Gallery of Wug-Shaped Curves

Bruce P. Hayes
Department of Linguistics
UCLA


A wug-shaped curve with color, beak, and feet


What is a wug-shaped curve and why is it of interest?

The wug-shaped curve is a pattern of frequency widely found in quantitative studies of variable phenomena in linguistics. Indeed, it is so widespread that I believe its appearance may be meaningful from the viewpoint of theoretical linguistics. Visually, the wug-shaped curve takes the form of two or more identical sigmoid (logistic) curves, spaced apart.

The wug-shaped curve is a natural consequence of probabilistic versions of Harmonic grammar, such as MaxEnt. Here is how the analysis is set up:  we divide the constraint set into two families, having different forms or teleologies:  Baseline constraints and Perturbers. We then plot the empirical data points, in the form of probabilities (zero to one) on the vertical axis, and Baseline probability on the horizontal axis. This is done  separately, in a different color, for the data series defined by violations of the Perturber constraints.  We also plot the sigmoid lines themselves, which show the model fit -- ideally, the data points will cling to their respective sigmoids. You can decide how often this happens by looking at the charts below.

To make the research agenda plain:  along with some of my colleagues, I suspect that MaxEnt, or something like it, is correct for natural language, and that is why we see wug-shaped curves all over the place when we look at quantitative language data. You can judge for yourself by browsing through the images in this gallery, or by analyzing your own data in this way (see last section for how).

Various people see various things in multiple sigmoids. It was Dustin Bowers who suggested to me that they look like wugs. The wug was invented and first drawn in 1958 by Jean Berko Gleason, in one of the most famous papers ever written in linguistics. In recent years, the wug has been adopted by the field of linguistics as a sort of mascot. The real wug is cuter than the mathematical one.

What is this web page for?

I've written a paper about wug-shaped curves in linguistics, which you can download here.  Due to length limitations it doesn't have all of the cases I've compiled, and it seems that a web site would be the best format to display them all together. I've included all the wug-shaped curves I have ever plotted for this project, include the ones where the data don't look entirely pretty.

Browsing hints

For references, please follow the links or look at the bibliography section of my paper.  In most cases, a spreadsheet is linked just above the curve, which will tell you  how I obtained the data, did the MaxEnt analysis, and plotted the curve. For a few, my spreadsheet is currently very messy and can't be shared yet (though you could ask).



Directory

Wug-shaped curves in phonology
Wug-shaped curves in phonetics
Wug-shaped curves in syntax
Wug-shaped curves in sociolinguistics
Wug-shaped curves in semantics/pragmatics
Wug-shaped curves in language change
Wug-shaped curves in sound symbolism

Appendices to the paper
Some graphs used to diagnose theories

How I made the curves


Appendices to the paper

A. Deriving a single data point for Kluender et al. (1988)
B. Recoding Harmony as a single value in a two-candidate system
C. Why language change normally occurs at a constant rate
D. Deriving the MaxEnt sigmoid from first principles
E. On interation terms/conjoined constraints
F. On statistics and linguistics



Wug-shaped curves in phonology

Hungarian vowel harmony

Sources:  Hayes and Londe (2006), Hayes et al. (2009), Zuraw and Hayes (2017), Hayes (in progress)
Y-axis:  how often a stem will take back suffixes in a wug experiment
Baseline constraints:  stem vowels influence harmony
Perturber constraints:  stem-final consonants influence harmony
Spreadsheet (forthcoming), plotting script

Wug shaped curve in Hungarian vowel harmony

French liaison

Sources:  Zuraw and Hayes (2017)Hayes (in progress)
Y-axis:  likelihood of elision or liason; e.g. use of [l] instead of [la] for feminine definite article
Baseline constraints:  lexical propensity of Word 1 to act as an h-aspire word
Perturber constraints:  lexical propensity of Word 2 to appear in its isolation form
Spreadsheet, plotting script

Wug shaped curve in French liaison

Tagalog Nasal Substitution

Scholarly sources:  Zuraw (2000, 2010), Zuraw and Hayes (2017)Hayes (in progress)
Y-axis:  how often a stem of a given type will under the process of Nasal  Subsitution
Baseline constraints:  place and manner of stem-initial consonant
Perturber constraints: propensity of a particular prefix to trigger the process
Spreadsheet, plotting script

Wug shaped curve for Tagalog Nasal Substitution

Inversion of  Final Devoicing in Dutch

Sources:  Ernestus and Baayen (2003)Hayes (in progress)
Y-axis:  how often speakers guess that a stem-final consonant will appear as voiced when suffixed
Baseline constraints:  place and manner of stem final consonants, neighboring segment
Perturber constraints:  based on three degrees of vowel length in the stem
Spreadsheet (forthcoming), plotting script

Wug shaped curve for inverted Dutch Final Devoicing


Finnish genitive plurals

Sources:  Anttila (1997), Boersma and Hayes (2001), Goldwater and Johnson (2003)Hayes (in progress)
Y-axis:  how often a stem will take the longer [-den] allomorph of the genitive plural
Baseline constraints:  whether allomorph choice will result in two consecutive light syllables
Perturber constraints:  based on vowel height and weight of stem syllables. One perturber is inviolable and therefore produces flat line, not a sigmoid.
Spreadsheet, plotting script

Wug shaped curve in Finnish genitive plurals


Schwa/zero alternations in French

Source:  Smith and Pater (2020)
Y-axis:  how often  zero shows up in French schwa/zero alternations
Baseline constraints:  whether schwa is inserted or deleted, consonants in environment
Perturber constraint:  whether deletion of a schwa creates clashing (adjacent) stressed syllables
Spreadsheet, plotting script

Deletion of schwa in French

Stress placement in Hupa

Source:  Ryan (2019)
Y-axis:  probability of initial stress rather than second syllable stress
Baseline constraints:  weight of initial syllable
Perturber constraints:  weight of second syllable
Spreadsheet, plotting script

Data for stress placement in Hupa



Wug-shaped curves in phonetics

Perception of voicing based on closure duration and length of preceding vowel

Scholarly sources:  Kluender et al. (1988)Hayes (in progress)
Y-axis:  likelihood an experimental participant will perceive a voiced instead of a voiceless stop
Baseline constraints:  closure duration
Perturber constraints:  long vs. short preceding vowel
Spreadsheet, plotting script

Wug shaped curve in voicing perception



Wug-shaped curves in syntax

Datives in English

Sources:  Szmrecsanyi et al. (2017)Hayes (in progress)
Y-axis:  how often the meaning of the dative construction will be expressed using NP NP rather than NP to NP
Baseline constraints:  governing various properties of the Recipient
Perturber constraints:  status of the Theme
Spreadsheet (forthcoming), plotting script

Wug shaped curve for English Dative constructions

Genitives in English

Sources:  Szmrecsanyi et al. (2017)Hayes (in progress)
Y-axis:  how often the meaning of the possessive will be expressed using NP NP rather than NP to NP
Baseline constraints:  an amalgam; see Hayes (in progress)
Perturber constraints:  based on length of possessor in words
Spreadsheet (forthcoming), plotting script

Wug shaped curve Szmrecsanyi et al Genitives

One can also plot the same data with length as the Perturber, like this:

Wug shaped curve in English genitives, length as perturber


Wug-shaped curves in sociolinguistics

Contraction of the copula in Black English

Sources:  Labov (1969), Cedergren and Sankoff (1974)Hayes (in progress)
Y-axis:  how often the  speaker uses a contracted (vowelless) allomorph of the copula
Baseline constraints:  left side environment, include pronominal portmanteaux like "he's"
Perturber constraints:  right side syntactic environment
Spreadsheet, plotting script

Wug shaped curve for Labov Black English contraction

Deletion of the copula in Black English


Sources:  Labov (1969), Cedergren and Sankoff (1974)Hayes (in progress)
Y-axis:  how often the  speaker uses a null allomorph of the copula, assuming they have already chosen to contract
Baseline constraints:  left side environment, include pronominal portmanteaux like he's
Perturber constraints:  right side syntactic environment
Spreadsheet, plotting script
This is perhaps the messiest case I have seen; perhaps the use of conditional probability is the problem?

Black English deletion

Deletion of [l] in Quebec French

Sources:  ms. by Gillian Sankoff, cited and discussed in Bailey (1973)Hayes (in progress)
Y-axis:  deletion rate of [l]
Baseline constraints:  varying propensity of various function words to lose their [l]
Perturber constraints:  sex and social class of speaker, perhaps a proxy for speaking style
Spreadsheet, plotting script

Wug shaped curve for Quebec French [l] deletion

This particular case stands out as problematic for Stochastic OT, critiqued in  Zuraw and Hayes (2017) and Hayes (in progress). Here is a graph of a best-fit model of these data in Stochastic OT:

Sankoff/Bailey French data, ill modeled in Stochastic OT

Omission of que in Quebec French

Source:   Cedergren and Sankoff (1974)
Y-axis: retention rate for que 
Baseline constraints:  surrounding consonants
Perturber constraints:  formality of style, as varied by type of speaker
Spreadsheet, plotting script

Data for que deletion in Montreal French



R-Spirantization in Panamanian Spanish

Source: Cedergren and Sankoff (1974)
Y-axis: probability of realizing /r/ as a spirant
Baseline constraints:  phrasal position, whether /r/ is part of the infinitive ending, speaking style
Perturber constraints:  following segment
This is a rather messy one, I admit, and in particular lacks extreme values of probability.
Spreadsheet, plotting script

Data for r-spirantization in Panamanian Spanish


R-Dropping in New York City English

Source:  William Labov, via  Cedergren and Sankoff (1974)
Y-axis: probability of deleting /r/ in syllable codas
Baseline constraints:  speaking context
Perturber constraints:  designating different dialects spoken in the same speech community. I'm not sure what the grammatical status of these would be.
Spreadsheet, plotting script

Frequency of r dropping in New York City English

Cluster Simplification in Detroit Black English

Source:  Wolfram (1969)Hayes (in progress)
Y-axis: probability of deleting one of a pair of adjacent consonants
Baseline constraints:  neighboring vowel/consonant, whether deleting consonant is part of past tense suffix
Perturber constraints:  social class, assumed to be proxy for speaking style
This curve has a puzzling too-close vertical grouping for the _ C/ -ed case.
Spreadsheet, plotting script

Data for cluster simpllification in Detroit English


Wug-shaped curves in language change

Portuguese definite articles

Source:  Kroch (1989), Hayes (in progress)
Y-axis: probability of use of a definite article when a NP also has an NP possessor "unevolved," smaller Pokemon creature 
Baseline constraints:  rising constraint preferring this usage, over centuries
No perturber
Spreadsheet, plotting script

Data for article appearance in Portuguese possessed noun phrases

Evolution of have from Aux to main verb in  English

Source Zimmermann (2017).  See Hayes (in progress) for discussion.
Y-axis: probability of employing have syntactially as a main verb rather than as an aux
Baseline constraints:  a preference constraint shifting over time
Perturber constraints:  governing various distinct uses of auxiliary verbs
Right graph plots same thing in different coordinates (harmony difference), showing identical slopes

Data on use of Have from Zimmermann 2017


Wug-shaped curves in semantics/pragmatics

Quantifier scope

Source:  AnderBois et al. (2012), Hayes (in progress)
Y-axis: probability subjects will prefer narrow scope 
Baseline constraint:  whether the target quantifier is in first or second position
Perturber constraint: whetherthe target quantifier is in subject or object position
Spreadsheet, plotting script

Data from quantifier scope experiment


Wug-shaped curves in sound symbolism

Classification of Pokemon character names

Source:  Kawahara (in press)
Y-axis: probability subjects will rate a Pokemon name as appropriate for an "unevolved, smaller Pokemon creature 
Baseline constraints:  length of name in moras
Perturber constraints: whether name includes an initial voiced obstruent (such as [d])
Spreadsheet (forthcoming), plotting script

Data for Kawahara Pokemon study



Graphs used to diagnose theories

The  MaxEnt sigmoid

This is discussed extensively in the main text of my paper and is plotted here to permit the comparisons that follow.

The MaxEnt sigmoid

The asymmetrical sigmoid of one version of Noisy Harmonic Grammar

In one version of Noisy Harmonic Grammar, discussed in section 7.2 of the article text, the "noise" added to make theory stochastic is added to the constraint weights. This ends up producing a sigmoid curve quite different from that of Maxent; it is asymmetrical, and the long tail can be shown to asymptote at a value above zero.

Asymmetrical sigmoid of constraint-noise NHG

The symmetrical, oddly-similar sigmoid of another version of Noisy Harmonic Grammar

If, in designing a Noisy Harmonic Grammar framework, you instead add the "noise" to the completed Harmony values of candidates, you get a sigmoid that is astonishingly similar to the MaxEnt (astonishing, since the math is completely different). Here are the MaxEnt and NHG sigmoids superposed, with constraint weights suitably scaled to make the resemblance clear.

Resemblance of candidate-noise NHG to MaxEnt



How I made the curves

I'm sure there are better ways (for instance, R is probably good) but this was the method I arrived at on an ad hoc basis. You can see examples of how all this works if you will download the spreadsheets and plotting scripts for individual cases above.

1. Obtain data.  Some authors web-post their data, other have the data printed in their article, and still others give just a graph. Even with the latter, it is not hard to use Microsoft Paint to get the values:  look at the bottom of the screen for vertical and horizontal coordinates of points in pixels.  Hover over the data points, and over the legend ticks, and put their values into a spreadsheet.  Then you can use arithmetic (or the handy Excel FORECAST() function) to convert pixels into real values.

2. If necessary, reduce data from tokens to types-plus-counts.  I do this by applying my little Typizer program to the rows of a spreadsheet, read in plain text form, containing just the constraint violations.

3. Do a MaxEnt analysis of the data, which is easily done in spreadsheet form.  The spreadsheets above show you do this; it helps also to know the basics of MaxEnt; see the paper. The key step requires you to deploy the Excel Solver (which is free, but must be activated), in order to calculate constraint weights.  During this stage, you should calculate Harmony in two columns, one for Baseline constraints and one for Perturber constraints, then use their sum to give the overall Harmony from which probabilties are calculted.

In doing the MaxEnt analysis, use this trick, assuming a particular input has two candidates A and B:  if Candidate B has one violation of Constraint X, record the violation in the spreadsheet not as a 1 in the Candidate B's row, but rather as a -1 in Candidate A's row. Then the B row ends up blank, other than the crucial frequency value for B.  The math will come out the same, and it gives you the harmony values in ways plottable as a single number, as described in my paper.

4. Collate the data, keeping only Candidate A for each pair.  I perform this collation with formulas in the space below the main MaxEnt analysis. You must also collate the  values for Observed Frequency, Base Harmony, and Perturber Harmony.  Optionally, you can include data for Counts, if you'd like the program to plot as small the datapoints that are not well-attested. It is also good to gather values for Predicted Frequency; then you can make a scattergram with Observed against Predicted, calculate correlation, and in general assess whether your MaxEnt model is a good model.

5. Within the spreadsheet, fill in the necessary fields to make a plot. These are shown in blue in the spreadsheets posted here, and also can be seen in the downloadable plotting scripts.

6. Clip the blue material out of your spreadsheet and save it as a text file, which is the plotting script.

7. Download my PlotSigmoids.exe program (Windows only, sorry!), put it in a new folder of your choice, click on it, drag a plotting script file onto the designated blank area of the interface. It will make a bmp image and put it into the "out" subfolder.  

Questions:   bhayes@humnet.ucla.edu.


Last updated March 2021