関: least squares、least-squares analysis

WordNet

the quality of excellence in thought and manners and taste; "a man of intellectual refinement"; "he is remembered for his generosity and civilization" (同)civilization, civilisation
the result of improving something; "he described a refinement of this technique" (同)elaboration

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〈U〉精製,精錬 / 〈U〉洗練,上品,優雅 / 〈C〉改良点,工夫 / 〈C〉(…の)極致《+『of』+『名』》

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/09/15 04:25:05」(JST)

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Rietveld refinement is a technique devised by Hugo Rietveld for use in the characterisation of crystalline materials. The neutron and x-ray diffraction of powder samples results in a pattern characterised by reflections (peaks in intensity) at certain positions. The height, width and position of these reflections can be used to determine many aspects of the material's structure.

The Rietveld method uses a least squares approach to refine a theoretical line profile until it matches the measured profile. The introduction of this technique was a significant step forward in the diffraction analysis of powder samples as, unlike other techniques at that time, it was able to deal reliably with strongly overlapping reflections.

The method was first reported for the diffraction of monochromatic neutrons where the reflection-position is reported in terms of the Bragg angle 2θ. This terminology will be used here although the technique is equally applicable to alternative scales such as x-ray energy or neutron time-of-flight. The only wavelength and technique independent scale is in reciprocal space units or momentum transfer Q, which is historically rarely used in powder diffraction but very common in all other diffraction and optics techniques. The relation is

Peak shape

The shape of a powder diffraction reflection is influenced by the characteristics of the beam, the experimental arrangement, and the sample size and shape. In the case of monochromatic neutron sources the convolution of the various effects has been found to result in a reflex almost exactly Gaussian in shape. If this distribution is assumed then the contribution of a given reflection to the profile y_i at position 2θ_i is:

$y_i = I_k \exp \left [ \frac{-4 \ln \left (2 \right )}{H_k^2} \left (2\theta_i - 2\theta_k \right )^2 \right ]$

where H_k is the full width at half peak height (full-width half-maximum), 2θ_k is the centre of the reflex, and I_k is the calculated intensity of the reflex (determined from the structure factor, the Lorentz factor, and multiplicity of the reflection)

At very low diffraction angles the reflections may acquire an asymmetry due to the vertical divergence of the beam. Rietveld used a semi-empirical correction factor, A_s to account for this asymmetry

where P is the asymmetry factor and s is +1,0,-1 depending on the difference 2θ_i-2θ_k being positive, zero or negative respectively.

At a given position more than one diffraction peak may contribute to the profile. The intensity is simply the sum of all reflections contributing at the point 2θ_i.

Peak width

The width of the diffraction peaks are found to broaden at higher Bragg angles. This angular dependency was originally represented by

where U, V and W are the halfwidth parameters and may be refined during the fit.

Preferred orientation

In powder samples there is a tendency for plate- or rod-like crystallites to align themselves along the axis of a cylindrical sample holder. In solid polycrystalline samples the production of the material may result in greater volume fraction of certain crystal orientations (commonly referred to as texture). In such cases the reflex intensities will vary from that predicted for a completely random distribution. Rietveld allowed for moderate cases of the former by introducing a correction factor:

where I_obs is the intensity expected for a random sample, G is the preferred orientation parameter and α is the acute angle between the scattering vector and the normal of the crystallites.

Refinement

The principle of the Rietveld Method is to minimise a function M which analyzes the difference between a calculated profile y(calc) and the observed data y(obs). Rietveld defined such an equation as:

where W_i is the statistical weight and c is an overall scale factor such that

References

H. M. Rietveld (1969). "A profile refinement method for nuclear and magnetic structures". Journal of Applied Crystallography 2 (2): 65–71. doi:10.1107/S0021889869006558.

External links

The Rietveld Method - the story.
The MAUD program - software for Rietveld refinement.

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English Journal

Determination of polyphenolic compounds of red wines by UV-VIS-NIR spectroscopy and chemometrics tools.

Martelo-Vidal MJ1, Vázquez M2.Author information 1Department of Analytical Chemistry, Faculty of Veterinary Science, University of Santiago de Compostela, Calle Carballo Calero, s/n, 27002 Lugo, Spain.2Department of Analytical Chemistry, Faculty of Veterinary Science, University of Santiago de Compostela, Calle Carballo Calero, s/n, 27002 Lugo, Spain. Electronic address: manuel.vazquez@usc.es.AbstractSpectral analysis is a quick and non-destructive method to analyse wine. In this work, trans-resveratrol, oenin, malvin, catechin, epicatechin, quercetin and syringic acid were determined in commercial red wines from DO Rías Baixas and DO Ribeira Sacra (Spain) by UV-VIS-NIR spectroscopy. Calibration models were developed using principal component regression (PCR) or partial least squares (PLS) regression. HPLC was used as reference method. The results showed that reliable PLS models were obtained to quantify all polyphenols for Rías Baixas wines. For Ribeira Sacra, feasible models were obtained to determine quercetin, epicatechin, oenin and syringic acid. PCR calibration models showed worst reliable of prediction than PLS models. For red wines from mencía grapes, feasible models were obtained for catechin and oenin, regardless the geographical origin. The results obtained demonstrate that UV-VIS-NIR spectroscopy can be used to determine individual polyphenolic compounds in red wines.
Food chemistry.Food Chem.2014 Sep 1;158:28-34. doi: 10.1016/j.foodchem.2014.02.080. Epub 2014 Feb 28.
Spectral analysis is a quick and non-destructive method to analyse wine. In this work, trans-resveratrol, oenin, malvin, catechin, epicatechin, quercetin and syringic acid were determined in commercial red wines from DO Rías Baixas and DO Ribeira Sacra (Spain) by UV-VIS-NIR spectroscopy. Calibratio
PMID 24731310

Application of a voltammetric electronic tongue and near infrared spectroscopy for a rapid umami taste assessment.

Bagnasco L1, Cosulich ME2, Speranza G3, Medini L4, Oliveri P5, Lanteri S5.Author information 1Department of Chemistry and Industrial Chemistry, University of Genoa, Via Dodecaneso 31, I-16146 Genoa, Italy; Department of Pharmacy, University of Genoa, Via Brigata Salerno 13, I-16147 Genoa, Italy. Electronic address: lucia.bagnasco@unige.it.2Department of Chemistry and Industrial Chemistry, University of Genoa, Via Dodecaneso 31, I-16146 Genoa, Italy; Department of Biology and Biotechnology "L. Spallanzani", University of Pavia, Via Ferrata 9, I-27100 Pavia, Italy.3Department of Chemistry, University of Milan, Via Golgi 19, I-20133 Milan, Italy; Italian Biocatalysis Center, Viale Taramelli 12, I-27100 Pavia, Italy.4Special Company for Professional Training and Technological and Commercial Promotion of the Chamber of Commerce of Savona, Regione Rollo, 98, I-17031 Albenga, SV, Italy.5Department of Pharmacy, University of Genoa, Via Brigata Salerno 13, I-16147 Genoa, Italy.AbstractThe relationships between sensory attribute and analytical measurements, performed by electronic tongue (ET) and near-infrared spectroscopy (NIRS), were investigated in order to develop a rapid method for the assessment of umami taste. Commercially available umami products and some aminoacids were submitted to sensory analysis. Results were analysed in comparison with the outcomes of analytical measurements. Multivariate exploratory analysis was performed by principal component analysis (PCA). Calibration models for prediction of the umami taste on the basis of ET and NIR signals were obtained using partial least squares (PLS) regression. Different approaches for merging data from the two different analytical instruments were considered. Both of the techniques demonstrated to provide information related with umami taste. In particular, ET signals showed the higher correlation with umami attribute. Data fusion was found to be slightly beneficial - not so significantly as to justify the coupled use of the two analytical techniques.
Food chemistry.Food Chem.2014 Aug 15;157:421-8. doi: 10.1016/j.foodchem.2014.02.044. Epub 2014 Feb 22.
The relationships between sensory attribute and analytical measurements, performed by electronic tongue (ET) and near-infrared spectroscopy (NIRS), were investigated in order to develop a rapid method for the assessment of umami taste. Commercially available umami products and some aminoacids were s
PMID 24679800

Beer fermentation: monitoring of process parameters by FT-NIR and multivariate data analysis.

Grassi S1, Amigo JM2, Lyndgaard CB3, Foschino R1, Casiraghi E1.Author information 1Department of Food, Environmental and Nutritional Sciences (DeFENS), Università degli Studi di Milano, via Celoria 2, 20133 Milano, Italy.2Department of Food Science, Faculty of Sciences, University of Copenhagen, Rolighedsvej 30, DK-1958 Frederiksberg C, Denmark. Electronic address: jmar@food.ku.dk.3Department of Food Science, Faculty of Sciences, University of Copenhagen, Rolighedsvej 30, DK-1958 Frederiksberg C, Denmark.AbstractThis work investigates the capability of Fourier-Transform near infrared (FT-NIR) spectroscopy to monitor and assess process parameters in beer fermentation at different operative conditions. For this purpose, the fermentation of wort with two different yeast strains and at different temperatures was monitored for nine days by FT-NIR. To correlate the collected spectra with °Brix, pH and biomass, different multivariate data methodologies were applied. Principal component analysis (PCA), partial least squares (PLS) and locally weighted regression (LWR) were used to assess the relationship between FT-NIR spectra and the abovementioned process parameters that define the beer fermentation. The accuracy and robustness of the obtained results clearly show the suitability of FT-NIR spectroscopy, combined with multivariate data analysis, to be used as a quality control tool in the beer fermentation process. FT-NIR spectroscopy, when combined with LWR, demonstrates to be a perfectly suitable quantitative method to be implemented in the production of beer.
Food chemistry.Food Chem.2014 Jul 15;155:279-86. doi: 10.1016/j.foodchem.2014.01.060. Epub 2014 Jan 27.
This work investigates the capability of Fourier-Transform near infrared (FT-NIR) spectroscopy to monitor and assess process parameters in beer fermentation at different operative conditions. For this purpose, the fermentation of wort with two different yeast strains and at different temperatures wa
PMID 24594186

Japanese Journal

Magnetic Properties of Layered Itinerant Electron Ferromagnet Fe

Chen Bin,Yang JinHu,Wang HangDong,Imai Masaki,Ohta Hiroto,Michioka Chishiro,Yoshimura Kazuyoshi,Fang MingHu
J Phys Soc Jpn 82(12), 124711-124711-7, 2013-12-15
… The rietveld refinement of the powder XRD data confirms that Fe<inf>3</inf>GeTe<inf>2</inf>crystallizes in the space group p6_{3}/mmc. …
NAID 150000107922

Preparation of Narrow Band-Gap Cu₂Sn(S,Se)₃ and Fabrication of Film by Non-Vacuum Process (Special Issue : Solid State Devices and Materials)

Nomura Takeshi,Maeda Tsuyoshi,Wada Takahiro
Jpn J Appl Phys 52(4), 04CR08-04CR08-4, 2013-04-25
… The crystal structure of Cu<inf>2</inf>SnS<inf>3</inf>(CTS) was characterized by Rietveld refinement of the powder X-ray diffraction data and determined to be a monoclinic crystal system. …
NAID 40019639538