Variance of spectrum

The first-order centred moment is equal to zero. The second one describes the variance of spectrum... 

The variance of spectrum consists of the contributions of spin-orbit and electrostatic interactions... 

Calculate the variance of the differences d between the measurements and the trial spectrum f,. 

The scalar-scalar transfer function Saoi(K q) appearing in the final term on the right-hand side of (A.6) denotes the contribution of scalar mode q to the scalar-variance transfer spectrum at k. (See Yeung (1994) and Yeung (1996) for examples of these functions extracted from DNS.) Similarly, the scalar-covariance transfer spectrum can be decomposed using... 

The forward and backscatter rate constants can be expressed explicitly in terms of the scalar-variance transfer spectrum decomposed into forward and backscatter contributions ... 

Usual procedures for the selection of the common best basis are based on maximum variance criteria (Walczak and Massart, 2000). For instance, the variance spectrum procedure computes at first the variance of all the variables and arranges them into a vector, which has the significance of a spectrum of the variance. The wavelet decomposition is applied onto this vector and the best basis obtained is used to transform and to compress all the objects. Instead, the variance tree procedure applies the wavelet decomposition to all of the objects, obtaining a wavelet tree for each of them. Then, the variance of each coefficient, approximation or detail, is computed, and the variance values are structured into a tree of variances. The best basis derived from this tree is used to transform and to compress all the objects. 

Having explicit formulas for a number of first moments we can approximately restore the envelope line of the radiation spectrum without its detailed calculations. If lines in the spectrum have one symmetric maximum, then its envelope line is approximated by a normal function whose reconstruction requires only the mean energy and variance of the spectrum. Such an approach is useful for the case of complex spectra consisting of many lines, which, due to low resolutions as well as Doppler and collisional broadening or large natural width, form continuous or quasi-continuous bands. Studies of variation of these statistical characteristics along isoelectronic sequences give a wealth of information on intra-atomic interactions. 

The pulser pulses are generated at a constant frequency, so that the variance of the pulser peak area that results from the statistical nature of spectrum acquisition is approximated at counting losses well below 50% by... 

But these distributions also hint at some fundamental limits to our measurements. At every , there are at most 21 + 1 quantities that we can measure, the individual. Even with a perfect measurement, we need to infer from these quantities the underlying variance from which these quantities are drawn the observed variance of the aim at best provides an estimate of the power spectrum, ( . With a gaussian distribution, we can compute the variance of a single coefficient, var( a m 2) = aim 4) - ( a m 2)2 = 3Cj - C2 2C. If there arc 2 + 1 measurements at a given , the variance of the estimatem is then lower limit, as well-instrument noise and systematic problems in realistic experiments will only increase the error. 

A sample can be classified by calculating the sum of the squares of the difference between its measured spectrum vector and the same spectrum reproduced using a principal component model. The residual variance, s, of a data vector i fitted to the training set for class q indicates how similar the spectrum is to class q. For data vectors from the training set, the residual variance of a sample is given by Equation 4.44. 

This equation of a straight line takes into account a slight difference of the base lines by the constant a. By applying least-squares regression the constants a and b are determined. The variance of A(R) indicates the degree of conformity between the two spectra. If sample and reference spectrum differ only slightly, the spectra should be compared blockwise, and the standard deviations of the different blocks should be used. This method is useful for spectra which show a sufficient number of sharp bands. It may fail if there are broad bands, in which case it is necessary to compare the second derivatives of the spectra. The algorithm is shown in Fig. 5.1-15. 

Figure 6. (a) Frequency distribution for wavelengths included in the best found subsets from 50 repetitive GSAMS/PRESS runs, (b) Mean calibration spectrum (-) and column variances of Y - (+). Note that both lines are normalized to the (0, 1) range and do not represent actual values. 

The general bandshapes are similar and are strongly reminiscent of those of PTS or DCHD. Significant structure to the blue of the most intense peak is observed for all of the crystals. Most striking, however, are the variances of the bandwidths of the entire reflection bands and frequencies of the first peaks. The frequencies range from 15,800 cm" for DDMU and DDU to 18,300 cm for POD, but the bandshape of the characteristic LT spectrum is seen in all cases. 

The colors of the objects to be discriminated are too similar the three color components (red/green/blue or hue intensity saturation coordinates) are not sufficiently discriminative to identify differences between an object of interest and any surrounding objects. This problem is enhanced by a wide variability in color typical of biological products variance of the colorimetric components inside an object may also be greater than the variance between this object and those to be removed. In that case, the solution is to seek other information, either in the visible spectrum or in the NIR spectrum. The NIR spectrum can provide information regarding the chemical composition and internal physical structure, and can be used to separate same-color objects of different composition. 

The spectrum denotes the variance of the process at a certain time b and scale a. With the chosen normalization of the wavelet transformation (Eq. (12.1)), white noise is given by Sg b,a) = m(6,a)p = const.. 

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