Hardware/Software Design Methodologies Introduction to DoE and Course Projects

Transcription

1 Hardware/Software Design Methodologies Introduction to DoE and Course Projects Vittorio Zaccaria Dipartimento di Elettronica e Informazione Politecnico di Milano zaccaria@elet.polimi.it for informations and course material go to:

2 Agenda Introduction to this lecture Design of experiments (DoE) Why design of experiments Full factorial design (two levels) Main effects, interaction effects for two factors FFD Graphic visualizations Fractional factorial designs Aliases, design resolution Plackett-Burman designs Three levels FFD, Central composite designs Response surface methods Linear regression Anova analysis

3 Agenda (continued) System Tuning Shell tool Overview Data types, environment variables Shell based scripting Control structures The driver plug-in subsystem The doe plug-in subsystem The optimization plug-in subsystem Response surface models Graphical visualisation utilities Library dependencies (GSL, MPI)

4 Why design of experiments Show the statistical significance of an effect that a particular factor exerts on the dependent variable of interest Extract the maximum amount of information regarding the factors affecting a production process from as few (costly) observations as possible. Gathered information can be used o build high-level models of the process (linear regression). The first statistician to consider a formal mathematical methodology for the design of experiments was Ronald A. Fisher.

5 Full factorial designs, 2**k Factors have two levels which are encoded with + or -, e.g.: Factor Levels - + A. Instruction cache size 2KB 32KB B. Data cache size 2KB 32KB C. Instruction issue width 1 8 D. Inst. cache associativity 1 8

6 Full factorial designs, 2**k All the 2**k combinations are excercised on the target process. Delay (MC)

7 Full factorial designs, 2**k, main effects plot Given a target metric y, we define the main effect for factor A on y as: ME(A) = z(a+)-z(a-) Where z(a+) = i y i (A i =+)/count(a i =+) z(a-) is defined analogously. Main effects plot for a set of factors: z(a-) z(a+)

8 Full factorial designs, factor interactions Conditional main effect of B at + level of A: ME(B A+)=z(B+ A+)-z(B- A+) Where z(b+ A+) = i y i (A i, B i =+)/count(a i, B i =+) Two factor interaction: INT(A,B) = 1/2{ ME(B A+) - ME(B A-)} = 1/2 {ME(A B+) - ME(A B-)}

9 Factor interactions, continued No interaction Mild interaction Strong interaction Synergistic strong interaction Antagonistic interaction

10 Factor interaction visualization

11 Factor interaction visualization, Pareto Diagram of interactions

12 Fundamental principles in Factorial Design The following principles have shown to be true on the majority of industrial processes EFFECT HIERARCHY: Lower order effects are likely to be more important than higher order effects. EFFECT SPARSITY: The number of relatively important effects is likely to be small EFFECT EREDITY: In order to be significant, an interaction should have at least one of its factors to be significant

13 Fractional factorial designs Effect hierarchy principle => 4fi s, 5fi and even 3fi s are not likely to be important. For a 2**5 design, the following runs would be required for estimating the associated effect: There are = 16 3FI, 4FI, 5FI effects, half of the total runs! Using a 2**5 design can be wasteful Use of a FF design instead of full factorial design is usually done for economic reasons.

14 Fractional factorial, an example 2**(5-1) Fractional Factorial Design E=BCD, E is said to be aliased with BCD Effect of E is aliased with interaction BCD E=BCD or I=BCDE I is the defining relation. The following aliases can be inferred: The resolution of the design is IV B, C, D, E are clear (no alias with 2FI, but alias with 3FI) Q is strongly clear (no alias with 2FI and 3FI)

15 Fractional factorial, generators Consider a 2**(6-2) design with the following design generators: 5=12, 6=134 I=125, I=1346 Defining Constrast Subgroup: { I, 125, 1346, 125*1346 = } Resolution is the shortest wordlength of the DCS, in this case is III.

16 Fractional factorials, fold-over Rationale: one can start with a set of experiments for estimating effects for aliased factors/interaction. If an interaction/factor alias is considered important, then procedures for de-aliasing should be introduced. Fold-over on factor F: The design is replicated. The column of factor F in the second design part has the signs inverted. This dissolves aliasing involving F. Complete fold-over: The design is replicated. The column of all factors are inverted in sign. The main effects of factors will not be aliased any more with interactions of second order (clear effects). Other techniques exists but are not covered here.

17 Fractional design tables, 16 runs

18 Fractional design tables, 32 runs

19 Plackett-Burman designs If interaction between variables are considered negligible, a placketburman design can be used to estimate main effects. It consists of generating orthogonal matrices whose elements are all either 1 or -1 (Hadamard matrices). N parameters are estimated in k = n+1 runs. Runs should be a multiple of 4. First row proposed by PB are: n k String Other rows are constructed by shifting the previous one. The last row consists of all -.

20 Three level factorial designs Three-level designs were proposed to model possible curvature in the response function and to handle the case of nominal factors at 3 levels. In fact, a third level for a continuous factor facilitates investigation of a quadratic relationship between the response and each of the factors. Three level full factorial analysis involves the analysis of the combinations of 3 levels, not just two. Three level fractional factorial design is not so intuitive, so we ll skip it.

21 Central composite designs CCD consists of three portions: a complete 2k or fractional 2k-m first-order factorial design in which the factor levels are coded into 1 and 1; Axial points at a distance α from the center point; One design center point Overall, 5 levels are needed to build a CCD Usually, most CCD have the following property: Where K is the number of points in the factorial design. A value of α=1 brings us to face-centered CCD, where only three levels are needed.

22 Response surface methodology Response Surface Methodology is a technique used to create mathematical models for the relationship between one or more responses and a set of input variables. The model functions which we investigate are polynomials of the first order: First order + interaction coefficients: Second order: Higher order polynomials can be introduced, depending on the complexity of the observed effects.

23 Response surface methodology Linear and linear+interaction models can be fitted from data coming from a two-level analysis (Full Factorial/Fractional Factorial) Quadratic and higher order models should be fitted with data coming from second order DoE like CCD and three level factorial designs. Once fitted, models can be used to perform various analysis and optimizations.

24 Response surface methods: Linear regression Observation vector can be written in matrix notation as: B is such that the following sum of square is minimum: Minimization gives us the following:

25 ANOVA Analysis Total sum of squares, indicates the total variability of the data with respect to its mean: The Regression sum of square is the variability included in the fitted model: The sum of square error is the variability not explained by the model:

26 ANOVA Analysis The following index is used to evaluate the goodness of the fit (coefficients of determination) ANOVA Table: p is the number of terms in the fit model, not counting the constant n is the number of samples

27 ANOVA Analysis Hipothesis H0, the MSE and MSR are the same, so the model correcly fits. To check this hypothesis at a confidence level alpha, compute If the following is true: Then the fitted model adequately describes, at the significance alpha, the behavior of the response over the experimental region. Alpha is the probability that this hypotesis is not correct. It is common to include the F value in the analysis of variance table.