Error-squared

The last column of Table 10-5 shows (1) that the long-term source of variation clearly overshadows the short-term, the ratio of variances exceeding 70 and (2) that the short-term variance is comparable with Arx, which is, of course, the standard counting error squared (Equation 10-8). 

For least squares estimation, p is the errors squared function, and the influence function ft =u. For a very large residual, u — oo, and r also grows to infinity this means that a single outlier has a large influence on the estimation. That is, for least squares estimation, every observation is treated equally and has the same weight. 

RMSE Root of mean squared errors (square root of MSE). 

The error term e in the above equation is the deviation of the value of Y, from the true value Y . A least square analysis was carried out for each dependent variable Y with the objective of finding the best linear equation that fits the data with respect to the criteria minimizing the sum of the error square (i.e., minimize... 

Reproduce the observation matrix in Section 1.8.7 using 1, 2, 3, and 4, respectively, primary factors. Compute the sum of reproduction error squares for each case. Compare these sums with the following sums 2 + 3 + 4 + 5 3 + M + 5> M + x5 5> respectively. 

The error square average of a trial has the well-known form ... 

Factor-variation intervals in simplex designing are as a rule chosen so that they are five to ten times greater than the error square mean of their measurements. 

The structure parameters rpq, 6pq, and npq in Eq. (9) are finally refined by the least-squares method to minimize the error square sum U,... 

In Equations 4 and 5, r is the multiple correlation coefficient, r2 is the percent correlation, SE is the standard error of the equation (i.e the error in the calculated error squares removed by regression to the mean sum of squares of the error residuals not removed by regression. The F-values were routinely used in statistical tests to determine the goodness of fit of the above and following equations. The numbers in parentheses beneath the fit parameters in each equation denote the standard error in the respective pa-... 

In contrast to the explicit analytical solution of least-squares fit used in linear regression, our present treatment of data analysis relies on an iterative optimization, which is a completely different approach as a result of the operations discussed in the previous section, theoretical data are calculated, dependent on the model and choice of parameters, which can be compared with the experimental results. The deviation between theoretical and experimental data is usually expressed as the sum of the errors squared for all the data points, alternatively called the sum of squared deviations (SSD) ... 

The usual criterion of best fit is the sum of errors squared (the SSD discussed above) rather than the absolute magnitude of the errors. This procedure is mathematically justified when the errors in the data follow the Gaussian (or normal) distribution. Under these conditions the error distribution function is given by Eqn. 9.6 in which % is the measurement, p the mean, and a the standard deviation cf. Sect. 

When the data are distributed according to this function, the frequency of occurrence of data falls according to the square of the deviation. In practice, the sum of error square (SSD) criterion is also used in cases where it has not been explicitly established that the errors are normally distributed, and it appears to function quite... 

Analysis of the shape of error surfaces. To conclude this section, we consider a more quantitative approach to error estimation. The first step is to estimate the accuracy of the individual data points this can either be done by analysis of the variability of replicate measurements, or from the variation of the fitted result. From that, one can assess the shape of the error surface in the region of the minimum. The procedure is straightforward the square root of the error, defined as the SSD, is taken as a measure of the quality of the fit. A maximum allowed error is defined which depends on the reliability of the individual points, for example, 30% more than with the best fit, if the points are scattered by about 30%. Then each variable (not the SSD as before) is minimised and also maximised. A further condition is imposed that the sum of errors squared (SSD) should not increase by more than the fraction defined above. This method allows good estimates to be made of the different accuracy of the component variables, and also enables accuracy to be estimated reliably even in complex analyses. Finally, it reveals whether parameters are correlated. This is an important matter since it happens often, and in some extreme cases where parameters are tightly correlated it leads to situations where individual constants are effectively not defined at all, merely their products or quotients. Correlations can also occur between global and local parameters. 

Situations arise very often where data need to be fitted to linear equations. Linear regression is one of the classical procedures in general regression analysis, and before the advent of accessible non-linear fitting methods it was the only one that could be readily used. For n data pairs in the form (x,y) where y is a function of x, the linear equation of the form y = a + bx that minimises the sum of errors squared (SSD) is given by ... 

Sillen L. G. (1962) High-speed computers as a supplement to graphical methods 1. The functional behavior of the error square sum. Acta Chem. Scand. 16, 159-172. 

It s possible to create an array with three dimensions by entering an array formula in each cell of a rectangular range of cells. The following example illustrates the use of a three-dimensional array to calculate an "error surface" curve such as the one shown in Figure 5-16. The error-square sum, i.e., the sum of the squares of the residuals, S(i/obsd J/calc) for a one-dimensional array of data points, was calculated for each cell of a two-dimensional array of trial values. The "best" values of the independent variables are those which produce the minimum error-square sum. 

Excel Tip. Don t introduce constraints (e.g., to force a constant to be greater than or equal to zero) if you re using the Solver to obtain the least-squares best fit. The solution will not be the "global minimum" of the error-square sum, and the regression coejficients may be seriously in error. 

Both the Precision and Tolerance options apply only to problems with constraints. The Precision parameter determines the amount by which a constraint can be violated. The Tolerance parameter is similar to the Precision parameter, but applies only to problems with integer solutions. Since adding constraints to a model that involves minimization of the error-square sum is not recommended, neither the Precision nor the Tolerance parameter is of use in non-linear regression analysis. 

The weights are proportional to the reciprocal of the errors squared in the experimental values. The quantity Vx and Vy are the x and y residuals. The principle of least squares is the minimization of S. The method of least squares is a rule or set of rules for proceeding with the actual computation. [Chap 4, 36, p. ]... 

To compute a nonlinear map, the distances between all pairs of descriptors are calculated. The initial positions of the compounds on the map are chosen randomly and then modified in an iterative algorithm until all distances are represented as well as possible. The core algorithm of NLM is a partial least-squares error minimization (PLS). The total error of mapping must be smaller than the distances between the molecules and is therefore given on the NLM, e.g., as sum of error squares, E2. 

It is important to incorporate this I/O model parameter uncertainty in the simulation of clinical trials. In order to implement parameter or model uncertainty in the simulation model, the typical values (mean values) of model parameters are usually defined as random variables (usually normally distributed), where the variance of the distribution is defined as standard error squared. The limits of the distribution can be defined at the discretion of the pharmacometrician. For a normal distribution, for example, this would be 0 + 2 SE, where 6 is the parameter. This would include 95% of the simulated distribution. When the simulation is performed, each replicate will have different typical starting values for the system parameters. The... 

Two methods are suggested for determining a either start with 1 and keep halving it until a smaller sum of error squares is obtained, or initiate a search for the value of a that minimizes the sum of error squares. This modification to the Newton-Raphson procedure helps stabilize the calculations, thereby highly enhancing the likelihood of convergence. 

Linear The constant K in the linear isotherm expression was determined by fitting that equation to the data. This was accomplished by minimizing the sum of the errors squared, X[q(data)-q(calculated)]T The resulting equation is... 

The following equation was obtained by minimizing the sum of errors squared based on the linearized equation. The resulting isotherm is... 

So the multiple imputation parameter estimate is the mean across all m-imputed data sets. Let U(0 ) be the variance of 0 , i.e., the standard error squared, averaged across all m-data sets... 

Table A-66 Equilibrium constants, -logm (Th (OH) ) +3ct in 3 M NaC104 and the standard deviation in the error-carrying variable E (in mV) from the least squares analysis of experimental data for the Th(lV)-hydroxide system. U is the error square sum for the different models.

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