Linear prediction backward

Fig. 5.26 Parameters for a backward Linear Prediction with ID WIN-NMR (above the...

Figure 4.36. O spectra of ethyl acetate recorded (a) with and (b) without the RIDE sequence. The severe baseline distortion in (b) arises from acoustic ringing in the probehead. Spectrum (c) was from the same FED of (b) but this had the first 10 data points replaced with backward linear predicted points, computed from 256 uncorrupted points. The spectra are referenced to D2O and processed with 100 Hz line-broadening.

An alternative approach now available with modem processing software is to collect the distorted FID with the simple one-pulse-acquire sequence and to replace the early distorted data points with uncormpted points generated through backward linear prediction. Fig. 4.36c was produced from the same raw data as 4.36b, but with the first 10 data points of the FID replaced with predicted points. The baseline distortion is completely removed and there is no signal-to-noise loss through experimental imperfections, as occurs for the RIDE data set. 

If we find that the ringdown delay required to produce an artifact-free spectrum yields an unacceptably low signal-to-noise ratio (this is a subjective decision), then we can use backwards linear prediction to correct the first few corrupted points in the FID. Linear prediction is generally accepted in the NMR community as a reasonable method for avoiding choosing between a spectrum with a low signal-to-noise ratio and a spectrum marred by ringdown artifacts. 

GHJCOSE 1D H GH 013001.FID. Note the baseline artifacts introduced by the truncated FID. In the Linear Prediction (LP) dialog box make sure that the Execute Backward LP option is enabled and the Execute Forward LP option disabled. Set LP backward to Point to 124. Following the rules given above vary the residual parameters First Point used for LP (recommended 196), Last Point used for for LP (recommended 2047) and Number of Coefficients (recommended 128 or larger). Carefully inspect the resulting spectra with respect to spectral resolution and signal shapes and compare it with the spectrum obtained without LP. 

A variety of algorithms exist for finding the coefficients and multipliers y3y, with which the experimental data can be extrapolated forwards or backwards all share some common problems. Linear prediction has difficulty distinguishing between positive and negative decay constants Ty, and so is best suited to time series in which all the decay constants are either positive or negative, allowing spurious values to be rejected. The number m of coefficients to be used has to be decided by the ex-... 

To provide a comparison, we also evaluated forecast errors and cost for the rules used by the plarmers at this retailer, the k median method based on store descriptors, alone and combined with sales mix differences, and two standard approaches to variable selection in linear regression, since the problem of choosing k test stores and a linear prediction function based on test sales at these stores can be viewed as choosing the best k out of n possible variables in a linear regression. Given actual sales Sp and test sales Sjp for i = 1,. . . n and p = 1,. . . m, we used the forward selection and backward elimination methods (Myers (1990)) to choose k out of the n test... 

Fuchs, 1995] Fuchs, H. (1995). Improving MPEG audio coding by backward adaptive linear stereo prediction. In Proc. of the 99th. AES-Convention. Preprint 4086. 

Linear models were generated using multiple linear regression analysis techniques. Several methods were utilized to develop models for evaluation, including stepwise addition, backward elimination, and leaps and bounds regression techniques. The models were evaluated with respect to the multiple correlation coefficient (r), the standard error (s), and predictive ability of the model. 

In this section, a stochastic compartmental model is employed to estimate the number of compartments (necessary to characterize the flow) and the intensities of forward and backward flows between compartments in an open-flow adsorber without adsorbate. In addition, this model and the linear batch model are combined into one for predicting breakthrough curves in an open-... 

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