Smoothing algorithm

All spectral plots shown were produced with a Digilab 3240 computer. Linear baselines have been subtracted in some cases to facilitate presentation. Some spectra have been interpolated by a factor of 4 for presentation purposes. No noise reduction or smoothing algorithms have been applied to any of the spectra used in these studies. 

Lowess normalization methods are based on lowess (loess) scatterplot smoothing algorithms. The lowess smoother attempts to smooth contours within a dataset. Typically the lowess will be robust to genes which are active in treatment as they will be observed as outliers (45). Some normalization methods include a print-tip normalization (46), since physical location on the array and the print-tip may contribute some effect and variance beyond the biological and treatment variation. 

The B score (Brideau et al., 2003) is a robust analog of the Z score after median polish it is more resistant to outliers and also more robust to row- and column-position related systematic errors (Table 14.1). The iterative median polish procedure followed by a smoothing algorithm over nearby plates is used to compute estimates for row and column (in addition to plate) effects that are subtracted from the measured value and then divided by the median absolute deviation (MAD) of the corrected measures to robustly standardize for the plate-to-plate variability of random noise. A similar approach uses a robust linear model to obtain robust estimates of row and column effects. After adjustment, the corrected measures are standardized by the scale estimate of the robust linear model fit to generate a Z statistic referred to as the R score (Wu, Liu, and Sui, 2008). In a related approach to detect and eliminate systematic position-dependent errors, the distribution of Z score-normalized data for each well position over a screening run or subset is fitted to a statistical model as a function of the plate the resulting trend is used to correct the data (Makarenkov et al., 2007). 

H. Verschelde, W. Schepens, A unified approach to smoothing algorithms, preprint... 

Figure 5.1 Structure of Savl866 (PDB code 2onj). The simplified shape ofthe protein surface was generated in ICM (Molsoft) using an FFT-based smoothing algorithm.

Feng, W.C., Rexford, J. Performance evaluation of smoothing algorithms for transmitting prerecorded variable-bit-rate video. IEEE Transactions on Multimedia 1(3), 302-313 (1999)... 

In the case of evaluating real data sets, the algorithm in Eqs. (6.126) and (6.127) only serves the purpose of providing the mathematical basis of how the algorithm should perform in principle. The operator of conditional expectations is replaced by an appropriate smoothing algorithm. 

Spectral filters The mathematical treatments of NIR spectra prior to regression used to remove variance that only adds to the error term in the calibration model, (i.e., variance not containing systematic information on the sample). Common spectral filters are multiplicative signal correction (MSC), derivatives and smoothing algorithms. 

Differentiation of a signal is commonly used to detect unresolved peak shoulders in ESR (see Chapter 4) and optical spectroscopies, and in many other contexts (see Smith Chapter, for example). However, the noise level is much increased in the derivative display. Ordinary smoothing of the derivative may be undesirable, because the smoothing algorithm may also obscure spectral details. 

Figure 11. Effects of simple and modified Fourier transform smoothing algorithms on noise-free exponential decay.

The deficiency of uniform weighting becomes apparent when smoothing over a peak. In this case the expected counts should vary rapidly with x. The counts in channels adjacent to channel x are poor representations of the expected counts in channel x. Consequently, the counts in channel x + i should receive less weight the larger the absolute value of i. Thus, the smoothing algorithm becomes... 

The decision-making model uses a trend-adjusted exponential smoothing algorithm [11]. It uses two smoothing parameters, 0 < a < 1 and 0 < < 1, as... 

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