Multivariate variables

Cjk Covariance between variables j and k C is the covariance matrix. d Distance between two objects in multivariate variable space for instance... 

It should be noted that there are other multivariate variable selection methods that one could consider for their application. For example, the interactive variable selection (IVS) method71 is an actual modification of the PLS method itself, where different sets of X-variables are removed from the PLS weights (W, see Equation 8.37) of each latent variable in order to assess the usefulness at each X-variable in the final PLS model. 

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

Multivariate statistics is the discipline to analyze data, to elucidate the intrinsic structure within the data, and to reduce the number of variables needed to describe the data. 

In multivariate least squares analysis, the dependent variable is a function of two or more independent variables. Because matrices are so conveniently handled by computer and because the mathematical formalism is simpler, multivariate analysis will be developed as a topic in matrix algebra rather than conventional algebra. 

We have already seen the normal equations in matrix form. In the multivariate case, there are as many slope parameters as there are independent variables and there is one intercept. The simplest multivariate problem is that in which there are only two independent variables and the intercept is zero... 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

Multivariable control strategies utilize multiple input—multiple output (MIMO) controUers that group the interacting manipulated and controlled variables as an entity. Using a matrix representation, the relationship between the deviations in the n controlled variable setpoints and thek current values,, and the n controUer outputs, is... 

Three examples of simple multivariable control problems are shown in Fig. 8-40. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate or Wg affects both of the controlled variables andi. For the pH neutrahzation process in Figure 8-40(Z ), liquid level h and the pH of the exit stream are to be controlled by adjusting the acid and base flow rates and w>b. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit strong process interacHons. In contrast, the process interactions for the gas-liquid separator in Fig. 8-40(c) are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirec t effect on one controlled variable, pressure P. 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequahty constraints are present and all equahty constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objec tive func tion. Veiy few reahstic problems in process optimization are unconstrained. However, it is desirable to have efficient unconstrained optimization techniques available since these techniques must be applied in real time and iterative calculations cost computer time. The two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

The for-end loop in examp88.m that employs equation (8.76), while appearing very simple, is in faet very powerful sinee it ean be used to simulate the time response of any size of multivariable system to any number and manner of inputs. If A and B are time-varying, then A(r) and B(r) should be ealeulated eaeh time around the loop. The author has used this teehnique to simulate the time response of a 14 state-variable, 6 input time-varying system. Example 8.10 shows the ease in whieh the eontrollability and observability matriees M and N ean be ealeulated using c t r b and ob s v and their rank eheeked. 

The relationship between a criterion variable and two or more predictor variables is given by a linear multivariate model ... 

The multivariable modeling/control package is able to hold more tightly against constraints and recover more quickly from disturbances. This results in an incremental capacity used to justify multivariable control. An extensive test run is necessary to measure the response of unit variables. 

Finally, process control systems allow the unit to operate smoothly and safely. At the next level, an APC package (whether within the DCS framework or as a host-based multivariable control system) provides more precise control of operating variables against the unit s constraints. It will gain incremental throughput or cracking severity. 

The Kd(X) values thus obtained (Table 3) were analyzed by the multivariant technique using such parameters as n, a0, Bt, Ibmch, and Ihb, where Bj is a STERIMOL parameter showing the minimum width of substituents from an axis connecting the a-atom of the substituents and the rest of molecule, and Ibrnch, an indicator variable representing the number of branches in a substituent. 

It should be noted that in this example the performance of only one variable, the three analysts, is investigated and thus this technique is called a one-way ANOVA. If two variables, e.g. the three analysts with four different titration methods, were to be studied, this would require the use of a two-way ANOVA. Details of suitable texts that provide a solution for this type of problem and methods for multivariate analysis are to be found in the Bibliography, page 156. 

Guasch H, Leira M, Montuelle B, Geiszinger A, Roulier JL, Tomes E, Serra A (2009) Use of multivariate analyses to investigate the contribution of metal pollution to diatom species composition search for the most appropriate cases and explanatory variables. Hydrobiologia 627 143... 

Analytical results are often represented in a data table, e.g., a table of the fatty acid compositions of a set of olive oils. Such a table is called a two-way multivariate data table. Because some olive oils may originate from the same region and others from a different one, the complete table has to be studied as a whole instead as a collection of individual samples, i.e., the results of each sample are interpreted in the context of the results obtained for the other samples. For example, one may ask for natural groupings of the samples in clusters with a common property, namely a similar fatty acid composition. This is the objective of cluster analysis (Chapter 30), which is one of the techniques of unsupervised pattern recognition. The results of the clustering do not depend on the way the results have been arranged in the table, i.e., the order of the objects (rows) or the order of the fatty acids (columns). In fact, the order of the variables or objects has no particular meaning. 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

When we consider the multivariate situation, it is again evident that the discriminating power of the combined variables will be good when the centroids of the two sets of objects are sufficiently distant from each other and when the clusters are tight or dense. In mathematical terms this means that the between-class variance is large compared with the within-class variances. 

The Mahalanobis distance representation will help us to have a more general look at discriminant analysis. The multivariate normal distribution for w variables and class K can be described by... 

UNEQ can be applied when only a few variables must be considered. It is based on the Mahalanobis distance from the centroid of the class. When this distance exceeds a critical distance, the object is an outlier and therefore not part of the class. Since for each class one uses its own covariance matrix, it is somewhat related to QDA (Section 33.2.3). The situation described here is very similar to that discussed for multivariate quality control in Chapter 20. In eq. (20.10) the original variables are used. This equation can therefore also be used for UNEQ. For convenience it is repeated here. 

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