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Modelu’r Gymraeg

Cafodd bapur academaidd gryn sylw yn y Wasg yn ddiweddar gydag adroddiadau amdano’n ymddangos yn y New Scientist, Nation.Cymru, y BBC, y Telegraph ac hyd yn oed y Daily Mail, ymhlith eraill, mae’n siŵr. “Whole nation ‘could speak Welsh’ within 300 years” oedd pennawd gwirion adroddiad y BBC. Dyma ychydig o gefndir, yn y gobaith y bydd rhannu ychydig o wybodaeth am dechnegau modelu yn y pendraw yn gwella safon y drafodaeth gyhoeddus ar waith fel hwn.

Model mathemategol oedd sail y papur. (Dydw i ddim am geisio egluro y gwahaniaeth rhwng model mathemategol a model ystadegol. Ymchwiliwch ar y we. Cafodd ei drafod yn ddiweddar gan Saltelli, A. A short comment on statistical versus mathematical modelling. Nat Commun 10, 3870 (2019). Os am eglurad manwl efallai hoffech fwrw golwg ar https://www4.stat.ncsu.edu/~davidian/stma810c/lectures/integrating.pdf.)

Erbyn hyn, mae cryn nifer o ymchwilwyr wedi defnyddio’r Gymraeg wrth geisio datblygu modelau mathemategol o sut mae nifer neu ganran siaradwyr iaith yn newid, a hynny am fod mwy o ddata ar gael am y Gymraeg na’r rhan fwyaf o ieithoedd sy dan fygythiad. Er bod mwy o ddata am y Gymraeg, dydy hynny ddim yn golygu fod digon o ddata ar gael i ddatblygu model dibynadwy,

Dyma rai papurau lle mae’r Gymraeg wedi ei defnyddio, yn ôl dyddiad:

1, Abrams, D., Strogatz, S. Modelling the dynamics of language death. Nature 424, 900 (2003) doi:10.1038/424900a https://www.math.uh.edu/~zpkilpat/teaching/math4309/project/nature03_abrams.pdf

Cafodd y papur yma sylw mawr ac esgorodd ar nifer o’r papurau canlynol a geisiodd ddatblygu’r model syfaenol a oedd, yng ngeirau’r awdurdon, yn trin “… languages as fixed, and as competing with each other for speakers. For simplicity, we also assume a highly connected population, with no spatial or social structure, in which all speakers are monolingual.”

Daw’r ffigur isod o’r papur. Fel sy’n gyffredin i’r maes hwn, defnyddiwyd hafaliadau differol i fodelu eu syniadau hwy o’r ffactorau oedd yn achosi i’r gyfan y siaradwyr newid. Roedd yn rhaid iddynt amcangyfrif  c, s, a ac x(0)  yn eu hafaliad:
Pyx(x,s)= cx^a s
Pxy(x,s)=c(1-x)^a (1-s)

I wneud hynny, defnyddiwyd ychydig iawn o bwyntiau data o gyfrifiadau hanesyddol a Chyfrifiad 1991, sef yr un diweddaraf oedd ag gael iddynt ar y pryd. Yn ôl eu dehongliad hwy o’r paramedr ‘s’, roedd yn cynrychioli statws. O gymharu â Gaeleg yr Alban (Ffigur 1 a) a Quechua (Ffigur 1 b) felly, roedd y Gymraeg yn elwa drwy fod ganddi statws uwch na’r rheini.

Ffigur 1 o bapur Abrams a Strogatz

Ffigur 1 o bapur Abrams a Strogatz

2. Sankoff, D. (2008). ‘How to predict the evolution of a bilingual community’, in Meyerhoff, M.
and Nagy, N.(eds.), Social Lives in Language – Sociolinguistics and Multilingual Speech
Communities Celebrating the Work of Gillian Sankoff, University of Edinburgh/University of
Toronto.

Nid model hafaliadau differol ddefnyddiodd Sankoff ond methodoleg carfan demograffig. Ar ôl amcangyfrif y ganran o bobl oed x ym mlwyddyn t, defnyddiwyd honno yn sail i amcangyfrif y ganran ymhlith yr un bobl – a fyddai’n x+1 oed erbyn hynny – yn y flwyddyn ganlynol (t+1) . Addasiad o hwn a ddefnyddiwyd gan Lywodraeth Cymru wrth lunio amcanestyniadau a’r hyn a alwasant yn ‘daflwybr’ at y targed o filiwn o siaradwyr yn eu strategaeth. Gweler Amcangyfrifon siaradwyr Cymraeg 2011 i 2050: adroddiad technegol.

Sankoff_Ffig 1

Sankoff_Ffig 1

Roedd yn dda gweld rhywun o brifysgol yng Nghymru – John Wyburn o Brifysgol De Cymru – yn cyfrannu at y maes yn y ddau bapur isod:

3. Wyburn, J,, Hayward, J. (2008) The Future of Bilingualism: An Application of the Baggs and Freedman Model, The Journal of Mathematical Sociology,32:4,267-284, DOI: 10.1080/00222500802352634

a

4. Wyburn, J,, Hayward, J. (2009) OR and language planning: modelling the interaction between unilingual and bilingual populations. Journal of the Operational Research Society, 2009, Volume 60, Number 5, Page 626 https://rdcu.be/b0xKe

Defnyddiwyd model tebyg i Abrams a Strogatz yn y papur cyntaf ac un dynameg system a methodoleg ymchwil gweithredol (OR) yn yr ail. Rhoddodd yr ail y crynodeb defnyddiol canlynol o’r maes adeg ysgrifennu’r papur:

“To date attempts to model the interaction of language-defined populations have relied on ordinary differential equations (ODEs). In particular Baggs and Freedman (1990) is much cited, and subsequent examinations of the problem (Baggs and Freedman, 1991, 1993; El-Owaidy and Ismail, 2002) are elaborations on this seminal paper. Abrams and Strogatz (2003), while not acknowledging Baggs and Freedman, adopt a similar approach. …. All such ODE-based models are limited in several ways. Their terms tend to be ‘black boxes’ obscuring the true
mechanisms of change;. …; and they are concerned with the long-term attainment of equilibria,
perhaps over centuries, during which time their parameters are unlikely to remain constant. For these reasons, and their inaccessibility to non-mathematicians, they are of limited use to the social planners, politicians, educational reformers, applied linguists, and market analysts to whom their predictions are of interest.”

5. Minett, J.W., Wang, W.S-Y. (2008) Modelling endangered languages: The effects of bilingualism and social structure. Lingua .188(1), 19-45. doi:10.1016/j.lingua.2007.04.001

Er i’r rhain grybwyll y Gymraeg wrth iddynt gyfeirio at waith Abrams a Strogatz, ni chafodd ei modelu ganddynt. Yn wir, er iddynt estyn fformula Abrams  fel y gallai “account for a number of factors that influence language competition and maintenance, including language status, proportions of speakers, population size, and the availability of monolingual and bilingual educational resources”, ni cheisiasant ffitio eu model i’r un iaith arall chwaith. Dylwn nodi fodd bynnag eu bod wedi defnyddio techneg arall, Yn ogystal â hafaliadau differol, defnyddion nhw efelychiadau yn seiliedig ar asiant. Gweler https://en.wikipedia.org/wiki/Agent-based_model os am wybod mwy am y dechneg.

6. Anne Kandler, Roman Unger and James Steele (2010) Language shift, bilingualism and the future of Britain’s Celtic languages.Phil. Trans. R. Soc. B 2010 365, 3855-3864 https://royalsocietypublishing.org/doi/full/10.1098/rstb.2010.0051

Unwaith eto, gadawaf i’r awduron egluro eu gwaith:

“We model the dynamics of language shift as a competition process in which the numbers of speakers of each language vary as a function both of internal recruitment (as the net outcome of birth, death, immigration and emigration rates of native speakers), and of gains and losses owing to language shift. Mathematical work on language shift dynamics has been stimulated by Abrams & Strogatz (2003), who proposed a simple two-language competition model in which the outcome (extinction of one or other language) is determined by the strength of innate
attraction to the higher status language and by the initial conditions (with preferential attachment—the nonlinear effect of initial concentrations on shift rates—capable of driving the higher status language to extinction when its speakers are rare). Our own basic model is very different. In addition to the status-related shift term, we model the changing sizes of speaker sub-populations as the balance of births and deaths, and of immigration and emigration, and
we model a bilingual transition state.”

Gan eu bod wedi cynnwys siartiau yn benodol i’r Gymraeg, cystal eu dangos yma. Mae Ffig.3(b) dipyn y fwy calonogol na 3(a) ond rhaid i chi droi at y papur am fanylion.

Ffigur 3 o Kandler

Ffigur 3 o Kandler

7. Menghan Zhang,Tao Gong (2013). Principles of parametric estimation in modeling language competition. 9698–9703.Proceedings of the National Academy of Sciences PNAS, 110(24) https://doi.org/10.1073/pnas.1303108110

Fe nodwyd diffygion model Abrams a Strogatz, a’u paramedr s a briododlwyd, fel y soniais uchod, i ‘statws’:

“…this abstract parameter lacked explicit sociocultural meanings; it remained unclear what were the characteristics of a language having a prestige value, say 1.2, and what was the sociocultural condition corresponding to the difference between two languages having prestige values, say 1.2 and 1.3, respectively. Lacking such empirical foundations, the prestige value had to be obtained via curve fitting, thus making this model useless in cases lacking sufficient empirical data.”

Aethon ymlaen felly:

“we define two concrete parameters, namely the impacts and inheritance rates of competing languages, and adopt the Lotka–Volterra competition model … in evolutionary biology  to study the dynamics of language competition. Meanwhile, we propose a language diffusion principle and two language inheritance principles to calculate these parameters based on the relevant data of population censuses and language surveys. The language diffusion principle, inspired by Fourier’s law of heat conduction, computes the impacts of competing languages from the population sizes of these languages and the geographical distances between the region where competition occurs and the population centers of these languages. …”

Ffig.1 o Menghan Zhang,Tao Gong

Ffig.1 o Menghan Zhang,Tao Gong

8. Isern N, Fort J. (2014). Language extinction and linguistic fronts. J. R. Soc. Interface 11: 20140028. http://dx.doi.org/10.1098/rsif.2014.0028

Cafodd y Gymraeg le blaenllaw yn y papur hwn er iddynt erbyn y diwedd ddweud “…the linguistic policies applied in Wales since the 1970s seem to have been able to raise the status of Welsh and stabilize (see figure 2d after 1971) and even reverse the languages shift (after 2001
[30,42]). In that sense, the data for the Welsh language after “1971 should probably have been omitted from the analysis” oherwydd eu bwriad oedd i:

“…define a new language competition model that can describe the historical decline of minority languages in competition with more advantageous languages. We then implement this non-spatial model as an interaction term in a reaction–diffusion system to model the evolution of the two competing languages. We use the results to estimate the speed at which the more advantageous language spreads geographically, resulting in the shrinkage of the area of dominance of the minority language. We compare the results from our model with the bserved retreat in the area of influence of the Welsh language in the UK, obtaining a good agreement between the model and the observed data.”

9. Wyburn, J. (2018) Media pressures on Welsh language preservation, The Journal of Mathematical Sociology,42:1,37-46, DOI: 10.1080/0022250X.2017.1396984 https://pure.southwales.ac.uk/files/1558311/JMS_JWyburn_Media_Pressures_17.pdf

Cyflwynwyd “a model explicitly addressing the interaction and evolution of populations distinguished by their language use, within modern Wales…” a’r model yn un epidemiolegol.  Defnyddiodd elfen o waith Kandler ynddo.

Yn wahanol iawn i’r rhan fwyaf o bapurau sydd wedi eu rhestru yma, daeth i gasgliad gwahanol iawn: “The model can be tested against census data (HMSO 1991–2011) as in Figure 5 [isod]. Values…adequately reproduce the observed behavior.
The recent fall is symptomatic of a rapid decline in the number of Welsh speakers, and effective
extinction in 60 years, i.e., by 2051.”

Fig5 Wyburn 2018

Fig5 Wyburn 2018

10. Sperlich, S., Uriarte, J-R.,(2019) The economics of minority language use: theory and empirical evidence for a language game model
https://arxiv.org/pdf/1908.11604.pdf

“We study three modern multilingual societies -the Basque Country, Ireland and Wales- which are endowed with two, linguistically distant, official languages: A, spoken by all individuals, and B, spoken by a bilingual minority. In the three cases it is observed a decay in the use of minoritarian B, a sign of diversity loss. However, for the Council of Europe the key factor to avoid the shift of B is its use in all domains. Thus, we investigate the language choices of the bilinguals by means of an evolutionary game theoretic model. We show that the language population dynamics has reached an evolutionary stable equilibrium where a fraction of bilinguals have shifted to speak A. Thus, this equilibrium captures the decline in the use of B…. “

Rwy’n gweld defnyddio’r model gêm yn ddiddorol ac yn berthnasol. Fe eglurir:

“We seek to understand the data of B use. To this end, we model the strategic manner by which the members of bilingual population choose, at the beginning of an interaction, the language to use with the interactive partners….” ….

“…the bilinguals’ language strategic behaviour is captured by the following pure strategies of bilinguals:
R: Reveal always your type, so that you will speak B whenever you meet a bilingual.
H: Hide your type, and reveal it only when matched with an R-player. That is, speak A, and switch to B only when you are addressed in B.”

Mae llawer o fathemateg yn y papur ond mae hefyd yn ceisio dangos perthynas y model at bolisïau ymarferol, e.e. bathodynau Iaith Gwaith, ac fe ddylai peth o resymu’r papur fod o ddiddordeb i gynllunwyr iaith yn y byd go iawn.

Gobeithio y bydd yr adolygiad byrfyfyr yma o faes modelu mathemateg mewn perthynas â’r Gymraeg o ddefnydd i rywun hefyd.


Dyma fanylion y papur a’m hysgogodd i ysgrifennu’r llith yma:

Barrett-Walker T, Plank MJ, Ka’ai-Mahuta R, Hikuroa D, James A.(2019) Kia kaua te reo e rite ki te moa, ka ngaro: do not let the language suffer the same fate as the moa. J. R. Soc. Interface 20190526. http://dx.doi.org/10.1098/rsif.2019.0526