ST 370 Probability and Statistics for Engineers Collecting Engineering Data Three ways of collecting data on the impacts of factors on a response in a system: Retrospective Study: Collect relevant data from historical records; Observational Study: Collect relevant data from current operations, without perturbing with the system; Designed Experiment: Perturb the system and observe the impacts. 1 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Example: Distillation column Engineers were interested in the concentration of acetone in the output stream from a distillation column. In particular, how this response was affected by three factors : Reboil temperature; Condenser temperature; Reflux rate. 2 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Example: Distillation column, retrospective study Collect data from operational records. Possible issues Missing data: records are often incomplete; Incompatible data: response may be hourly average, temperatures may be instantaneous. Some factors may not have changed much, so we cannot detect their impact; Some factors may vary together , so we cannot separate their impacts. Most importantly, we do not know what else might have been changing, and influencing the response. 3 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Example: Distillation column, observational study Collect data from current operations. Some improvement Data collection is more intensive than historical records, so no missing data and variables can be measured on compatible time scales. But some factors may still not have changed much, and other factors may still vary together, so we cannot detect or separate their impacts. With more intensive effort, we can sometimes monitor other factors that might influence the response. 4 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Example: Distillation column, designed experiment Engineers choose two levels of each factor, a low level labeled “-” and a high level labeled “+”. All possible combinations of these lead to 2 3 = 8 treatments : Reboil Temp. Condensate Temp. Reflux Rate - - - + - - - + - + + - - - + + - + - + + + + + 5 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Example: Distillation column, designed experiment (continued) If all other aspects of the distillation process are controlled, any differences in the response for different treatments can be attributed to the differences in the factor levels. Advantages of the designed experiment: All factors are varied, so all effects can be identified; Factors vary independently, so all effects can be separated, and identified with specific factors. If all treatments are used, we can identify interactions : when the level of one factor changes the effect of another factor. 6 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Factorial Designs Because the 2 3 factorial design has only 8 treatments, using all treatments is usually feasible: the complete factorial design . If 16 or 24 runs are feasible, the design can be replicated : each treatment used 2 or 3 times, or more. Often, especially in the early stage of an investigation, more factors need to be considered; because 2 k grows rapidly as k , the number of factors, increases, the complete factorial design may be infeasible, and a fractional factorial design may be used. These designs all have only two levels of each factor; when few factors are involved, three or more levels of each may be used. 7 / 12 The Role of Statistics in Engineering Collecting Engineering Data
ST 370 Probability and Statistics for Engineers Mechanistic and Empirical Models We usually express the effects of various factors on a response through a model . Sometimes basic science provides an idealized mechanistic model, like: Ohm’s Law: E = IR ; Hooke’s Law: F = kX ; Ideal gas law: PV = nRT . 8 / 12 The Role of Statistics in Engineering Mechanistic and Empirical Models
ST 370 Probability and Statistics for Engineers Most situations are too complex to use such simple models, and call instead for empirical models . For example, distillation is controlled by the variation of saturated vapor pressure with temperature, but simple models are for static equilibrium, not the dynamic environment of a continuous process. 9 / 12 The Role of Statistics in Engineering Mechanistic and Empirical Models
ST 370 Probability and Statistics for Engineers Example: Wire bond pull strength In semiconductor manufacturing, a semiconductor is wire-bonded to a frame. The pull strength is the force required to break the bond, and is affected by two factors: wire length; height of a die. No simple physical model exists. We often use a first-order approximation: � pull strength = f (wire length , die height) ≈ β 0 + β 1 × wire length + β 2 × die height . 10 / 12 The Role of Statistics in Engineering Mechanistic and Empirical Models
ST 370 Probability and Statistics for Engineers Example: Wire bond pull strength (continued) Twenty five parts were tested in an observational study. In R: wireBond <- read.csv("Data/Table-01-02.csv") pairs(wireBond) summary(lm(Strength ~ Length + Height, wireBond)) Pull strength increases strongly with wire length and weakly with die height. The fitted (empirical) model is � pull strength = 2 . 26 + 2 . 74 × wire length + 0 . 0125 × die height . 11 / 12 The Role of Statistics in Engineering Mechanistic and Empirical Models
ST 370 Probability and Statistics for Engineers We can use the fitted model to predict pull strength for other combinations of wire length and die height: x <- pretty(wireBond$Length, n = 40) y <- pretty(wireBond$Height, n = 40) strengthLm <- lm(Strength ~ Length + Height, wireBond) z <- predict(strengthLm, expand.grid(Length = x, Height = y)) persp(x, y, matrix(z, length(x), length(y)), xlab = "Wire length", ylab = "Die height", zlab = "Pull strength") 12 / 12 The Role of Statistics in Engineering Mechanistic and Empirical Models
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