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LOGO Regression Analysis Lecturer: Dr. Bo Yuan E-mail: yuanb@sz.tsinghua.edu.cn

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Regression To express the relationship between two or more variables by a mathematical formula. x : predictor (independent) variable y : response (dependent) variable Identify how y varies as a function of x. y is also considered as a random variable. Real-Word Example: Footwear impressions are commonly observed at crime scenes. While there are numerous forensic properties that can be obtained from these impressions, one in particular is the shoe size. The detectives would like to be able to estimate the height of the impression maker from the shoe size. The relationship between shoe sizes and heights 2

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Shoe Size vs. Height 3

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What is the predictor? What is the response? Can the height by accurately estimated from the shoe size? If a shoe size is 11, what would you advise the police? What if the size is 7 or 12.5? 4

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General Regression Model The systematic part m(x) is deterministic. The error ε(x) is a random variable. Measurement Error Natural Variations Additive 5

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Example: Sin Function 6

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Standard Assumptions 7

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A1 8

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A2 9

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A3 10

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Back to Shoes 11

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Simple Linear Regression 12

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Model Parameters 13

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Derivation 14

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Standard Deviations 15

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Polynomial Terms Modeling the data as a line is not always adequate. Polynomial Regression This is still a linear model! m(x) is a linear combination of β. Danger of Overfitting 16

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Matrix Representation 17

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Matrix Representation 18

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Model Comparison 19

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R2R2 20

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Example 21

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Summary Regression is the oldest data mining technique. Probably the first thing that you want to try on a new data set. No need to do programming! Matlab, Excel … Quality of Regression R 2 Residual Plot Cross Validation What you should learn after class: Confidence Interval Multiple Regression Nonlinear Regression 22

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