Vector of measurements

The matrix of measurements is rearranged into a stacked vector where each subsequent set of stream measurements follows the one above. As an example, the component flows in the Xf matrix are placed in the vector of measurements as follows ... 

In certain circumstances, the model equations may not have an explicit expression for the measured variables. Namely, the model can only be represented implicitly. In such cases, the distinction between dependent and independent variables becomes rather fuzzy, particularly when all the variables are subject to experimental error. As a result, it is preferable to consider an augmented vector of measured variables, y, that contains both regressor and response variables (Box. 

Now, if the vectors of measurement errors resulting from the resolution of each stage are added and multiplied by matrix A2, we obtain... 

As was indicated in Section 7.2, the vector of measurement adjustments, e, has a multivariate normal distribution with zero mean and covariance matrix V. Thus, the objective function value of the least square estimation problem (7.21), ofv = eT l> 1 e, has a central chi-square distribution with a number of degrees of freedom equal to the rank of A. 

As pointed out by Keller et al. (1992), if the process is truly at steady state, then estimation by the so-called direct method using the sample variance and covariance is adequate and simple to use. Let y(- be the ith element in a vector of measured variables, then the sample variance of the r repeated measurements of y, is given by... 

For a straight line, the function for ycaic describing a vector of measurements is ... 

A nonparametric approach can involve the use of synoptic data sets. In a synoptic data set, each unit is represented by a vector of measurements instead of a single measurement. For example, for synoptic data useful for pesticide fate, assessment could take the form of multiple physical-chemical measurements recorded for each of a sample of water bodies. The multivariate empirical distribution assigns equal probability (1/n) to each of n measurement vectors. Bootstrap evaluation of statistical error can involve sampling sets of n measurement vectors (with replacement). Dependencies are accounted for in such an approach because the variable combinations allowed are precisely those observed in the data, and correlations (or other dependency measures) are fixed equal to sample values. 

Figure 9.2 illustrates the inner workings of SHMPC during a given phase In this figure, let Xq be the initial state, d, be a vector of measured disturbances available after phase i, y, be a... 

X set of admissible state vectors y vector of measured output variables y set of admissible output vectors Euclidean norm... 

A general measurement protocol consists of collecting n noisy samples by changing the value of the explanatory variables. To a vector x then corresponds a vector of measurements y in each voxel of the image. This section presents several strategies for choosing x in order to optimize the quality of estimation a priori. 

---