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k_14slr.pdf
1.
14-1 ©2008 Raj Jain CSE567M Washington
University in St. Louis Simple Linear Simple Linear Regression Models Regression Models Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu These slides are available on-line at: http://www.cse.wustl.edu/~jain/cse567-08/
2.
14-2 ©2008 Raj Jain CSE567M Washington
University in St. Louis Overview Overview 1. Definition of a Good Model 2. Estimation of Model parameters 3. Allocation of Variation 4. Standard deviation of Errors 5. Confidence Intervals for Regression Parameters 6. Confidence Intervals for Predictions 7. Visual Tests for verifying Regression Assumption
3.
14-3 ©2008 Raj Jain CSE567M Washington
University in St. Louis Simple Linear Regression Models Simple Linear Regression Models ! Regression Model: Predict a response for a given set of predictor variables. ! Response Variable: Estimated variable ! Predictor Variables: Variables used to predict the response. predictors or factors ! Linear Regression Models: Response is a linear function of predictors. ! Simple Linear Regression Models: Only one predictor
4.
14-4 ©2008 Raj Jain CSE567M Washington
University in St. Louis Definition of a Good Model Definition of a Good Model x y x y x y x y Good Good Bad
5.
14-5 ©2008 Raj Jain CSE567M Washington
University in St. Louis Good Model (Cont) Good Model (Cont) ! Regression models attempt to minimize the distance measured vertically between the observation point and the model line (or curve). ! The length of the line segment is called residual, modeling error, or simply error. ! The negative and positive errors should cancel out ⇒ Zero overall error Many lines will satisfy this criterion.
6.
14-6 ©2008 Raj Jain CSE567M Washington
University in St. Louis Good Model (Cont) Good Model (Cont) ! Choose the line that minimizes the sum of squares of the errors. where, is the predicted response when the predictor variable is x. The parameter b0 and b1 are fixed regression parameters to be determined from the data. ! Given n observation pairs {(x1, y1), …, (xn, yn)}, the estimated response for the ith observation is: ! The error is:
7.
14-7 ©2008 Raj Jain CSE567M Washington
University in St. Louis Good Model (Cont) Good Model (Cont) ! The best linear model minimizes the sum of squared errors (SSE): subject to the constraint that the mean error is zero: ! This is equivalent to minimizing the variance of errors (see Exercise).
8.
14-8 ©2008 Raj Jain CSE567M Washington
University in St. Louis Estimation of Model Parameters Estimation of Model Parameters ! Regression parameters that give minimum error variance are: ! where, and
9.
14-9 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.1 Example 14.1 ! The number of disk I/O's and processor times of seven programs were measured as: (14, 2), (16, 5), (27, 7), (42, 9), (39, 10), (50, 13), (83, 20) ! For this data: n=7, Σ xy=3375, Σ x=271, Σ x2=13,855, Σ y=66, Σ y2=828, = 38.71, = 9.43. Therefore, ! The desired linear model is:
10.
14-10 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.1 (Cont) Example 14.1 (Cont)
11.
14-11 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14. (Cont) Example 14. (Cont) ! Error Computation
12.
14-12 ©2008 Raj Jain CSE567M Washington
University in St. Louis Derivation of Regression Parameters Derivation of Regression Parameters ! The error in the ith observation is: ! For a sample of n observations, the mean error is: ! Setting mean error to zero, we obtain: ! Substituting b0 in the error expression, we get:
13.
14-13 ©2008 Raj Jain CSE567M Washington
University in St. Louis Derivation of Regression Parameters (Cont) Derivation of Regression Parameters (Cont) ! The sum of squared errors SSE is:
14.
14-14 ©2008 Raj Jain CSE567M Washington
University in St. Louis Derivation (Cont) Derivation (Cont) ! Differentiating this equation with respect to b1 and equating the result to zero: ! That is,
15.
14-15 ©2008 Raj Jain CSE567M Washington
University in St. Louis Allocation of Variation Allocation of Variation ! Error variance without Regression = Variance of the response and
16.
14-16 ©2008 Raj Jain CSE567M Washington
University in St. Louis Allocation of Variation (Cont) Allocation of Variation (Cont) ! The sum of squared errors without regression would be: ! This is called total sum of squares or (SST). It is a measure of y's variability and is called variation of y. SST can be computed as follows: ! Where, SSY is the sum of squares of y (or Σ y2). SS0 is the sum of squares of and is equal to .
17.
14-17 ©2008 Raj Jain CSE567M Washington
University in St. Louis Allocation of Variation (Cont) Allocation of Variation (Cont) ! The difference between SST and SSE is the sum of squares explained by the regression. It is called SSR: or ! The fraction of the variation that is explained determines the goodness of the regression and is called the coefficient of determination, R2:
18.
14-18 ©2008 Raj Jain CSE567M Washington
University in St. Louis Allocation of Variation (Cont) Allocation of Variation (Cont) ! The higher the value of R2, the better the regression. R2=1 ⇒ Perfect fit R2=0 ⇒ No fit ! Coefficient of Determination = {Correlation Coefficient (x,y)}2 ! Shortcut formula for SSE:
19.
14-19 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.2 Example 14.2 ! For the disk I/O-CPU time data of Example 14.1: ! The regression explains 97% of CPU time's variation.
20.
14-20 ©2008 Raj Jain CSE567M Washington
University in St. Louis Standard Deviation of Errors Standard Deviation of Errors ! Since errors are obtained after calculating two regression parameters from the data, errors have n-2 degrees of freedom ! SSE/(n-2) is called mean squared errors or (MSE). ! Standard deviation of errors = square root of MSE. ! SSY has n degrees of freedom since it is obtained from n independent observations without estimating any parameters. ! SS0 has just one degree of freedom since it can be computed simply from ! SST has n-1 degrees of freedom, since one parameter must be calculated from the data before SST can be computed.
21.
14-21 ©2008 Raj Jain CSE567M Washington
University in St. Louis Standard Deviation of Errors (Cont) Standard Deviation of Errors (Cont) ! SSR, which is the difference between SST and SSE, has the remaining one degree of freedom. ! Overall, ! Notice that the degrees of freedom add just the way the sums of squares do.
22.
14-22 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.3 Example 14.3 ! For the disk I/O-CPU data of Example 14.1, the degrees of freedom of the sums are: ! The mean squared error is: ! The standard deviation of errors is:
23.
14-23 ©2008 Raj Jain CSE567M Washington
University in St. Louis Confidence Intervals for Regression Params Confidence Intervals for Regression Params ! Regression coefficients b0 and b1 are estimates from a single sample of size n ⇒ Random ⇒ Using another sample, the estimates may be different. If β0 and β1 are true parameters of the population. That is, ! Computed coefficients b0 and b1 are estimates of β0 and β1, respectively.
24.
14-24 ©2008 Raj Jain CSE567M Washington
University in St. Louis Confidence Intervals (Cont) Confidence Intervals (Cont) ! The 100(1-α)% confidence intervals for b0 and b1 can be be computed using t[1-α/2; n-2] --- the 1-α/2 quantile of a t variate with n-2 degrees of freedom. The confidence intervals are: And ! If a confidence interval includes zero, then the regression parameter cannot be considered different from zero at the at 100(1-α)% confidence level.
25.
14-25 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.4 Example 14.4 ! For the disk I/O and CPU data of Example 14.1, we have n=7, =38.71, =13,855, and se=1.0834. ! Standard deviations of b0 and b1 are:
26.
14-26 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.4 (Cont) Example 14.4 (Cont) ! From Appendix Table A.4, the 0.95-quantile of a t-variate with 5 degrees of freedom is 2.015. ⇒ 90% confidence interval for b0 is: ! Since, the confidence interval includes zero, the hypothesis that this parameter is zero cannot be rejected at 0.10 significance level. ⇒ b0 is essentially zero. ! 90% Confidence Interval for b1 is: ! Since the confidence interval does not include zero, the slope b1 is significantly different from zero at this confidence level.
27.
14-27 ©2008 Raj Jain CSE567M Washington
University in St. Louis Case Study 14.1: Remote Procedure Call Case Study 14.1: Remote Procedure Call
28.
14-28 ©2008 Raj Jain CSE567M Washington
University in St. Louis Case Study 14.1 (Cont) Case Study 14.1 (Cont) ! UNIX:
29.
14-29 ©2008 Raj Jain CSE567M Washington
University in St. Louis Case Study 14.1 (Cont) Case Study 14.1 (Cont) ! ARGUS:
30.
14-30 ©2008 Raj Jain CSE567M Washington
University in St. Louis Case Study 14.1 (Cont) Case Study 14.1 (Cont) ! Best linear models are: ! The regressions explain 81% and 75% of the variation, respectively. Does ARGUS takes larger time per byte as well as a larger set up time per call than UNIX?
31.
14-31 ©2008 Raj Jain CSE567M Washington
University in St. Louis Case Study 14.1 (Cont) Case Study 14.1 (Cont) ! Intervals for intercepts overlap while those of the slopes do not. ⇒ Set up times are not significantly different in the two systems while the per byte times (slopes) are different.
32.
14-32 ©2008 Raj Jain CSE567M Washington
University in St. Louis Confidence Intervals for Predictions Confidence Intervals for Predictions ! This is only the mean value of the predicted response. Standard deviation of the mean of a future sample of m observations is: ! m =1 ⇒ Standard deviation of a single future observation:
33.
14-33 ©2008 Raj Jain CSE567M Washington
University in St. Louis CI for Predictions (Cont) CI for Predictions (Cont) ! m = ∞ ⇒ Standard deviation of the mean of a large number of future observations at xp: ! 100(1-α)% confidence interval for the mean can be constructed using a t quantile read at n-2 degrees of freedom.
34.
14-34 ©2008 Raj Jain CSE567M Washington
University in St. Louis CI for Predictions (Cont) CI for Predictions (Cont) ! Goodness of the prediction decreases as we move away from the center.
35.
14-35 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.5 Example 14.5 ! Using the disk I/O and CPU time data of Example 14.1, let us estimate the CPU time for a program with 100 disk I/O's. ! For a program with 100 disk I/O's, the mean CPU time is:
36.
14-36 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.5 (Cont) Example 14.5 (Cont) ! The standard deviation of the predicted mean of a large number of observations is: ! From Table A.4, the 0.95-quantile of the t-variate with 5 degrees of freedom is 2.015. ⇒ 90% CI for the predicted mean
37.
14-37 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.5 (Cont) Example 14.5 (Cont) ! CPU time of a single future program with 100 disk I/O's: ! 90% CI for a single prediction:
38.
14-38 ©2008 Raj Jain CSE567M Washington
University in St. Louis Visual Tests for Regression Assumptions Visual Tests for Regression Assumptions Regression assumptions: 1. The true relationship between the response variable y and the predictor variable x is linear. 2. The predictor variable x is non-stochastic and it is measured without any error. 3. The model errors are statistically independent. 4. The errors are normally distributed with zero mean and a constant standard deviation.
39.
14-39 ©2008 Raj Jain CSE567M Washington
University in St. Louis 1. Linear Relationship: Visual Test 1. Linear Relationship: Visual Test ! Scatter plot of y versus x ⇒ Linear or nonlinear relationship
40.
14-40 ©2008 Raj Jain CSE567M Washington
University in St. Louis 2. Independent Errors: Visual Test 2. Independent Errors: Visual Test 1. Scatter plot of εi versus the predicted response ! All tests for independence simply try to find dependence.
41.
14-41 ©2008 Raj Jain CSE567M Washington
University in St. Louis Independent Errors (Cont) Independent Errors (Cont) 2. Plot the residuals as a function of the experiment number
42.
14-42 ©2008 Raj Jain CSE567M Washington
University in St. Louis 3. Normally Distributed Errors: Test 3. Normally Distributed Errors: Test ! Prepare a normal quantile-quantile plot of errors. Linear ⇒ the assumption is satisfied.
43.
14-43 ©2008 Raj Jain CSE567M Washington
University in St. Louis 4. Constant Standard Deviation of Errors 4. Constant Standard Deviation of Errors ! Also known as homoscedasticity ! Trend ⇒ Try curvilinear regression or transformation
44.
14-44 ©2008 Raj Jain CSE567M Washington
University in St. Louis Example 14.6 Example 14.6 For the disk I/O and CPU time data of Example 14.1 1. Relationship is linear 2. No trend in residuals ⇒ Seem independent 3. Linear normal quantile-quantile plot ⇒ Larger deviations at lower values but all values are small Number of disk I/Os Predicted Response Normal Quantile CPU time in ms Residual Residual Quantile
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University in St. Louis Example 14.7: RPC Performance Example 14.7: RPC Performance 1. Larger errors at larger responses 2. Normality of errors is questionable Predicted Response Normal Quantile Residual Residual Quantile
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University in St. Louis Summary Summary ! Terminology: Simple Linear Regression model, Sums of Squares, Mean Squares, degrees of freedom, percent of variation explained, Coefficient of determination, correlation coefficient ! Regression parameters as well as the predicted responses have confidence intervals ! It is important to verify assumptions of linearity, error independence, error normality ⇒ Visual tests
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University in St. Louis Exercise 14.7 Exercise 14.7 ! The time to encrypt a k byte record using an encryption technique is shown in the following table. Fit a linear regression model to this data. Use visual tests to verify the regression assumptions.
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14-54 ©2008 Raj Jain CSE567M Washington
University in St. Louis Exercise 2.1 Exercise 2.1 ! From published literature, select an article or a report that presents results of a performance evaluation study. Make a list of good and bad points of the study. What would you do different, if you were asked to repeat the study?
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14-55 ©2008 Raj Jain CSE567M Washington
University in St. Louis Homework 14 Homework 14 ! Read Chapter 14 ! Submit answers to exercise 14.7 ! Submit answer to exercise 2.1
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