Generalized additive modeling and dialectology

Lecture 5 of advanced regression for linguists

Martijn Wieling
Computational Linguistics Research Group

This lecture

  • Introduction
    • Logistic regression (recap)
    • Standard Italian and Tuscan dialects
  • Material: Standard Italian and Tuscan dialects
  • Methods: R code
  • Results
  • Discussion

Question 1

Logistic regression (recap)

  • Dependent variable is binary (1: success, 0: failure), not continuous
  • Transform to continuous variable via log odds: \(\log(\frac{p}{1-p})\) = logit\((p)\)
    • Done automatically in regression by setting family="binomial"
  • Generalized linear model: specific link function and error distribution
  • interpret coefficients w.r.t. success as logits: in R: plogis(x) plot of chunk unnamed-chunk-1

Standard Italian and Tuscan dialects

  • Standard Italian originated in the 14th century as a written language
  • It originated from the prestigious Florentine variety
  • The spoken standard Italian language was adopted in the 20th century
    • People used to speak in their local dialect
  • In this study, we investigate the relationship between standard Italian and Tuscan dialects
    • We focus on lexical variation
    • We assess which social, geographical and lexical variables influence this relationship

Material: lexical data

  • We used lexical data from the Atlante Lessicale Toscano (ALT)
    • We focus on 2060 speakers from 213 locations and 170 concepts
    • Total number of cases: 384,454
    • Dependent variable
      • 1: lexical form was different from standard Italian
      • 0: lexical form was identical to standard Italian

Geographic distribution of locations

Material: additional data

  • Speaker age
  • Speaker gender
  • Speaker education level
  • Speaker employment history
  • Number of inhabitants in each location
  • Average income in each location
  • Average age in each location
  • Frequency of each concept

Modeling geography's influence with a GAM

(R version 3.3.2 (2016-10-31), mgcv version 1.8.16)

geo <- bam(NotStd ~ s(Lon, Lat, k = 30), data = tuscan, family = "binomial", discrete = T)
summary(geo)

# 
# Family: binomial 
# Link function: logit 
# 
# Formula:
# NotStd ~ s(Lon, Lat, k = 30)
# 
# Parametric coefficients:
#             Estimate Std. Error z value Pr(>|z|)    
# (Intercept)  -0.2474     0.0033   -75.1   <2e-16 ***
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# 
# Approximate significance of smooth terms:
#             edf Ref.df Chi.sq p-value    
# s(Lon,Lat) 28.2     29   1591  <2e-16 ***
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# 
# R-sq.(adj) =  0.0042   Deviance explained = 0.312%
# fREML = 6.1609e+05  Scale est. = 1         n = 384454

First 15 two-dimensional basis functions

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First 15 two-dimensional basis functions

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Fitted surface

fvisgam(geo, view = c("Lon", "Lat"), too.far = 0.045, main = "", rm.ranef = T)

plot of chunk unnamed-chunk-5

Thin plate regression spline: scale-dependent

geo2 <- bam(NotStd ~ s(km.e, Lat), data = tuscan, family = "binomial", discrete = T)
fvisgam(geo2, view = c("km.e", "Lat"), too.far = 0.045, main = "", rm.ranef = T)

plot of chunk unnamed-chunk-6

Question 2

Solution: tensor product spline

geo3 <- bam(NotStd ~ te(km.e, Lat, k = c(6, 6)), data = tuscan, family = "binomial", discrete = T)
fvisgam(geo3, view = c("km.e", "Lat"), too.far = 0.045, main = "", rm.ranef = T)

plot of chunk unnamed-chunk-7

Varying geography's influence based on concept freq.

  • Wieling, Nerbonne and Baayen (2011) showed that the effect of word frequency varied depending on geography
  • Here we explicitly include this in the GAM with te(), which can model an \(N\)-way non-linear interaction:
    te(Lon, Lat, ConceptFreq, d=c(2,1))
  • As this pattern may be presumed to differ depending on speaker age, we can integrate this in the model as well:
    te(Lon, Lat, ConceptFreq, YearBirth, d=c(2,1,1))

Question 3

Full model specification

system.time(
  m <- bam(NotStd ~ te(Lon, Lat, ConceptFreq.log.z, SpeakerBirthYear.z, d=c(2,1,1)) +
    CommunitySize.log.z + SpeakerJob_Farmer + SpeakerEduLevel.log.z + SpeakerIsMale +
    s(Speaker,bs="re") + s(Location,bs="re") + s(Concept,bs="re") + 
    s(Concept,CommunityRecordingYear.z,bs="re") + s(Concept,CommunitySize.log.z,bs="re") +
    s(Concept,CommunityAvgIncome.log.z,bs="re") + s(Concept,CommunityAvgAge.log.z,bs="re") +
    s(Concept,SpeakerJob_Farmer,bs="re") + s(Concept,SpeakerJob_Executive_AuxiliaryWorker,bs="re") +
    s(Concept,SpeakerEduLevel.log.z,bs="re") + s(Concept,SpeakerIsMale,bs="re"), 
  data=tuscan, family="binomial", discrete=T, nthreads=4)
)

#    user  system elapsed 
#  2322.5    21.1   701.3

smry <- summary(m) # takes 10 minutes to calculate

Results: fixed effects and smooths

smry$p.table

#                       Estimate Std. Error z value Pr(>|z|)
# (Intercept)            -0.4282     0.1264   -3.39 7.08e-04
# CommunitySize.log.z    -0.0629     0.0223   -2.82 4.87e-03
# SpeakerJob_Farmer       0.0449     0.0169    2.66 7.81e-03
# SpeakerEduLevel.log.z  -0.0678     0.0126   -5.38 7.29e-08
# SpeakerIsMale           0.0378     0.0128    2.95 3.18e-03

head(smry$s.table, 1)

#                                                  edf Ref.df Chi.sq p-value
# te(SpeakerBirthYear.z,ConceptFreq.log.z,Lon,Lat) 221    265   3270       0

Interpreting logit coefficients (recap)

# chance for a male farmer in a
# very small village (z-scored
# population size = -2) for
# which the location is unknown
# with a very low education
# level (z-score = -2) to use a
# non-standard lexical form
(logit <- coef(m)["(Intercept)"] + 
    coef(m)["SpeakerIsMale"] + 
    coef(m)["SpeakerJob_Farmer"] + 
    -2 * coef(m)["CommunitySize.log.z"] + 
    -2 * coef(m)["SpeakerEduLevel.log.z"])

# (Intercept) 
#     -0.0841

plogis(logit)  # was: 0.438 (43.8%)

# (Intercept) 
#       0.479

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A complex geographical pattern

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Animation: increasing frequency for older speakers

Animation: increasing frequency for younger speakers

Results: random effects

tail(smry$s.table, 11)  # last 11 smooths are ranefs

#                                                   edf Ref.df Chi.sq   p-value
# s(Speaker)                                       97.1   2005    106  9.40e-03
# s(Location)                                     175.0    209   5642  1.33e-96
# s(Concept)                                      167.0    168 436864  0.00e+00
# s(CommunityRecordingYear.z,Concept)             158.9    170 155893 4.88e-181
# s(CommunitySize.log.z,Concept)                  149.9    169  29991 2.41e-111
# s(CommunityAvgIncome.log.z,Concept)             158.0    170 143207 1.75e-160
# s(CommunityAvgAge.log.z,Concept)                154.4    170 110722 5.80e-195
# s(SpeakerJob_Farmer,Concept)                     86.1    169  26572  1.27e-07
# s(SpeakerJob_Executive_AuxiliaryWorker,Concept)  53.3    170   3319  8.07e-04
# s(SpeakerEduLevel.log.z,Concept)                139.1    169   9377  8.05e-49
# s(SpeakerIsMale,Concept)                         85.4    169 112400  6.55e-11

By-concept random slopes for community size

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By-concept random slopes for speaker education level

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Discussion

  • Comparing Tuscan dialects to standard Italian revealed interesting dialectal patterns
  • GAMs are very suitable to model the non-linear influence of geography
  • The regression approach allowed for the simultaneous identification of important social, geographical and lexical predictors
  • By including many concepts, results are less subjective than traditional analyses focusing on only a few pre-selected concepts
  • The mixed-effects regression approach still allows a focus on individual concepts
  • More interested in Tuscan data and analysis? Paper package with all data and analyses available via http://www.martijnwieling.nl

Recap

  • We have applied GAMs to dialectometry data and learned how to:
    • use s() to model two-dimensional interactions on the same scale
    • model complex non-linear interactions using te()
    • use GAMs to conduct logistic regression (family="binomial")
  • After the break:
    • http://www.let.rug.nl/wieling/statscourse/lecture5/lab
      • We use a subset of Dutch dialect data in the lab (faster: no logistic regression)
      • Similar underlying idea: investigate the effect of geography, word frequency, and location characteristics on pronunciation distances from standard Dutch
  • Finally: please fill in the evaluation form of the course:
    http://www.let.rug.nl/wieling/statscourse/evaluation

Evaluation

Questions?

Thank you for your attention!

http://www.martijnwieling.nl
wieling@gmail.com