Multivariate distances

Highly correlated variables should not be included in columns of X. In this case, computation of multivariate distances becomes problematic because computation of the inverse of the variance-covariance matrix becomes unstable (see Equation 3.22). 

Treating a sample on a given day as a vector of values, x = (xj... xn), with one value for each of the measured biotic parameters, allows multivariate distance functions to be computed. 

The index is essentially a distance measure, Mahalanobis" squared distance (D ), which expresses the multivariate distance between the observation point and the common mean of the reference values, taldng into account the dispersion and correlation of the variables.More interpreta-tional guidance may be obtained from this distance by expressing it as a percentile analogous to the percentile presentation of univariate observed values. Also, the index of atypicality has a multivariate counterpart. ... 

Based on these multivariate distances or similarities, clusters of objects are generated. The distance between single objects within the clusters is minimized while the distance between clusters is maximized. The objective of the procedure is a clear representation of the objects with fewer dimensions. Cluster analysis is a multivariate explorative method which needs no a priori information and yields no statistical evidence about group memberships. 

The HCA method, which uses any of a variety of multivariate distance calculations to identify similar spectra, has found little use in Raman spectroscopy, although it could be of use in the growing analysis of complicated systems in which a large heterogeneous sample set is being analyzed. A study of spruce needles by Krizova et al. [54] and an investigation of cancerous skin lesions by Fendel and Schrader [55] are two examples showing the modest power of HCA. 

J. C. Gower. Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53 325, 1966. 

This relationship is of importance in multivariate data analysis as it relates distance between endpoints of two vectors to distances and angular distance from the origin of space. A geometrical interpretation is shown in Fig. 29.2. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

We consider an nxn table D of distances between the n row-items of an nxp data table X. Distances can be derived from the data by means of various functions, depending upon the nature of the data and the objective of the analysis. Each of these functions defines a particular metric (or yardstick), and the graphical result of a multivariate analysis may largely depend on the particular choice of distance function. 

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. 

Distance (in multivariate data) Discriminant function Discriminant variable... 

Some sort of estimation of the statistical distance to the overall model (see Section 16.5.4) should be reported for each compound to provide an estimate of how much an intra- or extrapolation in multivariate descriptor space the prediction actually is. 

Nearest neighbor distance inlier, n - a spectrum residing within a significant gap in the multivariate calibration space, the result for which is subject to possible interpolation error across the sparsely populated calibration space. 

The results show that DE-MS alone provides evidence of the presence of the most abundant components in samples. On account of the relatively greater difficulty in the interpretation of DE-MS mass spectra, the use of multivariate analysis by principal component analysis (PCA) of DE-MS mass spectral data was used to rapidly differentiate triterpene resinous materials and to compare reference samples with archaeological ones. This method classifies the spectra and indicates the level of similarity of the samples. The output is a two- or three-dimensional scatter plot in which the geometric distances among the various points, representing the samples, reflect the differences in the distribution of ion peaks in the mass spectra, which in turn point to differences in chemical composition of... 

In Chapter 2, we approach multivariate data analysis. This chapter will be helpful for getting familiar with the matrix notation used throughout the book. The art of statistical data analysis starts with an appropriate data preprocessing, and Section 2.2 mentions some basic transformation methods. The multivariate data information is contained in the covariance and distance matrix, respectively. Therefore, Sections... 

A fundamental idea in multivariate data analysis is to regard the distance between objects in the variable space as a measure of the similarity of the objects. Distance and similarity are inverse a large distance means a low similarity. Two objects are considered to belong to the same category or to have similar properties if their distance is small. The distance between objects depends on the selected distance definition, the used variables, and on the scaling of the variables. Distance measurements in high-dimensional space are extensions of distance measures in two dimensions (Table 2.3). 

If it can be assumed that the multivariate data follow a multivariate normal distribution with a certain mean and covariance matrix, then it can be shown that the squared Mahalanobis distance approximately follows a chi-square distribution... 

If an object has a larger squared Mahalanobis distance than the cutoff it is exceptionally high and can therefore be considered as potential multivariate outlier. 

The Mahalanobis distance used for multivariate outlier detection relies on the estimation of a covariance matrix (see Section 2.3.2), in this case preferably a robust covariance matrix. However, robust covariance estimators like the MCD estimator need more objects than variables, and thus for many applications with m>n this approach is not possible. For this situation, other multivariate outlier detection techniques can be used like a method based on robustified principal components (Filzmoser et al. 2008). The R code to apply this method on a data set X is as follows ... 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

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