19-1©2008 Raj JainCSE567MWashington University in St. Louis22kk--ppFractional Fractional Factorial DesignsFactorial DesignsRaj Jain Washington University in Saint LouisSaint Louis, MO [email protected] slides are available on-line at:http://www.cse.wustl.edu/~jain/cse567-08/19-2©2008 Raj JainCSE567MWashington University in St. LouisOverviewOverview! 2k-pFractional Factorial Designs! Sign Table for a 2k-pDesign ! Confounding! Other Fractional Factorial Designs! Algebra of Confounding! Design Resolution19-3©2008 Raj JainCSE567MWashington University in St. Louis22kk--ppFractional Factorial DesignsFractional Factorial Designs! Large number of factors⇒ large number of experiments⇒ full factorial design too expensive⇒ Use a fractional factorial design ! 2k-pdesign allows analyzing k factors with only 2k-pexperiments.2k-1design requires only half as many experiments2k-2design requires only one quarter of the experiments19-4©2008 Raj JainCSE567MWashington University in St. LouisExample: 2Example: 277--44DesignDesign! Study 7 factors with only 8 experiments!19-5©2008 Raj JainCSE567MWashington University in St. LouisFractional Design FeaturesFractional Design Features! Full factorial design is easy to analyze due to orthogonality ofsign vectors.Fractional factorial designs also use orthogonal vectors. That is:" The sum of each column is zero.∑ixij=0 ∀ jjth variable, ith experiment." The sum of the products of any two columns is zero.∑ixijxil=0 ∀ j≠ l " The sum of the squares of each column is 27-4, that is, 8.∑ixij2= 8 ∀ j19-6©2008 Raj JainCSE567MWashington University in St. LouisAnalysis of Fractional Factorial DesignsAnalysis of Fractional Factorial Designs! Model:! Effects can be computed using inner products.19-7©2008 Raj JainCSE567MWashington University in St. LouisExample 19.1Example 19.1! Factors A through G explain 37.26%, 4.74%, 43.40%, 6.75%, 0%, 8.06%, and 0.03% of variation, respectively.⇒ Use only factors C and A for further experimentation.19-8©2008 Raj JainCSE567MWashington University in St. LouisSign Table for a 2Sign Table for a 2kk--ppDesign Design Steps:1. Prepare a sign table for a full factorial design with k-p factors.2. Mark the first column I.3. Mark the next k-p columns with the k-p factors.4. Of the (2k-p-k-p-1) columns on the right, choose p columns and mark them with the p factors which were not chosen in step 1.19-9©2008 Raj JainCSE567MWashington University in St. LouisExample: 2Example: 277--44Design Design !19-10©2008 Raj JainCSE567MWashington University in St. LouisExample: 2Example: 244--11DesignDesign!19-11©2008 Raj JainCSE567MWashington University in St. LouisConfoundingConfounding! Confounding: Only the combined influence of two or more effects can be computed.19-12©2008 Raj JainCSE567MWashington University in St. LouisConfounding (Cont)Confounding (Cont)! ⇒ Effects of D and ABC are confounded. Not a problem if qABCis negligible.19-13©2008 Raj JainCSE567MWashington University in St. LouisConfounding (Cont)Confounding (Cont)! Confounding representation: D=ABCOther Confoundings:! I=ABCD ⇒ confounding of ABCD with the mean.19-14©2008 Raj JainCSE567MWashington University in St. LouisOther Fractional Factorial DesignsOther Fractional Factorial Designs! A fractional factorial design is not unique. 2pdifferent designs. ! Confoundings:Not as good as the previous design.19-15©2008 Raj JainCSE567MWashington University in St. LouisAlgebra of ConfoundingAlgebra of Confounding! Given just one confounding, it is possible to list all other confoundings.! Rules:" I is treated as unity. " Any term with a power of 2 is erased.Multiplying both sides by A:Multiplying both sides by B, C, D, and AB:19-16©2008 Raj JainCSE567MWashington University in St. LouisAlgebra of Confounding (Cont)Algebra of Confounding (Cont)and so on.! Generator polynomial: I=ABCDFor the second design: I=ABC.! In a 2k-pdesign, 2peffects are confounded together.19-17©2008 Raj JainCSE567MWashington University in St. LouisExample 19.7Example 19.7! In the 27-4design:! Using products of all subsets:19-18©2008 Raj JainCSE567MWashington University in St. LouisExample 19.7 (Cont)Example 19.7 (Cont)! Other confoundings:19-19©2008 Raj JainCSE567MWashington University in St. LouisDesign ResolutionDesign Resolution! Order of an effect = Number of termsOrder of ABCD = 4, order of I = 0. ! Order of a confounding = Sum of order of two termsE.g., AB=CDE is of order 5.! Resolution of a Design= Minimum of orders of confoundings! Notation: RIII= Resolution-III = 2k-pIII! Example 1: I=ABCD ⇒ RIV= Resolution-IV = 24-1IV19-20©2008 Raj JainCSE567MWashington University in St. LouisDesign Resolution (Cont)Design Resolution (Cont)! Example 2:I = ABD ⇒RIIIdesign. ! Example 3:! This is a resolution-III design.! A design of higher resolution is considered a better design.19-21©2008 Raj JainCSE567MWashington University in St. LouisCase Study 19.1: Latex vs. troffCase Study 19.1: Latex vs. troff19-22©2008 Raj JainCSE567MWashington University in St. LouisCase Study 19.1 (Cont)Case Study 19.1 (Cont)! Design: 26-1with I=BCDEF19-23©2008 Raj JainCSE567MWashington University in St. LouisCase Study 19.1: ConclusionsCase Study 19.1: Conclusions! Over 90% of the variation is due to: Bytes, Program, and Equations and a second order interaction.! Text file size were significantly different making it's effect more than that of the programs.! High percentage of variation explained by the ``program ×Equation'' interaction ⇒ Choice of the text formatting program depends upon the number of equations in the text. troff not as good for equations.19-24©2008 Raj JainCSE567MWashington University in St. LouisCase Study 19.1: Conclusions (Cont)Case Study 19.1: Conclusions (Cont)! Low ``Program × Bytes'' interaction ⇒ Changing the file size affects both programs in a similar manner.! In next phase, reduce range of file sizes. Alternately, increasethe number of levels of file sizes.19-25©2008 Raj JainCSE567MWashington University in St. LouisCase Study 19.2: Scheduler DesignCase Study 19.2: Scheduler Design! Three classes of jobs: word processing, data processing, and background data processing.! Design: 25-1with I=ABCDE19-26©2008 Raj JainCSE567MWashington University in St. LouisMeasured ThroughputsMeasured Throughputs19-27©2008 Raj JainCSE567MWashington University in St. LouisEffects and Variation ExplainedEffects and Variation Explained19-28©2008 Raj
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