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nonlindc.seq
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2009-10-10
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NONLIN DOC FILE
PURPOSE: NONLIN determines the best straight or curved line to fit a set of
data points by a process known as nonlinear least-squares curve-fitting. It
then prints the parameter values that define the curve and plots the results
using one of several formats. The program readily determines the best line
through a group of points, the rate constant of an exponential decay process,
the Km and Vmax (or Kd and N) of a saturable process, and solves related
problems for up to seven unknown parameters.
ANALYSES SUPPORTED: Data may be analyzed according to three basic equations:
POLYNOMIALS: Polynomial curves are defined by the equation:
Y = B(1) + B(2)*X + B(3)*X^2 + B(4)*X^3 + B(5)*X^4 + B(6)*X^5 + B(7)*X^6
Where B(1), B(2), etc. are the unknown parameters which will be optimized to
the data. From two to seven parameters can be fitted to the data. In its
simplist form (two parameters), the equation defines a straight line where
B(1) is the Y intercept and B(2) is the slope, and the analysis is called
linear regression. If more than two parameters are selected, the result is a
curve and the analysis is called nonlinear regression. Parameters are added
in the order that they appear in the above equation. Most smoothly varying
data can be fit to this equation if a sufficient number of parameters is used.
EXPONENTIALS: Exponential curves are defined by the equation:
Y = B(1)*e^(B(2)*X) + B(3)*e^(B(4)*X) + B(5)*e^(B(6)*X) + B(7)
where X is typically time and Y is typically a number, amount or
concentration. Again, from 2 to 7 parameters can be fitted. For a two
parameter fit, the computer determines B(1) (the initial value of Y for X = 0)
and B(2) (the exponential rate constant, which is negative for decay processes
and positive for exponential growth). The result is a single exponential
curve. When 4 or 6 parameters are fitted, the result is the sum of two or
three exponential curves. When an odd number of parameters is selected, the
final parameter is treated as a constant (e.g., its value does not change with
time, as in the case of the 7th parameter in the above equation). This
equation is useful for fitting exponential decay (e.g., radioactivity) and
growth (e.g., population) data.
HYPERBOLAS: This equation has the form:
Y = (B(1)*X)/(B(2)+X) + (B(3)*X)/(B(4)+X) + (B(5)*X)/(B(6)+X) + B(7)*X
where X is typically concentration and Y is typically the velocity of an
enzyme. When binding of ligand to a receptor or binding protein is studied, X
is the "free" and Y is the "bound" concentration respectively. The resulting
value of B(1) is the Vmax or concentration of binding sites, and B(2) is the
Km of the enzyme or Kd of the receptor or binding protein. Data can be fit to
the sum of two or three saturable processes by selecting four or six
parameters, respectively. If an odd number of parameters is selected, the
last parameter (e.g., B(3), B(5) or B(7)) is treated as a nonsaturable
process: B(i)*X. Examples are nonspecific binding or a nonenzymatic reaction
rate.
DATA ENTRY: After selecting which equation to use, you must enter the data. Up
to 100 data pairs can be entered. If a mistake is made, scan and correct the
data with the up/down cursor and delete keys. After entering the last data
pair, enter "E" in place of the next X value. You will be asked if you wish to
accept the data. If not, the computer will review each entry to allow errors to
be found and corrected. You will next be asked if you want to save the data.
Saved files can be loaded by entering "L" in place of an X value during data
entry, and will overwrite previous data. If you forget the name of your file,
enter "D" to display the directory. The Cardco numeric keypad is supported on
the C-64 version of this program.
NUMBER OF PARAMETERS: You will then specify the number of parameters (2 - 7) to
be fitted to the data. This number must be less (ideally, much less) than the
number of data points. Fits using large numbers of parameters typically take
much longer and require higher quality data for meaningful results. In general,
it is wise to start with two or three parameters, and then add parameters only
if needed.
WEIGHTING POWER: This number specifies the "weight" that is given to each Y
value in the analysis. A weighting power of two is used when the standard error
of the mean (SEM) of each Y value (i.e., the uncertainty in the true value of Y)
is proportional to Y, as is commonly true. A value of zero weights all points
equally, and is appropriate when the standard error is effectively constant
regardless of the value of Y. Use of zero weighting under other conditions
tends to overemphasize the importance of large values of Y. When high quality
data are available the weighting factor has little effect.
X OFFSET: In exponential time curves, it is sometimes useful to subtract a
constant from all measured times. The results will reflect this change,
although the data are printed and saved unchanged. Enter zero for no offset.
INITIAL ESTIMATES: Initial estimates of the expected values for each parameter
must be provided. These may be very approximate, but accurate estimates speed
the analysis and are essential when many parameters are being fitted. If the
experiment has been done previously, enter the earlier results. Extremely
inaccurate initial estimates may lead to an error condition during analysis.
When this happens, the screen border turns red and the program attempts to
recover by reducing the values of all parameters and restarting the analysis.
Such results may occasionally be inaccurate and should be validated by
reanalysis. Inaccurate extimates may rarely cause to program to home on a local
rather than global solution. If a result does not appear reasonable, repeat the
analysis with a better starting estimate.
ANALYSIS: The analysis is automatic. Current parameter values are displayed
along with the current sum of the squares. The data and current best-fit line
can be viewed during the analysis by holding down the space bar or F3, F5, F7 or
F2, but only if a Simon's Basic cartridge is installed. When further reduction
in the sum of the squares is not possible, the results are output to the screen.
If the analysis does not terminate within a reasonable time or the sum of the
squares is not decreasing, output can be forced by holding down the "F" key.
OUTPUT: Values are listed for each parameter along with the standard error
(uncertainty) of that value, presented both as an absolute amount and as a
percentage. Large uncertainties (over 40%) indicate that too many parameters
have been selected for fitting, that the data are being fitted to an
inappropriate equation, or that the quality of the data is poor. Also provided
are the weighted and unweighted sum of the squares, the standard error of the
analysis and the coefficient of determination (the square of the correlation
coefficient). These values provide a measure of the quality of the fit, and are
useful in determining the best equation and minimum number of parameters
necessary to fit the data.
PRINTOUT: Results can be printed by pushing the "P" key. A label of up to 60
characters can be added. This should work with most printers on the serial bus.
Pressing RETURN sends a line feed and the UP ARROW sends a form feed. If
everything is printed on the same line or other problems exist, try changing the
secondary address sent to the printer by pressing "S". If your printer is not
device #4, other numbers may be selected by pressing "N".
PLOTTING THE DATA: Graphic features (only) require that the Simon's Basic
cartridge be inserted in the cartridge slot. Push F3 for a linear plot, F5 for
a log plot, F7 for a double-reciprocal (Lineweaver-Burke) plot, and F2 for a
Scatchard plot of the data. The best fit line will be drawn automatically. If
you prefer that all points be joined by line segments, hold down "L" while the
data points are being plotted. To plot the data points without any line, hold
down "D". Holding down any key will suppress printing the numeric data to the
screen.
SETTING PLOT LIMITS: Plot scaling is automatic. The X = 0 and Y = 0 axes are
shown as dotted lines if they occur within the plot area. To set the plot
limits manually, hold down the function key used to select the plot until a
screen appears.
SCREEN DUMP: Pressing F2 will dump the hires Simon's screen to the printer.
This is known to work with the Gemini 10x and SG-10 printers with Cardco
interfaces, and may work with other Epson-like printers although this is
untested.
ENDING THE ANALYSIS: Three options are available when the analysis is complete.
1. Pressing "Q" will quit the program and return you to Basic (this is the
*only* method, and also works during the analysis). 2. Pressing F6 allows
reanalysis of the same data set (for example, with a different number of
parameters). 3. Pressing F8 resets the program for entry of new data. During
data entry, if any of the new entries is unchanged, pressing return will reenter
it automatically.
PRINCIPLES OF OPERATION: This program finds the best values of the parameters
by successive approximation. Matrix inversion is used to solve a system of
partial differential equations for the parameter values expected to give the
lowest values for the sum of the squares. The parameters are then adjusted to
these new values and minor adjustments made to further improve the fit (these
steps are labelled "iteration" and "subiteration" respectively). This process
is repeated until further improvement is not possible. Standard error estimates
for each parameter are then found by determining the sensitivity of the sum of
the squares to small changes in the value of the parameter.
SUMMARY OF COMMANDS:
C Disk command
D Plot only data (Hold down during plotting)
E Edit data set
F Force results (Hold down during analysis)
G Display graphic screen
L Connect points (Hold down during plotting)
N Change printer device number
P Send results to printer
Q Quit
S Change secondary address sent to printer
T Display prior text screen
<^> Send formfeed to printer
<F1> Redisplay parameter values of last analysis
<F3> Linear Plot (Hold down to set limits)
<F5> Log Plot (Hold down to set limits)
<F7> Double Inverse Plot (Hold down to set limits)
<F2> Scatchard Plot
<F4> Screen dump to printer
<F6> Reanalyze same data set
<F8> Reset to opening screen
<SPACE> Display current best-fit line (hold down during analysis)
<RETURN> Send linefeed to printer
COPYRIGHT NOTICE: Copyright 1985 by Richard A. Weisiger. All rights reserved.
Permission is granted to duplicate this program for personal use and to
distribute it through electronic data bases provided both the program and the
opening screen are not altered. This program may not be sold. Please notify
the author of any errors (Compuserve user # 75015,260 or write University of
California, San Francisco, CA 94143).