Mean squared deviation

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

This subroutine also prints all the experimentally measured points, the estimated true values corresponding to each measured point, and the deviations between experimental and calculated points. Finally, root-mean-squared deviations are printed for the P-T-x-y measurements. 

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

A typical molecular dynamics simulation comprises an equflibration and a production phase. The former is necessary, as the name imphes, to ensure that the system is in equilibrium before data acquisition starts. It is useful to check the time evolution of several simulation parameters such as temperature (which is directly connected to the kinetic energy), potential energy, total energy, density (when periodic boundary conditions with constant pressure are apphed), and their root-mean-square deviations. Having these and other variables constant at the end of the equilibration phase is the prerequisite for the statistically meaningful sampling of data in the following production phase. 

Example Crippen and Snow reported their success in developing a simplified potential for protein folding. In their model, single points represent amino acids. For the avian pancreatic polypeptide, the native structure is not at a potential minimum. However, a global search found that the most stable potential minimum had only a 1.8 Angstrom root-mean-square deviation from the native structure. 

In order to examine whether this sequence gave a fold similar to the template, the corresponding peptide was synthesized and its structure experimentally determined by NMR methods. The result is shown in Figure 17.15 and compared to the design target whose main chain conformation is identical to that of the Zif 268 template. The folds are remarkably similar even though there are some differences in the loop region between the two p strands. The core of the molecule, which comprises seven hydrophobic side chains, is well-ordered whereas the termini are disordered. The root mean square deviation of the main chain atoms are 2.0 A for residues 3 to 26 and 1.0 A for residues 8 to 26. 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

A test of the accuracy of the approximate wave function 0 for the ground state has been given by Eckart3 by considering the mean square deviation from the exact eigenfunction lP0 ... 

It seems as if an energy value of sufficiently high accuracy has now been found for the helium problem, but we still do not know the actual form of the corresponding exact eigenfunction. In this connection, the mean square deviation e = J — W 2 (dx) and criteria of the Eckart type (Eq. III.27) are not very informative, since s may turn out to be exceedingly small, even if trial function... 

Fig. 5.17. Time domain CARS of nitrogen under normal conditions. Points designate experimental data, solid line calculation with a = 6.0 A, b = 0.024, c = 0.0015. The insert depicts the dependences of the relative mean-square deviation on each of the parameters , b and c, the other two being fixed at their optimum values. The deviations are expressed as percentage of optimum parameters.

Sk and Sx are the mean square deviations of the variables log k and from their... 

Figure 3. A comparison of the aC backbone of PelC with PelE. The aCs which superimpose within a root-mean-square deviation of 1.5 A are shown in black and those, within 3.0 A are shown in dark gray. The remaining backbone regions are shown in light The largest structural differences occur in the loops capping one end of the parallel p as well as in those comprising the putative substrate binding groove.

One can hence think of (normal-mode composition factor) ej = ejaSja as the fractional involvement of atom j in normal mode a.The dimensionless vector eja also specifies the direction of the motion of atom j in the ot-th normal mode. Interestingly, the mode composition factors are also related to the magnitude of the atomic fluctuations. In a stationary state ) of a harmonic system, the mean square deviation (msd) of atom j from its equilibrium position may be expressed as a sum over modes of nonzero frequency ... 

The noise can be described by means of two approaches, namely, in the time domain and in the frequency domain. For the time domain approach the characteristic value is the well-known mean square deviation from the value over the time T,... 

Figure 2-4. Comparison of optimized and X-ray structures for the active site of RNR. The X-ray structure of R2met is superimposed on the optimized structures from active-site QM-only (left) and ONIOM2 0middle) models. The plot shows the quality of the optimizations evaluated as the root-mean-square deviations (in A) compared to the X-ray structures of RNR and MMO (right). (Adapted from Torrent et al. [24]. Reprinted with permission. Copyright 2002 Wiley Periodicals, Inc.)...

In all the studied systems addition of the surrounding protein in an ONIOM model clearly improves the calculated active-site geometries. This is clearly illustrated in Figure 2-13, which shows the root-mean-square deviation between calculated and experimental structures for four of the studied enzymes. 

Figure 13-6. The root mean square deviation (A) of the backbone of Orf2 over time (ps)...

H20, CH3OH-. . H20, CN-. . HzO, HCC-. . H20, HCOCT. . H20. The DFT(B3LYP) and the DFT(BLYP) results were in a fair agreement with the MP2 results. The root mean square deviation of the DFT and the MP2 complexation enthalpies amounted to 0.7 and 1.1 kcal/mol, for B3LYP and BLYP, respectively. From the basis set dependence of the DFT results, it was concluded that the nonlocal DFT calculations require diffuse atomic functions. 

To compute the results shown in Tables 34-3 and 34-4, the precision of each set of replicates for each sample, method, and location are individually calculated using the root mean square deviation equation as shown (Equations 34-1 and 34-2) in standard symbolic and MathCad notation, respectively. Thus the standard deviation of each set of sample replicates yields an estimate of the precision for each sample, for each method, and for each location. The precision is calculated where each ytj is an individual replicate (/ ) measurement for the ith sample yt is the average of the replicate measurements for the ith sample, for each method, at each location and N is the number of replicates for each sample, method, and location. The results of these computations for these data... 

The analytical results for each sample can again be pooled into a table of precision and accuracy estimates for all values reported for any individual sample. The pooled results for Tables 34-7 and 34-8 are calculated using equations 34-1 and 34-2 where precision is the root mean square deviation of all replicate analyses for any particular sample, and where accuracy is determined as the root mean square deviation between individual results and the Grand Mean of all the individual sample results (Table 34-7) or as the root mean square deviation between individual results and the True (Spiked) value for all the individual sample results (Table 34-8). The use of spiked samples allows a better comparison of precision to accuracy, as the spiked samples include the effects of systematic errors, whereas use of the Grand Mean averages the systematic errors across methods and shifts the apparent true value to include the systematic error. Table 34-8 yields a better estimate of the true precision and accuracy for the methods tested. 

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