Maximum-information method

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand. 

We briefly repeat now the essential parts of the maximum entropy method for details we refer to the literature [167-169]. We seek to obtain information on the dynamics of the internal degree of freedom of the model from PIMC simulations. The solution of this problem is not... 

Regardless of which assessment method you choose, assessors and auditors should take detailed notes using a common format to help capture maximum information in a consistent manner. Forms for questionnaires, topical outlines, and audit protocols (as shown in the exhibits) can perform double duty by providing interviewers with a format for notes as well as reporting. 

While publications on fluorescence lifetime imaging microscopy (FLIM) have been relatively evenly divided between time and frequency domain methods, a majority of the 10 most highly cited papers using FLIM have taken advantage of the frequency domain method [1, 2-9]. Both techniques have confronted similar challenges as they have developed and, as such, common themes may be found in both approaches to FLIM. One of the most important criteria is to retrieve the maximum information out of a FLIM... 

The management of an analytical chemistry laboratory involves a number of different but related operations. Analysts will be concerned with the development and routine application of analytical methods under optimum conditions. Instruments have to be set up to operate efficiently, reproducibly and reliably, sometimes over long periods and for a variety of analyses. Results will need to be recorded and presented so that the maximum information may be extracted from them. Repetitive analysis under identical conditions is often required, for instance, in quality assurance programmes. Hence a large number of results will need to be collated and interpreted so that conclusions may be drawn from their overall pattern. The progress of samples through a laboratory needs to be logged and results presented, stored, transmitted and retrieved in an ordered manner. Computers and microprocessors can contribute to these operations in a variety of ways. 

The first semi-high-throughput automated system to dispense crystallization trials of less than 1 jl1 was designed in 1990 to deliver batch trials imder oil (Chayen et ah, 1990). The method was named microbatch to define a microscale batch experiment. It was designed to obtain maximum information on the molecule to be crystallized while using minimal amounts of sample. In order to prevent the evaporation of such small volumes, the trials are dispensed and incubated under low density (0.87 g/cm ) paraffin oil (Fig. 3.2). The crystallization drops remain under the oil since the aqueous drops are denser than the paraffin oil. 

The maximum entropy method (MEM) is an information-theory-based technique that was first developed in the field of radioastronomy to enhance the information obtained from noisy data (Gull and Daniell 1978). The theory is based on the same equations that are the foundation of statistical thermodynamics. Both the statistical entropy and the information entropy deal with the most probable distribution. In the case of statistical thermodynamics, this is the distribution of the particles over position and momentum space ( phase space ), while in the case of information theory, the distribution of numerical quantities over the ensemble of pixels is considered. 

The maximum entropy method was first introduced into crystallography by Collins (1982), who, based on Eq. (5.47), expressed the information entropy of the electron density distribution as a sum over M grid points in the unit cell, using... 

Future development efforts in source characterization should 1) develop new source sampling methods including tethered balloon and ground based sampling 2) standardize data reporting and management procedures and store validated data in a central data base and 3) create a chemical component analysis protocol to obtain maximum information from each source test. 

Fig. 8.11 (c), and there is not even one full relevant oscillation in the frequency domain. But the maximum entropy method enables useful information to be obtained even from such poorly resolved data as this, and in the time-interval domain in Fig. 8.11(d) the MEM transform ofln S (/) — In So(/) has a pronounced peak from which 2d/v for the cell at that point can be determined. The time separation is about 2 ns, corresponding to a thickness of less than 2 pm. This may be a world record for acoustic distance resolution in this way. 

Several examples of the extraction of decay information from OLNO data are available, eg. for gamma decay in [vAN85b], and for alpha decay in [WOU85]. As a singles counting method OLNO is efficient at obtaining maximum information from weaker sources. 

In the Maximum Dissimilarity (MD) selection method described by Lajiness [40] the first compound is selected at random and subsequent compounds are then chosen iteratively, such that the distance to the nearest of the compounds already chosen is a maximum. This method is known as MaxMin. In this study, the compounds were represented by COUSIN 2-D fragment-based bitstrings. Polinsky et al. [41] use a similar algorithm in the LiBrain system. In this case, the molecules are represented by a feature vector that contains information about the following affinity types—aliphatic hydrophobic, aromatic hydrophobic, basic, acidic, hydrogen bond donor, hydrogen bond acceptor and polarizable heteroatom. 

Unfortunately, there is great scope for confusion, as two distinct techniques include the phrase maximum entropy in their names. The first technique, due to Burg,135 uses the autocorrelation coefficients of the time series signal, and is effectively an alternative means of calculating linear prediction coefficients. It has become known as the maximum-entropy method (MEM). The second technique, which is more directly rooted in information theory, estimates a spectrum with the maximum entropy (i.e. assumes the least about its form) consistent with the measured FID. This second technique has become known as maximum-entropy reconstruction (MaxEnt). The two methods will be discussed only briefly here. Further details can be found in references 24, 99, 136 and 137. Note that Laue et a/.136 describe the MaxEnt technique although they refer to it as MEM. 

More detailed information can be obtained from noise data analyzed in the frequency domain. Both -> Fourier transformation (FFT) and the Maximum Entropy Method (MEM) have been used to obtain the power spectral density (PSD) of the current and potential noise data [iv]. An advantage of the MEM is that it gives smooth curves, rather than the noisy spectra obtained with the Fourier transform. Taking the square root of the ratio of the PSD of the potential noise to that of the current noise generates the noise impedance spectrum, ZN(f), equivalent to the impedance spectrum obtained by conventional - electrochemical impedance spectroscopy (EIS) for the same frequency bandwidth. The noise impedance can be interpreted using methods common to EIS. A critical comparison of the FFT and MEM methods has been published [iv]. 

Statistical methods are based on the single concept of variability. It is through this fundamental concept that a basis is determined for design of experiments and analysis of data. Full utilization of this concept makes it possible to derive maximum information from a given set of data and to minimize the amount of data necessary to derive specific information. 

The maximum entropy method (MEM) is based on the philosophy of using a number of trial spectra generated by the computer to fit the observed FID by a least squares criterion. Because noise is present, there may be a number of spectra that provide a reasonably good fit, and the distinction is made within the computer program by looking for the one with the maximum entropy as defined in information theory, which means the one with the minimum information content. This criterion ensures that no extraneous information (e.g., additional spectral... 

Nuclear Magnetic Resonance Concepts and Methods by Daniel Canet42 contains particularly clear presentations on techniques and data processing for Fourier transform NMR and related methods. Articles in the Encyclopedia of NMR on Fourier Transform Spectroscopy,43 Fourier Transform and Linear Prediction Methods, 39 and Maximum Entropy Reconstruction44 are also very informative. A Handbook of NMR includes a very clear description of the maximum entropy method and its limitations.19... 

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