Bivariate data

Figure 6 Scatter plots for bivariate data with various values of correlation coefficient, r.

For bivariate data the simple scatter plot of variate x against variate y is popular and there are several ways in which this can be extended to accommodate fmther variables. Figure 9 iUustrates an example of a three-dimensional scatter plot. The data used are from Table 11, representing the results of the... 

How do we calculate these eigenvectors and eigenvalues In practice the calculations are always performed on a computer and there are many algorithms published in mathematical and chemometric texts. For our purposes, in order to illustrate their derivation, we will limit ourselves to bivariate data and calculate the eigenvectors manually. The procedure adopted largely follows that of Davis and Healy. ... 

Having determined the orthogonal axes or principal components of our bivariate data, it remains to undertake the projections of the data points on to the new axes. For the first principal components PCI,... 

In order to demonstrate the calculations and results associated with the different methods, the small set of bivariate data in Table 7 will be used. These data comprise 12 objects in two-dimensional space. Figure 8, and the positions... 

Table 7 A simple bivariate data set for cluster analysis ip), from Zupan, ° and the corresponding Euclidean distance matrix, (b)...

Figure 5 Contour plots of two groups of bivariate data with each group having identical variance-covariance matrices. Such groups are linearly separable...

Application of Equation (39) to the /sT-NN rule serves to define a sphere, or circle for bivariate data, about the unclassified sample point in space, of radius rjc which is the distance to the ATth nearest neighbour, containing K nearest neighbours. Figure 7. It is the volume of this sphere which is used as an estimate... 

Figure 10(a) illustrates a bivariate data set comprising two groups, each of two objects. These four objects are defined by their pattern vectors, including... 

Figure 10 A simple two-group, bivariate data set (a), and iterative discriminant analysis using the simple perceptron (b)...

Figure 12 A simple two-group, bivariate data set that is not linearly separable by a single function. The lines shown are the linear classifiersfrom the two units in the first layer of the multilayer system shown in Figure 13...

Scale of measurement Two independent groups Paired data Bivariate data... 

For bivariate data the simple scatter plot of variable y against variable a is popular and there are several ways in which this can be extended to accommodate further variables. Figure 1.12 illustrates an example of a three-dimensional scatter plot. The data used are from Table 1.11, representing the results of the analysis of nine alloys for four elements. The concentration of three analytes, zinc, tin, and iron, are displayed. It is immediately apparent from the illustration that the samples fall into one of two groups, with one sample lying between the groups. This pattern in the data is more readily seen in the graphical display than from the tabulated data. 

If it were necessary to reduce the display of our original bivariate data to a single dimension using only one variable, say chromium concentration, then a loss of 44% of the total variance would ensue. Using the first principal component, however, and optimally combining the two variables, only 5% of the total variance would be missing. 

To demonstrate the calculations and results associated with the different methods, the small set of bivariate data in Table 4.7 will be used. These data consist of 12 objects in two-dimensional space, Figure 4.9, and the positions of the points are representative of different shaped clusters, the single point (L), the extended group (B,C,D), the symmetrical group (A,E,F,G), and the asymmetrical cluster (H,I,J,K). ... 

Bivariate Data Tools and Tests for Regression and Correlation... 

Statistical methods for assessing bivariate data (two dimensional data where one parameter is measured as a function of another) are used to performr ression analysis of calibration data and to determine the goodness of fit of the calibration curve. A simple, linear equation is desirable when fitting quantitative calibration data for several reasons i) only two fitting parameters (A = intercept and B = slope) need to be calculated ii) it is straightforward to invert the equation so as to calculate an unknown value of x (e.g., analyte concentration) from a measured value of Y (e.g. mass spectrometer response), i.e. Xj = (Yj —A)/B and iii) relatively few experimental measurements (Xj,Yj) are required to establish reliable values of A and B in the catibration equation... 

---