CPE 619 2k-p Factorial DesignPART IV: Experimental Design and AnalysisIntroduction2k-p Fractional Factorial DesignsExample: 27-4 DesignFractional Design FeaturesAnalysis of Fractional Factorial DesignsExample 19.1Sign Table for a 2k-p DesignExample: 27-4 DesignExample: 24-1 DesignConfoundingConfounding (cont’d)Slide 14Other Fractional Factorial DesignsAlgebra of ConfoundingAlgebra of Confounding (cont’d)Example 19.7Example 19.7 (cont’d)Design ResolutionDesign Resolution (cont’d)Case Study 19.1: Latex vs. troffCase Study 19.1 (cont’d)Case Study 19.1: ConclusionsCase Study 19.1: Conclusions (cont’d)Case Study 19.2: Scheduler DesignMeasured ThroughputsEffects and Variation ExplainedCase Study 19.2: ConclusionsSummaryExercise 19.1Exercise 19.2HomeworkCPE 6192k-p Factorial DesignAleksandar MilenkovićThe LaCASA LaboratoryElectrical and Computer Engineering DepartmentThe University of Alabama in Huntsvillehttp://www.ece.uah.edu/~milenkahttp://www.ece.uah.edu/~lacasa2PART IV: Experimental Design and AnalysisHow to:Design a proper set of experiments for measurement or simulationDevelop a model that best describes the data obtainedEstimate the contribution of each alternative to the performanceIsolate the measurement errorsEstimate confidence intervals for model parametersCheck if the alternatives are significantly differentCheck if the model is adequate3Introduction2k-p Fractional Factorial DesignsSign Table for a 2k-p Design ConfoundingOther Fractional Factorial DesignsAlgebra of ConfoundingDesign Resolution42k-p Fractional Factorial DesignsLarge number of factors ) large number of experiments ) full factorial design too expensive ) Use a fractional factorial design 2k-p design allows analyzing k factors with only 2k-p experiments2k-1 design requires only half as many experiments2k-2 design requires only one quarter of the experiments5Example: 27-4 DesignStudy 7 factors with only 8 experiments!6Fractional Design FeaturesFull factorial design is easy to analyze due to orthogonality of sign vectorsFractional factorial designs also use orthogonal vectorsThat isThe sum of each column is zeroi xij =0 8 jjth variable, ith experimentThe sum of the products of any two columns is zeroi xijxil=0 8 j l The sum of the squares of each column is 27-4, that is, 8i xij2 = 8 8 j7Analysis of Fractional Factorial DesignsModel:Effects can be computed using inner products8Example 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 experimentation9Sign Table for a 2k-p Design Steps:1. Prepare a sign table for a full factorial design with k-p factors2. Mark the first column I3. Mark the next k-p columns with the k-p factors4. 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 110Example: 27-4 Design11Example: 24-1 Design12ConfoundingConfounding: Only the combined influence of two or more effects can be computed.13Confounding (cont’d)) Effects of D and ABC are confounded. Not a problem if qABC is negligible.14Confounding (cont’d)Confounding representation: D=ABCOther Confoundings:I=ABCD ) confounding of ABCD with the mean15Other Fractional Factorial DesignsA fractional factorial design is not unique. 2p different designsConfoundings:Not as good as the previous design16Algebra 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:17Algebra of Confounding (cont’d)Multiplying both sides by B, C, D, and AB:and so on.Generator polynomial: I=ABCDFor the second design: I=ABC.In a 2k-p design, 2p effects are confounded together18Example 19.7In the 27-4 design:Using products of all subsets:19Example 19.7 (cont’d)Other confoundings:20Design 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 = 24-1IV21Design Resolution (cont’d)Example 2:I = ABD  RIII design. Example 3:This is a resolution-III designA design of higher resolution is considered a better design22Case Study 19.1: Latex vs. troff23Case Study 19.1 (cont’d)Design: 26-1 with I=BCDEF24Case 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 equations25Case Study 19.1: Conclusions (cont’d)Low ``Program £ Bytes'' interaction ) Changing the file size affects both programs in a similar manner.In next phase, reduce range of file sizes. Alternately, increase the number of levels of file sizes.26Case Study 19.2: Scheduler DesignThree classes of jobs: word processing, data processing, and background data processing.Design: 25-1 with I=ABCDE27Measured Throughputs28Effects and Variation Explained29Case Study 19.2: ConclusionsFor word processing throughput (TW): A (Preemption), B (Time slice), and AB are important.For interactive jobs: E (Fairness), A (preemption), BE, and B (time slice).For background jobs: A (Preemption), AB, B (Time slice), E (Fairness).May use different policies for different classes of workloads.Factor C (queue assignment) or any of its interaction do not have any significant impact on the throughput.Factor D (Requiring) is not effective.Preemption (A) impacts all workloads significantly.Time slice (B) impacts less than preemption.Fairness (E) is important for interactive jobs and slightly important for background jobs.30SummaryFractional factorial designs allow a large number of variables to be analyzed with a small number of experimentsMany effects and interactions are confoundedThe resolution of a design is the sum of the order of confounded effectsA design with higher resolution is