Measurement vector

PCR is based on a PCA input data transformation that by definition is independent of the Y-data set. The approach to defining the X-Y relationship is therefore accomplished in two steps. The first is to perform PCA on the. Y-data, yielding a set of scores for each measurement vector. That is, if xk is the fcth vector of d measurements at a time k, then zk is the corresponding kth vector of scores. The score matrix Z is then regressed onto the Y data, generating the predictive model... 

Since the concept of observability was primarily defined for dynamic systems, observability as a property of steady-state systems will be defined in this chapter. Instead of a measurement trajectory, only a measurement vector is available for steady-state systems. Estimability of the state process variables is the concept associated with the analysis of a steady-state situation. 

In the absence of gross errors, the measurement vector can be written as... 

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... 

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 ... 

The measurement vector, the covariance matrix of the measurement errors, and the residuum vector are as follows ... 

An important question arises where in the 21-dimensional space (or 3-dimensional space) can all the measured vectors be found Is it possible to restrict the potential locations of the spectral vectors in a subspace A first restriction is obvious as absorbances can only be positive, only those parts of the space with positive coordinates are available to the spectral vectors. Is there anything more specific Figure 5-9 represents this question in a 3-dimensional space. 

Figure 5-25. The relationship between the original measurement vector y,its projection into the eigenvector space and the true ...

There is a basic difference between most of the better known methods of cluster analysis and the FCV family of clustering algorithms. That difference concerns the traditional requirement that every measurement vector e eventua11 y assigned to one, and 2iLiy one, o f h cluster classes. In FCV clustering. 

Figure 1. Orthagonal distance from measurement vector Xj to...

The computation of the shared membership values u j by Equation 6 usually results in relatively small values being assigned to outliers or noisy measurement vectors. It is not difficult to locate these values in the final membership matrix, and the corresponding data vectors can be singled out for deletion, or closer examination. 

Use of the subjective procedure may not be wholly Inappropriate for Investigations such as this one, where the total number of measurement vectors Is relatively small. Typically, the procedure required comparison of the membership of 25-35 vectors In 2-4 classes for the NILU data. For studies where the data consists of many more measurement vectors, and may require Investigating the existence of a number of classes In the data, such an approach may become Impractical and the accuracy of the results questionable. This should not be the case for the validity discriminant, where the Increase In measurement vectors and data classes results only In an Increase In computation time. 

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. 

Measurement Vector A term referring to a collection of measurement variables for a sh e sample. For example, the near-infrared spectrum of a sample is a measnrement vector. iSee also Spectrum.)... 

The residuals are the portion of the sample measurement vector that is not explained using a given number of PCs. 

When the measurement vectors are continuous in nature (e.g., infrared spectra) the loadings can help determine the number of relevant PCs. 

Raw Measurement Plot In Figure 4.48 the raw data for the four unknown samples are plotted with the mean of the training samples for the four classes (A-D). The mean is the average of the measurement vectors for all samples within a class and is used as a representation of the expected features for samples that belong to the respective class. The differences in the features may indicate the source of problems for unknowns where the classifications are suspect. Figure 4.48 confirms the other diagnostic tools, which indicate that ( ) unknown 1 is not a member of any class, ( ) unknown 2 is a member of class A, (c) unknown 3 is a member of class D, and (rf) unknown 4 is most like A but has different features. 

Independently exainining the different contributions to F can help determine wh) a sample is excluded from a particular class. The PCA contribution reflects structure in the residual spectrum, which is an indication of additional sources of variation present in the unknown measurement vector (e.g., increased noise level, an unmodeled interferent, or a noise spike). The distance contribution becomes significant when the magnitude of the features in the unknown are unlike the training set data. This can occur when additional sources of variation are present or wt en the concentrations of the expected components are outside the training set range. 

In Equation 5.28, s is a function of the concentration residuals observed during calibration, r is tlie measurement vector for the prediction sample, and R contains the calilxation measurements for the variables used in the model. Because the assumptions of linear regression are often not rigorously obeyed, the statistical pret ion error should be used empirically rather than absolutely. It is useful for validating the prediction samples by comparing the values for... 

Fraction of the total variation explained for given numbers of faaors. Calculated for both the concentrations and the measurement vectors. 

It is used to ideniifr- samples t dih unusual measurement vectors. 

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