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PC-SIG Diskette Library (Disk #925)
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Information about “LSTSQR”
LSTSQR carries out a least-squares fit of data to a variety of
functional forms including linear, logarithmic, polynomial and
exponential. It allows multiple fits of the same data using the same
or different functions without exiting the program. LSTSQR87 supports
the 8087 math co-processor.
The program has many measures of quality-of-fit, including standard
deviation and the 95% confidence interval. The program also graphs and
prints the fit if desired. You can input data by keyboard or from data
files.
FILES.DOC
FILES ON THIS DISK
(subset version)
----------------------------------------------------
TD1.COM - TURBO Designer program
TD1.DOC - TURBO Designer documentation.
GLIMPSE.COM - Gives Subset users a glimpse
of the advanced version.
PRACTICE.COM - Lets new users toy around while learning
how to use pull down menus. Also shows some
of the features of the GOODIES toolbox.
PRACTICE.PAS - Source code for PRACTICE.COM.
PRACTICE.DAT - Data for PRACTICE.
GLOBAL.PAS - Global variables for TURBO Designer output.
(For TURBO Pascal v4.0)
GLOBAL3.PAS - Global variables for TURBO Designer output.
(For TURBO Pascal v3.0)
PD.PAS - Pull down (spaghetti, uncommented) source code.
PRINTDOC.COM - Utility to put documentation on printer.
FILES.DOC - This file.
README. - Read this first for initial help.
---------------------------------------------------
FILES925.TXT
Disk No: 925
Program Title: LSTSQR version 1.00A
PC SIG version: 1.2
LSTSQR carries out a least-squares fit of data to a variety of
functional forms including linear, logarithmic, polynomial and
exponential. The program allows multiple fits of the same data using
the same or different functions without exiting the program. LSTSQR87
is nearly identical to LSTSQR except that it supports the 8087 math
co-processor. In addition, the program provides many measures of
quality-of-fit, including standard deviation and the 95% confidence
interval. The program also graphs and prints the fit if desired. Data
can be inputed by keyboard or from data files on disk.
TURBO DESIGNER has been moved to disk #1353.
Usage: Programming.
Special Requirements: 8087 math co-processor optional and CGA graphics.
How to Start: Type GO (press enter).
Suggested Registration: None specified.
File Descriptions:
LSTSQR87 COM Main program for use with 8087 co-processor.
LSTSQR COM Main program.
LSTSQR DOC Program documentation.
4X6 FON Printer font for graphing results.
ERROR MSG Contains error messages.
README Instructions on printing documentation.
DESCRIPT Brief description of program.
PC-SIG
1030D E Duane Avenue
Sunnyvale Ca. 94086
(408) 730-9291
(c) Copyright 1987,88,89 PC-SIG, Inc.
GO.TXT
╔═════════════════════════════════════════════════════════════════════════╗
║ <<<< Disk No 925 LSTSQR >>>> ║
╠═════════════════════════════════════════════════════════════════════════╣
║ To print the documentation for LSTSQR, Type: ║
║ COPY LSTSQR.DOC LPT1: (press enter) ║
║ ║
║ To run the program LSTSQR with a 8086 coprocessor, Type: ║
║ LSTSQR (press enter) ║
║ ║
║ To run the program LSTSQR with a 8087 coprocessor, Type: ║
║ LSTSQR87 (press enter) ║
╚═════════════════════════════════════════════════════════════════════════╝
LSTSQR.DOC
DOCUMENTATION FOR PROGRAM LSTSQR
GENERAL INFORMATION
AUTHOR: Arthur Ferguson
21 Upper Joclyn Avenue
Framingham, MA 01701
PURPOSE: To provide a least-squares fit of data to a variety of
functional forms.
FUNCTIONAL FORMS SUPPORTED:
A. Linear (Y = A + BX)
B. Polynomial (Y = A + BX + CX^2 ....) up to a
maximum power of X^5.
C. Exponential (Y = A*Exp(BX))
D. Ln(Y) = A + BX
E. Ln(Y) = A + B/X
F. Ln(Y) = A + B/X + C*Ln(X)
G. 1/Y = A + BX
H. Y = A + BX + C/(X^2)
I. Y = A + B/X
J. Y = A + B*Ln(X)
FITTING PROCEDURE USED: Since all of the functional forms except
the exponetial (C) are linear in the coefficients A, B,
etc., a single-pass "linear" procedure was used for those
functional forms (See the books by Miller and Hildebrand
referred to in the acknowledgements). For the exponential
functional form a first approximation obtained by
linearizing the equation was refined with an iterative
solution of the appropriate non-linear equation using
Newton's method (See Miller).
INTERACTIVE FLEXIBILITY: A system of menus combined with calls
for user input at key decision points allows multiple fits
of the same data set, using the same or different functional
forms, with the opportunity to change, add, or delete data
points between fits, and the fitting of several different
data sets, all without leaving the program.
MEASURES OF THE QUALITY OF THE FIT REPORTED (for all functional
forms except the exponential):
Standard Deviations of the fitting parameters
95% Confidence Intervals of the fitting parameters
Standard Deviation of the points
Correlation Coefficient of the fit
Sum of the Squares of the Residuals (the only
measure provided for the exponential fit)
PROGRAMMING LANGUAGE USED: Borland Turbo Pascal Version 3.0
FORMS OF PROGRAM PROVIDED: LSTSQR.COM and LSTSQR87.COM. These
are identical except that LSTSQR87 supports and requires
the 8087 numeric co-processor chip while LSTSQR neither
supports nor requires the 8087.
MAXIMUM NUMBER OF DATA POINTS: 200.
PRECISION OF CALCULATED QUANTITIES
Up to 11 significant digits (48-bit Reals) with LSTSQR.
Up to 16 significant digits (64-bit Reals) with LSTSQR87.
DATA WEIGHTING OPTIONS: Individual weighting of data points or
uniform (equal) weighting.
DATA INPUT OPTIONS: Keyboard entry from within the program or
entry in the form of previously prepared ASCII disk file.
For the format requirements of a disk data file see below.
DATA OUTPUT OPTIONS: Screen, printer, or disk file (DOS paths
supported), selectable from within the program.
RESULTS OUTPUT OPTIONS: Screen or printer, selectable from within
the program.
GRAPHING CAPABILITIES: After a fit is obtained the data and the
fitting curve can be graphed (requires CGA or compatible
graphics adapter). WHEN GRAPHING IS PLANNED, THE FILES
4X6.FON AND ERROR.MSG MUST BE IN THE CURRENT DISK DIRECTORY!
The absence of ERROR.MSG will produce an non-fatal error
when graphing is attempted. Graphs may be printed on IBM
compatible printers using the PrtSc or Shift PrtSc key.
ACKNOWLEDGEMENTS
The theoretical background and general procedure for linear and
polynomial least-squares fitting of data was gotten from
F.B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill,
New York (1956). The Gauss-Jordan Elimination procedure for inverting
matrices, the equations for determining standard deviations for the
fitting parameters and the correlation coefficients in the cases of
the "linear" fitting procedures, and the theoretical background for
the iterative non-linear fit to the exponential functional form using
Newton's Method were obtained from Arthur R. Miller, Pascal Programs
for Scientists and Engineers, Sybex (1981). The equations for the t
values used to calculate the 95% confidence intervals for the fitting
parameters from their standard deviations were based on those
presented by W.J. Blaedel and D.G. Iverson in Analytical Chemistry,
Vol. 48, No. 13, p. 2027 (1976). The graphics routines used were
from the Borland Turbo Graphix Toolbox.
RESTRICTIONS ON DATA VALUES AND FORMAT REQUIREMENTS FOR DISK DATA FILES
Data values must conform to the ranges and formats of Turbo
Pascal reals. This means that the exponents of numbers in scientific
notation must be represented by E followed by a positive or negative
integer (1.5E3 = 1.5 X (10 to the 3rd) = 1500). The E must follow the
last digit of the mantissa with no spaces (1.5E3, not 1.5 E3). For
LSTSQR numbers must be in the range 1E-38 to 1E+38, and the exponents
in data values may contain a maximum of two digits (1.5E3 and 1.5E03
are O.K.; 1.5E003 is not and will cause an I/O error and termination
of the program). For LSTSQR87 numbers must be in the range 4.09E-307
to 1.67E+308, and three digit exponents in data values are O.K.
(1.5E003 is acceptable to LSTSQR87). Though it seems not to be
necessary, it is probably a good idea to follow the convention of
including at least one digit before the decimal point in any data
value, even if it is only a zero (for example, 0.5 instead of .5).
Disk data files must be pure ASCII files with no symbols other
than those involved in the data values themselves. The first line of
the file must contain only one item, the letter W if the data points
are to be individually weighted, or the letter N if uniform, equal
weighting of the data points is to be used. Each succeeding line in
the file must contain the X value, the Y value, and, where individual
weighting factors are included, the weighting value for a single data
point. These values must appear in the order X, Y, weighting factor
and be separated by a blank space (NOT a comma). Below is a sample
data file for a set of equally weighted data points
N
0.055127 34.06
0.041545 54.21
0.033333 82.64
0.027832 131.6
0.023889 200.4
0.020925 318.3
0.018615 485.1
0.016764 765.8
0.015249 1188.1
0.013984 1845.2
When fit to the functional form Ln(Y) = A + B/X this set of data gives
A = 2.18386, B = 0.07466, and a correlation coefficient of 0.99997426.
REQUIREMENT OF THE USE OF THE "ENTER" (CARRIAGE RETURN) KEY WHEN
ENTERING INFORMATION OR RESPONDING TO QUERIES WITHIN THE PROGRAM
This program uses buffered input throughout, which means that all
data values and other information entered from within the program and
all menu selections and responses to program queries must be followed
by pressing the ENTER or carriage return key (Sometimes pressing the
ENTER key alone is sufficient). Some users used to the current trend
toward unbuffered input when single character responses are required
may find that this use of buffered input maddeningly slow, but it was
consciously chosen to provide the opportunity to correct mistakes or
change one's mind before the response is acted on by the program.
FITTING CRITERIA AND THE CHOICE OF WEIGHTING FACTORS
For all functional forms linear in the coefficients, which means
all forms except the exponential, the criterion for the fit was the
minimization of the sum of the squares of the ABSOLUTE residuals
[Y(calc) - Y(exp)]. Therefore, if individual weighting factors are to
be used, an appropriate basis for them would be the estimated ABSOLUTE
error of the individual data points. For the exponential functional
form the criterion for the fit was the minimization of the sum of the
squares of the RELATIVE residuals [(Y(calc) - Y(exp))/Y(exp)] (For the
rationale for this see Miller). Therefore, an appropriate basis for
weighting factors in this case would be the estimated RELATIVE error
of the individual data points.
INTERPRETATION OF THE 95% CONFIDENCE INTERVALS
The figures given for the 95% confidence intervals represent the
"distance" on either side of the value of a parameter to the
boundaries of the 95% confidence interval. Therefore, if the listing
of the results of a fit give the parameter A with a value of 1.507 and
a 95% confidence interval of 0.021, it means that there is a 95%
probability that A lies in the interval 1.507 + or - 0.021.
THINGS THAT HAVE BEEN FOUND TO CRASH THE PROGRAM
In the testing of this program only two things have been found to
make it crash. These are 1) a mistake in entering the name of a data
input file which causes the program to search for a file that does not
exist and 2) the entering of data, from either a data file or the
keyboard, that is out of the acceptable range or contains too many
digits in the exponent (See restrictions on data values above). Both
of these cause I/O errors and program termination.
STEPS TO TAKE IN RUNNING THE PROGRAM
1. Use your favorite text editor to set up a data input file as
described above, unless you prefer to enter the data from within
the program.
2. Be sure that either LSTSQR.COM or LSTSQR87.COM, depending on
which one you intend to use, is in the current disk directory and
that the files 4X6.FON and ERROR.MSG are also in the current
directory if you intend to graph the results.
3. Type LSTSQR or LSTSQR87.
4. Follow the menus and instructions in the program.
PROGRAM TESTING AND DISCLAIMERS
This program has been tested on an 8086-based COMPAQ DeskPro with
an 8087 and an 8088-based monochrome IBM PC without an 8087. Both
computers have two floppy disk drives and a RAM disk but no hard disk.
The tests involved fitting equally weighted data sets for each of the
functional forms which the program can handle. All tests produced the
expected results with no equipment problems. Most of the data sets
had previously been run on other computers using other programs, and
in all of these cases the results from this program were in excellent
agreement with the previous results. Therefore, the author believes
that the program will produce reliable results with no risk to
equipment. HOWEVER, HE ASSUMES NO RESPONSIBILITY FOR THE CONSEQUENCES
OF ITS USE.
Directory of PC-SIG Library Disk #0925
Volume in drive A has no label
Directory of A:\
4X6 FON 285 7-23-87 2:08p
DESCRIPT 1329 9-18-87 10:46p
ERROR MSG 567 7-23-87 2:08p
FILES DOC 1276 2-29-88 1:00a
FILES925 TXT 1366 3-02-89 3:35p
GO BAT 38 10-19-87 3:56p
GO TXT 925 2-24-89 2:58p
LSTSQR COM 56509 9-02-87 3:04p
LSTSQR DOC 12680 9-18-87 10:14p
LSTSQR87 COM 55006 9-02-87 3:02p
READ ME 189 9-04-87 12:36p
11 file(s) 130170 bytes
27136 bytes free