Filter polynomial

Leach, R. A., Carter, C. A., and Harris, J. M., Least-Square Polynomial Filters for Initial Point and Slope Estimation, Anal. Chem. 56, 1984, 2304-2307. 

Kahn, A., Procedure for Increasing the Accuracy of the Initial Data Point Slope Estimation by Least-Squares Polynomial Filters, Anal. Chem. 60, 1988,... 

With the aid of Eq. (48), we can show that 6ik (o) = (k + l)N(co) for t(co) = 0. The object estimate consists of noise at frequencies that t does not pass. The noise grows with each iteration. This problem can be alleviated if we bandpass-filter the data to the known extent of z to reject frequencies that t is incapable of transmitting. Practical applications of relaxation methods typically employ such filtering. Least-squares polynomial filters, applied by finite discrete convolution, approximate the desired characteristics (Section III.C.5). For k finite and t 0, but nevertheless small,... 

Noise is a serious problem where z(co) is small. We see that the estimate 6(cd) carries with it none of the sensible noise treatment that modified filters like those of the Wiener type provide. We may, however, supply the required noise suppression between iterations by smoothing. The polynomial filters described in Section III.C.5, for example, may be used. Very little has been done, however, to determine how suppression might be accomplished optimally. Usually it is treated in an ad hoc way. 

In Chapter 7, the high-frequency attenuation characteristics of polynomial filters are discusssed. These considerations are relevant to questions regarding the desirability of a single application of a filter of given length as opposed to multiple application of shorter filters. 

FIGURE 10.16 Polynomial least-squares filtering. A quadratic, five-point polynomial filter is shown. 

FIGURE 10.17 Strong filtering of noisy data using quadratic polynomials. Filtering was done with a five-point smoothing window. The true signal is shown as a dotted trace. 

D. C. Sorensen (1992) Implicit application of polynomial filters in a k-step Arnoldi method, SIAM, 7. Matrix Anal. Appl. 13, pp. 357. 

Fig. 3.3-2 A simulated 1000-point spectrum with five well-separated Lorentzian peaks of various widths, without (a) and with (b) Gaussian noise, and the noisy spectrum after filtering with a 25-point smoothingpolynomial of (c) first-order, (d) third-order, and (e) fifth-order respectively. Parameters used for the simulated spectrum ax = 0.001, b4 = 300, c, = 1, 2 = 0.003, b2 = 600, c2 = 1, a3 = 0.01, b3 = 750, c3=l,a4 = 0.03, b4 = 850, c4 = 1,<% = 0.1, b5 = 925, c5= 1, na = 0.1. Curve (f) shows the fit of the same noisy data set (b) after using Barak s adaptive-degree polynomial filter for a 25-point moving polynomial of order between 0 and 10.

To learn about digital filters, such as the nonrecursive moving-average and polynomial filters, as well as the recursive Kalman filter... 

The following data are to be smoothed by a quadratic 5-point polynomial filter ... 

For example, the first derivative, is obtained on the basis of a 5-point quadratic polynomial filter by... 

To characterize the corresponding noise, we consider the error propagation for the polynomial filter. For the Savitzky-Golay filter (see Eq. (3.2)), the result of error propagation (see Table 2.4) is expressed here by the standard deviation of the smoothed signal at point k, Sy ... 

Problems, however, arise if the intervals between the knots are not narrow enough and the spline begins to oscillate (cf. Figure 3.13). Also, in comparison to polynomial filters, many more coefficients are to be estimated and stored, since in each interval, different coefficients apply. An additional disadvantage is valid for smoothing splines, where the parameter estimates are not expectation-true. The statistical properties of spline functions are, therefore, more difficult to describe than in the case of linear regression (cf. Section 6.1). ... 

The method achieves self-consistency within a similar number of self-consistent field iterations as eigensolver-based approaches. However, the replacement of the standard diagonalization at each self-consistent iteration by a polynomial filtering step results in a significant speedup over methods based on standard diagonalization, often by more than an order of magnitude. Algorithmic details of a parallel... 

The proposed method combines the outer SCF iteration and the inner iteration required for diagonalization at each SCF step into one nonlinear subspace iteration. In this approach an initial subspace is progressively refined by a low degree Chebyshev polynomials filtering. This means that each basis vector is processed... 

Chebyshev polynomial filtering has long been utilized in electronic structure calculations (see e.g. [29,35-39]), focussing primarily on approximating the Fermi-Dirac operator. 

Methods used for preliminary examination of the data included smoothing the spectral data, multiplicative scatter correction, standard normal variance correction, baseline correction, and first- or second-derivative transformation of log (1/T) data. The smoothing and derivative transformations were based on the Savitzki-Golay second-order polynomial filter (22). 

The methods for data treatment included first-derivative transformation of log (1 /T) data with window size of 25 points, based on the Savitzki-Golay (22) polynomial filter. Calibration for quantitative determination of log SCC and absolute electrical conductivity was performed with PLS regression as described above. 

Frequency domain filtering is carried out by using the polynomial filters where each point x, is replaced by x ... 

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