<s>
Non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
is	O
the	O
form	O
of	O
least	B-Algorithm
squares	I-Algorithm
analysis	O
used	O
to	O
fit	O
a	O
set	O
of	O
m	O
observations	O
with	O
a	O
model	O
that	O
is	O
non-linear	O
in	O
n	O
unknown	O
parameters	O
(	O
m≥n	O
)	O
.	O
</s>
<s>
There	O
are	O
many	O
similarities	O
to	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
but	O
also	O
some	O
significant	O
differences	O
.	O
</s>
<s>
In	O
economic	O
theory	O
,	O
the	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
method	O
is	O
applied	O
in	O
(	O
i	O
)	O
the	O
probit	O
regression	O
,	O
(	O
ii	O
)	O
threshold	O
regression	O
,	O
(	O
iii	O
)	O
smooth	O
regression	O
,	O
(	O
iv	O
)	O
logistic	O
link	O
regression	O
,	O
(	O
v	O
)	O
Box	O
–	O
Cox	O
transformed	O
regressors	O
(	O
)	O
.	O
</s>
<s>
These	O
equations	O
form	O
the	O
basis	O
for	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
for	O
a	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
.	O
</s>
<s>
Each	O
element	O
of	O
the	O
diagonal	B-Algorithm
weight	O
matrix	O
should	O
,	O
ideally	O
,	O
be	O
equal	O
to	O
the	O
reciprocal	O
of	O
the	O
error	O
variance	O
of	O
the	O
measurement	O
.	O
</s>
<s>
The	O
normal	B-Algorithm
equations	I-Algorithm
are	O
then	O
,	O
more	O
generally	O
,	O
</s>
<s>
In	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
the	O
objective	O
function	O
,	O
,	O
is	O
a	O
quadratic	O
function	O
of	O
the	O
parameters	O
.	O
</s>
<s>
With	O
two	O
or	O
more	O
parameters	O
the	O
contours	O
of	O
with	O
respect	O
to	O
any	O
pair	O
of	O
parameters	O
will	O
be	O
concentric	O
ellipses	O
(	O
assuming	O
that	O
the	O
normal	B-Algorithm
equations	I-Algorithm
matrix	O
is	O
positive	B-Algorithm
definite	I-Algorithm
)	O
.	O
</s>
<s>
It	O
also	O
explains	O
how	O
divergence	O
can	O
come	O
about	O
as	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
is	O
convergent	O
only	O
when	O
the	O
objective	O
function	O
is	O
approximately	O
quadratic	O
in	O
the	O
parameters	O
.	O
</s>
<s>
A	O
good	O
way	O
to	O
do	O
this	O
is	O
by	O
computer	B-Application
simulation	I-Application
.	O
</s>
<s>
The	O
increment	O
,,	O
size	O
should	O
be	O
chosen	O
so	O
the	O
numerical	O
derivative	O
is	O
not	O
subject	O
to	O
approximation	O
error	O
by	O
being	O
too	O
large	O
,	O
or	O
round-off	B-Algorithm
error	I-Algorithm
by	O
being	O
too	O
small	O
.	O
</s>
<s>
Some	O
information	O
is	O
given	O
in	O
the	O
corresponding	O
section	O
on	O
the	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
page	O
.	O
</s>
<s>
See	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
algorithm	I-Algorithm
for	O
an	O
example	O
.	O
</s>
<s>
The	O
normal	B-Algorithm
equations	I-Algorithm
matrix	O
is	O
not	O
positive	B-Algorithm
definite	I-Algorithm
at	O
a	O
maximum	O
in	O
the	O
objective	O
function	O
,	O
as	O
the	O
gradient	O
is	O
zero	O
and	O
no	O
unique	O
direction	O
of	O
descent	O
exists	O
.	O
</s>
<s>
For	O
example	O
,	O
when	O
fitting	O
a	O
Lorentzian	O
the	O
normal	B-Algorithm
equations	I-Algorithm
matrix	O
is	O
not	O
positive	B-Algorithm
definite	I-Algorithm
when	O
the	O
half-width	O
of	O
the	O
band	O
is	O
zero	O
.	O
</s>
<s>
Graphically	O
this	O
corresponds	O
to	O
working	O
on	O
a	O
semi-log	B-Application
plot	I-Application
.	O
</s>
<s>
may	O
be	O
solved	O
for	O
by	O
Cholesky	O
decomposition	O
,	O
as	O
described	O
in	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
The	O
fraction	O
,	O
f	O
could	O
be	O
optimized	O
by	O
a	O
line	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
achieved	O
by	O
using	O
the	O
Marquardt	B-Algorithm
parameter	O
.	O
</s>
<s>
where	O
is	O
the	O
Marquardt	B-Algorithm
parameter	O
and	O
I	O
is	O
an	O
identity	O
matrix	O
.	O
</s>
<s>
is	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
vector	O
.	O
</s>
<s>
So	O
,	O
when	O
becomes	O
very	O
large	O
,	O
the	O
shift	O
vector	O
becomes	O
a	O
small	O
fraction	O
of	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
vector	O
.	O
</s>
<s>
Various	O
strategies	O
have	O
been	O
proposed	O
for	O
the	O
determination	O
of	O
the	O
Marquardt	B-Algorithm
parameter	O
.	O
</s>
<s>
When	O
reducing	O
the	O
value	O
of	O
the	O
Marquardt	B-Algorithm
parameter	O
,	O
there	O
is	O
a	O
cut-off	O
value	O
below	O
which	O
it	O
is	O
safe	O
to	O
set	O
it	O
to	O
zero	O
,	O
that	O
is	O
,	O
to	O
continue	O
with	O
the	O
unmodified	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
minimum	O
in	O
the	O
sum	O
of	O
squares	O
can	O
be	O
found	O
by	O
a	O
method	O
that	O
does	O
not	O
involve	O
forming	O
the	O
normal	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
Jacobian	O
is	O
subjected	O
to	O
an	O
orthogonal	B-Algorithm
decomposition	O
;	O
the	O
QR	O
decomposition	O
will	O
serve	O
to	O
illustrate	O
the	O
process	O
.	O
</s>
<s>
where	O
is	O
an	O
orthogonal	B-Algorithm
matrix	O
and	O
is	O
an	O
matrix	O
which	O
is	O
partitioned	B-Algorithm
into	O
an	O
block	O
,	O
,	O
and	O
a	O
zero	O
block	O
.	O
</s>
<s>
A	O
variant	O
of	O
the	O
method	O
of	O
orthogonal	B-Algorithm
decomposition	O
involves	O
singular	O
value	O
decomposition	O
,	O
in	O
which	O
R	O
is	O
diagonalized	O
by	O
further	O
orthogonal	B-Algorithm
transformations	O
.	O
</s>
<s>
where	O
is	O
orthogonal	B-Algorithm
,	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
singular	O
values	O
and	O
is	O
the	O
orthogonal	B-Algorithm
matrix	I-Algorithm
of	O
the	O
eigenvectors	O
of	O
or	O
equivalently	O
the	O
right	O
singular	O
vectors	O
of	O
.	O
</s>
<s>
The	O
relative	O
simplicity	O
of	O
this	O
expression	O
is	O
very	O
useful	O
in	O
theoretical	O
analysis	O
of	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
This	O
is	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
in	I-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
Davidon	B-Algorithm
–	I-Algorithm
Fletcher	I-Algorithm
–	I-Algorithm
Powell	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Steepest	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Although	O
a	O
reduction	O
in	O
the	O
sum	O
of	O
squares	O
is	O
guaranteed	O
when	O
the	O
shift	O
vector	O
points	O
in	O
the	O
direction	O
of	O
steepest	B-Algorithm
descent	I-Algorithm
,	O
this	O
method	O
often	O
performs	O
poorly	O
.	O
</s>
<s>
When	O
the	O
parameter	O
values	O
are	O
far	O
from	O
optimal	O
the	O
direction	O
of	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
vector	O
,	O
which	O
is	O
normal	O
(	O
perpendicular	O
)	O
to	O
the	O
contours	O
of	O
the	O
objective	O
function	O
,	O
is	O
very	O
different	O
from	O
the	O
direction	O
of	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
vector	O
.	O
</s>
<s>
This	O
makes	O
divergence	O
much	O
more	O
likely	O
,	O
especially	O
as	O
the	O
minimum	O
along	O
the	O
direction	O
of	O
steepest	B-Algorithm
descent	I-Algorithm
may	O
correspond	O
to	O
a	O
small	O
fraction	O
of	O
the	O
length	O
of	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
vector	O
.	O
</s>
<s>
When	O
the	O
contours	O
of	O
the	O
objective	O
function	O
are	O
very	O
eccentric	O
,	O
due	O
to	O
there	O
being	O
high	O
correlation	O
between	O
parameters	O
,	O
the	O
steepest	B-Algorithm
descent	I-Algorithm
iterations	O
,	O
with	O
shift-cutting	O
,	O
follow	O
a	O
slow	O
,	O
zig-zag	O
trajectory	O
towards	O
the	O
minimum	O
.	O
</s>
<s>
Conjugate	B-Algorithm
gradient	I-Algorithm
search	I-Algorithm
.	O
</s>
<s>
This	O
is	O
an	O
improved	O
steepest	B-Algorithm
descent	I-Algorithm
based	O
method	O
with	O
good	O
theoretical	O
convergence	O
properties	O
,	O
although	O
it	O
can	O
fail	O
on	O
finite-precision	O
digital	O
computers	O
even	O
when	O
used	O
on	O
quadratic	O
problems	O
.	O
</s>
<s>
They	O
offer	O
alternatives	O
to	O
the	O
use	O
of	O
numerical	O
derivatives	O
in	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
method	I-Algorithm
and	O
gradient	O
methods	O
.	O
</s>
<s>
Nelder	B-Algorithm
–	I-Algorithm
Mead	I-Algorithm
(	O
simplex	O
)	O
search	O
.	O
</s>
