Standard deviation spectrum

Fig. 34.39. Mixture spectra and the corresponding relative standard deviation spectrum.

Fig. 36.3. Summary of spectral information (a) average spectrum, (b) standard deviation spectrum.

Let us consider a data set containing Near-Infrared (NIR) spectra of 30 gasoline samples and five dependent variables (Y = [yl y2 y3 y4 ySj) [31], The original spectra and the so-called std spectrum are presented in Fig. 2. The std, i.e. standard deviation, spectrum it is a vector, the elements of which... 

To exercise the Averaging function, you should consider the ten RAMAN files SC1...SC10 as already shown in Fig. 10.58. These files represent the Raman spectra of stratum comeum or horny layer, the outermost layer of the mammalian skin. The samples studied were taken from the heels of 10 healthy Caucasian volunteers. Compute the averaged spectrum of these samples and the relative standard deviation spectrum. The report of this average is listed in Fig. 10.59. 

Plugge and van der Vlies have discussed the conformity index (Cl) for NIR analysis of ampicillin trihydrate [12,13]. The Cl is a metric used to determine the degree of conformity of a sample or batch with standards of known and acceptable quality. To use the Cl, reference spectra are first collected and baseline-corrected using a second-derivative or multiplicative scatter correction (MSC) spectrum. At every wavelength across the spectrum, the average absorbance and standard deviation are calculated for the baseline-corrected reference spectra, resulting in an average spectrum and a standard deviation spectrum. 

Fig. 14. Training set spectra (top) from Fig. 4 with the standard deviation spectrum (bottom). The large bands in the standard deviation indicate that most of the variation in the spectra is taking place in the three major bands at 1470,1380, and 725 cm". ...

The mean spectrum is subtracted from all the spectra in a teaching set (mean centering), with the additional step of dividing all the spectra in the teaching set by the standard deviation spectrum prior to calibration (Fig. 2). 

The noise spectrum in Figure 6.14 was obtained by reconstructing a spectrum from 100 pairs of Fourier coefficients and subtracting the reconstructed spectmm from the original. The standard deviation of the difference in this case is 82.8 /rOD. The standard deviation spectrum could serve as a real-time index of the noise. It could be recorded for every scan taken and a program could be written to alert the operator if the limits were exceeded. 

To better understand this, let s create a set of data that only contains random noise. Let s create 100 spectra of 10 wavelengths each. The absorbance value at each wavelength will be a random number selected from a gaussian distribution with a mean of 0 and a standard deviation of 1. In other words, our spectra will consist of pure, normally distributed noise. Figure SO contains plots of some of these spectra, It is difficult to draw a plot that shows each spectrum as a point in a 100-dimensional space, but we can plot the spectra in a 3-dimensional space using the absorbances at the first 3 wavelengths. That plot is shown in Figure 51. 

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

Chen used a second-derivative spectrophotometric method for the determination of miconazole nitrate in Pikangshuang [22]. Sample of miconazole nitrate was dissolved in anhydrous ethanol and the second-derivative spectrum of the resulting solution was recorded from 200 300 nm miconazole nitrate was determined by measuring the amplitude value between the peak at 233 nm and the trough at 228 nm. The recovery was 99.8% with a relative standard deviation (n = 6) of 0.2%. 

The expression under the radical gives the multiplying factor for the noise standard deviation for the computed derivative (or smoothed spectrum, but that is not our topic here, we will address only the question of the effect on derivatives), and can be computed solely from the convolution coefficients themselves, independently of the effect of the convolution on the signal part of the S/N ratio. 

SNV is the sample-wise equivalent to autoscahng, which was discussed earlier. The SNV correction parameters for a single spectrum x are simply the mean and standard deviation of the variable intensities in that spectrnm. With these parameters determined, SNV correction involves subtraction of the mean intensity from each of the variable intensities, followed by division of the resulting values by the standard deviation ... 

Like SNV, this pretreatment method [1,21] is a sample-wise scaling method, which has been effectively used in many spectroscopic applications where multiplicative variations are present. However, unlike SNV, the MSC correction parameters are not the mean and standard deviation of the variables in the spectrum x, but rather the result of a linear lit of a reference spectrum x f to the spectrum. The MSC model is given by the following equation ... 

Coppola [329] suggested that phenolic acid prohle could be used as an indicator of apple juice adulteration. Shui and Leong [330] reported the simultaneons determination in apple juice on an ODS column with methanol/snlfnric acid/water as the mobile phase within 80min. However, (-)-epicatechin and p-conmaric acid may be elnted with a similar retention time, so that the spectrum must be recorded in order to identify single compounds. The method was validated and intraday and interday relative standard deviation (RSD%) data resnlted within 0.3% and 1.6%, respectively. The paper also reports LOD data that range between 0.03 mg/L (syringic acid) and 0.18 mg/L (benzoic acid). 

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