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1
•
Issues of noise in experimental data: illustration
•
Projection operator
•
Writing the regression equations
Lecture 10: Regression
•
Getting the pseudo inverse
•
An example: computing pi
•
A least square error justification
Example of taxi data
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Example of taxi data
Example of taxi data
3
Example of taxi data
Example of taxi data
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Example of taxi data
Main problem of noisy data: regression
If I have a set of data points, and I try to ‘fit’ a line through the cloud of
points, what is the best way to fit that line?
5
Main problem of noisy data: regression
If I have a set of data points, and I try to ‘fit’ a line through the cloud of
points, what is the best way to fit that line?
Main problem of noisy data: regression
If I have a set of data points, and I try to ‘fit’ a line through the cloud of
points, what is the best way to fit that line?
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Main problem of noisy data: regression
Main question: “what does good mean”, i.e. what is the criterion for
being a “good” approximation.
•
Avector
Y
of “data”
•
ˆ
Y
projected along the red line
Main problem of noisy data: regression
What does “projected” mean? It means that
ˆ
Y
−
Y
is orthogonoal to
ˆ
Y
ˆ
Y
This means that from all the points along the red line, the point
is the one which minimizes the distance to
Y
7
Main problem of noisy data: regression
What does “projected” mean? It means that
ˆ
Y
−
Y
is orthogonoal to
ˆ
Y
ˆ
Y
This means that from all the points along the red line, the point
is the one which minimizes the distance to
Y
Orthogonality of vectors?
Vectors are orthogonal if their dot product is zero
ˆ
Y
−
Y
is orthogonoal to
ˆ
Y
Means that
dot(
ˆ
Y
−
Y
,
ˆ
Y
)=0
In other words:
ˆ
Y
T
ˆ
Y
−
Y
= 0
(
) = 0
Where as in the previous chapter
on linear algebra, the superscript T
means transpose
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So in summary: to find the projection
In order to find the projection
of a vector
on the red line, one
needs to check the following formula:
ˆ
Y
T
ˆ
Y
−
Y
= 0
ˆ
Y
Y
(
) = 0
Now we can define regression
We consider data points
(
x
i
,y
i
)
We want to construct an estimator,
which describes the data as a linea combination o functions
y
i
which describes the data as a linear combination of functions
with coefficients
x
i
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This note was uploaded on 09/07/2011 for the course ENGIN 7 taught by Professor Horowitz during the Spring '08 term at University of California, Berkeley.
 Spring '08
 HOROWITZ

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