Error function graph

Figure 4. Three dimensional graph of the error function, eqn.(8), for different combinations of the two peak maxima co-ordinates in the Kalman filtering of the combined model [1A+2A]. The error function has been inverted for graphical enhancement. (From Fresenius J Anal. Chem(1993) 345 490, with permission)...

Figure 7. Three dimensional graphs of the error functions corresponding to the iterative resolution of (a) the mixture peak c of figure 6, and (b) the mixturte peak a of figure 6, respectively. The error functions have been inverted for graphical enhancement. T j and T2 are the position parameters for AOH and ATS, respectively.(From Chromatographia (1992) 34(1/2) 56, with permission).

Figure 19-9 Power function graph presenting probability for rejection on the y-axis versus size of error on the x-axis.The different lines represent numbers of control observations, In this case n= 1,2, and 4.

Two power function graphs are necessary, one to describe the performance for random error (RE) and the other for the performance for systematic error. For RE, as shown in Figure 19-10, A, the x-axis is labeled ARE. A value of 1.0 corresponds with the original standard deviation of the analytical method, a value of 2.0 to a doubling of that standard deviation, 3.0 to a tripling, and so on. For systematic error (SE), the x-axis is labeled ASE (see Figure 19-10, B). A value of 1.0s corresponds to a systematic shift equivalent to the size of the standard deviation, a value of 2.0 s to a shift equivalent to two times s, and so on. 

Calculate the critical systematic error and draw a vertical fine showing its location on the power function graph, or plot the observed imprecision and inaccuracy of your method on the OPSpecs chart for the TE of interest. 

Figure 4.7 provides a graph of the error function and the complementary error function while Table 4.2 provides a look-up table of numerical values for the error function—you will find these to be handy when working diffusion problems that have error function solutions. 

FIGURE 4.7 Graph of error function erffte] and complimentary error function erfc[< ]. 

Comparison of Alignment Charts and Cartesian Graphs. There are typically fewer lines on an alignment chart as compared to Cartesian plots. This reduces error introduced by interpolation and inconsistency between scales. For example, to find a point (x,j) on a Cartesian graph one draws two lines, one perpendicular to each axis, and these reference lines intersect at the point x,j). This point (x,j) may correspond to some finite value found by rea ding a contour map represented by a family of curves corresponding to different values of the function. 

Equilibrium data are thus necessary to estimate compositions of both extract and raffinate when the time of extraction is sufficiently long. Phase equilibria have been studied for many ternary systems and the data can be found in the open literature. However, the position of the envelope can be strongly affected by other components of the feed. Furthermore, the envelope line and the tie lines are a function of temperature. Therefore, they should be determined experimentally. The other shapes of the equilibrium line can be found in literature. Equilibria in multi-component mixtures cannot be presented in planar graphs. To deal with such systems lumping of consolutes has been done to describe the system as pseudo-ternary. This can, however, lead to considerable errors in the estimation of the composition of the phases. A more rigorous thermodynamic approach is needed to regress the experimental data on equilibria in these systems. 

These results can then be compared to experimental values (at the same temperature) in a number of informative ways. First we can plot the calculated values as a function of temperature and represent results as a line, see Figure 4. The experimental results can be represented as an error band vs temperature plot. Real differences are readily apparent since they must lie outside the 95% confidence limits. Another way to represent the difference is to plot the difference between calculated and experimental values as a function of temperature. In the same graph an estimate and plot of the experimental errors can also be made, see Figure 5. 

Calibration graphs defined by data with non-negligible error have to be constructed by some kind of smoothing operation. In cases, in which the form of the underlying curve is known a priori, the latter can be approximated by minimizing the squares of deviations. Otherwise a spline function can be used (JJ[, 1 ). The spline function S(x) is constructed to minimize a measure of smoothness defined by... 

Fig. 9. Graph shows the logarithm of relative tumor volume as a function of days after treatment. Growth of tumors treated with adenovirus vector (dark square) or with radiation therapy (dark triangle) is diminished relative to that in the control group (light circle). Tumors receiving combined treatment (dark diamond) decreased in size over time. Error bars = SD.

In any case, the cross-validation process is repeated a number of times and the squared prediction errors are summed. This leads to a statistic [predicted residual sum of squares (PRESS), the sum of the squared errors] that varies as a function of model dimensionality. Typically a graph (PRESS plot) is used to draw conclusions. The best number of components is the one that minimises the overall prediction error (see Figure 4.16). Sometimes it is possible (depending on the software you can handle) to visualise in detail how the samples behaved in the LOOCV process and, thus, detect if some sample can be considered an outlier (see Figure 4.16a). Although Figure 4.16b is close to an ideal situation because the first minimum is very well defined, two different situations frequently occur ... 

One of the most useful tools to spot and eliminate errors is a spreadsheet, such as Excel or QuattroPro. QSAR modelers very frequently use spreadsheets to organize data into columns and rows of standardized values of the independent and dependent parameters. Spreadsheets allow easy sorting and filtering — two important functions used to find problem data and duplicates and other errors. In addition, spreadsheets have search and replace routines, plotting, and correlation functions, which allow the data to be reviewed in various comprehensive ways. The data can also be exported to other file types, which allow analysis by other software for statistics and any types of quantitative and qualitative relationships that may exist. It cannot be emphasized enough that the typical spreadsheet functions (including graphing functions) are excellent tools to find and eliminate erroneous or questionable values, duplicates, and other problem entries. 

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