RESPONSE SURFACE MODELLING, RSM

Transcription

1 CHEM-E3205 BIOPROCESS OPTIMIZATION AND SIMULATION LECTURE 3 RESPONSE SURFACE MODELLING, RSM Tool for process optimization

2 HISTORY Statistical experimental design pioneering work R.A. Fisher in 1925: Statistical Methods for Research Workers Randomized trials, repetitions, calculation principles, variance analysis Further development later by Box, Hunter, Scheffé, Taguchi

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4 Fisher took apart as a statistician at Rothamsted Experimental Station which was mostly involved in agricultural research. He started with a study on wheat and continued to work with all sorts of agricultural data, becoming well known in his field. Here he developed the analysis of variance, which brought into light the problems of not just the data but the actual experiments. This led to the science of experimental design and ultimately to his famous Statistical Methods for Research Workers.

5 THE REQUIREMENTS OF THE EMPIRICAL MODEL The model fits well to the measurement results The model interpolates the space between test points reliably The model reliably predicts the direction of improvement

6 WHAT DOES RESPONSE SURFACE MODELLING MEAN? A number of mathematical methods that are utilized in modeling and analyzing the processes by which responses are affected by several variables

7 THE GOALS OF RESPONSE SURFACE MODELLING To understand the impact of the factors (controlled, independent variables) on the responses (dependent variables) and get a description of the system To make predictions or to find a suitable operating area Optimization of the response variables

8 RESPONSE SURFACE PLOTS

9 TOOLS FOR RESPONSE SURFACE MODELLING Experimental designs (factorial designs, central composite designs(ccc and CCF), Box -Behnken and D-optimal designs) Regression analysis (MLR, PLS) Model descriptive parameters (ANOVA tables, coefficients, residuals, lack of fit terms, etc.) Graphical presentation (2D and 3D figures)

10 SOFTWARE SOLUTIONS Modde SPSS Systat SAS Statistix Design Expert Excel Matlab Toolboxes

11 DESIGN OF EXPERIMENTS Factors affecting: Number of independent variables Fitted model (linear, quadratic) Number of experiment Great benefits with lesser effort Full factorial vs. fractional factorial designs Factorial, fractional factorial, L-, D-optimal, Plackett-Burman-designs

12 OBJECTIVES OF MODELING AND EXPERIMENTAL DESIGN From Modde-help During an investigation one needs answers to the following questions: Which factors have a real influence on the responses (results)? Which factors have significant interactions (synergy's or antagonism)? What are the best settings of the factors to achieve optimal conditions for best performance of a process, a system or a product? What are the predicted values of the responses (results) for given settings of the factors? An experimental design can be set up to answer all of these questions.

13 FACTOR DEFINITION Name for the factor Low and high limits Type and use Typically quantitative and controlled Process vs. mixture (formulation) qualitative vs. quantitative Transformation (lin, log, exp, etc) E.g. when max/min >10 Scaling e.g.: Orthogonal (factors are perpendicular to each other i.e. non-correlating).mid-range Unit variance

14 PROCESS FACTOR AND MIXTURE FACTORS Process factors are variables like temperature, ph, etc. which don t act as part of mixture. They are expressed in quantities or in degrees and can their magnitude can be changed independently of each other. With the mixture factors the amount of the components is expressed as a part of the total amount of the mixture. Their experimental area is within the range

15 RESPONSE DEFINITION Naming the response variables Transforming possible but normally not needed MLR (Multiple Linear Regression) and PLS- (Partial Least Squares) scaling are possible

16 MLR AND PLS You can select Multiple regression or PLS to fit the model to the data. MODDE defaults to using Multiple regression when you have 1 to 3 responses, no missing values in the data, and no uncontrolled factors, and to using PLS otherwise. You may override the system default by selecting the desired Fit Method in the Option menu MLR: will separately but automatically fit all of the responses PLS: With PLS all responses are fitted simultaneously.

17 OBJECTIVE OF THE DESIGN Screening => find the most important factors from a larger number of variables RSM => optimization with the chosen variables

18 SEARCH FOR THE OPTIMUM OR DIRECTION OF INCREASING / DECREASING RESPONSE

19 ITERATIVE ADVANCING Process design includes many steps, which may include individual optimization steps The final process consists of a sequence of optimization steps PLAN MODELING OPTIMIZING

20 RSM (RESPONSE SURFACE MODEL) OBJECTIVE Select a response surface model objective when the goal of the study is known already relatively well, with the most important variables and their operational area The object is to evaluate the response with a mathematical model which can be used for predictions, optimization or to the determination of the operating range Suitable for the process factors, mixture factors and to their combinations

21 SELECTING THE MODEL AND DESIGN Model type selection linear interactions quadratic Design types Central composite designs (CCF vs. CCC) faktorisuunnitelmat D-optimal designs Number of replicates Star distance

22 EXAMPLE: CCC-DESIGN (CENTRAL COMPOSITE CIRCUMSCRIBED) FOR TWO FACTORS ph Temp

23 EXAMPLE: CCF-DESIGN (CENTRAL COMPOSITE FACE CENTERED) FOR TWO FACTORS ph Temp

24 EXPERIMENTAL DESIGN WORKSHEET WITH FACTORS AND RESPOSES

25 SECOND ORDER POLYNOMIAL FOR CCC-DESIGN Y a b X c X d X X e X 2 f X Here e.g. : Y is enzyme activity X 1 is ph X 2 is temperature T a f model parameters calculated with linear regression

26 MODEL EVALUATION Fit of the experimental points to the model R 2,Q 2, F-test Statistical significance of the model parameters Analysis of variance (ANOVA) Graphical and numerical presentation

27 COEFFICIENT OF DETERMINATION R 2 SS SS R 2 reg res (Ŷ (Y SSreg SS SS reg i i - Y) - Ŷ ) i res 2 2 (with regression explained sum of (residual sum of squares) (coefficient of determination) squares)

28 VARIANCES y = 9.5x + 7 R 2 = Y:n ka = Y Y Total variances Explained variances X X y = 9.5x + 7 R 2 = Y Residual variances X

29 F-VALUE CALCULATION MS MS F reg res MS MS DF SS DF reg SS res reg reg res res (Explained variance) (Residual variance) (Fisher distribution test value)

30 ANALYSIS OF VARIANCE Degrees of freedom Sum of squares Variances F-values Parameter significances Model significances Model lack of fit

31 DEGREES OF FREEDOM, RESIDUALS DF SS DF SS DF DF F lof reg resid resid pe pe lof p ki SS k SS SS lof pe lof n p 1 ( e ki ( n k n p / / SS e DF DF k k 1) lof pe ) pe ( n k 1) n = number of experiments p = number of parameters (including model constant) n k = number of replicates e k =average of residual in set k k = number of replicate sets (often 1) i= residual i in replicate set k pe=pure error; lof= lack of fit If DF lof = 0, model is saturated, and MODDE can not calculate or show R 2-, R 2 Adjusted- or Q 2 values when fitted with MLR-method.

32 COEFFICIENTS AND SIGNIFICANCES Activity Coeff. Std.Err. P Conf.int(±) Constant e ph Te ph ph Te Te e ph Te

33 Q 2 MODEL PREDICTIVE ABILITY Q 2 SS tot PRESS SS tot Measure the model prediction ability in the new experimental conditions PRESS is calculated as a sum of squares from the difference between of the observed Y values and the model predicted Ypred values using the original prediction area where the estimation points has been removed pointwise

34 GOOD MODEL Coefficient of determination R 2 >0.9 Prediction ability Q 2 >0.7 Model lack of fit not statistically significant

35 2D RESPONSE SURFACE Investigation: Kitiini (MLR) Contour of Activity Temp ph

36 3D RESPONSE SURFACE Investigation: Kitiini (MLR) Response Surface of Activity Activity ph Temp

37 MODEL COMBINATIONS

38 STATIONARY POINT LOCATIONS Response surface regression equation partial derivative zero points Min-min, max-max, or a saddle point (min-max or max-min)

39 THE IMPORTANCE OF THE DESIGN OF EXPERIMENTS A good experimental design methodology allow one to properly distribute the experiments within the factor space so that one can minimize the number of experiments required to develop a statistically sound relationship between factors and a response.

40 SCREENING EXPERIMENTS Select a screening experiments when you: start a new research and know only a little of o the effect of factors on the responses, o the behavior of the responses within the test area or o about the adequate size of the test area. aim to cut down the number of factors and find out the most important factors in relation to the responses Screening experiments fit for process factors, mixture factors or their combinations. For example, if 6 factors were tested with a 2 k experimental plan we would need 2 6 =64 experiments. But if with a screening design is used to test main effects (6) and pairwise interactions (15 cross products) only 22 experiments are needed

41 SUCCESSFUL FRACTIONAL FACTORIAL DESIGN IS BASED ON Small number of important effects The process or system is affected by just some of the main effects or the lower-order interactions form a group of several possibilities Projection feature Fractional factorial design can be projected onto a larger experimental design, in which the most significant main effects occur Consecutive experiment arrangements It is also possible to combine two or more fractional factorial designs, making it possible to find out more about the main effects or interactions

42 SCREENING DESIGNS AND DESIGN RESOLUTIONS (Full) factorial designs Factor values with two or more levels. In such experimental plans one tests every possible combinations of factorial levels. If there is p factors each in two levels one needs N =2 p experiments. Factorial design is orthogonal (balanced) in which everyfactor is independent of each other. Fractional factorial designs 2 level resolution III, IV, V or more. These designs are balanced subsets of a factorial design. Design resolution is dependent on the number of experiments. Resolution III designs: Main effects and interactions may be mixed (aliases) with two way interactions. Resolution IV designs: Two way interactions may be mixed with each other. In resolution III and IV designs only linear models. Resolution V designs: Main effects and two way interactions terms are not aliased. Plackett-Burman designs Can be used to test k=n-1 variables with N experiments. Fractional factorial designs (resolution III), Plackett-Burman designs is used with linear models and interactions are mixed with main effects.

43 TWO LEVEL FULL FACTORIAL DESIGNS

44 TWO LEVEL FRACTIONAL FACTORIAL DESIGNS Consider the main design for a 2 level full factorial experiment for three factors (a,b, and c) shown above. We have multiplied out the values of the interaction terms and split the design into two half fractions based on the value of a b c. One could take a half fraction of the full factorial design based on the runs that a b c = 1 (principle fraction) or a b c = -1 (complimentary fraction). These two fractions have different shadings in the table. In the principle fraction c = a b and in the complimentary fraction c = -a b. These equations are called the generators for the design. The word abc is called the defining word for the design. The defining relation is developed by setting the defining words equal to plus or minus one depending on the fraction of interest. So it can happen that when interpreting the behavior in Y with a model one cannot distinct here is the factor c or interaction ab causing behavior or changes in Y.

45 SCREENING DESIGN (FRACTIONAL FACTORIAL DESIGN) FOR IONS, VITAMIN AND TRACE ELEMENTS TEST

46 CONFOUNDINGS (MIXED INTERACTIONS) Main effect: Confoundings: CO 2-3 Cl - Na + NO - 3 Cl - Ca 2+ Sitraatti - Na + Ca 2+ Zn 2+ Cl - Na + Ca 2+ Mg 2+ Cl - SO2-4 Vitamin Na + SO2-4 Trace el. Ca 2+ SO 2-4 Mn 2+ Cl - Na + SO2-4 Fe 2+ /Fe 3+ Cl - Ca 2+ SO2-4 Cu 2+ Na + Ca 2+ SO2-4 Co 2+ Cl - Na + Ca 2+ SO2-4

47 FACTOR EFFECTS TO FINAL ABSORBANCE WITH THE P- VALUES, COEFFICIENT OF DETERMINATION (R2) AND STATISTICAL SIGNIFICANCE

48 CENTRAL COMPOSITE DESIGN AND THE RESULTING MODEL FOR SPECIFIC GROWTH RATE FOR ION CONCENTRATION OPTIMISATION Cl - 2- SO 4 Mg 2+ Mn 2+ nr: mm mm mm mm 1/h , , , , , , , , , , , , , , , , , , , , ,22 5 0, ,78 5 0, ,24 0, ,76 0, , , , , , ,

49 ITERATIVE OPTIMUM SEARCH WITH MODDE