Terms offset

Just as in the perceptron-Iike networks, an additional column of ones is added to the X matrix to accommodate for the offset or bias. This is sometimes explicitly depicted in the structure (see Fig. 44.9b). Notice that an offset term is also provided between the hidden layer and the output layer. 

In the above equation, the third term is introduced by the interference between the 7st RF field and the offset term SI, and it causes a small phase shift the fourth term is simply the BSFS introduced by the 1st RF field alone and the fifth term is created by the interference between the two RF fields. 

This model gives estimates of an offset term (Pq), a first-order effect (P,) of the first factor x, a first-order effect (Pj) of the second factor Xj, and a second-order... 

The first column of Table 14.3 gives the response notation (or, equivalently, the factor combination). The next eight columns list the eight factor effects of the model the three main effects (A, B, and C), the three two-factor interactions (AB, AC, and BC), the single three-factor interaction (ABC), and the single offset term (MEAN, analogous to PJ in the equivalent linear model). 

Note that for k factors, the model will have k + 1 parameters - one for each of the factors being investigated plus one for the offset term (Pq). After this model has been fitted to the data obtained from a screening design, the P s can be used to determine whether the factor is small and can be discarded, or is large and should be retained. 

This model gives estimates of an offset term (/ 0), a first-order effect (/ ,) of the first factor jcj, a first-order effect (/32) of the second factor x2, and a second-order interaction effect (/312) between the two factors. When the model of Equation 11.15 is fit to data from a 22 factorial design, the number of factor combinations (/ = 4) is equal to the number of parameters (p — 4), and the number of degrees of freedom... 

The difference here is that X is a matrix that contains responses from M (> 1) different X-variables, and b contains regression coefficients for each of the M X-variables. If X and b are augmented to include an offset term (as in Eqn. 8.7) the coefficients for MLR (bavg) are determined using the least squares method ... 

In heteronuclear Hartmann-Hahn experiments, a rf sequence is irradiated simultaneously at two frequencies vf and Vg. These experiments are conveniently analyzed in the corresponding doubly rotating frame (Ernst et al., 1987). In this frame, the free-evolution Hamiltonian contains offset terms and for / and 5 spins, isotropic homonuclear /-/ and 5-5 coupling terms and, and truncated heteronuclear coupling terms... 

In spin systems where the magnitudes of rf and offset terms are comparable, the condition (t 2tt ll Q° ir ) for Eq. (81) is violated and jog(O) tog yields only a poor approximation of the exact effective Hamiltonian even if the basis sequence is strictly cyclic. In this case, higher order terms with n > 1 must be taken into account and... 

As shown in Equation 20.9, drug action models based on the zero-asymptote model can be extended to include an offset term [EoFF (Ce,A)l the model of progress describing symptomatic benefit such as the relief of pain from a simple analgesic ... 

If we let (p Acu i, then eqn (9) corresponds to a calculation of the effect of a frequency offset term —Awl in the rotating frame Hamiltonian. Thus a modulation by exp (iqAcDt is caused by the offset, shifting the spectrum to Aco as seen in fig. 2. 

Some nomenclature is in order here bo is called the constant or offset term or intercept b is the regression coefficient x is the independent or predictor variable and y is the dependent or response variable is the residual. More regression nomenclature can be found, e.g., in Draper Smith [Draper Smith 1998],... 

If I > J, then in general 1 range(X). Although 1 e range( noiseless X), this property is destroyed upon adding noise to X. In the case I < J and if X has full rank I, then 1 is by definition in the range of X, whether or not there exists an offset term lp/. Hence, in the... 

OFFSET - Term used to describe the difference between the set point and the actual operating or control point. 

The bias is apparent from the offset term (1 — A)Cr, so that AT / 0 even when Cp = 0. Now two calibrating runs are needed. For instance, with an empty sample pan, a first run fixes (1 — X)Cr/KX from the average signal. In a second run, with Cp > 0, the average signal gives KX and the cyclic signal determines Cr/K. In a subsequent experiment, the (complex) cyclic heat capacity measurement is then... 

From this we can see that each field component consists of a term (C2 or C3) that is modulated through a phase angle and a constant offset term (Ci or C4) that remains fixed as the pixels switch. This implies that the Fourier transform will contain some form of undiffracted DC term... 

If the model contains an offset term bo then the X-matrix has to be augmented by a column containing a constant value of one. Furthermore, X and y have to be mean-centered before applying equation (18). Scores mmlr of the direction mlr possess the maximum possible correlation coefficient with y. Notice that calculation of Amlr requires the inversion of the covariance matrix C (equation 7). An inversion of C, however, is impossible if X contains highly correlated features or if the number of features is larger than the number of objects such conditions are common with chemical data. 

However, for many practical applications, data are at least mean centred (see Chapter 3), that is, an offset term is subtracted from both the X- and y-blocks from a regression standpoint, centring leads to model without intercept, so that Equation (17) becomes ... 

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