Bias errors

However, other bias errors are so substantial that their presence will significantly distort any conclusions drawn from the adjusted measurements. Rectification is the detection of the presence of significant bias in a set of measurements, the isolation of the specific measurements containing bias, and the removal of those measurements from subsequent reconcihation and interpretation. Significant bias in measurements is defined as gross error in the literature. 

The bias error is a quantity that gives the total systematic error of a measuring instrument under defined conditions. As mentioned earlier, the bias should be minimized by calibration. The repeatability error consists of the confidence limits of a single measurement under certain conditions. The mac-curacy or error of indication is the total error of the instrument, including the... 

The bias limit B. The bias limit is an estimate of the constant error. It is assigned with the understanding that the true value of the bias error, if known, would be less than B with 95% confidence. 

In Eq. (2.4) the temperatures Tout and Tin, denote the bulk mean air temperatures at the outlet and inlet cross-sections, respectively. Their representation by point measurements introduces a bias error equal to the difference between the latter and the corresponding bulk means. In evaluating it, allowance should be made for the residual uncertainty involved in the bias errors from the probe calibration, etc. 

Consider now a situation in which the bias limits in the temperature measurements are uncorrelated and are estimated as 0.5 °C, and the bias limit on the specific heat value is 0.5%. The estimated bias error of the mass flow meter system is specified as 0.25% of reading from 10 to 90% of full scale. According to the manufacturer, this is a fixed error estimate (it cannot be reduced by taking the average of multiple readings and is, thus, a true bias error), and B is taken as 0.0025 times the value of m. For AT = 20 °C, Eq. (2.9) gives ... 

Now consider the finite sampling systematic error. As discussed in Sect. 6.4.1, the fractional bias error in free energy is related to both the sample size and entropy difference 5e N exp(-AS/kB). With intermediates defined so that the entropy difference for each substage is the same (i.e., AS/n), the sampling length Ni required to reach a prescribed level of accuracy is the same for all stages, and satisfies... 

The primary goal of this series of chapters is to describe the statistical tests required to determine the magnitude of the random (i.e., precision and accuracy) and systematic (i.e., bias) error contributions due to choosing Analytical METHODS A or B, and/or the location/operator where each standard method is performed. The statistical analysis for this series of articles consists of five main parts as ... 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

In a variable-density PDF code, passing back the time-averaged particle fields should have an even greater effect on bias. For example, using a Lagrangian velocity, composition PDF code, Jenny et al. (2001) have shown that the bias error is inversely proportional to the product Vp K. 

The algorithms discussed earlier for time averaging and local time stepping apply also to velocity, composition PDF codes. A detailed discussion on the effect of simulation parameters on spatial discretization and bias error can be found in Muradoglu et al. (2001). These authors apply a hybrid FV-PDF code for the joint PDF of velocity fluctuations, turbulence frequency, and composition to a piloted-jet flame, and show that the proposed correction algorithms virtually eliminate the bias error in mean quantities. The same code... 

Bias = % error difference between measured value and true value Z score difference between measured value and certified reference value %... 

Use of CRMs calculate bias (% error) and/or Z score No CRMs spike typical matrix and blank sample with known amount of analyte calculate %recovery = 100 x [ sp i kellla,n,... 

Considerable bias errors may be due to the preparation of standards, as well as inaccuracies associated with the actual measurements. Undoubtedly, care in the selection and initial testing of electrodes will improve the quality of results. Most important, however, is the unambiguous description of preparation and measurement procedures, and the adoption of routine analytical quality control. The quality of pH data, like those from any other analytical determination, will be greatly improved by strict adherence to a rigorously defined proven routine. Although implementing a program of quality control will decrease errors, it will not ensure the accuracy of the determination. 

Dynamic response and bias errors of measurement (Section IV.C)... 

Remember 8.3 Impedance measurements entail a compromise balance behveen minimizing bias errors, minimizing stochastic errors, and maximizing the information content of the resulting spectrum. The optimal instrument settings and experimental parameters are not universal and must be selected for each system under study. 

The steps described in this section may be taken to reduce the role of systematic bias errors (see Section 21.3) in impedance measurements. Bias errors associated with nonstationary effects have greatest impact at low frequencies where each measurement requires a significant amount of time. 

Introduce a delay time As discussed above, the transient seen as the system adjusts to a changed modulation frequency yields a bias error in the measured... 

Select an appropriate modulation technique Proper selection of modulation technique, discussed in Section 8.2.3, can have a significant impact on presence of bias errors. Use of potentiostatic modulation for a system in which the potential changes with time can increase measurement time on autointegration. The user should consider what should be held constant (e.g., current or potential). 

Instrument bias errors are often seen at high frequencies, especially for systems exhibiting a small impedance. 

Use a faster potentiostat The influence of high-frequency bias errors can be mitigated by proper selection of potentiostat. The capability of potentiostats to perform measurements at high frequency differs from brand to brand. 

Use short shielded leads High-frequency bias errors can be seen when the cell impedance is of the same order as the internal impedance of the instrumentation. Under these circumstances, it is essential to minimize the impact of ancillary pieces such as wires. Use of short shielded cables is highly recommended. 

Check the results The presence of instrument bias errors can be difficult to discern. The Kramers-Kronig relations may provide a suitable guide, but as discussed in Chapter 22, some instrument-imposed bias errors are Kramers-Kronig transformable. If possible, high-frequency asymptotic values should... 

Regression problems in impedance spectroscopy may become ill-conditioned due to improper selection of measurement frequencies, excessive stochastic errors (noise) in the measured values, excessive bias errors in the measured values, and incomplete frequency ranges. The influences of stochastic errors and foequency range on regression are demonstrated by examples in this section. The issue of bias errors in impedance measurement is discussed in Chapter 22. The origin of stochastic errors in impedance measurements is presented in Chapter 21. 

While the nature of the error structure of the measurements is often ignored or understated in electrochemical impedance spectroscopy, recent developments have made possible experimental identification of error structure. Quantitative assessment of stochastic and experimental bias errors has been used to filter data, to design experiments, and to assess the validity of regression assumptions. 

A distinction is drawn in equation (21.1) between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The problem of interpretation of impedance data is therefore defined to consist of two parts one of identification of experimental errors, which includes assessment of consistency with the Kramers-Kronig relations (see Chapter 22), and one of fitting (see Chapter 19), which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models. The experimental bias errors, as referred to here, may be caused by nonstationary processes or by instrumental artifacts. 

In the absence of bias errors, the errors in the real and imaginary impedance are uncorrelated and the variances of the real and imaginary parts of the complex impedance are equal. Some specific identities are given in Table 21.1. 

Bias errors are systematic errors that do not have a mean value of zero and that cannot be attributed to an inadequate descriptive model of the system. Bias errors can arise from instrument artifacts, parts of the measured system that are not part of the system under investigation, and nonstationary behavior of the system. Some types of bias errors lead the data to be inconsistent with the Kramers-Kronig relations. In those cases, bias errors can be identified by checking the impedance data for inconsistencies with the Kramers-Kronig relations. As some bias errors are themselves consistent with the Kramers-Kronig relations, the Kramers-Kronig relations cannot be viewed as providing a definitive tool for identification of bias errors. 

---