Recurrence plot

We studied the response of heart rate variability to hemodialysis and orthostatism using traditional linear indexes and 9 recurrence quantification analysis indexes to reveal changes in the heart rate dynamics. Twenty healthy subjects and 19 chronic renal failure patients treated with hemodialysis thrice a week were included. Five-minute heart rate variability time series were obtained during supine position (clinostatism) and orthostatism from each participant; recordings in renal patients were repeated after hemodialysis. Linear indexes were consistent with sympathetic predominance in response to orthostatism in the control group. Renal patients before hemodialysis showed increased sympathetic predominance in clinostatism, with further increase in orthostatism and hemodialysis. In response to orthostatism, 4 recurrence indexes changed in the control group, while in renal patients any of them changed before hemodialysis and 1 changed after hemodialysis. In clinostatism, renal patients (both before and after hemodialysis) had higher laminarity, trapping time, and recurrence time than the control group. Recurrence indexes showed that the heart rate dynamics in renal patients are different from healthy subjects, suggesting loss of access to some regulatory conditions. These findings are consistent with reports of sympathetic stimulation induced by hemodialysis and active standing.

1. Introduction to observational methods, 2. Coding schemes and observational measurement, 3. Recording observational data, 4. Representing observational data, 5. Observer agreement and Cohen's kappa, 6. Kappas for point-by-point agreement, 7. The intraclass ...

The EEG (Electroencephalogram) signal indicates the electrical activity of the brain. They are highly random in nature and may contain useful information about the brain state. However, it is very difficult to get useful information from these signals directly in the time domain just by observing them. They are basically non-linear and nonstationary in nature. Hence, important features can be

El analisis de series temporales no lineales cuenta con un nuevo enfoque metodologico: el analisis de recurrencias. Sin embargo, este esquema de analisis precisa de nuevos programas informaticos que posibiliten su correcta implementacion. Este trabajo revisa los principales programas accesibles para realizar dicha tarea, y se centra en la presentacion detallada del VRA, un software de libre acceso util para el analisis general de series temporales no lineales.

In the process of rock cutting, the ripping head load measured as the supplied power is rapidly changing in a complex way difficult to describe mathematically. Among factors, which influence the ripping process, one should mention technologies and conditions of excavating, ripping head technical parameters, characteristics of the rock, and construction features of the tool. The scheme and specification of our test stand are presented in .

In this paper we study the fluctuations of mean indicated pressure (MIP) in a diesel engine. Using recurrence plots, recurrence quantification analysis and continuous wavelet transform, we investigate the cycle-to-cycle pressure variations for six rotational speeds of the crankshaft. We find that depending on the speed of rotation, the pressure variations may have a strongly periodic component and/or be intermittent. The strong periodicities appear in low-frequency bands and may persist over many cycles, whereas the intermittency is present at higher frequencies. The results may be useful to develop effective control strategies for efficient engine performance. r

Recurrence plot based methods are highly efficient and widely accepted tools for the investigation of time series or one-dimensional data. We present an extension of the recurrence plots and their quantifications in order to study recurrent structures in higher-dimensional spatial data. The capability of this extension is illustrated on prototypical 2D models. Next, the tested and proved approach is applied to assess the bone structure from CT images of human proximal tibia. We find that the spatial structures in trabecular bone become more recurrent during the bone loss in osteoporosis.

Recurrence plots were introduced to quantify the recurrence properties of chaotic dynamics. Hereafter, the recurrence quantification analysis was introduced to transform graphical interpretations into statistical analysis. In this spirit, a new definition for the Shannon entropy was recently introduced in order to have a measure correlated with the largest Lyapunov exponent. Recurrence plots and this Shannon entropy are thus used for the analysis of the dynamics underlying patient assisted with a mechanical noninvasive ventilation. The quality of the assistance strongly depends on the quality of the interactions between the patient and his ventilator which are crucial for tolerance and acceptability. Recurrence plots provide a global view of these interactions and the Shannon entropy is shown to be a measure of the rate of asynchronisms as well as the breathing rhythm.

Recurrence plots were introduced to help aid the detection of signals in complicated data series. This effort was furthered by the quantification of recurrence plot elements. We now demonstrate the utility of combining recurrence quantification analysis with principal components analysis to allow for a probabilistic evaluation for the presence of deterministic signals in relatively short data lengths. PACS: 05.40 05.45, 07.05Rm, 07.05Kf

Complex neurodynamical systems are quite difficult to analyze and understand. New type of plots are introduced to help in visualization of high-dimensional trajectories and show global picture of the phase space, including relations between basins of attractors. Color recurrence plots (RPs) display distances from each point on the trajectory to all other points in a two-dimensional matrix. Fuzzy Symbolic Dynamics (FSD) plots enhance this information mapping the whole trajectory to two or three dimensions. Each coordinate is defined by the value of a fuzzy localized membership function, optimized to visualize interesting features of the dynamics, showing to which degree a point on the trajectory belongs to some neighborhood. The variance of the trajectory within the attraction basin plotted against the variance of the synaptic noise provides information about sizes and shapes of these basins. Plots that use color to show the distance between each trajectory point and a larger number of selected reference points (for example centers of attractor basins) are also introduced. Activity of 140 neurons in the semantic layer of dyslexia model implemented in the Emergent neural simulator is analyzed in details showing different aspects of neurodynamics that may be understood in this way. Influence of connectivity and various neural properties on network dynamics is illustrated using visualization techniques. A number of interesting conclusions about cognitive neurodynamics of lexical concept activations are drawn. Changing neural accommodation parameters has very strong influence on the dwell time of the trajectories. This may be linked to attention deficits disorders observed in autism in case of strong enslavement, and to ADHD-like behavior in case of weak enslavement.

We propose a novel approach for analysing time series using complex network theory. We identify the recurrence matrix (calculated from time series) with the adjacency matrix of a complex network and apply measures for the characterisation of complex networks to this recurrence matrix. By using the logistic map, we illustrate the potential of these complex network measures for the detection of dynamical transitions. Finally, we apply the proposed approach to a marine palaeo-climate record and identify the subtle changes to the climate regime.

We have investigated plasma turbulence at the edge of a tokamak plasma using data from electrostatic potential fluctuations measured in the Brazilian tokamak TCABR. Recurrence quantification analysis has been used to provide diagnostics of the deterministic content of the series. We have focused our analysis on the radial dependence of potential fluctuations and their characterization by recurrence-based diagnostics. Our main result is that the deterministic content of the experimental signals is most pronounced at the external part of the plasma column just before the plasma radius. Since the chaoticity of the signals follows the same trend, we have concluded that the electrostatic plasma turbulence at the tokamak plasma edge can be partially explained by means of a deterministic nonlinear system.

Electroencephalogram (EEG)-based emotion recognition has been a rapidly growing field. However, accurate and sufficient performance rates are yet to be obtained. This paper presents the classification of EEG correlates on emotion using the relatively new non-linear feature extraction method, namely, Recurrence Plot analysis to extract thirteen non-linear features. This method is compared with feature extraction method based on spectral power analysis. The K nearest neighbor is applied to classify extracted features into the emotional states based on arousal-valence (high/low arousal, valence) plane with the addition of liking axis (positive/negative). Leading to performance rates of 58.05%, 64.56% and 67.42% for 3 classes of valence, arousal and liking; which confirm the advantage of a non-linear feature extraction method over previous frequency based feature extraction techniques.

[Some time after writing this article, I started to prefer and use the term 'siege narratives' instead of 'capture stories' (here, still in the title).]

The conclusion (p. 223) of this contribution reads:

We could end by suggesting that historiography as well as epic poetry can function along the established lines of (mono)mythic formulas.
Strikingly enough it seems that the more such lines – like Campbell’s – can be retraced, the further-reaching the consequences are on the ideological plane. This observation fits in well with the tenor of the
criticism (as Manganaro’s, cf. note 1) which Campbell’s work during the last decades had to take. Indeed, the monomyth is authority-confirming and, due to its roots in aristocratic patrimony and culture, usually supports the reigning ideology.

Campbell might have seemed rather blind to this. People, whether they are oppressed or not, will always like heroes (whether they are historical or not) whose deeds remind them of the mythic heroes of olden days. The fact that the components of these deeds are likely to echo those of the monomyth probably makes it easier to empathise with the
heroes, as well as with their sacrifices. We demonstrated how the latter, depending on focalisation and plot, could be situated either on a psychological (Rossiad) or on a more literal, bloodier (Tale of Constantinople) plane. Whatever the sacrifice may be, it should
stimulate the reader’s empathy so he can experience a kind of Aristotelian catharsis, and with respect to this it does not matter at all whether the genre in which the sacrificing leader is involved is (semi)historiographic or epical, or even (melo)dramatic or lyrical.

Diabetes is a chronic disease characterized by hyperglycaemia, which leads to specific long-term complications: retinopathy, neuropathy, nephropathy and cardiomyopathy. Analysis of cardiac health using heart rate variation (HRV) has become a popular method to assess the activities of the autonomic nervous system (ANS). It is beneficial in the assessment of cardiac abnormalities, because of its ability to capture fast fluctuations that may be an indication of sympathetic and vagal activity. This paper documents work on the analysis of both normal and diabetic heart rate signals using time domain, frequency domain and nonlinear techniques. The study is based on data from 15 patients with diabetes and 15 healthy volunteers. Our results show that non-linear analysis of HRV is superior compared to time and frequency methods. Non-linear parameters namely, correlation dimension (CD), approximate entropy (ApEn), sample entropy (SampEn) and recurrence plot properties (REC and DET), are clinically significant.

In this paper, we elaborate over the well-known interpretability issue in echo state networks. The idea is to investigate the dynamics of reservoir neurons with time-series analysis techniques taken from research on complex systems. Notably, we analyze time-series of neuron activations with Recurrence Plots (RPs) and Recurrence Quantification Analysis (RQA), which permit to visualize and characterize high-dimensional dynamical systems. We show that this approach is useful in a number of ways. First, the two-dimensional representation offered by RPs provides a way for visualizing the high-dimensional dynamics of a reservoir. Our results suggest that, if the network is stable, reservoir and input denote similar line patterns in the respective RPs. Conversely, the more unstable the ESN, the more the RP of the reservoir presents instability patterns. As a second result, we show that the Lmax measure is highly correlated with the well-established maximal local Lyapunov exponent. This suggests that complexity measures based on RP diagonal lines distribution provide a valuable tool to quantify the degree of network stability. Finally, our analysis shows that all RQA measures fluctuate on the proximity of the so-called edge of stability, where an ESN typically achieves maximum computational capability. We verify that the determination of the edge of stability provided by such RQA measures is more accurate than the one offered by the maximal local Lyapunov exponent. Therefore, we claim that RPs and RQA-based analyses can be used as valuable tools to design an effective network given a specific problem.

Identification of financial bubbles and crisis is a topic of major concern since it is important to prevent collapses that can severely impact nations and economies. Our analysis deals with the use of the recently proposed 'delay vector variance' (DVV) method, which examines local predictability of a signal in the phase space to detect the presence of determinism and nonlinearity in a time series. Optimal embedding parameters used in the DVV analysis are obtained via a differential entropy based method using wavelet-based surrogates. We exploit the concept of recurrence plots to study the stock market to locate hidden patterns, non-stationarity, and to examine the nature of these plots in events of financial crisis. In particular, the recurrence plots are employed to detect and characterize financial cycles. A comprehensive analysis of the feasibility of this approach is provided. We show that our methodology is useful in the diagnosis and detection of financial bubbles, which have significantly impacted economic upheavals in the past few decades.

In descriptive statistics and chaos theory, a recurrence plot is a plot showing, for each moment i in time, the times at which a phase space trajectory visits roughly the same area in the phase space as at time j. The purpose of this paper is to give an introduction to chaos theory and how it is applied in recurrence plots.

Metal cutting is complex nonlinear dynamical process. Cutting signals from turning operation exhibit a low dimensional chaos. When the depth of cut increases, with all other cutting parameters remaining constant, creation of chatter occurs. A recurrence plot based methodology is used to find the point of transition from normal cutting to chatter cutting. In this method two signals, one input signal (current) and one output signal (tool vibration) are recorded simultaneously at a constant sampling rate. A time series is generated from the recorded values and recurrence plot is prepared for the input signal and the output signal. This recurrence plot can be quantified using Recurrence Quantification analysis (RQA). Variations in the roughness of machined surface created by virtue of chatter, manifests as changes in the statistical properties of speckle images of the surface when examined frame by frame along the axis of the machined part. A significant parameter of such images, the frame wise average intensity value is extracted separately and arranged in sequence for constructing the time series. Since this time series is found to be non-stationary in nature, the nonlinear time series analysis methodology of RQA is used for analyzing the time series. The result obtained is compared to that of recurrence plot methodology. The experiment is repeated with different work piece materials. The present study ascertains that this methodology is capable of recognizing the transition from regular cutting to the chatter cutting irrespective of the work piece material and the type of signal.

The identification of complex periodic windows in the two-dimensional parameter space of certain dynamical systems has recently attracted considerable interest. While for discrete systems, a discrimination between periodic and chaotic windows can be easily made based on the maximum Lyapunov exponent of the system, this remains a challenging task for continuous systems, especially if only short time series are available (e.g., in case of experimental data). In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis (RQA) as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Using the well-studied Rössler system as a benchmark example, we find that the average path length and the clustering coefficient of the resulting recurrence networks (RNs) are particularly powerful discriminatory statistics for the identification of complex periodic windows. PACS numbers: 05.45.Tp, 89.75.Hc, 05.45.Ac

The knowledge of transitions between regular, laminar or chaotic behavior is essential to understand the underlying mechanisms behind complex systems. While several linear approaches are often insufficient to describe such processes, there are several nonlinear methods which however require rather long time observations. To overcome these difficulties, we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart rate variability data. For the logistic map these measures enable us not only to detect transitions between chaotic and periodic states, but also to identify laminar states, i. e. chaos-chaos transitions. The traditional recurrence quantification analysis fails to detect the latter transitions. Applying our new measures to the heart rate variability data, we are able to detect and quantify the laminar phases before a life-threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event.

A detailed nonlinear time series analysis of the hourly data of the geomagnetic horizontal intensity H measured at Kodaikanal (10.2° N; 77.5° E; mag: dip 3.5° N) has been carried out to investigate the dynamical behaviour of the fluctuations of H. The recurrence plots, spatiotemporal entropy and the result of the surrogate data test show the deterministic nature of the fluctuations, rejecting the hypothesis that H belong to the family of linear stochastic signals. The low dimensional character of the dynamics is evident from the estimated value of the correlation dimension and the fraction of false neighbours calculated for various embedding dimensions. The exponential decay of the power spectrum and the positive Lyapunov exponent indicate chaotic behaviour of the underlying dynamics of H. This is also supported by the results of the comparison of the chaotic characteristics of the time series of H with the pseudo-chaotic characteristics of coloured noise time series. We have also shown that the error involved in the short-term prediction of successive values of H, using a simple but robust, zero-order nonlinear prediction method, increases exponentially. It has also been suggested that there exists the possibility of characterizing the geomagnetic fluctuations in terms of the invariants in chaos theory, such as Lyapunov exponents and correlation dimension. The results of the analysis could also have implications in the development of a suitable model for the daily fluctuations of geomagnetic horizontal intensity.

This study investigates the presence of deterministic dependencies in international stock markets using recurrence plots and recurrence quantification analysis (RQA). The results are based on a large set of free float-adjusted market capitalization stock indices, covering a period of 15 years. The statistical tests suggest that the dynamics of stock prices in emerging markets is characterized by higher values of RQA measures when compared to their developed counterparts. The behavior of stock markets during critical financial events, such as the burst of the technology bubble, the Asian currency crisis, and the recent subprime mortgage crisis, is analyzed by performing RQA in sliding windows. It is shown that during these events stock markets exhibit a distinctive behavior that is characterized by temporary decreases in the fraction of recurrence points contained in diagonal and vertical structures.

A methodology based upon recurrence quantification analysis is proposed for the study of orthographic structure of written texts. Five different orthographic data sets (20th century Italian poems, 20th century American poems, contemporary Swedish poems with their corresponding Italian translations, Italian speech samples, and American speech samples) were subjected to recurrence quantification analysis, a procedure which has been found to be diagnostically useful in the quantitative assessment of ordered series in fields such as physics, molecular dynamics, physiology, and general signal processing. Recurrence quantification was developed from recurrence plots as applied to the analysis of nonlinear, complex systems in the physical sciences, and is based on the computation of a distance matrix of the elements of an ordered series (in this case the letters consituting selected speech and poetic texts). From a strictly mathematical view, the results show the possibility of demonstrating invariance between different language exemplars despite the apparent low-level of coding (orthography). Comparison with the actual texts confirms the ability of the method to reveal recurrent structures, and their complexity. Using poems as a reference standard for judging speech complexity, the technique exhibits language independence, order dependence and freedom from pure statistical characteristics of studied sequences, as well as consistency with easily identifiable texts. Such studies may provide phenomenological markers of hidden structure as coded by the purely orthographic level.

ObjectiÕe: Testing the hypothesis that alterations in non-linear dynamics of the cardiovascular variability appear in healthy, awake subjects during voluntary apnea. Subjects and Methods: Ten young subjects performed 20 apneas 60 s each separated by 1 min free Ž . Ž . Ž . breathing. Inter-beat interval IBI measured as RR-interval in ECG, systolic SYS and diastolic DIAS arterial blood pressure, stroke Ž . Ž . Ž . volume SV , cardiac output CO and total peripheral resistance TPR were non-invasively recorded and computed by Portapress 2 Ž . system. Correlation dimension according to Grassberger-Proccacia algorithm CD and ratio of % determinism versus % recurrence Ž . w Ž .

A detailed nonlinear time series analysis of the hourly data of the geomagnetic horizontal intensity H measured at Kodaikanal (10.2 • N; 77.5 • E; mag: dip 3.5 • N) has been carried out to investigate the dynamical behaviour of the fluctuations of H. The recurrence plots, spatiotemporal entropy and the result of the surrogate data test show the deterministic nature of the fluctuations, rejecting the hypothesis that H belong to the family of linear stochastic signals. The low dimensional character of the dynamics is evident from the estimated value of the correlation dimension and the fraction of false neighbours calculated for various embedding dimensions. The exponential decay of the power spectrum and the positive Lyapunov exponent indicate chaotic behaviour of the underlying dynamics of H. This is also supported by the results of the comparison of the chaotic characteristics of the time series of H with the pseudo-chaotic characteristics of coloured noise time series. We have also shown that the error involved in the short-term prediction of successive values of H , using a simple but robust, zero-order nonlinear prediction method, increases exponentially. It has also been suggested that there exists the possibility of characterizing the geomagnetic fluctuations in terms of the invariants in chaos theory, such as Lyapunov exponents and correlation dimension. The results of the analysis could also have implications in the development of a suitable model for the daily fluctuations of geomagnetic horizontal intensity.

We have examined a cutting process with dry friction and contact loss effects. To model the second pass of the tool through the workpiece, we used a harmonic function representing the surface corrugations. The mathematical model consists of a single differential equation of second order. By numerically solving the governing differential equation, we obtained the time series for the cutting depth for different excitation frequencies. The time series are analyzed using the methods of recurrence plots, recurrence quantification analysis and wavelet analysis. The results show an intermittent character of the system dynamics with transition to chaotic motion.

The analysis of combustion non-repeatability in the dynamical working conditions of a Diesel engine is presented. In the frame of statistical analysis, applied to a number of acceleration trails, basic determinants such as standard square deviation, skewness, and kurtosis have been calculated for the mean indicated pressure (MIP). The methods of return maps, delay phase portrait reconstruction, recurrence plots and quantification recurrence analysis have been also used to analyze a single chosen course of the measured pressure starting from the idle run of engine, through its acceleration process and ending on a stable engine run with a certain velocity. The results show that recurrence techniques are able to identify various phases of the engine non-stationary work.

We report our results on non-periodic experimental time series of pressure in a spark ignition engine. The experiments were performed for a low rotational velocity of a crankshaft and a relatively large spark advance angle. We show that the combustion process has many chaotic features. Surprisingly, the reconstructed attractor has a characteristic butterfly shape similar to a chaotic attractor of Lorentz type. The suitable recurrence plot shows that the dynamics of the combustion is a nonlinear multidimensional process mediated by stochastic noise.

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

We have undertaken a thorough dynamical investigation of five extrasolar planetary systems using extensive numerical experiments. The systems Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208 were examined concerning the question of whether they could host terrestrial like planets in their habitable zones (=HZ). First we investigated the mean motion resonances between fictitious terrestrial planets and the existing gas giants in these five extrasolar systems. Then a fine grid of initial conditions for a potential terrestrial planet within the HZ was chosen for each system, from which the stability of orbits was then assessed by direct integrations over a time interval of 1 million years. The computations were carried out using a Lie-series integration method with an adaptive step size control. This integration method achieves machine precision accuracy in a highly efficient and robust way, requiring no special adjustments when the orbits have large eccentricities. The stability of orbits was examined with a determination of the Renyi entropy, estimated from recurrence plots, and with a more straight forward method based on the maximum eccentricity achieved by the planet over the 1 million year integration. Additionally, the eccentricity is an indication of the habitability of a terrestrial planet in the HZ; any value of e>0.2 produces a significant temperature difference on a planet's surface between apoapse and periapse. The results for possible stable orbits for terrestrial planets in habitable zones for the five systems are summarized as follows: for Gl 777 A nearly the entire HZ is stable, for 47 Uma, HD 72659 and HD 4208 terrestrial planets can survive for a sufficiently long time, while for Gl 614 our results exclude terrestrial planets moving in stable orbits within the HZ.

During the last years, the concept of recurrence plots has received considerable interest as a tool for analysing nonlinear and non-stationary time series. However, in the case of discrete-valued observables or variations on very different time scales, problems may occur in direct interpretations of the results of recurrence quantification analysis (RQA). As a potential solution, we suggest combining this approach with ideas from symbolic time series analysis, which allows an arbitrary static or dynamic coarse-graining of the dynamics that goes beyond recent recurrence plot based methods. As an illustrative application, we discuss how the resulting symbolic recurrence plots may be used for a quantitative investigation of the dynamics of discrete-valued inventory levels of cooperating firms in a manufacturing network. Based on discrete-event simulations, measures from traditional RQA are used to evaluate the performance of the individual firms under different production strategies as well as order policies. The results of our investigations are an important step towards an anticipative knowledge about the performance of manufacturing systems under different conditions, which is of major importance for the planning and control of both production and logistics.

Composición algorítmica de música mediante herramientas de complejidad. La presentación no incluye pensamiento complejo, pues éste ha demostrado no servir para esos fines (o para algún otro propósito de carácter algorítmico).

Algunos algoritmos han demostrado ser casi tan buenos o tan malos como ciertos compositores humanos. De algunos se sabe que han superado la prueba de Turing, mientras que unos cuantos humanos no han podido hacerlo. Garantizo que unas pocas músicas fractales (usualmente fractales de Gingerbread o sistemas de Lindenmayer) pueden llegar a sonar como músicas humanas con ciertas inflexiones culturales, como p.ej. ragas. No toda la música algorítmica suena necesariamente artificial.

Se han experimentado diversos algotimos (por lo común recursivos) para modelar, describir y a veces explicar la música humana (o la música en general),

Igual que he procurado hacer con las gramáticas generativas de texto (disponibles en https://www.academia.edu/60807649/Cuatro_imperativos_metodol%C3%B3gicos_para_una_futura_pedagog%C3%ADa_de_la_complejidad) a medida que vaya refinando esta presentación trataré que los algoritmos de música y complejidad se pueden ejecutar en este mismo contexto.

Sobre (etno)música y complejidad puede leerse aquí mismo Antropología de la Música: De los géneros tribales a la globalización, Volumen II (2006) ( https://www.academia.edu/54897673/Antropolog%C3%ADa_de_la_M%C3%BAsica_De_los_g%C3%A9neros_tribales_a_la_globalizaci%C3%B3n_Volumen_II_2006_)

Aunque el tema se ha retorcido, simplificado y exagerado hasta el paroxismo, la música posee innegablemente características algorítmicas.  En ella se manifiestan una combinatoria, una red de relaciones, una pauta generativa, un conjunto de formas posibles y una lógica de series temporales cultural y cognitivamente modulada. Es por eso que se la puede analizar y sintetizar ricamente y desde puntos de vista novedosos mediante herramientas ligadas a las teorías de la complejidad y el caos.

Por añadidura -y por razones que aun no se comprenden bien- es más probable que la algorítmica compleja produzca música y no que engendre mero ruido.  Todavía falta bastante para el advenimiento de la máquina sensible que nos prometiera Ray Kurzweil; pero si se consideran válidos los experimentos que admiten géneros musicales de vanguardia, se podrá comprobar al cabo de los ejercicios que se proponen en esta página (y por poco empeño que se ponga en su ejecución ) que los algoritmos sonoros pueden superar con holgura una versión no trivial de la prueba de Turing. Aun en el terreno de las formas clásicas y convencionales de composición, los avances que han habido en la segunda década del siglo XXI en materia de inteligencia computacional compleja van mucho más allá de lo que en la época del heideggeriano Hubert Dreyfus [1929-2017] se habría estimado verosímil.

En esta presentación (preparada originalmente para los Cursos de Contexto de la Universidad Central de Bogotá, Colombia, dictados en el año 2010) se examinan algunos de esos instrumentos virtuales, basados en formalismos tales como los gráficos de recurrencia, la dimensión fractal, la geometría fractal propiamente dicha, los modelos basados en agentes, los autómatas celulares, la teoría de grafos y el algoritmo genético. Algunas variantes de complejidad desorganizada también integran estas estrategias: redes neuronales, sistemas expertos, métodos estocásticos, camino al azar, cadenas de Márkov. Algunas veces (como en GenJam) se combinan varias especies algorítmicas.

No puede asegurarse sin más que los recursos algorítmicos de la complejidad sean mejores en su desempeño compositivo que los que implementaban los métodos aleatorios o combinatorios más convencionales. Lo que sí es seguro es que las músicas generadas mediante algoritmos complejos se sienten casi invariablemente más “humanas” o “naturales” que las que surgen de métodos numéricos estándar, los cuales con frecuencia tienden ya sea hacia la rigidez o hacia el sinsentido.

Correspondientemente, la premisa dominante en la línea de investigación que aquí se adopta sugiere que si un método algorítmico o principio formal no puede sintetizar música entonces es dudoso que se trate de una genuina herramienta de complejidad. Más allá de que esa presunción resulte cierta o no, la creación musical puede constituir un recurso pedagógico excelente para introducir a los estudiosos de todas las ciencias en el tema de la complejidad y en las complicaciones de las técnicas que la integran. Para no hablar, por supuesto, de demixtificar  y tornar menos ominosas las matemáticas implicadas. Aun cuando los factores cuantitativos que aquí se despliegan son mínimos (y quizá por eso mismo) en pocos campos del conocimiento las matemáticas se sienten tan cercanas a la creación artística.

La presentación sobre Música y Complejidad (que se puede bajar, reproducir y modificar a total voluntad) puede verse en el documento aquí expuesto.

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the