M. S. Santhanam

Statistics of atmospheric correlations

For a large class of quantum systems, the statistical properties of their spectrum show remarkable agreement with random matrix predictions. Recent advances show that the scope of random matrix theory is much wider. In this work, we show that the random matrix approach can be beneficially applied to a completely different classical domain, namely, to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterize the state of atmosphere. We show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription. In particular, the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution.

Return interval distribution of extreme events and long-term memory

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.

Extreme events on complex networks.

A wide spectrum of extreme events ranging from traffic jams to floods take place on networks. Motivated by these, we employ a random walk model for transport and obtain analytical and numerical results for the extreme events on networks. They reveal an unforeseen, and yet a robust, feature: small degree nodes of a network are more likely to encounter extreme events than the hubs. Further, we also study the recurrence time distribution and scaling of the probabilities for extreme events. These results suggest a revision of design principles and can be used as an input for designing the nodes of a network so as to smoothly handle extreme events.

DIGITAL WATERMARKING IN THE SINGULAR VECTOR DOMAIN

Many current watermarking algorithms insert data in the spatial or transform domains like the discrete cosine, the discrete Fourier, and the discrete wavelet transforms. In this paper, we present a data-hiding algorithm that exploits the singular value decomposition (SVD) representation of the data. We compute the SVD of the host image and the watermark and embed the watermark in the singular vectors of the host image. The proposed method leads to an imperceptible scheme for digital images, both in grey scale and color and is quite robust against attacks like noise and JPEG compression.

Quantum spectrum as a time series: fluctuation measures.

The fluctuations in the quantum spectrum could be treated like a time series. In this framework, we explore the statistical self-similarity in the quantum spectrum using the detrended fluctuation analysis (DFA) and random matrix theory (RMT). We calculate the Hausdorff measure for the spectra of atoms and Gaussian ensembles and study their self-affine properties. We show that DFA is equivalent to the Delta3 statistics of RMT, unifying two different approaches. We exploit this connection to obtain theoretical estimates for the Hausdorff measure.

Spectral fluctuations and 1/f noise in the order-chaos transition regime

Level fluctuations in a quantum system have been used to characterize quantum chaos using random matrix models. Recently time series methods were used to relate the level fluctuations to the classical dynamics in the regular and chaotic limit. In this, we show that the spectrum of the system undergoing order to chaos transition displays a characteristic f(-gamma) noise and gamma is correlated with the classical chaos in the system. We demonstrate this using a smooth potential and a time-dependent system modeled by Gaussian and circular ensembles, respectively, of random matrix theory. We show the effect of short periodic orbits on these fluctuation measures.

Random matrix approach to categorical data analysis.

Correlation and similarity measures are widely used in all the areas of sciences and social sciences. Often the variables are not numbers but are instead qualitative descriptors called categorical data. We define and study similarity matrix, as a measure of similarity, for the case of categorical data. This is of interest due to a deluge of categorical data, such as movie ratings, top-10 rankings, and data from social media, in the public domain that require analysis. We show that the statistical properties of the spectra of similarity matrices, constructed from categorical data, follow random matrix predictions with the dominant eigenvalue being an exception. We demonstrate this approach by applying it to the data for Indian general elections and sea level pressures in the North Atlantic ocean.

Multichannel digital watermarking of color images using SVD

In this paper we propose a new invisible, non-blind digital watermarking scheme for color images based on singular vector domain (SVD). In this new approach, same or different watermarks can be embedded into the three color channels (R, G, B) of the host image in order to increase the robustness of the watermark. The proposed method demonstrates that color image watermarking is possible without affecting its visual quality and also allows for reasonable compromise between robustness and invisibility of watermarks. This research, based on our earlier work, consists of embedding a gray watermark into a gray host and a color watermark into a color host based on the singular vector domain. The main feature of our scheme is to use all channels of RGB color space to embed watermarks. Hence we are embedding maximum amount of watermark and also make it more secure. The other feature of our scheme is that it can be used for a variety of applications since we can perform single channel embedding to multichannel embedding. We then show our simulation results for one channel, two channel and three channel and also results for how it behaves under various attacks.

Effect of classical bifurcations on the quantum entanglement of two coupled quartic oscillators

We study entanglement in two coupled quartic oscillators. It is shown that the entanglement, as measured by the von Neumann entropy, increases with the classical chaos parameter for generic chaotic eigenstates. We consider certain isolated periodic orbits whose bifurcation sequence affects a class of quantum eigenstates, called the channel localized states. For these states, the entanglement is a local minima in the vicinity of a pitchfork bifurcation but is a local maxima near a anti-pitchfork bifurcation. We place these results in the context of the close connections that may exist between entanglement measures and conventional measures of localization that have been much studied in quantum chaos and elsewhere. We also point to an interesting near-degeneracy that arises in the spectrum of reduced density matrices of certain states as an interplay of localization and symmetry.

Financial time-series analysis: A brief overview

Prices of commodities or assets produce what is called time-series. Different kinds of financial time-series have been recorded and studied for decades. Nowadays, all transactions on a financial market are recorded, leading to a huge amount of data available, either for free in the Internet or commercially. Financial time-series analysis is of great interest to practitioners as well as to theoreticians, for making inferences and predictions. Furthermore, the stochastic uncertainties inherent in financial time-series and the theory needed to deal with them make the subject especially interesting not only to economists, but also to statisticians and physicists. While it would be a formidable task to make an exhaustive review on the topic, with this review we try to give a flavor of some of its aspects.