Error terms

This left 20 X 20 X 8 = 3200 classes, with some classes being very sparsely populated. For such classes, the error term is unacceptably large,... 

The error term in the various methods can be used to deduce a step size that will give a user-specified accuracy. Most packages today are based on a user-specified tolerance the step-size is changed during the calculation to achieve that accuracy. The accuracy itself is not guaranteed, but it improves as the tolerance is decreased. 

The validity of least squares model fitting is dependent on four prineipal assumptions eoneerning the random error term , whieh is inherent in the use of least squares. The assumptions as illustrated by Baeon and Downie [6] are as follows ... 

Assumption 3 The variance of the random error term is constant over the ranges of the operating variables used to collect the data. When the variance of the random error term varies over the operating range, then either weighted least squares must be used or a transformation of the data must be made. However, this may be violated by certain transformations of the model. 

Error term Inner ring h = 3.5 m) Outer ring h = 14. 5m)... 

The component-of-varlance analysis Is based upon the premise that the total variance for a particular population of samples Is composed of the variance from each of the Identified sources of error plus an error term which Is the sample-to-sample variance. The total population variance Is usually unknown therefore. It must be estimated from a set of samples collected from the population. The total variance of this set of samples Is estimated from the summation of the sum of squares (SS) for each of the Identified components of variance plus a residual error or error SS. For example ... 

Ej is determined by the weights (through Oj which is a function of NET, see eq. (44.8)). Note that this error is in fact the same as the error term used in a usual least squares procedure. 

The variable e has been described as an error term, but is not used in most applications of the equation. 

A single experiment consists of the measurement of each of the m response variables for a given set of values of the n independent variables. For each experiment, the measured output vector which can be viewed as a random variable is comprised of the deterministic part calculated by the model (Equation 2.1) and the stochastic part represented by the error term, i.e.,... 

If the mathematical model represents adequately the physical system, the error term in Equation 2.3 represents only measurement errors. As such, it can often be assumed to be normally distributed with zero mean (assuming there is no bias present in the measurement). In real life the vector e, incorporates not only the experimental error but also any inaccuracy of the mathematical model. 

Again, the measured output vector at time t denoted as y, is related to the value calculated by the mathematical model (using the true parameter values) through the error term,... 

As in algebraic models, the error term accounts for the measurement error as well as for all model inadequacies. In dynamic systems we have the additional complexity that the error terms may be autocorrelated and in such cases several modifications to the objective function should be performed. Details are provided in Chapter 8. 

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... 

Therefore, on statistical grounds, if the error terms (e,) are normally distributed with zero mean and with a known covariance matrix, then Q( should be the inverse of this covariance matrix, i.e.,... 

Case I Let us consider the stringent assumption that the error terms in each response variable and for each experiment (8,j, i=l,...N j=lare all identically and independently distributed (i.i.d) normally with zero mean and variance, a g. Namely,... 

In implicit estimation rather than minimizing a weighted sum of squares of the residuals in the response variables, we minimize a suitable implicit function of the measured variables dictated by the model equations. Namely, if we substitute the actual measured variables in Equation 2.8, an error term arises always even if the mathematical model is exact. 

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). 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

The unknown parameter vector k is obtained by minimizing the corresponding least squares objective function where the weighting matrix Q, is chosen based on the statistical characteristics of the error term e, as already discussed in Chapter 2. 

If on the other hand we wish to compute the(l-a) 100% confidence interval of the response of y, at t=to, we must include the error term (e0) in the calculation of the standard error, namely we have... 

The ability of the sequential design to discriminate among the rival models should be examined as a function of the standard error in the measurements (oe). For this reason, artificial data were generated by integrating the governing ODEs for Model 1 with "true" parameter values kt=0.31, k2=0.18, k3=0.55 and k4=0.03 and by adding noise to the noise free data. The error terms are taken from independent normal distributions with zero mean and constant standard deviation (oE). 

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 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... 

The above equation cannot be used directly for RLS estimation. Instead of the true error terms, e , we must use the estimated values from Equation 13.35. Therefore, the recursive generalized least squares (RGLS) algorithm can be implemented as a two-step estimation procedure ... 

Error term (general, in x, and in y, respectively), in concrete terms, ey may be the deviation of an individual value from the mean or the residual of a mathematical model, see next line (ex analogous)... 

In such cases it may be possible to check on robustness and ruggedness by means of statistical tests (see Sect. 4.3). All the variations to the measured signal, apart from that of the analyte, can be considered in form of error terms see Eq. (3.6) ... 

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