A Note on Entropy Estimation.

We compare an entropy estimator H(z) recently discussed by Zhang (2012) with two estimators, H(1) and H(2), introduced by Grassberger (2003) and Schürmann (2004). We prove the identity H(z) ≡ H(1), which has not been taken into account by Zhang (2012). Then we prove that the systematic error (bias) of H(1) is less than or equal to the bias of the ordinary likelihood (or plug-in) estimator of entropy. Finally, by numerical simulation, we verify that for the most interesting regime of small sample estimation and large event spaces, the estimator H(2) has a significantly smaller statistical error than H(z).

Liquid chromatography/electrospray time-of-flight mass spectrometry for the characterisation of cyclic phosphazenes.

Cyclic phosphazenes with different substituents were synthesised and investigated by liquid chromatography (LC) and electrospray ionisation mass spectrometry (ESI-MS). Hexachlorocyclotriphosphazene was functionalised with aliphatic substituents as alcohols and amines, leading to product mixtures, which were subsequently analysed. In contrast to classical methods of structural analysis such as nuclear magnetic resonance (NMR) spectroscopy or X-ray crystallography, which are restricted to pure compounds, these complex mixtures can favourably be analysed by means of LC/ESI-MS. The main products could be separated from by-products and, moreover, all the components of the unknown mixture were unambiguously identified by accurate mass measurements. For all compounds with different side-chain ratios, remaining chlorine atoms or hydroxyl groups and even for spiro or ansa products, molecular structures could be suggested.

Entropy estimation of symbol sequences.

We discuss algorithms for estimating the Shannon entropy h of finite symbol sequences with long range correlations. In particular, we consider algorithms which estimate h from the code lengths produced by some compression algorithm. Our interest is in describing their convergence with sequence length, assuming no limits for the space and time complexities of the compression algorithms. A scaling law is proposed for extrapolation from finite sample lengths. This is applied to sequences of dynamical systems in non-trivial chaotic regimes, a 1-D cellular automaton, and to written English texts. (c)1996 American Institute of Physics.

Enlarged scaling ranges for the KS-entropy and the information dimension.

Numerical estimates of the Kolmogorov-Sinai entropy based on a finite amount of data decay towards zero in the relevant limits. Rewriting differences of block entropies as averages over decay rates, and ignoring all parts of the sample where these rates are uncomputable because of the lack of neighbours, yields improved entropy estimates. In the same way, the scaling range for estimates of the information dimension can be extended considerably. The improvement is demonstrated for experimental data. (c) 1996 American Institute of Physics.