Mean square error expressed

The root-mean-square error of the model with only nearest-neighbor and next-nearest-neighbor interactions is 3.9 kJ/mol. The one of the model that also includes the linear 3-particle interaction is 2.9 kJ/mol. Using these errors as estimates for the errors in DFT calculations gives the error estimates of the lateral interactions in table 2. These should be regarded as lower estimates. If we would use a larger estimate for the errors in DFT then the errors of the lateral interactions would increase proportionally. Note that, because the term is always the same in the expressions for independent of the adlayer structure, possible systematic errors in DFT (cr in Section 3.4.2) only affect E 2 and not the lateral interactions. 

In this expression 1 is the cutoff frequency above which the data contain no information about o(x) that is, I( Q. We see that the sharpness criterion is a measure of the steepness of the solution o(x). The previous criterion, expression (40), is replaced with a sum of two terms. It includes both the mean-square-error criterion and sharpness. The filter is then sought that minimizes... 

We are permitted to specify the integrals for positive co only, because of the even property of the integrand. This simplication, in turn, stems from the real nature of all the x-space components of the integrand. Minimizing expression (9) is equivalent to asking that the physical solution conform to the Wiener inverse-filter estimate in the sense of minimum mean-square error after suitable weighting of the positive solution to ensure best conformance at frequencies of greatest certainty. 

In multivariate calibration, accuracy reports the closeness of agreement between the reference value and the value found by the calibration model and is generally expressed as the root mean square error of prediction (RMSEP, as described in section 4.5.6) for a set of validation samples ... 

Compare the mean squared errors of bj and bi.2 in Section 8.2.2. (Hint, the comparison depends on the data and the model parameters, but you can devise a compact expression for the two quantities.)... 

Some further simplification of this expression is possible, for example, by using the fact that Var[Z(a )] = a2, by assumption. We leave the mean squared error in this form, however, to facilitate comparison with its counterpart in Section 4 for the estimated effect of a group of variables. 

The mean square error is obtained by squaring both sides of this expression and averaging over an infinite population of sets of jVmeasurements. The averaging is done by multiplying the square of each side of Eq. (18) by the probability distribution function P and integrating from —to co ... 

Root mean square error (RMSE) RMSE is popular and often chosen by practitioners because of its ease of use and its theoretical relevance in statistical modeling. RMSE is expressed as follows ... 

The quality of the model is estimated by how well it performs the mapping between the descriptors and the targeted activity of compounds in the training set. This mapping is expressed by the coefficient of determination or the root-mean-square error (RMSE) between the experimental (Y) and the predicted (Y) activities (Eqns (2) and (3)) ... 

F(r ri) is the Foiuier-space contribution of the force between two unit charges at positions ri and ri -i- r as calculated by the P M method (note that due to broken rotational and translational symmetry this does in fact depend on the coordinates of both particles), and R(r) is the corresponding exact reference force (whose Fourier transform is just Eq. 23). The inner integral over r scans all particle separations, whereas the outer integral over ri averages over all possible locations of the first particle within a mesh cell. Obviously, up to a factor L this expression is just the mean-square error in the force for two unit charges, in other words, the quantity x from Eq. 28. This provides a link between the rms error of an N particle system and the error Q from Hockney and Eastwood. Using Eq. 32 one obtains... 

To discuss the prediction error, one must validate the calibration model [2]. There are two sorts of validation. One method is based on a new set of objects (external prediction). It requires a large and representative set of objects which have to be kept apart from the calibration for testing purposes only. The other validation method is based on the calibration data themselves (internal validation). In most cases, internal validation methods such as cross-validation and leverage correction [2] give sensible results with valuable information about the prediction ability. Cross-validation seeks to validate the calibration model with independent test data, but contrary to external validation it does not use data for testing only. The cross-validation is performed a number of times, each time with the use of only a few calibration samples as a test set. From the validation set it is possible to compare the prediction ability for the models, expressed by the estimated prediction mean square error. 

The process of decompression is completely the reverse of the compression process. The coded bit strings are first decoded, and the dequantization is performed by multiplying each element by the chosen quantizer step. The resulting block is then inverse DCT transformed to obtain the reconstructed block. The difference e of f and / denotes the loss due to compression. In the compression literature, the loss is expressed as the root mean square error (RMSE), and it is defined as... 

In a multivariate calibration, where a set of NIR spectra (Xnxk, N samples and K variables) is regressed onto a continuous variable (yivxi) such as the fat or moisture content, the statistical errors, the accuracy, are most often used as a quality measure of the calibration. The absolutely most common quality measure of a multivariate calibration is the prediction error, expressed either as root mean square error of prediction (RMSEP) or standard error of performance (SEP). Both are calculated and are the result of a validation process, such as test set or cross-validation. These prediction errors are defined as ... 

Once the ANN has been trained, its performance is evaluated separately in the training and testing dada sets. Correlation coefficients (R) between the experimental and predicted values, mean absolute percentage error (MAPE) and mean squared error (MSE) are the statistical parameters with which the performance of ANN is appraised. The expressions of these statistical parameters have been given below ... 

In principle, FCS can also measure very slow processes. In this limit the measurements are constrained by the stability of the system and the patience of the investigator. Because FCS requires the statistical analysis of many fluctuations to yield an accurate estimation of rate parameters, the slower the typical fluctuation, the longer the time required for the measurement. The fractional error of an FCS measurement, expressed as the root mean square of fluorescence fluctuations divided by the mean fluorescence, varies as 1V-1/2, where N is the number of fluctuations that are measured. If the characteristic lifetime of a fluctuation is r, the duration of a measurement to achieve a fractional error of E = N l,/- is T = Nr. Suppose, for example, that r = 1 s. If 1% accuracy is desired, N = 104 and so T = 104 s. 

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