Bivariate plots

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

Figure 1.31. Bivariable plot of oxygen versus carbon isotopic compositions of carbonates. Solid circle magnesite open circle dolomite open square calcite A oxygen and carbon isotopic compositions of igneous carbonates B oxygen and carbon isotopic compositions of marine carbonates (Shikazono et al., 1995).

A statistical study of the conversion with tetralin of 68 coals (60) must now be regarded as superseded by a later, more comprehensive paper (61), but it did show very clearly that bivariate plots are of little value in interrelating liquefaction behavior with coal properties at least two or three coal properties must be taken into account in seeking to explain the variance of liquefaction behavior, and some of these properties are not related to the rank of the coal. The paper implies strongly that any interrelationships of coal characteristics must necessarily be multivariate. Hence in any study of coal a large sample and data base is essential if worthwhile generalizations are to be made. 

In the absence of an assumed underlying normal distribution, simple bivariate plotting does not lead to an estimate of the true extent of the parent isotope field. This is particularly a problem if only relatively few samples are available, as is usually the case. Kernel density estimation (KDE Baxter et al., 1997) offers the prospect of building up an estimate of the true shape and size of an isotope field whilst making few extra assumptions about the data. Scaife et al. (1999) showed that lead isotope data can be fully described using KDE without resort to confidence ellipses which assume normality, and which are much less susceptible to the influence of outliers. The results of this approach are discussed in Section 9.6, after the conventional approach to interpreting lead isotope data in the eastern Mediterranean has been discussed. 

Analyze the data by constructing bivariate plots in which quadrants are set to enclose viable cells (annexin V-negative, 7-AAD-negative), early apoptotic cells (annexin V-positive, 7-AAD-negative), and late apo-ptotic or necrotic cells (annexin V-positive, 7-AAD-positive), as described in the legend of Figure 13-6. 

Ordination axes are determined sequentially. The first axis is calculated to account for the most prominent variance in the samples or species subsequent axes are calculated to account for variability not explained by the earlier axes. It is useful to visualize the distribution of samples or species as a function of their scores on the two or three most significant axes (as represented by their eigenvalues and testable by permutation tests). Plots of one ordination axis versus another (called bivariate plots or biplots) can be used to examine patterns of similarity and difference among species and samples. 

Figure 5. Bivariate plot of chromium and iron base-10 logged concentrations for Dragon Jar paste samples analyzed by INAA. Ellipses represent 90% confidence levels for group membership.

Figure 8. Bivariate plot of iron and lead log base-10 ppm oxide concentrations showing subgroup variation within the mineral and organic-paint groups. Ellipses represent 90% confidence interval for group membership. Unassigned specimens are not shown.

Multiple compositional groups recognized in the data are illustrated on a bivariate plot of log-transformed potassium and antimony (Figure 4). These groups are believed to represent different recipes and, consequently, different regions of manufacture. We compared the Pb/206 and Pb/ Pb ratios to... 

Figure 3. Bivariate plot ofHfand Al showing the negative correlation between most of the locally available clays and the Chapel bricks. Note that the correlation for the Chicamuxen Church Formation clays (Qc) is similarly negative. Ellipses indicate 95% confidence limits.

Figure 6. Bivariate plot ofln(Na2OIFe2Oi) versus ln(K20/Fe203)...

Figure 6. Bivariate plot of log-base 10 values for iron and cerium. The ellipses represent a 90% confidence interval for each group.

Figure 7. Bivariate plot of log-base 10 values of iron and cerium derivedfrom a previous INAA study of early Islamic period glazed ceramics(32). The ellipses represent a 90% confidence interval for each group. Note that the data for Group 2 fall in the same range as the Fe and Ce values for Group 2 in Figure 6.

Figure 5. Bivariate plot of zinc oxide and tin oxide base-10 logged concentrations showing distribution of the five main compositional groups.

Bivariate plots based on archaeological context were examined to investigate potential spatio-temporal contextually specific groupings (Figures 4-6). 

Figure 4. Bivariate plot oflogI0 [Sm/Fe] vs log I0 [Co/Fe]. Samples are plotted by groups as determined by context.

Figure 11.4 Separation of three grassland soils (seven replicates, 4 m apart at three grassland sites, unimproved and improved A and B) using bivariate plots of elements carbon (C) (by elemental analyzer) and phosphorus (P) (by ICP-OES) at three grassland sites.

Elemental results from soil forensic studies have been presented as spider diagrams (Pye and Blott 2004a) or as bivariate plots (Figure 11.4). Important considerations in using this type of information are how the values in the suspect sample may reflect a different fraction of the whole and, indeed, how the sample compares to every other sample. In addition, uncertainty increases as concentrations approach the limit of detection. A study of three soils using small sample sizes (0.05 g) showed that between sample variability... 

Fig. 8.2 Bivariate plot of zirconium (Zr) in parts-per-million on the K-axis and yttrium (Y) in parts per million on the x-axis, showing that the two elements neatly resolve four obsidian sources in New Mexico...

To analyse a simple plot of this type in order to distinguish between circles and squares , one would calculate an Euclidean distance measurement between clusters. Known samples would be a training set (calibration set) of samples in order to locate each cluster. An unknown sample can be identified based on its response to the two sensors and the relative distance to each of the known clusters. Already, with just the addition of one sensor, several analytes can be characterized using a bivariate plot. 

Figure 113 A bivariate plot of theoretical sample responses to a two-sensor array (top). The middle plot is the autoscaled data. The bottom plot is the sample data plotted in the PCA eigenvector space. As can be seen, the orientation of the data with respect to each other is maintained. Only scale and axis rotations occur with the PCA procedure.

However, this is not a reason for not analysing such data and this is an intelligent approach. Stephen Evans has described another attractive approach (Evans, 2000). This is to produce a bivariate plot in which each combination of drug and reaction is represented using one coordinate representing the PRR and another a standardized measure such as (23.7). (In fact, Evans uses a chi-square but the square of (23.7) will be approximately distributed as a chi-square with one degree of freedom.)... 

This may be done y plotting selected trace elements on bivariate plots. For... 

Figure 3.14 Bivariate plots of the oxides AI2O3, CaO, MgO, TlOj, Na20 vs Si02 in basaltic lavas from Kilauea Iki lava lake from the 1959-1960 eruption of Kilauea volcano, Hawaii (from Richter and Moore, 1966). The data are given in Table 3.3.

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