Estimation of the Error Term

To continue, the variance (tr ) of the error term (written as for a population estimate or for the sample variance) needs to be estimated. As a general 

The sample variance is then derived by dividing the sum of squares by the degrees of freedom (n — 1)  

This formulaic process is also applicable in regression. Hence, the sum of squares for the error term in regression analysis is 

The mean square error (MSe) is used to predict the sample variance or Hence, 

Two degrees of freedom are lost, because both bo and bi are estimated in the regression model bo + bi x,) to predict y. The standard deviation is simply the square root of MSe  

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The above equations suggest that the unknown parameters in polynomials A( ) and B() can be estimated with RLS with the transformed variables yn and un k. Having polynomials A( ) and B(-) we can go back to Equation 13.1 and obtain an estimate of the error term, e , as... 

If there is replication of the experiment then an independent estimate of the error terms can be calculated and valid statistical tests, such as ANOVA, can be constructed. 

The design matrix (including actual and model predicted responses) generated for the Box-Behnken study is shown in Table 3.1. Here, three center point experiments were incorporated to compute an estimate of the error term that does not depend on the fitted model. Figure 3.1a shows the whole model leverage plot of actual-versus-predicted responses (based on aU effects) with the quality of fit expressed by the coefficient of determination (r ). This coefficient is variation in the response around the mean that can be attributed to terms in the model rather than to random error. 

At this point let us assume that the covariance matrices (E,) of the measured responses (and hence of the error terms) during each experiment are known precisely. Obviously, in such a case the ML parameter estimates are obtained by minimizing the following objective function... 

Even if we make the stringent assumption that errors in the measurement of each variable ( >,. , M.2,...,N, j=l,2,...,R) are independently and identically distributed (i.i.d.) normally with zero mean and constant variance, it is rather difficult to establish the exact distribution of the error term e, in Equation 2.35. This is particularly true when the expression is highly nonlinear. For example, this situation arises in the estimation of parameters for nonlinear thermodynamic models and in the treatment of potentiometric titration data (Sutton and MacGregor. 1977 Sachs. 1976 Englezos et al., 1990a, 1990b). 

We shall present three recursive estimation methods for the estimation of the process parameters (ai,...,ap, b0, b,..., bq) that should be employed according to the statistical characteristics of the error term sequence e s (the stochastic disturbance). 

The entire cyclic voltammogram is no longer reversible according to the definition we have attached to this term so far. In other words, the symmetry and translation operations as in Figures 1.4 and 6.1 do no longer allow the superposition of the reverse and forward trace. It also appears that the midpoint between the anodic and cathodic peak potentials does not exactly coincide with the standard potential. The gap between the two potentials increases with the extent of the ohmic drop as illustrated in Figure 6.2 for typical conditions, which thus provides an estimate of the error that would result if the two potentials were regarded as equal. 

As the design stands we cannot determine whether there is any interaction between the terms, for though we can compare, say (Mi — Mj) for F == Fi with (Ml — Mj) for F = Fa, our comparison is pointless because we have no estimate of the error of either term and hence of theii difference. Without the estimate of the error we cannot estimate the significance of any apparent difference. 

The estimation of the error of a computed result R from the errors of the component terms or factors A, B, and C depends on whether the errors are determinate or random. The propagation of errors in computations is summarized in Table 26-2. The absolute determinate error e or the variance V = s for a random error is transmitted in addition or subtraction. (Note that the variance is additive for both a sum and a difference.) On the other hand, the relative determinate error ejx or square of the relative standard deviation (sJxY is additive in multiplication. The general case R = f A,. ) is valid only if A, B,C,... are independently variable it is... 

Often, an independent estimate of the error variance is not available and in such cases the error mean square, MSE, is used as an estimate of the error variance to assess the significance of the model terms. 

This technique to apportion the total sum of squares over the different sources of contribution and to compare the estimated mean squares thus obtained to estimates of the error variance is called analysis of variance. It is often abbreviated as ANOVA. The analysis of variance is usually presented as a table showing (a) the total sum of squares, the sum of squares due to regression (sometimes divided into the contribution of the individual terms in the model), the error sum of squares, (b) the degrees of freedom associated with the sums of squares, (c) the mean squares,... 

Because these parameters are linearly related in Equation 1, the statistical errors introduced by incorrect estimates are comparable for each term. The greatest cumulative error in many kinetic models lies in the estimating of natural systems reactive surface areas. This paper will review previous work, present additional data, and provide an estimate of the errors involved in using surface-area parameters in geochemical models. 

Equation (85) forms a very good, usually "chemical approximation (i.e. to about 1 kcal/mole) to E. It will be taken up again later as a basis for semi-empirical approaches. For non-empirical calculations, it is the part of Eq. (82) or (58) to be minimized in obtaining the 4,/s. Once the 4substitution back into R and D gives an upper limit to the exact E and an estimate of the error involved in dropping these minor terms. 

In experimental designs, usually the values of x are preselected at specific levels, and the y values corresponding to these are dependent on the x levels set. This provides y or x values, and a controlled regimen or process is implemented. Generally, multiple observations of y at a specific x value are taken to increase the precision of the error term estimate. 

Care must be exercised in relating the counting error (or indeed any intensity related error) with an estimate of the error in terms of concentration. Provided that the sensitivity of the spectrometer in counts per second per percent, is linear, a count error can be directly related to a concentration error. However, where the sensitivity of the spectrometer changes over the range of measured response, a given fractional count error may be much greater when expressed in terms of concentration. 

Another technique, that is more flexible than the rescaled expansion and appears to converge slightly better, is weighted truncation [11]. This method is based on the idea of an optimal asymptotic approximation [36]. A typical characteristic of divergent asymptotic expansions is that their partial sums at first steadily approach the correct value but then, after a certain point, become steadily worse. An estimate of the error in the nth partial sum is given by the term in the expansion of order n - -1, that is The partial sum for... 

It is a requirement of the GUM that all known systematic errors are corrected by an estimate of the correction term. This estimate will have an uncertainty even if... 

Perturbation theory offers some advantages in this respect. It is also subject to basis superposition error, but the contributions in which such errors may occur can be identified, and it is possible to make an estimate of the error and to correct for it to some extent. The charge-transfer energy is subject to basis superposition error, but it is possible to estimate the contribution of this error to the result. The extension correlation and double charge transfer terms are wholly due to basis superposition effects, to lowest order in overlap at least[l8], and can be discarded. The charge-transfer correlation and dispersion terms, on the other hand, can have no basis superposition error at all because they can only arise when occupied orbitals of both molecules are present[193. 

The key question is How good an approximation is it to neglect the terms involving dvr/dzV We can get an estimate of the error by evaluating the ratio of the two terms in Equation 6.14 as follows ilabel>(6.20b)i/label>imime type= graphic xlink href= 89%96eqn24 > j alt-text > j CD AT A j/sec > jsec... 

In practice, however, this does not pose serious problems, because one can derive useful estimates of the error introduced by dropping the second term of the series. One often defines the extent of a Gaussian distribution... 

This is the simplest form of finite difference derivative and is called the forward difference approximation to the derivative. E(f) represents the error in the approximation. In order to estimate the size of the error term, consider the Taylor expansion of/(x + Ax) in the neighborhood of x ... 

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