Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 295

problem. The regression problem is essentially an optimization problem: minimizing the
variance of data around the regression model by changing the regression parameters! In
this sense, we will not need any other special techniques to perform regression analysis.

7. 7. QUALITY OF FIT AND ACCURACY OF DAT A

When dealing with regression, we need to address the issues of accuracy , precision , and bias .
The term accuracy is commonly used, for example, as in reference to the accuracy of the
weather forecasting or to the accuracy of an opinion poll taken just before a national election.
Of interest here is the meaning of the term as it appears to the data as compared to what they
should be. It is directly applicable to the quality of fit.
Consider the four targets of Fig. 7.5 . Assuming that those shooting at the targets were
aiming at the center, the person shooting target A was successful. While not every bullet
went through the exact center, the distance between the holes and the center is small. The
holes in target B are similarly clustered as in target A, but they show large deviation from
the center. The large deviations or errors between the holes and the center of the target
suggest a lack of exactness or correctness. While all the holes deviate significantly from the
center, there is, however, a measure of consistency in the holes. In summary, the holes in B
show two important characteristics: they tend to agree with each other but they deviate
considerably from where the shooter was aiming.
The holes in target C are very different in characteristics from the holes in either target A or
target B. For one, most holes are not near the center, and two, they are not near to each other.
Thus, they lack both correctness and consistency. Because most of them are not near the
center of the target, the shooter is not exact. Because of the wide scatter, there is a lack of
consistency. In comparing the holes on targets B and C, they both lack exactness, but there
is a measure of consistency for the holes in target B that is missing in target C.
The fourth distribution of holes is shown as case D. Like case B, all of the holes are to the
lower left of the center; but unlike case B, the holes lack consistency. Both cases C and D show
considerable scatter, but with case D, the scatter is concentrated to one side of the target.
A comparison of the four targets indicates that there are three important characteristics
of data. The holes in targets A and B show a measure of consistency. In data analysis, this con-
sistency is more formally referred to as precision. Precision is defined as the ability to give
multiple estimates that are near to each other. In terms of cases A and B, the shooters were
precise. The shooters of cases C and D were imprecise since the holes show a lot of scatter.

Case A Case B Case C Case D

FIGURE 7.5 A schematic of accuracy, precision, and bias.