Linear prediction equation

This is known as the (forward) linear prediction equation, because the time series can be used to predict subsequent data points. For a single sinusoidal component, Eq. (101) reduces to xn = c exp[(Z> + iu>)n At, where the phase... 

If N data points are sampled during a FID, then it is possible to write a total of (N - M) linear prediction equations of the form of Eq. (102). The resulting matrix equation is... 

The linear prediction equations can be solved by the autocorrelation or covariance methods. Each involves solving the equations by means of efficient matrix inversion. 

Recall that Equation 13.18 is exactly the same as the linear prediction Equation 12.16, where = fli, 02,..., Op are the predictor coefficients and x[n] is the error signal e n. This shows that the result of linear prediction gives us the same type of transfer function as the serial formant synthesiser, and hence LP can produce exactly the same range of frequency responses as the serial formant S5mthesiser. The significance is of course that we can derive the linear prediction coefficients automatically fi om speech and don t have to make manual or perform potentially errorful automatic formant analysis. This is not however a solution to the formant estimation problem itself reversing the set of Equations 13.14 to 13.18 is not trivial, meaning that while we can accurately estimate the all-pole transfer function for arbitrary speech, we can t necessarily decompose this into individual formants. 

When structure-property relationships are mentioned in the current literature, it usually implies a quantitative mathematical relationship. Such relationships are most often derived by using curve-fitting software to find the linear combination of molecular properties that best predicts the property for a set of known compounds. This prediction equation can be used for either the interpolation or extrapolation of test set results. Interpolation is usually more accurate than extrapolation. 

Once the descriptors have been computed, is necessary to decide which ones will be used. This is usually done by computing correlation coelficients. Correlation coelficients are a measure of how closely two values (descriptor and property) are related to one another by a linear relationship. If a descriptor has a correlation coefficient of 1, it describes the property exactly. A correlation coefficient of zero means the descriptor has no relevance. The descriptors with the largest correlation coefficients are used in the curve fit to create a property prediction equation. There is no rigorous way to determine how large a correlation coefficient is acceptable. 

An example for the predictive power of the TS trajectory is shown in Fig. 4. This figure shows a random instance of the TS trajectory (black) and a reactive trajectory (red) for the linearized Langevin equation (15) with N = 2 degrees of... 

Using predictive models for measuring environmental chemodynamics of organic pollutants in complex mixtures requires literature data on partition coefficient values. In some cases the values cited are not strictly experimental, being derived from linear free energy relations, while in others wide variations are reported in experimental values. The main problem is how one should evaluate which values are correct. Thus, Table 2 provides some basis to discriminate between reported values of partition coefficients, as well as predictive equations for partition coefficient calculations [21,62,65-85]. 

Improvement in the accuracy of the linear yield equations might be obtained by using more independent variables in the prediction process. For instance, the use of aniline point and volumetric average boiling point (VABP) might lead to better accuracy. Unfortunately, the published data do not include these two variables. 

The compartmental model gives rise to a system of two linear differential equations whose forcing term (i.e., the drug intake) is a periodic function (ref. 9). After a transient period the solution of the differential equations is also a periodic function. This periodic solution predicts the drug concentration... 

One way around this problem is to restrict the potential changes to very small excursions. This approach is based on the principle that any curve can be approximated by a series of small straight line segments. Applying this principle to the exponential functions that cause all the trouble leads to a far more manageable set of linear differential equations, which can usually be solved, allowing a prediction of the expected response. 

It would be interesting to carry out experiments on S/F structures with non-collinear magnetization in order to observe this new type of superconductivity. As follows from a semiquantitative analysis, the best conditions to observe the Josephson critical current caused by the TC are high interface transparency (small y ) and low temperatures. These conditions are a bit beyond our quantitative study. Nevertheless, all qualitative features predicted here (angle dependence, etc) should remain in a general case when one has to deal with the non-linear Usadel equation. 

The scattering equations are illustrated in Figs. 10.6b and 10.7. In linear predictive coding of speech, this structure is known as the Kelly-Lochbaum scattering junction,... 

The retardation factors of the four radioelements for four hypothetical HLW compositions were derived using the prediction equations. (The retardation factor is the ratio of the solution velocity to the radioelement velocity in a system of solution flow through a porous medium and increases linearly with Kd.) The four hypothetical HLW solutions broadly represented dilute/non-complexed, dilute/complexed, concentrated/noncomplexed, and concentrated/complexed HLW. Dilute waste had low concentrations while concentrated waste had high concentrations of Na+, NaOH, and NaAlO,. Non-complexed waste had no HEDTA or EDTA while complexed waste had 0.1M HEDTA/0.05M EDTA. 

Some researchers have used approximate microscopic descriptions to develop more rigorous macroscopic constitutive laws. A microstructural model of AC [5] linked the directionality of mechanical stiffness of cartilage to the orientation of its microstructure. The biphasic composite model of [6] uses an isotropic fiber network described by a simple linear-elastic equation. A homogenization method based on a unit cell containing a single fiber and a surrounding matrix was used to predict the variations in AC properties with fiber orientation and fiber-matrix adhesion. A recent model of heart valve mechanics [8] accounts for fiber orientation and predicts a wide range of behavior but does not account for fiber-fiber interactions. 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Equation (96) is known as the linear diffusion equation since the lowest-order field dependence is linear. Thus we have a microscopic derivation of the Einstein relation, eqn. (98). This relation is normally derived from quite different considerations based on setting the current equal to zero in the linear diffusion equation and comparing the concentration profile C (x) with that predicted by equilibrium thermodynamics. 

Linear prediction and state-space methods are grouped together here because, although the philosophy behind the two methods is different, the mathematics involved is very similar. In both cases, the data-fitting problem is made linear by constructing a matrix from the observed data points, and the model equation is then solved by linear means. The nonlinear model parameters are... 

A potential concern in the use of a two-level factorial design is the implicit assumption of linearity in the true response function. Perfect linearity is not necessary, as the purpose of a screening experiment is to identify effects and interactions that are potentially important, not to produce an accurate prediction equation or empirical model for the response. Even if the linear approximation is only very approximate, usually sufficient information will be generated to identify important effects. In fact, the two-factor interaction terms in equation (1) do model some curvature in the response function, as the interaction terms twist the plane generated by the main effects. However, because the factor levels in screening experiments are usually aggressively spaced, there can be situations where the curvature in the response surface will not be adequately modeled by the two-factor interaction... 

These symmetry relationships do not depend on the specific features of any given model but follow quite generally from the linear phenomenological equations of nonequilibrium thermodynamics. Therefore, any linear model that does not predict these relations is likely to be incorrect. 

For a small step in temperature, the fictive temperature Tf is never far from the actual temperature T hence r, as given by the Narayanaswamy or the Adam-Gibbs equations, doesn t vary much with time. Equation (4-27) then simplifies to the ordinary linear KWW equation, Eq. (4-1). For large AT, varies during the relaxation, and the asymmetry discussed earlier is predicted. Note, however, that in Eq. (4-27) is assumed to be a constant this is not strictly valid for large changes in temperature, but is usually acceptable even when AT is a few tens of degrees. 

Many constitutive equations have been proposed in addition to those indicated above, which are special cases of fluids with memory. Most of these expressions arise from the generalization of linear viscoelasticity equations to nonlinear processes whenever they obey the material indifference principle. However, these generalizations are not unique, because there are many equations that reduce to the same linear equation. It should be noted that a determined choice among the possible generalizations may be suitable for certain types of fluids or special kinds of deformations. In any case, the use of relatively simple expressions is justified by the fact that they can predict, at least qualitatively, the behavior of complex fluids. 

The sign of V( Sn, H) across an aliphatic carbon atom is positive with the exception of [Me3Sn] and diorganostannylenes. The large data set of V( Sn, H) available for methyltin compounds [3] allows one to compare these data with values for V( Sn CMe) which gives an almost linear relationship [31]. A linear relationship also exists between V( Sn, H) and V( Pb, H) of related methyltin and -lead compounds [71]. Furthermore, it has been suggested that the values V( Sn, H) in methyltin compounds can be used to predict (equation (5)) the bond angles C-Sn-C in numerous methyltin derivatives [108]. 

In a time sequence of data points, the value of a particular data point, can be estimated from a linear combination (hence the name linear prediction) of the data points that immediately precede it (Hoch and Stem, 1996), as shown in the following equation, in which a, 02 the LP coejficients (also called the LP prediction filter) ... 

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