True value

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

For each experiment, the true values of the measured variables are related by one or more constraints. Because the number of data points exceeds the number of parameters to be estimated, all constraint equations are not exactly satisfied for all experimental measurements. Exact agreement between theory and experiment is not achieved due to random and systematic errors in the data and to "lack of fit" of the model to the data. Optimum parameters and true values corresponding to the experimental measurements must be found by satisfaction of an appropriate statistical criterion. 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

The algorithm employed in the estimation process linearizes the constraint equations at each iterative step at current estimates of the true values for the variables and parameters. 

Table 1 gives the measured data, estimates of the true values corresponding to the measurements, and deviations of the measured values from model predictions. Figure 1 shows the phase diagram corresponding to these parameters, together with the measured data. 

Measured Variables and Estimates of Their True Values for Acetone(1)/Methanol(2) System (Othmer, 1928)... 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

This subroutine also prints all the experimentally measured points, the estimated true values corresponding to each measured point, and the deviations between experimental and calculated points. Finally, root-mean-squared deviations are printed for the P-T-x-y measurements. 

PRINTS THE results OF THE REGRESSION-OP VARRIANCE-COVARPIANCE MATRIX, CORRELAT AND THE VARRIANCE OF THE FIT. ALSO, PP DATA, THEIR ESTIMATED TRUE VALUES, AND ALL NN DATA POINTS. FINALLY, THF ROOT-DEVIATIONS ARE GIVEN FOR EACH 0 = THE... 

Y - VECTOP OF TRUE VALUES CF FIRST DEPENDENT VARIABLE... 

C 2 - VECTOR OF TRUE VALUES OF SECOND DEPENDENT VARIAL0E C... 

INITIAL ESTIMATES OF TRUE VALUES OF THE VARIABLES ARE SET EQUAL TO THE MEASURED VALUES. 

CALCULATES THE SUM OF THE SQUARES OF THE DEVIATIONS OF ALL MEASURED VARIABLES FROM THEIR TRUE VALUES FOR REGRES. 

APRAV OF TRUE VALUES CORRESPCNDING TC THE MEASURED VALUES IN THE XM ARRAY... 

VECTOR OF TRUE VALUES CORRESPCNDING TO THE MEASURED VALUES IN TH ZM VECTOR... 

CALCULATES THE CONSTRAINT FUNCTIONS FOP BINARY VAaOR-LIOUIO EQUILIBRIUM OATA PEOUCTION. THE CCNSTRAINT FUNCTIONS RELATING THE TRUE VALUES QP THE MEASURED VARIABLES ARE (1) PRESS = F(LI0 COMP,TEMP,PARAMETERS)... 

When the sensitivities are performed the economic indicator which is commonly considered is the true value of the project, i.e. the NPV at the discount rate which represents the cost of capital, say 10%. 

Table 1. Computed results from the algorithm for sizing of the cracks from Fig.4a, Fig.4b, and Fig.4c and orientation angles d)=0°, 0=30° and 0=45°, and true values of the eorresponding crack parameters.

In Figure 4 the measured attenuation values (TT) and the corresponding estimates are plotted against each other. Ideally (with error free estimates) all sample points should lie on the straight line through the origin with unit slope. Clearly there is a strong correlation between the estimates and the true values. 

The shift makes the potential deviate from the true potential, and so any calculated thermodynamic properties will be changed. The true values can be retrieved but it is difficult to do so, and the shifted potential is thus rarely used in real simulations. Moreover, while it is relatively straightforward to implement for a homogeneous system under the influence of a simple potential such as the Lennard-jones potential, it is not easy for inhomogeneous systems containing rnany different types of atom. 

The true value of tk for a many-electron atom or a molecule is unknown. If we could set it equal ( expand it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of (k. Such a set of basis functions would be a complete set. 

It is an estimation of the unknown true value p of an infinite population. We can also define the sample variance s as follows ... 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

Example 5 The true value of a quantity is 30.00, and cr for the method of measurement is 0.30. What is the probability that a single measurement will have a deviation from the mean greater than 0.45 that is, what percentage of results will fall outside the range 30.00 0.45 ... 

Identifying Determinate Errors Determinate errors can be difficult to detect. Without knowing the true value for an analysis, the usual situation in any analysis with meaning, there is no accepted value with which the experimental result can be compared. Nevertheless, a few strategies can be used to discover the presence of a determinate error. 

Analytical chemists make a distinction between error and uncertainty Error is the difference between a single measurement or result and its true value. In other words, error is a measure of bias. As discussed earlier, error can be divided into determinate and indeterminate sources. Although we can correct for determinate error, the indeterminate portion of the error remains. Statistical significance testing, which is discussed later in this chapter, provides a way to determine whether a bias resulting from determinate error might be present. 

Table 5.2 demonstrates how an uncorrected constant error affects our determination of k. The first three columns show the concentration of analyte, the true measured signal (no constant error) and the true value of k for five standards. As expected, the value of k is the same for each standard. In the fourth column a constant determinate error of +0.50 has been added to the measured signals. The corresponding values of k are shown in the last column. Note that a different value of k is obtained for each standard and that all values are greater than the true value. As we noted in Section 5B.2, this is a significant limitation to any single-point standardization. 

When an analyst performs a single analysis on a sample, the difference between the experimentally determined value and the expected value is influenced by three sources of error random error, systematic errors inherent to the method, and systematic errors unique to the analyst. If enough replicate analyses are performed, a distribution of results can be plotted (Figure 14.16a). The width of this distribution is described by the standard deviation and can be used to determine the effect of random error on the analysis. The position of the distribution relative to the sample s true value, p, is determined both by systematic errors inherent to the method and those systematic errors unique to the analyst. For a single analyst there is no way to separate the total systematic error into its component parts. 

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