Model-free analysis in reverse

Figure 7.3: A schematic of the new model-free optimisation protocol. Initially models tm0 to tm9 (7.23.0-7.23.9) of the set $\mathfrak{T}_i$ for each spin system i are optimised, model elimination used to remove failed models, and AIC model selection used to pick the best model. Once all the $\mathfrak{T}_i$ have been determined for the system the the local τm parameter is removed, the model-free parameters are held fixed, and the global diffusion parameters of $\mathfrak{D}$ are optimised. These parameters are used as input for the central part of the schematic which follows the same procedure as that of Figure 7.2. Convergence is however precisely defined as identical models $\mathfrak{S}$, identical χ2 values, and identical parameters θ between two iterations. The universal solution $\widehat{{\mathfrak{U}}}$, the best description of the dynamics of the molecule, is determined using AIC model selection to select between the local τm models for all spins, the sphere, oblate spheroid, prolate spheroid, ellipsoid, and possibly hybrid models whereby multiple diffusion tensors have been applied to different parts of the molecule.
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A different approach was proposed in d'Auvergne and Gooley (2008c) for finding the universal solution $\widehat{{\mathfrak{U}}}$ of the extremely complex, convoluted model-free optimisation and modelling problem (d'Auvergne and Gooley, 2007), defined as

$\displaystyle \widehat{{\mathfrak{U}}}$ = $\displaystyle \hat{\theta}$$\displaystyle \left\{\vphantom{ \mathfrak{S}: \min_{\hat\theta \in \mathfrak{U}} \Delta_{\scriptscriptstyle \text{K-L}}(\hat\theta) }\right.$$\displaystyle \mathfrak{S}: \min_{\hat\theta \in \mathfrak{U}} \Delta_{\scriptscriptstyle \text{K-L}}(\hat\theta) $$\displaystyle \left.\vphantom{ \mathfrak{S}: \min_{\hat\theta \in \mathfrak{U}} \Delta_{\scriptscriptstyle \text{K-L}}(\hat\theta) }\right\}$,    s.t. $\displaystyle \hat{\theta}$ = argmin$\displaystyle \left\{\vphantom{\chi^2(\theta): \theta \in \mathfrak{S}}\right.$χ2(θ) : θ$\displaystyle \mathfrak{S}$$\displaystyle \left.\vphantom{\chi^2(\theta): \theta \in \mathfrak{S}}\right\}$. (7.36)

This notation says that the minimised parameter vector within the space $\mathfrak{S}$ which minimises the common Kullback-Leibler discrepancy ΔK-L is selected from the universal set $\mathfrak{U}$ as the universal solution $\widehat{{\mathfrak{U}}}$. The discrepancy of Kullback and Leibler (1951) is a measure of how well the model fits the data, in this case how well the global model $\mathfrak{S}$ of the diffusion tensor together with the model-free models of all residues fits the relaxation data. This selection is subject to the condition that $\hat{\theta}$ is the argument or specific parameter vector which minimises the chi-squared function χ2(θ) such that θ is an element of the space $\mathfrak{S}$. Whereas the minimisation of the continuous chi-squared function within the single space $\mathfrak{S}$ belongs to the mathematical field of optimisation (Nocedal and Wright, 1999), the selection of the universe $\mathfrak{S}$ which minimises the discrepancy belongs to the statistical field of model selection (d'Auvergne and Gooley, 2003; Schwarz, 1978; Akaike, 1973; Zucchini, 2000; Linhart and Zucchini, 1986).

This new model-free optimisation protocol incorporates the ideas of the local τm model-free model (Barbato et al., 1992; Schurr et al., 1994) and the optimisation of the diffusion tensor using information from these models, analogously to the linear least-squares fitting of the quadric model (Brüschweiler et al., 1995; Lee et al., 1997). The protocol also follows the lead of the model-free optimisation protocol presented in Butterwick et al. (2004) whereby the diffusion seeded paradigm was reversed. Rather than starting with an initial estimation of the global diffusion tensor from the set $\mathfrak{D}$ the protocol starts with the model-free parameters from $\mathfrak{F}$.

The first step of the Butterwick et al. (2004) protocol is the reduced spectral density mapping of Farrow et al. (1995). As Rex has been eliminated from the analysis, three model-free models corresponding to tm1, tm2, and tm5 (Models 7.23.1, 7.23.2, and 7.23.5 on page [*]) are employed. The model-free parameters are optimised using the reduced spectral density values and the best model is selected using F-tests. The spherical, spheroidal, and ellipsoidal diffusion tensors are obtained by linear least-squares fitting of the quadric model of Equation (7.36) using the local τm values (Brüschweiler et al., 1995; Lee et al., 1997). The best diffusion model is selected via F-tests and refined by iterative elimination of spins systems with high chi-squared values. This tensor is used to calculate local τm values for each spin system, approximating the multiexponential sum of the Brownian rotational diffusion correlation function with a single exponential, using the quadric model of Equation (7.36). In the final step of the protocol these τm values are fixed and m1, m2, and m5 (Models 7.22.1, 7.22.2, and 7.22.5 on page [*]) are optimised and the best model-free model selected using F-tests.

The new model-free protocol built into relax utilises the core foundation of the Butterwick et al. (2004) protocol yet its divergent implementation is designed to solve the universal equation of d'Auvergne and Gooley (2007) to find $\widehat{{\mathfrak{U}}}$ (Equation 7.37). Models tm0 to tm9 (7.23.0-7.23.9 on page [*]) in which no global diffusion parameters exist are employed to significantly collapse the complexity of the problem. Model-free minimisation (d'Auvergne and Gooley, 2008b), model elimination (d'Auvergne and Gooley, 2006), and then AIC model selection (Akaike, 1973; d'Auvergne and Gooley, 2003) can be carried out in the absence of the influence of global parameters. By removing the local τm parameter and holding the model-free parameter values constant these models can then be used to optimise the diffusion parameters of $\mathfrak{D}$. Model-free optimisation, model elimination, AIC model selection, and optimisation of the global model $\mathfrak{S}$ is iterated until convergence. The iterations allow for sliding between different universes $\mathfrak{S}$ to enable the collapse of model complexity, to refine the diffusion tensor, and to find the solution within the universal set $\mathfrak{U}$. The last step is the AIC model selection between the different diffusion models. Because the AIC criterion approximates the Kullback-Leibler discrepancy (Kullback and Leibler, 1951), central to the universal solution of Equation (7.37), it was chosen for all three model selection steps over BIC model selection (Chen et al., 2004; d'Auvergne and Gooley, 2003; Schwarz, 1978). The new protocol avoids the problem of under-fitting whereby artificial motions appear, avoids the problems involved in finding the initial diffusion tensor within $\mathfrak{D}$, and avoids the problem of hidden internal nanosecond motions and the inability to slide between universes to get to $\widehat{{\mathfrak{U}}}$ (see d'Auvergne and Gooley (2007) for more details). The full protocol is summarised in Figure 7.3.

The relax user manual (PDF), created 2020-08-26.