Data Fitting Basics
Watch the Data Fitting Primer video
This extract from the
GraFit user manual describes some of the basic
principles behind data fitting.
Regression Analysis
Regression analysis finds the "best fit" line or curve
through a series of data. This is considered to be the one that minimizes the sum of the
squares deviations of the experimental data points from the theoretical curve. GraFit uses
regression to allow the analysis of experimental data.
Practical Considerations
The Errors are in the y Data
It is the convention to assume that the x data
values represent the independent variable, and the y data values represent the
dependent variable, i.e. the x data are what is varied in the experiment (for
example time, concentration, etc.), and the y data are the experimentally
observed values.
 The vertical distance (the red lines in the above diagram) between the
data point and the curve or line is that which is minimized by normal regression analysis,
i.e., it is assumed that all errors are present in the y data. An important aspect of
experimental design is to ensure that the x data are as accurate as possible so that this
assumption is valid.
The Data Values Have A Suitable Numerical Range
A consequence of computer calculation is that rounding errors can
occur, particularly if very large or very small numbers are used. This is because
computers only store decimal numbers with a limited numerical precision. Although this
should not affect most calculations, it is advisable to scale any set of very large or
very small values prior to performing calculations. For example, if a series of time
values were entered as 0 to 10-15 seconds, the rounding errors could be
significant. Entering the data as 0 to 1 femtoseconds may be more appropriate.
Types of Regression Analysis
Regression acts to minimize the sum of the squares deviations of the
experimental values from values calculated using some theoretical equation. However, the
mechanics of the calculations are different for linear and non-linear equations.
Linear Regression
Linear regression is a technique that should be familiar to most
scientists. It is the mathematical equivalent of using a ruler to draw the
"best" line through a series of data points to obtain values for the slope and
intercept of this line. The calculations needed to perform linear regression are
relatively simple, and are even found on many scientific calculators.
Polynomial Regression
Polynomial regression is an extension of linear regression, where the
equation is
y = a + bx + cx2 +dx3 + ex4 + ...
Non-linear Regression
Non-linear regression allows data to be fitted in the more general case
to any equation where the y data value can be described as a function of the x
data and a series of parameters, i.e.
y = f(x, p1, p2, p3, ...)
As a result, it is applicable to many more situations than linear
regression, and is more generally useful for analyzing experimental data. The only
drawback to non-linear regression is that the calculations are too complex perform using a
calculator, and require a computer program such as GraFit.
Additional Considerations for Non-linear Regression
The more complex nature of the non-linear regression calculations do
require some further considerations compared with linear regression.
Equation to Use
Non-linear regression can employ a vast range of different equations.
It is therefore necessary to select an equation that is appropriate to the particular
experimental situation before fitting the data.
Initial Estimates
The mathematical method used to perform non-linear regression acts to
take starting estimates for the parameters in the equation, and optimize them during the
calculations. GraFit provides a mechanism for providing these initial estimates
automatically for some equations, but in principle it is necessary to give the
calculations some rough estimates prior to data fitting.
Convergence Criteria
Unlike linear regression, non-linear regression does not provide an
exact solution, but, by an iterative process, calculates successively better parameter
values until a suitable tolerance is reached. The criteria for defining this tolerance are
determined by the program, and can be specified when using GraFit.
Weighting Scheme
Both linear and non-linear regression analyses incorporate assumptions
about the accuracy of the data points. Normally, they assume that all data points are
equally accurate, and so carry equal weighting in the analysis. It is also possible to use
other weighting schemes that give different weighting to one or more data values.
Linear Regression of Transformed Data
A familiar method of analyzing non-linear data graphically is to
rearrange the data to a linear form, find the "best line" through the points,
and so derive the parameter values. This is illustrated in the following example. Enzyme
kinetic data are described by an equation of the form
which is hyperbolic. This can be linearized by plotting 1/y
versus 1/x, as
The ease of linear regression analysis makes it tempting to use in order
to fit data that have been rearranged in this way. However, the results that
are obtained are not statistically optimal. This is because the regression
calculations make assumptions about the error distribution that are not
usually valid after transformation. The distortion of the experimental
errors by the above transformation is easily demonstrated in the following
graphs.
 |
 |
|
Enzyme kinetics plot. The error is present only in the Rate (y
axis) measurement. Errors are simple, i.e. all data points have the same error.
|
Double reciprocal (Lineweaver Burk) plot of the data above. Note the large
distortion of the error bars resulting from this plot. |
In this case, the error involved in the high data points
of the transformed plot is much greater than in the low data points - this is simply a
consequence of the reciprocal transformation. Linear regression (unless suitable
compensation is made for the distorted errors) is therefore not an appropriate way to
analyze the data, and for best results the original data should be fitted to the
appropriate non-linear equation.
Information Obtained From Data Fitting
Data fitting provides the following information regarding the
experimental data that are analyzed.
The values of the parameters in the equation used that best describe
the data. These parameters may include (for example) rate constants, inhibition constants
etc. whose values need to be determined.
The accuracy of these calculated parameters.
From this information it is possible to judge how well the data fit the
experimental model (equation) that was used. In most cases, however, it is the numerical
values of the parameters that are important, and whose values are required from the
analysis. |
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