HP 49g+ User Manual
Page 617
Page 18-50
Suppose that we have n paired observations (x
i
, y
i
); we predict y by means of
∧
y = a + b
⋅x, where a and b are constant.
Define the prediction error as, e
i
= y
i
-
∧
y
i
= y
i
- (a + b
⋅x
i
).
The method of least squares requires us to choose a, b so as to minimize the
sum of squared errors (SSE)
2
1
1
2
)]
(
[
i
n
i
i
n
i
i
bx
a
y
e
SSE
+
−
=
=
∑
∑
=
=
the conditions
0
)
(
=
SSE
a
∂
∂
0
)
(
=
SSE
b
∂
∂
We get the, so-called, normal equations:
∑
∑
=
=
⋅
+
⋅
=
n
i
i
n
i
i
x
b
n
a
y
1
1
∑
∑
∑
=
=
=
⋅
+
⋅
=
⋅
n
i
i
n
i
i
n
i
i
i
x
b
x
a
y
x
1
2
1
1
This is a system of linear equations with a and b as the unknowns, which can
be solved using the linear equation features of the calculator. There is,
however, no need to bother with these calculations because you can use the
3. Fit Data … option in the ‚Ù menu as presented earlier.
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Notes:
• a,b are unbiased estimators of Α, Β.
• The Gauss-Markov theorem of probability indicates that among all
unbiased estimators for
Α and Β, the least-square estimators (a,b) are the
most efficient.
____________________________________________________________________