Andrews’ plot

Andrews plots. In Andrews plots (Andrews [1972]) the features (variables set) are used as coefficients of a Unear combination of trigonometric functions... 

Fig. 8.22. Andrews plot of five wines, according to Danzer et al. [2001]...

Figure 2.9 shows a graph of the quotient pV q- nRT (as y) against pressure p (as x). We sometime call such a graph an Andrews plot. It is clear from the ideal-gas equation (Equa tion (1.13)) that if pV = nRT then pV + nRT should always equal to one the horizontal line drawn through y = 1, therefore, indicates the behaviour of an ideal gas. 

Figure 2.10 shows a similar graph, and displays Andrews plots for methane as a function of temperature. The graph clearly... 

Figure 2.9 An Andrews plot of PV nRT (as y) against pressure p (as x) for a series of real gases, showing ideal behaviour only at low pressures. The function on the y-axis is sometimes called the compressibility Z...

A problem with the ANDREWS plot is that if variables of one object are permuted, a quite different picture arises because the amplitudes of the different harmonic oscillations change. It is, therefore, better if the functions are associated with principal components (see Section 5.4) of the set of features. Principal components are linear combinations of the features and explain the total variance of the data in descending order. In this manner the sequence of features, i.e. the sequence of the single principal components, should not be permuted in the ANDREWS plot. 

We conclude this section on multivariate methods by mentioning a number of additional techniques for obtaining graphic representations of multidimensional data sets. A good general review may be found in Everitt [82]. Amongst the techniques described there is the Andrews plot [83]. For the rth observation in a multivariate data set, the Andrews function is defined as ... 

Allison PD (2002) Missing data. Sage, Thousand Oaks, CA Andrews DF (1972) Plots of high dimensional data. Biometrics 28 125... 

The Analog Plot. The human eye is extremely good at comparing the size, shape and color of pictorial symbols (Anderson, 1960 Andrews, 1972 Davison, 1983 Schmid, 1983 Cleveland and McGill, 1985). Furthermore, it can simultaneously appreciate both the minute detail and the broad pattern. 

The hot-wire anemometer can, with suitable cahbration, accurately measure velocities from about 0.15 m/s (0.5 fl/s) to supersonic velocities and detect velocity fluctuations with frequencies up to 200,000 Hz. Eairly rugged, inexpensive units can be built for the measurement of mean velocities in the range of 0.15 to 30 m/s (about 0.5 to 100 ft/s). More elaborate, compensated units are commercially available for use in unsteady flow and turbulence measurements. In cahbrating a hotwire anemometer, it is preferable to use the same gas, temperature, and pressure as will be encountered in the intended apphcation. In this case the quantity I RJAt can be plotted against /v, where I = hot-wire current, = hot-wire resistance. At = difference between the wire temperature and the gas bulk temperature, and V = mean local velocity. A procedure is given by Wasan and Raid [Am. Inst. Chem. Eng. J., 17, 729-731 (1971)] for use when it is impractical to calibrate with the same gas composition or conditions of temperature and pressure. Andrews, Rradley, and Hundy [Int. J. Heat Mass Transfer, 15, 1765-1786 (1972)] give a cahbration correlation for measurement... 

Figure 5. Calibration curve for 6% agarose, prepared with unlabeled polypeptides (open circles), plotted according to the method of Andrews. The log M is plotted vs. Kd. Super-imposed on the curve are data points obtained with fluorescein (filled triangles) and rhodamine (filled squares) labeled proteins.

The curves which are obtained by plotting pressure against volume at various constant temperatures are known as isotherms (isos = equal therm = heat). Andrews obtained isotherms of carbon dioxide at different temperatures which are shown in figure (7). 

Analysis of these patterns of behaviour into single ion contributions suggests that, broadly, anions are destabilized and cations are stabilized when TA co-solvents are added to aqueous solutions. This behaviour is shown by plots of 8m (ion) for acetone + water mixtures as given by Bax et al. (1972) and for methyl alcohol + water mixtures by a number of workers (de Ligny and Alfenaar, 1965 Feakins et al., 1967 Andrews et al., 1968 Feakins and Voice,... 

It may be pointed out that the isotherms plotted in the figure given above are based on theoretical calcula tions of Vcorresponding to different values of P obtained by using the van der Waals equation. The isotherms for carbon dioxide, obtained by Andrews experimentally, were in close resemblance with these curves, with the difference that the wavelike portion LMNOQ was replaced by a horizontal line. Since then more careful experiments have shown that small portions corresponding to curves LM and OQ can be realised in practice also. These represent supersaturated vapour and superheated liquid, respectively. 

A means by which the effects of correlations can be visualized was presented by ANDREWS, who suggested plotting the function (N0 - Np) n s2(ji), whereN0 and Np are the numbers of observations and parameters, respectively, and s2 ) is a measure of the fit of the parameter set / , such as the median residual or the mean square residual. Contours of this function, plotted against two individual parameters, 0/ and fy, which are allowed to range over several standard errors from the fitted values, graphically present the correlation between 0/ and 0/ (Figure 1). For the linear case, the mean square contours are ellipsoidal, but they have been known to vary widely from this shape in nonlinear problems. 

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