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What explains power laws in language typology and language change? - - PowerPoint PPT Presentation

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What explains power laws in language typology and language change? Gerhard J ager gerhard.jaeger@uni-tuebingen.de UT ubingen Freiburg, January 19, 2011 Gerhard J ager (UT ubingen) Power laws Freiburg, January 19, 2011 1 / 112

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SLIDE 1

What explains power laws in language typology and language change?

Gerhard J¨ ager gerhard.jaeger@uni-tuebingen.de

UT¨ ubingen

Freiburg, January 19, 2011

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 1 / 112

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SLIDE 2

Overview

Typological distribution of color naming systems Power laws Computer simulations

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 2 / 112

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The psychological color space

physical color space has infinite dimensionality — every wavelength within the visible spectrum is one dimension psychological color space is only 3-dimensional this fact is employed in technical devices like computer screens (additive color space) or color printers (subtractive color space) additive color space subtractive color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 3 / 112

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SLIDE 4

The psychological color space

psychologically correct color space should not only correctly represent the topology of, but also the distances between colors distance is inverse function of perceived similarity L*a*b* color space has this property three axes:

black — white red — green blue — yellow

irregularly shaped 3d color solid

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 4 / 112

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SLIDE 5

The color solid

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 5 / 112

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SLIDE 6

The Munsell chart

for psychological investigations, the Munsell chart is being used 2d-rendering of the surface of the color solid

8 levels of lightness 40 hues

plus: black–white axis with 8 shaded of grey in between neighboring chips differ in the minimally perceivable way

J I H G F E D C B A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 6 / 112

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The World Color Survey

started by Paul Kay and co-workers; traces back to Berlin & Kay 1969 investigation of color vocabulary of 110 non-written languages from around the world around 25 informants per language two tasks:

the 330 Munsell chips were presented to each test person one after the

  • ther in random order; they had to assign each chip to some basic

color term from their native language for each native basic color term, each informant identified the prototypical instance(s)

data are publicly available under http://www.icsi.berkeley.edu/wcs/

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 7 / 112

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The World Color Survey

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 8 / 112

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Raw data

are irregular and noisy example: randomly picked test person (native language: Piraha) 1,771 such data points in total

J I H G F E D C B A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 9 / 112

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Raw data

partition of a randomly chosen informant from a randomly chosen language

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 10 / 112

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Raw data

partition of a randomly chosen informant from a randomly chosen language

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 11 / 112

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Raw data

partition of a randomly chosen informant from a randomly chosen language

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 12 / 112

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Raw data

partition of a randomly chosen informant from a randomly chosen language

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 13 / 112

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Raw data

partition of a randomly chosen informant from a randomly chosen language

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 14 / 112

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Statistical feature extraction

first step: representation of raw data in contingency matrix

rows: color terms from various languages columns: Munsell chips cells: number of test persons who used the row-term for the column-chip

A0 B0 B1 B2 · · · I38 I39 I40 J0 red · · · 2 green · · · blue · · · black · · · 18 23 21 25 white 25 25 22 23 · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rot · · · 1 gr¨ un · · · gelb 1 · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rouge · · · vert · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

further processing:

divide each row by the number n of test persons using the corresponding term duplicate each row n times

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 15 / 112

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Statistical feature extraction: PCA

technique to reduce dimensionality of data input: set of vectors in an n-dimensional space first step: rotate the coordinate system, such that

the new n coordinates are

  • rthogonal to each other

the variations of the data along the new coordinates are stochastically independent

second step: choose a suitable m < n project the data on those m new coordinates where the data have the highest variance

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 16 / 112

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Statistical feature extraction: PCA

alternative formulation:

choose an m-dimensional linear sub-manifold of your n-dimensional space project your data onto this manifold when doing so, pick your sub-manifold such that the average squared distance of the data points from the sub-manifold is minimized

intuition behind this formulation:

data are “actually” generated in an m-dimensional space

  • bservations are disturbed by n-dimensional noise

PCA is a way to reconstruct the underlying data distribution

applications: picture recognition, latent semantic analysis, statistical data analysis in general, data visualization, ...

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 17 / 112

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Statistical feature extraction: PCA

first 15 principal components jointly explain 91.6% of the total variance choice of m = 15 is determined by using “Kaiser’s stopping rule”

principal components proportion of variance explained 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 18 / 112

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Statistical feature extraction: PCA

after some post-processing (“varimax” algorithm):

J I H G F E D C B A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 19 / 112

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Projecting observed data on lower-dimensional-manifold

noise removal: project observed data onto the lower-dimensional submanifold that was obtained via PCA in our case: noisy binary categories are mapped to smoothed fuzzy categories (= probability distributions over Munsell chips) some examples:

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 20 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 21 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 22 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 23 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 24 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 25 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 26 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 27 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 28 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 29 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 30 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 31 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 32 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 33 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 34 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 35 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 36 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 37 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 38 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 39 / 112

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Projecting observed data on lower-dimensional-manifold

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 40 / 112

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Smoothing the partitions

from smoothed extensions we can recover smoothed partitions each pixel is assigned to category in which it has the highest degree of membership

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 41 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 42 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 43 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 44 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 45 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 46 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 47 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 48 / 112

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SLIDE 49

Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 49 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 50 / 112

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Smoothed partitions of the color space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 51 / 112

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Convexity

note: so far, we only used information from the WCS the location of the 330 Munsell chips in L*a*b* space played no role so far still, apparently partition cells always form continuous clusters in L*a*b* space Hypothesis (G¨ ardenfors): extension of color terms always form convex regions of L*a*b* space

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 52 / 112

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Support Vector Machines

supervised learning technique smart algorithm to classify data in a high-dimensional space by a (for instance) linear boundary minimizes number of mis-classifications if the training data are not linearly separable

green red −3 −2 −1 1 2 3 −3 −2 −1 1 2 3

  • o
  • SVM classification plot

y x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 53 / 112

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Convex partitions

a binary linear classifier divides an n-dimensional space into two convex half-spaces intersection of two convex set is itself convex hence: intersection of k binary classifications leads to convex sets procedure: if a language partitions the Munsell space into m categories, train m(m−1)

2

many binary SVMs, one for each pair of categories in L*a*b* space leads to m convex sets (which need not split the L*a*b* space exhaustively)

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 54 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 55 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 56 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 57 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 58 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 59 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 60 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 61 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 62 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 63 / 112

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Convex approximation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 64 / 112

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Convex approximation

  • n average, 93.7% of all Munsell chips are correctly classified by

convex approximation

  • 0.80

0.85 0.90 0.95 proportion of correctly classified Munsell chips

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 65 / 112

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Convex approximation

compare to the outcome of the same procedure without PCA, and with PCA but using a random permutation of the Munsell chips

  • 1

2 3 20 40 60 80 100 degree of convexity (%)

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 66 / 112

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Convex approximation

choice of m = 10 is somewhat arbitrary

  • utcome does not depend very much on this choice though
  • ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

10 20 30 40 50 50 60 70 80 90 100

  • no. of principal components used

mean degree of convexity (%)

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 67 / 112

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Implicative universals

first six features correspond nicely to the six primary colors white, black, red, green, blue, yellow according to Kay et al. (1997) (and many other authors) simple system of implicative universals regarding possible partitions of the primary colors

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 68 / 112

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Implicative universals

I II III IV V     white red/yellow green/blue black           white red yellow green/blue black       white/red/yellow black/green/blue

 white red/yellow black/green/blue       white red/yellow green black/blue             white red yellow green blue black             white red yellow black/green/blue           white red yellow green black/blue           white red yellow/green/blue black           white red yellow/green blue black           white red yellow/green black/blue    

source: Kay et al. (1997)

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 69 / 112

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Partition of the primary colors

each speaker/term pair can be projected to a 15-dimensional vector primary colors correspond to first 6 entries each primary color is assigned to the term for which it has the highest value defines for each speaker a partition over the primary colors

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 70 / 112

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Partition of the primary colors

for instance: sample speaker from Piraha (see above): extracted partition:     white/yellow red green/blue black     supposedly impossible, but

  • ccurs 61 times in the database
J I H G F E D C B A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 71 / 112

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SLIDE 72

Partition of primary colors

most frequent partition types:

1

{white}, {red}, {yellow}, {green, blue}, {black} (41.9%)

2

{white}, {red}, {yellow}, {green}, {blue}, {black} (25.2%)

3

{white}, {red, yellow}, {green, blue, black} (6.3%)

4

{white}, {red}, {yellow}, {green}, {black, blue} (4.2%)

5

{white, yellow}, {red}, {green, blue}, {black} (3.4%)

6

{white}, {red}, {yellow}, {green, blue, black} (3.2%)

7

{white}, {red, yellow}, {green, blue}, {black} (2.6%)

8

{white, yellow}, {red}, {green, blue, black} (2.0%)

9

{white}, {red}, {yellow}, {green, blue, black} (1.6%)

10 {white}, {red}, {green, yellow}, {blue, black} (1.2%) Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 72 / 112

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Partition of primay colors

87.1% of all speaker partitions obey Kay et al.’s universals the ten partitions that confirm to the universals occupy ranks 1, 2, 3, 4, 6, 7, 9, 10, 16, 18 decision what counts as an exception seems somewhat arbitrary on the basis of these counts

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 73 / 112

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Partition of primary colors

more fundamental problem:

partition frequencies are distributed according to power law frequency ∼ rank −1.99

no natural cutoff point to distinguish regular from exceptional partitions

  • ● ●
  • 1

2 5 10 20 50 1 2 5 10 20 50 100 200 500 rank frequency

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 74 / 112

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Partition of seven most important colors

frequency ∼ rank−1.64

  • ●
  • ● ● ●
  • 1

2 5 10 20 50 100 1 2 5 10 20 50 100 200 500 rank frequency

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 75 / 112

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Partition of eight most important colors

frequency ∼ rank−1.46

  • ● ●
  • 1

2 5 10 20 50 100 200 1 2 5 10 20 50 100 200 rank frequency

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 76 / 112

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Power laws

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 77 / 112

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Power laws

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 78 / 112

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Power laws

from Newman 2006

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 79 / 112

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Power laws are not everywhere

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 80 / 112

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Other linguistic power law distributions

number of vowels vowel systems and their frequency of occurrence 3 14 4 14 5 4 2 5 97 3 6 26 12 12 7 23 6 5 4 3 8 6 3 3 2 9 7 7 3

(from Schwartz et al. 1997, Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 81 / 112

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Other linguistic power law distributions

frequency ∼ rank−1.06

  • ●
  • ●
  • ●
  • ● ●●●
  • 1

2 5 10 20 2 5 10 20 50 100 rank frequency

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 82 / 112

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Other linguistic power law distributions

size of language families source: Ethnologue frequency ∼ rank−1.32

  • ● ● ●
  • 1

2 5 10 20 50 100 1 5 10 50 100 500 rank frequency

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 83 / 112

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Other linguistic power law distributions

number of speakers per language source: Ethnologue frequency ∼ rank−1.01

  • ●
  • ●●
  • 1

2 5 10 20 50 100 200 5 10 20 50 100 200 500 1000 rank frequency (in million)

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 84 / 112

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The World Atlas of Language Structures

large scale typological database, conducted mainly by the MPI EVA, Leipzig 2,650 languages in total are used 142 features, with between 120 and 1,370 languages per feature available online

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 85 / 112

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The World Atlas of Language Structures

Maslova 2008, “Meta-typological distributions” hypothesis:

pick a random value for each feature estimate the probability that a random language has this value the likelihood that an arbitrarily chosen feature value has a probability x is proportional to a power of x

  • nly holds for the most frequent 30% of

all types

  • ● ● ●●●●●●
  • 0.01

0.02 0.05 0.10 0.20 0.50 1.00 0.005 0.010 0.020 0.050 0.100 0.200 0.500 1.000 P(p(type)<=x) x

for the entire range of type frequencies, the hypothesis can be rejected

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 86 / 112

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The World Atlas of Language Structures

however, Maslova is perhaps right in the assumption that languages are power-law distributed across WALS types worth to test it within features rather than across features problem: number of feature values usually too small for statistic evaluation solution:

cross-classification of two (randomly chosen) features

  • nly such feature pairs are considered that lead to at least 30

non-empty feature value combinations

pilot study with 10 such feature pairs

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 87 / 112

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SLIDE 88

The World Atlas of Language Structures

Feature 1: Consonant-Vowel Ratio Feature 2: Subtypes of Asymmetric Standard Negation Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 88 / 112

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SLIDE 89

The World Atlas of Language Structures

Feature 1: Weight Factors in Weight-Sensitive Stress Systems Feature 2: Ordinal Numerals Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 89 / 112

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SLIDE 90

The World Atlas of Language Structures

Feature 1: Third Person Zero of Verbal Person Marking Feature 2: Subtypes of Asymmetric Standard Negation Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 90 / 112

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The World Atlas of Language Structures

Feature 1: Relationship between the Order of Object and Verb and the Order of Adjective and Noun Feature 2: Expression of Pronominal Subjects Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

3

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 91 / 112

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SLIDE 92

The World Atlas of Language Structures

Feature 1: Plurality in Independent Personal Pronouns Feature 2: Asymmetrical Case-Marking Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 92 / 112

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SLIDE 93

The World Atlas of Language Structures

Feature 1: Locus of Marking: Whole-language Typology Feature 2: Number of Cases Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 93 / 112

slide-94
SLIDE 94

The World Atlas of Language Structures

Feature 1: Prefixing vs. Suffixing in Inflectional Morphology Feature 2: Coding of Nominal Plurality Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

3

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 94 / 112

slide-95
SLIDE 95

The World Atlas of Language Structures

Feature 1: Prefixing vs. Suffixing in Inflectional Morphology Feature 2: Ordinal Numerals Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 95 / 112

slide-96
SLIDE 96

The World Atlas of Language Structures

Feature 1: Coding of Nominal Plurality Feature 2: Asymmetrical Case-Marking Kolmogorov-Smirnov test: positive

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 96 / 112

slide-97
SLIDE 97

The World Atlas of Language Structures

Feature 1: Position of Case Affixes Feature 2: Ordinal Numerals Kolmogorov-Smirnov test: negative

10 10

1

10

2

10

−2

10

−1

10 Pr(X ≥ x) x

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 97 / 112

slide-98
SLIDE 98

Why power laws?

Critical states Power laws are characteristic of critical states

  • nly small ice crystals in water above freezing point
  • ne big ice crystal in water below freezing point

during transition from liquid to solid state:

ice crystals of many sizes power-law distributed

similar effect for all kinds of phase transitions in physics power laws are considered finger print of criticality

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 98 / 112

slide-99
SLIDE 99

Why power laws?

Self-organized criticality some systems tend to return into a critical state due to their internal dynamics (see Bak et al. 1987) well-studied effect in computer simulations of cellular automata candidates for real-life examples are

earth quakes forest fires breakdowns of electricity networks landscape formation avalanches ...

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 99 / 112

slide-100
SLIDE 100

Why power laws?

The sandpile model cellular automaton; loosely inspired by real sand piles each cell has a certain value, its slope single grains are added at random, increasing the slope if the slope of a cell exceeds a critical value:

its slope is reduced by r the slope of the four neighboring cells is increased by 1

this may turn neighboring cells into the critical state, leading to further shifts see the computer simulation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 100 / 112

slide-101
SLIDE 101

The sandpile model

both avalanche sizes and avalance durations are distributed according to a power law

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 101 / 112

slide-102
SLIDE 102

The forest fire model

cellular automata model; inspired by behavior of wildfires each cell can be in either of three states: empty, tree, fire update rules:

fire → empty tree and fire in neighoring cell → fire with small probability: empty → tree with even smaller probability: tree → fire

simulation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 102 / 112

slide-103
SLIDE 103

The forest fire model

size of contiguous clusters of trees or clusters of empty space are power law distributed

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 103 / 112

slide-104
SLIDE 104

Simulating the evolution of color terms

Communication game game between a sender and a receiver two-dimensional conceptual space (n × n cells, periodic boundaries) small number of signals

  • ne round:

nature picks out a point in the concpetual space at random and shows it to the sender the sender sends a signal to the receiver the receiver has to guess which point the sender was referring to both receive the same payoff: payoff ∼ exp(−||ps − pr||2) if the distance between the sender’s point and the receiver’s guess is small, the payoff is high, and vice versa

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 104 / 112

slide-105
SLIDE 105

Simulating the evolution of color terms

Evolutionary dynamics each player has a memory for point-signal associations after each round, the association between the signal and the point which were used in this round are strengthened proportional to the payoff of this round amounts to an evolutionary dynamics of associations:

succesful associations have a high fitness and are selected unsuccesful associations have a low fitness and die out

simulation

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 105 / 112

slide-106
SLIDE 106

Simulating the evolution of color terms

Long-run dynamics players quickly evolve towards a local fitness maximum (neutrally stable states) induces a partition of the conceptual space into convex categories (each corresponding to one signal) most of the time evolution ends in one of the four global maxima (evolutionarily stable states)

  • nce a stable state has been reached, evolution comes to a standstill

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 106 / 112

slide-107
SLIDE 107

Spatial evolution

100 agents arranged on a 10 × 10 grid periodic boundaries in each round

a pair of neighbors is selected at random they talk to each other and update their point-signal associations accordingly

this is repeated thousands of times

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 107 / 112

slide-108
SLIDE 108

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-109
SLIDE 109

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-110
SLIDE 110

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-111
SLIDE 111

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-112
SLIDE 112

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-113
SLIDE 113

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-114
SLIDE 114

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-115
SLIDE 115

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-116
SLIDE 116

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-117
SLIDE 117

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-118
SLIDE 118

Spatial evolution

population does not reach a stable homogenous state “languages” of neighbors are similar, but not identical

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 108 / 112

slide-119
SLIDE 119

Spatial evolution

no regions of completely identical languages however, clear “isoglosses” for single “concepts”

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 109 / 112

slide-120
SLIDE 120

Spatial evolution

no regions of completely identical languages however, clear “isoglosses” for single “concepts”

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 109 / 112

slide-121
SLIDE 121

Spatial evolution

no regions of completely identical languages however, clear “isoglosses” for single “concepts”

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 109 / 112

slide-122
SLIDE 122

Spatial evolution

no regions of completely identical languages however, clear “isoglosses” for single “concepts”

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 109 / 112

slide-123
SLIDE 123

Spatial evolution

no regions of completely identical languages however, clear “isoglosses” for single “concepts”

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 109 / 112

slide-124
SLIDE 124

Spatial simulation produces power law

Simulation data produce similar data structure like World Color Survey each artificial agent is treated as test person points in conceptual space ≈ Munsell chips signal with strongest association to that point ≈ categorization judgment same method of data evaluation:

Principal Component Analysis dimensionality reduction automatic classification of speakers into categorization types

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 110 / 112

slide-125
SLIDE 125

Spatial simulation produces power law

  • ●
  • ● ●●
  • 1

2 5 10 20 50 1 2 5 10 20 50 100 rank frequency

frequency ∼ rank−1.3

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 111 / 112

slide-126
SLIDE 126

Conclusion

power laws are very common in cross linguistic variation indications that they are also characteristic of language change processes they are typical of self-organized criticality simulation of language evolution in a spatially structured population produces power law behavior (plus other characteristics that are

  • bserved in natural language)

still inconclusive, but encouraging

Gerhard J¨ ager (UT¨ ubingen) Power laws Freiburg, January 19, 2011 112 / 112