Fast Entropy Estimation for Natural Sequences

It is well known that to estimate the Shannon entropy for symbolic sequences accurately requires a large number of samples. When some aspects of the data are known it is plausible to attempt to use this to more efficiently compute entropy. A number of methods having various assumptions have been proposed which can be used to calculate entropy for small sample sizes. In this paper, we examine this problem and propose a method for estimating the Shannon entropy for a set of ranked symbolic natural events. Using a modified Zipf-Mandelbrot-Li law and a new rank-based coincidence counting method, we propose an efficient algorithm which enables the entropy to be estimated with surprising accuracy using only a small number of samples. The algorithm is tested on some natural sequences and shown to yield accurate results with very small amounts of data

A First Application of Independent Component Analysis to Extracting Structure from Stock Returns

This paper discusses the application of a modern signal processing technique known as independent component analysis (ICA) or blind source separation to multivariate financial time series such as a portfolio of stocks. The key idea of ICA is to linearly map the observed multivariate time series into a new space of statistically independent components (ICs). This can be viewed as a factorization of the portfolio since joint probabilities become simple products in the coordinate system of the ICs. We apply ICA to three years of daily returns of the 28 largest Japanese stocks and compare the results with those obtained using principal component analysis. The results indicate that the estimated ICs fall into two categories, (i) infrequent but large shocks (responsible for the major changes in the stock prices), and (ii) frequent smaller fluctuations (contributing little to the overall level of the stocks). We show that the overall stock price can be reconstructed surprisingly well by using a small number of thresholded weighted ICs. In contrast, when using shocks derived from principal components instead of independent components, the reconstructed price is less similar to the original one. Independent component analysis is a potentially powerful method of analyzing and understanding driving mechanisms in financial markets. There are further promising applications to risk management since ICA focuses on higher-order statistics.Information Systems Working Papers Serie

Determining the Number of Samples Required to Estimate Entropy in Natural Sequences

Calculating the Shannon entropy for symbolic sequences has been widely considered in many fields. For descriptive statistical problems such as estimating the N-gram entropy of English language text, a common approach is to use as much data as possible to obtain progressively more accurate estimates. However in some instances, only short sequences may be available. This gives rise to the question of how many samples are needed to compute entropy. In this paper, we examine this problem and propose a method for estimating the number of samples required to compute Shannon entropy for a set of ranked symbolic natural events. The result is developed using a modified Zipf-Mandelbrot law and the Dvoretzky-Kiefer-Wolfowitz inequality, and we propose an algorithm which yields an estimate for the minimum number of samples required to obtain an estimate of entropy with a given confidence level and degree of accuracy

Plasma concentrations of boosted and unboosted atazanavir are predicted by 63396C>T SNP in the PXR gene

Purpose of the study Atazanavir (ATV) is administered at the usual adult dose of 300 mg with 100 mg of ritonavir (RTV) once a day (boosted). However, 400 mg once a day (unboosted) is also used in some settings. ATV plasma concentrations are influenced by efflux transporters, influx transporters and metabolism enzymes. The expression of many of these proteins is regulated by nuclear receptors such as PXR. Recently polymorphisms in the regulatory region of the PXR gene have been reported to influence its expression and the activity of downstream genes, such as CYP3A4 and ABCB1. The aim of this study was to investigate whether polymorphisms in PXR influence trough concentrations (Ctrough) of boosted or unboosted ATV

Amplitude analysis of B0→D¯0K+π−decays

The Dalitz plot distribution of B0→D¯0K+π− decays is studied using a data sample corresponding to 3.0  fb−1 of pp collision data recorded by the LHCb experiment during 2011 and 2012. The data are described by an amplitude model that contains contributions from intermediate K∗(892)0, K∗(1410)0, K∗2(1430)0 and D∗2(2460)− resonances. The model also contains components to describe broad structures, including the K∗0(1430)0 and D∗0(2400)− resonances, in the Kπ S-wave and the Dπ S- and P-waves. The masses and widths of the D∗0(2400)− and D∗2(2460)− resonances are measured, as are the complex amplitudes and fit fractions for all components included in the amplitude model. The model obtained will be an integral part of a future determination of the angle γ of the Cabibbo-Kobayashi-Maskawa quark mixing matrix using B0→DK+π− decays

Model-independent measurement of mixing parameters in D0 → K S 0 π+π− decays

The first model-independent measurement of the charm mixing parameters in the decay D 0 → K S 0 π + π − is reported, using a sample of pp collision data recorded by the LHCb experiment, corresponding to an integrated luminosity of 1.0 fb−1 at a centre-of-mass energy of 7 TeV. The measured values are x=(−0.86±0.53±0.17)×10−2,y=(+0.03±0.46±0.13)×10−2, x=(−0.86±0.53±0.17)×10−2,y=(+0.03±0.46±0.13)×10−2, where the first uncertainties are statistical and include small contributions due to the external input for the strong phase measured by the CLEO collaboration, and the second uncertainties are systematic

Author Correction: A consensus-based transparency checklist.

An amendment to this paper has been published and can be accessed via a link at the top of the paper

Measurement of CP asymmetries and polarisation fractions in Bs0→K∗0K¯∗0 decays

An angular analysis of the decay Bs0→K∗0K¯∗0 is performed using pp collisions corresponding to an integrated luminosity of 1.0 fb−1 collected by the LHCb experiment at a centre-of-mass energy s=7 TeV. A combined angular and mass analysis separates six helicity amplitudes and allows the measurement of the longitudinal polarisation fraction fL = 0.201 ± 0.057 (stat.) ± 0.040 (syst.) for the Bs0→K∗8920K¯∗8920 decay. A large scalar contribution from the K0∗(1430) and K0∗(800) resonances is found, allowing the determination of additional CP asymmetries. Triple product and direct CP asymmetries are determined to be compatible with the Standard Model expectations. The branching fraction ℬBs→K∗8920K¯∗8920 is measured to be (10.8 ± 2.1 (stat.) ± 1.4 (syst.) ± 0.6 (fd/fs)) × 10−6

Study of η − η′ mixing from measurement of B (s) 0 → J/ψη(′) decay rates

A study of B and Bs meson decays into J/ψ η and J/ψ η′ final states is performed using a data set of proton-proton collisions at centre-of-mass energies of 7 and 8 TeV, collected by the LCHb experiment and corresponding to 3.0 fb−1 of integrated luminosity. The decay B0 → J/ψ η′ is observed for the first time. The following ratios of branching fractions are measured: B(B0→J/ψη′)B(B0s→ J/ψη′)=(2.28±0.65 (stat)±0.10 (syst)±0.13 (fs/fd))×10−2,B(B0→ J/ψη)B(B0s→ J/ψη)=(1.85±0.61 (stat)±0.09 (syst)±0.11 (fs/fd))×10−2, where the third uncertainty is related to the present knowledge of fs/fd, the ratio between the probabilities for a b quark to form a Bs or a B0 meson. The branching fraction ratios are used to determine the parameters of η − η′ meson mixing. In addition, the first evidence for the decay Bs → ψ(2S)η′ is reported, and the relative branching fraction is measured, B(B0s→ ψ(2S)η′)B(B0s→ J/ψη′)=(38.7±9.0 (stat)±1.3 (syst)±0.9(B))×10−2, where the third uncertainty is due to the limited knowledge of the branching fractions of J/ψ and ψ(2S) mesons