Error of estimate

Casado et al. have analyzed the error of estimating the initial rate from a tangent to the concentration-time curve at t = 0 and conclude that the error is unimportant if the extent of reaction is less than 5%. Chandler et al. ° fit the kinetic data to a polynomial in time to obtain initial rate estimates. 

The standard deviation gives the accuracy of prediction. If Y is related to one or more predictor variables, the error of prediction is reduced to the standard error of estimate S, (the standard deviation of the errors), where... 

The Mean Squared Error of Estimate (MSEE) is sometimes called the Mean Squared Error of Calibration (MSEC). It is supposed to refer uniquely to those situations when a calibration is generated with a data set and evaluated for its predictive performance on that same data set. Unfortunately, there are times when the term MSEC is wrongly applied to the errors in predicting the y... 

Figure 5. Ispoleth map of kriging error of estimations of lead concentrations (pg/g) in soil (5).

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

Firstly, various criteria for estimation, different from the least square E [P(x)-P (x)], may now be considered. Consider a general loss function L(e), function of the error of estimation e p(x) -p (x). The objective Is to build an estimator that would minimize the expected value of that loss function, and more precisely. Its conditional expectation given the N data values and configuration. 

Assessment of spatial distributions of pollutant concentrations is a very specific problem that requires more than blind mapping of these concentrations. Not only must the criterion of estimation be chosen carefully to allow zooming on the most critical values (the high concentrations), but also the evaluation of the potential error of estimation calls for a much more meaningful characteristic than the traditional estimation variance. Finally, the risks a and p of making wrong decisions on whether to clean or not must be assessed. 

The major disadvantage of the integral method is the difficulty in computing an estimate of the standard error in the estimation of the specific rates. Obviously, all linear least squares estimation routines provide automatically the standard error of estimate and other statistical information. However, the computed statistics are based on the assumption that there is no error present in the independent variable. 

At this point is worthwhile commenting on the computer standard estimation errors of the parameters also shown in Table 16.24. As seen in the last four estimation runs we are at the minimum of the LS objective function. The parameter estimates in the run where we optimized four only parameters (K2, kt, K k3) have the smallest standard error of estimate. This is due to the fact that in the computation of the standard errors, it is assumed that all other parameters are known precisely. In all subsequent runs by introducing additional parameters the overall uncertainty increases and as a result the standard error of all the parameters increases too. 

Once the standard error of estimate of the mean forecasted response has been estimated, i.e., the uncertainty in the total production rate, one can compute the probability level, a, for which the minimum total production rate is below some pre-determined value based on a previously conducted economic analysis. Such calculations can be performed as part of the post-processing calculations. 

Equation (15) ostensibly allows one to estimate permeability coefficients in units of cm/s. However, as with any parameter calculated from statistically drawn relationships, such estimates have to be taken with a grain of salt, because the absolute error of estimation for a single compound can be large. 

They include simple statistics (e.g., sums, means, standard deviations, coefficient of variation), error analysis terms (e.g., average error, relative error, standard error of estimate), linear regression analysis, and correlation coefficients. 

Now we come to the Standard Error of Estimate and the PRESS statistic, which show interesting behavior indeed. Compare the values of these statistics in Tables 25-IB and 25-1C. Note that the value in Table 25-1C is lower than the value in Table 25-1B. Thus, using either of these as a guide, an analyst would prefer the model of Table 25-1C to that of Table 25-1B. But we know a priori that the model in Table 25-1C is the wrong model. Therefore we come to the inescapable conclusion that in the presence of error in the X variable, the use of SEE, or even cross-validation as an indicator, is worse than useless, since it is actively misleading us as to the correct model to use to describe the data. 

Table 33-1 Summary of results obtained from synthetic linearity data using one PCA or PLS factor. We present only those performance results listed by the data analyst as Correlation Coefficient and Standard Error of Estimate...

A graphical comparison of the correlation coefficient (r) and the standard error of estimate (SEE) for a calibration model. 

You may be surprised that for our example data from Miller and Miller ([2], p. 106), the correlation coefficient calculated using any of these methods of computation for the r-value is 0.99887956534852. When we evaluate the correlation computation we see that given a relatively equivalent prediction error represented as (X - X), J2 (X - X), or SEP, the standard deviation of the data set (X) determines the magnitude of the correlation coefficient. This is illustrated using Graphics 59-la and 59-lb. These graphics allow the correlation coefficient to be displayed for any specified Standard error of prediction, also occasionally denoted as the standard error of estimate (SEE). It should be obvious that for any statistical study one must compare the actual computational recipes used to make a calculation, rather than to rely on the more or less non-standard terminology and assume that the computations are what one expected. 

A graphical comparison of the correlation coefficient (r) versus the standard error of estimate (SEE) is shown in Graphic 59-4. This graphic clearly shows that when the Sr is held constant (Sr = 4) the correlation decreases as the SEE increases. 

The attached worksheet from MathCad ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) is used for computing the statistical parameters and graphics discussed in Chapters 58 through 61, in references [b-l-b-4]. It is recommended that the statistics incorporated into this series of Worksheets be used for evaluations of goodness of fit statistics such as the correlation coefficient, the coefficient of determination, the standard error of estimate and the useful range of calibration standards used in method development. If you would like this Worksheet sent to you, please request this by e-mail from the authors. 

Standard Error of Estimate Syx = 0.4328 Slope Confidence Limits Method 1... 

A variety of statistical parameters have been reported in the QSAR literature to reflect the quality of the model. These measures give indications about how well the model fits existing data, i.e., they measure the explained variance of the target parameter y in the biological data. Some of the most common measures of regression are root mean squares error (rmse), standard error of estimates (s), and coefficient of determination (R2). 

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

Equation 8 has a standard error of estimate of 0.07 fraction gypsum saturation over the saturation range of 0.5-2.0. The equation is useful for monitoring the actual gypsum saturation... 

Equation 10 explains 95 percent of the variation in the data for SO2 removal with a standard error of estimate of 3.2 percent SO2 removal. Values of SO2 removal predicted by Equation 10 are plotted against the corresponding measured values in Figure 7. 

The first series of data includes five solutions with measurements by dynamic saturation at 25° to 168°C. As shown in Table II the ratio Psc HoO essentially independent of temperature. Yet from 25° to I50°C, Pg02 increases a factor of 150. For any given solution the standard deviation of PsOo l O over the temperature range was generally less than 10%. However, the error of estimate of Kc by equation (2) is as high as 26%, suggesting systematic errors from one experiment to the next. 

Flexible optimal descriptors have been defined as specific modifications of adjacency matrix, by means of utilization of nonzero diagonal elements (Randic and Basak, 1999, 2001 Randic and Pompe, 2001a, b). These nonzero values of matrix elements change vertex degrees and consequently the values of molecular descriptors. As a rule, these modifications are aimed to change topological indices. The values of these diagonal elements must provide minimum standard error of estimation for predictive model (that is based on the flexible descriptor) of property/activity of interest. 

Measurement of precision. Measurement of data quality is valuable for both the analyst and the data user. Least-squares curve-of-best-fit statistical programs usually provide some information on precision (correlation coefficient, standard error of estimate). However, these are not sufficiently quantitative and often overstate the quality parameters of the data. 

Calibration curve quality. Calibration curve quality is usually evaluated by statistical parameters, such as the correlation coefficient and standard error of estimate, and by empirical indexes, such as the length of the linear range. Using confidence band statistics, curve quality can be better described in terms of confidence band widths at several key concentrations. Other semi-quantitative indexes become redundant. Alternatively, the effects of curve quality can be incorporated into statements of sample analysis data quality. 

Light measurement offers the combined capability of rapidly predicting by nondestructive means dust and trash content in cotton and airborne dust level. Of course, the standard error of estimate is not a practical statistic based on only six cottons and is not reported in this feasibility paper. 

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