Least-Squares Using Normal Equations - HP -15C Advanced Functions Handbook

Section 4: Using Matrix Operations 1 31
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The output voltage is 5.3494 Z -2.4212
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Displays 7
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Calculates V
o
=(R
3
)\I
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\.
Deactivates User mode.
Least-Squares Using Normal Equations
The unconstrained least-squares problem is known in statistical
literature as multiple linear regression. It uses the linear model
Here, 6
1;
..., b
p
are the unknown parameters, x ^ , ..., x
p
are the
independent (or explanatory) variables, y is the dependent (or
response) variable, and r is the random error having expected
value E(r) = 0, variance a
2
.
After making n observations of y and x
1(
x%, ..., x
p
, this problem can
be expressed as
y - Xb + r
where y is an n -vector, X is an n X p matrix, and r is an n -vector
consisting of the unknown random errors satisfying E(r) = 0 and
If the model is correct and X X has an inverse, then the calculated
least-squares solution b = (X
r
X)"
1
X-
r
y has the following
properties:
• E(b) = b, so that b is an unbiased estimator of b.
Cov(b) = E((b - b)
T
(b - b)) = a
2
(X
r
X)"
1
, the covariance matrix
of the estimator b.