Paraphrase-this-paragraph-Put-it-in-your-own-words-
Paraphrase-this-paragraph-Put-it-in-your-own-words-
https://stats.stackexchange.com/questions/46185/qu…
Multiple regression can be obtained by sequential matching
Returning to the setting of the question, we have one target
y
and two matchers
x1
and
x2
. We seek numbers
b1
and
b2
for which
y
is approximated as closely as possible by
b1x1+b2x2
, again in the least-distance sense. Arbitrarily beginning with
x1
, Mosteller & Tukey match the remaining variables
x2
and
y
to
x1
. Write the residuals for these matches as
x2â‹…1
and
yâ‹…1
, respectively: the
â‹…1
indicates that
x1
has been “taken out of” the variable.
We can write
y=λ1x1+y⋅1 and x2=λ2x1+x2⋅1.
Having taken
x1
out of
x2
and
y
, we proceed to match the target residuals
yâ‹…1
to the matcher residuals
x2â‹…1
. The final residuals are
yâ‹…12
. Algebraically, we have written
y⋅1y=λ3x2⋅1+y⋅12; whence=λ1x1+y⋅1=λ1x1+λ3x2⋅1+y⋅12=λ1x1+λ3(x2−λ2x1)+y⋅12=(λ1−λ3λ2)x1+λ3x2+y⋅12.
This shows that the
λ3
in the last step is the coefficient of
x2
in a matching of
x1
and
x2
to
y
.
We could just as well have proceeded by first taking
x2
out of
x1
and
y
, producing
x1â‹…2
and
yâ‹…2
, and then taking
x1â‹…2
out of
yâ‹…2
, yielding a different set of residuals
yâ‹…21
. This time, the coefficient of
x1
found in the last step–let’s call it
μ3
–is the coefficient of
x1
in a matching of
x1
and
x2
to
y
.
Finally, for comparison, we might run a multiple (ordinary least squares regression) of
y
against
x1
and
x2
. Let those residuals be
yâ‹…lm
. It turns out that the coefficients in this multiple regression are precisely the coefficients
μ3
and
λ3
found previously and that all three sets of residuals,
yâ‹…12
,
yâ‹…21
, and
yâ‹…lm
, are identical.
