lecture_E7_10_regression_6p - Lecture 10: Regression...

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1 Issues of noise in experimental data: illustration Projection operator Writing the regression equations Getting the pseudo inverse An example: computing pi Lecture 10: Regression A least square error justification Example of taxi data Example of taxi data Example of taxi data Example of taxi data Example of taxi data
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2 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? 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? 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 This means that from all the points along the red line, the point is the one which minimizes the distance to ˆ Y Y
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3 Main problem of noisy data: regression What does “projected” mean? It means that ˆ Y Y is orthogonoal to ˆ Y This means that from all the points along the red line, the point is the one which minimizes the distance to ˆ Y 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 Where as in the previous chapter on linear algebra, the superscript T means transpose 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 ˆ Y Y Now we can define regression We consider data points ( x i ,y i ) y i We want to construct an estimator,
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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.

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lecture_E7_10_regression_6p - Lecture 10: Regression...

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