Canonical correlation

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

This is already a considerable improvement. The natural question then is Which linear combination of K-variables yields the highest R when regressed on the X-variables in a multiple regression Canonical correlation analysis answers this question. 

The particular linear combinations of the X- euid F-variables achieving the maximum correlation are the so-called first canonical variables, say tj = Xw, and u.-Yq,. The vectors of coefficients Wj and q, in these linear combinations are the canonical weights for the X-variables and T-variables, respectively. For the data of Table 35.5 they are found to be Wj = [0.583, -0.561] and qj = [0.737,0.731]. The correlation between these first canonical variables is called the first canonical correlation, p,. This maximum correlation turns out to be quite high p, = 0.95 R = 0.90), indicating a strong relation between the first canonical dimensions of X and Y. 

The next pair of canonical variates, t2 and U2 also has maximum correlation P2, subject, however, to the condition that this second pair should be uncorrelated to the first pair, i.e. t t2 = u U2 = 0. For the example at hand, this second canonical correlation is much lower p2 = 0.55 R = 0.31). For larger data sets, the analysis goes on with extracting additional pairs of canonical variables, orthogonal to the previous ones, until the data table with the smaller number of variables has been... 

Thus, we see that CCA forms a canonical analysis, namely a decomposition of each data set into a set of mutually orthogonal components. A similar type of decomposition is at the heart of many types of multivariate analysis, e.g. PCA and PLS. Under the assumption of multivariate normality for both populations the canonical correlations can be tested for significance [6]. Retaining only the significant canonical correlations may allow for a considerable dimension reduction. 

Computationally, canonical correlation analysis can be implemented using the following steps, where it is assumed that the data X and Y are mean-centered. 

It should be appreciated that canonical correlation analysis, as the name implies, is about correlation not about variance. The first step in the algorithm is to move from the original data matrices X and Y, to their singular vectors, Ux and Uy, respectively. The singular values, or the variances of the PCs of X and Y, play no role. 

C.J.F. ter Braak, Interpreting canonical correlation analysis through biplots of structure correlations and weights. Psychometrika, 55 (1990) 519-531. 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

Another way for BOD estimation is the use of sensor arrays [37]. An electronic nose incorporating a non-specific sensor array of 12 conducting polymers was evaluated for its ability to monitor wastewater samples. A statistical approach (canonical correlation analysis) showed a linear relationship between the sensor responses and BOD over 5 months for some subsets of samples, leading to the prediction of BOD values from electronic nose analysis using neural network analysis. 

T. Cserhati, A. Kosa and S. Balogh, Comparison of partial least-square method and canonical correlation analysis in a quantitative structure-retention relationship study. J. Biochem. Biophys. Meth., 36 (1998) 131-141. 

If more than one y-variable has to be modeled, a separate model can be developed for each y-variable or methods can be applied that work with an X- and a y-matrix, such as PLS2 (Section 4.7.1), or canonical correlation analysis (CCA) (Section 4.8.1). 

FIGURE 4.27 Canonical correlation analysis (CCA), x-scores are uncorrelated v-scores are uncorrelated pairs of x- and y-sores (for instance t and Ui) have maximum correlation loading vectors are in general not orthogonal. Score plots are connected projections of x- and y-space. 

The canonical correlation coefficients can also be used for hypothesis testing. The most important test is a test for uncorrelatedness of the x- and y-variables. This corresponds to testing the null hypothesis that the theoretical covariance matrix between the x- and y-variables is a zero matrix (of dimension mx x mY). Under the assumption of multivariate normal distribution, the test statistic... 

The diffusion of correlation methods and related software packages, such as partial-least-squares regression (PLS), canonical correlation on principal components, target factor analysis and non-linear PLS, will open up new horizons to food research. 

Cancer-risk-diet relationship, 262 Canonical correlation analysis, 104 Capsaicin, 15-16 N-(Carboxymethyl)chitosan, preservation of meat flavor, 73 Carrageenan, fat replacement in ground beef, 73-75 Carry-over, description, 57 Carry-through, description, 57 Carvone, headspace analysis, 24,25/ L-Carvone, chemicals resulting in anosmias, 211... 

Odor and taste quality can be mapped by multidimensional scaling (MDS) techniques. Physicochemical parameters can be related to these maps by a variety of mathematical methods including multiple regression, canonical correlation, and partial least squares. These approaches to studying QSAR (quantitative structure-activity relationships) in the chemical senses, along with procedures developed by the pharmaceutical industry, may ultimately be useful in designing flavor compounds by computer. 

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