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Rice COMP 528 - General Full Factorial Design with K Factors

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Dr. John Mellor-CrummeyDepartment of Computer ScienceRice [email protected] Full Factorial Designwith k FactorsCOMP 528 Lecture 18 24 March 20052Goals for TodayUnderstand• General Full Factorial Design with k Factors—motivation & model—model properties—estimating model parameters—estimating experimental errors—allocating variation to factors and interactions—analyzing significance of factors and interactions• Informal methods—observation method—ranking method3Full Factorial Design with k Factors• Motivation—previous designs we considered had either– only two levels per factor– only two factors—want designs for more than 2 factors, some with more than 2 levels withand without replications• General model—k factors yields 2k-1 effects—k main effects— j factor interactions, 2 ≤ j ≤ k! kj" # $ % & '4Symbol Factor Level 1 Level 2 Level 3A Page Replacement Algorithm LRUV FIFO RANDD Deck Arrangement GROUP FREQY ALPHAP Problem Program Small Medium LargeM Memory Pages 24P 20P 16PExample: Four Factor Paging StudyJain reports study from: Tsao, R.F. and Margolin, B.H. (1971). Multifactor Paging Experiment: II Statistical Methodology. in Freiberger, W., Ed. Statistical Computer Performance Evaluation, Academic Press, NY, 135-162.5Example: Model for 4 Factor Design! yijkln=µ+ (mean response) Ai+ Dj+ Pk+ Ml+ (main effects) "ADij+"APik+"AMil+"DPjk+"DMjl+"PMkl+ (1st order interactions) "ADPijk+"ADMijl+"APMikl+"DPMjkl+ (2nd order interactions) "ADPMijkl+ (3rd order interactions) eijkln (error) i = 1,...,a; j = 1,...,d; k = 1,..., p; l =1,...,m; n =1,...rn = replications; (no replications actually considered in following example)6• Effects: sums are 0• Interactions: sum to 0 along each dimension represented• Errors: sum to 0 among all replications of each experiment! Aii"= Djj"= Pkk"= Mll"= 0! "i #ADij=j=1d$ #APik=k=1p$#AMil=l=1m$#ADPijk=j=1d$#ADPijk=k=1p$0 #ADMijl=j=1d$#ADMijl=l=1m$#APMikl=k=1p$#APMikl=l=1m$0 #ADPMijkl=j=1d$#ADPMijkl=k=1p$#ADPMijkl=l=1m$0similarly for j, k, l dimensionsModel Properties for a 4 Factor Design! eijklnn=1r"= 0, #i, j,k,l7Symbol Factor Level 1 Level 2 Level 3A Page Replacement Algorithm LRUV FIFO RANDD Deck Arrangement GROUP FREQY ALPHAP Problem Program Small Medium LargeM Memory Pages 24P 20P 16PExample: Four Factor Paging StudyGROUP FREQY ALPHAAlgorithm Program 24P 20P 16P 24P 20P 16P 24P 20P 16PLRUV Small 32 48 538 52 244 998 59 536 1348Medium 53 81 1901 112 776 3621 121 1879 4639Large 142 197 5689 262 2625 10012 980 5698 12880FIFO Small 49 67 789 79 390 1373 85 814 1693Medium 100 134 3152 164 1255 4912 206 3394 5838Large 233 350 9100 458 3688 13531 1633 10022 17117RAND Small 62 100 1103 111 480 1782 111 839 2190Medium 96 245 2807 237 1502 6007 286 3092 7654Large 265 2012 12429 517 4870 18602 1728 8834 23134Data for Paging StudyShould we use a multiplicative or additive model?8Symbol Factor Level 1 Level 2 Level 3A Page Replacement Algorithm LRUV FIFO RANDD Deck Arrangement GROUP FREQY ALPHAP Problem Program Small Medium LargeM Memory Pages 24P 20P 16PExample: Four Factor Paging Studylog10 Transformed Data for Paging StudyAlgorithm Program 24P 20P 16P 24P 20P 16P 24P 20P 16PLRUV Small 1.51 1.68 2.73 1.72 2.39 3.00 1.77 2.73 3.13Medium 1.72 1.91 3.28 2.05 2.89 3.56 2.08 3.27 3.67Large 2.15 2.29 3.76 2.42 3.42 4.00 2.99 3.76 4.11FIFO Small 1.69 1.83 2.90 1.90 2.59 3.14 1.93 2.91 3.23Medium 2.00 2.13 3.50 2.21 3.10 3.69 2.31 3.53 3.77Large 2.37 2.54 3.96 2.66 3.57 4.13 3.21 4.00 4.23RAND Small 1.79 2.00 3.04 2.05 2.68 3.25 2.05 2.92 3.34Medium 1.98 2.39 3.45 2.37 3.18 3.78 2.46 3.49 3.88Large 2.42 3.30 4.09 2.71 3.69 4.27 3.24 3.95 4.369Estimating Model Parameters: 4 Factor Design• Organize measured data for two-factor full factorial design as— a x d x p x m matrix of cells: factor (A,D,P,M) at level (i,j,k,l)• Estimate model parameters—grand mean—main effects—first order interactions, e.g.! Ai= y i..." y ....Dj= y . j.." y ....Pk= y ..k." y ....Ml= y ...l" y ....! µ= y ....! "ADij= y ij..# (y ....+ Ai+ Dj)"APik= y i.k.# (y ....+ Ai+ Pk)"DMjl= y . j. l# (y ....+ Dj+ Ml)...10Estimating Model Parameters: 4 Factor Design• More model parameter estimates—second order interactions, e.g.—third order interaction! "ADPijk= y ijk.# (y ....+ Ai+ Dj+ Pk+"ADij+"APik+"DPjk)"ADMijl= y ij.l# (y ....+ Ai+ Dj+ Ml+"ADij+"AMil+"DMjl)...! "ADPMijkl= yijkl# (y ....+ Ai+ Dj+ Pk+ Ml+ "ADij+"APik+"AMil+"DPjk+"DMjl+"PMkl+ "ADPijk+"ADMijl+"APMikl+"DPMjkl)11Example: Four Factor Paging StudyMain EffectsLevelFactor 1 2 3A -0.16 0.02 0.14D -0.37 0.07 0.29P -0.46 -0.03 0.49M -0.69 -0.01 0.7012Degrees of FreedomComponent DOFy 81ybar… 1y-ybar… 80Main effects 8A 2D 2P 2M 2First order interactions 24AD 4AP 4AM 4DP 4DM 4PM 4Second-order interactions 32ADP 8ADM 8APM 8DPM 8Third-order interactions 16ADPM 1613Informal Methods I• Observation method—goal: find combination of factor levels yielding best response—how– inspect mean response column– identify• high values for HB metric• low values for LB metric– unique extreme: associated factor levels give desired combination– multiple extremes: common factor levels provide desired answer14A B C D E Tw Ti Tb-1 -1 -1 -1 1 15.0 25.0 15.21 -1 -1 -1 -1 11.0 41.0 3.0-1 1 -1 -1 -1 25.0 36.0 21.01 1 -1 -1 1 10.0 15.7 8.6-1 -1 1 -1 -1 14.0 63.9 7.51 -1 1 -1 1 10.0 13.2 7.5-1 1 1 -1 1 28.0 36.3 20.21 1 1 -1 -1 11.0 23.0 3.0-1 -1 -1 1 -1 14.0 66.1 6.41 -1 -1 1 1 10.0 9.1 8.4-1 1 -1 1 1 27.0 34.6 15.71 1 -1 1 -1 11.0 23.0 3.0-1 -1 1 1 1 14.0 26.0 12.01 -1 1 1 -1 11.0 38.0 2.0-1 1 1 1 -1 25.0 35.0 17.21 1 1 1 1 11.0 22.0 2.0Scheduler Throughput:Observation MethodA=-1 no preemptionB=1 large timesliceE=1 fairness enabled15Informal Methods II• Ranking method: similar to observation method—rank experiments in decreasing order of response value—observe factor levels that produce consistently good or badresponses16A B C D E Tw Ti Tb-1 1 1 -1 1 28.0 36.3 20.2-1 1 -1 1 1 27.0 34.6 15.7-1 1 -1 -1 -1 25.0 36.0 21.0-1 1 1 1 -1 25.0 35.0 17.2-1 -1 -1 -1 1 15.0 25.0 15.2-1 -1 1 -1 -1 14.0 63.9 7.5-1 -1 -1 1 -1 14.0 66.1 6.4-1 -1 1 1 1 14.0 26.0 12.01 -1 -1 -1 -1 11.0 41.0 3.01 1 1 -1 -1 11.0 23.0 3.01 1 -1 1 -1 11.0 23.0 3.01 -1 1 1 -1 11.0 38.0 2.01 1 1 1 1 11.0 22.0 2.01 1 -1 -1 1 10.0 15.7 8.61 -1 1 -1 1 10.0 13.2 7.51 -1 -1 1 1 10.0 9.1 8.4Scheduler Throughput: Ranking MethodA=-1 no preemption goodA=1 preemption badB=1 large


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