Procrustes analysis

Procrustes analysis is a method for relating two sets of multivariate observations, say X and Y. For example, one may wish to compare the results in Table 35.1 and Table 35.2 in order to find out to what extent the results from both panels agree, e.g., regarding the similarity of certain olive oils and the dissimilarity of others. Procrustes analysis has a strong geometric interpretation. The 

Physico-chemical quality parameters of the 16 olive oils 

The major problem is to find the rotation/reflection which gives the best match between the two centered configurations. Mathematically, rotations and reflections are both described by orthogonal transformations (see Section 29.8). These are linear transformations with an orthonormal matrix (see Section 29.4), i.e. a square matrix R satisfying = RR = I, or R = R . When its determinant is positive R represents a pure rotation, when the determinant is negative R also involves a reflection. 

Using a shorthand notation for a matrix sum of squares, l E p = lie = tr(E E), we may state the Procrustes optimization problem as  

The first term on the right-hand side represents the total sum of squares of Y, that obviously does not depend on R. Likewise, the last term represents the total sum of squares of the transformed X-configuration, viz. XR. Since the rotation/reflection given by R does not affect the distance of an object from the origin, the total sum of squares is invariant under the orthogonal transformation R. (This also follows from tr(R X XR) = tr(X rXRR T) = tr(X XI) = tr(X X).) The only term then in eq. (35.2) that depends on R is tr(Y XR), which we must seek to maximize. 

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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... 

It must be emphasized that Procrustes analysis is not a regression technique. It only involves the allowed operations of translation, rotation and reflection which preserve distances between objects. Regression allows any linear transformation there is no normality or orthogonality restriction to the columns of the matrix B transforming X. Because such restrictions are released in a regression setting Y = XB will fit Y more closely than the Procrustes match Y = XR (see Section 35.3). 

Procrustes analysis has been generalized in two ways. One extension is that more than two data sets may be considered. In that case the algorithm is iterative. One then must rotate, in turn, each data set to the average of the other data sets. The cycle must be repeated until the fit no longer improves. Procrustes analysis of many data sets has been applied mostly in the field of sensory data analysis [4]. Another extension is the application of individual scaling to the various data sets in order to improve the match. Mathematically, it amounts to multiplying all entries in a data set by the same scalar. Geometrically, it amounts to an expansion (or... 

The covariance criterion as such suggests a symmetrical situation, X and Y playing equivalent roles. In fact, up to here, there is little difference with Procrustes analysis which also utilizes the singular vectors of the covariance matrix (Section 35.2). The difference is that in PLS the first X-factor, say tj, is now used as a regressor to fit both the X-block and the T-block ... 

G. B. Dijksterhuis, Procrustes analysis in sensory research, Ch. 7 in Multivariate analysis of data in sensory science (T. Naes and E. Risvik, eds), Elsevier, Amsterdam (1996). 

A powerful technique which allows to answer such questions is Generalized Procrustes Analysis (GPA). This is a generalization of the Procrustes rotation method to the case of more than two data sets. As explained in Chapter 36 Procrustes analysis applies three basic operations to each data set with the objective to optimize their similarity, i.e. to reduce their distance. Each data set can be seen as defining a configuration of its rows (objects, food samples, products) in a space defined by the columns (sensory attributes) of that data set. In geometrical terms the (squared) distance between two data sets equals the sum over the squared distances between the two positions (one for data set and one for Xg) for each object. 

S. de Jong, J. Heidema and H.C.M. van der Knaap, Generalized Procrustes analysis of coffee brands tested by five European sensory panels. Food Qual. Pref., 9 (1998) 111-114. 

Rose VS, Rahr E, Hudson BD, The use of Procrustes analysis to compare different property sets for the characterisation of a diverse set of compounds, Quant. Struct-Act. Relat., 13 152-158, 1994. 

The goal of methods that standardize instrument response is to find a function that maps the response of the secondary instrument to match the response of the primary instrument. This concept is used in the statistical analysis procedure known as Procrustes analysis [97], One such method for standardizing instrument response is the piecewise direct standardization (PDS) method, first described in 1991 [98,100], PDS was designed to compensate for mismatches between spectroscopic instruments due to small differences in optical alignment, gratings, light sources, detectors, etc. The method has been demonstrated to work well in many NIR assays where PCR or PLS calibration models are used with a small number of factors. 

Anderson, C.E. and Kalivas, J.H., Fundamentals of calibration transfer through Procrustes analysis, Appl. Spectrosc., 53, 1268-1276, 1999. 

Another important facility is to be able to compare different types of measurements. For example, the mobile phase in the example in Figure 4.4 is methanol. How about using a different mobile phase A statistical method called procrustes analysis will help us here. 

Similarly, procrustes analysis in chemistry involves comparing two diagrams, such as two PC scores plots. One such plot is the reference and a second plot is manipulated to resemble the reference plot as closely as possible. This manipulation is done mathematically involving up to three main transformations. 

Rose, V.S., Rahr, E. and Hudson, B.D. The Use of Procrustes Analysis to Compare Different Property Sets for the Characterisation of a Diverse Set of Compounds. QSAR, 1994,13, 152-158. 

Le Fur, Y, Mercurio, V., Moio, L., Blanquet, J., and Meunier, J.M. (2003). A new approach to examine the relationships between sensory and gas chromatography-olfactometry data using generalized procrustes analysis applied to Six French Chardonnay wines. J. Agric. Food Chem., 5i, 443-452. 

Endeavors have been made to find a link between two data sets (sensory versus instrumental data). The common goal of these tools is to discover the components or parameters whose variation explains the variation of sensory characteristics. The most useful statistical methods used for such purpose are partial least squares regression and generalized procrustes analysis. From a practical point of view, the models can be used to complement sensory assessment in routine quality control or in product and process development work. Regression-based statistical techniques are often used in conjunction with GC to distinguish well-known brands of alcoholic beverages from less expensive ones to detect counterfeit products. 

Since much of the testing being done today, and for the foreseeable future, will involve scoring of a product characteristic, the AOV becomes an essential resource in support of data analysis and interpretation. Since there are many AOV models, one needs to be familiar with those most appropriate for sensory data for example, the AOV mixed model (fixed and random effects) with replication is appropriate. Other features should allow for the ability to test the main effect by interaction when interaction is significant, and so forth. Finally, one needs to be cautious when using software that allows for exclusion of some data but without providing the details of what was excluded. Procrustes analysis is one such system. See Huitson (1989) for more discussion on this topic when applied to sensory data. The problem with any computation that removes some data is an assumption that data are an aberration when it may not. How does one know that this does or does not represent a unique... 

The main result provided by FP is a sensory map. To this end, the individual data tables can be compiled and analysed by the means of a multi-block data analysis technique, such as generalized Procrustes analysis (GPA) (Gower, 1975 Dijksterhuis, 1996), multiple factor analysis (MFA) (Escofier and Pag6s, 2008) or STATIS (Lavit, 1988 Lavit et al., 1994). Figure 6.3 shows the map of the first two principal components obtained by GPA. 

Dijksterhuis, G. (1996). Procrustes analysis in sensory research. In Naes, T. and Risvik, E. (eds.) Mulitvariate Analysis of Data in Sensory Science. Amsterdam Elsevier Science, pp 185-219. 

King, B. M. and Arents, P. (1991). A statistical test of consensus obtained from generalized procrustes analysis of sensory data. Journal of Sensory Studies, 6, 37—48. 

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