Download presentation

Presentation is loading. Please wait.

Published byDavid Bates Modified over 4 years ago

2
Definition Regression Model Regression Equation Y i = 0 + 1 X i ^ Given a collection of paired data, the regression equation algebraically describes the relationship between the two variables Y i = 0 + 1 X i + ^ ^

3
y -intercept of regression equation 0 0 Slope of regression equation 1 1 Dependent Response Variable Independent Explanatory Variable Residuals (error) Population Parameter Estimate ^ ^ YiYi XiXi YiYi ^

4
Definition Regression Equation Given a collection of paired data, the regression equation Regression Line (line of best fit or least-squares line) is the graph of the regression equation algebraically describes the relationship between the two variables Y i = 0 + 1 X i ^^ ^

5
Definitions Residual (error) for a sample of paired ( x,y ) data, the difference ( y - y ) between an observed sample y -value and the value of y, which is the value of y that is predicted by using the regression equation. Least-Squares Property A straight line satisfies this property if the sum of the squares of the residuals is the smallest sum possible. ^ ^

6
x 1 2 4 5 y 4 24 8 32

7
x 1 2 4 5 y 4 24 8 32 y = 5 + 4 x 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 12345 x y Residual = 7 Residual = -13 Residual = -5 Residual = 11 ^

8
Total Deviation from the mean of the particular point ( x, y ) the vertical distance y - y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y Explained Deviation the vertical distance y - y, which is the distance between the predicted y value and the horizontal line passing through the sample mean y Unexplained Deviation the vertical distance y - y, which is the vertical distance between the point ( x, y ) and the regression line. (The distance y - y is also called a residual. ) ^ ^ ^

9
Total deviation ( y - y ) 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Unexplained deviation ( y - y ) Explained deviation ( y - y ) (5, 32) (5, 25) (5, 17) y = 5 + 4 x ^ y = 17 ^ ^ y x 0123456789 y = 25 y = 32 ^

10
( y - y ) = ( y - y ) + (y - y ) (total deviation) = (explained deviation) + (unexplained deviation) (total variation) = (explained variation) + (unexplained variation) Σ ( y - y ) 2 = Σ ( y - y ) 2 + Σ (y - y) 2 ^ ^ ^ ^ SST = SSR + SSE

11
Q=SSE=Σ (ε) 2 =Σ (y - y) 2 ^ =Σ (y - 0 - 1 X i ) 2 ^ ^ Minimize with respect to 1 and 0 ^^

12
0 = ( y) ( x 2 ) - ( x) ( xy) n( xy) - ( x) ( y) n( x 2 ) - ( x) 2 1 = n( x 2 ) - ( x) 2 ^ ^ ^ ^

13
Multiple Regression Models Polynomial Model Y k = 0 + 1 X 1k ……… k X nk + k Y k = 0 + 1 X+ 2 X 2 ……… k X k + k Y k = 1 X 1k ……… k X nk + k Multiple Regression Models (no intercept)

14
Y = XB + e Y is the n x 1 response vector (n x 1) X is the n x (k + 1) design matrix B is the n x 1 regression coefficients vector e is the n x 1 error (residual) vector 0 ≤ k ≤ n

15
Y k = 0 + 1 X 1k + 2 X 2k + k Describe Y, X, B & e for

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google