‡We describe the use of singular value decomposition in transforming genome-wide expression data from genes 3 arrays space to reduced diagonalized ‘‘eigengenes’’ 3 ‘‘eigenarrays’’ space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively.

Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling

: This thesis applies neural network feature selection techniques to multivariate time series data to improve prediction of a target time series. Two approaches to feature selection are used. First, a subset enumeration method is used to determine which financial indicators are most useful for aiding in prediction of the S&P 500 futures daily price. The candidate indicators evaluated include RSI, Stochastics and several moving averages. Results indicate that the Stochastics and RSI indicators result in better prediction results than the moving averages. The second approach to feature selection is calculation of individual saliency metrics. A new decision boundary-based individual saliency metric, and a classifier independent saliency metric are developed and tested. Ruck's saliency metric, the decision boundary based saliency metric, and the classifier independent saliency metric are compared for a data set consisting of the RSI and Stochastics indicators as well as delayed closing price values. The decision based metric and the Ruck metric results are similar, but the classifier independent metric agrees with neither of the other metrics. The nine most salient features, determined by the decision boundary based metric, are used to train a neural network and the results are presented and compared to other published results. (AN)

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Nonlinear Time Series Analysis.

Since the early 1980s, short-term traffic forecasting has been an integral part of most Intelligent Transportation Systems (ITS) research and applications; most effort has gone into developing methodologies that can be used to model traffic characteristics and produce anticipated traffic conditions. Existing literature is voluminous, and has largely used single point data from motorways and has employed univariate mathematical models to predict traffic volumes or travel times. Recent developments in technology and the widespread use of powerful computers and mathematical models allow researchers an unprecedented opportunity to expand horizons and direct work in 10 challenging, yet relatively under researched, directions. It is these existing challenges that we review in this paper and offer suggestions for future work.

Short-term traffic forecasting: Where we are and where we’re going

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

/pdf/analysis-of-observed-chaotic-data-3spe9uweeb.pdf

Analysis of observed chaotic data

Multifractal behavior in traffic time series usually connected with different long-range (time-) correlations of the small and large fluctuations or (and) a broad probability density function for the values of the time series Multifractal detrended fluctuation analysis (MFDFA) is used to study the traffic speed fluctuations It is demonstrated that the speed time series, observed on the Beijing Yuquanying highway over a period of about 40 months, has a crossover time scale s x , where the signal has different correlation exponents in time scales s  >  s x and s s x  Finally, the long-range correlation was validated to be dominant by the method of comparing the MFDFA results for original series to those obtained via the MFDFA for shuffled series

Detecting long-range correlations of traffic time series with multifractal detrended fluctuation analysis

In this paper, we applied non-linear time series modeling techniques to analyze the traffic data collected from the Beijing Xizhimen. The results indicated that chaotic characteristics obviously exist in the traffic system; techniques based on phase space dynamics can be used to analyze and predict the traffic speed.

Chaotic analysis of traffic time series

We introduce a new method, multifractal cross-correlation analysis based on statistical moments (MFSMXA), to investigate the long-term cross-correlations and cross-multifractality between time series generated from complex system. Efficiency of this method is shown on multifractal series, comparing with the well-known multifractal detrended cross-correlation analysis (MFXDFA) and multifractal detrending moving average cross-correlation analysis (MFXDMA). We further apply this method on volatility time series of DJIA and NASDAQ indices, and find some interesting results. The MFSMXA has comparative performance with MFXDMA and sometimes perform slightly better than MFXDFA. Multifractal nature exists in volatility series. In addition, we find that the cross-multifractality of volatility series is mainly due to their cross-correlations, via comparing the MFSMXA results for original series with those for shuffled series.

Multifractal cross-correlation analysis based on statistical moments

Multifractal detrended cross-correlation analysis (MF-DXA) has been developed to detect the long-range power-law cross-correlation of considered signals in the presence of non-stationarity. However, crossovers arising from extrinsic periodic trends make the scaling behavior difficult to analyze. We introduce a Fourier filtering method to eliminate the trend effects and systematically investigate the multifractal cross-correlation of simulated and real traffic signals. The crossover locations are found approximately corresponding to the periods of underlying trend. Traffic velocity on one road and flows on adjacent roads show strong cross-correlation. They also present weak multifractality after periodic trends are removed. The traffic velocity and flow are cross-correlated in opposite directions which is accordant to their actual evolution.

Multifractal Fourier detrended cross-correlation analysis of traffic signals

In this paper, we propose a new two-stage methodology that combines the ensemble empirical mode decomposition (EEMD) with multidimensional k -nearest neighbor model (MKNN) in order to forecast the closing price and high price of the stocks simultaneously. The modified algorithm of k -nearest neighbors (KNN) has an increasingly wide application in the prediction of all fields. Empirical mode decomposition (EMD) decomposes a nonlinear and non-stationary signal into a series of intrinsic mode functions (IMFs), however, it cannot reveal characteristic information of the signal with much accuracy as a result of mode mixing. So ensemble empirical mode decomposition (EEMD), an improved method of EMD, is presented to resolve the weaknesses of EMD by adding white noise to the original data. With EEMD, the components with true physical meaning can be extracted from the time series. Utilizing the advantage of EEMD and MKNN, the new proposed ensemble empirical mode decomposition combined with multidimensional k -nearest neighbor model (EEMD–MKNN) has high predictive precision for short-term forecasting. Moreover, we extend this methodology to the case of two-dimensions to forecast the closing price and high price of the four stocks (NAS, S&P500, DJI and STI stock indices) at the same time. The results indicate that the proposed EEMD–MKNN model has a higher forecast precision than EMD–KNN, KNN method and ARIMA.

Pengjian Shang

Papers

Detecting long-range correlations of traffic time series with multifractal detrended fluctuation analysis

Chaotic analysis of traffic time series

Multifractal cross-correlation analysis based on statistical moments

Multifractal Fourier detrended cross-correlation analysis of traffic signals

Multidimensional k-nearest neighbor model based on EEMD for financial time series forecasting