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.
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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.
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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.
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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.
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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 23 = 8 treatments:
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Reboil Temp.
Condensate Temp.
Reflux Rate
+
+
+
+
+
+
+
+
+
+
+
+
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.
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The Role of Statistics in Engineering
Collecting Engineering Data
ST 370
Probability and Statistics for Engineers
Factorial Designs
Because the 23 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 2k 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.
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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 .
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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.
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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)
≈ β0 + β1 × wire length + β2 × die height.
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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.
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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")
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The Role of Statistics in Engineering
Mechanistic and Empirical Models