Biomedical Engineering Reference
In-Depth Information
encountered in bioprocess engineering. For example, finding the kinetic constants of a chem-
ical reaction with known mechanism, finding the exact relationship between flow rate and
pressure drop for laminar flow of a liquid through complex network, etc. Parametric estima-
tion is the main topic of this chapter.
In other occasions, one may have no access to the fundamental knowledge locking the
underlying problem to be modeled. Sets of data are collected or available for the relations
of the variables in question. Although no sound physical or engineering background is
required, a mathematical model is needed for process optimization, prediction of outcome
at some untested input, or process control. In this case, one must first determine if the vari-
ables are related. If they are related, then determine how they are related. This leads to the
classic problem of
correlation
or
correlation analysis
. Still, if we know for sure that the variables
are related, a mathematical model must be determined from the data set. This latter case is
a subset of the former:
correlation analysis
. We call the latter problem
regression analysis
or
curve-fitting
. Correlation and/or regression analysis is a classic subject area in applied math-
ematics and has been an active subject of discussion for centuries.
The three formal problems or analyses:
parametric estimation
,
regression
, and
correlation
are
interrelated.
Regression
or
curve-fitting
consists of selecting a mathematical model, usually of
polynomials and then applying
parametric estimation
to determine the unknown coefficients.
The mathematical model is thus frequently called regression model.
Correlation
is the most
general terminology among these three terminologies. It consists of determining whether
the variables are related (or correlated) and then regression/curve-fitting is applied to find
a best relationship relating the variables. Because of the close relation among the three termi-
nologies, it is often unclear what the differences are among these three terminologies. Very
often, one uses the term correlation or curve-fitting without qualification to substitute all
the three terminologies. Because a set of data can be fitted with many different functions
with similar quality in the representation of the data, one is not to confuse the correlation
with a physically correct mathematical model. As the data contain error, the best-fit model
does not necessarily mean the true relationship. Nevertheless, correlation can be applied
as the best aid to engineering research and process design and/or modeling.
7.1. REGRESSION MODELS
Regression analysis is a statistical/numerical technique for investigating and/or modeling
the relationship between two or more variables. For example, in a bioprocess, suppose that
the yield of the product is related to the process-operating temperature. Thermodynamics
and chemical/bioprocess engineering principles (including heat transfer, mass transfer,
and reaction engineering) can be applied to relate the yield to the operating temperature.
However, because of the complexity and uncertainty involved, we often attempt to approach
the problem in a heuristic manner. The derived model expression is thus a simplified version
of the true underlying relationship. There are model parameters in the derived expression
that can only be determined by experimentation. The evaluation of the unknown parameters
is thus called parametric estimation. In other occasions, we may not know how the variables
are related and model expressions are selected simply based on simplicity. In any case, the
experimental data need to be generated for the underlying problem. Regression analysis
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