Linear prediction singular value decomposition

Figure 8. Top Transient dichroism signal of I2 in n-hexane. Bottom Linear prediction singular-value decomposition (LP-SVD) for the pump and probe wavelengths of 580 and 475 nm, respectively. Note the overall delay of the negative signal, which is modulated by the B state frequency. (From Ref. 28.)...

Linear prediction singular value decomposition Linear prediction total least-squares Least-squares... 

A similar zone folding also occurs at Cs/Pt(lll) and the phonon mode appears as a small dip in the Fourier spectrum in Figure 19.4. A detailed analysis of the time domain data by linear prediction singular value decomposition has been performed and a decomposition of the time-domain data to phonon modes and alkali-substrate stretching modes has been carried out. Coherent nuclear motions have been observed on substrates other than Pt. Figure 19.5a shows time-resolved SHG traces... 

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 nm (b) frequency analysis of the oscillations obtained using a linear prediction, singular-value-decomposition method.

The multiple linear regression (MLR) method was historically the first and, until now, the most popular method used for building QSPR models. In MLR, a property is represented as a weighted linear combination of descriptor values F=ATX, where F is a column vector of property to be predicted, X is a matrix of descriptor values, and A is a column vector of adjustable coefficients calculated as A = (XTX) XTY. The latter equation can be applied only if the matrix XTX can be inverted, which requires linear independence of the descriptors ( multicollinearity problem ). If this is not the case, special techniques (e.g., singular value decomposition (SVD)26) should be applied. 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a true concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. 

The states in Eqn (25.2) are now being formed as linear combinations of the -step ahead predicted outputs k= 1, 2,. ..). The literature on state space identification has shown how the states can be estimated directly from the process data by certain projections. (Verhaegen, 1994 van Overschee and de Moor, 1996 Ljung and McKelvey, 1996). The MATLAB function n4sid (Numerical Algorithms for Subspace State Space System Identification) uses subspace methods to identify state space models (Matlab 2000, van Overschee and de Moor, 1996) via singular value decomposition and estimates the state x directly from the data. 

---