Spectrum distribution entropy

A spectrum is built up from at least a million lifetime measurements from individual positrons. Standard computer codes are available for the decomposition of the spectra including POSITRONFIT in which a least squares fit is used to fit a model spectrum with a given number of decay components to the observed spectrum. Maximum entropy techniques have also been used to determine the most probable underlying distribution of trap lifetimes. As the number of different traps increases, interpretation becomes increasingly difficult. Several research groups have published positron annihilation lifetime data on irradiated RPV steels (see Section 9.11.1). 

The central engine of this data workflow is the process of spectral deconvolution. During spectral deconvolution, sets of multiply charged ions associated with particular proteins are reduced to a simplified spectrum representing the neutral mass forms of those proteins. Our laboratory makes use of a maximum entropy-based approach to spectral deconvolution (Ferrige et al., 1992a and b) that attempts to identify the most likely distribution of neutral masses that accounts for all data within the m/z mass spectrum. With this approach, quantitative peak intensity information is retained from the source spectrum, and meaningful intensity differences can be obtained by comparison of LC/MS runs acquired and processed under similar conditions. 

The choice of the preferred solution within the feasible set can be achieved by maximizing some function F[a(r)] of the spectrum that introduces the fewest artefacts into the distribution. It has been proved that only the Shannon-Jaynes entropy S will give the least correlated solution 1. All other maximization (or regularization) functions introduce correlations into the solution not demanded by the data. The function S is defined as... 

Application of this technique to measurements of the spectral distribution of tight scattered from a pure SF fluid at its critical point was present by Ford and Benedek The scattering is produced by entropy fluctuations which decay very slowly in the critical region. Therefore the spectrum of the scattered light is extremely narrow (10 - lO cps) and can only be observed by this light beating technique 240a)... 

A spectrum or chromatogram can be considered as a probability distribution. If the data are sampled at 1000 different points, then the intensity at each datapoint is a probability. For a flat spectrum, the intensity at each point in the spectrum equals 0.001, so die entropy is given by... 

Figure 3. Statistical analysis of the intensity distribution of the high-resolution spectrum of C2 H2 at about 26,500 cm 1, including nearly 4000 lines. (Adapted from Ref. 55.) The solid line is the maximum entropy distribution (cf. Ref. 56) given by Eq. (3) with v = 3.2.

Experimental and computational results often do deviate from the distributions of maximal entropy subject only to a given total strength. Consider the following simple modification. The spectrum can be regarded as the set of expectation values of the state Pi... 

The probability of an entire spectrum will be determined by the procedure of maximum entropy.49 The purpose is to show that under the constraints considered so far, including the various sum rules, the distribution of intensities of different lines are independent of one another. When considered as a function of the ratio of the actual intensity to its mean, local value, all transitions have the same distribution, which is -square in the general case. Hence, one can group intensities together into bins, and consider the distribution of intensity irrespective of line position provided one takes proper account of the variation of the envelope with energy. The technical backup for these statements and the specific conditions under which they will fail are the subjects of this subsection. 

The amplitudes of the different lines in the spectrum can be regarded as the components of the optically prepared bright state in the basis of the eigenstates of the system, cf. Eqs. (2)-(3). Already in Sec. II we had several occasions to note that the bright state behaves not unlike a random vector. One can therefore ask what the spectrum will be if we make the approximation that these components are truly random. This requires us to specify, in a technical sense, what we mean by random. This is where entropy comes in. By random we will mean that the distribution of the amplitudes be as uniform as possible and, as such, be a distribution of maximal entropy. 

The case of a truly random bright state is likely to be a limiting one. In reality, the dynamics do constrain the time evolution of the bright state. What one therefore needs is a prescription for dealing with situations which are not fully random. Entropy continues to provide a convenient tool because rather than look for that distribution which is as random as possible one can instead specify that distribution which is as uniform as it can be subject to the input from the dynamics. In technical terms, we seek that distribution of amplitudes whose entropy is maximal, but the maximum is subject to auxiliary conditions. These additional conditions (or constraints) are to be provided by the dynamics. The result is a spectrum that is consistent with the given dynamical input and is otherwise the result of a maximally uniform set of amplitudes. Section IV.A.2 provides a more technical version of these considerations. 

The LSQ method finds a solution that gives the best fit to the expraimental data. However, it suffers from the problem that the noise in the experimental data are carried through to the final result. To minimize the noise in the deconvoluted distribution, a maximum entropy method (MEM) can be applied [40], Figure 7.9 shows the isotope deconvolution of the experimental spectrum of a deuterated peptide with both LSQ method and MEM method. While LSQ gives a noisy distribution with some negative abundances, MEM produces a smoother and more realistic distribution of the trimodal isotope distribution. Similar performances were also obtained by a fast Fourier transform method [6] and an LI regularization method [8] with significantly improved computation speed. 

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