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PubMed2006

Informational entropy of Fourier maps.

An analysis on what is known as the interpretation of Fourier maps has been done from the information theory point of view: determining the nature of the peaks in the map (in order to assign them a suitable scattering factor) and allocating bonds between some of the possible peak pairs. Before interpreting the map, a quantitatively measurable entropy (uncertainty, unknowingness) relating to the molecular structure is known. After the interpretation, this entropy becomes amount of information. This analysis allows us, for the first time in crystallography, to quantify these parameters and analyse the contributions of the different information sources.

Crystallography, X-Ray↗

PubMed1991

A multisolution method of phase determination by combined maximization of entropy and likelihood. V. The use of likelihood as a discriminator of phase sets produced by the SAYTAN program for a small protein.

The use of a likelihood criterion associated with maximum-entropy (ME) extrapolation for selecting phase sets as part of a new multisolution phasing strategy, already applied to solving small crystal structures from single-crystal data [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297-308] and X-ray powder diffraction data [Gilmore, Henderson & Bricogne (1991). Acta Cryst. A47, 830-841], has been tested on the small protein avian pancreatic polypeptide (APP) with 301 non-H atoms in the asymmetric unit in space group C2. A collection of 50 phase sets for APP were provided by Woolfson & Yao. They had been generated from random starting phases by the SAYTAN procedure [Woolfson & Yao (1990). Acta Cryst. A46, 409-413] using data to a resolution of 0.98 A. Six of these had an unweighted mean absolute phase error, mean value of magnitude of delta phi, of less than 50 degrees, the remainder having phase errors of 60 degrees or more. However, none of the conventional figures of merit were able to identify these preferred sets. Each phase set was subjected to our standard procedure of entropy maximization and of evaluation of the log-likelihood gain resulting from the associated ME extrapolation. With only a small subset of data (to 2 A resolution), the likelihood criterion identified unambiguously the phase sets with mean value of magnitude of delta phi less than 50 degrees. In contrast, conventional figures of merit showed no such ability.(ABSTRACT TRUNCATED AT 250 WORDS)

Pancreatic Polypeptide↗

PubMed1993

Experience with phase extension and ab initio phase determination in macromolecular crystallography using maximum-entropy methods.

Three procedures, or 'tools', have been developed and tested for applying maximum-entropy methods to phase extension and to ab initio phase determination. The phase expander tool has been used in connection with the solution of two previously unknown macromolecular structures. An efficient algorithm for the determination of an electron-density distribution that is everywhere positive and that agrees with observed structure amplitudes (tools II and III) has been used to determine the phases of X-ray diffraction data from recombinant bovine chymosin, a protein with 323 amino-acid residues in the molecular chain, the structure of which was recently determined using replacement methods. By use of the same maximum-entropy methods, the structure amplitudes from the unknown structure of bovine heart creatine kinase, a protein with 381 amino-acid residues, have been phased ab initio to 2.7 A resolution. The phases of the centric reflections have also been confirmed by a satisfactory solution of the Patterson map of a mercury derivative. The current status of the structure interpretation is presented. This technique has also been applied to a test case where 48 centric reflections from bovine prothrombin fragment 1 data were phased ab initio and subsequently used in the determination of Patterson solutions for a heavy-atom derivative data set.

Journal Article↗

PubMed1993

Direct phase determination by entropy maximization and likelihood ranking: status report and perspectives.

A new multisolution phasing method based on entropy maximization and likelihood ranking, proposed for the specific purpose of extending probabilistic direct methods to the field of macromolecules, has been implemented in two different computer programs and applied to a wide variety of problems. The latter comprise the determination of small crystal structures from X-ray diffraction data obtained from single crystals or from powders, and from electron diffraction data partially phased by image processing of electron micrographs, the ab initio generation and ranking of phase sets for small proteins; and the improvement of poor quality phases for a larger protein at medium resolution under constraint of solvent flatness. These applications show that the primary goal of this new method - namely increasing the accuracy and sensitivity of probabilistic phase indications compared with conventional direct methods - has been achieved. The main components of the method are (1) a tree-directed search through a space of trial phase sets; (2) the saddle-point method for calculating joint probabilities of structure factors, using entropy maximization; (3) likelihood-based scores to rank trial phase sets and prune the search tree; (4) efficient schemes, based on error-correcting codes, for sampling trial phase sets; (5) a statistical analysis of the scores for automatically selecting reliable phase indications. They have been implemented to varying degrees of completeness in a computer program (BUSTER) and tested on two small structures as well as on the small protein crambin. The main obstructions to successful ab initio phasing in the latter case seem to reside in the accumulation of phase sampling errors and in the lack of a properly defined molecular envelope, both of which can be remedied within the methods proposed. A review of the Bayesian statistical theory encompassing all phasing procedures, proposed earlier as an extension of the initial theory, shows that the techniques now available in BUSTER bring closer a number of major enhancements of standard macromolecular phasing techniques, namely isomorphous replacement, molecular replacement, solvent flattening and non-crystallographic symmetry averaging. The gradual implementation of the successive stages of this 'Bayesian programme' should lead to an increasingly integrated, effective and dependable phasing procedure for macromolecular structure determination.

Journal Article↗

PubMed2000

Wavelet time entropy, T wave morphology and myocardial ischemia.

Using wavelets, we computed the entropy of the signal at various frequency levels (wavelet time entropy) and, thus, find an optimal measure to differentiate normal states from ischemic ones. This new indicator is independent from the ST segment and yet provide a conclusive detection of the ischemic states.

Algorithms↗

PubMed2006

Nonparametric supervised learning by linear interpolation with maximum entropy.

Nonparametric neighborhood methods for learning entail estimation of class conditional probabilities based on relative frequencies of samples that are "near-neighbors" of a test point. We propose and explore the behavior of a learning algorithm that uses linear interpolation and the principle of maximum entropy (LIME). We consider some theoretical properties of the LIME algorithm: LIME weights have exponential form; the estimates are consistent; and the estimates are robust to additive noise. In relation to bias reduction, we show that near-neighbors contain a test point in their convex hull asymptotically. The common linear interpolation solution used for regression on grids or look-up-tables is shown to solve a related maximum entropy problem. LIME simulation results support use of the method, and performance on a pipeline integrity classification problem demonstrates that the proposed algorithm has practical value.

Algorithms↗

PubMed1985

Enthalpy-entropy relationship in drug-cholinoceptor interaction: a new approach.

The partial molal volume at infinite dilution, V2, was determined in toluene, benzene and acetonitrile for fifteen different drug molecules comprising muscarinic agonists, partial agonists and antagonists. The difference in V2 between a given drug, X, and hyoscine, expressed as (V2x - V2h) was then multiplied by the internal pressure of the holding phase (Pi approximately cohesive energy density) in order to obtain an estimate of the excess enthalpy (delta H) over hyoscine in the interaction of drug molecule X with a common cholinoceptor. As a working hypothesis, delta H for hyoscine is taken as zero, hyoscine having the lowest V2/affinity ratio of any drug in the series investigated. The corresponding change in entropy (delta S) was then calculated from the relationship: RT ln Kx = Pi(V2x - V2h) - T delta S, where Kx is the affinity constant of drug molecule X to the common cholinoceptor, obtained independently. Linear regression of Pi (V2x - V2h) congruent to delta H from the data in acetonitrile over delta S gave a satisfactory isoequilibrium plot, r2 = 0.954, slope (beta) = 231 degrees K. The present approach offers a new course for the study of the enthalpy-entropy relationship in the interaction of drug molecules in a given series with a common receptor. It could provide an alternative to the Van't Hoff procedure for the estimation of relative delta H, and is independent of the free energy of binding (delta G).

Acetonitriles↗

PubMed1982

Thermodynamic parameters of molecular complexes in aqueous solution: enthalpy-entropy compensation in a series of complexes of caffeine with beta- naphthoxyacetic acid and drug-related aromatic compounds.

Stability constants and thermodynamic parameters have been evaluated for the complexation reaction in aqueous solution of caffeine with beta-naphthoxy acetic acid. The values were higher than those previously reported for the complexation of other ligands with methyl xanthines. In nearly all aromatic ligands complexing with caffeine and theophylline for which data are available, both entropy and free energy of complexation were linearly related to the enthalpy, giving an isoequilibrium relationship. Salicylamide, sodium benzoate and cis-methyl cinnamate exhibited slight deviations on the delta G-delta H plot; the non-aromatic dehydroacetic acid showed the largest deviation. The isoequilibrium relationship was shown to be valid statistically (349-365 K, caffeine systems; 353-372 K, caffeine and theophylline systems) indicating underlying chemical causation. Thermodynamic equations are presented for analysis of the factor involved, which are attributed to a combination of substrate-ligand interactions and solvent effects. The substrate-ligand overlap area is considered as a common parameter through which the solvent and interaction forces might cooperate to give rise to linearity in the isoequilibrium relationship. The increasingly negative experimental values of the enthalpy and entropy with increase in ligand planar overlap area are discussed in relation to the underlying forces involved in the complexation.

Caffeine↗

PubMed2001

Minimum entropy deconvolution of pulse-echo signals acquired from attenuative layered media.

In this article deconvolution of ultrasonic pulse-echo data acquired from attenuative layered media is considered. The problem is divided in two subproblems: treating the sparse reflection sequence caused by the layered structure of the media and treating the frequency-dependent attenuation. The first subproblem is solved by means of joint maximum a posteriori estimation of the assumed zero mean, white, nonstationary reflection sequence and its corresponding sequence of unknown standard deviations. This approach leads to an algorithm that seeks minimum entropy solutions for the reflection sequence and therefore the algorithm serves as a novel link between the classical Wiener filter and methods for sparse or minimum entropy deconvolution. The second subproblem is solved by introducing a new signal processing-oriented, linear discrete-time model for frequency-dependent attenuation in isotropic and homogeneous media. The deconvolution algorithm is tested using simulated data and its performance for real normal incidence pulse-echo data from a composite material is also demonstrated. The results show that the algorithm, in combination with the attenuation model, yields estimates that reveal the internal structure of the composite and, thus, simplify the interpretation of the ultrasonic data.

Journal Article↗

PubMed2004

Multivariate entropy distance method for prokaryotic gene identification.

A new simple method is found for efficient and accurate identification of coding sequences in prokaryotic genome. The method employs a Shannon description of artificial language for DNA sequences. It consists in translating a DNA sequence into a pseudo-amino acid sequence with 20 fundamental words according to the universal genetic code. With an entropy-density profile (EDP), the method maps a sequence of finite length to a vector and then analyzes its position in the 20-dimensional phase space depending on its nature. It is found that the ratio of the relative distance to an averaged coding and non-coding EDP over a small number (up to one) of open reading frames (ORFs) can serve as a good coding potential. An iterative algorithm is designed for finding a set of "root" sequences using this coding potential. A multivariate entropy distance (MED) algorithm is then proposed for the identification of prokaryotic genes; it has a feature to combine the use of a coding potential and an EDP-based sequence similarity analysis. The current version of MED is unsupervised, parameter-free and simple to implement. It is demonstrated to be able to detect 95-99% genes with 10-30% of additional genes when tested against the RefSeq database of NCBI and to detect 97.5-99.8% of confirmed genes with known functions. It is also shown to be able to find a set of (functionally known) genes that are missed by other well-known gene finding algorithms. All measurements show that the MED algorithm reaches a similar performance level as the algorithms like GeneMark and Glimmer for prokaryotic gene prediction.

Algorithms↗

PubMed1983

Detection of aortic porcine valve dysfunction by maximum entropy spectral analysis.

A high-resolution method of spectral analysis, of the class generally called "maximum entropy method," was used in a study of aortic porcine valve closing sounds in 37 patients (ages 19 to 76). Spectra from 27 normal xenografts, implanted from 2 weeks to 61 months previously, were characterized by a dominant frequency peak, F1, at 89 +/- 15 Hz (mean +/- SD), with a lower amplitude peak, F2, at 154 +/- 25 Hz. Eight of nine patients with aortic porcine valve dysfunction were proved surgically to have leaflet degeneration or infection and had either F1 (139 +/- 54 Hz) and/or F2 (195 +/- 74 Hz) significantly higher than normal (p less than .001). In two patients with paravalvar leak but no leaflet abnormality, F1 and F2 were in the normal range. Estimation of F1 and F2 was highly reproducible and was unaffected by duration of implant up to 5 years. Spectral analysis of aortic porcine valve closing sounds by the maximum entropy method may be useful for detection of intrinsic xenograft dysfunction.

Adult↗

PubMed1998

Probability Density Estimation Using Entropy Maximization.

We propose a method for estimating probability density functions and conditional density functions by training on data produced by such distributions. The algorithm employs new stochastic variables that amount to coding of the input, using a principle of entropy maximization. It is shown to be closely related to the maximum likelihood approach. The encoding step of the algorithm provides an estimate of the probability distribution. The decoding step serves as a generative mode, producing an ensemble of data with the desired distribution. The algorithm is readily implemented by neural networks, using stochastic gradient ascent to achieve entropy maximization.

Journal Article↗

PubMed1988

Ultrasound attenuation estimation in soft tissue using the entropy difference of pulsed echoes between two adjacent envelope segments.

In this paper, we describe a new method of estimating the ultrasound attenuation coefficient of soft tissue based on the entropy difference between two adjacent envelope segments of narrowband ultrasound pulse echoes. Assuming uniform attenuation in the region under investigation, the attenuation is estimated by minimizing the difference of entropies in the two segments as the attenuation is continuously compensated for. The simulation and experimental results with a tissue equivalent (TE) phantom show that the proposed estimator is quite robust to receiver noise and requires considerably less data length as compared with conventional methods. At present, an experimental result with a TE phantom shows that a resolution of 10 mm x 10 mm can be attained with echoes from a depth of 3 cm using a 2.25 MHz transducer.

Computer Simulation↗

PubMed1993

Quantification of alpha rhythm desynchronization using the acceleration spectrum entropy of the EEG.

Desynchronization of the alpha rhythm is usually quantified as a loss of power in the alpha band. In this study we describe an alternative approach where desynchronization is characterized as an increase in the irregularity of the EEG signal. A new measure, the acceleration spectrum entropy (ASE), is used to quantify EEG irregularity. The ASE is the normalized information entropy of the amplitude spectrum of the second derivative of a time series. The ASE is a measure of the randomness or irregularity of a time series and ranges from 0 (straight line) to 100 (discrete random signal). The ASE was calculated for short EEG epochs under eyes closed and eyes open conditions in 15 subjects. There was a significant increase in ASE in the eyes open condition, reflecting an increase in EEG complexity. The ASE is a sensitive measure of EEG desynchronization.

Alpha Rhythm↗

PubMed2005

Polarization degree of optical waves with non-Gaussian probability density functions: Kullback relative entropy-based approach.

The definition of degree of polarization for non-Gaussian partially polarized light is analyzed. A general framework based on the Kullback relative entropy is developed, and properties that enlighten the physical meaning of the degree of polarization are established. In particular, it is shown how the degree of polarization is related to the measure of proximity between probability density functions and to the measure of disorder provided by the Shannon entropy.

Journal Article↗

PubMed2006

Transformation of time-resolved spectra to lifetime-resolved spectra by maximum entropy inversion of the laplace transform.

We present a method for the analysis of time-resolved spectroscopic data following first-order kinetics. The time traces at all the available spectroscopic channels (e.g., wavelength or wavenumber) are inverse Laplace transformed. The transformation is stabilized by the maximum entropy method generalized for solutions without sign-restriction. In this way, time-resolved spectra can be converted to lifetime-resolved spectra, where bands appear at coordinates corresponding to their spectroscopic maxima and time constant of appearance (negative amplitude) or disappearance (positive amplitude). From the lifetime-resolved spectra, the number of exponentially decaying components, their time constants, and their decay-associated spectra are readily available. Moreover, since bands are spread in two dimensions extra band-resolution is possible. We named this method of transforming time-resolved spectra into lifetime-resolved spectra multi-spectroscopic channel maximum entropy inversion of the Laplace transform (M-MaxEnt-iLT). The basis of M-MaxEnt-iLT is presented in detail and its properties and limitations are thoroughly discussed. We also show how the combination of M-MaxEnt-iLT with spectral smoothing or deconvolution can improve the appearance and/or band resolution of the obtained lifetime-resolved spectra.

Journal Article↗

PubMed1993

Twenty-four-hour power spectral analysis by maximum entropy method of blood pressure in primary hyperaldosteronism.

In the present study we estimated the periodic profiles and variance structure of systolic blood pressure, diastolic blood pressure, heart rate and mean arterial pressure by using an autoregressive model of power spectrum, Maximum Entropy Method (MEM) in 8 patients with primary aldosteronism, during long-term therapy with nicardipine slow release. The four blood pressure variables were measured at 30-min intervals, using a noninvasive device (Spacelabs 90202) in 8 hypertensive patients of whom 6 with idiopathic aldosteronism (IHA) and 2 with dexamethasone-suppressible aldosteronism (DSH), before and after 24 weeks of 80 mg nicardipine daily. Blood pressure data were processed by MEM and spectral profiles were obtained. During nicardipine therapy all patients showed a significant decrease of 24-h ambulatory blood pressure values (p < 0.01). Before therapy, spectrum analysis by MEM indicated the presence of high frequency distribution of peaks for SBP, DBP, MAP and HR. The MEM power spectrum showed an increase in amplitude of sharp peaks of systolic, diastolic, MAP and heart rate in all patients after therapy at 24 h corresponding to the circadian rhythm blood pressure. Furthermore, the trend of these variables synchronized themselves in the same period after 24 weeks of nicardipine therapy, with spectral patterns of blood pressure similar to those of normotensive subjects. This chronobiologic approach, by Maximum Entropy Method, may be used as an alternative statistical analysis to search for possible rhythmic behavior of ambulatory blood pressure data before and after pharmacological treatment in secondary hypertensive patients.

Activity Cycles↗

PubMed1976

Mechanoelectrical transduction in hyaluronic acid salt solution is an entropy-driven process.

An electrical potential develops between the ends of a column of hyaluronic salt solution displaced from a resting position by gentle pressure. A previous study demonstrated that such displacement changes the optical rotary dispersion properties of the salt, either increasing the rotation in the direction already shown by the salt before displacement or changing and increasing the rotation in the opposite direction, depending on the direction of the displacement. The present investigation demonstrates that the loss of bound water component across a membrane separating the solution and water is corelated with the extent of the column displacement. In addition, a return of the column to the position before displacement is correlated with a return of the water component across the membrane-but not at the same rate as the exodus. The data seem consistent with the hypothesis that the hyaluronic acid salt, when strained, adopts a less entropic configuration, releasing bound water and thus increasing the entropy of water component. This change in the distribution of entropy is reversible; i.e., Eddington's "time's arrow" is reversible with respect to the water component of the solution.

Female↗