In the name of GOD Zeinab Mokhtari 1

In the name of GOD

Zeinab Mokhtari 1 -Mar-2010

In data analysis, many situations arise where plotting and visualization are helpful or an absolute requirement for understanding. Plotting = visualization = graphing scatter plots plotting some measured result against some parameter in a Cartesian co-ordinate system the entries from two vectors of the same size are plotted pairwise in the Cartesian co-ordinate system. Geographical maps satellite images Cartesian plotting Contour plots Scatter plots Line plots Images Bar plots Loading plots Score plots Biplots Joint plots

PCA Latent variable methods PLS regression factor analysis scores and loadings PARAFAC … The items plotted against each other (scores, loadings) are based on the same measured data but projected differently. principal component analysis unit-free plots orthogonal scores orthonormal loadings the co-ordinate systems used for the score plots

PLOTTING IN COMPONENT MODELS Sign inversion PCA scores and loadings are mirrored together.

Figure 1. The mean-centered data Figure 2. The score plot after PCA Figure 3. The normalized scoreplot

PLOTTING IN PARTIAL LEAST SQUARES REGRESSION Outliers non-linearity Grouping of data … a linear relationship with slope one

Biplots For almost equal number of objects and variables The singular value is distributed equally among the u and v parts (scores and loadings) for the purpose of forming new variables h and g to be plotted. c=1 row metric-preserving version Euclidean distances between objects and Mahalanobis distances between variables The special cases of c=0 and 1 c=0 column metric-preserving version Euclidean distances between variables and Mahalanobis distances between objects

A compensation for number of objects (I) and variables (K) is made by introducing a fudge or zoom factor z: Biplots can be expanded to the use of threeway loadings, especially for Tucker 3 models. Then they get the name joint plots.

Principal component analysis (PCA), a versatile and easy-to-use multivariate mathematical–statistical method suitable for the elucidation of the similarities and dissimilarities among the columns and rows of two-dimensional data matrices cannot be employed for the evaluation of arrays of higher dimensions

Three-way analysis by PARAFAC c a c b b a OLBOX O T Y A N-W not orthogonal loadings

Tucker 3 model or three-way PCA analysis of the similarities and dissimilarities among N-dimensional data arrays C X A G B OLBOX O T Y A N-W The Tucker 3 model computes three orthogonal matrices with lower dimensions than the original data arrays such a manner that the variance explained by the reduced matrices being as high as possible.

Cluster analysis The reduction of the dimensionality of multidimensional arrays Projection of the points scattered in the multidimensional space on a plane, such a manner that the distances among the points in the multidimensional space on the plane are as similar as possible The objectives of the study were the measurement of the microbiological effect of benzimidazolium salts containing various anions, the application of the combination of Tucker 3 model and cluster analysis for the evaluation of the dependence of the microbiological effect on the type of test organism, chemical structure of the free benzimidazolium base and the type of cation.

The free base and the salts formed with Cl−, SO 42−, PO 43− and NO 3−

Species tested for the microbiological activity (altogether 15 species)

The Tucker 3 model has been employed for the three dimensional data matrix consisting of the inhibitory activity of seven benzimidazole derivatives (factor I), the presence and type of anion (factor II) and the 15 test organisms (factor III) (3 -way array with dimensions 7, 5, 15). Arrays of the largest possible dimensions (6, 4, 14) The arrays explaining more than 0. 28% of the total variance (in this case 3, 2, 3)

the total variance explained : 99. 75%

the total variance explained : 98. 05%

Fig. 1. Plot of the first two elements of component matrix I The distribution of benzimidazole derivatives is highly similar on both figures. Similarity and dissimilarity of microbiological activity Fig. 2. Cluster dendogram of component matrix I

Fig. 3. Plot of component matrix II The presence of sulfate anion may have a considerable impact on the biological efficacy of benzimidazole derivatives.

Fig. 4. Plot of the first two elements of component matrix III Fig. 5. Cluster dendogram of component matrix III

It can be concluded from the results that a Tucker 3 model combined with cluster analysis can be successfully used for the study of the microbiological activity of benzimidazolium salts and separates the effect of the type of benzimidazole derivatives and saltforming anions.

Five different breads were baked in replicates giving a total of ten samples. Eight different judges assessed the breads with respect to eleven different attributes. The data can be regarded as a three-way array (10 × 11 × 8) or alternatively as an ordinary two-way matrix (10 × 88).

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