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


y

and two matchers

x1



x


1


and

x2



x


2


. We seek numbers

b1



b


1


and

b2



b


2


for which

y


y

is approximated as closely as possible by

b1x1+b2x2



b


1




x


1



+



b


2




x


2


, again in the least-distance sense. Arbitrarily beginning with

x1



x


1


, Mosteller & Tukey match the remaining variables

x2



x


2


and

y


y

to

x1



x


1


. Write the residuals for these matches as

x2â‹…1



x



2


â‹…


1



and

yâ‹…1



y



â‹…


1



, respectively: the

â‹…1





â‹…


1



indicates that

x1



x


1


has been “taken out of” the variable.

We can write



y=λ1x1+y⋅1 and x2=λ2x1+x2⋅1.


y


=



λ


1




x


1



+



y



â‹…


1




and



x


2



=



λ


2




x


1



+



x



2


â‹…


1




.

Having taken

x1



x


1


out of

x2



x


2


and

y


y

, we proceed to match the target residuals

yâ‹…1



y



â‹…


1



to the matcher residuals

x2â‹…1



x



2


â‹…


1



. The final residuals are

yâ‹…12



y



â‹…


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.






y



â‹…


1






=



λ


3




x



2


â‹…


1




+



y



â‹…


12




;


whence






y




=



λ


1




x


1



+



y



â‹…


1




=



λ


1




x


1



+



λ


3




x



2


â‹…


1




+



y



â‹…


12




=



λ


1




x


1



+



λ


3




(




x


2



−



λ


2




x


1




)



+



y



â‹…


12









=



(




λ


1



−



λ


3




λ


2




)




x


1



+



λ


3




x


2



+



y



â‹…


12




.




This shows that the

λ3



λ


3


in the last step is the coefficient of

x2



x


2


in a matching of

x1



x


1


and

x2



x


2


to

y


y

.

We could just as well have proceeded by first taking

x2



x


2


out of

x1



x


1


and

y


y

, producing

x1â‹…2



x



1


â‹…


2



and

yâ‹…2



y



â‹…


2



, and then taking

x1â‹…2



x



1


â‹…


2



out of

yâ‹…2



y



â‹…


2



, yielding a different set of residuals

yâ‹…21



y



â‹…


21



. This time, the coefficient of

x1



x


1


found in the last step–let’s call it

μ3



μ


3


–is the coefficient of

x1



x


1


in a matching of

x1



x


1


and

x2



x


2


to

y


y

.

Finally, for comparison, we might run a multiple (ordinary least squares regression) of

y


y

against

x1



x


1


and

x2



x


2


. Let those residuals be

yâ‹…lm



y



â‹…


l


m



. It turns out that the coefficients in this multiple regression are precisely the coefficients

μ3



μ


3


and

λ3



λ


3


found previously and that all three sets of residuals,

yâ‹…12



y



â‹…


12



,

yâ‹…21



y



â‹…


21



, and

yâ‹…lm



y



â‹…


l


m



, are identical.