Reduce Bias Errors

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. 

Avoid the line frequency and harmonics As discussed above, measurements made at the line frequency or its first harmonic typically have a significant error that will appear as an outlier when compared to the rest of the spectrum. Measurement within 5 Hz of the line quency and its first harmonic should be avoided. 

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. 

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Work should therefore begin well in advance of any controller design work and certainly before any MVC implementation project is awarded. Many of the questions can be answered by the plant owner, supported by a specialist if required, by first attempting to develop regression type inferentials. It will quickly become apparent whether further data collection is required. For example it may be necessary to operate under different test run conditions with accurate time-stamping of samples taken at steady state. Automatic laboratory updating can be explored to see if this significantly reduces bias error or worsens random error. 

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 ... 

Note that bias does not affect the optimum n (since replication does not reduce bias). The major difficulty with applying this model lies In Identifying the cost of estimation error, 2 ... 

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... 

Reduce bias and stochastic errors The efforts described in Sections 8.3.1 and 8.3.2 to reduce buis and stochastic errors will also improve the information content of the data. 

Typically, the number of line-shapes that can be determined in a complex fit will increase when data inconsistent with the Kramers-Kronig relations are removed. Deletion of data that are strongly influenced by bias errors increases the amount of information that can be extracted from the data. In other words, the bias in the complete data set induces correlation in the model parameters, which reduces the number of parameters that can be identified. Removal of the biased data results in a better-conditioned data set that enables reliable identification of a larger set of parameters. 

SD itself does not yield useful information, unless compared to the mean (%CV Section 15.1). Between-assay variation is generally greater than within-assay variation, but care should be taken with the latter to reduce bias by randomly distributing the duplicates. The between-assay variation is usually of more value to estimate the accuracy of the procedure and, plotted on a quality-control chart, may indicate trends, such as deterioration of reagents (bias-type error Section 1.3). An example of the calculation of the within-assay and between-assay variability is given in Table 15.3. In this example, the SD of between-assay results is about 5-6 times higher than for the within-assay results. For a satisfactory EIA, this factor should be less than 2-3 and the between-assay variability should be less than 10%. 

The goal of randomization is to eliminate bias or, in practical terms, to reduce bias to the greatest extent possible. Bias is the difference between the true value of a particular quantity and an estimate of the quantity obtained from scientific investigation. Various influences can introduce error into our assessment of treatment effects, and these are discussed at various points in the following chapters. At this point we discuss an example of systematic error, or bias. 

In the process of experimentation, there exist two types of errors random errors and bias errors. Random error is experimental error for which the numerical vtilues change from one run to another without a consistent pattern. It can be thought of as inherent noise in measured responses. Bias error is experimental error for which the numerical values tend to follow a consistent pattern over a number of experimental runs. It is attributed to an assignable cause. To reduce the effects of both types of errors, it is strongly adviced that the following good experimented practices be taken into consideration. 

The source of bias error in an experiment may accompany differences among blocks, namely batches of raw materials, production machine, hours within a day, or seasons of the year. It is necessary to reduce their influence by proper design of the experiment. Blocking means running the experiment in a specially chosen subgroup that allows removal of the effect of bias errors that are confounded with the main factors. 

An illustration of the sampling bias (i.e., due to discretization error) is shown in Eig.7.1. As the stepsize is increased, the error in sampling is increased as well, limiting the effectiveness of numerical methods. This bias can be dramatically different for different numerical methods. As we shall show, with the right choice of numerical method it is often possible to substantially reduce this error, and it is also possible to calculate (under some assumptions) the perturbation introduced by the numerical method, and to correct for its presence. 

The factors that contribute to Nmc have been examined previously by Ceperley [159]. First, the variance of the local energy increases with molecular size, so the number of independent MC points must increase to reduce the error to the same tolerance. Second, the MC points will not be independent of each other until the random walk has been given sufficient time for the walker positions to decorrelate. Third, the time step bias of a DMC calculation increases with system size, so a larger number of steps must be taken before the decorrelation time is achieved. 

Provided the results are in the correct sequence, there is no need for the sample interval to be fixed. Thus if samples are taken at irregular intervals, such as repeat tests, they may still be included. Figure 9.11 shows the CUSUM trend. If the error were 100 % random, the trend would be noisy but horizontal and no bias update is required. If a bias error is present then the gradient of the CUSUM trend is the amount by which the inferential is overestimating and so the amount by which the bias should be reduced. In our example this value is 0.49. Since it already includes several historical values the correction can applied immediately with confidence. 

Our case study provided evidence of automation bias effects in the use of CAD effects which could not be attributed to complacency and could actually coexist with users reported mistrust towards the tool [35]. Previous studies had concluded that on average using CAD was either beneficial or ineffectual. Our analyses indicated instead that CAD reduced decision errors by some readers on some cases but increased errors by other readers on some cases. In short, this simple computer-assisted task hid subtle effects, easy to miss by designers and assessors [37]. 

It is worth noting that traditional correlation approaches, such as MRCI and CCSD(T), are bia.sed toward states with fewer d electrons while the DFT approaches are bia.sed toward states with higher d occupations. That is, the errors in the atomic state separations for the traditional and DFT approaches are in the opposite direction. If the atomic eiror is accounted for, the agreement between the traditional and DFT methods is improved, as well as the agreement with experiment. Ideally, improved methods will reduce the errors in the atomic description and hence reduce, if not eliminate, the need to consider such residual errors. However, at the present time, one must still account for possible d occupation biases in molecular calculations. 

To reduce the bias error Vbias necessary to reduce the total measurement time (to avoid the system nonstationarity however, that leads to higher stochastic errors as integration becomes restricted), introduce delay or quiet... 

Each observation in any branch of scientific investigation is inaccurate to some degree. Often the accurate value for the concentration of some particular constituent in the analyte cannot be determined. However, it is reasonable to assume the accurate value exists, and it is important to estimate the limits between which this value lies. It must be understood that the statistical approach is concerned with the appraisal of experimental design and data. Statistical techniques can neither detect nor evaluate constant errors (bias) the detection and elimination of inaccuracy are analytical problems. Nevertheless, statistical techniques can assist considerably in determining whether or not inaccuracies exist and in indicating when procedural modifications have reduced them. 

The Math As is immediately apparent, a mean cannot coincide with a SL if all measurements that go into it must also conform (cf. Fig. 2.13, distribution for p = 0.5), unless = 0. Any attempt to limit the individual measurements to the specification interval will result in a narrowing of the available margin for error in Xmean, be it manufacturing bias or inhomogeneity. This may be acceptable as long as one has the luxury of 90-110% release limits, but becomes impracticable if these are reduced to 95-105%. 

In the statistics literature, one usually distinguishes between the estimated mean () and the true (unknown) mean (). Here, in order to keep the notation as simple as possible, we will not make such distinctions. However, the reader should be aware of the fact that the estimate will be subject to statistical error (bias, variance, etc.) that can be reduced by increasing the number of notional particles (Vp. 

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