A Comparison of natural (english) and artificial (esperanto) languages. A Multifractal method based analysis

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by  J. Gillet

APS/123-QED
A Comparison of natural (english) and artificial (esperanto) languages. A Multifractal
method based analysis
J. Gillet∗ and M. Ausloos†
GRAPES, B5a Sart-Tilman, B-4000 Li`ege, Belgium
(Dated: February 3, 2008)
Wepresentacomparisonoftwoenglishtexts,writtenbyLewisCarroll,one(Aliceinwonderland)
and the other (Through a looking glass), the former translated into esperanto, in order to observe
whether natural and artificial languages significantly differ from each other. We construct one
dimensional time series like signals using either word lengths or word frequencies. We use the
multifractalideasforsortingoutcorrelationsinthewritings. Inordertochecktherobustnessofthe
methodswealsowrite(!) (consider?) thecorrespondingshuffledtexts. Wecomparecharacteristic
functions and e.g. observe marked differences in the (far from parabolic) f(α) curves, differences
whichweattributetoTsallisnonextensivestatisticalfeaturesinthefrequencytimeseriesandlength
time series. The esperanto text has more extreme vallues. A very rough approximation consists in
8 modeling the texts as a random Cantor set if resulting from a binomial cascade of long and short
0
words(orwordsandblanks). Thisleadstoparameterscharacterizingthetextstyle,andmostlikely
0
in fine the author writings.
2

n PACSnumbers:
a
J
As soon as modern fractals appeared in order to de- itcanbedecomposedthroughlevelthresholdswhichcan
6
1 scribe physical objects, it was evident that some gen- be thought to be a set of characters taken from an al-
eralization was in order: multifractals spurred up, e.g., phabet. Here below we take real texts in fact as the

] since obvioulsy a fractal dimension D is not enough to source of experimental observations and follow the mul-
L
describe an object [1, 2]. The more so in non equilib- tifractal ideas to make an analysis of such texts. The
C
rium systems, characterized by some unusual dynamics. main question concerns whether multifractals are indeed
s. Through a generator and from an initiator one can pro- found in real texts; a question raised in [20]; another is
c duce a fractal object with a given dimension. How to whether the technique of analysis can give some insight
[
produce realistic and meaningful multifractal models is on a logical construction [21], from which stems the pos-

1 a challenge. Do they really exist [3]? Do multifractal sibleconnectionofsuchideaswithcoding, transcription,
v model exist nowadays [4]? These questions come in par- machinetranslation, socialdistances, networkproperties
0 allel with the measurement of the fractal dimension, ... of languages, ... [21] [22] [23] and more classically in
1 and its distribution. One question of interest is whether physicsaboutidentifyingcoherentstructuresinspatially
5
the apparently multifractal nature of an object is due to extended systems [24]. We examine two classical english
2
itsfinitesizeortoacomplexdynamicalfeatureorsome- texts but also one translation, into an unusual language,
.
1 thing else! Some attempt in this direction results from and the corresponding shuffled texts. We focus on how
0
observation of multifractal features in meteorology and local/structural properties develop into global ones.
8
climatestudies[5, 6, 7], butalsoinmanyotherfields[8],
0
like mathematical finance [4, 9, 10, 11, 12, 13] Since Shannon [25], writings and codings are of inter-
:
v estinstatisticalphysics. Writingsaresystemspractically
Let us recall that one has basically to obtain a D(q)
i composed of a large number of internal components (the
X functionorthef(α)spectrum,whereqrepresentsthede-
words, signs, and blanks in printed texts). In terms of
r greeofsomemomentdistributionofsomevariable,andα
a issomesortofcriticalexponentatphasetransitions,also complexity investigations, writings which are a form of
recordedlanguages,likelivingsystems,belongtothetop
called the Holder exponent; f(α) being its distribution.
levelclassinvolvinghighlyoptimizedtolerancedesign,er-
There is a need for experimental work leading to re-
ror thresholds in optimal coding, and financial markets
liable D(q) and f(α) data, before modeling. Interest-
[21]. Relevant questions pertain to the life time, concen-
ing pioneering data should be here recalled : see work
tration, distribution, .. complexity of these. One should
on DLA [14, 15], DNA [16, 17], SOI [18], NAO [19], ....
distinguish two main frameworks. On one hand, lan-
It appears that most of the time some “signal is either
guage developments seem to be understandable through
directly a time series or is transformed into a time se-
competitions, like in Ising models, and in self-organized
ries; more generally, the signal is called a text, because
systems. Theirdiffusionseemssimilartopercolationand
nucleation-growthproblemstakingintoaccounttheexis-
tenceofdifferenttimescales, forinter-andintra-effects;
thisistherealmofanthropology. Thesecondframeorig-
∗Electronic address: [email protected]; Present address : Institut
inates from more classical linguistics studies; it pertains
de Physique Nucl´eaire, Atomique et de Spectroscopie, Universit´e
to the content and meanings of words and texts. Con-
deLi`ege,B-4000Li`ege,Belgium
†Electronicaddress: [email protected] cerningtheinternalstructureofatext, supposedlychar-
2
acterized by the language in which it is written, it is tion or to the specificity of this author work. Previous
well known that a text can be mapped into a signal, of work on the english AWL version (AWLeng) should be
course through the alphabet characters. However it can mentioned [40], but pertains to a mere Zipf analysis.
be also reduced to less abundant symbols through some In Sect. I, we present some elementary facts on these
threshold, like a time series, which can be a list of +1 texts and briefly expose the methodology, i.e. we em-
and -1, or sometimes 0. In fact, laws of text content and phasize that we distinguish between “frequency time se-
structureshavebeensearchedforalongtimeagobyZipf ries (FTS) and “length time series (LTS). We recall the
and others, see many refs. in [26], through the least ef- multifractal technique for this specific application. In
fort (so called ranking) method. The technique is now order to check the robustness of the method we also in-
currently applied in statistical physics as a first step to vent (write !) (or consider) the corresponding shuffled
obtain, when they exist, the primary scaling law. Yet texts, to which we apply the same technique of analysis.
long range order correlations (LROC) between words in Therefore, in Sect. II, we present the somewhat unusual
text are searched for. In [27] it was claimed that LROC results, and discuss them in Sect. III. For the simplicity
express an author’s ideas, and in fine consist in some of the discussion we very roughly approximate the texts
author’s signature. as resulting from a binomial cascade of short and long
Interestingly writings can be thought as social net- words. We obtain parameters in fine characterizing the
works [23]. Social networks have fractal properties [28]; text style and the author’s writings. We observe that
most usually they should be multifractals; one can thus such multifractals have a deep connection to Tsallis non
imagine/consider that a text which is a form of partially extensive statistics as pointed out in ref. [41] in another
self-organized social network (for words) due to gram- framework.
matical and style constraints present multifractal fea-
tures. The properties of such texts taken as signals have
already been examined, e.g., a multifractal analysis of I. DATA AND METHODOLOGY
Moby Dick letter distribution can be found in [20].
Even though we recognize such a pioneering paper, For our empirical considerations we have selected the
we stress that sentences made of words, not letters, are two texts here above mentioned downloading them from
translated. Thus we present below an original consider- a freely available site [37], resulting obviously into three
ation in this respect, i.e. the analysis and results about files, called. Next, we have removed the chapter heads.
a translation between one of the most commonly used All our analyses are carried over this reorganized file for
language, i.e. english, and a relatively recent language, eachtext. Thereafter,wehaveshuffledthesetexts,inthe
i.e. esperanto. Esperanto is an artificially constructed files, without taking into account the punctuation [42].
language [29], which was intended to be an easy-to-learn There are two ways to construct a time series from
lingua franca. Statistical analyses seem to indicate that such documents
esperanto’s statistical proportions are similar to those of 1. Take a document of N words. Select all the differ-
other languages [30]. It was found that esperanto’s sta- entwords. Countthefrequencyf ofappearanceof
tisticalproportionsresemblemostlythoseofGermanand each word in the document. The time of “appear-
Spanish,andsomewhatsurprisinglyleastthoseofFrench ance is played by the rank position of the word in
and Italian. English seems to be the intermediary case. the file. We map the word frequency to a time se-
Comparison of different languages (writings) arising ries f(t). Such a time series is called the frequency
from apparently different origins or containing different time series (FTS).
signs, e.g. greek [31], turkish [32], chinese [33], ... even
2. Take a document of N words. Consider the length
somewhatartificiallanguages,likethoseusedforsimula-
l (number of letters) of each word. Record where
tioncodesoncomputers[34]hasalsobeenmade. Toour
each word of length l is located in the text; the
knowledge few comparisons have been presented about
time is played by the position of the word in the
written texts translated from one to another language
document, i.e. the first word is considered to be
[35,36]andinparticularfromthepointofviewofLROC
emittedattimet=1,thesecondattimet=2,etc.
in words.
A time series l(t) is so constructed. We refer to
The text used here was chosen for its wide diffusion,
such a time series as the length time series (LTS).
freelyavailablefromtheweb[37]andasarepresentative
one of a famous scientist, Lewis Caroll, i.e. Alice in Obviously there is a large number of ways to map a
wonderland (AWL) [38]. Moreover knowing the special text to a time series, but in the present study we only
(mathematical)qualityofthisauthormind,andsome,as consider the above two since some physical meaning can
we thought a priori, some possibly special way of writ- be thought to arise in the mapping. As indicated in [43]
ing, another text has been chosen for comparison, i.e. the length of the word is associated with speaker effort,
Through a looking glass (TLG) [39]; - to our knowledge meaning that the longer the word the higher the effort
only available in english (on the web). This will allow required to pronounce it. The frequency of the word is
us to discuss whether the differences, if any, between es- also associated with the hearer effort as frequently used
peranto and english, are apparently due to the transla- words require less effort to be understood by the hearer.
3
1.4 AWLeng 1 AWLeng
AWLesp AWLesp
Generalized fractal dimension D(q)1.2 TLGeng Generalized critical exponent f()α–01 TLGeng
1.0
1.0 1.2 1.4 1.6
–20 –10 0 10 20 α
q
1.4 1
AWLeng
)
neralized fractal dimension D(q)111...123 ATLWGLenesgp Generalized critical exponent f(α–01 AWLeng
Ge1.0 AWLesp
TLGeng
1.0 1.2 1.4
α
–20 –10 0 10 20 FIG. 3: f(α) for (a) FTS (b) LTS of the indicated texts
q
FIG. 1: D(q) for (a) FTS (b) LTS of the original texts
Shuffled AWLeng
1.10 Shuffled AWLesp 1.0 Shuffled AWLeng
Shuffled TLGeng Shuffled AWLesp
neralized fractal dimension D(q)11..0005 Generalized critical exponent f()α 00..05 Shuffled TLGeng
Ge
0.95 –0.5
–20 –10 0 10 20 0.9 1.0 1.1 1.2
q α
1.0
Shuffled AWLeng
1.2 Shuffled AWLesp
D(q) Shuffled TLGeng nt f()α 0.5
Generalized fractal dimension 11..01 Generalized critical expone–00..05 SShhuufffflleedd AAWWLLeenspg
Shuffled TLGeng
–1.0
0.9 1.0 1.1 1.2 1.3
α
–20 –10 q0 10 20 FIG. 4: f(α) for (a) FTS (b) LTS of the shuffled indicated
FIG. 2: D(q) for (a) FTS (b) LTS of the shuffled indicated texts
texts
4
These time series are thereby analyzed along the mul- II. RESULTS
tifractal ideas, for which we briefly recall the formulae of
interest in order to set the notations. The results of the FTS and LTS multifractal analysis
forthethreemaintextsandtheirshuffledcorresponding
ones are shown in Figs. 1-4 (a-b).
A. Multifractal Analysis
Amereperusingofthegraphsindicatethatthemulti-
fractalapproachisingoodorder,e.g. sinceD(q)isnota
Letthe(LTSouFTS)timeserieshavingN datapoints single point, and should allow one to observe interesting
(words, here), i.e. yi (1≤i≤N). LRO correlations and local ones.
Transformtheseriesasfollows: ifthelengthofaword
mot in LTS (or its frequency in FTS) is smaller than the
nextone,theformerwordgetsavalue=2;ifitisgreater, A. D(q) plots: Figs. 1-2
it gets the value = 1; and 0 if both are equal.
The new series is called M (1 ≤ i ≤ N −1). Each
i InFTS,thegeneralizedfractaldimensionhasasimilar
M is cut into N subseries of size s , where N is the
i s s setofvaluesforbothenglishtexts,decayingfromca. 1.2
smallest integer in N/s. The ordering starts from the
to 1.0 for q increasing but negative; D(q) decays slowly
beginningofthetext(contrarytoanalysesinwhichsome
forqpositive,barelyreachingavalue0.95forq=80(Fig.
forecastingisexpected,andforwhichtheend“pointsare
1). ThevalueofD(q)ismuchgreateralongthenegative
more relevant).
q axis, for AWLesp but is identical to the other two for
Next one calculates the probability
q ≥ 0. In LTS, even though the form of D(q) is that to
Σs M be expected, it has to be stressed that the AWLeng and
P(s,ν)= ΣNsi=Σ1s (Mν−1)s+i (1) AWLesp are very quantitatively similar, but markedly
ν=1 i=1 (ν−1)s+i differfromTLGeng. Thisalreadyindicatesthatonecan
for every ν and s. Thereafter one calculates observe the high creativity of the author through these
two books. Moreover the translation effect on style is
χ(s,q)=ΣNν=s1P(s,ν)q (2) much better seen on FTS than LTS.
The shuffled texts (Fig.2) remarkably have the same
for each s value. A power law behaviour is expected
D(q) values; their range and variations being similar to
χ(s,q)∼sτ(q); (3) those of the real texts. Slight quantitative differences
occur, more markedlyforthe shuffledAWLesp FTS,but
where τ(q) plays the role of the partition function [1].
along a Baeysian reasoning these can be attributed to
The generalized fractal dimension D(q) [1, 2] is defined
the finite size of the sample.
from
By the way,
(cid:12)
τ(q) dτ(q)(cid:12)
D(q)= q−1 (4) C1 = dq (cid:12)(cid:12) (8)
q=1
and the generalized Hurst exponent, h(q), from
isameasureoftheintermittencylyinginthesignaly(n);
1+τ(q) it can be numerically estimated by measuring τq around
h(q)= . (5) q =1. InallcasesthevalueofC isclosetounity. Some
q 1
conjectureontherole/meaningofC isfoundinRef.[45].
1
Let From some financial and political data analysis it seems
dτ(q) that H1 is a measure of the information entropy of the
α= , (6) system. The same can be thought of here.
dq
fromwhichoneobtainsthegeneralizedcriticalexponent,
f(α) curve [44] from B. f(α) plots: Figs. 3-4
The f(α) spectra are shown in Figs. 3-4. They are
f(α)=qα−τ(q). (7)
markedly non symmetric, as was found for DLA [14, 15],
In the present work, we have calculated χ(s,q) for s withveryhighpositiveskewness, i.e. forq ≤0. Interest-
between 2 and 200. The τ(q) values were calculated by ingly, the esperanto text curve behaves differently from
a linear best fit on a log-log plot of χ(s,q) and s, for the english texts, in FTS, though TLGeng is different in
q values ranging from -25 till 25. As one may expect theLTScase;theshuffledtextsf(α)spectrabehavingin
the q and q − 1 values at the denominator in Eq.(4)- averysimilarqualitativeandquantitativeway. However
Eq.(5)wereleadingtonumericalsingularities. Asmooth the shuffling does not fully symmetrize the spectra.
interpolationcanbevisuallymadewithoutdifficulty. For It is interesting to observe that the f(α) curve is very
conciseness we don’t show χ(s,q). sharp: it originates from negative values for α less than
5
1.0; reaches a maximum (=1.0) at 1.0, at the maximum Series D(q) Ds(q) f(α) fs(α)
so called box dimension, and decays rapidly for α posi- AWLeng AWLeng AWLeng AWLeng
FTS (cid:39) (cid:39) (cid:39) (cid:39)
tive;f(α)=0atα=1.2and1.3respectivelyforAWLeng
TLGeng TLGeng TLGeng TLGeng
and TLGeng; the maximum is also reached at (1.0, 1.0) AWLeng AWLeng AWLeng AWLeng
and the spectrum spans the (narrow) interval 0.90: 1.25 LTS (cid:39) (cid:39) (cid:39) (cid:39)
for AWLesp on the f(α) =0 line. It is worth noticing AWLesp TLGeng AWLesp TLGeng
thatthevaluesarereasonableinviewoftheircorrespon-
dence to the fractal dimensions. On the other hand the
TABLEI:Comparingoriginaltextsquasiidenticalbehaviors
sharpness indicates a high lack of uniformity of the texts
through functions D(q) and f(α), and their counterpart for
LROC. shuffled (s) cases, i.e. Ds(q) and fs(α)
Originaltexts Q α− α+ w1 w2 r1 r2
III. DISCUSSION WITH CONCLUSION AWLeng FTS 5.71 0.95 1.19 0.96 0.04 0.97 0.03
AWLesp FTS 4.39 0.94 1.30 0.99 0.01 0.99 0.01
TLGeng FTS 5.71 0.95 1.19 0.99 0.01 0.99 0.01
In summary, one can observe similarities between the
AWLeng LTS 4.65 0.92 1.23 0.89 0.11 0.91 0.09
original andshuffledtexts and their translations; see Ta- AWLesp LTS 4.83 0.92 1.21 0.87 0.13 0.89 0.11
ble 1 for summarizing the similarities seen through D(q) TLGeng LTS 3.94 0.92 1.34 0.96 0.04 0.97 0.03
and f(α). The english texts look more similar with each Shuffledtexts Q α− α+ w1 w2 r1 r2
otherthanwithrespecttotheesperantotranslation. On AWLeng FTS 6.94 0.95 1.13 0.89 0.11 0.90 0.10
the other hand, one physical conclusion arises from the AWLesp FTS 6.57 0.96 1.16 0.97 0.03 0.97 0.03
TLGeng FTS 6.59 0.94 1.13 0.82 0.18 0.84 0.16
above : the existence of a multifractal spectrum found
AWLeng LTS 4.35 0.91 1.25 0.88 0.12 0.90 0.10
for the examined texts indicates a multiplicative process
AWLesp LTS 4.56 0.92 1.24 0.90 0.10 0.92 0.08
intheusualstatisticalsenseforthedistributionofwords TLGeng LTS 4.35 0.91 1.25 0.88 0.12 0.9 0.10
length and frequency in the text considered as a time se-
ries. Thus linguistic signals may be considered indeed
TABLE II: Tsallis Q-parameter, derived from α and α
as the manifestation of a complex system of high di- − +
values,readonFigs. 3-4,fromEq. (11),fortheoriginaltexts
mensionality, different from random signals or systems
and for shuffled or translated corresponding texts, according
of low dimensionality such as the financial and geophys-
to the type of series; the weights and ratios of the binomial
ical (climate) signals. Finally, the f(α) curve represents
cascade approximation (see text) are given
the measurable aspects of the word networks, be they
consideredthroughLTSorFTS.Ourworkconfirmsthat
texts could be seen as networks indeed [23].
tervals the generalized fractal dimension (or rather τ(q))
Before suggesting a physical model describing the
is obtained from
construction of a writing, let us consider implications
from the f(α) spectrum in some detail. The not fully n
(cid:88)
parabolic, to say the least, f(α) curve indicates non uni- wiqri−τ =1. (9)
formity and strong LROC between long words and small i=1
words, - evidently arising from strong short range corre-
The formula is easily generalized for random contraction
lations between these. In some sense this is expected for
ratios and weights. However it simplifies for the case of
classicalwritings. Itisusuallyknownthattheleft(right)
a simple binomial cascade, i.e.
hand side of the f(α) curve corresponds to fluctuations
oftheq ≥0(q ≤0)-correlationfunction. Inotherwords, wqr−τ +wqr−τ =1. (10)
theycorrespondtofluctuationsinsmall(large)worddis- 1 1 2 2
tributions. Therefore the distribution of small and long
Whence the extremal α values read α =
−
wordsshouldbeexaminedinfurtherworkinordertoob-
log(w )/log(r ) and α = log(w )/log(r ), from
2 2 + 1 1
serve these local correlations, e.g. through a detrended
which the weights and ratios can be estimated by
fluctuation analysis.
inversion (table I), thereby suggesting the author’s
Moreover, in order to characterize the writings, texts
somewhat systemic way used in his/her writings.
and/or authors, we propose a very rough approxima-
The physics connection can be obtained if one relates
tion/model, i.e. let us consider (assume !) that the writ-
the f(α) curve extremal points through their physical
ings are made of only two types of words : small and
meanings [48], i.e.
large [46, 47], appearing through some recursive process.
In so doing one can consider the behavior of the atypi-
cal f(α) curve as originating form a binomial cascade of 1 1 1
= − (11)
short and long words, on a support [0,1], with an arbi- 1−Q α α
+ −
trarycontractionratior andaweightw forthewordin
i i
each successive subinterval, as for a multifractal Cantor whereα andα aretheextremesoftherangeofsup-
− +
setconstruction[1]. Foranarbitrarynumbernofsubin- port for the (positive) multifractal spectrum f(α) and Q
6
(instead of the usual q) is used to represent the parame- of these correlations, thus a new measure of an author’s
ter arising in the non extensive description of statistical style. This suggests a (binomial, at first) cascade model
physics [49]; see values in table I. By extension it is a containing parameters characterizing (or reflecting, at
measure of the attractor dimension or the number of so least) the text style, and most likely in fine the author
called degrees of freedom. It is obvious from the table writings. It remains to be seen whether the f(α) curve
that Q varies between 4 and 7, with interesting differ- andthe(tobegeneralized)binomialcascademodel,with
ences between the LTS and FTS cases, LTS’s Q being the weight and ratio parameters hold through in other
systematically smaller, in the original or shuffled texts. cases, and can characterize authors and texts, - and in
Notice that the value of Q is more extreme though with general time series. Moreover the multifractal method
thesameorderofmagnitudeinthecaseoftheesperanto should additionally be able to distinguish a natural lan-
text for both types of series. guagesignalfromacomputercodesignal[43]andhelpin
Finally we re-emphasize the remarkable difference for improving translations by suggesting perfection criteria
theesperantotext(Fig. 3a)withtheenglishtextsinthe and indicators of text qualitative values.
FTS analysis. Linguistics input should be searched at
this level and is left for further discussion. The origin of AcknowledgementsTheauthorswouldliketothank
differences between TLG and AWL needs more work at D.Staufferforasusualfruitfuldiscussions.... Thiswork
the linguistic level. However we have indicated the in- has been supported by European Commission Project
terest of the multifractal scheme in providing a measure CREEN FP6-2003-NEST-Path-012864.
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