I wish I could post or attach scan copy of the problem outline here but it = is not possible due to limited functionality of group discussion. It is for= the earlier reasons that I quoted the exact section, page and example no. = so that someone using the book can refer to. However, I'll try to explain b= elow where possible though it can be difficult to type legible equations in= this group discussion.
The example provided several equations as below;
a0S00 + a1S01 =3D T0
a0S10 + a1S11 =3D T1
where,
S00 =3D sum of square 1 for data i=3D1 to 5
S01 =3D sum of 1 time xi for data i=3D1 to 5
S10 =3D sum of xi time 1 for data i=3D1 to 5
S11 =3D sum of square of xi for data i=3D1 to 5
T0 =3D sum of 1 times yi for data i=3D1 to 5
T1 =3D sum of xi times yi for data i=3D1 to 5
Data refers to;
xi 1.0 2.3 2.9 4.0 4.9 yi 2.0 4.4 5.4 7.5 9.1
When I followed the example question, I go the following results;
S00 =3D 1
S01 =3D 15.1
S10 =3D 15.1
S11 =3D 54.71
T0 =3D 28.4
T1 =3D 102.37
On substituting these numbers into first two equations above and solve alge= braically, I got different result than stipulated on the book.
Hah. It took me a while to realize what exactly you're doing. (you _could_ give an overview).
They're asking you to do a 1st-order least-squares fit of x and y; this is the optimal result if y = a*x + b + n, where a and b are constant and n is a vector of independent Gaussian variables of zero mean and constant variance.
Taking x and y as column vectors, what you really want to solve for is
[x^0 x] * A = y,
where A = [a b]^T ("^T" meaning "transform of", so A is a two-element column vector)
They don't want you wrapped around the axle with higher-order linear math like singular value decomposition, so they're defining
X = [x^0 x] (note that x^0 is just a column of ones),
then you want to solve
X * A = y
but X is 2 x 5 and y is 1 x 5, so you don't have the tools to do it. However, thanks to some clever mathematicians in the 18th and 19th century, you know that you can multiply through by X^T to get
X^T * X * A = X^T * y
This reduces the problem to a 2x2 times a 1x2 = a 1x2 -- and you can solve that with "ordinary" linear algebra, at the expense of some numerical precision from squaring X. (as X gets bigger, you change the reading to "at the expense of lots of numerical precision..., and you investigate the use of better linear algebra techniques).
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