Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 296

Bias is a second important characteristic of data. The holes in targets B and D are consis-
tently to the lower left of the center; that is, there is a systematic distribution of the wholes
with respect to the center of the target. If we found a point that could represent the center
of the holes in any target, then the difference between the center of the holes and the center
of the target would be a measure of the shooter's bias. More formally, bias is a systematic
deviation of values from the true value. The holes in cases B and D show a systematic devi-
ation to the lower left, while cases A and C are considered to be unbiased because there is no
systematic deviation.
Precision and bias are elements of accuracy. Specifically, bias is a measure of systematic
deviation and precision is a measure of random deviations. Inaccuracy can result from either
a bias or a lack of precision. In referring to the targets, the terms correctness and exactness were
used. More formally, accuracy is the degree to which the data deviate from the true value. If the
deviations are small, then the method or process that led to the measurement is accurate.
Table 7.2 shows a summary of the concepts of accuracy, precision, and bias as they char-
acterize the holes in the targets of Fig. 7.5 . Of course, terms like high, moderate, and low are
somewhat objective.
Approximation of one single value of point can be illustrated as neatly as Fig. 7.5 . One can
observe the precision and bias. For the approximation of a continuous line, one cannot show
as clear as for one single point. For the case where the continuous line is not known for sure
and the data contain error, the accuracy becomes more difficult to define. For regression or
parametric estimation analysis, this is exactly the case.
Similar to Fig. 7.5 for the approximation of a true value by experimentation or shooting
a target, Fig. 7.6 shows the regressions of a set of data. In this case, the data contain error
but with a true value manifests in the mean. Regression analysis can be applied to reveal
the relationship or continuous line. Case A shows a perfect fit to the data: the line represents
the mean of the data. The data scatter around the line and there is no systematic bias in
appearance. One can therefore observe the error distribution in the data.
In case B, the continuous line shows the same trend as the data qualitatively. However, the
data are consistently off the line: the mean of the data is shifted away from the continuous
line. There is a systematic error in the parameters of the line. This is similar to case B of
Fig. 7.5 . Corrections to the parameters are needed.
In case C, the line represents the overall mean of the data. However, the line lacks the
detailed agreement with the data: a hump in the middle of the range cannot be explained
as simply the experimental error. Therefore, the regression model is biased and unable to
reveal the detail of the data. In case D, the regression model has the same problem as

TABLE 7.2 Summary of Bias, Precision and Accuracy of the Cases Shown in Fig. 7.5

Case Bias Precision Accuracy

A None High High

B High High Low

C None Low Low

D Moderate Low Low