Likelihood‐based refinement. I. Irremovable model errors
Identifieur interne : 008786 ( Main/Exploration ); précédent : 008785; suivant : 008787Likelihood‐based refinement. I. Irremovable model errors
Auteurs : V. Y. Lunin ; P. V. Afonine ; A. G. UrzhumtsevSource :
- Acta Crystallographica Section A [ 0108-7673 ] ; 2002-05.
Descripteurs français
- Pascal (Inist)
English descriptors
- KwdEn :
- Teeft :
- Acentric, Acentric cases, Acentric reflections, Acta, Acta cryst, Asymptotic, Asymptotic formula, Atomic centres, Atomic coordinates, Atomic model, Atomic parameters, Calc, Ccp4 newslett, Centric, Centric case, Centric reflections, Cryst, Current model, Current values, Different types, Eobs, Exact coordinates, Exact values, Experimental data, Experimental errors, Ffsobs, Fscalc, Fsobs, Gaussian, Gaussian distribution, Gsps, Imaginary parts, Irremovable, Irremovable errors, Joint probability distribution, Likelihood function, Likelihood value, Lunin, Lunin skovoroda, Lunin urzhumtsev, Magnitudes fscalc, Mathematical statistics, Minimization, Model parameters, Modelling, Negative value, Other words, Pannu, Parameter, Partial model, Partial models, Petrova, Probabilistic, Probabilistic model, Probabilistic modelling, Probabilistic models, Probability distribution, Protein crystallogr, Quadratic, Quadratic approximation, Relative magnitudes, Research papers, Residual, Scale factor, Search model, Skovoroda, Srinivasan parthasarathy, Structure factor, Structure factors, Target values, True positions, Unit cell, Unit weights, Urzhumtsev, Variable parameters, Water molecules, Whole structure.
Abstract
In conventional structure refinement, the discrepancy between the calculated magnitudes and those observed in X‐ray experiments is attributed to errors inherent in preliminary assigned values of the model parameters. However, the chosen set of model parameters may not be adequate to describe the structure factors precisely. For example, if some atoms are not included in the current model, then the structure factors calculated from such a partial model contain `irremovable errors'. These errors cannot be eliminated by any choice of the parameters of the partial structure. Probabilistic modelling suggests a way to take irremovable errors into account. Every trial set of values of the model parameters is now associated with the joint probability distribution of the calculated magnitudes, rather than with a particular set of magnitudes. The new goal of the refinement is formulated as the search for the distribution that is the most consistent with the observed data. The statistical likelihood is a possible measure of the consistency. The suggested quadratic approximation of the likelihood function allows the likelihood‐based refinement to be considered as a kind of least‐squares refinement that uses appropriate weights and modified targets for the calculated magnitudes. This in turn enables the analysis of tendencies of the likelihood‐based refinement in comparison with the classical least‐squares refinement.
Url:
DOI: 10.1107/S0108767302001046
Affiliations:
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I. Irremovable model errors</title><author><name sortKey="Lunin, V Y" sort="Lunin, V Y" uniqKey="Lunin V" first="V. Y." last="Lunin">V. Y. Lunin</name></author><author><name sortKey="Afonine, P V" sort="Afonine, P V" uniqKey="Afonine P" first="P. V." last="Afonine">P. V. Afonine</name></author><author><name sortKey="Urzhumtsev, A G" sort="Urzhumtsev, A G" uniqKey="Urzhumtsev A" first="A. G." last="Urzhumtsev">A. G. Urzhumtsev</name></author></analytic><monogr></monogr><series><title level="j" type="main">Acta Crystallographica Section A</title><title level="j" type="alt">ACTA CRYSTALLOGRAPHICA A</title><idno type="ISSN">0108-7673</idno><idno type="eISSN">1600-5724</idno><imprint><biblScope unit="vol">58</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="270">270</biblScope><biblScope unit="page" to="282">282</biblScope><publisher>Munksgaard International Publishers</publisher><pubPlace>5 Abbey Square, Chester, Cheshire CH1 2HU, England</pubPlace><date type="published" when="2002-05">2002-05</date></imprint><idno type="ISSN">0108-7673</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0108-7673</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Crystal structure</term><term>Errors</term><term>Likelihood function</term><term>Modelling</term><term>Parameterization</term><term>Refinement</term><term>Structure resolution</term><term>Theoretical study</term><term>XRD</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>6110N</term><term>Affinement</term><term>Diffraction RX</term><term>Erreur</term><term>Etude théorique</term><term>Fonction vraisemblance</term><term>Modélisation</term><term>Paramétrisation</term><term>Résolution structurale</term><term>Structure cristalline</term></keywords><keywords scheme="Teeft" xml:lang="en"><term>Acentric</term><term>Acentric cases</term><term>Acentric reflections</term><term>Acta</term><term>Acta cryst</term><term>Asymptotic</term><term>Asymptotic formula</term><term>Atomic centres</term><term>Atomic coordinates</term><term>Atomic model</term><term>Atomic parameters</term><term>Calc</term><term>Ccp4 newslett</term><term>Centric</term><term>Centric case</term><term>Centric reflections</term><term>Cryst</term><term>Current model</term><term>Current values</term><term>Different types</term><term>Eobs</term><term>Exact coordinates</term><term>Exact values</term><term>Experimental data</term><term>Experimental errors</term><term>Ffsobs</term><term>Fscalc</term><term>Fsobs</term><term>Gaussian</term><term>Gaussian distribution</term><term>Gsps</term><term>Imaginary parts</term><term>Irremovable</term><term>Irremovable errors</term><term>Joint probability distribution</term><term>Likelihood function</term><term>Likelihood value</term><term>Lunin</term><term>Lunin skovoroda</term><term>Lunin urzhumtsev</term><term>Magnitudes fscalc</term><term>Mathematical statistics</term><term>Minimization</term><term>Model parameters</term><term>Modelling</term><term>Negative value</term><term>Other words</term><term>Pannu</term><term>Parameter</term><term>Partial model</term><term>Partial models</term><term>Petrova</term><term>Probabilistic</term><term>Probabilistic model</term><term>Probabilistic modelling</term><term>Probabilistic models</term><term>Probability distribution</term><term>Protein crystallogr</term><term>Quadratic</term><term>Quadratic approximation</term><term>Relative magnitudes</term><term>Research papers</term><term>Residual</term><term>Scale factor</term><term>Search model</term><term>Skovoroda</term><term>Srinivasan parthasarathy</term><term>Structure factor</term><term>Structure factors</term><term>Target values</term><term>True positions</term><term>Unit cell</term><term>Unit weights</term><term>Urzhumtsev</term><term>Variable parameters</term><term>Water molecules</term><term>Whole structure</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">In conventional structure refinement, the discrepancy between the calculated magnitudes and those observed in X‐ray experiments is attributed to errors inherent in preliminary assigned values of the model parameters. However, the chosen set of model parameters may not be adequate to describe the structure factors precisely. For example, if some atoms are not included in the current model, then the structure factors calculated from such a partial model contain `irremovable errors'. These errors cannot be eliminated by any choice of the parameters of the partial structure. Probabilistic modelling suggests a way to take irremovable errors into account. Every trial set of values of the model parameters is now associated with the joint probability distribution of the calculated magnitudes, rather than with a particular set of magnitudes. The new goal of the refinement is formulated as the search for the distribution that is the most consistent with the observed data. The statistical likelihood is a possible measure of the consistency. The suggested quadratic approximation of the likelihood function allows the likelihood‐based refinement to be considered as a kind of least‐squares refinement that uses appropriate weights and modified targets for the calculated magnitudes. This in turn enables the analysis of tendencies of the likelihood‐based refinement in comparison with the classical least‐squares refinement.</div></front></TEI><affiliations><list></list><tree><noCountry><name sortKey="Afonine, P V" sort="Afonine, P V" uniqKey="Afonine P" first="P. V." last="Afonine">P. V. Afonine</name><name sortKey="Lunin, V Y" sort="Lunin, V Y" uniqKey="Lunin V" first="V. Y." last="Lunin">V. Y. Lunin</name><name sortKey="Urzhumtsev, A G" sort="Urzhumtsev, A G" uniqKey="Urzhumtsev A" first="A. G." last="Urzhumtsev">A. G. Urzhumtsev</name></noCountry></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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