Linear prediction analysis

A model which has found application in many areas of time series processing, including audio restoration (see sections 4.3 and 4.7), is the autoregressive (AR) or allpole model (see Box and Jenkins [Box and Jenkins, 1970], Priestley [Priestley, 1981] and also Makhoul [Makhoul, 1975] for an introduction to linear predictive analysis) in which the current value of a signal is represented as a weighted sum of P previous signal values and a white noise term ... 

The preceding sections showed the basic techniques of source filter separation using first cepstral then linear prediction analysis. We now turn to the issue of using these techniques to generate a variety of representations, each of which by some means describes the spectral envelope of the speech. 

Here Y z) is the speech, U z) is the source, V z) is the vocal tract and R z) is the radiation. Ideally, the transfer function H z) found by linear prediction analysis would be V(z), the vocal tract transfer function. In the course of doing this, we could then find U(z) and R z). In reality, in general H z) is a close approximation to F(z) but is not exactly the same. The main reason for this is that LP minimisation criterion means that the algorithm attempts to find the lowest error for the whole system, not just the vocal tract component. In fact, H z) is properly expressed as... 

Here Y(z) is the speech, U(z) is the source, V(z) is the vocal tract and R(z) is file radiation. Ideally, the transfer function H(z) found by linear-prediction analysis would... 

Figure B2.1.8 Dynamic absorption trace obtained with the dye IR144 in methanol, showing oscillations arising from coherent wavepacket motion (a) transient observed at 775 mn (b) frequency analysis of the oscillations obtained using a linear prediction, smgular-value-decomposition method.

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

Other data support the above picture. Hexanol adsorbs very weakly on Ag(l 10), more weakly than expected, and in any case less than on the (100) face.440 Such a poor adsorption on (110) faces has been explained in terms of steric hindrance caused by the superficial rails of atoms. Consistently, adsorption on the (110) face of Cu is vanishing small.587 Predictions based on a linear regression analysis of the data for pentanol (nine metals) give a value of-12 kJ mol 1 for Cu(l 10) and about -16 kJ mol 1 for Au(110). No data are available for polycrystalline Au, but Au(l 11) is placed in the correct position in the adsorption of hexanol.910 Thus, these data confirm the hydrophilicity sequence Hg < Au < Ag and the crystal face sequence for fee metals (111) < (100) < (110). 

Most of the supervised pattern recognition procedures permit the carrying out of stepwise selection, i.e. the selection first of the most important feature, then, of the second most important, etc. One way to do this is by prediction using e.g. cross-validation (see next section), i.e. we first select the variable that best classifies objects of known classification but that are not part of the training set, then the variable that most improves the classification already obtained with the first selected variable, etc. The results for the linear discriminant analysis of the EU/HYPER classification of Section 33.2.1 is that with all 5 or 4 variables a selectivity of 91.4% is obtained and for 3 or 2 variables 88.6% [2] as a measure of classification success. Selectivity is used here. It is applied in the sense of Chapter... 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

A drawback of the method is that highly correlating canonical variables may contribute little to the variance in the data. A similar remark has been made with respect to linear discriminant analysis. Furthermore, CCA does not possess a direction of prediction as it is symmetrical with respect to X and Y. For these reasons it is now replaced by two-block or multi-block partial least squares analysis (PLS), which bears some similarity with CCA without having its shortcomings. 

More recently, another linear discriminant analysis (LDA) model was constructed for a set of 157 compounds for which Pcaco-2 was measured [43]. This model, which applied DRAGON descriptors, achieved an accuracy of classification at 91 % for the training set and 84% for the test set. When this model was applied to predict a set of 241 drugs for which HIA data were available, good correlation (>81%) was achieved between the two ADME-Tox properties. 

It appears that a number of complications await the recovering bipolar patient after an episode of mania. For example, Lucas et al. ( 44) reported on a retrospective linear discriminant analysis of 100 manic episodes (1981 to 1985) during the recovery phase and found that the incidence of subsequent depression was 30% in the first month. Many episodes were transient, however, and did not necessarily require treatment. This phenomenon could be successfully predicted in 81% of cases in which there is a premorbid history of cyclothymia with either a personal or a family history of depression. The highly significant association between family history and postmanic depression again supports the hypothesis of a genetic basis for bipolar disorder. 

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