Noisy data, figures

Continuing, we find the second eigenvector for the noisy data. Figures 43 and 44 contain plots of the first two eigenvectors for the noisy data. Again, the second eigenvector for the noisy data is nearly identical to that of the noise-free data. 

Completing the cycle, we calculate the third eigenvector for the noisy data. Figures 45 and 46 contain the plots of all three eigenvectors for the noisy data. 

Next, we find the first eigenvector of the noisy data set and plot it in Figures 41 and 42. We see that it is nearly identical to the first eigenvector of the noise... 

Figure 41. The noisy data from Figure 39 plotted together with the first eigenvector (factor) for the data.

Figure 77. Projections of the concentration data onto each concentration factor vs. the corresponding projections of the spectral data onto each spectral factor for the noisy data using eigenvectors as factors.

Figure 5. L-Alanine. Fit to noisy data. Calculation A. Distribution of residual structure factor amplitudes at the end of the MaxEnt calculation on 2532 noisy data up to 0.463A. Residuals plotted ... 

We briefly discuss in this section the results of the valence MaxEnt calculation on the noisy data set for L-alanine at 23 K we will denote this calculation with the letter A. The distribution of residuals at the end of the calculation is shown in Figure 5. It is apparent that no gross outliers are present, the calculated structure factor amplitudes being within 5 esd s from the observed values at all resolution ranges. 

Figure 6. l-Alanine. Fit to noisy data. Calculation A. MaxEnt deformation density and error map in the COO- plane Map size, orientation and contouring levels as in Figure 2. (a) MaxEnt dynamic deformation density A uP. (b) Error map qME - Model. 

Figures, l-Alanine.Fits to noisy data Calculations A (experimental noise) and B (10% experimental noise). MaxEnt, deformation and error density profiles along the Cl-01 bond. Solid line Model valence density. Dashed line MaxEnt density A. Dot-dashed line MaxEnt density B. Dotted line valence-shells non-uniform prior.

Figure 4-2. Noisy data scattered around the underlying straight line...

Figure 4-24. In all panels the true data are represented by the line marker. The top panel displays the noisy ( ) data the middle panel shows the result of a 4th degree polynomial fitted through 11 noisy data points ( ) and the bottom panel, the result of a 2nd degree smoothing through 21 noisy data points ( ).

Figure 4-52. Original (—) and calculated (-) emission spectra as the result of linear regression of very noisy data. Top panel x2 -fitting bottom panel traditional least-squares fitting.

Consider the noisy data given in Figure 10.1. Our task is to enhance the signal-to-noise ratio, if we can. To process these spectra, we must consider what is already known about the data. When a chemical measurement r(t) is obtained, we presume that this measurement consists of a true signal sit) corrupted by noise n(t). For simplicity, the linear additivity of signal and noise is usually assumed, as depicted by the measurement in Equation 10.1... 

FIGURE 10.12 Time-domain smoothing of the noisy data in Figure 10.1 with the impulse response function of Figure 10.7, processed from left to right in this spectrum. The true signal is shown as a dotted line. Note the significant filter lag in this example. 

FIGURE 10.18 Weak filtering of noisy data using quadratic polynomials. Weak filtering was done with a 20-point smoothing window. The true signal is shown as a dotted trace. 

---