Bias vector

When we import the feed type, Aspen HYSYS shows the details of the feed type as shown in Figure 4.57. The Kinetic Lump Weight Percents indicate the starting composition of the kinetic lumps and the Methyls and Biases indicate how various bulk properties affect the final lump composition. Aspen HYSYS uses the biases to calculate actual kinetic lumps with the bias vectors. The bias vectors essentially correct the kinetic lump composition for the measured bulk properties (which we will enter) from the reference bulk properties in the feed type. We will not modify any information in this window and simply close it to continue the feed configuration process. 

To comphcate matters further, because of the time dependence, leaks, or accumulation, the constraints might not actually apply such that there is a vector of unknown plant bias associated with the constraints. 

If the mathematical model represents adequately the physical system, the error term in Equation 2.3 represents only measurement errors. As such, it can often be assumed to be normally distributed with zero mean (assuming there is no bias present in the measurement). In real life the vector e, incorporates not only the experimental error but also any inaccuracy of the mathematical model. 

Figure 20 shows more definitively how the location and orientation of a hyperplane is determined by the projection directions, a and the bias, o- Given a pattern vector x, its projection on the linear discriminant is in the a direction and the distance is calculated as d(x ) / cf The problem is the determination of the weight parameters for the hyper-plane ) that separate different pattern classes. These parameters are typically learned using labeled exemplar patterns for each of the pattern classes. 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

In the following discussion, one or several sensor failures are assumed, so a constant bias of magnitude mb is added to the measurement vector y. In the presence of a failure in the sensors, the measurement equation takes the form... 

Case 1. A bias present in the measurement of /2 was identified by the sequential processing of the measurements (see Example 7.2). We augment, in consequence, the vector of parameters of the original problem by adding an additional component to represent the uncertain parameter (bias term). 

Brm (gxj) matrix with e, column vectors indicating bias positions... 

Consequently, in the following discussion a sensor failure that affects one or more sensors will be assumed to add a constant bias of magnitude Sy to the measurement vector, y. In the presence of a sensor failure, let us consider the following models for the process and measurement ... 

Note This is a more restricted formulation than the one posed in Eqs. (8.35) and (8.36), since only bias in the measurements is considered and an autonomous system is assumed. Also, here vector Sy stands for vector g in Eq. (8.36). 4k... 

The behavior of the detection algorithm is illustrated by adding a bias to some of the measurements. Curves A, B, C, and D of Fig. 3 illustrate the absolute values of the innovation sequences, showing the simulated error at different times and for different measurements. These errors can be easily recognized in curve E when the chi-square test is applied to the whole innovation vector (n = 4 and a = 0.01). Finally, curves F,G,H, and I display the ratio between the critical value of the test statistic, r, and the chi-value that arises from the source when the variance of the ith innovation (suspected to be at fault) has been substantially increased. This ratio, which is approximately equal to 1 under no-fault conditions, rises sharply when the discarded innovation is the one at fault. 

Fig. 3. The multilayered structure considered. The arrows show the bias current. In the case of positive (negative) chirality the magnetization vector M of the layer F3 makes an angle 3a (—a) with the z- axis, i.e. in the case of positive chirality the vector M rotates in one direction if we go over from one F layer to another whereas it oscillates in space in the case of negative chirality.

From Eq. (G.6) we obtain appropriate probability information via the system operators T = jl T2), while the transformation formulas (G.4) correspond to proper truth-values consistent with Eq. (G.7). The new eigenvectors here are obtained as a superposition of vectors corresponding to legitimate input values for p = 1. For T2 = 0, Eq. (G.6) gives the classical result p =, i.e., no information at all. Consequently r yields a bias to the no information platform. Note that the operator T, or the truth matrix T, is a nonclassical quantity (operator), which will play a crucial role below serving as the square root of the relevant bias" part of the system operator transforming the input information accordingly. 

The optimal model is determined by finding the minimum error between the extracted concentrations and the reference concentrations. Cross-validation is also used to determine the optimal number of model parameters, for example, the number of factors in PLS or principal components in PCR, and to prevent over- or underfitting. Technically, because the data sets used for calibration and validation are independent for each iteration, the validation is performed without bias. When a statistically sufficient number of spectra are used for calibration and validation, the chosen model and its outcome, the b vector, should be representative of the data. 

Equations (4.60) and (4.61) are also able to yield the vector recurrence relations for the case of a skew bias field, that is, when vectors h and n are not parallel. In this case one should ascribe to each bi as many as 21+1 components, corresponding to different values of the azimuthal index m. Another problem, involving vector recurrence relations, is a steady-state nonlinear oscillations of bi in a high-AC field. To study the harmonic content of the nonlinear response, one has to expand all the moments b t)l in the Fourier series. Then the Fourier coefficients may be treated as components of a... 

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