Fit Legendre polynomials to f0 contours, token by token

For each token, normalises the time axis to [-1, 1], constructs a Legendre polynomial basis of the requested degree, and fits ordinary least squares. Returns one row per token with the fitted coefficients c0, c1, ..., c{degree}. Useful as a compact, interpretable summary of contour shape, or as features for downstream classifiers and regression models (Mirman 2014; Xu & Zhang 2024).

Usage

fit_polynomial(
  data,
  f0 = "f0",
  token = "token",
  time = "time",
  speaker = "speaker",
  tone = "tone",
  degree = 2
)

Arguments

data

A long-format data frame with one row per f0 sample.

f0

Column name of f0 values. Default "f0". Using normalised f0 (e.g. "f0_st" from normalise_f0()) is recommended for cross-speaker comparison.

token

Column name of token ID. Default "token".

time

Column name of time within each token. Default "time".

speaker

Column name of speaker ID, carried through to the output. Default "speaker".

tone

Column name of tone category, carried through to the output. Default "tone".

degree

Polynomial degree. One of 1, 2, or 3. Default 2.

Value

A token-level data frame with one row per token, containing the token, speaker, tone, and coefficient columns c0, c1, ..., c{degree}.

Details

What the function does internally

For each token:

1. Compute the per-token range of time and rescale it linearly to the interval [-1, 1], which is where Legendre polynomials are orthogonal.

2. Build the Legendre basis up to the requested degree.

3. Solve the ordinary least-squares problem with stats::lm.fit() to recover the coefficients.

Interpreting the coefficients

Because Legendre polynomials are orthogonal on [-1, 1], the fitted coefficients have a tidy shape interpretation:

• c0: token-level mean f0.

• c1: linear slope across the contour. Positive means rising.

• c2: quadratic curvature. Positive means U-shaped (dipping), negative means inverted-U (peaking).

• c3: cubic shape. Differentiates sigmoid from reverse-sigmoid contours.

Edge cases

Tokens with fewer than degree + 1 valid samples receive NA for all coefficients. Tokens with a single unique time point keep c0 as the mean f0 and set the higher-order coefficients to NA.

References

Mirman, D. (2014). Growth Curve Analysis and Visualization Using R. Chapman and Hall/CRC.

Xu, C., & Zhang, C. (2024). A cross-linguistic review of citation tone production studies: Methodology and recommendations. The Journal of the Acoustical Society of America, 156(4), 2538–2565. doi:10.1121/10.0032356

See also

• normalise_f0() for the upstream normalisation step.

• fit_gca() for a population-level mixed-effects extension of polynomial contour modelling.

Examples

data(sample_f0)
normed <- normalise_f0(sample_f0,
                       f0      = "f0_Hz",
                       speaker = "speaker",
                       tone    = "tone")
coefs <- fit_polynomial(normed,
                        f0      = "f0_st",
                        token   = "token",
                        time    = "time",
                        speaker = "speaker",
                        tone    = "tone",
                        degree  = 2)
head(coefs)
#> # A tibble: 6 × 6
#>   token            speaker  tone    c0    c1    c2
#>   <chr>            <chr>   <int> <dbl> <dbl> <dbl>
#> 1 dc102一1s22.3525 dc102       6 -1.53  3.86 0.114
#> 2 dc102一2s22.9425 dc102       6 -1.07  2.62 0.282
#> 3 dc102一3s23.5725 dc102       6 -1.37  2.79 0.474
#> 4 dc102一4s24.1425 dc102       6 -1.14  1.76 0.732
#> 5 dc102一5s24.7925 dc102       6 -1.09  1.90 0.296
#> 6 dc102一6s25.3125 dc102       6 -1.87  2.14 1.28