Lecture13 - Simple Linear Regression With Thanks to My...

Info icon This preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
With Thanks to My Students in AMS572 – Data Analysis Simple Linear Regression 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
1. Introduction Response (out come or dependent) variable ( Y ): height of the wife Predictor (explanatory or independent) variable ( X ): height of the husband Example: George Bush :1.81m Laura Bush: ? David Beckham: 1.83m Victoria Beckham: 1.68m Brad Pitt: 1.83m Angelina Jolie: 1.70m To predict height of the wife in a couple, based on the husband’s height 2
Image of page 2
Regression analysis: The earliest form of linear regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809. The method was extended by Francis Galton in the 19th century to describe a biological phenomenon. This work was extended by Karl Pearson and Udny Yule to a more general statistical context around 20th century. ●1 regression analysis is a statistical methodology to estimate the relationship of a response variable to a set of predictor variable. History: ●1 when there is just one predictor variable, we will use simple linear regression . When there are two or more predictor variables, we use multiple linear regression . when it is not clear which variable represents a response and which is a predictor, correlation analysis is used to study the strength of the relationship 3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
A probabilistic model We denote the n observed values of the predictor variable x as n x x x ..., , , 2 1 We denote the corresponding observed values of the response variable Y as n y y y ..., , , 2 1 4
Image of page 4
Notations of the simple linear Regression - Observed value of the random variable Y i depends on x i i y 0 1 ( 1, 2, ..., ) i i i Y x i n β β ε = + + = i ε 2 ) ( 0 ) ( σ ε ε = = i i Var E 0 1 ( ) i i i E Y x μ β β = = + - random error with unknown mean of Y i Unknown Slope True Regression Line Unknown Intercept 5
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
6
Image of page 6
i Y 2 σ Linear function of the predictor variable Have a common variance, Same for all values of x . i ε Normally distributed 4 BASIC ASSUMPTIONS – for statistical inference Independent 7
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Comments: 1. Linear not in x But in the parameters and 2. Predictor variable is not set as predetermined fixed values, is random along with Y. The model can be considered as a conditional model x Y E log ) ( 1 0 β β + = 1 0 β β Example: x x X Y E 1 0 ) | ( β β + = = linear, logx = x* Example: Height and Weight of the children. Height (X) – given Weight (Y) – predict Conditional expectation of Y given X = x 8
Image of page 8
2. Fitting the Simple Linear Regression Model 2.1 Least Squares (LS) Fit 9
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example 10.1 (Tires Tread Wear vs. Mileage: Scatter Plot. From: Statistics and Data Analysis; Tamhane and Dunlop; Prentice Hall. ) 10
Image of page 10
0 1 0 1 ( ) ( 1,2, ..... ) i i y x y x i n β β β β = + - + = 2 0 1 1 [ ( )] n i i i Q y x β β = = - + 11
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The “best” fitting straight line in the sense of minimizing Q: LS estimate One way to find the LS estimate and Setting these partial derivatives equal to zero and simplifying, we get 1 β 0 1 1 1 2 0 1 1 1 1 n n i i i i n n n i i i i i i i n x y x x x y β β β β = = = = = + = + = 0 1 1 0 0 1 1 1 2 [ ( )] 2 [ ( )] n i i i n i i i i Q y x Q x y x β β β β β β = = = - - + = - - + 0 β 12
Image of page 12
Solve the equations and we get 2 1 1 1 1 0 2 2 1 1 1 1 1 1 2 2 1 1 ( )( ) ( )( ) ( ) ( )( ) ( ) n n n
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern