Linear combination of variables

In addition to the graphical representations we also obtain a set of simple linear combinations of variables that enable us to... 

Differences between PIS and PCR Principal component regression and partial least squares use different approaches for choosing the linear combinations of variables for the columns of U. Specifically, PCR only uses the R matrix to determine the linear combinations of variables. The concentrations are used when the regression coefficients are estimated (see Equation 5.32), but not to estimate A potential disadvantage with this approach is that variation in R that is not correlated with the concentrations of interest is used to construct U. Sometiraes the variance that is related to the concentrations is a verv... 

Various approaches can be taken for constructing the U matrix. With PCR, a principal components analysis is used because PCA is an efficient method for finding linear combinations of variables that describe variation in the row space of R (See Section 4.2.2). With analytical chemistry data, it is usually possible to describe the variation in R using significantly fewer PCs than the number of original variables. This small number of columns effectively eliminates the matrix inversion problem. 

In the field of chemometrics, PCR and PLS are the most widely used of the inverse calibration methods. Tliese methods solve the matrix inversion problem inherent to the inverse methods by using a linear combination of variables in... 

The ILDM technique proposed by Maas and Pope overcomes this problem by describing geometrically the optimum slow manifold of a system. The criterion for reduction is based on the time-scales of linear combinations of variables and not on species themselves. The main advantage of the technique is that it requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steady-state. The only inputs to the system are the detailed chemical mechanism and the number of degrees of freedom required for the simplified scheme. The ILDM method then tabulates quantities such as rates of production on the lower-dimensional manifold. For this reason, it is necessarily better suited to numerical problems since it does not result in sets of rate... 

The first factor, defined as the linear combination of the original variables, obtained by the above described procedure will account for more of the variance in the data set than any other combination of variables. The second factor will be the linear combination of variables that accounts for most of the residual variance after the effect of the first factor has been removed from the data. Subsequent factors are defined similarly until all variance in the data is exhausted. In case the original variables are uncorrelated, the factor analysis solution requires as many factors as there are variables. However, in most data sets, many variables are more or less correlated and the variance in the data can be described by a smaller set of factors than there are variables. Therefore the data reduction is applicable. 

For any linear combination of variables defining a new variable AT given by... 

As in the previous section, we are interested in linear combinations of variables, with the goal of determining that combination which best summarizes the n-dimensional distribution of data. We are seeking the Unear combination with the largest variance, with normalized coefficients applied to the variables used in the linear combinations. This axis is the so-called rst principal axis or first principal component. Once this is determined, then the search proceeds to find a second normalized linear combination that has most of the remaining variance and is uncorrelated with the first principal component. The procedure is continued, usually imtil all the principial components have been calculated. In this case, p = n and a selected subset of the principal components is then used for further analysis and for interpretation. 

Least squares models, 39, 158 Linear combination, normalized, 65 Linear combination of variables, 64 Linear discriminant analysis, 134 Linear discriminant function, 132 Linear interpolation, 47 Linear regression, 156 Loadings, factor, 74 Lorentzian distribution, 14... 

Linear discriminant analysis (LDA) is aimed at finding a linear combination of descriptors that best separate two or more classes of objects [100]. The resulting transformation (combination) may be used as a classifier to separate the classes. LDA is closely related to principal component analysis and partial least square discriminant analysis (PLS-DA) in that all three methods are aimed at identifying linear combinations of variables that best explain the data under investigation. However, LDA and PLS-DA, on one hand, explicitly attempt to model the difference between the classes of data whereas PCA, on the other hand, tries to extract common information for the problem at hand. The difference between LDA and PLS-DA is that LDA is a linear regression-like method whereas PLS-DA is a projection technique... 

An important and commonly used procedure which generally satisfies these criteria is principal components analysis. Before this specific topic is examined it is worthwhile discussing some of the more general features associated with linear combinations of variables. 

The so-called factor loadings P are calculated from the matrix by singular value decomposition. T is called the scores matrix. The inner workings of this algorithm are not of interest here. Suffice to say that, if X has n variables, P will contain less than n linear combinations of variables which yet retain the information present in the original data. 

Formally, the purpose of PCA is to create or find such correlations and create linear combinations of variables that reduce the dimensionality of the data. In this example with three variables, three principal components exist ... 

Principal component analysis (PCA) A technique us to reduce the dimensionality of data and to reveal groups works based on creating new linear combinations of variables. 

On the surface, factor analysis and principal component analysis are very similar. Both rely on an eigenvalue analysis of the covariance matrix, and both use linear combinations of variables to explain a set of observations. However, in PCA the quantities of interest are the observed variables themselves the combination of these variables is simply a means for simplifying their analysis and interpretation. Conversely, in factor analysis the observed variables are of little intrinsic value what is of interest is the underlying factors. 

Although the previous paragraph describes the manipulated variables as control valves, there are many choices available other than just the individual valves. For example, many columns have reflux ratio as a manipulated variable for either inventory or composition control. When ratios and linear combinations of variables are included, the choice of a manipulator for a given loop broadens considerably for a simple two-product column. However, the steady-state and dynamic degrees of freedom remain unchanged as two and three respectively, totalling five. 

---