Selecting an Effective Entropy Estimator for Short Sequences of Bits and Bytes with Maximum Entropy

Entropy makes it possible to measure the uncertainty about an information source from the distribution of its output symbols. It is known that the maximum Shannon’s entropy of a discrete source of information is reached when its symbols follow a Uniform distribution. In cryptography, these sources have great applications since they allow for the highest security standards to be reached. In this work, the most effective estimator is selected to estimate entropy in short samples of bytes and bits with maximum entropy. For this, 18 estimators were compared. Results concerning the comparisons published in the literature between these estimators are discussed. The most suitable estimator is determined experimentally, based on its bias, the mean square error short samples of bytes and bits.     

An Entropy-Based Approach to Measurement of Stock Market Depth

The aim of this study is to investigate market depth as a stock market liquidity dimension. A new methodology for market depth measurement exactly based on Shannon information entropy for high-frequency data is introduced and utilized. The proposed entropy-based market depth indicator is supported by an algorithm inferring the initiator of a trade. This new indicator seems to be a promising liquidity measure. Both market entropy and market liquidity can be directly measured by the new indicator. The findings of empirical experiments for real-data with a time stamp rounded to the nearest second from the Warsaw Stock Exchange (WSE) confirm that the new proxy enables us to effectively compare market depth and liquidity for different equities. Robustness tests and statistical analyses are conducted. Furthermore, an intra-day seasonality assessment is provided. Results indicate that the entropy-based approach can be considered as an auspicious market depth and liquidity proxy with an intuitive base for both theoretical and empirical analyses in financial markets.     

Intelligent Fault Identification for Rolling Bearings Fusing Average Refined Composite Multiscale Dispersion Entropy-Assisted Feature Extraction and SVM with Multi-Strategy Enhanced Swarm Optimization

Rolling bearings act as key parts in many items of mechanical equipment and any abnormality will affect the normal operation of the entire apparatus. To diagnose the faults of rolling bearings effectively, a novel fault identification method is proposed by merging variational mode decomposition (VMD), average refined composite multiscale dispersion entropy (ARCMDE) and support vector machine (SVM) optimized by multistrategy enhanced swarm optimization in this paper. Firstly, the vibration signals are decomposed into different series of intrinsic mode functions (IMFs) based on VMD with the center frequency observation method. Subsequently, the proposed ARCMDE, fusing the superiorities of DE and average refined composite multiscale procedure, is employed to enhance the ability of the multiscale fault-feature extraction from the IMFs. Afterwards, grey wolf optimization (GWO), enhanced by multistrategy including levy flight, cosine factor and polynomial mutation strategies (LCPGWO), is proposed to optimize the penalty factor C and kernel parameter g of SVM. Then, the optimized SVM model is trained to identify the fault type of samples based on features extracted by ARCMDE. Finally, the application experiment and contrastive analysis verify the effectiveness of the proposed VMD-ARCMDE-LCPGWO-SVM method.     

Pathwise superhedging for time-dependent barrier options on càdlàg paths—Finite or infinite tradeable European,One-Touch,lookback or forward starting options

We establish pathwise duality using simple predictable trading strategies for the robust hedging problem associated with a barrier option whose payoff depends on the terminal level and the infimum of a càdlàg strictly positive stock price process, given tradeable European options at all strikes at a single maturity. The result allows for a significant dimension reduction in the computation of the superhedging cost, via an alternate lower-dimensional formulation of the primal problem as a convex optimization problem, which is qualitatively similar to the duality which was formally sketched using linear programming arguments in Duembgen and Rogers [10] for the case where we only consider continuous sample paths. The proof exploits a simplification of a classical result by Rogers (1993) which characterizes the attainable joint laws for the supremum and the drawdown of a uniformly integrable martingale (not necessarily continuous), combined with classical convex duality results from Rockefellar (1974) using paired spaces with compatible locally convex topologies and the Hahn–Banach theorem. We later adapt this result to include additional tradeable One-Touch options using the Kertz and Rösler (1990) condition. We also compute the superhedging cost when in the more realistic situation where there is only finite tradeable European options; for this case we obtain the full duality in the sense of quantile hedging as in Soner (2015), where the superhedge works with probability $1?\epsilon$ where $\epsilon$ can be arbitrarily small), and we obtain an upper bound for the true pathwise superhedging cost. In Section 5, we extend our analysis to include time-dependent barrier options using martingale coupling arguments, where we now have tradeable European options at both maturities at all strikes and tradeable forward starting options at all strikes. This set up is designed to approximate the more realistic situation where we have a finite number of tradeable Europeans at both maturities plus a finite number of tradeable forward starting options.¹     

Randomness and Nonlinear Evolution Equations

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.     

Evaluating continuous chirality measure as a 3D descriptor in chemoinformatics applied to asymmetric catalysis

Continuous Chirality Measure (CCM) is a computational metric by which to quantify the chirality of a compound. In enantioselective catalysis, prior work has postulated that CCM is correlated to selectivity and can be used to understand which structural features dictate catalyst efficacy. Herein, the investigation of CCM as a metric capable of guiding catalyst optimization is explored. Conformer-dependent CCM is also explored. Finally, CCM is used with Sterimol parameters to significantly improve the performance of Random Forest models.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.