Spectral noise

The whole idea behind PLS is to try to restore, to the extent possible, the optimum congruence between the each spectral factor and its corresponding concentration factor. For the purposes of this concept, optimum congruence is defined as a perfectly linear relationship between the projections, or scores, of the spectral and concentration data onto the spectral and concentration factors as exemplified in Figure 73. Since the spectral noise is independent from the concentration noise, a perfectly linear relationship is no longer possible. So, the best we can do is restore optimum congruence in the least-squares sense. 

In general, because the noise in the concentration data is independent from the spectral noise, each optimum factor, W, will lie at some angle to the plane that contains the spectral data. But we can find the projection of each W, onto the plane containing the spectral data. These projections are called the spectral factors, or spectral loadings. They are usually assigned to the variable named P. Each spectral factor P, is usually organized as a row vector. 

A noise power equivalent to one photon generates an interference signal which has an amplitude equals to twice the rms photon noise of the source. But as only the in-phase components of the source generates an interference with the local oscillator, the result is that the spectral Noise Equivalent Power of the heterodyne receiver is hv. 

At this point we have completed our analysis of spectral noise for the case where the noise is constant (or at least independent of the signal level). Having completed this part of the analyses originally proposed in Chapter 40 (referenced as [1]) we will continue by doing a similar analysis for a complicated case. 

As we mentioned, the two-point first derivative is equivalent to using the convolution function -1, 1. We also treated this in our previous chapter, but it is worth repeating here. Therefore the multiplying factor of the spectral noise variance is — l2 + l2 = 2,... 

This value depends on the spectral width of the resonant mode, amplitude noise (e.g., thermal and shot noise), spectral noise (e.g., thermo-optic variations in the sensing device), and the spectral resolution of the measurement technique21. Minimizing these parameters will improve the DL. 

The randomisation test proposed by Wiklund et al. [34] assesses the statistical significance of each individual component that enters the model. This had been studied previously, e.g. using a t- or F-test (for instance, Wold s criterion seen above), but they are all based on unrealistic assumptions about the data, e.g. the absence of spectral noise see [34] for more advanced explanations and examples. A pragmatic data-driven approach is therefore called for and it has been studied in some detail recently [34,40]. We have included it here because it is simple, fairly intuitive and fast and it seems promising for many applications. 

Fig. 8.3. A Acquired high SNR data and simulated noisy spectra (peak-to-peak noise = 0.001, 0.01, 0.1, and 0.4 a.u.), showing the degradation in data quality. Spectra are offset for clarity. B Spectra after noise reduction demonstrate the dramatic gains possible by chemometric methods. C Noise reduction was implemented to classify breast tissue and application of noise rejection allowed the same quality of classification (accuracy) to be recovered at higher noise levels. D In another example, image fidelity (here the nitrile stretching vibrational mode at 2227 cm-1) is much enhanced as a result of spectral noise rejection A and C are reproduced from Reddy and Bhargava, Submitted [165], D is reproduced from [43]...

Correlated noise data can be transformed to the frequency domain using a fast Fourier transform (EFT) or the maximum entropy method (MEM) (151) A spectral noise impedance, Rsn(f), can then be calculated (152) ... 

In this expression, E and / are the magnitude of the potential and current noise at any given frequency, /. RRe and Rlm are the real and imaginary components of Rsn. Plots of spectral noise impedance versus frequency resemble Bode magnitude plots of EIS data as shown in Fig. 58. Meaningful phase angle information is not usually obtained, as this is not preserved by the MEM transform, and data are usually of insufficient quality for accurate phase information to be obtained from the EFT. 

The spectral noise resistance, determined from the DC limit in a spectral noise impedance plot, is defined as... 

Figure 58 Frequency dependence of the spectral noise resistance, Rm, for iron in aerated, and aerated and inhibited 0.5 M NaCl after exposure for (a) 1 h and (b) 24 h. (From F. Mansfeld, H. Xiao. p. 59, Electrochemical Noise Measurements for Corrosion Applications, ASTM STP 1277. ASTM, Philadelphia, PA (1996).)...

Under the proper conditions there are important relations among the PSDs, the noise resistance, and the spectral noise resistance that make PSD measurement useful in corrosion studies (159-161). The variance of a random signal, x, is the integral of its PSD in the frequency domain ... 

If Rsn(f) is frequency-independent in the range of/max to fan, then Rn will equal Rsn. If the spectral noise resistance is equal to the magnitude of the impedance, which is reasonable at low frequencies, then... 

One goal of the research in the Moerner group at IBM was the exploration of ultimate limits to the spectral hole-burning optical storage process. A particularly interesting limit on the SNR of a spectral hole results from the finite number of molecules that contribute to the absorption profile near the hole. Due to unavoidable number of fluctuations in the density of molecules in any spectral interval, there should exist a spectral noise on an inhomogeneous... 

The threshold curve is a plot of retention time versus a similarity factor threshold, below which the presence of an impurity cannot be distinguished from spectral noise. The threshold trace may be computed automatically from the standard deviation of a number of user-selected pure noise spectra. Alternatively, the threshold may be set at a fixed value. Similarity and threshold curves tend to rise at the extremities of the eluted peak, even when no impurity is present. As signal strength decreases, a larger proportion of the spectral response is caused by noise. If an impurity is present at a detectable concentration, the similarity curve will intersect the threshold (Fig. 5). 

A moderate line broadening (<2 Hz, not enough to cause signal overlap) reduces spectral noise. 

An evaluation of the performance of these algorithms in predictions using inside model space and outside model space was conducted. In principal component regression, principal axes highly correlated with sample constituents of interest are considered to be inside model space, while axes typically attributed to spectral noise are termed outside model space. 

If the experimental errors are underestimated, it can lead to tight but inaccurate distance bounds. Conversely, the overestimated errors can lead to unnecessarily imprecise distance bounds. RANDMARDI takes into account two types of experimental errors relative integration errors and absolute errors due to spectral noise. The first kind can be estimated, for example, by comparing intensities of symmetric peaks below and above the diagonal, and the second type can be estimated as 50-200% of the lowest quantifiable peak, depending on the spectrum quality. 

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