Wen-Feng Hsiao wrote:
>
> Dear Hartig,
>
> Thanks for your reply. I am sorry for my poor knowledge in statistics.
> But I wonder why the definition of 'linearity' of statistics is different
> from that of engineering mathematics, which defines 'linear' as:
>
> Each unknown xj appears to the first power only, and that there are no
> cross product terms xi*xj with i!=j.
>
Your initial model
Y = X11 * X12 + X21 * X22.
is bi-linear in some sense. It is linear on both X1 (with
fixed X2) and X2 (with fixed X1). But it is not a
statistical model, or at least not a regression one: there
is no unknown parameter(s) to estimate.
On other hand, the model
Y = X1*A*X2,
where A is a unknown matrix of parameters (4 unknown
parameters if X1, X2 are 2-dimensional) is a linear
regression model. Linear in the sense that it is liner on
the unknown parameters A. I think, you want to ask about
this model.
Another example: regression on polynomial:
Y = A0 * X^n + A1 * X^(n-1) + ... + An
is a linear regression: linear on the unknown parameters
A0,...An.
Yes, Y depends on X polynomially not linearly, but it is not
important. Given a sample { (xj, yj) }, j=1,100, we can
estimate {A0...A5} using linear LSQ. For this we will
calculate { xj^2 }, ... { xj^5 } and then will use the
obtained numbers the same way as if they were be
(undependent) variables in the textbook regression
Y = A0*Xn + A1*X(n-1)+ ... + An
(after all, nobody gives us warranty that the 1, X1, ... Xn
are undependent: they can turn to be multicollinear).
As to concerning "engineering mathematics", it reminds me
the old Russian joke: "for the blue whales pi~=3.14".
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