Calibration sample distance

For SAXS and USAXS experiments it is sufficient to measure the distance R by hand or by a laser distance-meter. For MAXS and WAXS, calibration by means of samples with known sharp reflection are required. Such calibration samples are available at synchrotron beamlines. Different materials are used for application in different angular regions (isotropic crystalline materials for WAXS, Ag-behenate for MAXS, diverse biological samples3 for SAXS and USAXS). Calibration sheets are frequently published on the home pages of synchrotron beamlines. 

In MLR, if m is the vector (dimension by 1) of the selected absorbance values obtained from a spectral vector a, and M is the matrix of selected absorbance values for the calibration samples, then the Mahalanobis Distance is defined as equation 74-6a ... 

The distances of the N calibration samples from the prediction sample in the space are ranked from least to greatest. 

Although the SIMCA method is very versatile, and a properly optimized model can be very effective, one must keep in mind that this method does not use, or even calculate, between-class variability. This can be problematic in special cases where there is strong natural clustering of samples that is not relevant to the problem. In such cases, the inherent interclass distance can be rather low compared to the mtraclass variation, thus rendering the classification problem very difficult. Furthermore, from a practical viewpoint, the SIMCA method requires that one must obtain sufficient calibration samples to fully represent each of the J classes. Also, the on-line deployment of a SIMCA model requires a fair amount of overhead, due to the relatively large number of parameters and somewhat complex data processing instructions required. However, there are several current software products that facilitate SIMCA deployment. 

P jjj has already validated the predictions to some extent. A very large P value (e.g., unknown 2 on class A has a calc of 11,000) indicates a very large difference ijetween the unknown sample and the calibration samples from that class. Unknown 1 has large P j values for all of the classes with most of the conisSbution coming from the PCA measurement residual. Unknown 2 is within the box of all SIMCA models but has large PCA contributions with classes A and C. Unknown 3 is excluded only from class C primarily because of a large contribution from the distance term (the expected value is zero). 

Calculate the exact sample film distance for the wide angle camera from the calibration sample, NaF (Note 1). 

Once the classification space has been defined, it is then necessary to define the distance in the space that will be used to assess the similarity of prediction samples and calibration samples. The most straightforward distance that can be used for this purpose is the Euclidean distance between two vectors (Dab), which is defined as ... 

The leverage, / , of the z th calibration sample is the z th diagonal of the hat matrix, H. The leverage is a measure of how far the z th calibration sample lies from the other n - 1 calibration samples in X-space. The matrix H is called the hat matrix because it is a projection matrix that projects the vector y into the space spanned by the X matrix, thus producing y-hat. Notice the similarity between leverage and the Mahalanobis distance described in Chapter 4. 

Distance measures, such as Cook s distance, combine the concept of leverage and residuals to compute an overall measure of a calibration sample s influence on the calibration model. Cook s distance is computed as... 

The amorphous material induces a broad hump or halo, in to the diffraction pattern. The width and the position of the halo indicate the distribution of interatomic distances in the structure of the material. The area under halo depends on the content of amorphous material, and therefore it is possible to semiquantitatively determine the content of amorphous material from the x-ray diffraction pattern, Mixtures of synthetic ash and common glass were used as calibration samples. 

In multivariate regression the samples are grouped into a defined space much like the clusters in figure 11.3. An outlier or unknown interference (which was not present during calibration) will place the sample outside the region of the calibration samples. By measuring the distance away from the... 

The Rayleigh ratio eliminates the geometry-dependent factors in I (k)/Io such as the scattering volume V(9) and the detector-sample distance r and retains the factors related to the solution only. In the actual measmement system, /q and V d) cannot be measured correctly. Therefore, a pure solvent such as benzene and toluene is used as a calibration standard. See Refs. 21 and 22 for details. With Rg, Eq. 2.114 is rewritten to... 

The root mean squared error of calibration (RMSEC) has been defined above. The leverage, ha, quantifies the distance of the predicted sample (at zero concentration level) to the mean of the calibration set in the -dimensional space Hq can be estimated as an average value of the leverages of a set of validation samples having zero concentration of the analyte. For a model calculated from mean-centred spectra its calculation was presented in Section 5.3 in matrix notation /zo=l//+to (T T) 4o, where to is the (.4x1) score vector of the predicted sample and T is the (7x4) matrix of scores for the calibration set. Finally, A(a,p,v) is a statistical parameter that takes into account the a and (3 probabilities of falsely stating the presence/absence of analyte, respectively, as recommended elsewhere. When the number of degrees of freedom i.e. the number of calibration samples) is high (v>25), as is usually the case in multivariate calibration models, and a =) , then A(a,(S,v) can be safely approximated to 2 ... 

Shin, M., Jeon, I. C., Frequency-distance responses in SECM-EQCM—A novel method for calibration of the tip-sample distance. Bull. Kor. Chem. Soc. 1998, 19, 1227-1232. 

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