Simplex:
A Manual
for Easy
Nonlinear
and Interpretation
Anthony
Mittertreiner
and Software
Package
Parameter
Estimation
In Fishery
Research
and Jon Schnute
Department
of Fisheries and Oceans
Fisheries Research Branch
Pacific Biological
Station
Nanaimo, British Columbia
V9R SK6
July
1985
Canadian
Fisheries
No.1384
Technical
Repo'rt
'of
and Aquatic
'Sciences
Canadian
Technical
and
Report
Aquatic
Sciences
July
SIMPLEX:
A MANUAL
NONLINEAR
AND
PARAMETER
IN
of
Fisheries
No.1384
1985
SOFTWARE
ESTIMATION
FISHERY
PACK~GE
AND
FOR
INTERPRETATION
RESEARCH
by
Anthony
Mittertreiner
Department
of
Fisheries
f1sher1es
Pacif1c
Nanaimo.
*
Send
reprint
requests
to
this
and
Research
Biological
British
author.
Columbia
EASY
Jon
Schnute*
and
Oceans
Branch
Station
V9R
5K6
(c)Minister
Correct
fishery
and
Services
for
this
C.
easy
and J.
Schnute.
1985.
SIMPLEX:
nonlinear
parameter
estimation
research.
97-6/1384E
Canada
No.
A.
for
Fs
Supply
Cat.
citation
Mittertreiner,
package
of
JSSN
1985
0706-6457
publication:
Can.
Tech.
Rep.
Fish.
p'Juat.
A manual
and
and
interpretation
Sc1.
1384:
90
software
in
p.
-111
-
TABLE OF CO~ITENTS
LIST
LIST
OF TABLES.
OF FIGURES.
LIST
OF COMPUTER
Page
v
v1
LISTINGS.
v11
ABSTRACT.
RESUME.
FOREWDRD
v111
BY JD~
1x
x
SCHNUTE
,
7.6. 5.
4.
3.
2. 1.
7.2.
PLOTTING
7.1.
PLOTTING
6.1.
5.1.
5.2.
PLOTTING
4.3.
4.2.
4.1.
SIMPLEX
3.4.
3.5.
3.3.
3.2.
3.1.
PREPARATIONS.
2.5.
2.4.
2.2.
2.3.
THE
2.1.
INTRODUCTION.
Profiling
Profiler
MINSUM
Plotter
C.
Plot
A.
B.
Minimization.
Menu
System
SIMPLEX
MINFUN
Preparations
Linking
A.
B.
MINSUM
Two
Algorithm
Convergence.
Final
Simplex
Background. Minimum
Control
Interpreting
OPERATION:
PROFILES
OBSERVATIONS,
CreatIng
Preparing
FACILITIES.
types.
methods:
options.
plotter axial
initialization.
and
hardware
and METHOD.
description.
SIMPLEX.
instructions.
iteratIon.
Cfound.
data.UFUN
TEMPLATE.
ANDa
(AC)
search.
summary.
the
MINSUM
MINIMIZING.
options.
module
the SECTIONS
requirements.
ETC.
data
output. and
from
(MINSUM
file.
MINFUN(MINSUM
TEMPLATE.
ONLY)
AND
MINFUN)
1
12
12
12
18
18
19
21
21
21
21
24
24
26
28
29
29
29
31
31
32
32
33
33
38
39
39
41
41
46
-iv
TABLE
-
OF CONTENTS
Page
APPENDIX
ACKNOWLEDGEMENTS.
REFERENCES.
12.
11.
10.
9.
8.
TROUBLE
9.2.
9.3.
9.1.
CALCULATING
B.l.
R.2. WORKED
VARIATIONS
11.3.
11.1.
11.2.
ADDITIONAL
10.3.
10.4.
10.2.
10.1.
'.'u
Overfll>ws
Slow
Theory.
Software
A.
B.
C.
Multiple
Mult1-po1nt
Adjustable
Eliminating
Chi-square
Holrfing
Imposing
1
EXAMPLE
tip
SHOOTING
CHOICES
GLOSSARY
SIMPLEX
convergence.
le
OFPOSSIBILITIES.
COVARIANCES
mioperation.
THE
some
extens1ons
constraints.
ni
FOR
ITERATION.
THE OF
action
and
ma
reflect1on',
SIMPLEX
linear
parameters
MINSUM
TERMS.
SIMPLEX
weighted
coeff1c1ents
parameters.
SEARCH.PLOTTING.
SEARCH.
f1xed
least
squares.
47
47
49
51
51
51
52
53
53
53
54
54
55
!)5
55
55
57
81
82
83
86
89
-v
LIST
-
OF TABLES
Table
2.1
Page
Values
for
for
various
editing
distribution
and standard
algorithm
function
deviation
example
values.
data.
13
4.1
Codes
5.1
Options
for
high
and
low
resolution
5.2
Options
for
high
resolution
plot
characters.
34
5.3
Options
for
high
resolution
line
types.
34
5.4
Options
for
high
resolution
(flatbed)
patterns
for
normal
REan
30
plot
types.
panel.
34
34
-vi
LIST
-
OF FIGURES
Figure
1.1
Page
Two
predicted
mussel
curves
length
1.2
Normalized
1.3
Profile
(plotting
data
and
observed
{plotting
residuals
for
andexample)
section
along
example)
~Jssel
y-
length
data
(profile
example)
8
1.4
Profile
and
section
along
K
1.5
Prof1le
and
section
along
to
1.6
Opt1mal
K V5.
2.1
Contour
plot
Y.
of
(prof1le
(profile
example)
(prof1le
10
exarnple)
example)
l1kelihood
11
surface
for
normal
distribut1on
14
2.2
Simplex
search
5.1
Histogram
mussels
show1ng
(plott1ng
7.1
Contour
plot
7.2
Prof1le
and
the
7.3
Profile
the
response
and
response
act1ons
of
15
age-frequency
example)
hatch
sect1on
surface
section
surface
fa1lure
along
of
along
of
distr1but1on
36
rate
xI
F1g.
x2
Fig.
for
response
(sal1n1ty)
surface.
for
7.1
(temperature)
7.1
43
44
for
45
-v11
LIST
-
OF COMPUTER
LISTINGS
L1st1ng
Page
1.1
Low
resolution
length
1.2
at
Low
(P1ottingresolution
data
plot
age
for
of
example) plot
(Plotting
from
observed
freshwater
Jf
and
predicted
mussels.
residuals
for
mussel
matrix
calculation
3
example)
3
1.3
Output
covariance
3.1
TEMPLATE
(Covariance (withexample)
example
3.2
UFUN
(skeletal
l1st1ng)
3.3
UFUN
(complete
example)
3.4
GOMINSUM.COM
27
3.5
GOMINFUN.COM
28
5.1
4.1
Histogram
SIMPLEX
5.2
Low
resolut1on
plot
example
(1ncorrect)
5.3
Low
resolut1on
plot
example
(corrected)
6.1
Plotter
code)
6
22
menu
examples.
iten5
25
25-26
high
and
low
resolution.
...
29
35
menu
{create.
store.
recall.
37
37
etc)
39
-ix
,
,
RESUME
Mittertreiner,
A.
C.
package
for
fishery
research.
Le pr~sent
dans
lequel
mod~les
d'interpr~ter
mod~les.
le
and
easy
les
r~sultats
donne
de
similaires.
A manual
Rep.
Fish.
and
AI1uat.
l'analyse
a fin
des
sur
et
de calculer
commode
quand
la
la
covariance
des
fonction
~conomique
c'est
le
etre
cas.
fa~on
exprim~
par
sous
exemple.
forme
avec
plus
g~n~rale.
ou la
fonction
Le
pr~sente
guide
l'1nterpr~tat1on
donne
~galement
son utilisation.
minimalisation
non
toute
de
l'appliquer
!
des
qui
notamment
.(2)
l'utilit~
l'ajustement
des
de la d~viation
suggestions
pour
de
th~orie
mod~les
standard
utiliser
situations
des
(2)
desdits
SIMPLEX.
estimation
pr~diction
param~tres.
(c.-A-d.
d'une
et
par
de
SIMPLEX
crit~re
le
somme
l'ajustement
outil
de
termes
la
~thode
SIMPLEX
peut
~galement
etre
~conomique
est
analytique.~ent
n'a
~ fournir
d~sir~:
cela
sur-le-champ
qu'une
fait.
pour
mod~les.
seulement
la
On y d~crit
des
fonctions
l'~valuation
covariance.
donne
des
des
in
p.
validit~
logiciel
d'obtenir
la
meill~ure
divers
diagrammes
de
peut
software
90
d'ajuster
et
la
le
l'ut1l1sateur
de produire
carr~s.
De
cas
d'utilisation
et
1384:
(I)
lin~aires
d'~valuer
d~tails
and
interpretatio"
Sci.
une m~thode
permettant
des
param~tre~
non
de
~
SIMPLEX:
estililation
complexe.
De fa~on
caract~ristique.
l'utilisateur
section
du code
FORTRAN pour
d~crire
le mod~le
les
caract~ristiques
du SIMPLEX
sont
disponibles
l'ajustement
pour
Tech.
~galement
mod~les)
comme
des moindres
adapt~
S des
plus
courte
toutes
mod~les.
1985.
parameter
guide
d~crit
on introduit
qui
permet
facilement
possible
des
param~tres.
d'ajustement
Schnute.
Can.
guide
probab1lit~
de
est
particuli~rement
J.
nonlinear
d~tails
sur
le
a
de
base
servi
(I)
S
son
SIMPLEX.
mais
~laboration
et
la m~thode
de recherch~
SIMPLEX
de divers
diagrammes
dans
et
et
de la
efficacement
sp~ciales.
logiciel
(3)
le
calcul
corr~lation
l'algorithme
num~rique
des
de
param~tres.
de recherche
!
de
la
On
et
-x
-
FOREWORD
By
Jon
Schnute
-xi
complete
w1th
1tself.
At
fresh
th1s
1ns1ghts
p01nt.
1nto
1 became
(and
project
myself
and wrote
extens1ve
the
ent1re
manuscr1pt.
Both
Tony
everyth1ng
in
the
effort
viewp01nt:
worthwhile
he thought
jaded
to
here.
which
tine
1mag1ne.
perhaps
for
h1s
departure
only
readers
a
at
w111
Jon
the
1 was
things
student
about)
engaged
1n
rev1s10ns:
and
1 burned
partly
becausa
of
describing
1 hope
quest10ns
deeply
end
of
Nana1mo,
from
bring
to
B1olog1cal
B.C.
V9R
Canada
SeptenDer.
s1mplex
19B4.
by
which
Tony's
1 had
the
a
1984
part1cular
problam.
Stat1on
5K6
method
phases
August.
fasc1nated
1n a way
Schnute
Pac1f1c
the
f1nal
1ndeed.
1 essent1ally
the
m1dn1ght
011 to
prof1t
can
the
of
the
rewrote
complete
1
cons1dered
fresh
become
too
parspect1ve
1. INTRODI.JCTION
Readers
parameter
nonl1near
of
surface
analys1s,
an ab1l1ty
to
find
mathemat1cal
est1mat1on.
and s1ze-frequency
analys1s,
estimates
of parameters
in
Unfortunately,
computer
derivative
calculations,
the
der1vative
of the
Because
l1near
turn
of
the
provided
by
the
1s
to
free
t1me
1nvolved,
many
which
mayor
thus
l1mit1ng
package
has
S1nce
SIMPLEX
Because
the
experiment
capab1~1t1es.
opt1on.
non11near
or
models,
programs,
SIMPLEX
above.
calcuated.
a
f
, rely
equ~t1ons.
practit1oners
may not
themselves
prove
to
to
heavily
code
task.
resort
on
for
to
adequate;
or users
models
and features
may
package.
This
out11ned
need
be
to
name
nonlinear
algorithms
for
th1s
purpose
typically
1nvolve
wh1ch
may mean that
the
user
has
to
prov1de
funct1on
be1ng
considered
--often
a nontr1vial
diff1culty
or quadrat1c
to
Mcanned"
b1ology
may not
requ1re
an 1ntroduct1on
Top1cs
such
as growth
model1ng,
response
a
with
des1gned
var1ations.
calculat1on
Hopefully
parameter
been
to
circumvent
the
1nvolves
a d1rect
search
method.
user
specif1es
the
funct1on
or
of
these
features
est1mat1on
The
parameter
package
also
covariances.
g1ve
the
user
can be performed
offers
and
enough
almost
problems
no der1vativcs
model.
he or
she
plotting
an
1n1t1al
flex1b1l1ty
as eas1ly
search
that
as l1near
regress1on.
SIMPLEX
1s
wr1tten
transportab1l1ty.
The
11-780;
1t
1s enhanced
procedure.
and the
high
10 for
a TEK 4105
color
1nterested
Pac1f1c
1n
a
copy
B1ological
1n
ANSI
standard
FORTRAN
77.
1n
the
1nterest
of
version
descr1bed
here
1s wr1tten
for
the
VAX/VMS
with
some system
ser~ices
to aid
1n the
min1m1zat1on
resolut1on
software
1s written
w1th
Tektron1x
IGL Plot
graph1cs
term1ndl
and a 4662
flatbed
plotter.
Readers
of
the
Stat1on.
software
should
contact
Nana1mo.
B.
V9R
C..
the
5K6.
Computer
for
Centre.
1nformat1on.
To illustrate
briefly
the capabilities
of this
package.
consider
some data*
for freshwater
mussels
(Anodonta
kennerylii
). consisting
of mean
lengths
(mm) at ages one to sixteen
years
as follows:
7.36.
14.3.
21.8.
27.6.
31.5.
35.3,
39.0.
41.1.
43.8.45.1.
47.4.
48.9.
50.1.
51.7.
51.7.
and 54.1.
Suppose that
one wished
to fit
these
data to the von Bertalanffy
curve
(Ricker
1975.
p. 221):
(1.1)
where
y(t)
y(t)
is
.y
the
-
[1-
e
length
at
-K(t-tO)]
age
t
and
(y
,K,t
is
these
data,
-0
a
vector
of
three
*.**~~***********************************.***********************
*For
deta1ls
Enr1chment
Canada.
on
Project,
In
th1s
appl1cat1ons,
acknowledgements.
the
acqu1s1t1on
Pac1fic
manual,
w1thout
the
regard
of
B1olog1cal
data
to
are
Stat1on,
used
underly1ng
only
see
Nana1mo,
to
spec1es
1llustrate
b1ology.
Dr.K1m
B.C.
Hyatt,
V9R
Lake
SK6,
software
See
also
the
-2
parameters.
fit,
If
this
the
model
additional
lines
(A)
N .3
(8)
PRED
(C)
RES.
(D)
TERM.
Here
w1th
Sum
could
of
be
of
squares
of
residuals
incorporated
FORTRAN
-
into
.PA~S(l)*(l
(8)
and
to
-EXP(-PARS(Z)*(XO(l)
suppl1ed
by
best
only
fnur
-PARS(3))))
(YO
-YP)**2
.y..
(1.1)
PARS(2)
define
the
.K.
same
parameters.
PARS(3)
model,
These
.K.t
.0
the
.(50.0.
user.
1.00.
SIMPLEX
are
assoc1ated
.to.
where
length
y(t))
to
the
variable
XO(l)
because
SIMPLEX
allows
predictions
(y
the
with
'.~
YP
(i.e.,
to
is
the
the
depend
variable.
Line
(C)
defines
the
residual
as the
difference
(YO)
and predicted
(YP)
responses,
and
finally
(0)
defines
the
objective
function
as the
square
of a resIdual.
(1.2)
determine
package
YO -YP
PARS(l)
(1.
e.,
the
XO is
lndexed
used
SIMPLEX
code:
l1ne
(A) def1nes
the number of m~del
the vector
PARS 1n (8) as follows:
Thus,
is
the
predicted
age
on
response
t).
The
variable
more
than
one
between
a single
observed
term
in
1.00)
takes
a
few
seconds
to
arr1ve
at
the
opt1mum
est1mate
(1.3)
The
(y
.K.t
.o
complete
resolution
analysis
SIMPLEX
plots
resolution
plot
for
and
Fig.
the
residuals
residuals.
at
or
resolution.
level
1.2
0.15).
problem
is
the
user
to
model
fit.
and
along
with
residuals
example
plots
shows
normalized
each
this
observations
plot,
Model
resolution)
0.16,
also
allows
depicting
the
of
high
resolution
estimate
(1.2).
(low
) .(57.3.
(high
in
a
later
section.
construct
both
low
For
example,
Listing
predictions:
a graph
can
also
themselves,
In fact,
given
or
be
Fig.
the
curve
plotted,
resolution).
while
the
either
choice
Note
high
of
1.1
gives
defined
as shown
that
and high
1.1
shows
the
rf$olution
residuals
a
by
in
more
a
the
initial
Listing
1.2
low
low
precise
resolution
Fig.
1.2
is
available
-3
Listing
1.1.
age for
estimate
freshwater
(1.3)
of
Low
resolution
plot
of
-
OOservt!d
mussels.
Predictions
the
parameter
vector.
are
(0)
on
predicted
the
(P)
optlhlal
length
1
60+
1
POPPOPPPOPPOPPPOPP
l
N
POPPOPPPOP
40
POPPOPP
+
M
M
PPOPP
i
PPPo
20
PPO
+
I
PPO
poP
P-+---+--+---+--+---+--+---+--+---+--+--+---+--+---+-S
10
AGE
1.2.
These
predictions
parameter
Low resolut1on
are
computed
based
estimate
on the
(1.3).
as
plot
th~
van
of
residuals
difference
Bertalanffy
IN
IS
YEARS
for
between
model
the
mussel
observations
length
and
with
optimal
(1.1)
the
2.0+
1
R
E
s
I
D
U
A
L
S
at
least-squares
+
80+
L
E
N
G
T
H
L1st1ng
data.
and
based
1.01
1
.01
1
-1.01
1
-2.0!
+
+
10
+
+
20
+
+
30
PREDICTED LENGTH (MM)
+
+-40
50
at
age
00,
~,
.-,
...
"
"
.
..-,
.,..
f
,/V
..
...
.
.
.e.
I\
.
.
.
.
.
.
.
.
~,
r
r
v
.
I
.
.
.
t~
.
.
.
.
.
.
~1
z
1&1
.J
.
.
19.
.
.
.
.
-t-.-,-.
s
..~18'-,-,-
Kf.~
Fig.
1.1.
High resolution
plot
of observed
data
(0).
and two
von Bertalanffy
curves
(1.1)
for
freshwater
mussel
data.
The dotted
(upper)
curve
corresponds
to the first
estimate
of (y~.K,tQ)
= (50.0,
1.00,
1.00),
while
the solid
(lower)
curve
represents
the final
estimate
(y~,K,to)
8 (57.3,
0.16,
0.15),
optimal
for
the sum of
squares
criterion.
~
-5
-
3.81
I;
II
t
i'
2.8
If
I
c.'1
.i
t
.
1 .a.
t
! .
1
..
..t
.8.
U)
-t
(
.1
a
11
U)
1&1
a.
..
..
t
..t
:
t
:
c
.
..t
..
-f
-I
.81
.
-2.9-
-.1.8.J
-I
ia
I
~
.~
.:e
I
~~
fRDLflt1H
Fig.
1.2.
Normalized
residuals
were
and predictions
estimate
(1.3).
residuals
calculated
based
on
for
mussle
as the
difference
model
(1.1)
with
length
at
between
the
optimal
age
data.
observations
parameter
These
-6
SIMPLEX
asymptot1c
der1vat1ves
option
output
1n
also
from
L1sting
The
S1nce
est1mates
gr1ds
L1st1ng
~ssel
prov1des
the
1n
method
these
are
1nvolves
GRID'
SCALE FACTOR:
from
the
1
2
3
PARI.
18.13788632
14.82079593
0.81711814
PARI
-0.92494678
-0.56895712
0.76523097
1.2
1.
3
2.3
PARI.
for
calculat1ng
and correlations.
mussel
example
the
estimates
sens1t1ve
result.
based
of
to
on
1.3
2.2
3
COY MATRIX:
COR MATRIX:
the
of
second
m1ni~m
po1nt.
covar1ance
matr1x
calculat1on
second
applied
3) 0.15506405
3) 0.30000001
* 0.5 * SIGMA**-2
2
.lOOE-Ol
3
.lOOE-O2
FUNCTION VALUES
4.121189160
0.1479410426
0.35897659E-01
3.981329099
0.1660836190E-02
0.41715622E-03
STANDARD DEVIATIONS
0.6521245397
0.5801261418E-02
0.6949696734E-01
0.6524315524
0.5804290173E-02
0.6951788838E-01
CORRELATIONS
-0.93158925
-0.59740172
0.77986622
Th1s
F1nal
is
shown
the
choic.e
of scale.
three
1nc.reas1ngly
-0.93165596
-0.59769399
0.78001727
0.3889559619
-.3184103008E-02
-.2397742590E-01
0.3046774398E-04
0.2854207191E-03
0.4566102042E-02
COVARIANCES
0.4252664153
-.3524337152E-02
-.2707465094E-01
0.3365463404E-04
0.3144187211E-03
0.4829828469E-02
0.4256669306
-.3528089866E-02
-.2710880770E-01
0.3368978441E-04
0.3147385262E-03
0.4832736805E-02
0.9569502553E-08
0.17684917
DETERMINANTS
O.1057089487E-07
0.15292345
0.1058193995E-07
0.15268768
PARI
1
1.2
2.3
3.
option
COEFF ICIENTS OF VARIAT ION
0. 11388000E-01
0.11382641E-01
0.10885859E-01
0.33572087E-01
0.35302650E-01
0.35284228E-0I.
0.44831725
0.44818233
0.43577437
PAR6
1.
1
.IDO
0.6236633402
O.5519759413E-02
O.6757293868E-O]
3
an
space.
POINT
1)
57.291145
2) 0.16441514
STEP FOR DIFFERENCING
1)
1O.0OO0OO
2) 0.1OOOOOOO
COVARIANCE CONSTANT 1.0OOOOOOOO
PARI
1
2
w1th
numer1cal
can be numerically
given
for
eac.h
parameter
1.3.
Output
example.
AVERAGE F:
STD DEV F:
COF VAR F:
user
matr1x.
based
on the
matr1x
the
OOject1ve
funct1on
at
the
tabulates
parameter
standard
dev1at1ons
the
covar1ance
calculator
app11ed
to
1.3.
der1vat1ves.
three
suc.h
smaller
also
parameter
covar1ance
(the
Hess1an)
of
-
to
the
-7
F1nally,
the
object1ve
by
the
lett1ng
surface
sl1c1ng
SIMPLEX
funct1on.
w1th
a
kn1fe
s1multaneously
prof1le
represent
various
values
direct1on
and
elevat'ion,
curvature
prof1le
1s
scann1ng
to
vary
funct1on
(1.3).
The
var1at1on
K
final
Y.
of
-0.93
and
respect
feature
stored
plots
both
of
as the
showing
1.5
each
for
the
1s
of
indicates
that
increases.
the
(gr1~
13)
The
above
to
the
best
between
Show
a
hor1zon
vary
of
obta1ned
example,
seen
by
Mprof11eM
1s
a
1f
more
and
Thus
points
function
for
minimum
with
on
the
always
represents
(We ignore
Statistically,
the
perspect1ve
a
model
statist1cal
theory
of
theor1es
relative
prof1le
K and
estimates
prof11e
parameter
for
1.2
have
that
values
Fig.
the
lowest
and
1.3
for
K and
Y.
sum
1s
sect1on
values
poss1ble
represent
in
non-profile
shows
each
are
the
one
Indeed.
Listing
the
user
parameter
optimal
value
of
negatively
K decreases.
2
of
Consequently.
value
of
1.6
vary)
1 and
the
Y.
respect1vely.
performed.
in
the
optimal
to
of
represents
a
the1r
opt1mum
F1gs.
to.
example.
of
parameters
and
The
represent1ng
is
is
allowed
estimate
sect1on
example.
narrower
curve
held
f1xed
at
profiler
For
a
Y..
value
The
correlated:
the
correlation
1.3
corroborates
features
available
this
observation.
the
user
profiler.
in
complete
adventurous)
examples
of
SIMPLEX:
a
The
remainder
detail
task
and
of
illustrate
the
minimizer,
of this
plotter,
report
is
offering
suggestions
building
models
for
four
main
covariance
devoted
to
for
fisheries
calculator,
describing
their
use
data.
a
Of a profile:
1n a given
overall
whole
the
S1m1larly.
analysis
variation
are
A
parameter
1llustrates
mussel
yto
changes.
K and
For
be
prof11es
(1971).
curve
w1th
f1xed.
could
with
an example
range.
Looking
Indeed,
example,
The
h1gher.
K and to
are
value
funct1on
parameters.
the
objective
the
idea
of
the
it
and
the
d1stance
away.
th1s
analogy.)
because
each
parameter
(where
figure
1.4.
rema1n
sect1on
one
r1ght.
to
the
stating
Kalbfleish
broader
for
are
obtain
to
of
cross-sect1on.
lett1ng
parameter.
See,
for
funct1on
lower.
funct1on
another
left
sect1ons
graph
a
provides
us
a mountain
of
regard
earth
1n
1n F1g.
1.3.
along
wh1ch
value
as
by
1.3.
funct1on
A
the
by
1nteresting.
object1ve
allowed
of
the
parameters
can
also
from
w1thout
of the
F1gs.
squares
exam1ne
a
m1n1m1z1ng
with
respect
to
other
the
lowest
possible
value
of
of
one parameter.
If
we replace
descr1bed
of
at
obta1ned
to
a part1cular
around
profiles.
l1kel1hood
to
1s
wh1le
the
others
by a clay
model,
look1ng
1s
extremely
sens1t1vity
been
dev1sed
as
It
user
Msect1onM
then
everyday
exper1ence
formed
by the
horizon
highest
and the
of
and
object.
the
a
one parameter
vary
were
represented
compl1cated
max1mum.
the
curve
allows
Here,
-
in
and
these
the
(rather
to
features
-12
2.
2.1.
The
the
method
s1mplex
to
had
m1n1m1z1ng
METHOD
a
search
and
potent1al
are
and
a
d1fferences
2.2.
funct1on
This
search
user
to
package.
take
The
of
section
m is
any
(See
Schnute
of
parentheses
a
F.
standard
52
1n
and
algor1thm
a
Hext.
max1m1z1ng
su1table
for
Schnute
(1982).
As
b1olog1cal
model
such
pred1cted
as
the
wh1ch
1s
sum
SIMPLEX
prof1ler.
g1ve~
Schnute
a
expla~ns.
typ1cally
of
that
or
computers
FORTRAN program.
The
the
min1m1zer.
ed1tor.
by
and
des1gned
to
(1965)
not1ced
for
m1crocomputers.
parameters.
intended
to
A good
the
solved
squares
by
of
values.
acquaint
the
understanding
following
funct1on
exp{[(4-m)2
number
+
and
s>o.
of
the
likelihood
normal
distribution
p.
17.
Schnute's
should
(4
of
user
the
with
the
iterative
method
will
allow
and tools
example.
so
available
that
the
the
within
this
main
ideas
+
20)/2.
at
the
two
var1ables
(20-m)2]/2s2}
It
m and
s:
.
turns
function
with
although
equation
occur
of
out
that
F(m.s)
is
for
a sample
of
mean m and standard
note
a
(3.2).)
typographical
The maximum
sample
mean
proportional
two
values.
deviation
error
in
likelihood.
the
4 and
s.
position
that
is.
12
deviation
s.
and
.s2
real
m.
2.
is
the
1982.
in
minimum
in
report
w1th
Spendley.
concretely.
to
the
reciprocal
20.
drawn
from
s
an
a
by
exper1ments
and Mead
techn1que
full
advantage
of the
information
explanation
here
is
based
on an
F(m.s)
for
the
the
procedure.
Consider
where
conce1ved
numer1cal
descr1bed
use
observed
illustrated
(2.1)
f1rst
iteration
simplex
be
a
parameters
between
Simplex
can
for
est1mat1ng
as
they
from
of
was
forma11zed
1nto
manual.
1nclud1ng
SIMPLEX
of
m1n1m1z1ng
later
th1s
adapted
vers1on
problem
value
funct1on.
plotter.
BASIC
method
opt1m1ze
control
var1ables
1n
for
an opt1mum
response.
Nelder
wh1ch
O'Ne1l
(1971)
package
outl1ned
1n
and
SIMPLEX
Background
H1msworth
(1962)
locate
cond1t1ons
the
THE
-
{[(4-12)2
Incidentally.
this
case
should
is
be
+
it
biased
corrected
(20-12)2]/2}1/2.8.
is
well
known
that
for
small
samples.
by
the
factor
the
maximum
Here
the
21(2-1).
(The
likelihood
sample
size
reader
may
estimate
is
only
be
-13
familiar
with
Thus.
ignore
the
strictly
in
values
relief
which
map.
the
The
represented
the
lowe,.t
for
general
correction
an unbiased
estimate
this
limitation
of
n/(n-l).
for
s
maximum
problem
of
in
of
s
near
above).
m-20;
These
as a first
considerations
bottom
based
situated
near
on an expanded
m:
s:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
guess.
0.
it
By
is
8
times
here.
m and
s.
We can
visualize
the
horizontal
of
the
function
inspection
apparent
the
Indeed.
table
then.
boils
F(m.s)
coordinates
map.
is
of
that
suppose
that
the
suggest
a map shaped
the
origin.
version
of
n is
the
sample
the
square
since
our
size.)
root
of
interest
2.
We
is
(2.1).
estimating
by the
vertical
height
point
on the
map.
Our
cited
where
would
be
likelihood
defined
minimize
F(m.s).
where
m and s are
values
answer
function
-
The
large
Fin
the
(2.1)
4
8
12
16
20
+
+
+
+
+
1 4E55 1 6E34 1 6E27 1 6E34 1 4E55
1 3E14 1 2E9 1 4E7 1 2E9 1 3E14
1 1E6 165259 111030 165259 1 1E6
147695 1 2375 1 874 1 2375 147695
1 4183 1 613 1 323 1 613 1 4183
1 1260 1 332 1 213 1 332 I 1260
1 668 1 251 1 181 1 251 1 668
1 473 I
223 1 174 1 223 I
473
1 393 1 217 1 178 1 217 1 393
1 360 1 223 1 190 1 223 1 360
1 349 1 234 1 205 1 234 1 349
1 350 1 251 1 225 1 251 1 350
1 360 1 271 1 24.' 1 271 1 360
1 377 1 295 1 272 1 2J5 1 377
1 397 1 321 1 299 1 .~,1 1 397
the
between
of
with
shows
listed
finding
F(m.s)
values
(ignoring
occurs
value
a bowl
Fig.
2.1
of
values
to
a three
dimensional
and F(m.s)
is
minimum
of
for
large
minimum
desired
like
down
as
s
a
would
be
of
sand
known
ma4
is
close
to
steep
sides.
contour
below:
map
and
1.
the
for
F
+
1
I
1
1
1
1
1
I
1
I
I
1
1
1
1
Clearly
our initial
guess of s-l
is poor.
but for
the sake of our example we
will
start
off
at (m.s)-(4.1)
to illustrate
the simplex
search
method.
We
continue
as if we did not knC7lf the actual
minimum location.
which
from the
above table
occurs
at the point
(12.8).
as theory
suggests.
The simplex
search
method begins
with
three
arbitrary
points
on the
map: an initial
point
and two others
nearby.
As a wild
guess.
try
(m.s)
equal
to (8.1)
and (4.2).
in addition
to the starting
point
(4.1).
Evaluating
our
function
at these
points.
we find
that
F(~.l)
> F(8.1)
> F(4.2);
that
is.
(4.2)
is the best
(lC7lfest)
point.
and (4.1)
is the worst
(highest)
point.
It
is reasonable
to look for
a lower
point
far
from (4.1)
but close
to (8.1)
and
(4.2).
The method chooses
the point
(8.2).
which
is found by stepping
from the
highest
point
(4.1)
to the centroid
(average)
of the other
two points.
and
then taking
another
step of the same size
in the same direction
to get to the
new point.
This
process
is called
the reflection
of (4.1)
across
the two
lC7lfer points.
See Fig.
2.2A.
where the labelled
points
for our example
are
listed
below:
-14
400
33
250
15
le
6
6
8
STANDARD
10
12
14
DEVIATION
F1g.2.1.
Contour
plot
of the surface
(2.1).
Th1s funct1on
essent1ally
1s the rec1procal
of the l1ke11hood
funct1on
for
two samples,
4 and 20,
drawn from a normal
d1str1but1on
w1th mean m and standard
deviat1on
s.
Note the m1n1mum value
found at m .12
and s ~ 8, the sample mean and
standard
dev1ation,
respect1vely.
-16
Label
A.
B.
map.
traded
C
(4.1)
the
(8.1)
(4.2)
an
the
D
C'
(6.3)
(8.2)
the
the
value
points.
we
The method
requires
funct1on
evaluations
a simp11ficat1on
Successful
in
To
genera11zed
chosen
for
eventually
about
to
what
of
finding
the
begin
this
to
1nclude
po1nts
in
figure
search
that
N+1
The
from
1n
previous
to
As
labeled
A.
one
the
min1mum
the
B.
F(C)
example
and
C
> F(B)
find
the
1terations
the
centroid
relection
bottom
(i.e..
po1nts.
The
search
algorithm.
let
many
N-dimensional
determined
when
is.
a
a
way
point
of A and R
of C through
value
of
direction.
m1n1mum
happens.
rJf
point
F
than
up on the
s1de
of
one.
By Successive
N 1s 3
triangular
points
1s called
key
idea
1s to
such
highest
intermediate
lowest
point
a lower
the
right
points
lower
description.
functions
N-space
determined
by
the
method
its
name.
the
a
locate
actually
of
S1milarly.
a
by 4 points.
converge
30
lower
uses
N+l
2-dimensional
tr1angle.
determined
C' gives
mov1ng
in
point
descr1ption
parameters
2.
and a
Significance
B
A
have
initially
the
highest
complete
(m.s)
The
new point
C. so we are
and
we
Point
-
that
of
the
30
the
initial
Referring
to
the
bowl
movements
bowl
at
different
with
reasonable
because
reflections
following
us see
variables.
by
D
prec1sion.
are
simp11ces
~
more
above.
a
3-dimensional
In general.
a NsimplexN.
the
term
1terate
by constructing
successive
60
can be
with
N
In the
example
1s.
of Course.
involves
a
pyramid.
and
This
1s
always
not
provide
the
method
The
search
space.
3 points
and then
from
high
(m.s)
.(12.8).
triangles)
paragr3phs
how
guesses
our
relief
N 15
f1gure
a figure
which
a new
1n
gives
simplex
gradually
point.
above
where
> F(A).
shows.
when
N-2
the
three
simplex
po1nts
can
be
-17
With
precisely
simplex
however,
necessarily
1n
the
that
problem,
is,
Call
value
this
of
Because
per
2:
The
by
mov1ng
each
than
to
one
us
C
the
course
of
To
action
C'
is
and
B
the
other
find
out
near
a
is
the
s1de
how
simplex
MaN+l,
point
A
onto
original
cons1der:
new
with
up
itself.
in
begins
with
Notice,
C than
A or B does
not
our
steep
sided
valley
s1de,
take
points
the
a
h1gher
Mal,
than
of
good
determ1ned
MaO,
lower
case,
we
s1gn1f1cant
1mprovement,
another
step
step,
1llustrated
1n
the
in
eliminate
C from
the
we
all
C'
than
by
and
the
l<M<N+l.
points
attempt
d1rection
Fig.
2.2B,
be mov1ng
toward
the
m1nimum.
new simplex
point,
otherwise
In
th1s
that
case
1s,
step
has
actua"y
past
the
from
C halfway
to
the
new
1t
been
If
we
s1mplex.
po1nt
C'
1s
in
from
is
an
C to
based
C'
on
for
the
F(C")
take
is
less
than
C' as the
new
Dnly
one
h1gher
1nvolves
the
1nter1or
D.
the
the
half
of
the
Such
extens1on
than
all
1s
po1nts
descr1bed
earl1er,
determ1ne
a
a
step
step
(F1g.
of
1n
towards
rarely,
the
1s
h1gher
that
C'
1mproves
1s
CM
called
the
lower
replace
we are
s1mplex
1t
C w1th
forced
to
by mov1ng
lowest
all
po1nt
2.2C)
rema1n1ng
1s,
we
because
we
new
contract1ng
the
that
1mprove
C,
the
ent1re
distance
s1mplex)
as
we
of
not
of
happens
the
case
cC!ntro1d
contract1on,
1s,
CM does
or reduct1on
s1tutat1on
po1nt
th1s
conservat1ve
toward
accept
po1nts
Th1s
In
centro1d
~ F(C),
that
contract1on,
2.2£.
detr1mental;
m1n1mum.
the
the
s1mplex
< F(C)
we
N h1ghest
F1g.
po1nt,
occurs
as
only
vertices
of
the
s1mplex.
Case
only.
for
to
gone
CM (an
current
higher
subsequent
h3ve
the
1n
when
a
case
F(CM)
a general
of
shown
on
describe
> F(C).
point
of
If
F(CM)
CM. In
perform
led
still
CM as
because
h1ghest
po1nts.
up
to
always
determined.
furt.her
from
Remembering
to
cases
which
1nvolves
to
DC'.
This
M-O.
simplex,
contract1on
C'
is
point.
of
possible
1terat1on.
reflect1on
to
Our
four
has
F(C')
appear
point
now
The
process
C'
is
then
likely
In this
case
that
is.
e1ther
Case
current
is
< F(A).
to CM,
equal
In
is
rumber
M.
M-N+l.
simplex.
that
we will
we accept
po1nt.
a
are
reflect1on
attempted
it
C h1gh
C'
the
There
1:
extens1on
a distance
the
to
count
F(C')
hope
F(C'),
picture
number
M.
Case
current
the
we
mind,
fact
that
is
a good
to
poss1bly
actually
in
of
simplex
iteration.
the
reflected
point
A reflection
valley,
C'.
concepts
the
simple
mean that
C'
sample
miniRkJm.
the
these
the
process
ABC from
which
-
3:
M-l.
Here
We might
be
the
next
po1nt
Instead
1mproves
Otherw1se,
process
reflected
we
C'
of
try
we
CM,
the
tempted
would
the
accept
po1nt
1t,
we are
forced
reflect1on-reduct1on
vers1ons
of
reflected
to
then
the
reduce
us
by
a
contract1ng
h1ghest
po1nt,
where
from
reflect1on-contract1on
the
entire
(F1g.
2.2F).
correspond1ng
the
C' as our
new h1gh
us r1ght
back
to C,
obta1ned
g1v1ng
to
po1nt
accept
br1ng
C'
po1nt
but
the
we started.
to
D.
(F1g.
s1mplex
towards
the
Both
these
operations
operat1ons
performed
when
If
C
search
C"
2.2D).
low
point,
are
just
M-O.
a
-18
Case
not
at
4:
l<M<N+l.
improve
least
the
one
This
lowest
other
point
shown
in Fig.
2.2A
by
is
not
the
new highest
proceed
in
simplex
iteration,
another
as
simplest
On
the
well.
removing
point.
case.
other
In
this
C and
so that
The
hand.
case.
we
adding
C'
,eflection
to
reflected
C'
point
improves
accept
C'
not
the
does
only
C.
but
reflection
the
simplex.
for
the
next
Note
iteration
that
C'
will
B gives
a
compact
beginning
with
algorithmic
one
description
simplex
and
proceeding
of
the
process
to
the
next.
of
Convergence
The
simplex
contracting,
and
algorithm
reducing
SIMPLEX
package
here
and
lowest
function
by
the
A.
direction.
Appendix
2.3
is
point
-
the
user.
simplex
When
points
To
consider
m1nimum.
While
the
shr1nk
unt1l
small.
(2)
near
a
lowest
extentions.
of
point
is
near
The
a
lies
outs1de
which
m1nimum.
do
the
reductions
cond1t1ons
be
highest
specified
that
all
cond1tion
toward
most
only
(1)
actions
(3)
reasonable
to
test
ideal
s1mplex
not
s1mplex
and
rather
the
does
the
taken
The
when
close.
distracted
by
are
discussed
for
a
size.
typ1cally
that
are
this
simplex
s1mplex
1s
The
between
a limit
proceeds
simplex.
and
1t
satisfied.
course,
of
it
decrease
describes
algorithm
may
These
difficulties
of
not
values
so
paragraph
as
the
The
equ1valent.
preceding
the
algorithm
extending,
is
together.
significance
the
funct1on
often
however.
minima.
and
reflecting,
difference
less
than
implies,
close
con~equently.
corresponding
are
this
of
condition
that
the
must
be
contract1ons
po1nt;
process
fairly
validity
action
performs
1t
the
smal',
values
and
the
convergence
criterion
the
simplex
is
the
m1nimum
m1n1mum
problems.
and nultiple
limit
the
normally
encloses
some
the
on
function
the
reflections
algorithm
the
understand
stopping.
are
uses
values
have
repeats
until
the
(2)
conditions.
is
s1mplex
1s
only.
In
some
narrow
canyons.
ridges.
in
greater
detail
later.
2.4.
F1nal
axidl
When
convergence
At
this
search
the
algorithm
criterion.
stage
convergence.
small
distance
an
The
the
axial
search
search
tests
(both
positive
determines
low
point
is
the
and
a
is
used
to
proposed
negative)
simplex
taken
trap
found
by
the
axial
search.
which
an
passes
estimate
conditions
minimum
along
2N test
points;
however.
the
axial
search
is
at any
such
point.
When this
happens.
it
is
converged
prematurely.
and the
entire
process
minimum
as
by
each
of
the
above
minimum.
premature
stepping
axis.
stopped
if
a
assumed
that
is
restarted
the
of
away
from
This
gives
it
up
a
to
lower
value
is
found
the
algorithm
has
from
the
new
-19
2.5.
Algor1thm
data
Before
the
operating
parameters.
performance,
criteria,
and
ca"ed
SIMPLEX
package
These
such
as the
instructions
algorithm
data,
initial
for
and
user
needn't
program
so
is
far.
spend
running,
On
All
spec1f1ed
explained
Obviously,
the
(2)
then
In1t1al
mutually
perpen~1cular
r1ght
step
a
our
(3)
low
be
be
the
Here
less
l1m1ts,
however,
caus1ng
zero-d1v1de
(4)
fu~~t1on
convr:rge
opt:ton
of
on
of
so
the
point
point
(4,1),
ed1ts
on
vector
(8,1),
As
each
the
found
vector.
1.0
1f
not
s1mplex.
value
1n
If
typ1cal
of
h1gh
value.
some
the
1n1t1al
1n1t1al
requ1res
po1nt
II
rema1n1ng
s1nce
1ts
po1nt
1n
also
are
(Th1nk
should
and
the
value.
(2.1)
w1th
taken
II
s1des
po1nt.
coord1nate
earl1er,
to
the
e1ther
d1fference
user
an
has
no
s1tuat1on),
the
be
1n1t1al
(4,1),
g1v1ng
and
low
values,
user
must
the
s1mplex
l1m1t
absolute
between
or
the
pr1or
the
The
1nstances
the
simplex
po1nt
as
(4,2).
the
(a
w1th
the
the
correspond1ng
1n1t1al
po1nt
size,
F(m,s)
and
expla1ned
the
1n1t1al
1s
the
obta1n
step
1n
the
used
s1mplexM,
by
The user
can
select
looks
only
at
the
the
1s
S1nce
1n1t1al
to
Mr1ght
of
based
spec1f1ed
low
knowledge
relat1ve
d1v1ded
be
value
1s
of
l1mit
by
careful
the
w1th
m1ght
used
relat1ve
h1gh
and
be
the
may
low,
relat1ve
zero,
errors.
default
value
for
Max1mum
funct1o~~~.
calls
perforlr.ed
1n
qu1tt1ng
s1ze
user
d1fference
the
a
uncerta1nty
to
IO~ of
step
l1m1t.
the
of
problem,
because
poorly.
of
a bit
of
robust,
lowest
value
vector
determ1ned
The
the
funct1on
than
The
search
is to
Once
initial
default
step
vertex
1n1t1al
po1nts
values
useful.
must
of
example
S1mplex
of
a
2.2),
the
estimate
choice.
the
the
the
search.
by tak1ng
the
called
Na2.)
convergence.
absolute
method
funct1on
magn1tude
in
1nd1v1dually
be
the
t1me
the
SIMPLEX
for
The
to
1n1t1al
magn1tude
s1ze
defaults
~
(m,s)-(4,1),
to
test
l1m1t.
contains
various
paragraphs.
this
the
up
size,
convergence
parameters
are
converge,
so
is
remarkably
over
lies
set
coord1nates
m1ght
when
the
step
altered
1n1t1al
are
The
at
tr1angle
In
po1nt
step
s1mplex
roughly,
The
1s
step.
the
Such
a
vector
the
first
s1mplex
for
can be def1ned
s1mply
add1ng
reflect,
parameter.
point
s~t
algorithm
following
(section
closer
will
algorithm
answer
est1mates
the
earlier
the
agonizing
must
of
simplex
these
in
time
the
user
user.
po1nts.
of
described
the
algorithm
however,
the
initial
the
aspects
the
initial
Collectively,
As
point.
then,
1n1t1al
by
po1nt
to
define
N+1 po1nts,
1t
and
the
point,
output.
much
run,
example,
are
too
completion,
be
for
they
the
actual
minimum,
the
faster
thought
here
is
recommended;
the
can
affect,
(1)
Initial
point.
needs
a starting
algorithm
-
When
or
th1s
the
a
number
cont1nu1ng.
s1mplex
run.
of
The
Th1s
as a
calls
default
l1m1t
1s
sets
dev1ce
1s
0.1E-05
relat1ve.
an upper
l1m1t
for
stopp1ng
reached.
value
the
1s
N*lOO
on
runs
user
1s
+
100.
the
number
wh1ch
g1ven
the
(5)
often
Terminal
summary
informed
only
the
receives
user
If
T>l.
is
T
output
information
when
the
I.
2.
3.
number
number
current
4.
current
items
an
frequency.
is
output
error
condition
following
output
of
function
calls,
of
values
improved
mi ni mum poi nt ,
maximum
1.3,
and
4
File
output
This
to
number,
the
terminal.
exists
or
every
minimum
value.
are
printed
every
T,
controls
If
T-O,
a minimum
is
the
how
user
is
located.
If
T-l,
iteration:
iterations,
and action
and
say
restarts,
taken,
T
th
The
iteration.
default
value
.1.
(6)
terminal
output
default
frequency.
frequency,
value
is
This
except
O (so
no
file
that
has
an
effect
information
is
opened),
reduction
s1zes
Dy
fraction.
multlplylng
is
and
the
similar
output
to
to
a
the
file.
default
file
name
When SIMPLEX
each
initial
does
step
an axial
value
The
is
SIMPOUT.DAT.
it
step
size
specify
a
(7)
Step
t~eax1ai
finds
step
size
step
reduction
this
may
fraction.
parameter
pass
points
other
hand,
too
small
algorithm
frustratingly
discussion
of
this
The
with
at
some
which
a step
slow
point.
can
and
default
care,
the
value
because
algorithm
cause
multiple
inefficient.
is
an
should
0.01.
axial
The
search
be
restarts,
See section
restarted.
which
9.3
user
with
by
search
the
should
too
On
large
the
make the
for
further
-21
3.
3.1.
Two
methods:
MINSUM
Early
in
the
arose
for
1ncorporat1ng
to
keep
the
user's
work
wh1ch
can
eas1ly
pract1cal
enough
to
be
completed
made
cover
wh1ch
general
can
be
sum
the
of
sim1lar
composite
the
names
users
entered
3.2A.
called
dur1ng
final
part
least
they
least
squares
1dea.
funct1on
be
of
we
can
used.
the
user
thought
to
SUM FUN w1th
example
example.
worked
length
then.
must
is
be
in
be
not
the
versat1le
m1n1m1zer:
MINFUN.
wh1ch
Essent1ally.
expressed
Thus.
SIMPLEX
1s 1nvolved
ourselves.
as
a
refers
to
only
with
was
that
SIMPLEX.
the
the
introduction
response
incorporated
relat1onships.
the
objective
in
conta1ns
from
the
to
into
such
as
function.
represents
the
the
factor
MINSUM.
predicting
Second.
the
the
funct1on
1s
to
various
age.
a
Several
First.
there
funct1on
data
must
and
be
the
TEMPLATE
is
shown
1n
the
paragraphs
RES,
used,
may
run.
needed
and
of
three
var1ous
funct1on
(wh1ch
minimization
not
object1ve
have
was
vers1ons
calculat1on.
TERM
PRED,
are
two
which
MINSUM can be applied.
The objective
predicted
and observed
values
of a response
for
operat1ons
TEMPLATE
fully
1n
to
Unfortunately.
template
TEMPLATE
a module
funct1on
a
able
TEMPLATE
cons1sts
MINSUM to
1n1t1al1ze
by
second
The
be
the
code.
TEMPLATE.
and
(2)
wr1tten
by the
user.
The
the
Creat1ng
a pre-def1ned
rout1ne
UFUN
to
MINFUN.
and
funct1onal
each
term
1nto
led
MINFUN
must
but
1n pract1ce
1nformat1on.
are
var1ous
nature
of
Th1s
of
a s1mple
whenever
Here.
of
l1nes
Otherw1se
and
(2).
situat1on
in
terms
involv1ng
factors.
items
1ncorporates
funct1on
few
that
terms.
of
(I)
and
The
typical
sum of
a
clear
s1tuat1o!1.
always
MINSUM
w1th
1t
app11ed
MINSUM
should
3.2.
every
MINFUN
development
of
this
project.
two
d1st1nct
ph1losoph1es
the
user's
object1ve
funct1on
1nto
SIMPLEX.
A des1re
to a m1n1mum
1nsp1red
the
concept
of
a "template".
app11cat1ons
(I)
MINSUM.
uses
a more
MINSUM
and
PREPARATIONS
evaluat1on
depend
The
by
the
on
third
user,
TERM.
INITM
skeleton
rout1nes
PRED
and
such
List1ng
3.1.
follow1ng.
the1r
d1stinct
sections.
operat1ng
variables,
and
f1nal
as
SUMMAR are
forms
must
PRED,
1s
be
f1rst
1s INITM,
1f
necessary.
The
RES,
called
section,
summary
Its
subrout1nes
The user
1s
and
RES)
The
and
many
SUMMAR,
calculations
left
however,
with1n
performs
and
and
funct1ons
respons1ble
for
optional;
TERM.
t1mes
any
output.
are
descr1bed
complet1ng
at
even
TEMPLATE.
1f
-22
L1st1ng
3.1.
The
1nit1alization
final
summary
code
for
t~e
module
section
section
model
TEMPLATE
(INITM).
(SUMMAR).
of
mussel
for
MINSUM.
an evaluation
Underlined
growth
It
includes
section
comments
discussed
1n
a model
(PHED.
RES.
show
the
four
the
TERM).
lines
1ntroduction.
C*******************************************************************
SUBROUTINE
INITM
DOUBLE PRECISION
(N.AUX.NAUX.X.YOBS.NVAR.NDAT)
X(2O.50OO)
!INDEPENDENT
DOUBLE
PRECISION
YOBS(5O00)
!DEPENOENT
DOUBLE
PRECISION
AUX(50)
INAUX
C Subroutine
C 1n1t1alize
C
(if
INITM
is
the
vector
necessary).
used
of
and
to
set
auxilary
perform
the
any
AUXILARY
number
parameters.
other
VARIABLES(NVAR.NDAT)
VARIABLE
of
VALUES
(NDAT)
PARAMETERS
parameters
define
the
1nitializations
N.
model
needed
C by the
user.
C EXAMPLE:
N-3
RETURN
END
C*******************************************************************
FUNCTION
PRED
(XO.
DOUBLE
DOUBLE
PRECISION
PRECISION
DOUBLE
PRECISION
NVAR.
PAR~.
N.
XO(2O)
AUX(5O)
PARS(5O)
DOUBLE PRECISION
C PRED calculates
the
C EXAMPLE:
PRED .PARS
AUX.
NAUX.
ICASE.
NDAT)
IOBSERVED
POINT(NVAR)
!AS DEFINED
ABOVE
PRED
predicted
1
*
I PARAMETERS(
1-
N)
JPREDICTED
VALUE OF Y
of
the
passed
data
point.
-PARS
2 * XO 1 -PARS
3
value
EXP
I
END
C
FUNCTION
C TERM
C to
TERM
(YO.
YP.
XO.
NVAR.
PARS.
N.
DOUBLE
PRECISION
AUX(5O).XO(2O).YO.YP.PARS(5OJ
DOUBLE
PRECISION
TERM
INTEGER
calculates
the
ICASE
the
passed
term
data
C EXAM~~~6K~ERM
!TERM
OF
NAUX.
IAS
OBJECTIVE
IDATA
POINT
the
object1ve
of
AUXt
ICASE.
NDAT)
ABOVE
FUN~TION
COUNTEH
function
corresponding
point.
.(YO-YP)**2
END
C
C RES
FUNCTION
RES
(YO.
YP.
XO.
NVAR.
PARS.
DOUBLE
PRECISION
AUX(5O).XO(2O).PARS(5O)
DOUBLE
DOUBLE
PRECISION
PRECISION
YO.
RES
calculates
the
C EXAMPLE:
RES.
KtlUKN
model
YP
N.
AUX.
NAUX.
!
I OBSERVED
J RESIDUAL
residual
at
the
passed
Y.
ICASE.
AS ABOVE
PREOICTED
data
point.
YO-YP
END
C*******************************************************************
SUBROUTINE
SUMMAR (AUX.NAUX.X.YOBS.NDAT.NVAR.PARS.N.YPRED.
.TERM.
OOUBLE
DOUBLE
C SUMMAR allows
the
C function
evaluation
RETURN
END
RESIO.
F.
NF.
HI.
NR)
AUX(5O).X(2O.5OOO).YOBS(5OOO).PARS(5O)
YPRED(5OOO).TERM(5O0O).RESID(5O0O).
PRECISION
PRECISION
user
at
to
the
wr1te
current
information
parameter
associated
vector
NDAT)
F
with
PARS.
Y
and
of
a
-23
parameters
the
user.
user
will
or
SUBROUTINE
INITM.
(described
and to
set
below).
to
perform
any
the
number
of
parameters
be
file
has
determine
to
supply
already
been
any
required
maxima
of
parametersM.
are
not
to
utilized
the
observed
several
PRED
X of
values
for
data
NDAT
response
YO to
Such
For
of
and
a
NVAR
in
any
response.
and
the
maximum
based
on the
observation
auxiliary
parameter
vector
value
of N is
50.
FUNCTION
runct1on,
object1ve
correspond1ng
the
number
TERM.
based
Usually,
least
squares
parameter
th1s
report
can
be
RES.
from
MINSUM
SUMMAR)
rout1nes,
vector
NI
number
m1n1m1zat1on
-the
number
NR -the
number
SUMMAR can
automat1cally
dev1ces,
as
calculates
data
very
s1mple
funct1on
a
20.
the
an
observed
NVAR.
The
maximum
predicted
value
value
of
the
model
parameter
vector
dimension
N. where
the
funct1on,
of
at
term
and
of
the
PARS.
the
parameters,
for
the
such
(wh1ch
as
has been
of
1terat1ons
of
a
restarts
be
used,
l1sted
for
by
a
means
of
values
vector
funct1on
at
the
1n
and
(YO-YP)**2
1n
performed;
1n the
record1ng
the
most
most
access
the
to
current
the
s1mple.
current
the
user.
Typ1cally,
m1n1m1zat1on,
but
1t
evaluat1on
all
PRED)
current
vector
recent
recent
recent
on
on
the
usually
funct1on
var1ables
1n
a
used
funct1on
parameter
parameter
vector
PARS;
m1n1m1zat1on,
m1n1m1zat1on,
m1n1m1zat1on,
to
l1st
any
pert1nant
SUMMAR can
also
wr1te
data
at
1s
PARS;
calculat1on;
w1th
most
a
(from
current
the
report
does
parameter
current
calls
example,
MINSUM.
of
a
des1red
by
a successful
SUMMAR has
the
following:
assoc1ated
1n
when
1s
response
value
funct1on
called
res1dual
the
funct1on
1s just
YO-YP.
automat1cally
any
t1me.
as well
as
res1duals
funct1on
of
1s
the
Aga1n,
RES
quant1t1es
at
the
end
pred1cted
of
calculates
TERM.
errors,
Th1s
at
the
current
of
TERMs 1n
vector
the
Th1s
SUMMAR.
and
be
PARS;
F -the
object1ve
NF-
a
same data
as
of
add1t1ve
associated
requested
-the
the
XO. the
PARS has
and
Maximum
dimension
where
be
to
available.
as
aux1l1ary
ava1l~ble
1s
several
and
YP. The model
parameters,
current
data
po1nt
are
also
TERM
the
case
calculat1on
-the
vector
RESID
50DO
NAUX.
flags
responses
a typ1cal
(XO,YO)
SUBROUTINE
the
has
po1nt
vector
would
cal11ng
prev1ous
YPREDTERM
on
1n
called
before
1n the
Th1s
funct1on
on the
current
user
as
variables.
calculates
vector
AUX.
used
observation
dimension
function
"auxiliary
be
XO of
used
to
minima.
model
parameters.
parameters
can
also
are
data
can be
deviations.
observed
particular
vector
the
by
case.
FUNCTION
po1nt,
based
For
example,
has
This
pred1ct1on
ICASE
of
the
calculat1on.
the
AUX
PRED.
NDAT
auxiliary
desired
here.
the
set
called
explanatory
of
called.
unlike
Auxiliary
the
Y of
NVAR
explanatory
v~ctor
are
if
is
themselves
standard
but.
implementation
parameter
is
50.
INITM
might
use.
of
auxiliary
of
NAUX
FUNCTION
to
treats
observed
When
they
vector
initializations
If
N is
not
quantities
vector
this
software
the
other
N.
run
time.
minimization.
model
vectors
initialize
the
data
as means.
example.
which
consist
the
to
time.
at
observed
The
run
variables.
ways.
NDAT
respectively.
N at
used
determined
by function
function
The
matrix
is
read.
Consequently.
quantities.
such
They
are
be estimated
in
tell
asked
This
-
values
or
or
or
1f
no
0;
0.
1nformat1on
to
external
of
a
spec1al
not
f1les
and
1nterest.
-24
3.28.
Preparing
the
The
data
f11e
of
file.
observed
data
read
by
MINSUM
must
have
the
follow1ng
format:
L1ne
1:
Number
L1ne
2:
FORTRAN
format;-ablank
can
each
the
MINSUM
po1nt
MINFUN
la_st:
sun1l1ary
Data.
1n
to
on
the
TEMPLATE
section.
Listing
on
the
basis
(maximum
(4).
MINFUN.
run
of
which
50).
and
here
labelled
UFUN itself
is
started.
to
the
initialization
are
declared
in this
During
minimization.
simply
evaluate
NF--l;
MINFUN
in this
case
UFUN
saves
the
parameters
use
order
descr1bed
the
these
function.
values
prove
3.2
parameters
can
data
read
can
be
read
free
as
above.
(the
max1mum
value
funct1on
evaluat1on.
be
precision)
UFUN
the
are
YO
the
method
UFUN must
initialization
UFUN references
be returned.
(2)
branch
later
here.
If
l1nes
of
NDAT)
data
the
amount
po1nts.
of
data
speed.
that
to
Item
data.
Data
5000
every
Notice
value
called
by
minimization
the
UFUN
Should
final
g1ven.
program
more
general
subroutine
distinct
sections:
an
UFUN.
function
for
be
XO(NVAR).
affect
and
var1ables
format
should
can
read
up
1s accessed
substant1ally
3.3.
explanatory
XO(2).
,~o
Lines
of
read
l1ne
XO(I).
S1nce
NVAR
(4)
inadequate
written.
section.
shows
four
the
the
the
function
number
NF.
should
MINFUN
section.
section.
MINFUN
tamper
NF-O;
If
then
keeps
(without
branch
function
to
the
user's
structure
(1)
vector
is
of
determines
not
sets
needs.
calculated.
function
of
the
value
recalculation)
of
(3)
calls
which
with
UFUN.
a
section
NF.
when
the
so
typical
of
Before
called.
UFUN
the
should
available
be included
should
MINFUN
final
is
a
must
be
should
and
UFUN
a minimization.
number
far.
final
summary
section.
associated
with
the
in
a
has
three
and a
the
double
precision
(also
double
variables
that
a SAVE statement
track
of
NF>O.
On completion
should
and
skeletal
variables:
parameter
the
for
Like
TEMPLATE.
UFUN
a function
evaluator.
sets
Since
minimum.
section.
of
-25
listing
example
estimating
based
allowing
(2.1)
on
two
user
function
sample
Input
(2.1)
(3.1)
As
3.3
the
for
Section
small
s.
values:
of
the
is
2
.s
2.
we
complete
sample
depending
the
again
(a
SUBROUTINE
2
+
2
(b-m)
interested
factor
of
subroutine
of
2
)
2
]/2s
in
}
UFUN(F.
PARS.
N.
DOUBLE
PRECISION
PARS(*)
1 Parameter
three
the
function
by
and
we
ignore
estimate
subroutine
which
are
depends
referenced
on
value
vector
N
1 Number
of
parameters
NF
1 Number
of
function
ca11s
So
far
THEN
1:
In1t1al1zat1on
when
(NF.GT.O)
THEN
2:
runct1on
evaluat1on
ELSE
Sect1on
END IF
F.
likelihood
UFUN.
The
sections
INTEGER
c
function
maximum
INTEGER
ELSE IF
Sect1on
idea
the
NF)
1 Returned
(NF.EQ.O)
this
Thus
b.
NF.
F
Sect1on
the
.
in
the
a typical
It
has
PRECISION
IF
with
concerned
with
normal
distribution.
3.3
generalizes
called
a and
only
1/2
DOUBLE
c
associated
we were
s for
a
10.
listing
values.
here
1isting
for
as shown.
value
UFUN
2.
There.
deviation
by
exp{[(a-m)
bias
A skeleta1
declared
on
here
are
L1st1ng
3.2.
four
variables.
c
a
4 a~d
sample
replaced
F(m.s)
in
shows
discussed
in
Section
the
mean m and standard
-
3:
F1nal
summary
when
NF-O;
may
when
NF>O.
1nclude
a
SAVE.
NF<O.
RETURN
END
Listing
3.3.
A complete
sample
and standard
deviation
s
values,
a and b.
Maximum
in
This
can
be
are
used
to
estimate
based
equivalent
on
UFUN
(F,
DOUBLE PRECISION
INTEGER
N, NF
IF
(NF.EQ.O)
N .2
PRINT
PRINT
*.
*.
READ (5.
PRINT *.
PARS,
F,
THEN
I Please
I A: I
10) A
I B: '
READ (5. 10) B
FORMAT (F18.0)
SAVE
N,
PARS(*),
NF)
A,
B,
EXPTRM
1 Inftfalfzatfon
fnput
sectfon
*****
1 Two parameters:
m and s
values
for A and B.'
1 Get a
I Get
the
mean
two
sample
to minimizing
(3.1).
SUBROUTINE
10
UFUN.
for
a normal
distribution,
likelihood
estimates
b
m
F
-26
ELSE
IF
(NF.GT.O)
EXPTRM.
F .PARS(2)
20
END
IF
RETURN
END
THEN
I
.5
*
(PARS(2)
**
«
A -PARS(1»**2
**
2 * EXP
(
-
Evaluat1on
(-2»
sect1on
*
+ (B -PARS(1»**2)
EXPTRM )
*********
-27
.MINSUM
CALLS
>
-
.TEMPLATE
.
..I.
.
I
1--
CALLS
)
...IMSL
.
.COMMON
.--CALLS
--)
.AND
.
...IGL
1--
..I.
.
CALLS
>
The
linking
procedure
can
GOMINSUM.COM
and GOMINFUN.COM.
files
object
resulting
Typical
.
)
I
.MINFUN
an
CALLS
file
will
result
executable
calls
image
would
$ @ SIMPlEX:GOMINFUN
in
that
being
.UFUN
.
be simplified
Calling
these
file
given
being
the
same
somewhat
programs
linked
name
into
as
with
with
the
the
the
command
the
name of
package,
object
and
the
file.
be:
UFUN
or
$ ,
SIMPLEX:GOMINSUM
TEMPLATE
Note that
the .OBJ extens1ons
must not be added.
The f1le
names TEMPLATE and
UFUN are not sacred
here,
as long as the spec1f1ed
files
conta1n
the requ1red
funct1ons
and subrout1nes
w1th the standard
names.
L1st1ngs
3.4 and 3.5 show
the command f1les
GOMINSUM and GOMINFUN.
List1ng
3.4.
The
command file
S
SI
SET VERIFY
GOMINSUM.COM.
******.
SI USEO TO LINK USER'S TEMPLATE ROUTINE INTO SIMPLEX PACKAGE WITH
SI MIHSUM.
TO USE ENTER:
SI
S'
GOMINSUM file
SI WHERE file
IS THE NAME OF THE USER ROUTINE.
$1*****..*.***.*.*.*..*.*.**.********.**********.*...****
**.
S
LINK/NOMAP 'P1,
SIMPLEX:COMMOH, MINSUM, S
$
SYS$SHARE:IMSLIBD/LIB,
SET NOVERIFY
EXIT
PBS$IGLOPT:/OPT
-28
L1sting
3.5.
$
SET
$1
USED
$1
TO USE
command
LINK
USER'S
f11e
IS
LINK/NOMAP
$
$
SET
f~TT
'PI,
user
Steps
USER
ROUTINE.
SIMPLEX:MINFUN,
1.
has
the
option
required
Prepare
for
COMMON,
-
PBS$IGLOPT:IOPT
a
file
of
the
of
using
first
MINSUM
option
observed
INITM.
As
data
PRED.
Rname.FORR.
described
TERM.
Step
3.
FORTRAN
compile
and
are
in
TEMPLATE
listed
the
or
else
MINFUN
below:
format
Step
4.
Execute
the
will
SIMPlEX
result
in
options
The
similar.
steps
A data
-name.FOR-
described
above
in
4
rather
still
apply.
than
file
chosen
will
by the
contain
SUMMAR.
to
obtain
GOMINSUM
Rname.EXER,
in
contain
and
a prefix
this
file
Rname.OBJR.
by
entering:
name
file
as
RES.
command
this
requirf!d
file.
should
3 and
GOMINFUN.
a
described
where
RnameR is
in
Section
3.2A.
Rname.FORR
$ 'SIMPLEX:GOMINSUM
Steps
MINFUN.
3.28.
routines
This
WITH
f11e
NAME OF THE
Step
2.
Create
a file
based
on TEMPLATE.
user.
PACKAGE
summary
UFUN.
Step
the
THE
Preparations
Section
SIMPLEX
SYS$SHARE:IMSLIBD/LIB,
NOVERIFY
The
and
INTO
ENTER:
$
$1
3.5.
GOMINFUN.COM.
ROUTINE
$ @ GOMINFUN
WHERE
file
VERIFY
TD
$1
$1
The
-
in
for
the
Step
1.
the
second
mayor
subroutine
except
GOMINSUM.
which
can
be
RUN to
obtain
all
the
report.
that
option
may
UFUN.
the
not
as
command
with
be
MINFUN
described
f11e
and
necessary.
in
in
Step
UFUN
In
Section
4
are
Step
should
2
3.3.
be
-29
4.
4.1.
System
SIMPLEX
-
OPERATION:
MINIMIZING
1n1t1al1zat1on
To
1n1t1ate
SIMPLEX
operat1on,
the
user
enters
S RUN f11ena~
where
Mf1lenameM
1n Sect1on
MINSUM
or
The
user
file.
or
f1le
1s
then
data
Listing
1)
la)
Ib)
2)
3)
4)
5)
6)
7)
8)
9)
l° l
SIMPLEX
checks
matches
main
.EXE
name
of
data
to
the
a
ensure
that
the
the
name
value
of
from
described
algorithm
of
an
or
4.1
shows
section.
data
algor1thm
N fronl
TEMPLATE
L1st1ng
following
as
INITM
from
init1alizations.
created
If
obta1ned
d1splayed.
1n the
created
either
user's
previously
values.
value
file,
calls
the
the
UFUN.
the
At
menu
th1s
1tems.
menu items.
Options
1.
RE-INITIALIZE.
is
called
from
MINFUN
in
data
only,
2.
the
algorithm
Simplex
appropr1ate
COMPUTE COVARIANCE
MATRIX
SEARCH GLOB~LLY
FOR MINIMUM
EXIT
Menu
reading
E\lIT
new
part1cular
p~rticular
f1elds
algor1thm
With
ALGORITHM
data,~
the
e1ther
default
f1le
an
SIMPLEX
immediately
from
MINFUN
to perform
RE-INITIALIZE
COMPLETELY
RE-INITIALIZE
DATA (MINSUM ONLY)
RE-INITIALIZE
MODEL (MINSUM ONLY)
EOIT ALGORITHM DATA
LOAD NEW ALGORITHM DATA
SAVE ALGORITHM DATA
MINIMIZE
REPORT ON MINIMUM
FUNCTION CALL AT CURRENT POINT
PLOT PREDICTIONS AND RESIDUALS (MINSUM ONLY)
PLOT PROFILE
II
12
UFUH
of
ma1n SIMPLEX
menu
1s
covered
1n more
deta1l
4.1.
4.2.
name
started,
NF-O)
provides
entered.
the
are
the
Once
(with
accepts
algor1thm
point
These
1s
3.4.
UFUN
data
this
option,
with
(18)
DATA.
The
these
is
user
or,
INITM
~y
in
variables.
presented.
INITM
MINSUM
calling
var1ables,
of
either
NF-O.
only,
choose
the
When
A menu
users
allows
case
the
is
called
have
or
the
(I)
to
edit
of
POINT
editor
selection
MINSUM
of
or
(lA)
both.
all
1s
from
choice
the
and
algorithm
STEP,
sunmoned,
of
the
only
a
various
l1st
of
-30
-
editing
alternatives.
as itemized
in
the
top
of Table
4.1.
Algorithm
data
types
are
listcd
in
the
lower
portion
of Table
4.1.
along
with
command
codes
for
the
editing
functions.
A value
selected
for
editing
is
displayed
before
the
user
<CR>.
changes
it;
Ordinarily.
the
when
however.
field.
there
is
one
the
corresponding
value.
evell
Table
4.1.
if
the
Codes
FUNCTIONS
Edit
all
Edit
particular
value
value
is
exception
field
user
for
EDIT
current
a
can
to
this
in
STEP
accepts
ed1t1ng
be
changed.
the
retained
this
has
no
rule.
Whenever
is
automatically
default
algor1thm
POINT
simply
by
pressing
effect
on
other
the
user
changed
values;
edits
a POINT
to
the
derau1t
field.
data.
ENTER:
values
Al
value(s)
Edit
code.
as
below
H
Help
Review
current
values
Return
Exit
(
<CR>
)
Q
VARIABLE
TO
EOIT
EDIT
CDDE
IP
IPn
IS
ISn
SL
MF
FREQUENCY
OF TERMINAL
OUTPUT
FREQUENCY
OF FILE
OUTPUT
OUTPUT FILE
(ACCESSED
IF
STEP
3.
REDUCTION
lOAD
FRACTION
4.
spec1t1ea
SAVE
by
5. MINIMIZE.
m1n1m1zat1on
of
cond1t1ons
are
out11ned
1s
the
user
OATA.
wh1ch
along
ON MINIMUM.
user
has
an
pred1cted.
and
norma11zed
res1duals.
FN
RESTART
SR
The
can
The
new
request
current
data
may
automatic
algor1thm
res1dual
must
w1th
be resolved
suggest1ons
SIMPlEX
add1t1onal
values.
calls
along
the
opt1on
w1th
by
for
be
read
user
user
an
which
15
wr1tten
1n1t1ated
1terates.
to
1nput.
act1on.
user
summary
of
d1splay1ng
standard
from
~efaults.
data
The
s1mplex
search
algor1thm
-15
user
funct1on.
As the
algor1thm
may ar1se
later.
6. REPORT
act1ve.
the
> 0)
DATA.
the
AlGORITHM
the
user.
FF
FF
FOR
NEW AlGORITHM
file.
Alternatively.
be edited.
TF
can
to
attempt
var1ous
These
cond1t1ons
rout1ne.
If
the
OOserved.
dev1at1ons
a
of
the
MINSUM
-31
7.
EVALUATE
its
function
can
plot
8.
PLOT
PREOICTIONS.
observed
values.
and
Sections
and
6
PROFILE.
parameters
can be sent
Detailed
information
point.
11.
is
output,
along
with
MATRIX.
SEARCH.
function
(1)
value
at
being
points
optionally
printed
the
hard
res)
both
parameters
deciding
As in
terminal,
in
the
previous
plotter.
MINFUN.
It
from
option.
or file.
7.
of
numerical
and
discusses
the
and
are
varied
how well
the
properties
obtain
correlations.
MINSUM
Section
Theoretical
to
(high
display.
detail.
with
given
user
(low
plotter
later
when
in
is
The
response
terminal
copy
for
values
very
useful
here
below
on
a
in
by the
data.
resolution
exploited
8
displayed
only.
predicted
available
function
can
be
COVARIANCE
MINSUM
versus
the
log
estimates
covariances
underlying
at
theory
of
the
current
and
the
available.
values
which
is
with
res).
or
(low/high
options
plots
Section
available:
search.
distributed
option
of
available
be
various
deviations.
GLOBAL
calculates
calculate
can
profile
are
options
option
point
observed
(high
to
a file
determined
low
or high
on
standard
software
a
grid
over
an
uses
uniformly
each
Given
a set
of
values
over
point
a
search.
which
evenly
is
and
the
ranges.
range.
the
parameter
number
space.
calculated
as
employs
spaced
pseudo-random
in
parameter
retained
to a file.
parameter
the
search
IMSL
grid.
generator
In either
displayed.
current
point.
the
global
Two types
routine
and
ZSRCH
(2)
a
to
calculate
(1)
or
(2).
with
the
The
user
to
random
random
points
the
function
minimum
can
search
of search
point
also
have
found
the
M1n1m1zat1on
4.3A.
Interpreting
As
consistent
descr1bed
the
SIMPLEX
w1th
the
1n Sect1on
1nformat1on,
1t
1nput.
The
occurs
when
two
(1)
the
This
function
parameter
plots
the
are
to
a
CALCULATE
likelihood
4.3.
describe
is
curves.
terminal
written
to
obt~~n
plots
point.
Such
plots
model's
plots
are
These
graphics
also
be
PLOT
minimum
minimum
option
predicted
a
can
be used
minimum
10.
current
This
data.
residuals.
5
9.
The
value.
resolution).
res).
They
can
the
FUNCTION.
-
algor1thm
saved,
mak1ng
data.
counter
Should
(NF)
the
may
output
1terates
term1nal
2.5
on
also
pr1nt
poss1b1l1t1es
"MAXIMUM
number
data.
th1s
the
user
1s reset
the
user
opportune
elect
to
1,
a
to
to
stop
produces
output
1nd1cate
problems
as
requ1r1ng
user
belQw.
REACHED,
CONTINUE?
evaluat1ons
has
elects
1t
and f1le
output
frequency,
Bes1des
this
1terative
wh1ch
d1scussed
t1me
to
and
m1n1m1zat1on,
frequency
data.
messages
are
FUNCTIONS
of
funct1on
If
an
through
output
alg~r1thm
qu1t,
and
continue
w1thout
m1n1m1zat1on
1s
the
«Y>,N)".
reached
ent1re
perhaps
rev1s1ons,
resumed.
the
Th1s
message
max1mum
set
current
rev1se
the
s1mplex
the
algor1thm
funct1on
1n
1s
-32
(2j
MCONSECUTIVE
1nformat1onal
or
a
only.
large
problems.
opt1on
of
Control
pause
(AC).
1n
some
no
changes)
ma1n.
the
current
and
the
the
numbers
returned
to
Th1s
message
consecut1ve
1nd1cat1ve
of
be halted
rev1sed.
w1th
Sect1on
the
may
be
user
1s
On
the
po1nt
1nterrupt
the
s1gnal.
to
1s
saved.
Then.
values.
the
algor1thm
tun1ng
control
d1scusses
9
1s
restarts.
C ("C)
the
the
1nterrupt
comp1etes
s1mplex.
and
the
menu.
kept
1f
ma1n
as
the
the
new
user
procedure
offers
If
the
current
opt1on
the
returns
1s
the
1n1t1a1
resumes
1mplemented
s1gnal
to
user
to
1terat1on.
of
cont1nu1ng
returns
to
po1nt.
and
the
m1n1m1zer
exactly
where
a1low
a control-C
the
1t
the
ent1re
w1thout
1eft
off.
found
funct1on
the
opt1on
VAX
SIMPLEX
current
When SIMPLEX
f1nds
a point
ax1al
search
test.
1t d1splays
of
PROBLEMSM.
more
1s
return1ng
low
po1nt
M1n1mum
a
th1s
of
or
s1mplex
4.3C.
or
should
should
be
execut1on.
status
(w1th
chang1ng
two
further.
systems.
rece1v1ng
the
TUNING
hav1ng
restarts.
algor1thm
data
convergence
program
After
outputs
current
of
INDICATE
of
C (AC)
On
a
number
slow
RESTARTS
cond1t1on
Usually.
the
and the
algor1thm
problem
4.38.
total
A
-
calls.
main
menu.
1terat1ons
w1th
the
wh1ch
the
and
final
passes
po1nt.
restarts
po1nt
both
its
the
s1mplex
l1m1t
test
funct1on
value.
and the
requ1red.
retained
Control
as
the
current
is
po1nt.
-33
5.
5.1.
Plot
types
PLOTTING
-
FACILITIES
-34
Table
5.1.
Low
Options
for
low
and
high
resolution
o
1-
Scatter
Plot
2-
Step
Function
Histogram
3-
Open
Solirl
5-
Vertical
6-
Sequential
Table
5.2.
Histogram
Bar
Histogram
Counting
Options
0-
small
1-
medium
open
2-
~dium
3-
medium
4-
~dium
+
5-
small
open
6-
~dium
X
7-
medium
open
Table
5.3.
Options
plot
High
Resolut1on
4-
-
Plot
for
solid
high
Resolution
-Scatter
Plot
1-
Connected
2-
Step
Function
Histogram
3-
Open
4-
Solid
5-
Vertical
6-
Sequential
resolution
types.
Points
Plot
Histogram
Bar
Histogram
Counting
markers
(plot
Plot
type
a
square
8-
~dium
X
in
open
octagon
square
9-
medium
+
in
open
square
open
octagon
A
-~dium
inverted
open
triangle
B
-medium
open
C
-~dium
diamond
D
-large
sign
sign
in
open
square
in
for
high
square
E
-up
X
F
-down
sign
resolution
only).
triangle
star
asterisk.
X
arrow
arrow
line
types
(plot
types
I.
2.
and
3
only).
O
-solid
line
1-
closely
2-
single
3-
small
4-
nedium
Table
(plot
the
5.4.
types
video
dotted
line
dotted
close
dashed
line
dashed
line
spaced
dashed
Options
4 and
for
cross-hatching
5 only).
These
screen
may
line
be
somewhat
O -close
lines.
12-
close
close
lines.
lines.
O degrees
90
-45
degrees(vertical)
degrees
34-
close
close
lines.
lines.
45
-30
degrees
degrees
5-
close
lines.
30
degrees
6-
close
lines.
7-
close
lines.
-60
60
degrees
degrees
5-
large
dashed
6-
large
small
7-
small
8-
very
9-
nedium
apply
dashed
large
high
to
d>l.
dash
d>l.
dashed
spaced
dotted
dotted
line
l1ne
dotted
resolution
the
flatbed
line
line
line
histogram
plotter.
different.
(horizontal)
8.9.A-F
-like
G-N
0-7
but
medium
spaced
-like
0-7
but
wide
O -solid
spaced
filled
panel
patterns
Results
on
-35
L1st1ng
5.1
functions
available
For
example.
F1g.
L1stings
5.2
included
to
Listing
5.1.
and
1llustrates
1n
5.1
5.3.
1llustrate
resolut1on.
a high
also
created
one
Histogram
the
low
shows
of
the
-
var1ous
types
High
resolution
with
resolution
open
the
trick1er
of
versions
histogram.
SIMPLEX
aspects
plotting
of
VERTICAL
****
*
******
*
*
*
*
*
*
*
*
*
*
SOLID
HISTOGRAM:
I
I
I
I
I
******
I****
I
*
I
*
I
*
resolution
*
*
*
*
*
*
*
I
OPEN
software.
low
BAR HISTOGRAM:
I
*****
*
*********
and
******
*
*
*
*
*
****
*
*******
* ****
*
*
***
*
*
step
are
similar.
Finally.
examples.
STEP FUNCTION:
I
I
I
I
I******
I
histograms
HISTOGRAM:
*
*
*
are
plott1ng.
-36
-
15
1
>u
z
w
:J
o
w
!l
I&.
IS
All:
Fig.
5.1.
of 76 fish.
capabilities,
observations
6, 3, 4, 6,
Histogram
showing
the age-frequency
The figure
was created
with
the
using
the open histogram
option.
for
ages 3 to 16 are,
respectively:
2, and 5.
distribution
high resolution
for a sample
plotting
The frequencies
7, 6, 4, 3,
9,
of
12,
6,
-37
l
e
n
9
t
h
-
1
80+
1
60+
1
n
0
PPPPPPPPPPPPPPPPPPP
1
40
m
m
PPPPPPPPPP
PPPPPP
+
PPPPP
1
PPPP
20
PPP
+
PPP
I
PPP
P-+---+--+---+--+--+---+--+--+---+--+---+--+--+---+--+
5
AGE
L
e
n
9
t
h
in
10
Years
1
80+
1
60+
0
POPPPOPPOPPOPPPOPPP
1
n
40
1
OPPPOP
+
m
m
15
POPPOPPPOP
PPOPP
1
PPPO
20
PPO
+
I
PPO
POP
P-+---+--+---+--+--+---+--+--+---+--+---+--+--+-5
AGE
in
10
Years
-38
5.2.
Plotter
requ1re
the
and a Tek
terminal.
higher
4662
With
onto
and
Flatbed
a small
the
VT100,
our
w1th
$
term1nal
term1nal.
rate
should
BAUD
1200
that
wh1le
the
on
th1s
w1ll
plott1ng
features
4105
allow
the
and
on
user
4105
of
turn
the
term1nal
yourself)
have
to
w11l
refuse
1s
logged
1n
the
the
SIMPLEX,
and
load
beg1n
have
to
CRT
w1th
the
the
the
user
4662
need
w1th
plott1ng
been
left
respond
onto
w1th
or
If
at
Computer
and
rate
SETUP
an
Next
key
baud
leave
mode
problems
command.
SETUP
If
To
setup
OWNER-NONE.
the
1200.
command.
s1multaneously.
a member
of
the
however,
press1ng
be
FACTORY
plotting
regular
VT100
or VT125
the
4105
can be replaced
in
a state
the
at
all,
4105.
f1rst
Th1s
can
ensure
easily
be
TK:
should
by
If
the
the
package
Term1nal
command:
SHOW TERMINAL
The
use
a
of
the
SIMPLEX
Color
Graphics
purpose.
(1nclud1ng
the
along
with
modification
turn
Somet1mes,
for
Should
done
To
Normally
1mmediately.
unsuitable
no-one
Plotter,
software
4006.
paper.
that
requirements
The
high
resolution
plotting
capab1l1t1es
user
to have
access
to
a Tektronix
4105
resolut1on
onlylog
pen
hardware
-
aster1sk
the
leave
1s
mode
(*)
plotter
SETUP
th1s
po1nt
Center.
the
check
press
appears
press
term1nal
baud
5iATUS
1ncorrect.
pers1st.
and
the
enter1ng
rate
BAUD
reset
the
1t
SETUP
as
st1ll
key
the
return
SHIFT
on
us1ng
aga1n.
prompt
to
and
w1ll
the
<CR>.
The
baud
the
SET
Not1ce
symbol.
SETUP
mode
and
try
CANCEL
not
respond.
contact
-39
6.
The
plotting
assessing
observations,
resolut1on
part1cularly
and
for
work,
software
h~
well
the
predictions,
plots,
which
but
the
and
how
Sect1on
the
var1ous
provides
accuracy
makes
of
12
MINStX-I
the
can
plotter
1s
plots
more
also
be
the
be
a
convenient
tool
and
1nvolve
su1table
for
for
1nvolves
a
reader
may
of
to
study
w1sh
a
the
the
fa1r
to
data
b1t
MINSUM
1nput.
by step
procedure
ahead
growth
set
of
step
plott1ng
sk1p
s1mple
pred1ct1ons,
a b1t
more
study1ng
when
sl1ghtly
d1fferently.
C. However.
the
analys1s
used
observat1ons
plots
1mportant
generally
the
can
them
prepared
as Append1x
and
out11n1ng
(MINSUM ONLY)
fits
the
data.
Plots
can
involve
var1ables,
and res1duals.
Low
on a regular
CRT screen,
are
qu1ck
plots
of
H1gh
resolution
stra1ghtforward.
1n
shows
use
ETC.
creat1ng
patterns.
each
type
of plot
have
been
1ncluded
reasonably
in
High
resolution
points.
Because
and because
1nstruct1ons
examp)e
included
1ncreased
curves.
many
data
OBSERVATIONS.
proposed
model
explanatory
are
displayed
useful
for
study1ng
res1dual
predicted
1ncludes
1s
PLOTTING
-
to
curve.
results
the
Th1s
qu1ckly
eff1c1ently.
6.1.
MINSUM
plotter
options
Options
6.1.
Of
simply
.display
these.
act
as
while
from
option
toggles.
feature
off.
available
available
only
the
MINSUM
3 merits
reversing
is
especially
creating
high
lengthy
the
status
needed
resolution
the
most
recently
created
plot
being
while
option
5 allows
that
same
plot
for
a file
name for
the
stored
plot.
plots;
setting
again
does
resolution
Listing
the
not
plots
6.1.
1)
2)
3)
4)
5)
6)
7)
following
user
affect
are
will
the
always
have
to
operation
stored
The MINStJ1 plotter
plotter
are
displayed
description.
of the
when
no
plots.
Options
and
main
a file
options
displayed
Selecting
name.
The
4 through
at
listing
and
option
that
current
6; i.e.
4 results
option
list
of
3,
available
"Create
plots:
plot",
resolution
high
or
resolution.
menu.
is
chosen,
MINSUM
displays
The
is
in
device.
be prompted
recall
saved
Set to high or low resolution.
Set display
on or off.
Create
plot.
Display
current
picture.
Save current
picture
on file.
Load and display
saved picture.
Exit.
When
2
display
or resolution.
high
resolution
terminal
displayed
on the
appropriate
to be stored.
The
user
will
Option
6 allows
the
user
to
supply
of
in
1
the
low
-40
1)
2)
3)
4)
5)
For
t1tle,
each
y vs. X
y vs. y
X vs. X
RESIDUALS
RESIDUALS
vs.
vs.
of
these
plots,
and the
codes
for
y
X
the
user
1s asked
to
marker
and
l1ne
types,
supply
as
ax1s
11sted
ranges
and labels,
1n Tables
5.1
to
a
5.3.
The
allows
more
explanatory
user
Y versus
than
var1able
chooses
repeated
to
plot
unt1l
whether
values
X plots
one
an
plot
on
X to
use
are
(1f
Y-observed.
ex1t
1s
probably
each
the
set
of
the
model
selected.
For
the
more
nunber
of
po1nts
to
than
one X 1s used).
spec1f1es
a
type
and
MINSUM
plot
also
pred1cted
(F1g.
1.1)
character
allows
curves
1n
Creat1on
the
of
or
the
can
user
be
a
user
(observed
user
may
or
also
pred1cted)
request
and the
ranges
that
a 45-degree
are
1n
X plots
F1nally.
elect
to
have
hor1zontal
resolut1on
user
may
l1ghtly
prepared
we
to
the
be
hor1zontal
each
l1ne.
the
res1dual
l1ne
code.
ed1t
the
us1ng
way.
of
res1dual
plott1ng.
Y-pred1cted.
po1nt
at
dec1des
connected
1s
the
so
that
h1gh
resolut1on
somewhat
hor1zontal
ax1s
For
h1gh
resolut1on
through
the
plot.
plots.
and
or
res1dual
Xthe
opt1on.
plot
for
same
l1ne
the
1s
user
po1nt.
th1s
Y-pred1cted
before
X).
values
for
the
case.
the
user
also
The
of the
axes.
l1ne
be drawn
the
one
cho1ce
the
current
var1able
creat1on
Y-observed.
the
have
vert1cal
much
norma11zed
plots.
opt1onally
dotted
cover
res1duals
var1able
h1gh
spec1fy
MINSUM
po1nts
or to
generate
second
opt1on
means
that
and set
e1ther
created
versus
The
than
plots.
that
wh1ch
Th1s
s1multaneously.
was
Y-observed
only
a
to
s1mpler.
versus
need
plot
In
plotted
1ntroduct1on
more
ex1t.
Y-pred1cted
spec1fy
Xs (1f
d1fferent
or
1n
select1ng
1nvolves
1n the
plotted
Choos1ng
the
user
must
rema1n1ng
plot
complex.
After
Y-pred1cted.
to
use
the
observed
X-values
evenly
across
the
range
of
X.
most
axes.
a
The
may
user
choose
part1cular
zero
to
1s
th1s
the
The
may
the
X.
drawn.
l1ne
For
and
w1th
the
a
X
-41
7.1
Prof11er
(7.1)
white
(7.2)
descr1pt1on
PJ.
the
step
P2 + (P2 -PI).
is
STEP.
set
equat
P2 -PI.
to
the
difference:
-
-42
(7.3)
t~~
-
-46
7.2.
Profiling
instructions
-
-47
8.
8.1.
to
It
reader.
the
may be
Don't
Suppose
COVARIANCES
the
(xl.XZ
parameter
11ke11hood
descr1bes
the
(8.1)
F(X)
c
1s
vector
to
for
est1mated.
Suppose
parameters.
of
.-109
then
The
be
the
probab111ty
the
L(X)
a constant
F.
previous
paragraph
contains
The practical
1mplications
terminology
can be stated
unknown
quite
xN)
funct10n
parameters).
m1n1m1zes
that
the
despair.
that
X.
where
CALCULATING
Theory
simply.
1s
-
+
(POSS1bly
(Th1s
observed
C
also
1s
data.
that
L(X)
1s
just
the
funct10n
g1ven
X.)
If
the
that
.
dependent
the
max1mum
11kel1hood
so.called
Hess1an
for
on
the
est1mate
F 1s the
d~ta.
but
not
on
the
X for
X 1s the
vector
that
NxN matr1x
of
second
part1al
der1vat1ves
(8.2)
H.
evaluated
at
asymptotic
inverse
(8.3)
((32
~.
(that
of
H.
V.
F(X)/(3X1
where
is.
that
H-I.
i
valid
is.
3Xj)]]
and
j
for
range
large
from
samples)
i
to
N.
According
covariance
to
matrix
the
theory.
V
for
X
the
is
the
-48
(8.4}
Fab
.F(xl+adl,x2+bd2)
where
a and
b
-.0.
and
on
values
+
of
-
can
take
the
F shown
the
left
values
side
-I.
of
0.
and
(8.4)).
diagramatically
+1
(with
We are
on
the
corresponding
particularly
xlx2-grid
subscripts
interested
in
the
below:
*
F--
In
terms
of
IFO-
these
values.
numer1cal
approx1mat1ons
to
the
second
der1vat1ves
are
(8.5a)
a2F/ax12.
(8.5b)
a2F/(ax1ax2)
(8.5c)
a2F/ax22.
S1nce
(x1
each
and
element
Xj).
F(X)
of
~
as
descr1bed
at
a
(Fo+
the
formulas
-2Foo
F_o)/d12
.
2Foo-F-o-Fo-+F__)/(2d1d2)
+
Hess1an
(8.5)
Schnute
(1983)
negat1ve
109
the
essent1allyequal
bu11d
general
S(X).
+
Fo-)/d22
p»tr1x
have
.
.
(8.2)
natural
1nvolves
only
extens1ons
to
two
the
der1vat1ves
general
case.
be
constant
-2Foo
.(F++-F+o-Fo++
of
the
N-d1mens10nal
(F+o
sum
above.
Schnute
to
tw1ce
software-wnrch
run
of
t1me.
squares
1nvest1gates
11ke11hood
the
and
res1duals.
1n
pract1cal
comput1ng
Fourn1er
negat1ve
allows
the
Furthermore.
of
a
and
log
user
some
In
f1sher1es
X and the
(1980)
s1m11arly
11ke11hood.
to
1nclude
problems
part1cular.
problem
covar1ance
use
by
a
lett1ng
p»tr1x
funct10n
so 1t
1s conven1ent
a mult1p11cat1ve
the
objective
function
the
object1ve
funct10n
to
1s
-49
takes
this
where
0
case,
the
form
is
an
n is
sample
L(X)
the
bias,
residuals
log
-log
where
the
parameter
negative
(8.6)
.n
of
regard
(8.8)
this
F(X)
represents
from
(8.8)
.(n-N)
the
that
of
in
and
because
the
8.2.
estimate
the
point
the
to
has
02,
X.
In
th1s
of
02,
adjusted
for
small
may
in
(8.6).
happen
but
a
(1984.
p.
946.
of
(8.7)
for
the
02
is
to
02.
then
$.
5.4;
by
the
eq.
function
have
and
will
be
not
the
the
objective
also
will
$(X)
affected
computing
is
The
see
objective
this
(8.8)
in
function
of
eq.
on
be used
user
multiple
between
based
actual
function
5.5)
provides
should
no effect
$(~).
be
on
Consequently.
correct
in
multiplicative
any
case.
constant.
operation
The
To
previous
cancel
been
the
of
will
that
is
then.
the
$;
however.
calculation
with
the
relationships
includes
value
5(~)]
In
accordance
investiqate
in
correct
it
constant
matrix
Software
used
be
the Hessian
of 5(X)
parameter
estimates.
McKinnell
expression
matrix
V
coefficients
the
var1ance
vector
S(X)/(2o2}
An
example.
Technically.
by a constant
to
obtain
covariance
Only
to
if
the
required
$chnute
(8.8)
grids
with
parameter
negative
log likelihood.
except
for a constant.
It follows
the sum of squares
function
5(X) must be adjusted
by the
squares
a practical
multiplied
be
normal
the
be
5(X)![2
Incidentally.
the
+
observations.
to
estimate
factor
(n-N)![2
5(~)]
covariance
matrix
of
A
to
is
log(2wo)
out
presumed
addit1on
~2 .S(X)/(n-N)
we
sum
are
1n
l1kel1hood
number
turns
(8.7)
If
when
extra
-
foutld.
(8.3)
of
and.
variation.
by
correlation
set
varying
up
section.
calculator
the
begins
covariance
covariance
on
The
theory.
the
among
parameter
the
usual
formulas.
and correlations
and
sizes.
above
which
base
by
matrix
grid
Hessian
is
computing
calculator
after
the
a
can
minimum
covariance
computes
standard
of the
estimates.
deviations.
Output
also
determinants.
calculation.
the
covariance
estimates
the
is
specified
user
must
calculated.
using
supply
as
the
three
described
current
point
in
and
-50
step
((X1,X2)
three
user
gr1ds
for
and
(d1,d2),
spec1f1ed
actual
respect1vely,
factors
use.
The
user
values,
0.010,
and
1s then
asked
to
and 0.001,
1f
needed.
values
unless
select
constant.
an
a
For
the
user
1)
2)
previous
will
1s
explained
factor
with
receive
the
(8.4)).
to
Th1s
(a,b)
allowed
revise
The
analys1s
reasons
adjustment
If
MINFUN
1n
(analogous
to
1n
gr1d
1s
to
obta1n
(8.4))
ed1t
both
the
scaled
po1nt
and
the
default
sca11ng
factors
user
w1ll
probably
not
want
to
w1th
the
defaults
results.
the
end
of
at
gave
section
poor
8.1,
for
the
a IJFUN is
objective
function,
being
used
to
define
following
choices:
of
the
called
the
by
three
step
0.10,
change
user
these
may
the
user's
also
covariance
function,
Constant.
I
User chosen
constant
The user should
enter
whatever
constant
would be needed to make the object1ve
funct1on
be the negat1ve
log l1ke11hood.
The default
value
1n th1s case 1s
cho1ce
1.
If MINSUM w1th a TEMPLATE 1s used the user w11l also
rece1ve
the
cho1ces:
3)
4)
In
Constant.
Constant
0.5
.user
accordance
of
the
calculated
the
objective
squares.
the
estimate
to
to
have
the
run
the
before
SIGMA ** (-2)
chosen
constant
(8.7)-(8.8),
automat1cally
that
*
for
explained
of
02
may
in
0.5
default
the
function
As
*
is
actually
4
be
last
(-2)
choice
Choice
3;
allows
of
times
and
user
if
all
the
can
assume
because
of
the
scale;
the
main
(statistical
minimum
only
SIMPLEX
a
scale
8.1,
required
this
are
completed,
the
or a file.
The
user
for
TEMPLATE
or NF--l
menu.
of
the
grids
~
give
we
use
based
three
can
arise
encompassed.
numerical
estimate
determine
the
of
sum
effects
true
more
(8.5)
minimum.)
same
results.
sensitive
remedy
then
the
can
occur
to
the
choice
deviations
so small
that
the
"minimimum
apply
vo.l\A.~.elT
than
one.
results
standard
To
or
may elect
also
elect
UFUN)
cc~('e-\0..~V\~
the
are
Negative
precisely
~I(
and
ess
Conflicting
on
grids.
user
may
for
a~~.0\IA."\;.L
ately
when
the
grid
is
(Remenber
that
the
minimum
approxi
are
positive
successful.
calculations
why
nonsense)
is
not
the
actual
found"
is
this
slightly
situation.
larger
grid
factors.
The
of
two
standard
deviations
the
analysts
was
is
point
either
least
difference
this
1s
possib1lity
the
section
~
at
constant
the
only.
to
If
the
for
a constant
paragraph
When the
covariance
calculations
results
printed
to
the
screen
final
summary
routine
(SUMMAR
returning
SIGMA **
here
user.
the
*
a subset
of
calculator
also
the
parameters.
prescr1bed
values.
remain1ng
parameters
calculator
matr1x.
1s
standard
w1th
the
would
entered
at
dev1at1ons.
supports
Th1s
1nvest1gat1on
would
beg1n
corresponding
then
this
be
po1nt.
etc..
1nitial
est1mated
1t
only
by
w1ll
on
by
of
the
f1x1ng
steps
to
m1f1;mizat1on.
automat1cally
the
set
covar1ance
matr1x
some parmeters
at
par.lmeters
zero.
If
report
allowed
The
the
a
to
covariance
vary.
~
~i
-51
9.
Hopefully.
parameteric
TROUBLE
this
models
ease
PROMISED
to
THE
package
relative
WE NEVER
Although
we've
tr1ed
have
some understand1ng
SHOOTING
SIMPLEX
with
-
YOu
will
and
A
SIMPLEX
SEARCH
enable
users
confidence.
ROSE
fact
1s that
We hope
expla1ns
software
frustrat1on.
th1ngs
clearly
enough
to
allow
you.
dear
reader.
knowledgeably.
and thus
avo1d
a certa1n
amount
This
sect1on
deals
w1th
some of
the
hazards
1nvolved
1n
9.1.
Mult1ple
because
the
exist
The
of
the
the
plot
should
this
sort
search
user
should
feature
the
of
the
inappropriate
model.
why
this
the
ambiguity
any
remaining
minima
exists.
minima
and
attempt
the
model.
and residuals.
should
manual
If
a
by
regard;
pattern
(as
some
trying
the
some
plotting
proposed
minimum.
predicted
to
a
line
uniform
random
fit
in
understand
parameter
by
searches
"pockets"
incorrect
values.
or
the
minima
and the
an
model
biologically
estimates,
Then
applying
with
multiple
at the
model
the
the
ambiguity
to
best
parameter
Again,
an
final
other
the
of
indicate
eliminate
and
that
opposed
indicative
from
problems
hard
look
initial
accepting
this
determine
in model
building,
global
minimum.
The
the
first
look
at
the
knowledge
outside
persist,
data,
with
it
features
can
be
crashes.
perhaps
an
useful
may be
eye
for
Overflows
common
is
crash
long
minima
benefit
to
towards
studying
a
is
unrealistic
model.
and take
revising
the
data
any
plot
To
with
of the
mathematical
time
to sit
down
9.2.
user
th1s
to
apply
the
gr1ef
and
and p1tfalls
of
indicate
before
in
and
often
reduced
searches
them
data,
can
be
the
useful
I!IJltiple
user
eliminate
can
quite
residual
Essentia"y,
and
an obvious
difficulty
minimum
point
as
If
investigate
also
observed
in
error
option.
is
distribution)
itself,
the
that
est1mat1on.
minima
represent
may accept
a local
wide
fit
nonlinear
remember:
mi n1ma
Multiple
user
occurrence
using
build
please
GARDEN!
keep
th1ngs
s1mple.
the
of
the
processes
1nvolved.
non-l1near
to
but
Of
all
an
overflow
can
search
the
problems
be
particularly
or
profile
in
the
leading
math
frustrating
calculation.
to
program
library
during
if
the
a
user
function
has
the
nDst
evaluation.
just
waded
Such
through
a
-52
9.3.
Slow
convergence
.i?';~~,
-53
10.
10.1.
Chi-square
and
An
limited
to
discussed
in
least
feature
sum
the
ADDITIONAL
weighted
important
the
of
-
squares
of
squares
POSSIBILITIES
the
TEMPLATE
objective
introduction.
For
concept
function,2used
example,
the
X
is
that
in
the
it
is
not
example
statistic
can
be
used
by
letting
TERM.
(10.1)
where
YO and
respectively.
predicted
YP are
Thus,
and
(10.2)
obtain
the
the
values
parameters
least
on
the
parameter
one
might
vector
PARS.
observed
and
frequencies,
PRED would
be
Similarly,
following
a
let
,
log
likelihood
function
TERM
YO and
AUX.
and
squares.
(10.3)
one
of
weight1ng
parameter
used.
1ncluded
1n
10.2.
the
one
select
for
the
Holding
It
predicted
frequencies
set
simplex
parameters
is
easily.
the
and
*
observed
data.
parameters.
of
INITM
by
MINSUM
the
data
observed
the
can
is
object1ve
particular
and
auxiliary
also
handle
actua"y
could
used
function.
explanatory
weight1ng
AUX(l)
The
schemes.
1nclude
sett1ng
1nterest
to
values.
in1tial
simplices
The
f1nal
the
on
XOs.
weighted
code
as
a
aux1liary
variable
both
is
could
allowing
the
f1xed
study
SIMPLEX
user
accord1ngly.
the
prescr1bed
steps
to
to
lie
minimum
values
of
the
model
allows
values
zero.
Th1s
w1th
the
1n
the
rema1ning
to
do
th1s
1n1t1al
forces
1n a hyperplane
reflects
opt1mal
the
certa1n
user
the
ones.
po1nt.
in1t1al
w1th
the
values
so
be
fixed
include
subsequent
constant.
in
possible
and
great
S1mply
only
.
squares
which
two
prescr1bed
g1ven
PARS.
-YP)**2
scheme
correspond1ng
all
held
(YO
parameters
at
observed
var1ables
cons1der
often
not
the
definition
we1ghting
f1xed
depend
on
parameters
the
to
some
parameters
model
explanatory
wished
can
also
with1n
a sum of
1s used
to
select
either
extremely
the
.XO(AUX(l»
factor
AUX(l)
If
YP but
Consider
TERM
Here.
free
observed
and expected
the
TEMPLATE
function
LOG(YO/YP)
negative
Since
and
as
case,
(1980),
YO *
YP
YP.
predicted
to
I
based
Fournier
TERM.
YO and
-YP)**2
interpreted
in
this
frequency
Schnute
to
(YO
preset
for
the
-54
10.3.
Impos1ng
that
constra1nts
At t1mes
the
s1mplex
search
make b1olog1cal
sense
-l1ke
don't
we1gh1ng
s1xteen
the
s1mplex
to
case,
the
pounds.
Normally
avo1d
unreasonable
user
example,
can
UFUN
IF
eas1ly
m1ght
The
(PARS(l).GT.LIMIT)
F .F
+ LARGE
important
thing
here
added
penalty
goes
not
be d1fferentiable.
gives
the
user
objective
10.4.
opt1on
function
the
is
zero
In
of
has
this
that
ffsherfes
add1ng
*
a
penalty
(PARS(l)-LIMIT)
objective
the
funct1on
approaches
parameter
penalty
when
research,
w1ll
cause
1s not
the
funct1on.
For
only
once,
**
2
F
1s
continuous
the
l1m1t),
1n the
TERM
at
the
end
(the
but
it
rout1ne
of
need
the
ICASE-NDAT.
one
and can
parameters,
be
often
encounters
estfmated
The
earlfer
by
models
lfnear
example
fn
which
some
regressfon,
gfven
(7.3),
repeated
here
convenfence:
a
(100-y)
.a+
feature.
Here.
bx1
if
a.
B
S.
and
than
the
full
this
technique
Y, rather
d1SCUSS
show
estimates.
that
it
leads
2a
+ cx2
S. and
(1984)
and
by
parameter
the
ICASE
by linear
regression.
simplex
search
needs
parameter
:1=.
the
can be estimated
UFUH.
then
the
models.
~
~.
THEN
CONSTANT
as the
MINSUM,
y
(10.4)
1ssue
topology
1f th1s
parameters
enter
lfne~rly
the
remafnfng
reader's
to
avo1d
par~meter
values
age or a salmon
smolt
code
add1ng
l1near
In
-
calculat1on
El1m1nat1ng
parameters
values
of
for
to
the
the
ENDIF
be told
negat1ve
the
object1ve
funct1on
parameter
values,
but,
force
1nclude
must
a
set
fully
to
+ dx1
yare
to
2B
+ ex2
given.
then
If
the
regression
deal
with
only
a
+ fx1
a.
the
B
x2
b.
c.
itself
three
of
nine
parameters.
Schnute
for
(lU.4)
and a more
general
efficient
and
robust
searches
.
d.
e,
and
is
included
parameters
and McKinnell
class
of
for
optimal
f
in
a.
-55
11.
VARIATIONS
Mead's
Numerous
articles
and
(1965)
paper.
suggesting
of
s1mplex
the
1nterested
improve
not
search
in
algorithm
provided
tried.
no
The
1n
th1s
alterations.
11.1.
Adjustable
the
Nelder
control
the
SIMPLEX
package
coeff1c1ent
Mead
of
the
uses
BETA.
suggested
by
the1r
for
a
Nelder
extens1on
and
Mead(1965)
each
11.2
Multiple
gives
problem
but
be
can
were
becau$e
they
1n the
cases
we
themselves
to
1.
a
Nash(1979).
cho1ce
the
of
2.
These
Walmsley
a
start1ng
and
was
he
one quarter
the
for
th1s
package.
po1nt
d1fference
values
(1981).
contract1on.
po1nt
would
move
only
l1m1ted
test1ng
done
the
Th1s
contract1on
GAMMA.
drast1c
to
contract1ons.
ALPHA.
too
overall
greatly
affected
1ns1gn1f1cant.
so
we
extensions
Walmsley
also
advocates
cont1nu1ng
funct1on
values
are
obta1ned.
not
accepted,
result1ng
1n a
accepted,
the
updated
several
1terat1ons
to
from
th1s
mod1f1cat1on.
Mult1-po1nt
Evans
by
s1mplex
reshape
the
extens1on
Unfortunately,
wasted
funct1on
1s
d1storted
1tself.
(long
We found
process
the
second
evaluat1on.
and
no
narrow)
benef1t,
as
extens1on
Also,
and
and
long
as
1s
when
1t
requ1res
some
loss,
reflect1ons
and
modified
Nelder-Mead
are
reflected
through
1dentif1ed
tney
coeff1c1ents
and
coeff1cient
and
that
BETA was super1or.
BETA at 0.5.
adjustable
coeff1c1ent
an
we
greatest
m1ght
1f
suggest1ons
usually
expans1ons.
and
BETA-0.5
which
kept
11.3
see
These
reasons.
contemplated
reflect1on
0.5.
that
(1965)
BETA-O.75.
so the
contracted
towards
the
centro1d.
In the
1s
app11cat1on.
reflections.
suggests
lower
often
exper1ence
to
(or
even
h1ndered
progress)
that
certa1n
problems
lend
proposes
distance
for
programm1ng
suggest1ons
var1ous
however.
found
w1th
follow1ng
1n
SEARCH
coeff1cients
and
s1zes
SIMPLEX
have
been
wr1tten
since
Nelder
and
methods
for
1mprov1ng
the
performance
Users
of
package
action
THE
reports
various
cons1stent
improvement
user.
however.
may f1nd
these
are
some
performance
implemented
OF
algor1thm.
try1ng
-
Craig
determ1n1ng
d1fference
(1978)
pub1ished
algorithm
in
the
centroid
in
wh1ch
mean
an
interesting
which
all
the
of
the
"best"
group1ng
funct1on
of
values.
h1gh
In
paper
"worst"
po1nts.
and
our
low
points
The
po1nts
exper1ence.
proposing
of
two
a
the
simplex
groups
are
g1ves
th1s
the
method
-56
f1nds
the
min1mum
with
nuch
fewer
but
w1th
about
the
sane
number
of
algor1thm
can
Users
qu1te
fasc1n~~1ng;
be
found
are
1n
warned
don't
1terat1ons
funct1on
COMMO2.FOR.
that
lose
wh1ch
f1ddl1ng
s1ght
than
does
evaluat1ons.
of
can
w1th
the
s1mply
the
or1g1nal
the
Nelder-Mead
The code
for
replace
s1mplex
problem!
nethod.
th1s
COMMON.FOR.
algor1thm
can
become
-57
12.
$ ~SIMPLEX:GOMINSUM
would
all
result
MINSUM
squares
(F4.0.T7.FlO.0)
1
7.36
2
14.33
21.86
4
27.61
5
31.59
6
35.38
7
39.02
8
41.19
9
10
43.89
45.08
11
47.41
12
48.95
13
50.14
J4
51.79
15
lli
51.77
!\41~
EXAMPLE
VONB
the
functions
objective
1
3
in
WORKED
executable
applied
function.
file
to
the
VONB.EXE.
von
which
Bertalanffy
is
curve
capable
(1.1)
of
performing
and
a
Sum
of
-58
Both
section
progr~s
consists
responses
in column
and
of
appear
one.
COMPUTER
an
data
are
annotated
in thc
left
column.
Notes
and comments
INPUT
AND
now
ex~ple
with
appear
ready
to
run.
The
go.
The
renalnder
of
this
program
prompts
and
user
user
responses
in t~e
right
OUTPUT
denoted
by
hand
column.
COMMENTS
S RUN VONB
FUNCTION
N,A1'1E OF
DATA
MINIMIZER
FILE?
Entel'
IMUSSELS.DAT
ALGORITHM
DATA
run
First
run
accept
DATA SET TO DEFAUlT
SIMPLEX
for
so
defaults
VAlUES.
MENU
Main
menu
for
SIMPLEX
Re-1nft1a11ze
la)
New
Ib 2
this
data
TO
I<CR>
AlGORITHM
of
prepared
FILE?
ENTER FILE
NAME OR <CR>
ACCEPT
DEFAULTS.
I)
n~e
file
!
data
Edft,
Model
5
MINIMIZE.
7
Functfon
8)
Predfctfon
Load,
parameters 4)
6)
Covarfance
II)
Search
12)
Quft.
Save
User
call
10)
ENTER
only
3)
and
or
only. Alg
data.
summary.
at
current
low.
9)
Proffle
plots.
matrfx.
Globally.
CODE:
Try
a global
search
to get
an idea
of
the
III
GLOBAL
SEARCH
par~eter
SET SEARCH BOUNDS:
Search
within
values
is done
this
range
CHOOSE:
1)
2)
3)
Review
Set
Take
current
bounds.
bounds.
default
bounds
of
POINT +-
STEP.
I
Displays
I
for
I
Allows
I
specify
I
Gives
I
val
max
all
user
max
default
ues
and
mln
par~s
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&
min
bound
a I
symbol
-59
4)
Edit
ENTER
12
Enter
POINT
COOE:
,nd
(
STEP with
1-
4
or
-
editor.
<CR>
if
I
Allows
!
defaults
value
for
range
of
to
reset
finish~d.)1
I
low
user
parCJneter
1
140
I
Set
mln
Enter
and
max
values
I
Enter
high
value
for
range
of
parCJneter
11
160
I
Enter
low
value
for
range
of
parCJneter
2
#.1
I
I
Enter
high
value
for
range
of
parCJneter
21
11
!
Enter
low
value
for
range
of
parCJneter
3
10
!
!
Enter
high
value
for
range
of
parCJneter
31
#.5
I
CHOOSE:
1)
2)
3)
4)
Review current
bounds.
Set bounds.
Take default
bounds of POINT +Edit
POINT and STEP with
editor.
STEP.
i
!
I
I
I
ENTER CODE: ( 1 -4
or
I<CR>
Enter
Q. the ;1il6nber of
Default
of Q = 10 * N.
164
Enter
Enter
I<CR>
Select
Enter
#6
file
narle to write
<CR> if no output
Random
R or G
or
Grid
<CR>
points
to.
file
If
flnished.)1
to
Is
search:
search:
desired.
I
I
I
I
I
I
I
I
I
I
Bounds
are
4x4x4=64
grid
No
451.86045
906.05667
2311.9458
4212.6078
228.90367
1496.4592
3565.6761
1423.8246
361.64637
2504.2889
1135.2687
1896.5065
850.0885'~
733.08140
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
44.000000
48.000000
52.000000
56.000000
48.000000
52.000000
56.000000
44.000000
52.000000
56.000000
44.000000
48.000000
56.000000
44.000000
gives
nice
values
output
file
desired
Request
I
I output
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
set
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
a grid
looks
search
like:
0.10000000
0.20000000
0.30000000
0.40000000
0.l0000000
0.20000000
0.30000000
0.40000000
0.l0000000
0.200JOO00
0.30000000
0.40000000
0.10000000
0.20000000
60
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
VAL:
1501.8100
2826.1009
443.75653
825.54508
2172.9144
4804.7660
201.33166
1388.4657
3393.2262
1732.6273
311.48438
2365.6177
1053.3355
2290.2783
774.21468
676.85592
1392.9508
3314.3245
440.00958
749.95912
2598.8948
4608.6269
178.43574
1285.6173
3918.8406
1628.9343
266.30477
2232.2817
1308.7050
2158.8255
703.61666
625.30724
1728.8496
3152.0564
441.00752
990.54082
2454.4983
4410.4264
160.66312
1608.5078
3741.5148
1525.4549
226.61803
2647.0595
1220.8147
2026.9390
638.87225
793.15865
1614.2649
2988.5962
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
POINT:
VALUE ON ENTRY OF:
FOUND AT POINT:
1.00000
1.00000
48.000000
52.000000
44.000000
48.000000
52.000000
56.000000
48.000000
52.000000
56.000000
44.000000
52.000000
56.000000
44.000000
48.000000
56.000000
44.000000
48.000000
52.000000
44.000000
48.000000
52.000000
56.000000
48.000000
52.000000
56.000000
44.000000
52.000000
56.000000
44.000000
48.000000
56.000000
44.000000
48.000000
52.000000
44.000000
48.000000
52.000000
56.000000
48.000000
52.000000
56.000000
44.000000
52.000000
56.000000
44.000000
48.000000
56.000000
44.000000
48.000000
52.000000
25207.363
1.00000
0.64000000
0.82000000
0.2800')000
0.460~O000
0.640~0000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.2bQ00000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64(100000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.82000000
0.28000000
0.46000000
0.64000000
0.R2000000
0.28000000
0.46000000
0.64000000
0.82000000
I
SOS
I
SIMPLEX
I
values
0.30000000
0.40000000
0.20000000
0.30000000
0.40000000
0.10000000
0.20000000
0.30000000
0.40000000
0.l0000000
0.20000000
0.30000000
0.40000000
0.10000000
0.20000000
0.30000000
0.40000000
0.10000000
0.30000000
0.40000000
0.l0000000
0.20000000
0.30000000
0.40000000
0.l0000000
0.20000000
0.30000000
0.40000000
0.10000000
0.20000000
0.30000000
0.40000000
0.10000000
0.20000000
0.40000000
0.l0000000
0.20000000
0.30000000
0.40000000
0.10000000
0.20000000
0.30000000
0.40000000
0.la0000aa
0.20000000
0.30000000
0.40000000
0.l0000000
0.20000000
0.30000000
mlnlmun
using
default
-61
WHILE SEARCH GIVES
AT POINT:
48.0000
0.28000
VALUE OF:
-
160.663
0.40000
Note
improvement-
this
gives
a
starting
ACCEPT
NEW POINT
FROM
SEARCH?
«V>,
N)
Point
IV
found
search
point
Repeat
IN
with
same
bounds1
Is
SIMPLEX
In
Choose
al gorl
POINT:
INITIAL
1)
0.280000
3)
2)
O.100000
3)
0.400000
REQUIRED
SIMPLEX
0.l00000
OF
TF)
TERMINAL
FF)
FILE
WRITE
SR)
STEP
REDUCTION
lists
the
the
data
!
LIMIT.
I
REL.
MAX.
edit
data
!
0.10000000E-05
MF)
menu
I
STEP:
0.l00000
SL)
2)
to
ttJn
algorithm
I
48.00000
majn
Editor
.
1)
main
above
3 VARIABLE(S)
IS)
SIMPLEX
MENU
ENTER CODE:
'2
INITIAL
global
CURRENT
SIMPLEX
to
SIMPLEX
IP)
In
new
(V/<N»:
Return
as
likely
~Int
400
I
FUNCTION
CALLS
I
I
DISPLAY
FREQUENCY:
1
I
I
FREQUENCY:
0
I
I
FRACTION:
0.10000000
PARTICULAR
FIELD,
I
I
ENTER
CODE
TO
"AL"
TO
EDIT
<CR>
TO
SEE
OR
"Q"
TO
EDIT
ALL
FIELDS,
CURRENT
EXIT
"H"
FOR
I
HELP,
I
VALUES,
I
EDITOR
I
IIS
F1rst
I
<CR>
TO
ACCEPT
CURRENT
OR DEFAULT
CURRENT VALUE OF STEP'
NEW VALUE?
'10
CURRENT
VALUE
OF
1 :
VALUE.
0.10000000
fnitfal
!
!
!
STEP'
2
:
O.lO00OO00
Gives
50
+1-
10
for
! YINF
1
1
I<CR>
1.3
reasonable
steps.
is okay
!
r~EW VALUE?
CURRENT
VALUE
NEW VALUE?
set
.fnitfal
! pofnt
1
OF
STEP
1
3
:
Accept
+/0.1
default
for
K
0. 10000000
I
Gives
1+/-
TZERO
0.3
.0.25
of
0.25
.,
{~
-62
ENTER
CODE
TO
"AL"
TO
EDIT
<CR>
TO
SEE
OR
ISL
"Q"
<CR>
TO
TO
EDIT
ALL
CURRENT
EXIT
FOR
HELP,
VALUES,
CURRENT
SIMPLE
NEW TYPE?
I<CR>
FIELD,
"H"
EDITOR
ACCEPT
CURRENT
PARTICULAR
FIELDS,
-
LIMIT
(RLL
or
CURRENT SIMPLEX
NEW VALUE?
1.001
00
OCFAULT
TYPE
IS
1
Now
1
limit
I
Accept
I
relatIve
reset
SIMPLEX
VALUE.
RELATIVE
.
ABS):
LIMIT:
0. 10000000E-05
default
of
I
ENTER CODE TO EDIT PARTICULAR FIELD,
"AL" TO EDIT ALL FIELDS,
"H" FOR HELP,
<CR> TO SEE CURRENT VALUES,
OR "Q" TO EXIT EDITOR
I<CR>
I
Set
I
lE-3
SIMPLEX
limit
at
I
I
I
I
Review
current
values
3 VARIABLE(S)
IP
} INITIAL
1
IS
1)
SL)
POINT:
48.00000
2) 0.280000
INITIAL STEP:
l0.00000
2) 0.100000
REQUIRED SIMPLEX LIMIT
0.10000000E-02
ABS.
3) 0.400000
3) 0.300000
.
I
I
.
1
MF) MAX. OF
400 FUNCTION CALLS
I
TF) TERMINAL DISPLAY FREQUENCY:
1
FF) FILE WRITE FREQUENCY:
0
SR)
STEP
REDUCTION
FRACTION:
0.10000000
PARTICULAR
FIELD.
1
1
1
ENTER
CODE
"ALM
TO
<CR>
TO
OR "QM
#Q
TO
EDIT
SEE
TO
EDIT
ALL
FIELDS.
CURRENT
EXIT
"H"
FOR
HELP.
VALUES.
1
EDITOR
SIMPLEX
1
1
1
1 Values
are
1 return
to
okay
main
-menu
MENU
SIMPLEX
main
menu
1
Attempt
m1n1m1zat1on
1
Initial
IX>lnt
as above
ENTER CODE:
15
I'tITIAL
POINT:
48.000000
1
0.28000000
0.400000001
In
I
global
search
as
found
,
-63
INITIAL
FUNCTION
INITIAL
HIGH:
INITIAL
VAL:
-
160.66312
FunctIon
InItIal
974.86160
Worst
value
poInt
value
In
constructed
LOW:
143.03085
Best
SIMPLEX
value
In
constructed
First
6
0
CALLS,
VALUES
CURRENT
1
ITERS,
IMPROVED.
MINIMUM
0
RESTARTS.
ACTION
ACCEPTED:
FOUND
48.000000
CURRENT
MINIMUM:
CALLS.
0.28000000
143.03086
2
ITERS.
RESTARTS.
2 VALUES
IMPROVED.
ACTION
CURRENT
MINIMUM
FOUND AT:
ACCEPTED:
48.000000
CONTRACTION
143.03086
Second
9
CALLS.
VALUES
CURRENT
3
MINIMUM
51.055556
CURRENT
ITERS.
IMPROVED.
FOUND
0
RESTARTS.
ACTION
MAXIMUM:
ACCEPTED:
375.89989
Iteration
CONTRACTION
AT:
0.27351852
MINIMUM:
iteration:
0.70000001
CURRENT
Third
0
441.70366
REFLECTION
0.28000000
MINIMUM:
iteration:
0.70000001
CURRENT
MAXIMUM:
0
CURRENT
SIMPLEX
AT:
I
7
126.60105
0.51388889
CURRENT
MAXIMUM:
182.53134
~
0
81 CALLS,
42 ITERS,
0
VALUES
IMPROVED.
ACTION
CURRENT
MINIMUM
57.291145
CURRENT
MINIMUM:
FOUND
at
RESTARTS.
ACCEPTED:
I
Max
I
close,
I
be
and
min
so
near
are
we
the
very
should
minimum
CONTRACTION
AT:
0.16441514
3.9799164
0.15506405
CURRENT
MAXIMUM:
3.9807709
-65
J)
RESET
HIGH
-
OR LOW RE5ULUTION
CURRENTLY-LOW
Z)
RESET
DISPLAY
ON OR OFF
CURRENTLY-ON
3)
RESET
LOW RESOLUTION
CURRENT
~
4
CREATE
HEIGHT-
SIZE
20.
WIDTH-
5
6
SAVE CURRENT
PICTURE
ON FILE
DISPLAY
CURRENT
PICTURE
7)
LOAD
8)
<EXIT>
ENTER
60
PlOT
AND
DISPLAY
A
SAVED
PLOT
L'ODE :
14
PLOT TYPES
CHOOSE
ONE
1)
2)
3)
4)
5)
6)
OF
THE
Menu for
various
plot s a-t a 11 ab 1e
FOLLOWING
y VS. X
y VS. y
X VS. X
RESIDUALS
RESIDUALS
<EXIT>
VS.
VS.
OPTIONS:
y
X
ENTER CODE:
11
Plot
Y observed
predicted
SET
RANGES
FOR
X OATA:
Select
Default
COMPUTED MIN1.0000000
COMPUTED MAX16.000000
DO YOU WISH TO CHANGE THESE VALUES1
max
&
on
X
range
of X
Is mln
and
of
X val
ues
Y,<N>1
IY
PLEASE E~TER MINIMUM
Reset
axis
range
is
for
nice
10
PLEASE
ENTER
MAXIMUM
120
SET RANGES FOR Y DATA:
COMPUTED MIN7.3600001
COMPUTED MAX54.160000
DO YOU WISH TO CHANGE THESE VALUES1
Y,<N>1
IY
PLEASE
ENTER
MINIMUM
ENTER
~AXIMUM
ENTER
THE
I
Range
I
to
I
I
Defaults
predicted
I
observed
now
fran
0
20
1 pred
of
mfn
and
and
and
Reset
for
Range
is
max
of
of
obs
nicer
axis
10
PLEASE
1100
PLEASE
PLOT
TITLE
to
100
now
from
0
-66
laBs.
AND
PRED.
LENGTH
ENTER THE LABEL
AGE IN YEARS
FOR
,
ENTER THE
LENGTH
IN
FOR
I
CHOOSE
OR EXIT
y
LABEL
MM
AT
THE
THE
PREOICTED,
y
TO MAIN MENU
AGE
FaR
-
FRESHWATER
HORIZONTAL
VERTICAL
AXIS
MUSSELS
Enter
title
and
labels
AXIS
OBSERVED,
P,0,<E>1
10
CHOOSE
PLOT
CHARACTER
PLOT
TYPE
<*>
I
Plot
I
first
I<CR>
CHOOSE
«1>-6)
11
ENTER
STOP
CHARACTER
L
E
N
G
T
H
I
N
M
M
AND
PRED.
lengths
I
Oata
points
I
with
a"*"
marked
I
Scatter
clot
I
Default
of
(2-9,A-Z,a-<z»
I<CR>
OBS.
observed
LENGTH
AT
AGE
FOR
I
stop
I
Resultinn
z
for
charactEr
plot:
FRESHWATER
MUSSELS
!
801
+
6ol
*
+
*****
40+
* * *
I
* *
.,.
* *
20+
*
! * *
-+--+-+--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+~-+-+
5
10
15
AGE
IN
YEARS
1
PRESS <CR> TO CONTINUE
I<CR>
CHOOSE Y PREDICTED. YOBSERVED.
OR EXIT TO MAIN MENU
1
Pause
1
graph
to
inspect
1
1
P.O.<E>?
IP
PLOT PREDICTED POINTS AT DATA POINTS OR
DISTRIBUTED
THRDUGH RANGE? (P.<D»
I<CR>
HOW MANY POINTS ON THE PREDICTED CURVE?
< 50>
I<C?>
This
time
predicted
plot
points
Default
gives
points
Accept
default
many
-67
-
CHOOSE PLOT CHARACTER <*>
IP
Plotting
CHOOSE PLOT TYPE «1>-6)
I<CR>
ENTER STOP CHARACTER (2-9,A-Z,a-<z»
I<CR>
a
Scatter
I
OBS.
AND
PRED.
LENGTH
character
AT
AGE
FOR
Is
"P"
New
plot
plot
looks
FRESHWATER
like:
MUSSELS
!
L
E
N
G
T
H
801
+
6ol
+
40+
I
N
*
pp*p*pp*
pp*p*
1
20+
M
M
1
pppppppp
p*p*pp*p*pp*pppp
p*p*
p*p
p*
*P
P+--+-+--+-+--+-+--+-+--+-+-+--+-+--+-+--+-+--+-+
5
10
AGE
IN
15
YEARS
Fit
PRESS <CR> TO CONTINUE
I<CR>
DO YOU WANT A NEW PREDICTION7
appears
to
be
good
I
I
Allows
us
predicted
We don't
Y,<N>7
IN
to
get
new
curve
want
one
CHOOSE Y PREDICTED, Y OBSERVED,
OR EXIT TO MAIN MENU
I
P,O,<E>7
I<CR>
I
Exit
to
plotter
MaIn
menu
menu
I
PLOTTER
as
ENTER CODE:
14
PLOT TYPES
ME.NU
I
for
plotter
above
1
Create
1
try
1
1
plot:
residuals
this
time
(.:;
-68
CHOOSE
ONE
ENTER
Of
THE
FOLLOWING
1)
y
VS.
2)
y
VS.
y
3)
X
VS.
X
4)
RESIDUALS
VS.
y
5) RESIDUALS
6)
<EXIT>
CODE:
VS.
X
-
OPTIONS:
X
!
14
Call
up
residuals
vs
y
CHOOSE
a.
Y OBSERVED
OR <Y
PREDICTED>.
<P>1
I<CR>
Plot
SET
RANGES
FOR
. pred
1
Y DATA:
COMPUTED
MIN-
7.4308090
COMPUTED
MAX-
53.057774
DO YOU WISH
TO
CHANGE
1
THESE
VALUES?
1
1
1
IY
PLEASE
ENTER
MINIMUM
1
1
PLEASE
160
ENTER
MAXIMUM
1
1
10
YOU
Y,
WISH
TO
y
1
Y,<N>?
00
against
Icted
NORMALIZE
THE
RESIDUALS?
Reset
range
1
<N>?
1
,y
Look
at
nonnalized
residuals
SET RANGES FOR RESIDUALS :
COMPUTED MIN-1.7272640
COMPUTED MAX2.2032764
DO YOU WISH TO CHANGE THESE VALUES?
y.<N>?
IV
PLEASE
ENTER
MINIMUM
PLEASE
ENTER
MAXIMUM
PLEASE
ENTER
THE
1-2
'2
I
RESIDUALS
FOR
ENTER
LABEL
THE
I
PRED
LEN.
I
ENTER THE
RESIDUALS
CHOOSE
PLOT
LENGTH
.
I
TITLE
AT
AGE
FITTED
FOR
THE
HORIZONTAL
FOR
THE
VERTICAL
TO
VONB
AXIS
(MM)
LABEL
1
PLOi
CHARACTER
PLOT
TYPE
AXIS
<*>
1 Plot
(SCATTER-O
OR <COUNT-1>1
0,11
I<CR>
ENTER
I<CR>
1
1
1
I<CR>
CHOOSE
1
1
1
ST~P
CHARACTER
(2-9,A-Z,a-<z»
1
1
usIng
default
"*-
:..
-69
RESIDUALS
FOR
LENGTH
AT
-
AGE
FITTED
TO
VONB
2.0+
R
E
S
I
D
U
A
L
S
*
1
1.0+
1
.01
*
1
-1.0+
*
1
-2.01---+---+---+---+---+
10
+---+---+---+:--+---+30
40
20
PRED
LEN.
(MM)
PRESS <CR> TO CONTINUE
I<CR>
PLOTTER
as
50
Pause
MENU
above
ENTER CODE:
I<CR>
Exit
plotter
.
I
I
NOTE:
I
similar
I
different
The
to
1 course
have
I See Section
procedure
for
the
above
ccdes
for
access
5 for
creating
method,
plot
to
a
details.
types
high
high
except
and
resolution
resolution
that
the
characters,
plots
user
and
plotting
Is
specifies
must
of
terminal.
I
SIMPLEX
as
ENTER CODE:
110
MENU
SIMPLEX
main
Look
par~eter
above
at
covariances
COVARIANCE
CALCULATOR.
menu
-70
Three
MgrldsM
are
used
Information,
the base
using
the current
point
scale
factors
must be
Initial
steps
will
be
scale
factors
to adjust
grid.
Please
edit
the
(If
desired).
to
calculate
-
the!
Instructions
grid
being
formed!
and step.
Three
supplied,
and the!
multiplied
by these!
the size
of the!
point
and step
first!
!
3 VARIABLE(S)
IP)
INITIAL
1) 57.29114
IS) INITIAL
1) 10.00000
and
Information
3 grids
are used
because
the method
used to calc
the
covarlances
Is
! affected
by scale.
If
I
!
Is
two grids
agree ~
assume that
answer
!. correct
POINT:
2) 0.164415
STEP:
2) 0.100000
3)
0.155064
3)
0.300000
Edit
used
point
and step
to build
base
grid
ENTER CODE TO EDIT PARTICULAR FIELD,
MALM TO EDIT ALL FIELDS,
"ti"
FOR HELP,
<CR> TO SEE CURRENT VALUES, OR MQ" TO
EXIT EDITOR
IQ
We don't
PDINT:
1) 57.29114
2)
0.164415
3)
0.155064
INITIAL
STEP:
1) 10.00000
2)
0.100000
3)
0.300000
0.99E-02
3)
0.10E-02
GRID
1)
FACTORS:
0.100000
INPUT
PRESS
I<CR>
INPUT
PRESS
I<CR>
INPUT
PRESS
I<CR>
2)
NEW VALUE OF FACTOR'
2 (>
RETURN TO ACCEPT DEFAULT
Default
yr1d
factors
grid
size
defaults
NEW VALUE OF FACTOR I
3 (> 0)
RETURN TO ACCEPT DEFAULT
3) 0.10£-02
ok?
Factors
choose
(inverse
this
values
Accept
0)
.
Please
to
these
NEW VALUE OF FACTOR I
1 (> 0)
RETURN TO ACCEPT DEFAULT
New values for factors:
1) 0.100000
2) 0.99£-02
Factors
<Y>/N
I<CR>
want
change
of
number
correction
Hessian
to
obtain
constant
will
be
Covariance
scaled
1
by
matrix).!
1
are
okay
-71
-
CHOOSE:
1)
Constant.
2)
3)
function
is
log
likeljhood
User
chosen
constant
Constant.
0.5
* SIGMA
Use
4)
1
if
;
if
minimizing
on
Constant.
user
0.5
* SIGMA
**
ENTER
objective
**
sum
chosen
(-2).
(-2);
of
squares.
constant
*
1.2.<3>.4:
'<CR>
This
COVARIANCE
OBTAINED
AND
CONSTANT.
FROM MINIMUM
ESTIMATED
Calculating
AVERAGE
STD DEV
Now
1.63320013
STAN.
F on
F:
F:
OF
3.97991640
DEV.
grid'
Slln
of
problem,
take
OF 0.91844225
1
Crunching
18.137886
14.820796
calculating
Is
squares
default
Hessian'
1
Inverting
Hessian'
1
DETERMINANT:
0.1045E+09
Calculating
F on grid
12
AVERAGE F:
4.1211892
STD DEV F:
0.14794104
Now calculat;ng
Hessian
I
Inverting
Hessian
I
2
D~TERMlNANT:
O.9460E+O8
Calculating
F on grid
I 3
AVERAGE F:
3.9813291
STD DEV F:
O.16608362E-O2
Now calculating
Hessian
#
Inverting
Hessian
I
3
DETERMINANT:
O.9450E+08
1
2
1
I
I
3
1
1
I
1
I
Calcu13tions
Ddisplay
Fprint
R -run
Q -quit
canpleted.
Enter
results
on screen,
results
to
file,
user
and
final
return
routine,
to
main
choice:
or
merlu.
Choice:
ID
I
I
Displayed
I
results
I
and
1
close.
I
these
I
values.
I
with
I
.01
below.
fran
3
are
so
accept
correct
grid
check
OUTPUT FROMCOVARIANCEMATRIX FROM SIMPLEX MINIMIZER
POINT
1)
57.291145
2) 0.16441514
STEP FOR DIFFERENCING
1)
10.000000
2) 0.10000000
~
the
We
to
3) 0.15506405
3) 0.30000001
2
fairly
as
a
The
grids
could
re-do
factor
val
of
ues
-72
COVARIANCE
GRID
CONSTANT
1
.100
,
SCALE
* 0.5
1.OOOOOOOOO
FACTOR:
2
.100E-01
FUNCTION
AVERAGE F:
STD DEV F:
COF VAR F:
18.13788632
14.82079593
0.81711814
PARI
1
2
3
.100E-O2
VALUES
4.121189160
0.1479410426
0.35897659E-01
STANDARD DEVIATIONS
0.6521245397
0.5801261418E-02
0.6949696734E-01
0.6236633402
0.5519759413E-02
0.6757293868E-01
PARI
1
2
3
* SIGMA**-2
3.981329099
O.1660836190E-02
O.41715622E-O3
0.6524315524
0.5804290173E-02
0.6951788838E-01
CO£FF ICIENTS OF VARIAl rION
0.10885859£-01
0.11382641E-01
0. 11388000E-01
0.33572087£-01
0.35284228E-01
0.35302650£-01
0.43577437
0.44818233
0.44831725
PARI,
PARI
1, 2
1, 3
2, 3
CORRELATIONS
-0.93158925
-0.59740172
0.77986622
-0.92494678
-0.56895712
0.76523097
-0.93165596
-0.59769399
0.78001727
PARI,
PARI
1, 1
1, 2
1, 3
2, 2
2, 3
3, 3
0.3889559619
-.3184103008£-02
-.2397742590£-01
0.3046774398£-04
0.2854207191£-03
0.4566102042£-02
COVARIANCES
0.4252664153
-.3524337152E-02
-.2707465094E-01
0.3365463404E-04
O.31441S7211E-03
0.48298~8469E-02
0.4256669306
-.3528089866E-02
-.2710880770E-01
0.3368978441E-04
0.3147385262E-03
0.4832736805E-02
COY MATRIX:
COR ~TRIX:
0.9569502553E-08
0.17684917
(:I:TERMINANTS
0.1057089487E-07
0.15292345
0.1058193995E-07
0.15268768
Calculations
D -display
F-
print
R-
run
Q -quit
Choice:
completed.
results
on
results
user
and
final
return
to
Enter
screen.
choice'
file.
routine.
to
or
main
menu.
IQ
F1n1shed.
return
SIMPLEX
SIMPLEX
MENU
SIMPLEX
as above.
ma1n
menu
to
"1.
-73
ENTER CODE:
19
'-
!
Create
profile
plots
I
PROFILER
I
Please
choose
profile
parameter:
11
Input
gives
point
function
Enter
low
57.291J45
v~lue:
value
of
3.9799164
search
high
value
of
along
Minimum
we
YINF
found
with
minimizer
range:
150
Enter
Profile
search
range.
!
Vary
I
70
YINF
fran
50
to
170
57.291145
50.000000
70.000000
Input
m!nimlJn
is at:
Low end of range
is:
High en<J of range
is:
Check
bounds
I
Range
of
profile
par~eter
okay?
«Y>.N)
I<CR>
How
many
points
fran
low
to
many
.points
fran
mlnlmlJn
mlnlmlJn?
I
110
How
I
IOkay
to
high?
110
I
Number
I
high
of
and
points
low
on
profiles
I
Initial
poInt
is:
57.291145
Minimizing
on point:
56.562031
Please
~it
alg data:
do not edit
.profile
para;neter.
STEP 1 will
be set
to
1
MlnlmlJ1l
1
SIMPLEX
point
from
1
2nd
point
of
1
edit
alg
data
1
minimizing
profile
before
on
it
zero.
Press
<CR> to
continue.
.I<CR>
1
1 SIMPLEX
3 VARIABLE(S)
IP) INITIAL
POINT:
1) 56.56203
2) 0.164415
3) 0.155064
IS) INITIAL
STEP:
1) 0.00E+OO
2) 0.100000
3) 0.300000
SL) REQUIREO SIMPLEX LIMIT
.
0.10000000E-02
ABS.
MF) MAX. OF 400 FUNCTION CALLS
TF) TERMINAL DISPLAY FREQUENCY:
10
FF) FILE WRITE FREQUENCY:
0
SR) STEP REDUCTION FRACTION:
0.10000000
ENTER OODE TO EDIT PARTICULAR FIELD.
.AL.
TO EDIT ALL FIELDS.
.H- FOR HELP.
<CR> TO SEE CURRENT VALUES.
OR .Q. TO EXIT EDITOR
ISL
Note
step
parameter
TO
ACCEPT
CURRENT
OR DEFAULT
VALUE.
for
is
profile
set
to
I
1
I
I
I
I
1
Raise
to
<CR>
editor
1
I
I
SIMPLEX
speed
up
limit
search
O
-74
CURRENT SIMPLE LIMIT
TYPE IS ABSOLUTE .
NEW TYPE? (REL nr ASS):
I<CR>
CURRENT SIMPLEX LIMIT:
0. 10000000E-02
NEW VALUE?
1.0001
ENTER EDIT CODE:
ITF
CURRENT TERMINAL OUTPUT FREQUENCY:
10
NEW VALUE?
10
ENTER
EDIT
CODE:
IQ
***
Calculating
68
CALLS.
MINIMUM
OF
56.562031
***
point
31
44
CALLS,
MINIMUM
Cf"
55.8329~6
I
ITERSt
2
**********..**1
a
RESTAR1S.
4.3987860
FOUND
0.17065154
0.20033245
Calculating
point
18
Suppress
output
I
I
ITERS,
AT:
3
~*************!
0
RESTARTS.
5.7301709
FOUND
0.17727307
0.24586886
AT:
I
Begin
I
profl
2nd
calculating
let
point
the
startlllg
at
1
1
1
1
!
!
Note
hEre:
Increase~
!
steps,
!
Increases
!
TZERO
YINF
In eve~
MI n 11:!i 11!71
and
also
K
and
respond
!
!
!
***
Calculatfng
48 CAlLS.
MIrlIMUM OF
50.729115
***
pofnt
20 ITERS.
58.430474
0.23545740
Calculating
45
CALLS.
MINIMUM
OF
50.000000
***
point
CALLS.
MINIMUM
58.562031
OF
1
1 We ,~sked for
1 pofnts...
I
onl y 10
**************!
0
RESTARTS.
I
I
but
75.553139
FOUND
I
calculates
odd
number
0.24556838
0.')4614626
I
of
points.
Note
val
I
of
YINF
and
half
of
19
#
0 RESTARTS.
FOUND AT:
0.51588916
1
11
Calculating
68
# 10 **************1
ITERS.
point
2
*~.************I
0
RESTARTS.
I
1 2nd
4.9965620
FOUND
I
0.15438531
0.71809889E-011
34
ITERS.
#
AT:
AT:
profiler
always
minimum
profile
ues
-75
...Calculating
point
,
3
-
**************1
I
***
Calculating
55
point'
1
1
50.378141
FOUND
1
0.973781E-01
-0.67002972
26
Of
7D.000000
ITERS.
0
CREATE PLOT FROM RESULTS?
IY
PLOT
1)
RESET
RESOLUTION
RESET
LOW RESOLUTION
CURRENT
HEIGHT-
3)
<CREATE
PLOT>
4)
DISPLAY
PLOT
5)
SAVE
PLOT
SAVE
PROFILE
AT:
Plot
SIZE
22.
WIDTH-
80
VALUES
PLOT
O~t1un
val ues
values
6 writes
and func
to file
param
Select
high
plot
Create
plot
res
MENU
above
ENTER CODE;
PROFILE
PARAMETER
IS NUMBER
1.
ENTER PARAMETER
NO. F~K HORIZONTAL
<
profiles
CURRENTLY-LOW
ENTER CODE:
11
'3
pt)int
<Y>.N}
) EXIT
as
Final
1
MENU
2)
)
.
**************1
RESTARTS.
CALLS.
MINIMUM
11
AXIS.
1>
I<CR>
ENTER
YINF
axis
PARAMETER
VERTICAL
I<CR>
AXIS,
OR RESPONSE
RESPONSE
IS
NO.
on horizontal
FOR
< 4>
SET RANGES FOR PARAMETER DATA:
COMPUTED MIN50.000000
COMPUTED MAX70.000000
DO YOU WISH TO CHANGE THESE VALUES?
Y,<N>?
I<CR>
Func val ues on
vertical
axis
YINF
okay
ax1s
range
1s
-76
SET RANGES FOR FUNCTION VALUES:
COMPUTED MIN3.9799163
COMPUTED MAX75.553139
DO YOU WISH TO CHANGE THESE VALUES?
Y.<N>1
IY
PLEASE ENTER MINIMUM
10
PLEASE ENTER MAXIMUM
1100
PLEASE ENTER THE PLOT TITLE
IPROFILE
ON YINF
PLEASE ENTER HORIZONTAL AXIS LABEL
IYINF
PLEASE ENTER VERTICAL AXIS LA~[L
ISUM OF SQUARES
CHOOSE PLOT TYPE «0>-6)
11
CHOOSE LINE TYPE (0-9
<0>-solid
line)
I<CR>
CHOOSE A <SMOOTH-O> OR JAGGED-1 LINE
0.11
I<CR>
PLOT
MENU
Reset
functIon
ax Is
range
1
Line
grJph
with
1
1
solid
line...
I
smoothed
I
See
I
of
results
in
Fig
introduction
as above
ENTER CODE:
13
I Now look
at relation
I between
YINF and K
PROFILE
ENTER
AXIS.
PARAMETER
PARAMETER
< 1>
IS
NO.
NUMBER
FOR
1.
HORIZONTAL
I<CR>
YINF
ENTER PARAMETER OR RESPONSE NO. FOR
VERTICAl
AXIS.
RESPONSE IS < 4>
'2
SET RANGES FOR PARAMETER DATA:
COMPUTED MIN50.000000
COMPUTED MAX70.000000
DO YOU WISH TO CHANGE THESE VALUES?
Y,<N>?
I<CR>
1
Par~eter
1
plotted
1
axis
1
YINF
on
horizontal
2
on
ranges
(K)
vertical
okay
1
FUNCTION VALUES:
0.97365782E-01
C(W-IPUTED MAX0.24556838
DO YOU WISH TO CHANGE THESE VALUES1
SET
RANGES
COMPUTED
FOR
MIN-
Y,<N>1
IY
PLEASE
Reset
ENTER
MINIMUM
ax Is
for
nicer
1.3
,.,~
-77
-
10
PLEASE
ENTER
MAXIMUM
1.4
I
PLEASE
ENTER THE PLOT TITLE
CORRELATION
OF YINF
AND K
#
YINF
PLEASE
ENTER
HORIZONTAL
PLEASE
ENTER
VERTICAL
CHOOSE
PLOT
TYPE
«0>-6)
CHOOSE
LINE
TYPE
(0-9
CHOOSE
A
AXIS
AXIS
LABEL
1
LABEL
IK
I
II
Line
<0>-solid
line)
OR JAGGED-1
LINE
10
graph
solid
<SMOOTH-O>
0.11
I<CR>
with
line...
smoothed
See
similar
plot
introduction.
PLOT
in
Fig.
1.6
MENU
as above
ENTER CODE:
17
Exit
SIMPLEX
as
plotter
MENU
above
ENTER CODE:
'9
Profiler
again.
time
PROFILER
a
this
slice
I
ENTER CODE TO EDIT PARTICULAR FIELD,
-AL. TO EDIT ALL FIELDS,
-H- FOR HELP,
<CR> TO SEE CURRENT VALUES,
OR -Q- TO EXIT EDITOR
IIS
<CR> TO ACCEPT CURRENT OR DEFAULT VALUE.
CURRENT
do
VALUE
OF
STEP
#
1
:
0.00000000
OF
STEP
#
2
:
0.10000000
I
Input
I
profIle
I
edIt
is
sMne
above.
as
for
untIl
phase
1
1
Set
non-prof11e
par~eter
steps
to
NEW VALUE7
#<CR>
CURRENT
VALUE
NEW VALUE7
Keep
prof11e
as 1s
par~eter
O
-78
10
-
1
CURRENT
VALUE
OF
STEP
1
3
:
0.30000001
Now
NEW VALUE?
1 And
EDIT
is
con;tant
1
10
ENTER
K
1
TZERO
is
constant
1
1
CODE:
ITF
1
<CR>
TO
CURRENT
ACCEPT
CURRENT
TERMINAL
OR DEFAULT
OUTPUT
VALUE.
I
10
1
1
FREQUENCY:
NEW VALUE?
1
10
1
ENTER
EDIT
CODE:
1
IQ
Suppress
output
1
***
Calculating
point
15 CALLS.
MINIMUM Cf"
56.562031
8.2943907
FOUND
0.16441514
0.15506405
***
1
Calculatfng
1
ITERS.
point
#
2
**************~
0
RESTARTS.
AT:
3 **************1
1
1
1
1
1
.1
.1
.1
***
Calculating
15
point
CALLS.
MINIMUM
1
OF
70.000000
CREATE
PlOT
111
RESTARTS.
K
1301.0090
FOUND
changed,
0.16441514
0.15506405
FROM
ITERS.
**************!
0
RESUlTS?
I<CR>
«Y>.N)
AT:
and
very
TZERO
have
minimlln
not
is
high
1
1 Create
plot
of
slfce
Select
high
resolution
Create
plot
1
PI.Or MENU
1
1
1
as
'1
above
ENTER CODE:
I
PLOT
as
ENTER CODE:
I<CR>
1
MENU
above
.,
~
-79
PROFILE
ENTER
AXIS.
PARAMETER
PARAMETER
< 1>
IS
NO.
NUMBER
FOR
1.
HORIZONTAL
11
ENTER
PARAMETER
VERTICAL
I<CR>
OR RESPONSE
AXIS.
-
RESPONSE
IS
NO.
I
YINF
I
axis
on
hor1zontal
FOR
< 4>
Function
values
(response)
SET RANGES FOR PARAMETER DATA:
COMPUTED MIN50.000000
COMPUTEO MAX70.000000
DO YOU WISH TO CHANGE THESE VALUES?
Y,<N>?
I<CR>
ENTER MINIMUM
10
PLEASE
ENTER
MAXIMUM
ENTER
THE
The higher
val ues are
! Interesting;
I SOS-100
PLEASE
I
PLOT
and
TITLE
SLICE
ON YINF.
ENTER
THE
LABEL
FOR
THE
HORIZONTAL
TZERO
fixed
THE
LABEL
FOR
THE
VERTICAl
at
!
!
SUM OF
SQUARES
CHOOSE
PLOT
TYPE
«0>-6)
CHOOSE
LINE
TYPE
(0-9
CHOOSE
A <SMOOTH-O>
AXIS
!
!
I
11
14
opt.!
AXIS
YINF
ENTER
I
K
slIce
not too
stop
at
!
1100
I
ax Is
1
1
1
1
SET RANGES FOR FUNCTION VALUES:
COMPUTED MIN3.9799163
COMPUTED MAX1301.0090
DO YOU WISH TO CHANGE THESE VALUES?
Y,<N>?
IY
PLEASE
on
vertical
!
<0>-solid
OR
JAGGED-l
line)
!
LINE
!
-
Line
graph
dashed
with
line,...
I
0,11
I<CR>
PLOT
ENTER CODE:
SIMPLEX
smoothed.
1
See
!
introduction,
MENU
I
I
as above
results
I
!
!
!
!
!
! Return
!
MENU
as above
'1
I
in
Fig.
to
SIMPLEX
1.3
main
-81
-
ACKN(J.ILEDGEMENTS
We would
Computer
Centre
SIMPLEX
package
incentive.
Most
of
the
interrupt
and
us
K1m
the
to
the
from
give
Pac1f1c
1ts
special
support.
software
capabilit1es
were
Hyatt.
benef1t
who
of
and
was
adv1ce
wr1tten
prov1ded
and
Laura
improvement.
grac1ously
h1s
critical
to
Stat1on.
beginn1ngs.
by
adv1ce
rece1ved
both
from
We extend
spec1al
thanks
John
Jensen.
suggest1ons
for
cred1t
B1ological
ear11est
technical
plott1ng
programm1ng
the
project.
Schwe1gert.
numerous
like
at
needed
to
by Donna
Marc
Hamer.
R1chards.
F1nally.
prov1ded
comments
for
He
Marc
and
to
Cra1g
some
on
McK1nnell.
Head
of
the
h1s
work
on
the
act1ve
prov1ded
much
produce
Sweeney.
We appreciate
useful
helped
we are
the
thoughout
Jake
SIMPLEX
and made
to
J1m Radz1ul
data.
and
product.
the
Marshall
Shelbourn.
test
grateful
sample
SIMPLEX
of
the
f1nal
wh1le
the
from
Mar1l,Yn
Clarke.
John
who
both
Sk1p
th1s
J1m
report.
also
gave
-82
REFERENCES
Evans.
J.W.
and
R.J.
Nelder-Mead
Kentucky.
Technical
Kalbfleisch.
J.
G.
Mathematics
York)
J.C.
and
A.
Report
130:
and
Dept.
of
Stuart.
and
748
Dept.
statistical
1979.
The
relationship.
Compact
inference
University
(lecture
University
advanced
4th
numerical
and
R.
theory
Edition.
methods
Halsted
Mead.
1965.
Computer
J.
0'Ne11l.
R. 1971.
Algor1thm
procedure.
Appl.
Stat.
R1cker,
W. E. 1975.
fish
populat1ons.
for
Press.
A
7:
of
notes
for
cf
Waterloo.
Waterloo.
of
statistics.
Vol.
MacMillan
Publishing
Co.
ccxnputers:
John
linear
Wiley
and
algebra
Sons
(New
simplex
AS 47, funct1on
20:
338-345.
function
us1ng
a simplex
of b1olog1cal
191: 382 p.
stat1st1cs
of
1980.
A new approach
to length-frequency
Can. J. F1sh.
Aquat.
Sc1. 37:1337-1351.
A new approach
J. F1sh.
Aquat.
and S. ~Kinnell.
surface
analysis.
for
m1n1m1zat1on
Computat1on
and 1nterpretation
Fish.
Res. Board Can. Bull.
and D. Fournier.
growth
stucture.
1983.
Can.
method
308-313.
Schnute,
J. 1982.
A manual
for
easy non11near
research
w1th 1nteract1ve
m1crocanputer
Aquat.
Sci.
1140:
xvi + 115p.
Schnute.
J..
response
a modified
Statistics.
227p.
J.A..
Schnute,
J.
method.
using
of
22p.
Statistics.
minimization.
minimization.
Schnute,
J.,
analysis:
maximization
p.
1979.
function
York):
Nelder.
and
Function
procedure.
(approx).
Inference
(New
Nash.
250p
M.
2.
1978.
search
Probability
233).
Ontario.
Kendall.
Craig.
simplex
parameter
programs.
est1mat1on
Can. Tech.
to estimating
populations
Sci.
40: 2153-2169.
1984.
Can.
A biologically
J. Fish.
Aquat.
by
the
in f1shery
Rep. F1sh.
renoval
meaningful
approach
Sci.
41:936-953.
Walmsley.
D.A. 1981.
The simplex
method
for
minimization
of
function.
Transport
and Road Research
Laboratory.
Dept.
and Dept.
of Transport.
Crowthorne.
Berkshire.
England.
Supplementary
Report
686: 9p.
to
a general
of Environment
TRRL
-83
APPENDIX
ALGORITHM
DATA
AUXILARY
PARAMETERS
perform
AXIAL
...controls
SEARCH
to
for
the
TERMS
minimization
procedure.
(TEMPLATE
simplex
search
a
number
of
of
the
N lowest
in
particular)
algorithm
to
points.
common
covariance
COMMOM.FOR.
simplex
search
...used
in
UFUN
and
high
point
In
to
finds
SIMPLEX.
points
subroutines
minimizer.
modified
CONSTRAINTS
of
function
the
containing
...similar
(1978)
of
values
editor.
COMM02.FOR
OF
a mlnimlln.
convergence.
value
...file
includes
user
l\11en
parameter
COMMON. FOR
operation
in
premature
...average
average
GlOSSARY
tasks.
...performed
check
CENTROID
the
...used
special
A.
-
of
to
the
both
MINSUM
calculator.
~xcept
refers
wide
minimizer
to
the
simplex.
and
MINFUN;
search.
uses
Evans
etc.
and
Craig
algorithm.
TEMPLATE
to
restrict
parameters
to
desired
values.
CONTRACTION
...Simplex
CONTRACTION
COEFF~t:IENT
centroid;
here
...dictates
set
at
CONVERGENCE
...occurs
some
limit
defined
COVARIANCE
MATRIX
EDITOR
ESTIMATES
to
...of
simplex
from
EXTENSION
GLOBAL
FIXED
the
grid
PARAMETERS
constant
as
HYPERSURFACE
INTERRUPT
...a
...on
minimization
with
C.
a
rest
p.)fnt
set
all
moves
points
of
routine;
of
minimize
moves
across
size
of
for
points.
toward
tt,e
the
the
par~eter
the
object
simplex
large
are
withfn
what
the
sample
estimates.
function;
centroid
to
point
twice
as
far
parameter
of
extension;
bounds,
here
GLOBAL
set
SEARCH
at
2.0.
performs
a
a minimun.
of
are
minimization,
where
scxne
parameters
user
to
are
held
minimized.
systems,
procedure
point
lowe..-
~oint.
multidimensional
some
of
data.
~ich
...variation
the
at
matrix
...dete~ines
search
high
covariance
algorithm
reflected
...Given
or
far
centroid
finds.
high
as
COEFFICIENT
SEARCH
the
par~eters
...Simplex
towards
value
covariance
algorithm
centroid
randcxn
frum
the
-edit-
search
EXTENSION
when
function
by the
user.
to
...used
how
in
0.5.
...output
approximation
moves
.surface..
the
to
make
capability
changes.
for
On
the
the
v ax,
Interrupt
Interrupt
the
Is
achieved
-84
LIKELIHOOD
"'
data,
the
given
estimates
are
negative
MINFUN
MINSUM
function
that
values
of
those
describes
the
~'.I
-
the
par5neters,
which
probability
MaxlmlJn
maximize
this
of
the
llkehood
function
observed
par5neter
(or
minimize
Its
logarithm),
...central
progr~
wtllch
minimizes
user's
...central
program
which
minimizes
the
UFUN.
sum
of
TERMs
In
the
user's
TEMPLATE.
MULTIPLE
MINIMA
...condition
points.
This
of
can
be
a
function
indicatve
of
having
a poor
several
model.
nolsy
local
minimum
data.
or
be
dealt
too
little
data.
NON-LINEAR
ESTIMATION
ordinary
OBJECT
FUNCTION
PARAMETERS
function
...constants
in
~ich
variables
the
object
of
to
...a
the
for
~ich
~
are
or
set
typically
typically
cannot
trying
with
part
of
if
the
the
are
user
by
a minimlJn.
for
user);
by
which
takes
on
auxiliary
dependent
estimated
routine
a paraneter
obtain
(except
by
Par~eters
function
to
unknown
known
function.
special
object
~ich
a model,
are
the
object
function.
FUNCTIONS
value
estimation
regression.
...the
par~eters,
PENALTY
...par~eter
linedr
minimizing
assigns
an
a high
undesirable
value.
POINT
...in
this
parameter
values
PROFILE
of
PLOT
other
object
...a
of
are
...The
REFLECTION
search
and
high
poInt
the
...If
as
one
chosen
algorithm
point,
space
to
defined
high
all
par~eter
is
optimally
can
so
find
and
varied
for
no
points
the
restarts
po1nt
ax1al
sImplex
how
search
of
that
better
are
by
N
low
and
par~eter.
points
between
toward
the
moved
far
fInds
1s
tun1ng
problems
as outlined
po1nt
rema1n1ng
s1mplex
a better
restarted
a new
the
reflect1ng;
procedure
1nd1cates
fInds
centroId
when
...several
contexts:
The minimization
method,
(1971).
algor1thm
the
...controls
h1gh
the
search,
few
correspond
values
search
across
COEFFICIENT
from
RESTARTS
N-dimensional
respectively.
function
simplex
the
in
points
simultaneously
sImplex
h1ghest
away
"low"
point.
REFLECTION
the
a point
and
function,
plot
centroid
minim\ln
SIMPLEX
1)
the
...The
the
usually
"High"
par~eters
REDUCTION
a
ma~l,
values.
here
search
set
m1n1mlln
fran
w1th
the
the
by Nelder
oppos1te
low
algor1tl1n
at
fran
poInts.
steps
1.0.
than
new
dId
the
po1nt.
algor1tl1n
and Mead
sImplex
More
data.
than
-85
2)
The
gecxnetrlc
to
3)
entire
SIMPLEX
This
LIMIT
high
STEP
actual
surface
and
...part
the
of
structure
~Ich
Is
including
software
manipulated
along
the
function
minimum.
package.
...the
low
-
value
defining
function
the
the
values
a1gorlthm
for
data.
and
maximum
documentation.
allowable
difference
between
convergence.
STEP
defines
the
Initial
size
of
the
si:rlplex.
STEP
SIZE
REDUCTON
dfstances
TEMPLATE
FRACTION
for
...the
the
file
axfal
fnto
...Multiplying
STEP
by
the
SSRF
gives
axfal
search.
which
users
can
fnsert
the
code
needed
for
applfcatfon.
UFUN
...the
user
routine
written
by
the
user
giving
the
object
function.
thefr
-86
APPENDIX
This
sImplex
appendIx
gIves
an
B.
SIMPLEX
algorithmIc
ITERATION
description
of
one
IteratIon
search.
INPUT/OUTPUT
VARIABLES:
SIMPLEX
(N.N+l):
Contains
N+l
points
Updated
VALUES (N+l):
Contains
simplex,
that
by N parameters.
Is,
Iteration.
function
values
Updated
during
at
each
simplex
Iteration.
VARIABLES:
CENTROID
L.
entire
defined
during
point.
INTERNAL
the
(N):
Location
of
simplex
Indices
H:
the
centroid
points.
of
the
high
respectively.
CPRIME (N):
CPVAL:
CPP (N):
CPPVAL:
M:
in
Location
of
Function
value
Location
of
Function
of
and
SIMPLEX
the
of
the
reflected
the
in
VALUES.
point.
the
of
N lowest
point.
and
extension
points
the
low
reflected
value
Number
of
point.
point.
extension
SIMPLEX
point.
improved
by
CPRIME.
PREDEFINED
CONSTANTS:
ALPHA.
1.0
BETA.
0.5
GA1.v.1A.2.0
reflection
coefficient
contraction/reduction
extension
coefficient
coefficient
SEARCH AlGORITtt1:
1)
Find
H,
L.
and
High
CENTROID:
a)
Find
b)
Compute
CENTROID
not
included):
DO I.
1,
and
Low
N lowest
in
VALUES,
points
set
(i.e.
H and
L.
SIMPLEX(I,H)
.0.0
N+1
CENTROID(I)
END
of
points
N
CENTROID(I)
DO J .1.
END
of
.CE~tTROID(I)
+ SIMPLEX(I.J)
DO
CENTROID(I)
DO
.(CENTROID(I)
-SIMPLEX(I.H))
/
N
is
of
the
1
-87
2)
Reflection
to
DO I.
Find
CALL
UFUN
M,
M .0
the
00
1-1
If
DO
M.
N+l
Find
(CPVAL,
then
Improved
new
UFUN
extension
M -M
point
Is
accept
M >
F1n,j
1
1
accept
new
point
.CPPVAL
new
point
(l...N,H)
VALUE(H)
is
good.
.CPRIME,
.CPVAL
Accept
VALUE(H)
reflection.
.CPVAL
contraction.
point
is
(l...N,H)
just
OK.
Perform
.CPRIME,
reflection
VALUE(H)
before
.CPVAL
point:
N
-(I-BETA)
*
SIMPLEX(I.H)
*
CENTROID(I)
+
00
CALL
UFUN
(CPVAL.
contraction
SIMPLEX
Else
point,
VALUE(H)
.CPRIME,
then
I -1
TO
CPRIME(I)
EHD
NF)
reflected
BETA
,
+
reflection.
contraction
DO
N,
.CPP,
(l...N,H)
M .1,
If
reflection.
excellent.
CPP,
improves
contracting.
SIMPLEX
d)
the
N
(l...N,H)
attempt
c)
by
(CPPVAL,
If
b)
points
NF)
CALL
SIMPLEX
a)
N,
.GAMMA*CPRIME(I)+(l-GAMMA)*CENTRIOD(I)
if
Else
of
1 TO
S[MPLEX
6)
CPVAL:
N+1
I.
Else
Else
CPRIME,
number
S[MPLEX
5)
value
CPP(I)
END 00
If
c)
function
extension.
DO
b)
get
(CPVAL.LT.VALUE(I))
END
a)
and
.(l+ALPHA)*CENTROID(I)-ALPHA*SIMPLEX(I,H)
TO
IF
4)
CPRIME,
TO N
CPRIME(I)
DO
END
3)
1
CPRIME.
point
(1...N.H)
reduce
entire
CPVAL
N.
is
lower
-CPRIME.
simplex.
NF)
than
VALUE(H)
find
all
VALUE(H).
-CPVAL
new
values.
DO J -1
TO N+l
IF JOL
THEN
DO I -1
TO N
SIMPLEX(I.J)END
CALL
BETA
-(BETA-
1)
*
*
00
UFUN
(VALUE(I).SIMPLEX(I.I).N.NF)
SIMPLEX(I.J)
SIMPLEX(I.L)
accept.
This
appendix
describes,
in algorithmic
form,
the various
choices
available
for
plotting
observations.
predictions.
explanatory
variables.
and
residuals.
These options
are discussed
generally
in Section
5.2;
they
are
available
to MINSUM only.
There
are 5 main ~ssibilities:
(1) V vs.
X. (2) V
vs.
V, (3) X vs.
X, (4) residuals
vs.
V, and (5) residuals
vs.
X.
1)
V vs.
X
a) Select
X subscript.
if more than one X.
b) Set X and V ranges.
c) Input
axis
labels
and title.
d) Choose OBS. PRED. or quit
(return
to main,
keep
LOW RES:
I)OBS:
i) Choose 1) plot
character;
plot).
2) plot
type
(1-6).
Create
plot.
return
to d).
II)PRED:
i) Select:
predictions
at data
~ints
or
spread
over
range.
ia)
If
(spreOO
over
range)
then:
set number of points
on curve;
set remaining
X's.
i1)
Choose 1) plot
character;
2) plot
type
(1-6).
111) If new prediction
wanted,
call
editor
to change
para11eters;
go to 1);
else
go to d).
I)OBS:
1) Choose 1) plot
type
(0.2-6);
2) plot
character.
Return
to d).
II)PRED:
1) Select
predictions
at data
~ints
or spreOO over ran<)e.
1a) If (spread
over
range)
then:
set number of ~ints
on curve;
set remaining
X's.
11) Choose 1
plot
type
(0-6).
2
connect1ng
l1ne,
or
3 marker.
1i1)
If new predict1on
wanted.
call
ed1tor
to change
par5neters;
go to i);
else
goto d).
ii)
HIGH RES:
!
2)
y vs. y
a) Choose y PRED or Y OBS for
horizontal
axis.
b) Set range
of Y's.
Default
is lowest
of OBS and
highest
of OBS and PRED.
c) Input
title
and axis
labels.
d) Plot
Y OBS vs. Y PRED.
LOW RES: I) Choose plotting
character.
Default
II)
Choose plot
type
(count
or overprint).
HIGH RES I) Choose marker.
Default
*.
11) Indicate
if 45 degree
line
needed.
e) Return
to main menu.
PRED to
*.
-90
3)
X vs. X
a) Select
b) Select
c)
d)
e)
horizontal
vertical
X subscript
X subscript
-
and range.
and range.
Input
title
and labels.
Plot
Xl vs. X2.
LOW RES: I) Choose plotting
character.
II)
Choose plot
type
(count
HIGH RES I) Choose marker.
Default
Return
to main menu.
or
*.
Default
overprint).
*.
4)
Residuals
vs.
y
a) Select
y PRED or y OOS.
b) Set y range.
c) Indicate
if
residuals
should
be nonnalized.
d) Set residual
range.
e) Input
title
and labels.
f) Plot
residual
vs. Y.
LOW RES:
I) Choose plotting
character.
Default
*.
II)
Choose plot
type
(count
or overprint).
HIGH RES: I) Choose marker.
Default
*.
g) Return
to menu.
5)
Residuals
vs. X
a) Select
Xi.
b) Set X range.
c) Indicate
if residuals
should
be nonnalized.
d) Set residual
range.
e) Input
title
and labels.
f) Plot
residual
vs. X.
LOW RES:
I) Choose plotting
character.
Default
*.
II)
Choose plot
type
(count
or overprint).
HIGH RES: I) Choose marker.
Default
*.
g) Return
to menu.
~
Jon
Schnute
Department
of Fisheries
and Oceans
Fisheries
Research
Branch
Pacific
Biological
Station
Nanaimo,
British
Columbia
V9R SK6
December
1982
Canadian
Technical
and
Report
Aquatic
Sciences
December
A
MANUAL
ESTIMATION
FOR
IN
EASY
FISHERY
of
Fisheries
No.1140
1982
NONLINEAR
PARAMETER
RESEARCH WITH INTERACTIVE
MICROCOMPUTER
PROGRAMS
by
Jon
Schnute
((
.P7"'jc)
Department
of
Fisheries
Pacific
Nanaimo,
Fisheries
Research
Biological
British
Columbia
15"'
and Oceans
Branch
Station
v9R SK6
~
-1.1.1.
TABLE OF CONTENTS
Page
List
of
tables
V1
List
of
figures
VI.
List
of
computer
V11
listings
Ab 5 t r ac t
Vi i i
Resume
1.X
x
Preface
Preface
I.
2.
3.
to
the
second
X11
edition
1
Introduction.
The
Extension.
Reflection.
Cont
Reduction.
simplex
The
Auxiliary
Variables
Tasks
TRY
TEMPLATE.
First
Second
user
r
ac
t
(Figure
level
ion.
program.
level
search
parameters.
(Table
7
9
9
9
9
method.
rules.
rules.
3.1)
10
3.1)
11
12
13
14
15
15
17
4.
MICRO
Input
Tuning
Simplex
5.
SIMPLEX.
4.6.
2.
3.
5.
7.
1.
PROFILE
Data
21
data
Maximum
Required
Printer
Restart
Console
Convergence
Initial
swmnary
entry.
algorithm
number
display
display
point
step
standard
check
alone.
reduction.
and
of
code.
code.
deviation.
calls.
code.
step.
21
24
24
24
24
24
26
30
30
30
32
-iv
PROTECT.
P,
Prefix
E,
6.
7.
8.
S, PRO.
L,.
or
35
35
35
Q
PREDICT.
PROTECT.
Data
Prefix
entry.
PRED.
Problems
Additional
7.1
7.2
Multiple
MULTIMIN
GROW
GROwrH
Choosing
REGRE
8.1
8.2
8.3
8.5
8.4
References.
Appendix
to
watch
PARS
SS
Arbitrary
Bayes
Analysis
MINV1,
Some
Some
Constrained
Nonlinear
A.
minima.
the
possibilities. parameters
parameters estimates.
MINV2
Technical
A.l
A.2
A.3
A.4
A.6
A.5
Applesoft
General
RENUMBE
Graphics
HELLO.
PROTECT.
Daki
Hard
Compressed
Computer
Vector
Data
Applesoft
TEMPLATE
COMPUSHOP
LENGTH
for.
of
information
equat
right
36
39
39
42
42
42
43
parameters.
v~riance
problems.
ions.
held
linear.
fixed.
43
43
43
43
44
46
46
47
48
functions.
requirements
n
copy
files.
36
36
36
36
~ ...
5
R DATA,
list
Appendix
B.
Version
Appendix
c.
Worked
requirements.
CRUNCHER
memory
Station Toolkit
graphics.
Toolkit
DATABASE.programs.
1.1
subrout
and
WEIGHT
protection.
files
example
Paper
APA EDIT.
ine
DATA.
.
Tiger
suggestions.
software.
..
49
50
50
50
50
50
50
50
50
51
51
51
51
51
51
51
51
51
52
53
-v
-
Page
Appendix
GROW
PREDICT.
Dakin
MICRO
MINV2
PROTECT.
TRY.
TEMPLATE
GROWTH.
HELLO.
MULTIMIN
REGRESS.
PROFILE.
Dakin
MINVl
D.
PARS.
55
SIMPLEX. Program
VARIABLE
LISTER.
listings.
CROSS
REFERENCE
55
55
55
56
63
68
71O
83
85
87
9!0
97
10. 4
10 6
1110
11 3
-viii
ABSTRACT
Schnute,
Jon.
fishery
Rep.
1982.
research
Fish.
A manual
for
with
interactive
Aquat.
Sci.
1140:
easy
nonlinear
microcomputer
xvi
+
115
parameter
estimation
programs.
Can.
in
Tech.
p.
This
manual
describes
in detail
how to solve
many practical
problems
encountered
in nonlinear
parameter
estimation.
In addition,
it
presents
software
to aid
the
user
with
three
tasks:
(1)
finding
optimal
parameter
estimates,
(2)
plotting
observations
and model
predictions,
and (3)
displaying
graphically
the
variation
in likelihood
(or
sum of squares)
when the
parameters
are
varied
from
their
optimal
estimates.
This
software
is coded
in
BASIC
for
the
Apple
II
microcomputer,
and it
is available
00 a suitable
5 1/4"
diskette.
or her
heart
In
particular
of
Although
cases,
problem
The simplex
the
discussion
a simplex
derivative-based
human
time
emphasis
many
search
ease
of
time.
It
also
shows
and how to apply
it
discussion
literature,
Key
words:
to
be
less
how to
in other
adjust
the
simplex
contexts
besides
is
the
known
of
process
of linear
minimization,
microcomputers,
likelihood,
even
il~ustrated
methods
purpose
parametric
software
here
BASIC code.
for
a function
describes
the
development,
the
broad
searching
This
manual
program
although
with
procedure
is
of
can adapt
the
general
just
a few lines
of
minimum
method
efficient
of
with
have
this
manual
models,
parameter
function
Apple
fisheries
the
for
is
to
estimation
estimation,
minimization,
II
microcomputer,
models.
of
examples
make
the
as
BASIC,
great
fields.
as
the
nonlinear
simplex
than
fisheries
many
reader
with
the
efficiency
The
from
in
his
computer
optimal
estimation.
application
time
of minimizing
manual
places
expense
method
nonlinear
numerous
obvious
of nonlinear
regression.
at
to
lies
at
completely.
compute
it
has
the
considerable
advantage
~oding
a particular
problem.
The
throughout
The
comfortable
standard
method
here.
methods,
required
for
on
the
user
by adding
much
simpler
estimation,
search,
likelihood,
maximum
~
-ix
\-'
RESUME
I
Schnute,
Jon.
fishery
Rep.
problemes
De plus,
1982.
research
Fish.
A manual
for
with
interactive
Aquat.
Le present
pratiques
il
presente
Sci.
1140:
easy
nonlinear
microcomputer
xvi
+
115
manuel decrit
en detail
recontres
dans l'estimation
le logiciel
necessaire
parameter
estimation
programs.
Can.
in
Tech
p.
la
maniere
de resoudre
plusieurs
non lineaire
des parametres.
pour aider
l'usager
dans les trois
taches
suivantes:
(1) la recherche
OOs estimations
optimales
de parametres,
(2) le tra9age
des observations
et des predictions
de modeles,
et (3) la
visualisation
graphique
00 la variation
00 la possibilite
(au sqmme OOs
carres)
quand les parametres
different
de leurs
estimations
optimales.
Ce
logiciel
est code en BASIC pour le micro-ordinateur
Apple
II et..il
est
disponible
sur disquette
appropriee
de 5 1/4".
Dans plusieurs
cas,
l'usager
peut adapter
le logiciel
general
a ses besoins
particuliers
en ajoutant
quelques
lignes
en BASIC.
Cette
d'un
minimum
discussion.
methode
J:xJur
route
de
recherche
par
temps
sur
considerable
ordinateur
que
de minimiser
transmission
probleme.
Le manuel
vise
meme aux depens
du temps
methode
par
transmission
maximale
non line
d'ouvrages
d'autres
beaucoup
Mots-cles:
unidirectionnelle
constitue
est
decrite
la
partie
dans
le
unidirectionnelle
les
le
sur
les
utilise
elle
doit
la
recherche
de la
Quoique
mains
methodes
a base
derivee,
temps
qu'une
personne
surtout
a faciliter
sur
ordinateur.
unidirectionnelle
peches,
pour
principale
manuel.
la
efficacement
le
a l'avantage
passer
~ coder
un
l'elaboration
de programmes,
11 mantre
aussi
comment
ajuster
pour
obtenir
une efficacite
et comment
l'appliquer
dans
des contextes
aires.
route
la discussion
est
illustree
mais
les
methodes
ce
manuel
est
autres
que
de nombreux
peuvent
la
les
estimations
exemples
tires
evidemment
s'appliquer
a
domaines.
Le
procede
transmission
la fonction
la methode
but
general
d'estimation
plus
simple
modeles
lineaire,
non
de
de
lineaire
regression
parametriques,
minimisation
unidirectionnelle,
BASIC,
probabilite,
autant
de
familiariser
qu'avec
la
le
procedure
lecteur
avec
le
classique
lineaire.
estimation
de fonction,
micro-ordinateurs,
probabilite
de
parametre,
recherche
pqr
micro-ordinateur
maxima~e,
modeles
estimation
transmission
de
Apple
peche.
non
II,
-x
PREFACE
Fisheries
literature
of
analyses
based
on nonlinear
been
authored
or co-authored
III
paper,
I've
been
left
the
ideas
into
practice.
parameter
parametric
estimates
models
the
likelihood
optimized.
If
not
Which
require
the
feeling
actually
just
linear
translate
tHe user
appropriate
that,
no
be
attempted
microcomputers
like
nonlinear
are
readily
to a computer
to program
an information
smallanalysis.
to
medium-sized
The ease
of
the
highly
inconveniences,
portable
hardware
information
matter
how
the
problems
program
of
errors,
with
an
for
any
user
programs
prove
with
described
useful
computers
to
simply
The
as easy
nonlinear
in
it
for
a single
does
as
operation
performs,
'errors.
can be
at
variety
an Apple
I
particularly
fishery
intent
management
visible
and (3)
of
this
occur
BASIC
example,
with
one
to
take
it
some
methods.
a large
numerical
on many
of the
immediate
the
for
small
systems
Department
of
there
are
diskette,
that
developing
he or
looks,
in fishery
data
helps
compensate
Also,
these
the
Canadian
hope
in
report
enough
providing
so
needs
such
Extra
conditions,
added
easily.
The
is
to
to estimate
ones,
where
I propose
to
the
practitioner's
the
user
function,
II
it's
now several
Apple
makes
it
possible
advantage
programs
countries
some
have
of
here
Where
all
the
might
large
available.
practitioner
as in linear
step.
much of
specified
techniques
to
here.
aren't
the
models
Essentially,
information
access
as
effort.
environment
for
perform
acceptably
like
slow
operation.
and software.
In
a
to put
the
Nonlinear
such
not,
then
analysis
technology
pose
required
optimization
which
commonly
development
in
Fisheries
and Oceans,
Pacific
Region,
for
II
microcomputers.
This
report,
together
program.
function,
If
the
enormous
perfect
they
can
able
for
Which
must
then
be
this
process,
then
function.
interesting
in microcomputing
implementing
the
hardly
estimation,
an increasing
number
of
these
papers
have
participated
in such
how easily
readers
might
be
models,
one can give
formulas
without
Current
developments
possibilities
for
remarkable
While
project
wondering
For
years
includes
models.
Some
Each
time
I've
or a sum of
squared
model
the
reader
feels
comfortable
enough
to specify
she may be left
could
in recent
parametric
by myself.
that
easy
achieve
present
a tool
parameters
ordinary
which
this
by (I)
tailoring
work
as possible,
(2)
he or she can see clearly
graphics
display
to assess
to supply
only
a program
as (but
not
necessarily)
such
as parameter
tool
readily
allows
makes
it
in IOOderate-sized
regression
gives
a
constraints
selected
almost
the
the
tool
rendering
answer
so
its
that
how the
tool
the
outcome.
to calculate
the
sum of squares
of
or Bayesian
parameters
to
model
priors,
be fixed
values.
The
for
good
underlying
function
algorithms
methods
minimization
employed
have
exist
FORTRAN
in
here
been
to
do
certainly
studied
are
not
extensively,
the
effectively
job
new.
and
on
a
General
a
large
-XI.
~~{
;l~ computer.
:," by Nelder
i
This
report
employs
and Mead (1965)
.The
more
flexible
counterpart
and much more
(O'Neill,
1971).
experience
over
is
or
totally
of
problems
optimal
discussion
I
The
writing
Since
ef£icient
a year
or
resource:
in BASIC
a Data
General
with
about
9000
programs
and
the
In
Station,
enlarged
of
handy
patience.
half-an-hour,
especially
here
may
Sometimes
or
just
letting
hours
computer
a
This
files
for
do
the
report
was
work
the
on
the
suggestions,
on
the
best
large
along
that
in
file
even
mind,
a few
has
involves
Literature
tailoring
to
and
at
the
routines
to
to
more
than
five
on the
poorly
with
more
than
four
to 15.
It's
partly
a
up
be
left
overnight,
alone
while
possible
proposals
on
system
diskette,
in Appendix
A.
diskette,
is discussed.
a complete
I emphasize
and
£or
the
for
it
ease
run-time.
searches
of
for
program
If
a model
minutes.
SIMPLEX
appear
1.1
operated
exclusively
to the
user
£or
then
it
can easily
be tried,
even
if
it
a while.
On the
other
hand,
if
a model
then
it
may not
be tried,
even
if
the
programs
version
£or
available
workspace,
with
graphics
In a research
context,
more
important
than
computer
work
only
regard.
performs
with
even
reasons.
not .(as
computing
almost
essential
available
when
the model
quite
slowly.
or
program
graphics)
were
originally
microcomputer
became
available
In the
context
0£ computing
a very
form,
computer
hours,
describes
this
report
where
each
versions,
and possibly
these
possibilities
in
corrections,
run
the
several
and with
the MICRO
system
requirements
available
my
in
few minutes
to program,
the
computer
work
for
or days
of programming,
could
48K memory
details
on
method
the
any
peculiar
1976,
does
interactive
and use
is
2 microcomputer
which
is
in central
memory
available
capabilities
optimal
parameter
estimates.
development
is often
much
takes
means
needs
began
that
welcome
improvement.
commonly
states
that
it
have
used
it
successfully
I
imply
this
project
for
some rather
where
I have
worked
since
this
felt
like
to their
present
some cases,
the
programs
simplex
algorithm
parameters,
but
matter
simplex
not
proposed
somewhat
referenced
FORTRAN
on my practical
gratefully
and the
programs
here
(wihout
Nova 2.
Later,
an Apple
II
user
space
of 36000
bytes.
Apple's
parameters,
does
would
for
I
Nova
bytes
The
these
contraints,
developed
for
the
with
an e£fective
This
I
originally
here
is
commonly
is based
1981)
have
a full-scale
program
development
estimation,
data.
than
its
version
a half.
and
involved
in
Station,
nonlinear
exploit
and
suggestions
in early
interactive
the
Biological
programs
were
interactive
The current
error-free,
became
Biological
Pacific
0£ this
system.
about
the
simplex
search
method
BASIC
algorithm
developed
together
the
Apple
II
Plus
Version
Appendix
1.1.
B lists
with
sections
the
with
Further
the
I anticipate
future
enhanced
FORTRAN counterpart.
Naturally,
again
that
I ~uld
welcome
all
for
future
software
design.
Jon
Schnute
Pacific
Biological
April,
1981
Station
of
with
user
-X
11
PREFACE TO THE SECOND EDITION
except
(Schnute
for
Strictly
speaking,
this
document
never
appeared
in a fir$t
edition,
some preprints
given
to a few trial
users.
A paper
of mine
1981)
refers
to this
manual
as "Can.
Spec.
Publ.
Fish.
Aqu..at.
Sci.
(in
press)".
Technical
Special
Indeed,
although
Report
series,
the
Publications
series.
r~view,
the
referee
and additions.
As
wish
I had time
to
Unfortunately,
takes
priority.
did
not
I explain
produce
methods
here,
series
so
without
to address
active
use
discussion
I
I
speci~ic
almost
(underlined),
1.
that
a
ver
make this
job
formula
gives
2.
subset
The
almost
as
the
answer
PASCAL
here,
s
to
practitioners
to
make
also
find
available
in
changes
and
of
it.
continually
since
the
value
the
the
a
requests
from
several
for
this
paper
based
to
find
the
programs
may
them
to
seek
without
significant
some of
the
criticism
that
dealt
with
all
,which
should
can
easy
in
be
except
stem
it
the
user
continually
reasonably
solved
my reply.
the
roblem
be
of
model
case
Just
applied,
matter
of
of
as
in
only
of
the
Technical
so too
selecting
Report
discussion
have
been
written
in
a higher
rather
than
BASI~.
to
but
true.
subroutines.
worry
it
APPLE
useless
in
,
,
(four-byte)
The
about
can't
PASCAL
be
most
All
suffers
this
context.
precision,
models
apprpopriate
level
names.
from
with
section
annoying
variables
completely
fit
regression,
which
depend
in
variable
and
tin
.
is
to document
a technology
that
the
user
has defined
estimates.
I'm
trying
to
can
limited
that's
true
selection
linear
models
some
compiled,
here,
Unfortuntely,
1n the
step.
for
has
it
awkward
and even
supports
only
single
as
one
wrongly
The
In
principle,
doesn't
support
that
of
with
The whole
point
of this
manual
estimates
easily.
I assume
and simply
wants
to find
the
parameters
can
enter
nonlinearly.
addressed
many
together
agree.
parameter
properly
getting
problem
their
which
decided
59
the
delay.
small
I
for
the
editor
for
instead
In order
to clarify
this
manual's
purpose
and limitations,
I'd
like
the referee's
main comments here.
Readers
interested
in making
of this
material
will
do well
to take a few minutes
to follow
the
below.
In each case I give
a short
paraphrase
of the referee's
comment
is
written
submit
it
problems
of
fishery
management
a year
and a half
has
gone
by
decided
further
the
I have
had dozens
of
owners
in remote
places)
(1981),
and I continue
believe
have
when
originally
that
I
recommend
publication
below,
I agree
with
a manual
and software
first
edition
was completed.
countries
(including
APPLE
on its
reference
in Schnute
use.
manual
was
requested
However,
research
on
MeanWhile,
considerable
the
editor
where
a
linearly
on
parameters
models
isn't
7.
language,
like
feature
of
are
global
That
APPLE
BASIC
is
so that
problem
is
circumvented.
serious
flaws
The fatal
flaw
is
and that
just
isn't
which
render
that
it
enough
for
-X1.11.
it:~
!
1:1
~,!
0
"cc'
most
practical
fishery
system
is conceptually
Debugging
is
a serious
PASCAL imposes
on the
in
hardware
rather
and
large
estimation
problems.
elegant
on the
APPLE
problem
compared
with
user
significant
extra
software.
audience
The
of
Also,
although
the
operating
II,
it's
slow
and awkward
to use.
APPLE BASIC.
Finally,
APPLE
costs
(at
least
for
the APPLE II)
material
here
II
without
APPLE
users
is
primarily
intended
a particularly
for
the
enhanced
system.
In contrast
and that
precision,
language
runs
there
are
now
programs,
after
the
speed
of
"compiled"
code
is
interpretively,
several
good
debugging,
compiled
("P-code"),
because
isn't
without
too
system.
to put
will
3.
the
much
made
Mar
same way
Furthermore.
as
biologis_~~
higher
for
here
for
the
the
simplex
the
detailed
BASIC
varieS
the
than
works
been
than
that
on ly
to
of
machine
well
language.
on
the
converted
considerably
so-called
Pseudo-machine
to
APPLE
II,
it
VAX BASIC
from
system
to
is FORTRAN~ and
plans
are
underway
language.
When this
is done,
it
of
have
similar
of the
non-APPLE
been
im
users.
lemented
simplicit
sim
lex
to
method
in
exactl
the
isn't
the
user.
necessar
for
.
main
convenience
Whicht
consequentlyt
commentt
he
different
(happily
people.
The
or unhappily)
instances
required
where
biologists
for
gradient-based
by
clearly
using
thought
the
Another
understand
complicated
of
the
simplex
method
is
that
doesn't
require
the
user
methods
want
to know
of
the
derivatives.
of
trouble
often
the
ist
ends
biologist
simplex
nice
come
Figure
terms
with
code
is downhillt
that
referee
made his
programmer
as
that
the
programs.
two
biologist
I know
complex
labor
could
of
derivatives
have
been
method.
feature
won't
are
the
days
calculating
and this
whole
of
the
how it
works.
Gradient-based
analysis,
while
the
simplex
who
and
it
is a direct
to compute
or
which
way
When the
in my experiencet
up writing
his
own
have
spent
methodst
be that
the
biologist
there
will
be some
robust:
sometimes
deciding
here
portability
into
that
could
greater
compiled
has
five-byte
Since
easy
to debug.
Furthermore,
on the market,
so that
run
at high
speed.
In
fact,
language
It
method.
with
discussion
Gradient-based
depend
on a knowledge
to
is
implemented
with
practical
problems.
is
convenience
method
derivatives.
ist
they
spared
level
but
language
software
II
actually
presented
difficulty~
gradient
The
method
search
APPLE
is
transportable.
available
uardt's
the
software
easily
The ideal
much of
the
be
on
PASCAL
a slightly
Although
APPLE BASIC
enough
for
it
is remarkably
compilers
available
can be compiled
and
BASIC
PASCAL,
certainly
with
PASCAL,
usually
just
really
sufficiently
care
simplex
method
is
that
it's
easy
to
methods
are motivated
by some
method
is entirely
geometric.
to follow
curious
the
details,
that
they
read
but
I
section
rather
It
may
hope
2 and
2.1.
The
simplex
method
has
the
added
advantage
that
it
is reasonably
sooner
or later
it
usually
finds
a minimum.
Gradient
methods
fail
for
rather
mysterious
reasons
and leave
the
user
helpless
what
to do next.
This
rarely
happens
with
the
simplex
method,
in
and
-X1.V
users
be
who
able
problem;
down
have
to
see
when
taken
the
it
converge.
get
in
The
the
to
trouble
particular
biggest
nUmber
the
eventually
so
the
Readers
interested
excellent
comparisons
wrote
this
readable
among
manual
description
methods.
originally,
algorithms.
candidate
for
fisheries
so that
His variable
a general
estimation
practical
how
it
includes
wrks
will
detailed
sunmary"
on
page
the
simplex
method
becomes
large.
I
method
does
4
almost
always
discussion
of
this
30.
is that
it
tends
to bog
have
run
this
software
successfully,
although
sometimes
slowly.
My
let
the
computer
spend
some extra
time
than
derivatives.
Still
there
is no question
that
a gradient-based
however,
understand
"tuning
drawback
of
of parameters
with
15 and even
20 parameters
philosophy
is
that
I'd
rather
devote
my own time
to coding
diminish,
to
Section
becomes
need
in
to
essential.
employ
studying
other
method
method
computing
costs
method.
minimization
in Nash
(1979),
If
I had known
that
I would
have
cast
metric
gradient
As
a gradient
methods
will
find
along
with
some interesting
Nash's
book
existed
when I
parts
of the coding
around
(Algorithm
on a small
21)
would
computer.
his
be an excellent
Incidentally,
problems
frequently
cannot
be reduced
to least
squares,
software
must
be capable
of
finding
minima
of arbitrary
functions.
4.
No
provision
.~
est1.mates.
parameter
software
is easy
The
the
is
made
for
calculating
the
variance-covariance
matrix
That's
true.
Again,
the main
purpose
of
this
manual
is
to
to find
the
estimates
themselves.
However,
this
additional
to implement,
and software
to do so will
be included
in future
referee
matrix
suggests
of
second
using
partial
Newton's
divided
difference
derivatives
{also
called
method
to
the Hessian)
of
provide
feature
work.
approximate
of the
negative
log-likelihood
at
its
minimum.
The inverse
of
this
matrix
approximates
the
covariance
matrix
of
parameters.
To the
reader
who isn't
familiar
with
ali
this
terminology,
that
may sound
like
an imposing
task,
it
requires
only
a few lines
of code,
once
the
function
itself
has already
been
5.
coded
~~~
context.
~sed.
main
for
use
simplex
ma~~_~~.~~~e~~~~~~~~~~~e
Pivoting
point
with'the
This
of
--- isn
is an
section
t
enter
linearly,
they
function
calculation.
can be
This
potentially
a great
simplex
gradient
disputes
leads
method
inverting
operating
method
is my
pivoting
isn't
and secondly
to
this
~s.~~~
Choleski
op~~~~!
decomposition
comment;
however,
idea
there
is
that
improvement
this
operating
on
sledge
hammer
of
~.~.
the
it
if
fashion
the
full
approach
in the
matrix
decomposition
kind.
in
The
can
set
to
as
of
part
the
of each
problem,
Recent
work
be
more
efficient
much
inversion,
is known
shows
What the
First
in
8.3
will
the
and
that
than
as suggested
in
to be the
optimal
described
be
detract
from
the
parameters
speed.
of parameters.
linear
regression.
methods
should
shouldn't
some of
found
by linear
regression
reduces
the
dimensionality
in
necessary
the
Choleski
matrices
and
excellent
8.3
The
but
algorithm.
~n.s~~~ig~
needed,
the
the
a
referee
of all,
section
method
8.3,
of
work;
they
~
I
-xv
just
aren't
method
6.
optimal.
The
(Algorithm
The
s
7)
stem
book
for
doesn't
by
Nash
Choleski
permit
(1979)
cited
above
gives
a compact
decomposition.
wei
hted
residuals
in
cam.
the
sum of
~
function
!-quares_~
That's
.i~
;t:,;l:,."
.;l:~
note
fish
;4fi$
~!
~'
not
strictly
true.
Since
the
system
permits
be minimized,
in particular
it
allows
for
a weighted
it
is true
that
the
module
TEMPLATE
for
the
particular
doesn't
include
the
possibility
of variable
weights.
that
the
at various
there
is
age.
The
justification
correct
the
population
differently,
for
procedure
not
demonstration
problems
for
of
weighting
depends
growth
curve.
because
that
problem
practical
of
squares.
However,
case
of
sums of
squares
A sensitive
reader
will
data
used
in the
example
of section
1 represent
mean
ages.
These
means
come from
samples
of differ~sizes,
SQme
I have
is the
the
the
on
the
estimation,
residuals
model,
Once
not
on
;is,
this
complete
at
on
weight
rather
again,
the
lengths
differently
that
chosen
not
to
only
way,
but
software.
to
s~
how
one
of
and
each
defines
the
residuals
to obtain
a
simple
is
on
a manual
science
of
the
model
building.
7.
There
is
no
provision
It
is
certainly
for
residual
plots,
as
re
uired
for
valid
model
selectl.on.
whether
behavior
point
A
made
fully
but
true
or not
the
model
of
the
residuals
that
this
manual
integrated
this
system
document
that
residual
is
of
doesn't
directed
at
software
attempt
such
This
of
linear
manual
but
I hope
estimation
mathematics
and
Although
building,
and skill.
It
may be
model
works,
in
be
practitioner
problem
estimation
can
it
certainly
isn't
As the
referee
used
iteratively:
it,
pro.per
freed
and
model
to
in
problem
building
deciding
fact,
strange
I repeat
the
would
of
estimation.
be
valuable,
project.
biologists
They
the
who
are
may
feel
parameters
here
that
understood
troubled
enter
there
are
completely
by
comfortable
the
the
with
model
straightforward
without
sophisticated
ease.
here
to give
If
the
process
is
of
with
modify
some detail
involved.
when
useful
Remarks
for
helpless
be
limited
ambitious
estimation.
to demonstrate
which
can be
used
model
thought
way.
intended
parameter
regression,
nonlinearly.
methods
of
is
nonlinear
to
model
an
can
to the
data.
In
modify
the
model.
the
for
ConclOOing
problem
plots
is really
appropriate
can suggest
ways
devote
design..
the
most
always
points
estimate
estimate
the
of
be
again.
reader
at
est.imation
his
or
her
technically
difficult
problem
the
part
that
requires
out,
this
is only
one
the
parameters,
see
I
have
least
itself
creative
included
a glimpse
can be
powers
of
the most
step
of
the
how well
the
referee's
remarks
of what
else
might
made easy
enough,
the
to
the
deeper
~
To
my apologies
those
for
who
have
waited
for
months
to
receive
Schnute
Pacific
Nanaimo,
Biological
B.C.
Canada
November,
Il,il
~I
i!
,
;
i
I
,
;'I
I'i
"I
I'
I",
'
11!
I ,
l .i!
'I
I
:'
i ii
I
f
1
!;
manual,
the delay.:
Jon
Ii
this
1982
Station
V9R SK6
I
extend
2.
THE
SIMPLEX
SEARCH
METHOD
From the previous
section
it
is clear
that
a key purpose
of the
software
here is to minimize
a function
F of several,
say N, variables.
(In
the example
above,
F is the criterion
function
S or S*, and N is 3.)
There
are many known algorithms
designed
for
this
purpose,
and the most efficient
ones require
the user to code the derivatives
of F, as well
as F itself.
Other
algorithms,
called
direct
search
methods,
require
information
on F
alone.
The simplex
method is a direct
search
method,
and the Preface
to this
report
gives
some background
on my reasons
for
choosing
it.
If the
practitioner
wants to minimize
programming
~,
as opposed
to computer
~
_tt~e., then a direct
search
method is perhaps
the best.
Users will
have to
judge
from practical
operation
whether
or not this
software
meets their
particular
needs.
The simplex
method was first
conceived
by Spendley,
Hext,
and
Himsworth
(1962)
primarily
as a technique
for
designing
experiments
to locate
points
of optimal
response
for a system
being
actively
measured.
Later,
Nelder
and Mead (1965)
pointed
out that
the method could
also
be implemented
as a computer
search.
O'Neill
(1971)
formalized
the procedure
into
a computer
FORTRAN algorithm,
and the BASIC software
here is a variation
of O'Neill's
algorithm
designed
for greater
flexibility
and interactive
input/output.
A
N +
1 faces.
simplex
When
in N dimensions
N = 2, the
simplex
3 sides.
N = 3,
is
it
polyhedron
a triangle,
verti~es
and
tetrahedron
involves
simplex.
with
4 vertices
and 4 triangular
inspecting
the
values
of
the
function
In order
to minimize
F,
the
vertex
rejected
remaining
in favor
N vertices
of
When
is
some new point.
defines
a new
is
a
This
simplex,
with
N +
which,
triangular
1 vertices
of course,
pyramid,
that
and
has
is,
faces.
The simplex
search
Fat
the
N + 1 vertices
with
the
highest
value
of
new point,
and the
together
procedure
3
a
method
of a
F is
with
the
continues
by
iteration.
Figure
ABC
at
represents
C.
2.1
illustrates
a two-dimensional
the
process
simplex
on
when
which
N = 2.
F is
Here
lowest
the
at
triangle
A and
highest
Symbolically,
F(A)
< F(B)
< F(C).
Since
F is highest
at C, the idea is to move away from C towards
the line
AB
on which F is lower,
at least
at the end points.
This
is done by the process
of reflection
(Figure
2.1,
diagram
1) in which
a new point
C' is determined
as
the mirror
image of C across
the line
AB.
The point
c' is defined
so that
the
lines
CC' and AB both have midpoint
D.
Hopefully,
the value
of F is lower
at C' than at C.
Indeed,
one can
simply
count
the number of original
simplex
points
A, B, C which have higher
F-values
than the new point
C'.
Call
this
number M.
By definition
of M, the
B
.
./
~\\\
\
c'
..
.:;...C'
~
function
the value
possibilities:
points
value
F(C')
at the
reflected
point
C'
improves,
i.e.
, is
lower
than,
of
Fat
M points
on the
original
simplex
ABC.
There
are
four
M is 0,
1, 2, or 3.
The next
action
taken
depends
on M.
Case
1:
M = 3.
In
the
original
simplex.
on
found
so
therefore
t
gives
F(C")
to
C'
C"
is
in diagram
lower
than
give
a new
is accepted
process
is
called
the
ABC"
.This
a new simplex
M = 0.
If
The reflected
simplex
midpoint
ABC.
D on
diagram
Hopefully
F(C'
In
the
')
is
C, then
drastic
the
whole
simplex
ADC' , , is half
called
reduction
Case
3:
original
point
the reflection
have
would
the
property
just
undo
This
would
attempted
process
be
at
is
to
the
this
wasted
of
Case
In
lower
the
best
is actually
than
This
generalized
F(C),
regarded
as
once
case
It
1),
the
might
but
think
the
low
at which
represents
the
centroid
a
reflected
this
is
any vertex
the
on
to move
from
C
This
gives
C' , in
small
instead
contraction
that
obtain
the
in
is,
step
has
been
if
is
simplex
and high
F takes
a face
C"
to
Cdiagram
improves
C' ,
the
ADC"'
the
simplest
the
accept
ABC'
would
4)
then
C'.
is
the
algorithm
Cdiagram
case.
has
been
N.
In
face.
The
good
(since
reflection
Incidentally,
step
will
performs
6),
analogous
new
point
C'
it
doesn't
(diagram
notice
take
place
is
improve
1) in which
that
B is
in the new
AC'.
set
the
in
two
original
points;
however,
let
values
between
F(A)
of
the
N-dimensional
this
improves
2.
B across
arbitrary
to
C"
C'
simply
new simplex
the
next
iteration
mirror
image
of
Otherwise
new
case
This
of
point
seem.appropriate
ahead.
The
-contraction.
description
to
since
then
than
required.
The decision
is
then
made to
the
lowest
point
A.
Each
side
of
the
new
of
its
original
counterpart,
and the
process
reflecting
whole
at
again
denote
N-l
vertices
Fig.
2.1 now
by
point
is
so
M = 2.
defined
the
the
the
lower
points
A and B.
If
so,
then
ABC' , (diagram
3) , and the
process
is
quirk
in the
topography,
C' , doesn't
good
(since
it
improves
Band
C) but
not
v~ry
A).
Consequently,
the
algorithm
proceeds
with
C' is accepted
to give
the
new simplex
ABC'.
now the high
point
on the
simplex,
so the
next
direction
possible,
worse
highest
at C';
consequently
give
the
old
point
C as the
reduction
4.
case
the
choice
is made
AB, but
only
half-way.
successful,
to
either
accepted
is
1).
this
reflection
-reduction
process
1.
F is
one to
If
F
is
5).
effort,
once.
C' ,
It
This
extension.
Otherwise,
and,
as mentioned
above,
this
line
no other.
diagram
that
called
reflection
M =
C, but
Cas in
called
ABC' ,
C towards
new simplex
some weird
(diagram
C' has been very
productive.
by,
say,
the
distance
DC'.
1 is
C'
action
towards
the
size
C'
improves
all
lowest
value
of
the
case
point
is
taken
from
the
highest
point
C"
is accepted
to give
the
called
contraction.
If,
by
improve
shrink
simplex
is
point
gives
There
are
two possibilites:
so,
then
the new point
(diagram
the
original
towards
the
3.
to
step
Figure
2.1.
or not.
If
reflection
2:
case
the
reflected
Thus
the
point
C'
step
this
extend
2 of
F(C')
simplex
to give
Case
possible.
worst
Ilf1
~\
~
;,~
far;
consequently,
reasonable
to
this
The
reflected
dimensions,
simplex
B represent
and F(C).
simplex,
point
but
ABC,
let
it
can
A and
the
remain~n'~
The
line
AB 1.n
and D can be
,
C'
is
defined
,bf
be
C
reflection
points
of
the
of C across
in the
original
N + 1 values
0,
Case
D, exactly
as
simplex
which
1,
...,
N.
before.
Again
let
M denote
the
are
improved
by c'.
Then M can
The four
cases
to consider
are:
1
M = N.
This
case
is
the
best
2.
M =
This
case
is
the
worst
possible,
and
it
number
of
take
any
might
lead
to extension.
Case
I~I
contraction
or
deceptive,
even
because
midpoints
of
reflection
the
the
Case
0.
reduction.
centroid
lines
3.
from
M =
This
or
interpretation
Dof
A to
I.
-contraction
needs
Incidentally,
3.
face
vertices
is
and
5
is
the
leads
5
from
is
the
to
slightly
N-l
B.
as
in
it
diagram
bad,
and
it
-reduction.
diagram
This
N>2,
AB differs
almost
reflection
to
< M < N.
the
N-l
case
even
similar
Case
the
possible,
when
the
leads
When
N>2,
reduction
step.
case,
leading
average
to
diagram
6
to
reflection.
The software
presented
of M, the
action
taken
the
value
low
simplex
values.
extensions,
because
it
:~I
Typically,
reflections,
first
finds
the
contractions,
successful
forced
into
discovered.
a reduced
scale
The practitioner
process,
into
as
a TV
show.
To
may go still
seems
linked
be
moves
back
the
ability
the
resulting
through
directions
herself
or
algorithm
at
to
cycles
to extensions.
exploits
them,
successful
himself
the
and
model
and the
data.
during
the
search.
further
during
a
with
program
problem.
minimized,
(II)
the
(1.1),
here
(1.2),
requires
locates
interface
write
particular
to
new
find
Watching
3.
must
and
directions,
option
and
is
For example,
These
are
This
then
occurs
is
this
not
a bad
the
parameters
ones
which
the
and
of
can be
drawn
into
work
exhibit
high
way
rather
quickly,
while
the
algorithm
continues
to make
larger
to others.
Even covariances
become
somewhat
evident.
It's
quite
see the
algorithm
consistently
adjust
one parameter
upward
whenever
another
parameter
in some particular
direction,
up or down.
The
sensitive
user
whose
behavior
user
algorithm
before
may
to learn
something
about
the
best
determined
become
evident
stabilize
adjustments
common to
it
adjusts
here
has
as an
on each
iteration,
function
minima,
the
as
which
or
(1.3).
information
references
USE R PROGRAM
general
software
rather
this
the
sum
defines
whole
groups
of
for
lower
F-values.
THE
(usually
Typically,
such
and detect
the
search
of
the
described
short)
involved
squares
model,
which
two
Sin
such
the
function
(I)
this
describes
(1.4)
as
one
to
be
manual,
his
functions:
Each of
the
three
major
software
~bout
one of
these
functions.
only
in
parameters
(I)
or
S*
of
the
in
or
the
(1.8),
growth
the
her
function
and
models
packages
discussed
MICRO SIMPLEX,
which
minimized.
Similarly
49
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imizat
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