Rotating Principal Components to explore change over the lifespan in New Zealand English monophthongs

Elena Sheard & Joshua Wilson Black

Te Kāhui Roro Reo | New Zealand Institute of Language, Brain and Behaviour
Te Whare Wānanga o Waitaha | University of Canterbury

Methods in Dialectology, July 4

Background

• QuakeBox 1 (2011-2012) and QuakeBox 2 (2020-2021) contain monologues from the same speakers (n=51).
• Two questions:
  1. Are the patterns of covariation similar?
  2. Do speakers change their position with respect to these patterns?
• Yes to (1): Hurring et al. (Under review)
• Yes to (2): Sheard and Wilson Black (2023)
• Methodological challenge: compare PCA across datasets.

• NZILBB has a very valuable resource for studying speaker change over time in the QuakeBox 1 and QuakeBox 2 corpora. We use a subset of 51 speakers who pass a filter for data quantity at both QuakeBox 1 and QuakeBox 2.

• With covariation in mind. We can ask two questions:

1. Are the patterns of covariation in vocalic variables the same in QuakeBox 1 and QuakeBox 2? This is to ask about stability between the two corpora.

2. If the patterns of covariation remain the same, do individual speakers or groups of speakers shift with respect to these patterns?

• Hurring et al. (under review) answers “yes” to the first question.

• We presented some evidence towards a “yes” answer to the second question at last year’s ALS. Our results remain pretty much unchanged, although with a touch more methodological sophistication.

• Indeed, today’s focus is methodological. There are a series of issues which arise when trying to compare PCAs applied to distinct data sets.

• We can borrow tools from other disciplines, but there are practical issues which we have to figure out for the kind of data used in sociophonetics, too.

Especially important for difference

Overview

1. When is PCA unstable?
2. Rotating and Scaling PCs
   • Manual rotation
   • Procrustes rotation
3. Change over the life span in QuakeBox.

1. It is worth starting with an intuitive, geometrical, explanation of what PCA is.
   • Get on the same page as one another.
   • Rotation is an idea which you will have a geometrical handle on.
2. PCA is unstable in the sense that small differences in the input data can have large consequences for PC loadings and scores.
3. We’ll then introduce the idea of rotating (and scaling) PC loadings and scores.
   • We’ll look at ‘manual’ rotation — i.e., deciding as a researcher that you’d like to rotate the loadings and scores by 45 degrees, or some other value.
   • We’ll then introduce a more systematic approach: procrustes rotation, which minimises the sum of squared differences between two PCA analyses.

PCA Instability

PCA Instability

• Small changes in data → big changes in PCs.
  • Outlier sensitivity.
  • Competing patterns of covariation:
    • PC2 in one sample might be PC3 in another.
  • Axis flipping.
    • Direction along a PC is arbitrary.

Competing PCs in QB2

Overlap in variance explained.

PC2 and PC3 are unstable.

QB2 Instability

Points to hit:

• Sometimes the axes flip (i.e. sign of the loadings changes)
• Sometimes there’s straight forward rotations
• The red ones are the ones most associated with PC2 in QB1.
• Almost all of the time, they are in the same relative configuration
• So there is something being captured with PC2 and PC3 together.
• How can we get at the structure which is in common across all of these difference PCAs?

Comparing QB1 and QB2

We can flip.

But can we rotate?

PCA Rotation

Rotating PCs

• Rotation is a technique from Factor Analysis.
• Why?
  • Increase ‘interpretability’ (Jolliffe 2002)
  • Compare across analyses
• Rotated components are no longer “PCs”
  • They don’t maximise variance explained.
  • But the ‘space’ is maintained.
• Call this PCA + rotation.

How to Rotate?

1. Manual rotation
   1. Decide where you want an arrow to go,
   2. work out the required angle,
   3. rotate.
2. Procrustes rotation
   • Rotate (and scale) to minimise sum of squared differences with reference shape.

For ‘interpretability’, other methods are used.

Manual Rotation

We do this to the loadings, but you can do it to the scores as well.

I find it helpful for my memory to know where the names of these techniques come from. So please forgive a brief aside into Ancient Greek Mythology.

The name “Procrustes Rotation”, and the phrase “Procrustean bed”, come from the story of Procrustes and Theseus.

• Procrustes would invite tired travellers to stay at his house along the way.
• Procrustes had a very awkwardly shaped bed.
• If the traveller was too small, he would stretch them out to the dimensions of the bed.
• If the traveller was too large, he would cut them down to the “correct” size.
• Theseus defeats Procrustes by subjecting him to his own treatment (as in the image on the slide.)

Procrustes Rotation

• Procrustes rotation stretches and rotates the traveller to fit the bed.
  • The bed: the first PCA (QB1).
  • The traveller: the second PCA (QB2).
  • i.e., it minimises the overall difference between shapes.
• It is commonly used in community ecology (Peres-Neto and Jackson 2001)
  • We use the procrustes() function from the vegan package.

Procrustes Rotation (cont.)

• Applied to the first \(n\) PCs (your \(n\) may vary!).
• Less arbitrary than manual rotation.

Procrustes Rotation (cont.)

Change Over the Lifespan

(In)Stability (PC1)

(In)Stability (PC2)

Shifting Speakers

What is PC2, Anyway?

• QB1: PC2 is dominated by start F2, strut F2, and thought F1.
  • Positive PC2 = Backer start /strut (and lot), lower thought
  • Negative PC2 = Fronter start /strut (and lot), higher thought
• Does weaker correlation (compared to PC1) suggest instability in the back vowels for some individuals?
• Return to the original data!

Who is Shifting?

• We identified the ten speakers who have the largest difference between QB1 PC2 and QB2 + Procrustes rotation:
  • 8 women, 3 Māori, 18-35 (n=2), 36-55 (n=7), 56-65 (n=1)
• We explore their back vowel midpoints (n = 3581)
  • thought, lot, strut and start.

Who is Shifting? (cont.)

Vowel Space Analysis

Vowel Space Analysis (cont.)

• Speakers with QB1 → QB2 score increase (positive shift):
  • Higher/backer thought, fronter lot, start, strut
  • 3 Māori, all 35+
• Speakers with QB1 → QB2 score decrease (negative shift):
  • Lower/fronter thought, backer lot, start, strut
  • Non-Māori, 2 18-35.
• Consistent with broader Positive/Negative PC loading patterns

Conclusions

• PCA validated as:
  • A means of tracking vocalic covariation in the community
  • A diagnostic for changes over the lifespan across covarying vowels
• Importance of moving between PCA and original data.
• Language change over the lifespan can encompass changing relationships within the vowel space
• Rotation in an important part of the PCA tool kit.

References

Brand, James, Jen Hay, Lynn Clark, Kevin Watson, and Márton Sóskuthy. 2021. “Systematic Co-Variation of Monophthongs Across Speakers of New Zealand English.” Journal Article. Journal of Phonetics 88: 101096.

Hurring, Gia, Joshua Wilson Black, Jen Hay, and Lynn Clark. Under review. “How Stable Are Patterns of Covariation Across Time?” Under review.

Jolliffe, Ian T. 2002. Principal Component Analysis. Springer.

Peres-Neto, Pedro R., and Donald A. Jackson. 2001. “How Well Do Multivariate Data Sets Match? The Advantages of a Procrustean Superimposition Approach over the Mantel Test.” Oecologia 129 (2): 169–78. https://doi.org/10.1007/s004420100720.

Sheard, Elena, and Joshua Wilson Black. 2023. “Change over the Lifespan Across Covarying New Zealand English Monophthongs.” Presented at the Annual Conference of the Australian Linguistic Society.

Appendix: data and models

• 51 speakers
  • Satisfy data quantity and quality filtering steps at both recording points
• Generalised Additive Mixed Models (GAMMs) fit for F1 and F2 of same 10 monophthongs
  • kit, dress, trap, fleece, nurse, goose, thought, lot, strut and start
  • n = 22529 (QB1), 43403 (QB2)
  • Gender, age (at QB1), and articulation rate as predictors