Entropy correlation distance method. The Euro introduction effect on the Consumer Price Index

The idea of entropy was introduced in thermodynamics, but it can be used in time series analysis. There are various ways to define and measure the entropy of a system. Here the so called Theil index, which is often used in economy and finance, is applied as it were an entropy measure. In this study the time series are remapped through the Theil index. Then the linear correlation coefficient between the remapped time series is evaluated as a function of time and time window size and the corresponding statistical distance is defined. The results are compared with the the usual correlation distance measure for the time series themselves. As an example this entropy correlation distance method (ECDM) is applied to several series, as those of the Consumer Price Index (CPI) in order to test some so called globalisation processes. Distance matrices are calculated in order to construct two network structures which are next analysed. The role of two different time scales introduced by the Theil index and a correlation coefficient is also discussed. The evolution of the mean distance between the most developed countries is presented and the globalisation periods of the prices discussed. It is finally shown that the evolution of mean distance between the most developed countries on several networks follows the process of introducing the European currency — the Euro. It is contrasted to the GDP based analysis. It is stressed that the entropy correlation distance measure is more suitable in detecting significant changes, like a globalisation process than the usual statistical (correlation based) measure.     

Globalization — Entropy unification through the Theil index

A time series is remapped onto an entropy concept, based on the Theil index. The Manhattan distance between these surrogate series is calculated, and contrasted to the usual correlation distance measure. The idea is applied to several Gross Domestic Product (relative increments) of rich countries. Such distances are calculated for various time window sizes. The role of time averaging in such finite size windows is discussed. We construct the locally minimum spanning tree (LMST) corresponding to the distance matrix. Another hierarchical network structure (Unidirectional Minimal Length Path) is compared with the LMST for confirming that the mean distance between the most developed countries on different networks actually decreases in time, — which we consider as a proof of economy globalization. It is stressed that this entropy distance measure seems more suitable in detecting some “phase transition” in time series, like a globalization process than the usual correlation based measure.     

Interplay between topology and dynamics in the World Trade Web

We present an empirical analysis of the network formed by the trade relationships between all world countries, or World Trade Web (WTW). Each (directed) link is weighted by the amount of wealth flowing between two countries, and each country is characterized by the value of its Gross Domestic Product (GDP). By analysing a set of year-by-year data covering the time interval 1950–2000, we show that the dynamics of all GDP values and the evolution of the WTW (trade flow and topology) are tightly coupled. The probability that two countries are connected depends on their GDP values, supporting recent theoretical models relating network topology to the presence of a `hidden' variable (or fitness). On the other hand, the topology is shown to determine the GDP values due to the exchange between countries. This leads us to a new framework where the fitness value is a dynamical variable determining, and at the same time depending on, network topology in a continuous feedback.     

Clusters in weighted macroeconomic networks: the EU case. Introducing the overlapping index of GDP/capita fluctuation correlations

GDP/capita correlations are investigated in various time windows (TW), for the time interval 1990–2005. The target group of countries is the set of 25 EU members, 15 till 2004 plus the 10 countries which joined EU later on. The TW-means of the statistical correlation coefficients are taken as the weights (links) of a fully connected network having the countries as nodes. Thereafter we define and introduce the overlapping index of weighted network nodes. A cluster structure of EU countries is derived from the statistically relevant eigenvalues and eigenvectors of the adjacency matrix. This may be considered to yield some information about the structure, stability and evolution of the EU country clusters in a macroeconomic sense.     

Clusters or networks of economies? A macroeconomy study through Gross Domestic Product

We study correlations between web-downloaded gross domestic product (GDP)'s of rich countries. GDP is used as wealth signatures of the country economical state. We calculate the yearly fluctuations of the GDP. We look for forward and backward correlations between such fluctuations. The correlation measure is based on the Theil index. The system is represented by an evolving weighted network, nodes being the GDP fluctuations (or countries) at different times.In order to extract structures from the network, we focus on filtering the time delayed correlations by removing the least correlated links. This percolation idea-based method reveals the emergence of connections, that are visualized by a branching representation. Note that the network is made of weighted and directed links when taking into account a delay time. Such a measure of collective habits does not readily fit the usual expectations, except if an economy globalization framework is accepted.     

Power law for the duration of recession and prosperity in Latin American countries

Ormerod and Mounfield [P. Ormerod, C. Mounfield, Power law distribution of duration and magnitude of recessions in capitalist economies: Breakdown of scaling, Physica A 293 (2001) 573] and Ausloos et al. [M. Ausloos, J. Mikiewicz, M. Sanglier, The durations of recession and prosperity: Does their distribution follow a power or an exponential law? Physica A 339 (2004) 548] have independently analyzed the duration of recessions for developed countries through the evolution of the GDP in different time windows. It was found that there is a power law governing the duration distribution. We have analyzed data collected from 19 Latin American countries in order to observe whether such results are valid or not for developing countries. The case of prosperity years is also discussed. We observe that the power law of recession time intervals, see Ref. [1], is valid for Latin American countries as well. Thus an interesting point is discovered: the same scaling time is found in the case of recessions for the three data sets (ca. 1 year), and this could represent a universal feature. Other time scale parameters differ significantly from each other.     

Structure of a financial cross-correlation matrix under attack

We investigate the structure of a perturbed stock market in terms of correlation matrices. For the purpose of perturbing a stock market, two distinct methods are used, namely local and global perturbation. The former involves replacing a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series while the latter reconstructs the cross-correlation matrix just after replacing the original return series with Gaussian-distributed time series. Concerning the local case, it is a technical study only and there is no attempt to model reality. The term ‘global’ means the overall effect of the replacement on other untouched returns. Through statistical analyses such as random matrix theory (RMT), network theory, and the correlation coefficient distributions, we show that the global structure of a stock market is vulnerable to perturbation. However, apart from in the analysis of inverse participation ratios (IPRs), the vulnerability becomes dull under a small-scale perturbation. This means that these analysis tools are inappropriate for monitoring the whole stock market due to the low sensitivity of a stock market to a small-scale perturbation. In contrast, when going down to the structure of business sectors, we confirm that correlation-based business sectors are regrouped in terms of IPRs. This result gives a clue about monitoring the effect of hidden intentions, which are revealed via portfolios taken mostly by large investors.     

Revisiting the role of correlation coefficient to distinguish chaos from noise

The correlation coefficient vs. prediction time profile has been widely used to distinguish chaos from noise. The correlation coefficient remains initially high, gradually decreasing as prediction time increases for chaos and remains low for all prediction time for noise. We here show that for some chaotic series with dominant embedded cyclical component(s), when modelled through a newly developed scheme of periodic decomposition, will yield high correlation coefficient even for long prediction time intervals, thus leading to a wrong assessment of inherent chaoticity. But if this profile of correlation coefficient vs. prediction horizon is compared with the profile obtained from the surrogate series, correct interpretations about the underlying dynamics are very much likely. Received 8 March 1999     

Time series of ac conductivities of terbium nitrate crystals at low temperatures

Time series of ac conductivities are measured at 2 kHz along three crystal directions in terbium nitrate crystal below 50 K. The anisotropy is found in the structure of the fluctuation observed in the time series of the conductivities. The bursts with non-periodic oscillations are found in only the measurements of the time series of the ac conductivities for one direction perpendicular to the c-axis (c2-axis). The anisotropy is seen in probability distribution functions and power spectral ones derived from the time series of the conductivities. The characteristic behaviors are found in the time series of the conductivity for the one crystal direction perpendicular to the c-axis. Asymmetric non-Gaussian line shape of the probability distribution function is seen for the c2-axis. Chaotic behavior is also found in the function of the correlation exponent to the embedding dimension derived from the time series of the conductivities.     

Impact of acoustic pressure on ambient pressure estimation using ultrasound contrast agent

Local blood pressure measurements provide important information on the state of health of organs in the body and can be used to diagnose diseases in the heart, lungs, and kidneys. This paper presents an approach for investigating the ambient pressure sensitivity of a contrast agent using diagnostic ultrasound. The experimental setup resembles a realistic clinical setup utilizing a single array transducer for transmit and receive. The ambient pressure sensitivity of SonoVue (Bracco, Milano, Italy) was measured twice using two different acoustic driving pressures, which were selected based on a preliminary experiment. To compensate for variations in bubble response and to make the estimates more robust, the relation between the energy of the subharmonic and the fundamental component was chosen as a measure over the subharmonic peak amplitude. The preliminary study revealed the growth stage of the subharmonic component to occur at acoustic driving pressures between 300 and 500 kPa. Based on this, the pressure sensitivity was investigated using a driving pressure of 485 and 500 kPa. At 485 kPa, a linear pressure sensitivity of 0.42 dB/kPa was found having a linear correlation coefficient of 0.94. The second measurement series at 485 kPa showed a sensitivity of 0.41 dB/kPa with a correlation coefficient of 0.89. Based on the measurements at 500 kPa, this acoustic driving pressure was concluded to be too high causing the bubbles to be destroyed. The pressure sensitivity for these two measurement series were 0.42 and 0.25 dB/kPa with linear correlation coefficients of 0.98 and 0.93, respectively.     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Visibility graph analysis on quarterly macroeconomic series of China based on complex network theory

The visibility graph approach and complex network theory provide a new insight into time series analysis. The inheritance of the visibility graph from the original time series was further explored in the paper. We found that degree distributions of visibility graphs extracted from Pseudo Brownian Motion series obtained by the Frequency Domain algorithm exhibit exponential behaviors, in which the exponential exponent is a binomial function of the Hurst index inherited in the time series. Our simulations presented that the quantitative relations between the Hurst indexes and the exponents of degree distribution function are different for different series and the visibility graph inherits some important features of the original time series. Further, we convert some quarterly macroeconomic series including the growth rates of value-added of three industry series and the growth rates of Gross Domestic Product series of China to graphs by the visibility algorithm and explore the topological properties of graphs associated from the four macroeconomic series, namely, the degree distribution and correlations, the clustering coefficient, the average path length, and community structure. Based on complex network analysis we find degree distributions of associated networks from the growth rates of value-added of three industry series are almost exponential and the degree distributions of associated networks from the growth rates of GDP series are scale free. We also discussed the assortativity and disassortativity of the four associated networks as they are related to the evolutionary process of the original macroeconomic series. All the constructed networks have “small-world” features. The community structures of associated networks suggest dynamic changes of the original macroeconomic series. We also detected the relationship among government policy changes, community structures of associated networks and macroeconomic dynamics. We find great influences of government policies in China on the changes of dynamics of GDP and the three industries adjustment. The work in our paper provides a new way to understand the dynamics of economic development.     

Correlation between detrended fluctuation analysis and the Lempel–Ziv complexity in nonlinear time series analysis

We focus on the study of the correlation between the detrended fluctuation analysis (DFA) and the Lempel-Ziv complexity (LZC) in nonlinear time series analysis in this paper. Typical dynamical systems including logistic map and Duffing model are investigated. Moreover, the influences of the Gaussian random noise on both DFA and LZC are analyzed. The results show a high correlation between DFA and LZC, which can quantify the non-stationarity and the nonlinearity of the time series, respectively. With the enhancement of the random component, the exponent α and the normalized complexity index C show increasing trends. In addition, C is found to be more sensitive to the fluctuation in the nonlinear time series than α. Finally, the correlation between DFA and LZC is applied to the feature extraction of vibration signals for a reciprocating compressor gas valve, and an effective fault diagnosis result is obtained.     

Pyroelectric effects in magnetoelectric multilayer composites

Magnetoelectric and pyroelectric properties have been investigated in heterostructures of nickel zinc ferrite (NZFO)-lead zirconate titanate (PZT) and lanthanum calcium manganite (LCMO)-PZT. The magnetoelectric (ME) coupling, mediated by mechanical strain, is found to be two orders of magnitude stronger in NZFO-PZT than in LCMO-PZT. The pyroelectric effect is investigated by measuring the current through the sample as the temperature is varied at 0.1 K/s. For NZFO-PZT the pyroelectric coefficient is in the range 0.2-15 nC/cm² K, depending on the temperature. A much weaker current is observed in LCMO-PZT. A reversal in the current direction is detected when the thermal cycle is switched from heating to cooling. The pyroelectric coefficient is found to scale with the strength of ME interactions. A clear correlation between pyroelectric current and ME interactions is evident from the results.     

时间序列三维荧光光谱的数据压缩

时间序列三维荧光光谱数据量大和信息丰富的特点虽然有利于有机物的定性和定量分析,但是大量冗余信息的存在增加了计算的复杂度和计算量。在分析了时间序列三维荧光光谱的时频特征后,分别利用聚类分析和二维小波变换从时间维和光谱维对三维荧光光谱进行了压缩。在聚类分析中探讨了样本距离、类间距离、复合相关系数和R平方统计量等关键因素。相关系数和R平方统计量的结合不仅提高了聚类分析的精确度,而且减少了二维小波在光谱维进行数据压缩的工作量。相关的数值实验表明压缩后的数据保留了原有时间序列三维荧光光谱的重要信息。     

The “S” curve relationship between export diversity and economic size of countries

The highly detailed international trade data among all countries in the world during 1971-2000 shows that the kinds of export goods and the logarithmic GDP (gross domestic production) of a country have an S-shaped relationship. This indicates that all countries can be divided into three stages accordingly. First, the small economies always export very few kinds of products as we expect. Second, once the economic size magnitude (log(GDP)) of a country is beyond a threshold, its export diversity may increase dramatically. However, this is not the case for large economies because a ceiling on the export diversity is observed when their logarithmic GDPs are higher than another threshold. This pattern is very stable for different years although the concrete parameters of the fitting sigmoid functions may change with time. In addition, we also discussed other relationships such as import diversity with respect to logarithmic GDP, diversity of exporters with respect to the number of export goods etc.; all of these relationships show S-shaped or power law patterns which can be derived by the “S” curve relations. Although this paper does not explain the origin of the S-shaped curve, it may provide a basic empirical fact and insights for economic diversity.     

Rotating high current arc

A high current arc (1–16 kA) of 5–20 ms duration is investigated, which rotates between two annular electrodes of variable distance (5–16 mm) at elevated pressures (0.1–1 MPa) in SF₆. The are is driven by the magnetic field of two ring-shaped coils in cusp geometry, one on either side of the electrodes. Both experiments and theoretical models concentrate on the investigation of the hot wake. The temperature, velocity, and size of the wake are determined by schlieren photography, streak records, and shock waves. A simple model is developed which yields a quantitative picture. The immediate surroundings of the transversely blown are can be described in terms of sucking in cold gas with associated heating and expansion, whereas the far-field region is governed by turbulent entrainment. The model involves only one empirical parameter: the entrainment coefficient α, which turns out to have a value of about 10⁻³.     

Spartan random processes in time series modeling

A Spartan random process (SRP) is used to estimate the correlation structure of time series and to predict (interpolate and extrapolate) the data values. SRPs are motivated from statistical physics, and they can be viewed as Ginzburg-Landau models. The temporal correlations of the SRP are modeled in terms of ‘interactions’ between the field values. Model parameter inference employs the computationally fast modified method of moments, which is based on matching sample energy moments with the respective stochastic constraints. The parameters thus inferred are then compared with those obtained by means of the maximum likelihood method. The performance of the Spartan predictor (SP) is investigated using real time series of the quarterly S&P 500 index. SP prediction errors are compared with those of the Kolmogorov-Wiener predictor. Two predictors, one of which is explicit, are derived and used for extrapolation. The performance of the predictors is similarly evaluated.     

Detecting community structure from coherent oscillation of excitable systems

Many networks are proved to have community structures. On the basis of the fact that the dynamics on networks are intensively affected by the related topology, in this paper the dynamics of excitable systems on networks and a corresponding approach for detecting communities are discussed. Dynamical networks are formed by interacting neurons; each neuron is described using the FHN model. For noisy disturbance and appropriate coupling strength, neurons may oscillate coherently and their behavior is tightly related to the community structure. Synchronization between nodes is measured in terms of a correlation coefficient based on long time series. The correlation coefficient matrix can be used to project network topology onto a vector space. Then by the K-means cluster method, the communities can be detected. Experiments demonstrate that our algorithm is effective at discovering community structure in artificial networks and real networks, especially for directed networks. The results also provide us with a deep understanding of the relationship of function and structure for dynamical networks.