Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data Samples Different experiments

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data... 1 set of experiments Repeated set Comparison of Alternatives Common case – one sample point for each. Conclusions only about this set of experiments.

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data 1 repeated experiment Samples......

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data 1 repeated experiment Samples Characterizing this sample data set Central tendency – means, mode, median Variability – range, std dev, COV, quantiles Fit to known distribution Sample data vs. Population Confidence interval for mean Significance level a Sample size n given ! r % accuracy

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data 1 experiment Samples Different experiments

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data 1 experiment Samples set of experiments Comparison of Alternatives Paired Observations As one sample of pairwise differences a i - b i Confidence interval A B

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data 1 experiment Samples set of experiments Unpaired Observations As multiple samples, sample means and overlapping CIs t-test on mean difference: x a - x b x a, s a, CI a x b, s b, CI b

Data Analysis Overview Experimental environment prototype real sys exec- driven sim trace- driven sim stochastic sim Workload parameters System Config parameters Factor levels Raw Data Samples Predictor values x, factor levels Samples of response... y1y2yny1y2yn Regression models response var = f (predictors)

© 1998, Geoff Kuenning Linear Regression Models What is a (good) model? Estimating model parameters Allocating variation (R 2 ) Confidence intervals for regressions Verifying assumptions visually

© 1998, Geoff Kuenning Confidence Intervals for Regressions Regression is done from a single sample (size n) –Different sample might give different results –True model is y =  0 +  1 x –Parameters b 0 and b 1 are really means taken from a population sample

© 1998, Geoff Kuenning Calculating Intervals for Regression Parameters Standard deviations of parameters: Confidence intervals are b i t s bi where t has n - 2 degrees of freedom !

© 1998, Geoff Kuenning Example of Regression Confidence Intervals Recall s e = 0.13, n = 5,  x 2 = 264, = 6.8 So Using a 90% confidence level, t 0.95;3 = 2.353

© 1998, Geoff Kuenning (0.004) = (0.28,0.30) Regression Confidence Example, cont’d Thus, b 0 interval is –Not significant at 90% And b 1 is –Significant at 90% (and would survive even 99.9% test) ! ! (0.16) = (-0.03,0.73)

© 1998, Geoff Kuenning Confidence Intervals for Nonlinear Regressions For nonlinear fits using exponential transformations: –Confidence intervals apply to transformed parameters –Not valid to perform inverse transformation on intervals

© 1998, Geoff Kuenning Confidence Intervals for Predictions Previous confidence intervals are for parameters –How certain can we be that the parameters are correct? Purpose of regression is prediction –How accurate are the predictions? –Regression gives mean of predicted response, based on sample we took

© 1998, Geoff Kuenning Predicting m Samples Standard deviation for mean of future sample of m observations at x p is Note deviation drops as m  Variance minimal at x = Use t-quantiles with n–2 DOF for interval y mp N S

© 1998, Geoff Kuenning Example of Confidence of Predictions Using previous equation, what is predicted time for a single run of 8 loops? Time = (8) = 2.67 Standard deviation of errors s e = % interval is then ypyp N S ! (0.14) = (2.34, 3.00)

© 1998, Geoff Kuenning Multiple linear regression – more than one predictor variable Categorical predictors – some of the predictors aren’t quantitative but represent categories Curvilinear regression – nonlinear relationship Transformations – when errors not normally distributed or variance not constant Handling outliers Common mistakes in regression analysis Other Regression Methods

© 1998, Geoff Kuenning Multiple Linear Regression Models with more than one predictor variable But each predictor variable has a linear relationship to the response variable Conceptually, plotting a regression line in n- dimensional space, instead of 2-dimensional

© 1998, Geoff Kuenning Basic Multiple Linear Regression Formula Response y is a function of k predictor variables x 1,x 2,..., x k y = b 0 + b 1 x 1 + b 2 x b k x k + e

© 1998, Geoff Kuenning A Multiple Linear Regression Model Given sample of n observations model consists of n equations (note typo in book):

© 1998, Geoff Kuenning Looks Like It’s Matrix Arithmetic Time y = Xb +e...

© 1998, Geoff Kuenning Analysis of Multiple Linear Regression Listed in box 15.1 of Jain Not terribly important (for our purposes) how they were derived –This isn’t a class on statistics But you need to know how to use them Mostly matrix analogs to simple linear regression results

© 1998, Geoff Kuenning Example of Multiple Linear Regression Internet Movie Database keeps popularity ratings of movies (in numerical form) Postulate popularity of Academy Award winning films is based on two factors - –Age –Running time Produce a regression rating = b 0 + b 1 (length) +b 2 (age)

© 1998, Geoff Kuenning Some Sample Data TitleAgeLengthRating Silence of the Lambs Terms of Endearment Rocky Oliver! Marty Gentleman’s Agreement Mutiny on the Bounty It Happened One Night

© 1998, Geoff Kuenning Now for Some Tedious Matrix Arithmetic We need to calculate X, X T, X T X, (X T X) -1, and X T y Because We will see that b = (8.373,.005, ) Meaning the regression predicts: rating = *age – 0.009*length

© 1998, Geoff Kuenning X Matrix for Example

© 1998, Geoff Kuenning Transpose to Get X T

© 1998, Geoff Kuenning Multiply To Get X T X

© 1998, Geoff Kuenning Invert to Get (X T X) -1

© 1998, Geoff Kuenning Multiply to Get X T y

© 1998, Geoff Kuenning Multiply (X T X) -1 (X T y) to Get b

© 1998, Geoff Kuenning How Good Is This Regression Model? How accurately does the model predict the rating of a film based on its age and running time? Best way to determine this analytically is to calculate the errors or

© 1998, Geoff Kuenning Calculating the Errors

© 1998, Geoff Kuenning Calculating the Errors, Continued So SSE = 1.08 SSY = SS0 = SST = SSY - SS0 = = 1.4 SSR = SST - SSE =.33 In other words, this regression stinks

© 1998, Geoff Kuenning Why Does It Stink? Let’s look at the properties of the regression parameters Now calculate standard deviations of the regression parameters

© 1998, Geoff Kuenning Calculating STDEV of Regression Parameters Estimations only, since we’re working with a sample Estimated stdev of

© 1998, Geoff Kuenning Calculating Confidence Intervals At the 90% level, for instance Confidence intervals for Only b 0 is significant, at this level b 0 = 8.37 (2.015)(1.29) = (5.77, 10.97) b 1 =.005 (2.015)(.01) = (-.02,.02) b 2 = (2.015)(.008) = (-.03,.01) ! ! !

© 1998, Geoff Kuenning Analysis of Variance So, can we really say that none of the predictor variables are significant? –Not yet; predictors may be correlated F-test can be used for this purpose –E.g., to determine if the SSR is significantly higher than the SSE –Equivalent to testing that y does not depend on any of the predictor variables

© 1998, Geoff Kuenning Running an F-Test Need to calculate SSR and SSE From those, calculate mean squares of the regression (MSR) and the errors (MSE) MSR/MSE has an F distribution If MSR/MSE > F-table, predictors explain a significant fraction of response variation Note typos in book’s table 15.3 –SSR has k degrees of freedom –SST matches

© 1998, Geoff Kuenning F-Test for Our Example SSR =.33 SSE = 1.08 MSR = SSR/k =.33/2 =.16 MSE = SSE/(n-k-1) = 1.08/( ) =.22 F-computed = MSR/MSE =.76 F [90; 2,5] = 3.78 (at 90%) So it fails the F-test at 90% (miserably)

© 1998, Geoff Kuenning Multicollinearity If two predictor variables are linearly dependent, they are collinear –Meaning they are related –And thus the second variable does not improve the regression –In fact, it can make it worse Typical symptom is inconsistent results from various significance tests

© 1998, Geoff Kuenning Finding Multicollinearity Must test correlation between predictor variables If it’s high, eliminate one and repeat the regression without it If the significance of regression improves, it’s probably due to collinearity between the variables

© 1998, Geoff Kuenning Is Multicollinearity a Problem in Our Example? Probably not, since the significance tests are consistent But let’s check, anyway Calculate correlation of age and length After tedious calculation, -.25 –Not especially correlated Important point - adding a predictor variable does not always improve a regression

© 1998, Geoff Kuenning Why Didn’t Regression Work Well Here? Check the scatter plots –Rating vs. age –Rating vs. length Regardless of how good or bad regressions look, always check the scatter plots

© 1998, Geoff Kuenning Rating vs. Length

© 1998, Geoff Kuenning Rating vs. Age

© 1998, Geoff Kuenning Regression With Categorical Predictors Regression methods discussed so far assume numerical variables What if some of your variables are categorical in nature? Use techniques discussed later in the class if all predictors are categorical Levels - number of values a category can take

© 1998, Geoff Kuenning Handling Categorical Predictors If only two levels, define b i as follows –b i = 0 for first value –b i = 1 for second value This definition is missing from book in section 15.2 Can use +1 and -1 as values, instead Need k-1 predictor variables for k levels –To avoid implying order in categories

© 1998, Geoff Kuenning Categorical Variables Example Which is a better predictor of a high rating in the movie database, winning an Oscar,winning the Golden Palm at Cannes, or winning the New York Critics Circle?

© 1998, Geoff Kuenning Choosing Variables Categories are not mutually exclusive x 1 = 1 if Oscar 0 if otherwise x 2 = 1 if Golden Palm 0 if otherwise x 3 = 1 if Critics Circle Award 0 if otherwise y = b 0 +b 1 x 1 +b 2 x 2 +b 3 x 3

© 1998, Geoff Kuenning A Few Data Points Title Rating Oscar Palm NYC Gentleman’s Agreement 7.5 X X Mutiny on the Bounty 7.6 X Marty 7.4 X X X If 7.8 X La Dolce Vita 8.1 X Kagemusha 8.2 X The Defiant Ones 7.5 X Reds 6.6 X High Noon 8.1 X

© 1998, Geoff Kuenning And Regression Says... How good is that? R 2 is 34% of the variation –Better than age and length –But still no great shakes Are the regression parameters significant at the 90% level?

© 1998, Geoff Kuenning Curvilinear Regression Linear regression assumes a linear relationship between predictor and response What if it isn’t linear? You need to fit some other type of function to the relationship

© 1998, Geoff Kuenning When To Use Curvilinear Regression Easiest to tell by sight Make a scatter plot –If plot looks non-linear, try curvilinear regression Or if non-linear relationship is suspected for other reasons Relationship should be convertible to a linear form

© 1998, Geoff Kuenning Types of Curvilinear Regression Many possible types, based on a variety of relationships: Many others

© 1998, Geoff Kuenning Transform Them to Linear Forms Apply logarithms, multiplication, division, whatever to produce something in linear form I.e., y = a + b*something Or a similar form If predictor appears in more than one transformed predictor variable, correlation likely

© 1998, Geoff Kuenning Transformations Using some function of the response variable y in place of y itself Curvilinear regression is one example of transformation But techniques are more generally applicable

© 1998, Geoff Kuenning When To Transform? If known properties of the measured system suggest it If the data’s range covers several orders of magnitude If the homogeneous variance assumption of the residuals is violated

© 1998, Geoff Kuenning Transforming Due To Homoscedasticity If spread of scatter plot of residual vs. predicted response is not homogeneous, Then residuals are still functions of the predictor variables Transformation of response may solve the problem

© 1998, Geoff Kuenning What Transformation To Use? Compute standard deviation of the residuals Plot as function of the mean of the observations –Assuming multiple experiments for single set of predictor values Check for linearity - if it is, use a log transform

© 1998, Geoff Kuenning Other Tests for Transformations If variance against mean of observations is linear, use square root transform If standard deviation against mean squared is linear, use inverse transform If standard deviation against mean to a power is linear, use a power transform More covered in the book

© 1998, Geoff Kuenning General Transformation Principle For some observed function if transform to

© 1998, Geoff Kuenning For Example, A log transformation: If the standard deviation against the mean is linear, then g(y) = ay So

© 1998, Geoff Kuenning Outliers Atypical observations might be outliers –Measurements that are not truly characteristic –By chance, several standard deviations out –Or mistakes might have been made in measurement Which leads to a problem: Do you include outliers in analysis or not?

© 1998, Geoff Kuenning Deciding How To Handle Outliers 1. Find them (by looking at scatter plot) 2. Check carefully for experimental error 3. Repeat experiments at predictor values for the outlier 4. Decide whether to include or not include outliers –Or do analysis both ways Question: Is the first point in the example an outlier on the rating vs. age plot?

© 1998, Geoff Kuenning Common Mistakes in Regression Generally based on taking shortcuts Or not being careful Or not understanding some fundamental principles of statistics

© 1998, Geoff Kuenning Not Verifying Linearity Draw the scatter plot If it isn’t linear, check for curvilinear possibilities Using linear regression when the relationship isn’t linear is misleading

© 1998, Geoff Kuenning Relying on Results Without Visual Verification Always check the scatter plot as part of regression –Examining the line regression predicts vs. the actual points Particularly important if regression is done automatically

© 1998, Geoff Kuenning Attaching Importance To Values of Parameters Numerical values of regression parameters depend on scale of predictor variables So just because a particular parameter’s value seems “small” or “large,” not necessarily an indication of importance E.g., converting seconds to microseconds doesn’t change anything fundamental –But magnitude of associated parameter changes

© 1998, Geoff Kuenning Not Specifying Confidence Intervals Samples of observations are random Thus, regression performed on them yields parameters with random properties Without a confidence interval, it’s impossible to understand what a parameter really means

© 1998, Geoff Kuenning Not Calculating Coefficient of Determination Without R 2, difficult to determine how much of variance is explained by the regression Even if R 2 looks good, safest to also perform an F-test The extra amount of effort isn’t that large, anyway

© 1998, Geoff Kuenning Using Coefficient of Correlation Improperly Coefficient of determination is R 2 Coefficient of correlation is R R 2 gives percentage of variance explained by regression, not R E.g., if R is.5, R 2 is.25 –And the regression explains 25% of variance –Not 50%

© 1998, Geoff Kuenning Using Highly Correlated Predictor Variables If two predictor variables are highly correlated, using both degrades regression E.g., likely to be a correlation between an executable’s on-disk size and in-core size –So don’t use both as predictors of run time Which means you need to understand your predictor variables as well as possible

© 1998, Geoff Kuenning Using Regression Beyond Range of Observations Regression is based on observed behavior in a particular sample Most likely to predict accurately within range of that sample –Far outside the range, who knows? E.g., a regression on run time of executables that are smaller than size of main memory may not predict performance of executables that require much VM activity

© 1998, Geoff Kuenning Using Too Many Predictor Variables Adding more predictors does not necessarily improve the model More likely to run into multicollinearity problems So what variables to choose? –Subject of much of this course

© 1998, Geoff Kuenning Measuring Too Little of the Range Regression only predicts well near range of observations If you don’t measure the commonly used range, regression won’t predict much E.g., if many programs are bigger than main memory, only measuring those that are smaller is a mistake

© 1998, Geoff Kuenning Assuming Good Predictor Is a Good Controller Correlation isn’t necessarily control Just because variable A is related to variable B, you may not be able to control values of B by varying A E.g., if number of hits on a Web page and server bandwidth are correlated, you might not increase hits by increasing bandwidth Often, a goal of regression is finding control variables

For Discussion Today Project Proposal 1.Statement of hypothesis 2.Workload decisions 3.Metrics to be used 4.Method