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Procrustes Problems (Oxford Statistical Science Series)

In contrast there are many popular presentations of the inherent 'weirdness' of the quantum world that are light on jargon and contain no mathematics. Together these presentations serve to create the impression that there are two theories - the 'serious, one and the 'wierd' one. Baggott successfully bridges the gulf between these presentations by grounding the discussion of the theory's profound problems directly in its mathematical formalism in a way that under-graduate students and interested individuals can follow.

Informed lay readers the proverbial "average New Scientist reader" - they will have read many popular treatments but are interested to penetrate more of the detail. Academic scientists and philosophers - because this is their field, the book represents a possible course text or, if this is not their field, because they are also curious. Hardback Publication date: 18 December pages, none, mm x mm Series: Oxford Lecture Series in Mathematics and Its Applications Concise and clear coverage of analysis in metric spaces Based on lecture notes from the Scuola Normale Supplemented with exercises of varying difficulty Coverage of classical and recent results Description ''This would be an excellent basis for a one-semester graduate course.

This is very much a good time for this book. The selected topics provide a good start in the field.

Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers. Some application areas, future and conclusion A1. Configurations A2. Rotations and reflections A3. Orthogonal projections A4. Oblique axes A5. A minimisation problem A6.

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See All Customer Reviews. Shop Textbooks. Add to Wishlist. USD Sign in to Purchase Instantly. Temporarily Out of Stock Online Please check back later for updated availability. Overview Procrustean methods are used to transform one set of data to represent another set of data as closely as possible.

About the Author John C. Symmetric matrix products References. An alignment of three proteins with missing residues.

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Both indicator matrices, though redundant, are useful for simplifying the presentation of the solutions. In our probabilistic model, each structure X i is considered to be a randomly rotated and translated Gaussian perturbation of a mean structure M :.

For simplicity, we assume that the variance about each atom is spatially spherical. Extensions to higher and lower dimensions are trivial e. For the non-isotropic solution, the covariance matrix is diagonal, with all offdiagonal, covariance elements constrained to 0. For the isotropic solution, which corresponds to least squares, the covariance matrix is constrained to be diagonal and to have identical diagonal elements i. The full joint PDF for our likelihood superposition problem is thus obtained from a multivariate matrix normal distribution Dutilleul, ; Gupta and Nagar, corresponding to the perturbation model described by equation 6.

The Jacobian for the transformation from Y i to X i is the product of the Jacobians for the translation and rotation, which are each simply unity [see Chapter 1. Detailed background and justification of this likelihood treatment can be found elsewhere Theobald and Wuttke, a , b , Note that columns of the alignment that contain all gaps except for one lone sequence have no influence on the likelihood and should be excluded from the maximization calculations.

For the E-step of the EM algorithm, one first finds the expected log likelihood, where the expectation is over the missing data conditional on the observed data and current estimates of the other parameters. In practice, these conditional expectations can be cast in terms of the current parameter estimates, and hence the expectations can be combined with other terms in the log likelihood containing those parameters. For the M-step, the expected log likelihood is maximized over a given parameter by taking the derivative as explained above.

In the following sections, the conditional ML estimates are provided for both the complete data and missing data cases. Detailed derivations are provided in the Supplementary Material. Where is the estimate of the translation:. The optimal rotations are calculated using a singular value decomposition SVD. The ML rotations are estimated by.

The mean structure is estimated as the arithmetic average of the optimally translated and rotated structures. The following equations for the covariance matrix estimates are only valid when the translations are known Theobald and Wuttke, a. In general, this is not the case, and thus a constrained regularized estimator of the covariance matrix is necessary. For instance, the estimates given below may be modified to give a hierarchical estimator as shown in equation 6 of Theobald and Wuttke, a or equation 10 of Theobald and Wuttke, The estimators of the isotropic variance are already adequately constrained and do not need to be adjusted.

To simplify the following formulae, we first define the matrix D i :. The unconstrained estimate of the diagonal, non-isotropic covariance matrix is. The Hadamard operation simply sets all offdiagonal elements of the covariance matrix to 0. Note that in equation 24 , one needs to only calculate the diagonal elements. The ML estimate of the non-isotropic, diagonal covariance matrix is.

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The summation terms may be calculated easily by noting that. The simultaneous solution of the optimal parameters must be solved numerically, as each of the unknown parameters is a function of some of the others. Our iterative algorithm is an extension of similar algorithms proposed previously, and it is based on nested rounds of EM cycles and conditional maximization Dempster et al.

Initialize : Set for all i. Estimate the mean structure by embedding the average of the distance matrices, including gaps, for each structure Crippen and Havel, ; Lele, ; Lele and Richtsmeier, Rather than embedding, one may simply choose one of the structures preferably with the fewest gaps to serve as the mean for the first iteration, setting missing coordinates to zeros convergence may be hindered in cases with a large fraction of gaps. Translate : Translate i.

Rotate : Calculate each rotation and rotate each translated structure:. Estimate the mean : Recalculate the average structure. Return to Step 2 and loop until convergence.

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In the isotropic case, this step can be omitted if desired. When superpositioning homologous proteins with different sequences or identical proteins with missing portions, a sequence alignment must be provided to THESEUS.

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THESEUS will superposition any number of structures within the limits set by the operating system and memory capability. Via command line options, users can choose to superposition assuming an isotropic covariance matrix i. Specified alignment columns can also be excluded from the calculation. On modern personal desktop computers, convergence is usually very fast within seconds for even very large problems. To demonstrate the advantages of the method, we constructed three test sets of structures based on four NMR models of a zinc finger domain protein.

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In each of the three test sets, different portions of the the four structures were removed. In the first test set, the C-terminal helix of the zinc finger is the only region fully shared among all four partial structures indicated by asterisks above the alignment columns in Fig. For comparison, the modified structures with various portions deleted were then superpositioned using the EM algorithm and using the traditional method which omits columns with gaps.

The EM superposition is also largely independent of which portions were fully aligned. Least-squares isotropic superpositions with missing data. In each pane, four protein structures are superpositioned, each with a different conformation. Other regions of the structures, e. The left-most column, a and d , shows superpositions found using the EM method described here. For ease of comparison, in these images, the missing residues are not displayed, even though all of the original data were included in the superposition calculation.

The right-most column, c and f , shows conventional superpositions based on only the subset of fully shared residues.

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Maximum-likelihood non-isotropic superpositions with missing data. Aside from the optimization criterion, all other details, structures and alignments are as in Figure 4. Least-squares superpositions when no residues are completely shared among the four proteins. Panel a shows the results of the EM missing data algorithm, based on the alignment shown in Figure 2 c. Corresponding non-isotropic ML superpositions. The reference data green line is in fact the same in all three panes; it is re-plotted in each for convenience.

For details see the legend to Figure 7. Because the conventional superpositions ignore regions of the structure that have missing data, they are biased to closely superposition only the regions that are fully shared.