17-1©2006 Raj JainCSE567MWashington University in St. Louis22kkFactorial Factorial DesignsDesignsRaj Jain Washington University in Saint LouisSaint Louis, MO [email protected] slides are available on-line at:http://www.cse.wustl.edu/~jain/cse567-06/17-2©2006 Raj JainCSE567MWashington University in St. LouisOverviewOverview! 22Factorial Designs! Model! Computation of Effects! Sign Table Method! Allocation of Variation! General 2kFactorial Designs17-3©2006 Raj JainCSE567MWashington University in St. Louis22kkFactorial DesignsFactorial Designs! k factors, each at two levels.! Easy to analyze.! Helps in sorting out impact of factors.! Good at the beginning of a study.! Valid only if the effect is unidirectional. E.g., memory size, the number of disk drives17-4©2006 Raj JainCSE567MWashington University in St. Louis2222Factorial DesignsFactorial Designs! Two factors, each at two levels.17-5©2006 Raj JainCSE567MWashington University in St. LouisModelModelInterpretation: Mean performance = 40 MIPSEffect of memory = 20 MIPS; Effect of cache = 10 MIPSInteraction between memory and cache = 5 MIPS.17-6©2006 Raj JainCSE567MWashington University in St. LouisComputation of EffectsComputation of Effects17-7©2006 Raj JainCSE567MWashington University in St. LouisComputation of Effects (Cont)Computation of Effects (Cont)Solution:Notice that effects are linear combinations of responses.Sum of the coefficients is zero ⇒ contrasts.17-8©2006 Raj JainCSE567MWashington University in St. LouisComputation of Effects (Cont)Computation of Effects (Cont)Notice:qA= Column A × Column yqB= Column B × Column y17-9©2006 Raj JainCSE567MWashington University in St. LouisSign Table MethodSign Table Method17-10©2006 Raj JainCSE567MWashington University in St. LouisAllocation of VariationAllocation of Variation! Importance of a factor = proportion of the variation explained! For a 22design:! Variation due to A = SSA = 22qA2! Variation due to B = SSB = 22qB2! Variation due to interaction = SSAB = 22qAB2! Fraction explained by A = Variation ≠ Variance17-11©2006 Raj JainCSE567MWashington University in St. LouisDerivationDerivation! Model:Notice1. The sum of entries in each column is zero:2. The sum of the squares of entries in each column is 4:17-12©2006 Raj JainCSE567MWashington University in St. LouisDerivation (Cont)Derivation (Cont)3. The columns are orthogonal (inner product of any two columns is zero):17-13©2006 Raj JainCSE567MWashington University in St. LouisDerivation (Cont)Derivation (Cont)!17-14©2006 Raj JainCSE567MWashington University in St. LouisDerivation (Cont)Derivation (Cont)! Variation of y17-15©2006 Raj JainCSE567MWashington University in St. LouisExample 17.2Example 17.2! Memory-cache study:! Total variation= 2100Variation due to Memory = 1600 (76%)Variation due to cache = 400 (19%)Variation due to interaction = 100 (5%)17-16©2006 Raj JainCSE567MWashington University in St. LouisCase Study 17.1: Interconnection NetsCase Study 17.1: Interconnection Nets! Memory interconnection networks: Omega and Crossbar.! Memory reference patterns: Random and Matrix! Fixed factors:" Number of processors was fixed at 16." Queued requests were not buffered but blocked." Circuit switching instead of packet switching." Random arbitration instead of round robin." Infinite interleaving of memory ⇒ no memory bank contention.17-17©2006 Raj JainCSE567MWashington University in St. Louis2222Design for Interconnection NetworksDesign for Interconnection Networks17-18©2006 Raj JainCSE567MWashington University in St. LouisInterconnection Networks ResultsInterconnection Networks Results! Average throughput = 0.5725! Most effective factor = B = Reference pattern⇒ The address patterns chosen are very different.! Reference pattern explains ∓ 0.1257 (77%) of variation.! Effect of network type = 0.0595Omega networks = Average + 0.0595Crossbar networks = Average - 0.0595! Slight interaction (0.0346) between reference pattern and network type.17-19©2006 Raj JainCSE567MWashington University in St. LouisGeneral 2General 2kkFactorial DesignsFactorial Designs! k factors at two levels each.2kexperiments.2keffects:17-20©2006 Raj JainCSE567MWashington University in St. Louis22kkDesign ExampleDesign Example! Three factors in designing a machine:" Cache size" Memory size" Number of processors17-21©2006 Raj JainCSE567MWashington University in St. Louis22kkDesign Example (cont)Design Example (cont)17-22©2006 Raj JainCSE567MWashington University in St. LouisAnalysis of 2Analysis of 2kkDesignDesign! Number of Processors (C) is the most important factor.17-23©2006 Raj JainCSE567MWashington University in St. LouisSummarySummary! 2kdesign allows k factors to be studied at two levels each! Can compute main effects and all multi-factors interactions! Easy computation using sign table method! Easy allocation of variation using squares of effects17-24©2006 Raj JainCSE567MWashington University in St. LouisExercise 17.1Exercise 17.1Analyze the 23design:" Quantify main effects and all interactions." Quantify percentages of variation explained." Sort the variables in the order of decreasing importance.17-25©2006 Raj JainCSE567MWashington University in St. LouisHomeworkHomeworkModified Exercise 17.1 Analyze the 23design:" Quantify main effects and all interactions." Quantify percentages of variation explained." Sort the variables in the order of decreasing
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