COMBINATORICS & SYNCHRONIZATION

IN NATURAL SEMIOTICS

Franco Orsucci ♦*, Alessandro Giuliani †, Charles Webber Jr ‡,

Joseph Zbilut •, Peter Fonagy ♦ and Marianna Mazza *

♦ Department of Psychology, University College London, 1-19 Torrington Place, London WC1E 7HB, UK

* Institute of Psychiatry and Clinical Psychology, Catholic University of Rome and A. Gemelli University Hospital, L.go F. Vito, I-00100 Rome, Italy,

† Environment and Health Dept. Istituto Superiore di Sanità, Viale Regina Elena 299, 00161 Rome, Italy

• Department of Molecular Biophysics & Physiology, Rush Medical College, 1750 W. Harrison St.,Chicago, IL 60612, USA

‡Department of Physiology, Loyola University Chicago, Stritch School of Medicine 2160 South First Avenue, Maywood, IL 60153 USA

Corresponding author:

Prof. Franco F. Orsucci, Sub-Department of Clinical Health Psychology, University College London, 1-19 Torrington Place, London WC1E 7HB, UK Tel +44(0)2076791943 Fax +44(0)2079168502 Email: f.orsucci@ucl.ac.uk

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Abstract. In this study the derivation of an objective metrics to appreciate the

degree of structuring of written and spoken texts is presented. The proposed

metrics is based on the scoring of recurrences inside a text by means of the

application of Recurrence Quantification Analysis (RQA), a non linear technique

widely used in other fields of sciences.

The adopted approach allowed us to create a ranking of different poems strictly

related to their prosodic structure and, more importantly, the possibility to

recognize the same structure across different languages, to define a level of

structuring typical of spoken texts and identifying the progressive synchronization

of a dyadic relation between two speakers in terms of relative complexity of their

speeches. These results suggest the possibility of introducing objective

measurement methods into humanities studies.

Classification codes (MSC): 68R05 Combinatorics, 68T50 Natural language

processing, 91E10 Cognitive psychology, 91E45 Measurement and performance,

94A17 Measures of information, entropy

Keywords: Natural language, recurrence quantification analysis,

synchronization, information, text processing.

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INTRODUCTION

The study of language is differentiated on its complex levels of structuring: from

informational microscopic systems, to morphological mesoscopic patterns, to semantic

and narrative macroscopic frameworks.

Claude E. Shannon posed the foundations of modern studies on the informational

structure of texts and speech (1) His less famous work on the prediction and entropy of

printed English (2) is a source for inspiring new research.

While we could place Shannon’s contributions at the micro-semiotic level, George K.

Zipf (3) was mostly working at a different semiotic level: he described laws of word

frequencies.

More recent studies used different approaches. Schenkel (4), for example, studied

long range correlation in various human writings by mapping them into a simple 1-D

random walk model. This approach allowed to obtain better quality scaling data than the

traditional power spectrum methods. Optimally written computer programs, for instance,

seem to have the highest scaling exponent, i.e., close to that of ideal 1/f noise. Literature,

on the other hand, leads to smaller exponents. Amit (5) studied long-range correlations in

various translations of the Bible by mapping them onto a one dimensional random walk

model. He concluded that an exponent α, which characterizes the algebraic decay law of

these correlations, depends on both language and coding. In general terms, it seems that

translations tend to decrease complexity of texts.

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Maturana & Varela (6, 7), from another perspective, had suggested that structural

coupling is a process which occurs when two structurally plastic systems repeatedly

perturb one another’s structure in a non-destructive way over a period of time. This leads

to the development of structural ‘fit’ between systems. There is an intimate relationship

between this process and the emergence of ‘appropriate’ behaviour from the interplay

between interacting systems, because the structure of a system determines its responses to

perturbing environmental events. Pecora & Carroll (8), Ott, Grebogi & Yorke (9) and

Pyragas (10) opened a new and reliable way to contemporary research on control and

synchronization of complex systems. Comprehensive reviews of this research field have

been offered by Boccaletti et al. (11, 12) and Pikovsky, Rosenblum & Kurths (13). This

perspective is crucial in text analysis as for the study of synchronization of the speech

structuring of two dialoguing persons.

Our approach differentiates from the majority of the above mentioned studies by its

peculiar operative character as opposed to mainly theoretical perspectives: we chose a

method (Recurrence Quantification Analysis, RQA) (14) that does not make any specific

assumption on the mathematical structure of the data, does not rely on assumptions of

stationarity and does not need to consider the studied data set as the output of a

dynamical system. Our basic aim was to develop a statistical tool to allow for quantitative

and reliable measures of the degree of structuring (in the basic sense of recurrence of

motifs) of a flow of symbols . We demonstrate how this can be done with a relatively

simple mathematical formalism. The importance of deriving a measure of the prosodic

structuring of the text (in the sense of repetitions of words and phrases irrespective of

their meaning) could be very important for the attainment of a common ‘style’ of talking

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correspondent to a sort of ‘synchronization’ of the two speaking persons . This

synchronization is a proxy for a non directly verbal interaction (and thus flow of

information) linking the two actors.

MATERIALS AND METHODS

Recurrence plots (RP) were first introduced in physics by Eckmann, Kamphorst &

Ruelle (15). Later, Webber and Zbilut (14) enhanced this technique by defining nonlinear

variables diagnostically useful in the quantitative assessment of RPs, so developing

Recurrence Quantification Analysis (RQA). Since then, RQA has been used in different

fields, ranging from molecular dynamics (16) to physiology (14) and bioinformatics (17).

In order to be processed by RQA, the original series must be transformed into its

Embedding Matrix by the agency of the method of delays transforming the original n

elements column vector correspondent to the symbol series into a p dimensional matrix

having as columns the original Xn series plus its lagged copies Xn+1, Xn+2...Xn+p-1,

the value p being the embedding dimension.

The application of RQA is based on the calculation of the Euclidean distance between all the

pairs of rows of the embedding matrix. If the distance between two generic rows (i.e. windows of

predefined length along the sequence) falls below the radius, we obtain a recurrence.

The concept of recurrence is straightforward: for any ordered series (time or spatial), a

recurrence is a point which repeats itself. In this respect, the statistical literature points out that

recurrences are the most basic of relations shaping a given system, since they are strictly local and

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independent of any mathematical assumption regarding the system itself. Furthermore, it is worth

stressing that recurrences computation requires no transformation of the data and can be used for

both linear and nonlinear systems . The concept of a recurrence can be expressed as follows: given

a reference point, X0, and a ball of radius r, a point X is said to recur (with reference to X0) if : { X :

|| X - X0 || ≤ r }.

In the case of a time series, i.e., of a system occupying in different times different positions

along a trajectory in a suitable state space, the recurrences correspond to the time points where the

system passes nearby to already visited states. In the case of texts , time corresponds to the order in

which the different letters appear the recurrences are patches, with a length equal to the embedding

dimension, sharing their letter profile with other patches along the chain. Depending on the

embedding dimension these patches will correspond to specific words or phrases.

The number and relative positions of recurrences are expressed by recurrence plots (RP),

that are symmetrical N x N arrays in which a point is placed at (i, j) whenever a point Xi on the

trajectory is close to another point Xj. The closeness between Xi and Xj is expressed by calculating

the Euclidian distance between these two normed vectors, i.e., by subtracting one from the other

obtaining the expression || Xi - Xj || ≤ r , where r is a fixed radius. If the distance falls within this

radius, the two vectors are considered to be recurrent, and graphically this can be indicated by a dot

.In the case of alphabetical strings, the radius is constrained to be zero and the similarity collapses

to the identity of motifs along the text.

Thus, recurrence plots correspond to the distance matrix between the different epochs (rows

of the embedding matrix) filtered, by the action of the radius, to a binary 0/1 matrix having a 1 (dot)

for distances falling below the radius and a 0 for distances greater than radius.

An important feature of such matrixes is the existence of short line segments parallel to the

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main diagonal, which correspond to sequences (i, j), (i + 1, j + 1), ..., (i + k, j + k) such that the

fragment X(j), X(j + 1), X(j + k) is close to X(i), X(i + 1), ..., X(i + k). The absence of such patterns

suggests randomness . For texts these deterministic lines correspond to contiguous patches of

letters repeating themselves in different portions of the text. They can represent identical words or

phrases appearing in different moments of a speech. . The “line” is a measurement parameter which

states the minimum number of adjacent recurrent points required to define a deterministic line. We

set our line to 2, that corresponds to say that we need two contiguous recurrent patches of 3 to get a

deterministic line.

The quantification of recurrences is obtained by many different ‘counts’ of recurrences

disposition on the matrix, here we will limit ourselves to the two most basic descriptors:

- Recurrence (REC) : percentage of recurrence points in the recurrence plot.

- Determinism (DET) : percentage of recurrent points which form diagonal lines.

Basic RQA refers to the autocorrelation structure of a given series, Zbilut et al. (18) demonstrated

the applicability of the method to the quantification of the ‘cross-correlation’ structure of different

series, this possibility was further exploited and refined by Marwan et al. (19) that developed the

concept of Cross-Recurrence Quantification Analysis (CRQA).

The basic formalism is identical, but in this case the X and Y axis of the recurrence plot are no

more coincident but refer to the two different signals to be compared. This obviously eliminates the

recurrent main diagonal and shift the meaning of the dots to ‘repetitions’ inside the same text to

concordances scored between different texts.

Our basic material was constituted by series of letters corresponding to the analysed texts coming

by the following data sets:

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1. Italian poems of the 20th century : 9 Italian poems of the last century were chosen

with the aim of maximizing the range of variation of the degree of prosodic

structuring. The texts include complex poems with a minimum level of redundancy,

as well as very structured poems using long repetitive motifs.

2. American (English) poems of the 20th century : this set is composed by 11 American

poems chosen with the same criteria adopted for set 1.

3. 12 Swedish contemporary poems and their correspondent Italian translations . It is

important to note that the translation was a poetical one, i.e. the translator had the

goal to reproduce the prosodic structure of the original poem, the correlation between

the RQA description of the poem with its translation is the test of the language

invariance of the method and of its sensitivity to the prosodic structure of the text.

4. Italian speech samples : transcriptions of the speech samples of 7 people with

different cultural backgrounds .

5. American (English) speech samples : transcriptions of the speech samples of 10

university students. The recorded speech samples correspond to periods of time in

which the individuals are fluently speaking with no external constraints.

6. A natural dialogue between two friends: they are talking about some general topic as

“the meaning of life”. The subsequent speech samples correspondent to the two

friends ‘talk portions’ are each other correlated.

7. A clinical dialogue between a patient and a psychiatrist. They didn’t know each other

before, because this is a first interview. As in the previous case, here the dynamics of

a progressive synchronization between the two actors (in terms of similarities in their

recurrence structure) is investigated.

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These different data sets were used for exploiting different possibilities of the proposed

methodology with the general aim of proposing a quantitatively sound approach for

humanities and psychiatric studies (20).

RESULTS AND DISCUSSION

In our application texts were considered as continuous streams of letters with no

interruptions.. The choice of a 3-dimensional embedding maximizes the sensitivity of the

method by increasing the number of scored recurrences without disturbance from low-

level statistical features of language (asymmetrical distribution of couples of letters

typical of each language).

Another basic requirement is to choose the measurement parameters in a way to

have a relatively low recurrence so to minimize the risk of scoring a deterministic line for

the pure juxtaposition of two randomly occurring contiguous recurrences.

A recurrence is scored whenever paired 3-letter patterns are detected along the

sequence. The departure of the analysed texts from random sequences was evaluated by

comparing real sequences with their randomly shuffled counterparts. The shuffling

procedure keeps invariant the statistical features of the studied sequence so as to evaluate

the amount of pure dynamical (i.e. phasic) information present in the text at the

orthographic level.

Fig. 1 shows a combined analysis of American poems (AMP), Italian poems (ITP),

American speech transcriptions (AMS), and Italian speech transcriptions (ITS). It is

worth noting the common scaling of texts along a linear relationship between REC and

DET ( r = 0.87, p < 0.001). This correlation refers to all the text considered as a whole

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(49 statistical units), the position on the REC-DET plane is then a simple numerical

index of the relative complexity of a text despite its relative length, in fact poems are

much shorter than speeches (100-300 letters for poems, 3000-7000 letters for speeches).

Moreover the linear relation spanning almost the entire range of DET (from around zero

to 68 for a theoretical maximum of 100) tells us that it does not exist a ‘privileged scale’

of text periodicity.

Fig. 1: Plot of structure distribution of all described linguistic samples: a combined analysis of American

(AMP) and Italian poems (ITP); American (AMS) and Italian speech transcriptions (ITS). It is worth noting

DET corresponds to the percentage of recurrent points appearing in lines thus, in principle, we can reach a

maximum DET even with an extremely low number of recurrences if all the recurrent points are

contiguous.

As a matter of fact , when the same procedure was applied to a protein data set (21)

a privileged (maximum determinism) length of 6 consecutive residues along the chain

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came out , corresponding to the average length of proteins folding core. In the case of

texts, the contemporary presence of repetitions at different scales (words, phrases)

prevents the identification of such a scale.

For both Italian and English poems (sets 1 and 2) the more repetitive poems (e.g.

Dr. Suess Green Egg and Ham, Aldo Palazzeschi La Fontana Malata) with evident

rhymes and recurrences occupy the most deterministic and recurrent corner of the graph.

The majority of poems are in the central area of the graph, corresponding to natural

speeches. This is remarkable if we think to the very short length of poems with respect to

speeches, in some sense poems ‘condensate’ in few words the same ‘prosodic meaning’

of relatively long speeches.

Almost all the poems, when shuffled, gave very similar results in terms of REC and

DET with respect to the native series (data not shown), and the same was true for the

natural speeches, nevertheless we know both the speeches and poems convey a specific

meaning. This is a striking concordance between texts and proteins (21): both convey a

specific meaning (that for the proteins can be identified with their 3D structure giving

rise to the peculiar physiological role) by means of quasi-random string of symbols (in

the case of protein the amino-acid sequence). The point is now to check if this ‘global

randomness’ of letters juxtaposition conveys some relevant information (23).

The analysis of set 3 (Swedish poems and their Italian translations) allowed us to answer

this question. The analysis of this data set permitted to investigate the issue of the

recognition of a given prosodic structure across two different languages and thus its

information relevance despite the apparent randomness of the internal structuring of

texts. A demonstration of the possibility to recognize the same structure in terms of RQA

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variables when translated into another language, is both a stringent test of the language

invariance of this technique, as well as of the consistency of dynamical descriptors. The

obvious linguistic differences between Swedish and Italian languages makes the test

more cogent. Swedish texts were transliterated using the Italian 21 letter standard

alphabet (ch for k, and so on) maintaining the same phonetic character of the original.

The determinism of the Swedish poem is strictly related with the determinism of the

correspondent Italian translation (r = 0.91, p < 0.01).

Fig.2: Points in figure correspond to different poems. Determinism of the Swedish version is indicated by

det(swe), determinism of the corresponding Italian translation as det(it).

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This means that, despite the lack of statistically meaningful periodicities (shuffling

resistance) , the peculiar prosodic structure of letter juxtaposition of the poems maintains

its individuality in terms of RQA descriptors allowing for the mutual recognition of

Swedish-Italian translations.

It is worth noting RQA was able to discover the prosodic linkage between texts without

being affected by the obvious morphological differences between Italian and Swedish

languages. This means that different relations between spoken language (phonetics) and

written language (orthography) does not affect in a meaningful way the robustness of this

methodology.

Notwithstanding the ability of the method to recognize the same prosodic structure even

when expressed by a different coding (language), not all the languages seem to have the

same average complexity, the comparison between Italian and American speech samples

points to a greater complexity (lesser degree of structuring) of American speeches.

[American Mean (REC) = 0.416, SD = 0.03, Mean (DET) = 20.00, SD = 1.36; Italian

Mean (REC) = 0.676, SD= 0.06; Mean (DET) = 27.94, SD = 2.66 ] (t-test corrected for

variance non-homogeneity: p < 0.0001 for both REC and DET). American speakers

displayed less variance than the Italian ones, demonstrating a statistical significance for

REC (F-test for homogeneity of variances: F = 4.69, p < 0.04), pointing to the possibility

of using this methodology to discriminate sub-sets of speakers.

This could be a very interesting point for the use of speech transcriptions

complexity as diagnostic tool in psychopathology. We applied this method to the

comparison of control and patient (suffering of a range of psychopathological syndromes

from schizophrenia to depression) groups. Results are still too preliminary, due to the

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confounding of other elements as culture, registration conditions, length of speech, but

they show some interesting perspective for further investigation.

As final stage of our explorations, we used RQA and CRQA (Cross Recurrence) to

measure coupling and synchronization of two subjects during a conversation (semiotic

interaction). The existence of such a synchronization is related to a level of

communication coexistent but not coincident with the pure verbal information exchange

mediated by the meaning of the words. This pre-verbal (and largely unconscious)

communication channel could be highlighted by the common appearance of

progressively similar prosodic structures (in terms of stereotyped phrases, words, pauses)

during an ongoing dialogue between two agents.

We measured a series of text samples derived from transcriptions of a natural,

spontaneous, conversation (NAT) and a clinical conversation (CLIN) held in an

organized and stable setting. While the first one was supposed to evolve following its

inner co-evolutionary dynamics; the second one is supposed to be finalized towards

controlled and partially pre-defined dynamics by one of the agents (therapist).

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NAT

45505560657075808590

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Turn

Det

CLIN

455055606570758085

1 2 3 4 5 6 7 8 9 10 11

Turn

Det

Fig. 3-A and 3-B: both represent a semiotic interaction each dot being a turn in conversation. In 4-A red

and blue dots are for friend J or K; in 3-B red dots are for patient P and blue dots for therapist T. The

abscissa report the subsequent order of the dialogue (turn), the ordinate the amount of determinism of the

speeches.

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Figure 3-A reports a natural dialogue between two friends: they are talking about

some general topic as “the meaning of life”. While in the beginning of conversation they

seem quite well synchronized, as they are going to approach the end of conversation and

separation they tend to decouple.

Fig. 3-B is a clinical dialogue between a patient and a psychiatrist. They didn’t

know each other before, because this is a first interview. They start in synchrony

(probably linked to initial short stereotypical introduction phrases), they soon loose their

initial coupling , to find slowly a new synchronization level including some crossing.

Finally, to highlight global differences in structure of each dynamic interaction we

used CRQA which compares the dynamical behaviour of two interacting time series,

simultaneously embedded in the same phase space (19). Figures 4-A and 4-B are CRQA

plots for NAT and CLIN dialogues respectively.

Fig. 4-A Cross Recurrence Quantification Analysis (CRQA) of semiotic dynamics in natural conversation ;

Fig. 4-B, CRQA of semiotic dynamics in clinical conversation (sample 2).

The visual impression of a greater synchronization for natural conversation with respect

to the clinical one is confirmed by the CRQA quantitative values: cross-recurrence was

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0.14 for NAT and 0.12 for CLIN, whiled cross-determinism scored 41.15 for NAT and

36.23 for CLIN. These data globally confirm the RQA analysis reported in Fig.3, where

the synchronization between the two agents was evaluated in terms of global invariants of

the speeches (DET) without any explicit reference to the actual disposition of recurrences

in time (phasic correlations). This situation can be equated by a correlation of two

spectral content (Fig.3, RQA) and the direct cross-correlation of two series (Fig 4,

CRQA), the general resemblance of the two analyses is a proof of the reliability of the

method. This research can go along the same way of the study by Shockley et al (24), in

which interpersonal coordination during conversation was based on recurrence strategies,

to evaluate the shared activity between two postural time series in a reconstructed phase

space.

In a study on speech and rhythmic behavior Port et al. (25) found that animals and

humans exhibit many kinds of behavior where frequencies of gestures are related by

small integer ratios (like 1:1, 2:1 or 3:1). Many properties like these are found in speech,

as an embodied activity considered as an oscillator prone to possible synchronizations.

All these findings are related to interpersonal and intrapersonal embodied

dynamics. Measures of the influence of these variables could be of great interest for

example in psychometrics, which is now largely based on inferential and non-direct

measures. A differentiation of cultural, personality and psychopathology traits based on

direct quantitative measures of semiotic samples could be of practical, theoretical and

clinical value.

Our findings in the synchronization of conversation dynamics can be relevant for

the general issue of structural coupling of psychobiological organizations. Implications

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are related with psycho-chrono-biology research and the clinical field. Data on

synchronization suggest that this dynamic behaviour can be evident also in semiotic and

cognitive dynamics, besides the well established research on biological oscillators.

CONCLUSIONS

We demonstrated the possibility of a pure operational use of concepts and techniques derived by

complex systems dynamics for developing quantitative and reliable observables in neuro-cognitive

science. The freedom of the proposed techniques (RQA, CRQA) from any kind of consideration of

the nature of the analysed data together with the lack of theoretical assumptions about the systems

at study greatly enlarge the applicability domain of the proposed approach.

In the above case studies we limited ourselves to letter repetitions, but the proposed method can be

applied as well at any user defined atomic level: as a matter of fact we can study the recurrence

structures of words, phrases or even more ‘erratic’ entities like concepts and ideas along a written

text or a speech.

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ACKNOWLEDGEMENTS

The beginning of this line of research, some years ago, has been encouraged with

suggestions and comments by Giorgio Parisi, Luciano Pietronero and Angelo Vulpiani.

We are grateful for contributions from Marco Lucciola, Kimberly Walters and Stefano

Boccaletti.

Some parts of this research have been presented at the 8th Experimental Chaos

Conference (Florence, June 2004), and at the Tandem Workshop sponsored by the Max

Planck Institute for Human Cognitive and Brain Sciences, University of Potsdam (Berlin,

March 2005).

Questions and comments received in these occasions were very useful: we wish to thank

Tito Arecchi, Douglas Saddy, Juergen Kurths, Peter beim Graben, Mamen Romano,

Norbert Marwan and Marco Thiel.

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