Prediction error – HP 50g Graphing Calculator User Manual
Page 619
Page 18-52
From which it follows that the standard deviations of x and y, and the
covariance of x,y are given, respectively, by
,
, and
Also, the sample correlation coefficient is
In terms of
⎯x, ⎯y, S
xx
, S
yy
, and S
xy
, the solution to the normal equations is:
,
Prediction error
The regression curve of Y on x is defined as Y =
Α + Β⋅x + ε. If we have a set
of n data points (x
i
, y
i
), then we can write Y
i
=
Α + Β⋅x
i
+
ε
I
, (i = 1,2,…,n),
where Y
i
= independent, normally distributed random variables with mean (
Α +
Β⋅x
i
) and the common variance
σ
2
;
ε
i
= independent, normally distributed
random variables with mean zero and the common variance
σ
2
.
Let y
i
= actual data value,
^
y
i
= a + b
⋅x
i
= least-square prediction of the data.
Then, the prediction error is: e
i
= y
i
-
^
y
i
= y
i
- (a + b
⋅x
i
).
An estimate of
σ
2
is the, so-called, standard error of the estimate,
Confidence intervals and hypothesis testing in linear regression
Here are some concepts and equations related to statistical inference for linear
regression:
1
−
=
n
S
s
xx
x
1
−
=
n
S
s
yy
y
1
−
=
n
S
s
yx
xy
.
yy
xx
xy
xy
S
S
S
r
⋅
=
x
b
y
a
−
=
2
x
xy
xx
xy
s
s
S
S
b
=
=
)
1
(
2
1
2
/
)
(
)]
(
[
2
1
2
2
2
2
1
2
xy
y
xx
xy
yy
i
n
i
i
e
r
s
n
n
n
S
S
S
bx
a
y
n
s
−
⋅
⋅
−
−
=
−
−
=
+
−
−
=
∑
=