<s>
The	O
iterative	B-General_Concept
proportional	I-General_Concept
fitting	I-General_Concept
procedure	O
(	O
IPF	O
or	O
IPFP	B-General_Concept
,	O
also	O
known	O
as	O
biproportional	O
fitting	O
or	O
biproportion	O
in	O
statistics	O
or	O
economics	O
(	O
input-output	O
analysis	O
,	O
etc	O
.	O
</s>
<s>
)	O
,	O
RAS	B-General_Concept
algorithm	I-General_Concept
in	O
economics	O
,	O
raking	O
in	O
survey	O
statistics	O
,	O
and	O
matrix	O
scaling	O
in	O
computer	O
science	O
)	O
is	O
the	O
operation	O
of	O
finding	O
the	O
fitted	O
matrix	O
which	O
is	O
the	O
closest	O
to	O
an	O
initial	O
matrix	O
but	O
with	O
the	O
row	O
and	O
column	O
totals	O
of	O
a	O
target	O
matrix	O
(	O
which	O
provides	O
the	O
constraints	O
of	O
the	O
problem	O
;	O
the	O
interior	O
of	O
is	O
unknown	O
)	O
.	O
</s>
<s>
in	O
relation	O
to	O
telephone	O
traffic	O
(	O
"	O
Kruithof	O
’s	O
double	O
factor	O
method	O
"	O
)	O
,	O
Deming	O
and	O
Stephan	O
in	O
1940	O
for	O
adjusting	O
census	O
crosstabulations	B-Application
,	O
and	O
G.V.	O
</s>
<s>
(	O
Deming	O
and	O
Stephan	O
proposed	O
IPFP	B-General_Concept
as	O
an	O
algorithm	O
leading	O
to	O
a	O
minimizer	O
of	O
the	O
Pearson	B-General_Concept
X-squared	I-General_Concept
statistic	I-General_Concept
,	O
which	O
Stephan	O
later	O
reported	O
it	O
does	O
not	O
)	O
.	O
</s>
<s>
Bishop	O
's	O
proof	O
that	O
IPFP	B-General_Concept
finds	O
the	O
maximum	O
likelihood	O
estimator	O
for	O
any	O
number	O
of	O
dimensions	O
extended	O
a	O
1959	O
proof	O
by	O
Brown	O
for	O
2x2x2	O
...	O
cases	O
.	O
</s>
<s>
Fienberg	O
's	O
proof	O
by	O
differential	B-Language
geometry	I-Language
exploits	O
the	O
method	O
's	O
constant	O
crossproduct	O
ratios	O
,	O
for	O
strictly	O
positive	O
tables	O
.	O
</s>
<s>
the	O
EM	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
most	O
cases	O
,	O
IPFP	B-General_Concept
is	O
preferred	O
due	O
to	O
its	O
computational	O
speed	O
,	O
low	O
storage	O
requirements	O
,	O
numerical	O
stability	O
and	O
algebraic	O
simplicity	O
.	O
</s>
<s>
Applications	O
of	O
IPFP	B-General_Concept
have	O
grown	O
to	O
include	O
trip	O
distribution	O
models	O
,	O
Fratar	O
or	O
Furness	O
and	O
other	O
applications	O
in	O
transportation	O
planning	O
(	O
Lamond	O
and	O
Stewart	O
)	O
,	O
survey	O
weighting	O
,	O
synthesis	O
of	O
cross-classified	O
demographic	O
data	O
,	O
adjusting	O
input	O
–	O
output	O
models	O
in	O
economics	O
,	O
estimating	O
expected	O
quasi-independent	O
contingency	B-Application
tables	I-Application
,	O
biproportional	O
apportionment	O
systems	O
of	O
political	O
representation	O
,	O
and	O
for	O
a	O
preconditioner	O
in	O
linear	O
algebra	O
.	O
</s>
<s>
Assume	O
the	O
same	O
setting	O
as	O
in	O
the	O
classical	O
IPFP	B-General_Concept
.	O
</s>
<s>
This	O
property	O
is	O
sometimes	O
called	O
structure	O
conservation	O
and	O
directly	O
leads	O
to	O
the	O
geometrical	O
interpretation	O
of	O
contingency	B-Application
tables	I-Application
and	O
the	O
proof	O
of	O
convergence	O
in	O
the	O
seminal	O
paper	O
of	O
Fienberg	O
(	O
1970	O
)	O
.	O
</s>
<s>
operations	O
being	O
at	O
least	O
one	O
order	O
in	O
magnitude	O
faster	O
than	O
classical	O
IPFP	B-General_Concept
.	O
</s>
<s>
IPFP	B-General_Concept
can	O
be	O
used	O
to	O
estimate	O
expected	O
quasi-independent	O
(	O
incomplete	O
)	O
contingency	B-Application
tables	I-Application
,	O
with	O
,	O
and	O
for	O
included	O
cells	O
and	O
for	O
excluded	O
cells	O
.	O
</s>
<s>
For	O
fully	O
independent	O
(	O
complete	O
)	O
contingency	B-Application
tables	I-Application
,	O
estimation	O
with	O
IPFP	B-General_Concept
concludes	O
exactly	O
in	O
one	O
cycle	O
.	O
</s>
<s>
Similar	O
to	O
the	O
IPF	O
,	O
the	O
NM-method	B-General_Concept
is	O
also	O
an	O
operation	O
of	O
finding	O
a	O
matrix	O
which	O
is	O
the	O
“	O
closest	O
”	O
to	O
matrix	O
(	O
)	O
while	O
its	O
row	O
totals	O
and	O
column	O
totals	O
are	O
identical	O
to	O
those	O
of	O
a	O
target	O
matrix	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
differences	O
between	O
the	O
NM-method	B-General_Concept
and	O
the	O
IPF	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
NM-method	B-General_Concept
defines	O
closeness	O
of	O
matrices	O
of	O
the	O
same	O
size	O
differently	O
from	O
the	O
IPF	O
.	O
</s>
<s>
Also	O
,	O
the	O
NM-method	B-General_Concept
was	O
developed	O
to	O
solve	O
for	O
matrix	O
in	O
problems	O
,	O
where	O
matrix	O
is	O
not	O
a	O
sample	O
from	O
the	O
population	O
characterized	O
by	O
the	O
row	O
totals	O
and	O
column	O
totals	O
of	O
matrix	O
,	O
but	O
represents	O
another	O
population	O
.	O
</s>
<s>
If	O
unique	O
MLEs	O
exist	O
,	O
IPFP	B-General_Concept
exhibits	O
linear	O
convergence	O
in	O
the	O
worst	O
case	O
(	O
Fienberg	O
1970	O
)	O
,	O
but	O
exponential	O
convergence	O
has	O
also	O
been	O
observed	O
(	O
Pukelsheim	O
and	O
Simeone	O
2009	O
)	O
.	O
</s>
<s>
a	O
closed	O
form	O
of	O
)	O
exists	O
,	O
IPFP	B-General_Concept
converges	O
after	O
2	O
iterations	O
.	O
</s>
<s>
If	O
unique	O
MLEs	O
do	O
not	O
exist	O
,	O
IPFP	B-General_Concept
converges	O
toward	O
the	O
so-called	O
extended	O
MLEs	O
by	O
design	O
(	O
Haberman	O
1974	O
)	O
,	O
but	O
convergence	O
may	O
be	O
arbitrarily	O
slow	O
and	O
often	O
computationally	O
infeasible	O
.	O
</s>
<s>
For	O
executing	O
the	O
classical	O
IPFP	B-General_Concept
,	O
we	O
first	O
adjust	O
the	O
rows	O
:	O
</s>
<s>
The	O
R	O
package	O
mipfp	O
(	O
currently	O
in	O
version	O
3.1	O
)	O
provides	O
a	O
multi-dimensional	O
implementation	O
of	O
the	O
traditional	O
iterative	B-General_Concept
proportional	I-General_Concept
fitting	I-General_Concept
procedure	O
.	O
</s>
