Loss function, unbiasedness, and optimality of Gaussian graphical model selection
A Gaussian graphical model is a graphical representation of the dependence structure for
a Gaussian random vector. Gaussian graphical model selection is a statistical problem that
identifies the Gaussian graphical model from observations. There are several statistical
approaches for Gaussian graphical model identification. Their properties, such as unbiasedeness
and optimality, are not established. In this paper we study these properties.
We consider the graphical model selection problem in the framework of multiple decision
theory and suggest assessing these procedures using an additive loss function. Associated
risk function in this case is a linear combination of the expected numbers of the two types
of error (False Positive and False Negative). We combine the tests of a Neyman structure for
individual hypotheses with simultaneous inference and prove that the obtained multiple
decision procedure is optimal in the class of unbiased multiple decision procedures.
Research into the market graph is attracting increasing attention in stock market analysis. One of the important problems connected with the market graph is its identification from observations. The standard way of identifying the market graph is to use a simple procedure based on statistical estimations of Pearson correlations between pairs of stocks. Recently a new class of statistical procedures for market graph identification was introduced and the optimality of these procedures in the Pearson correlation Gaussian network was proved. However, the procedures obtained have a high reliability only for Gaussian multivariate distributions of stock attributes. One of the ways to correct this problem is to consider different networks generated by different measures of pairwise similarity of stocks. A new and promising model in this context is the sign similarity network. In this paper the market graph identification problem in the sign similarity network is reviewed. A new class of statistical procedures for the market graph identification is introduced and the optimality of these procedures is proved. Numerical experiments reveal an essential difference in the quality between optimal procedures in sign similarity and Pearson correlation networks. In particular, it is observed that the quality of the optimal identification procedure in the sign similarity network is not sensitive to the assumptions on the distribution of stock attributes.
The loss functions of deep neural networks are complex and their geometric properties are not well understood. We show that the optima of these complex loss functions are in fact connected by simple curves over which training and test accuracy are nearly constant. We introduce a training procedure to discover these high-accuracy pathways between modes. Inspired by this new geometric insight, we also propose a new ensembling method entitled Fast Geometric Ensembling (FGE). Using FGE we can train high-performing ensembles in the time required to train a single model. We achieve improved performance compared to the recent state-of-the-art Snapshot Ensembles, on CIFAR-10, CIFAR-100, and ImageNet.
In this paper we address the problem of forecasting the target events of a time series given the distribution ξξ of time gaps between target events. Strong earthquakes and stock market crashes are the two types of such events that we are focusing on. In the series of earthquakes, as McCann et al. show [W.R. Mc Cann, S.P. Nishenko, L.R. Sykes, J. Krause, Seismic gaps and plate tectonics: seismic potential for major boundaries, Pure and Applied Geophysics 117 (1979) 1082–1147], there are well-defined gaps (called seismic gaps) between strong earthquakes. On the other hand, usually there are no regular gaps in the series of stock market crashes [M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314 (2002) 749–755]. For the case of seismic gaps, we analytically derive an upper bound of prediction efficiency given the coefficient of variation of the distribution ξξ. For the case of stock market crashes, we develop an algorithm that predicts the next crash within a certain time interval after the previous one. We show that this algorithm outperforms random prediction. The efficiency of our algorithm sets up a lower bound of efficiency for effective prediction of stock market crashes.
The users of today are already about to enter the era of modern smart wearable devices, a time when smart accessories will in turn push aside regular old phones and Tablets bringing a variety of new security challenges. The number of bio-sensors both integrated in smart wearables and connected over wireless interface allows a new way of multi-factor authentication (MFA) of the user being him temporary or even unwanted. This manuscript proposes a solution for conguring the MFA based on the average direct and indirect losses risk analysis. The example application of Bayesian function for MFA is proving the applicability of the proposed framework for the utilization with wearable devices.
Graphical models are used in a variety of problems to uncover hidden structures. There is an important number of different identification procedures to recover graphical model from observations. In this paper, undirected Gaussian graphical models are considered. Some Gaussian graphical model identification statistical procedures are compared using different measures, such as Type I and Type II errors, ROC AUC.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.