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Robert Ricci
Evaluating Networked Systems
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b9642f00
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b9642f00
authored
Mar 03, 2014
by
Robert Ricci
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Finish part of lecture about parameter esimation
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lectures/lecture14/lecturenotes.tex
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lectures/lecture14/lecturenotes.tex
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b9642f00
...
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@@ 20,8 +20,43 @@
\begin{outline}
\1
From last time
\1
Today: How well does your data fit a line?
\2
More complicated regressions exist, of course, but we'll stick with this one for now
\2
Eyeballing is just not rigorous enough
\1
Basic model:
$
y
_
i
=
b
_
0
+
b
_
1
x
_
i
+
e
_
i
$
\2
$
y
_
i
$
is the prediction
\2
$
b
_
0
$
is the yintercept
\2
$
b
_
1
$
is the slope
\2
$
x
_
i
$
is the predictor
\2
$
e
_
i
$
is the error
\2
\textit
{
Which of these are random variables?
}
\3
A: All but
$
x
_
i
$
the
$
b
$
s are estimated from random variables,
$
e
$
is difference between random variables
\3
So, we can compute statistics on them
\1
Two criteria for getting
$
b
$
s
\2
Zero total error
\2
Minimize SSE (sum of squared errors)
\2
Example of why one is not enough: two points, infinite lines with zero total error
\2
Squared errors always positive, so this criterion alone could overshoot
or undershoot
\1
Deriving
$
b
_
0
$
is easy
\2
Solve for
$
e
_
i
$
:
$
y
_
i

(
b
_
0
+
b
_
i x
_
i
)
$
\2
Take the mean over all
$
i
$
:
$
\overline
{
x
}
=
\overline
{
y
}

b
_
0

b
_
1
\overline
{
x
}$
\2
Set mean error to 0 to get
$
b
_
0
=
\overline
{
y
}

b
_
1
\overline
{
x
}$
\2
Now we just need
$
b
_
1
$
\1
Deriving
$
b
_
1
$
is harder
\2
SSE = sum of errors squared over all
$
i
$
\2
We want a minimum value for this
\2
It's a function with one local maximum
\2
So we can differentiate and look for zero
\2
$
s
_
y
^
2

2
b
_
1
s
^
2
_{
xy
}
+
b
_
1
^
2
s
_
x
^
2
$
, then take derivative
\2
$
s
_{
xy
}$
is correlation coefficient of
$
x
$
and
$
y
$
(see p. 181)
\2
In the end, gives us
$
b
_
1
=
\frac
{
s
^
2
_{
xy
}}{
s
_
x
^
2
}$
\3
Correlation of
$
x
$
and
$
y
$
divided by variance of
$
x
$
\3
$
\frac
{
\sum
{
xy
}

n
\overline
{
x
}
\overline
{
y
}}{
\sum
{
x
^
2
}

n
(
\overline
{
x
}
)
^
2
}$
\1
For next time
\end{outline}
...
...
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