y[1]) - c0

c0 c c2

# Calculate the values of the parabola

F2par = plafit(frames(vowlax.fdat[1,2]), fit=T)

ylim = range(c(F2par, frames(vowlax.fdat[1,2])))

xlab="Time (ms)"; ylab="F2 (Hz)"

# Plot the raw values

plot(vowlax.fdat[1,2], type="b", ylim=ylim, xlab=xlab, ylab=ylab)

# Superimpose the smoothed values

par(new=T)

plot(as.numeric(names(F2par)), F2par, type="l", ylim=ylim, xlab=xlab, ylab=ylab, lwd=2)

The task now is to explore a worked example of measuring formant curvatures in a larger sample of speech. The analysis of the vowel spaces in the F2 × F1 plane, as well as the Euclidean distance measurements in 6.5 have suggested that the female speaker produces more distinctive vowel targets, or in terms of Lindblom's (1990) H&H theory, her vowels show greater evidence of hyperarticulation and less of a tendency to be undershot. Is this also reflected in a difference in the extent of formant curvature?

In order to address this question, the two speakers' [ɛ] vowels will be compared. Before applying an algorithm for quantifying the data, it is always helpful to look at a few plots first, if this is possible (if only because a gross inconsistency between what is seen and what is obtained numerically often indicates that there is a mistake in the calculation!). A plot of all of the F2 trajectories lined up at the temporal midpoint and shown separately for the two speakers does not seem to be especially revealing (Fig. 6.19, left panel), perhaps in part because the trajectories have different durations and, as was mentioned earlier, in order to compare whether one trajectory is more curved than another, they need to be time normalized to the same length. The argument norm=T in the dplot() function does just this by linearly stretching and compressing the trajectories so that they extend in time between 0 and 1. There is some suggestion from the time-normalized data (Fig. 6.19 centre) that the female's F2 trajectories are more curved than for the male speaker. This emerges especially clearly when the linearly time-normalzed, male and female F2-trajectories are separately averaged (Fig. 6.19, right). However, the average is just that: at best a trend, and one that we will now seek to quantify by calculating the c₂ coefficients of the fitted parabola. Before doing this, here are the instructions for producing Fig. 6.19:
temp = vowlax.l == "E"

par(mfrow=c(1,3)); ylim = c(1500, 2500)

# F2 of E separately for M and F synchronised at the midpoint

dplot(vowlax.fdat[temp,2], vowlax.spkr[temp], offset=.5, ylab="F2 (Hz)", ylim=ylim, xlab="Time (ms)",legend=F)

# As above with linear time normalization

dplot(vowlax.fdat[temp,2], vowlax.spkr[temp], norm=T, xlab="Normalized time", ylim=ylim, legend=F)

# As above and averaged

dplot(vowlax.fdat[temp,2], vowlax.spkr[temp], norm=T, average=T, ylim=ylim, xlab="Normalized time")

plafit(r)

temp = vowlax.l == "E"

# Matrix of coefficients, one row per segment.

coeffs = trapply(vowlax.fdat[temp,2], plafit, simplify=T)

boxplot(coeffs[,3] ~ factor(vowlax.spkr[temp]), ylab="Amplitude", ylim=ylim)

As foreshadowed at various stages in this section, the fit=T argument applies the function in synthesis mode: it works out the corresponding fitted formant parabola as a function of time. In order to smooth an entire trackdata object, trapply()can once again be used but this time with the argument returntrack=T to build a trackdata object (see 5.5.2):

vow.sm2 = trapply(vowlax.fdat[,2], plafit, T, returntrack=T)

Although fitting a parabola is an effective method of data reduction that is especially useful for measuring formant curvature and hence undershoot, there are two disadvantages as far as obtaining a smoothed contour are concerned:

• 
  not every formant trajectory has a parabolic shape
  
• 
  parabolic fitting of the kind illustrated above forces a fit at the segment onset, offset, and midpoint as is evident from Figs. 6.21.
  

One way around both of these problems is to use the discrete cosine transformation (see e.g., Watson & Harrington; 1999; Harrington, 2006; Harrington et al., 2008) which will be discussed more fully in Chapter 8. This transformation decomposes a trajectory into a set of coefficients (this is the analysis mode) that are the amplitudes of half-cycle cosine waves of increasing frequency. The number of coefficients derived from the discrete cosine transformation (DCT) is the same as the length of the trajectory. If, in synthesis mode, all of these cosine waves are summed, then the original, raw trajectory is exactly reconstructed. However, if only the first few lowest frequency cosine waves are summed, then a smoothed trajectory is derived. Moreover, the fewer the cosine waves that are summed, the greater the degree of smoothing. Therefore, an advantage of this type of smoothing over that of fitting parabolas is that it is possible to control the degree of smoothing. Another advantage is that the DCT does not necessarily force the smoothed trajectory to pass through the values at the onset, offset, and midpoint and so is not as prone to produce a wildly inaccurate contour, if formant onsets and offsets were inaccurately tracked (which is often the case especially if there is a preceding or following voiceless segment).

There is a function in the Emu-R library for computing the DCT coefficients, dct()which, just like plafit() and euclid() takes a vector of values as its main argument. The function can be used in an exactly analogous way to the plafit() function in synthesis mode for obtaining smoothed trajectories from the coefficients that are calculated in analysis mode. In the example in Fig. 6.21, a smoothed F2 trajectory is calculated from the first five DCT coefficients. The DCT coefficients were calculated as follows:
Fig. 6.21 about here
# Calculate a smoothed-trajectory based on the lowest 5 DCT coefficients

vow.dct2 = trapply(vowlax.fdat[,2], dct, fit=T, 4, returntrack=T)

# A label vector to identify the trajectories

lab = c("raw", "parabola", "DCT")

# Row-bind the three trajectories into one trackdata object

dat = rbind(vowlax.fdat[j,2], vow.sm2[j,], vow.dct2[j,])

dplot(dat, lab, ylab="F2 (Hz)", xlab="Time (ms)")