When analysing relaxation and/or diffusion data acquired with various Nuclear Magnetic Resonance (NMR) techniques followed by the use of the Inverse Laplace Transform (ILT) [1–9], the smoothing of the data is essential to produce distributions of relaxation times and/or diffusion coefficients. Even though there has been some work in non-uniform smoothing the most common way to smooth the data is to apply a uniform smoothing on the data, i.e. a constant smoothing throughout the processing irrespective of the content of the data. Consequently, there will be a different broadening of the peaks depending on the peak position or fraction of the data in which the peak contributes to an attenuating signal [10]. Alternatively, discrete methods have been developed that fits the data to a limited number of components [11–13]. However, these methods suffer from the lack of information on the distributivity of the dataset.

Recently a method was proposed that combines a discrete processing of the dataset together with the ILT, namely the Anahess distribution [10]. Here the number of components in the solution is limited to a minimum, which makes it possible to divide the solution into sub-groups that can be transformed and processed as superimposed datasets using the ILT with conditions set by the discrete solution. This approach has shown to reproduce synthetic distributions better than the ILT only. An extension of this method has now been developed and will be presented here. It aims at finding the expectation values for the fitted discrete components and the corresponding distribution. This should provide a measure allowing to evaluate the quality of the fit. In short, the procedure is as follows: After fitting the exponentially decaying data to a limited number of components according to the Bayesian Information stop Criterion (BIC) [14], a set of residual data or noise from the fit is produced. The residuals are then rearranged with respect to position. That is, the residuals or a group of such are interchanged in a random way so that new noise data is produced but with same expectation value and standard deviation as the original set of residuals. A new exponentially decaying data set can then be produced from the regrouped noise and the fitted components are determined from the discrete fitting procedure. The new raw data set is then analysed using the discrete Anahess, resulting in a new set of fitted components. If there is an impact of noise on the fitted components, the values will vary due to the different noise present. This procedure is repeated until enough data with rearranged residuals are produced to find an expectation value and a standard deviation for the fitted components.

In the following we will recapture the combination of the discrete Anahess approach with ILT [10] and provide the method for determining the expectation values of the fitted components and distributions.

1.1 The Anahess approach

In experimental data, the noise, ε, is superimposed on the exponentially decaying signal. Data arising from relaxation and/or diffusion NMR experiments can be described by a multi-exponential decaying signal S(t)

$$\:\varvec{S}\left({\varvec{t}}_{\varvec{i}}\right)={\sum\:}_{\varvec{k}}A\left({\varvec{T}}_{\varvec{k}}\right)\varvec{exp}\left(-\frac{{\varvec{t}}_{\varvec{i}}}{{\varvec{T}}_{\varvec{k}}}\right)+{\varvec{\epsilon\:}}_{\varvec{i}}$$

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Here *A(T**k**)* is the distribution of intensities to be fitted to the corresponding *T**k*, which could be the corresponding relaxation time or the diffusion coefficient. The most common way to fit the equation above to experimental data is to use the Inverse Laplace Transform initially developed by Provencher [1–3]. Then, a predefined grid of a fixed number of points are defined, on which the solution (*A(T**k**)*) have to be found. Another approach is to use a discrete method where the number of components is limited [11]. In this work we apply the discrete Anahess approach, where the number of points (components) in the solution is minimized according to a Bayesian Information criterion and the points are allowed to move anywhere in the space of solutions. As in the ILT routine, a function involving the sum squared relationship is to be minimized [11, 15]

$$\:S{S}_{res}\hspace{0.33em}={\sum\:}_{j}^{NX}\{{a}_{0}+{\sum\:}_{p}^{NCO}{a}_{p}{e}^{(-(1/{T}_{p}\left){t}_{j}\right)}-{R}_{j}{\}}^{2}$$

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Where

$$\:{R}_{j}={a}_{0}+{\sum\:}_{p}^{NCO}{a}_{p}{e}^{(-(1/{T}_{p}\left){t}_{j}\right)}+{\epsilon\:}_{j}$$

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*a* 0 is a baseline offset which may be positive or negative. The data matrix *R* have the corresponding number of data points *NX*. The parameters *a**p*, *T**p* are thus characteristic properties of the component with index *p* out of all the *NCO* components. As *SS**re*s will decrease with increasing number of fitted components, a stop criterion is needed. The Bayesian information criterion (*BIC*) [14] is such a criterion. Let *n* be the number of observed data points, let *p* be the number of free model parameters, and *SS**res* be the sum of squared residuals. Then if the residuals are normally distributed, the *BIC* has the form:

$$\:BIC\hspace{0.33em}\hspace{0.33em}=\hspace{0.33em}n{ln}(\frac{s{s}_{res}}{n})+p{ln}(n)$$

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In this equation, a good model fit gives a low first term while few model parameters give a low second term. When comparing a set of models, the model with the minimal *BIC* value is selected.

The discrete solution fits to a relatively small number of components that provides a satisfactorily fit of the raw data. This is due to the discreteness of the fitting routine when using the BIC as a stop criterion [14]. As most of data reflects continuous distributions of components, a method for probing the distributivity using the discrete Anahess results has been developed. This is done by applying the ILT where the results from the discrete Anahess fit is fed into the ILT as initial and restricting conditions [8]. Because of the limited number of components in the Anahess, one may group various regions in the solution and provide a superimposed fit of each group. Consequently, prior to the use of the ILT the data is grouped and then transformed in *t* according to the following equation [10]

$$\:{t}^{k}\to\:\:\:t\bullet\:\left(\frac{5{T}_{k}}{{t}_{max}}\right)\:\:$$

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where *t* is the original observation time, *T**k* is the result from the Anahess fit for component *k*, and *t**max* is the longest observation time in the data set. With this approach one may process data with the discrete Anahess method and probe the distributivity.

## 1.2 The method for determining the expectation value and its standard deviation

When the exponentially decaying data are fitted using the Discrete Anahess [11, 15], a resulting set of residuals is produced. These residuals are the difference between the fitted and the original data. As the noise is the crucial part that affects the result of the fitting procedure, both using discrete and continuous fittings, we here propose to produce new raw data to be subjected for processing through redistributing the noise as follows; divide the residual noise into several compartments for both 1- and 2-dimensional noise, as shown in Fig. 1. The compartments should be set so small that when moving them around in a random way such that the end product is a different noise dataset but with the same expectation value and standard deviation.

$$\:{\varvec{n}\varvec{o}\varvec{i}\varvec{s}\varvec{e}}^{\varvec{N}\varvec{E}\varvec{W}}\left({\varvec{c}\varvec{o}\varvec{m}\varvec{p}\varvec{a}\varvec{r}\varvec{t}\varvec{m}\varvec{e}\varvec{n}\varvec{t}}_{\varvec{i}}\right)={\varvec{n}\varvec{o}\varvec{i}\varvec{s}\varvec{e}}^{\varvec{O}\varvec{R}\varvec{I}\varvec{G}\varvec{I}\varvec{N}\varvec{A}\varvec{L}}\left({\varvec{c}\varvec{o}\varvec{m}\varvec{p}\varvec{a}\varvec{r}\varvec{t}\varvec{m}\varvec{e}\varvec{n}\varvec{t}}_{\varvec{j}}\right)$$

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where *j* is a random number within the number of compartments. Using the fitted components and intensities from the Discrete Anahess method one may then perform a Laplace Transform and produce new data sets where the noise is different due to the random interchanging of the compartments in the original noise data.

$$\:{\varvec{S}}^{\varvec{N}\varvec{E}\varvec{W}}\left({\varvec{t}}_{\varvec{i}}\right)={\sum\:}_{\varvec{p}}{a}_{p}\varvec{exp}\left(-\frac{{\varvec{t}}_{\varvec{i}}}{{\varvec{T}}_{\varvec{p}}}\right)+{\varvec{n}\varvec{o}\varvec{i}\varvec{s}\varvec{e}}^{\varvec{N}\varvec{E}\varvec{W}}$$

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These datasets can then be analysed using the discrete Anahess approach and a variation in the fitted intensities and new relaxation times and/or diffusion coefficients will be found. When this procedure has been repeated until the effect of the noise has been probed properly, it will result in an expectation value for the various components, and these values can then be used as initial and restricting values to probe the distributivity of the original dataset.

This approach assumes that there is no dependency of the noise as a function of observation time and/or applied gradient strength. That is, the discrete Anahess fit returns a Gaussian distribution of residuals that has an expectation value of 0 and the same insignificant skew [16]. Also, this method provides a way to check if the fit to the original raw data is a stable one. That is, the minimum BIC number is achieved at the same number of components and the variation in the expectation values is acceptable.