First and Second Derivatives of a Data Set

The simplest method to obtain the first derivative of a function represented by a table of x, y data points is to calculate Ay/Ax. The first derivative or slope of the curve at a given data point x, yn can be calculated using either of the following formulas  

The second derivative, (fiyfdx, of a data set is calculated in a similar manner, namely by calculating A(Ay/Ax)/Ax. 

Calculation of the first or second derivative of a data set tends to emphasize the noise in the data set that is, small errors in the measurements become relatively much more important. 

Points on a curve of x, y values for which the first derivative is either a maximum, a minimum or zero are often of particular importance and are termed 

There are more sophisticated equations for numerical differentiation. These equations use three, four or five points instead of two points to calculate the derivative. Since they usually require equal intervals between points, they are of less generality. Their main advantage is that they minimize the effect of noise . 

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Derivs.xls illustrates how to obtain the first and second derivatives of a data set. 

Plots of the correlation coefficients obtained from the absorbance and its first and second derivatives (D1 and D2) are shown superimposed on the absorbance spectrum of the 100% Fe SWy in Figure 4a. Correlation coefficients for the full set, superimposed on the absorbance spectrum for 100% Fe Otay sample, are shown in Figure 4b. The quality of the correlation is significantly improved using 1st derivative data and improved still further using 2nd derivative data. This can be seen from Table III where a summary of the correlation coefficients obtained from absorbance and first and second derivatives of absorbance at several prominent wavelengths, including the computer optimum, are displayed. 

Typically, the parameterization of the intramolecular (valence terms) force constants is done iteratively based on a set of ab initio data [131-133]. Such sets might include the electrostatic potentials, the energies and the first and second derivatives of the energies. 

The experimental use of flame theory is a simpler problem because the measured profiles allow the replacement of the coupled differential equations with a pointwise set of algebraic equations. For example, to determine the rate of appearance (or disappearance) of a particular species at various points (i.e., at various temperatures in the flame) requires a knowledge of the first and second derivatives of the composition of a species and the temperature and gas velocity at the point in question. These are experimental quantities available from the flame structure measurements (see, e.g.. Fig. 8). The usual method of data reduction involves two steps (1) the calculation of the species flux [Eq. (2)] (i.e., the amount of the species passing through a unit area per unit time)—this takes the effect of diffusion into account quantitatively (2) from this flux curve the rate of species production is obtained by differentiation [Eq. (3)]. 

Figure 4. Correlation coefficients of total iron content as a function of wavelength calculated from absorbance and its first and second derivatives, a) data for the homologous sample set. b) the full sample set. The correlation coefficients are superposed on the spectrum of the 100 % Fe SWy in a and on the 100% Fe Otay in b for comparison.

Kikic et al. (41) employed only the values of P and T at the UCEP to evaluate these two constants ky and /y). The location of the UCEP was estimated from the experimental data by locating the intersection of the S-L-V line and (L = V) critical locus curve. For a type IP-Ttrace (with no temperature minimum with increasing pressure, e.g., a naphthalene-ethylene system), the solubility isotherm at Tucep provides an inflection point at P = PucEP (13). By setting the first and second derivatives to zero at this point, one can obtain the two equations needed for the two binary interaction parameters, kij and ly, respectively. When this approach was used for the inter-... 

Consider a countercurrent mass-transfer device for which the equilibrium-distribution relation consists of a set of discrete values XiD,YiD instead of a continuous model such as Henry s law. An analysis to determine minimum flow rates similar to that presented in Problem 3.14 is still possible using the cubic spline interpolation capabilities of Mathcad. Cubic spline interpolation passes a smooth curve through a set of points in such a way that the first and second derivatives are continuous across each point. Once the cubic spline describing a data set is assembled, it can be used as if it were a continuous model relating the two variables to predict accurately interpolated... 

The SVD derived data for the second process, i.e, the conversion of species 2 to species 3, could only be compared with electronic spectral measurements that indicate that the SVD overestimated the rate by a factor of -- 2. This process also has the largest associated error. The first process (A B) is approximately ten times faster than the second process (B -> C), as determined by single wavelength studies, and it is not optimal to collect data for a period long enough to observed significant production of species 3, while maintaining sufficient observation of the conversion of species 1 into species 2. A reasonable period of observation for the second process would require an approximate 33 h time frame and the size of the data set required to simultaneously account for the more rapid conversion of species 1 to species 2 and the species 2 to species 3 conversions would approach 1,000 spectra. 

At this point, a considerable amount of theory on Hansch analysis has been presented with almost no examples of practice. The next three Case Studies will hopefully solidify ideas on Hansch analysis that have already been discussed. Each Case Study introduces a different idea. The first is an example of a very simple Hansch equation with a small data set. The second demonstrates the use of squared parameters in Hansch equations. The third and final Case Study shows how indicator variables are used in QSAR studies. If you are unfamiliar with performing linear regressions, be sure to read Appendix B on performing a regression analysis with the LINEST function in almost any common spreadsheet software. A section in the appendix describes in great detail how to derive Equations 12.20 through 12.22 in the first Case Study. 

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