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Statistical methods for
EngineeringEngineering
Yun-Hsuan Yeh
Software JMP
2014/12/29 1Yun-Hsuan Yeh
Outline
TYPE OF ANALYSIS APPLICATION
Sample Plan Reduce excessive sampling
Capability Study (SPC) Cpk analysis of Machine,...
DMAIC (6 sigma)
•Xs/Ys SPC selection
•Identify problem
•Customer ‘s requirement
•Select CTQ (Critical to Quality)
•Goal se...
DOE (WLB RDL Short Issue)
Defect mode: RDL short (Width 15um / Spacing 10um)
Factors: PR thickness, Exposure energy, Devel...
Key factor: PR thickness
Optimal solution:
Actually, we just select one or two factors for this problem
2014/12/29 Yun-Hsu...
DOE
Design of experiments are a structured development strategy for
product/process engineering in order to characterize, ...
DOE (Levels)
If unsure of levels, run a few pre-experiments to find an operating
range, then begin the experiment.
3 level...
DOE (Full factorial designs)
A class of designs that use k factors at 2 levels.
t = nk
where
t = the number of tests
n = n...
DOE (Fractional factorial designs)
We have a k factors and 2k-p fractional factorial designs, then the design
follows the ...
DOE (Custom Design)
D or I-optimal designs are the forms of design provided by a computer
algorithm. The reasons for using...
D-Optimal (emphasize the corners) I-Optimal (emphasize the centers)
DOE (Custom Design)
2014/12/29 Yun-Hsuan Yeh 47
Evaluation of DOE
Prediction Variance Profile is used to determine the relative
variability in the response depending on t...
Prediction Variance
By definition we have,
Then , so that different x has distinct variance.
xxxx
xx
xy
1TT2
T
][ˆ
)ˆVar(
...
DOE (Correlation of Xs check)
If Xs were correlated to any great degree
(r > 0.5) then experiment was not well
designed an...
Regression
Learn more about the relationship between several independent or
predictor variables and a dependent or criteri...
R2
A data set has values yi each of which has an associated modeled value f
the yi are called the observed values and the ...
Adjusted R2
To correct the R2 for such situations, an adjusted R2 takes into account the
degrees of freedom of an equation...
Regression (Parameter estimates)
2014/12/29 Yun-Hsuan Yeh 54
: the coefficient is zero.0H
Therefore, the terms, Correlatio...
DOE (Replication)
Replication improve the chance of detecting a statistically significant
effect in the midst of natural p...
DOE (Residuals)
Residuals can tell us whether our assumptions are reasonable and
our choice of model is appropriate.
Resid...
DOE (Response Surface Methods)
High-resolution fractional or full factorial to understand interactions (
add center points...
Postscript
Nowadays, the problem can be solved by the experiences that will not be a
problem. Furthermore, those methods h...
Statistical methods for
EngineeringEngineering
Yun-Hsuan Yeh
Software JMP
2014/12/29 1Yun-Hsuan Yeh
Outline
TYPE OF ANALYSIS APPLICATION
Sample Plan Reduce excessive sampling
Capability Study (SPC) Cpk analysis of Machine,...
6 sigma (DMAIC)
•Xs/Ys SPC selection
•Identify problem
•Customer ‘s requirement
•Select CTQ (Critical to Quality)
•Goal se...
Population, Sample and Variable
A population includes each element from the set of observations that can
be made.
A sample...
Sample Plan
Simple Random Sampling: Every member of the population has an equal
chance of being selected for your sample.
...
Sample Plan
Alpha is the probability of mistakenly detecting a difference even when
there is no difference.
Beta is the pr...
Sample Plan (Difference)
D
The D means that we want to defect the difference under the power.
D is the accuracy of the est...
Sample Plan (K means)
D1 D3
If we fix the Di, then the sample size increases as the total variances
increase. When the Di ...
Sample Size (Traditional industrial)
Compute the UCI and LCI for the instant-in-time variation at 95%
UCL=x+1.96xS, LCI=x-...
Sample Size (Traditional industrial)
Determine the range of the Cl at various sample sizes and compare it as a
percentage ...
Sample Frequency (Traditional industrial)
Use regression to correlate the run order to the data. Determine the change
in t...
Attribute sampling (GO/NG)
Using the Poisson to calculate probabilities, the probability of exactly x
defects or defective...
Attribute sampling (GO/NG)
As an example, consider a wafer manufacturing process with a target of 4
defects per wafer. You...
Postscript
Sample plan is a comparative calculation that depends on the
detection and acceptable risk .
How long should th...
Gr&R
True value True value
Step 1: Set 3-10 parts and 3-5 operators for measuring 3-10 times .
Step 2: Check the tool is r...
Gr&R
The calculation parameters, repeatability, reproducibility, and total
measurement precision ,
22
& RrrR SSS +=
where,...
Gr&R
We have range and average to imply the K factors, since the range and
estimated standard deviation have a relationshi...
Gr&R
Trail K1 Operations K2 Parts K3
2 4.5656 2 3.6525 2 3.6524
3 3.0419 3 2.6963 3 2.6963
4 2.5012 4 2.2991 4 2.2991
5 2....
Gr&R
GRR Rating
The precision-to-tolerance ratio, GRR, shows percent of the specification
window is consumed by measuremen...
Attribute Gauge (GO/NG)
The result of the checks made by attribute gauges is not a numerical value,
but a good/scrap statu...
Gauge Linearity and Bias Study
The gauge linearity and bias study is used extensively in quality control in
manufacturing ...
Degrees of Freedom
The number of degrees of freedom is the number of values in the final
calculation of a statistic that a...
Central Limit Theorem
The distribution of an average tends to be Normal, even when the
distribution from which the average...
SPC (Cpk and Ppk)
The Ca, Cp, and Cpk.
where is estimated by range or sigma of subgroup.
2/)( LCLUSL
x
Ca
−
−
=
µ
σˆ6
LCLU...
On the other hand, Cpk is
we have
SPC (Cpk and PPM of OOS)





 −−
=
σσ ˆ3
,
ˆ3
min
LSLxxUSL
Cpk
3
xUSL
Cpkz
−
=×=
...
SPC (Cpk Level)
Level Cpk PPM
A >1.67 <0.5443
Process capability measures the ability of the process to meet the design
re...
t-Test (σ is unknown)
Populations are normally distributed.
Samples are independent, The hypothesis is defined as:
If judg...
p-value
p-value is the probability of getting an effect of the size or greater by random
chance alone (tail probability), ...
Test Standard Deviations
An F-test is used to test if the variances of two populations are equal. The F
hypothesis is defi...
Step 1: Unequal Variances Test
p > .05 p < .05
t-Test Stdevs are unequal
When the variation by group was not equal and the...
Chi-squared test
There are basically two types of random variables and they yield two types of
data: integral and categori...
Chi-squared test
(Lot-yield comparison)
A test for independence
Test statistic:
t.independenaretionsclassificatwoThe:0H
20...
One way ANOVA (Means Test)
The distribution of the response variable follows a normal distribution..
All populations have ...
Two way ANOVA
The distribution of the response variable follows a normal distribution..
All populations have a common vari...
Two way ANOVA
An experiment with a levels with factor A and b levels with factor B and
sample size is r.
1 … a
1 1…r…
1…r
...
Two way ANOVA
0H :The pains do not depend on gender.
0H :The pains do not depend on drug.
0H :The pains do not depend on g...
Hierarchical Clustering
(Apply to tool mapping or product difference)
The basic process of hierarchical clustering is:
1. ...
Hierarchical Clustering
Single-link : is the distance between clusters and .
Complete-link:
),(),( min,
ba
ba
dCCd
ji CC
j...
K-means Clustering
The process of K-means clustering is:
1. Define the initial groups' centroids. This step can be done us...
DOE (WLB RDL Short Issue)
Defect mode: RDL short (Width 15um / Spacing 10um)
Factors: PR thickness, Exposure energy, Devel...
Key factor: PR thickness
Optimal solution:
Actually, we just select one or two factors for this problem
2014/12/29 Yun-Hsu...
DOE
Design of experiments are a structured development strategy for
product/process engineering in order to characterize, ...
DOE (Levels)
If unsure of levels, run a few pre-experiments to find an operating
range, then begin the experiment.
3 level...
DOE (Full factorial designs)
A class of designs that use k factors at 2 levels.
t = nk
where
t = the number of tests
n = n...
DOE (Fractional factorial designs)
We have a k factors and 2k-p fractional factorial designs, then the design
follows the ...
DOE (Custom Design)
D or I-optimal designs are the forms of design provided by a computer
algorithm. The reasons for using...
D-Optimal (emphasize the corners) I-Optimal (emphasize the centers)
DOE (Custom Design)
2014/12/29 Yun-Hsuan Yeh 47
Evaluation of DOE
Prediction Variance Profile is used to determine the relative
variability in the response depending on t...
Prediction Variance
By definition we have,
Then , so that different x has distinct variance.
xxxx
xx
xy
1TT2
T
][ˆ
)ˆVar(
...
DOE (Correlation of Xs check)
If Xs were correlated to any great degree
(r > 0.5) then experiment was not well
designed an...
Regression
Learn more about the relationship between several independent or
predictor variables and a dependent or criteri...
R2
A data set has values yi each of which has an associated modeled value f
the yi are called the observed values and the ...
Adjusted R2
To correct the R2 for such situations, an adjusted R2 takes into account the
degrees of freedom of an equation...
Regression (Parameter estimates)
2014/12/29 Yun-Hsuan Yeh 54
: the coefficient is zero.0H
Therefore, the terms, Correlatio...
DOE (Replication)
Replication improve the chance of detecting a statistically significant
effect in the midst of natural p...
DOE (Residuals)
Residuals can tell us whether our assumptions are reasonable and
our choice of model is appropriate.
Resid...
DOE (Response Surface Methods)
High-resolution fractional or full factorial to understand interactions (
add center points...
Postscript
Nowadays, the problem can be solved by the experiences that will not be a
problem. Furthermore, those methods h...
Reliability (Censor)
During the T hours of test we observe r failures (where r can be any number
from 0 to n). The failure...
Reliability (Weibull)
The most general expression of the Weibull pdf is given by the two-
parameter Weibull distribution e...
Reliability (Life Distribution)
The Life Distribution platform helps you discover distributional properties
of time-to-eve...
Reliability (Sample Size)
From historical information an
engineer knows that component's
life follows a Weibull distributi...
Reliability (Acceleration)
Failure may be due to mechanical fatigue, corrosion, chemical reaction,
diffusion, migration, e...
Reliability (Arrhenius)
This empirically based model is known as the Arrhenius equation. The acceleration
factor AF betwee...
Reliability (Thermal Cycling Test)
The AF is given by
where f is temperature-cycling frequency m=1/3, n=1.9 (for SnPb sold...
Postscript
AF (Arrhenius) can be estimated through the experiments and mode
fitting.
Acceleration experiment can reduce th...
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DMAIC

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DMAIC

  1. 1. Statistical methods for EngineeringEngineering Yun-Hsuan Yeh Software JMP 2014/12/29 1Yun-Hsuan Yeh
  2. 2. Outline TYPE OF ANALYSIS APPLICATION Sample Plan Reduce excessive sampling Capability Study (SPC) Cpk analysis of Machine, Process or System Gr&R For all measuring equipment Comparative Analysis Compare of two or more groupsComparative Analysis Compare of two or more groups DOE For all new system & process (material, parameters) Correlation Correlation between two objectives Regression X & Y correlation (input and output relationship) Reliability Risk assessments of long term All the items linked with one another Yun-Hsuan Yeh2014/12/29 2
  3. 3. DMAIC (6 sigma) •Xs/Ys SPC selection •Identify problem •Customer ‘s requirement •Select CTQ (Critical to Quality) •Goal setting •Time line •Sampling Plan •Measuring technology •Screening DOE •Optimization (Regression ) •Validation •Xs/Ys SPC selection •SPC control •Documentation •Reliability monitor Yun-Hsuan Yeh2014/12/29 3 •Hypothesis testing •ANOVA •Contingency & Cluster analysis
  4. 4. DOE (WLB RDL Short Issue) Defect mode: RDL short (Width 15um / Spacing 10um) Factors: PR thickness, Exposure energy, Developer time, and PR ash time. 2014/12/29 Yun-Hsuan Yeh 40 Factors: PR thickness, Exposure energy, Developer time, and PR ash time. Faults: We should remove the trials that were anticipated. On the others side, we should not put the depending variables in the DoE. It is waste!!
  5. 5. Key factor: PR thickness Optimal solution: Actually, we just select one or two factors for this problem 2014/12/29 Yun-Hsuan Yeh 41 Actually, we just select one or two factors for this problem that will save many resources.
  6. 6. DOE Design of experiments are a structured development strategy for product/process engineering in order to characterize, optimize and control performance with minimal waste. This is accomplished by experimenting with many factors at the same time. Main Effects or Linear Interactions Powers or Quadratics Correlations between factor and respond. Yun-Hsuan Yeh2014/12/29 42
  7. 7. DOE (Levels) If unsure of levels, run a few pre-experiments to find an operating range, then begin the experiment. 3 levels are used if the factor range is wide. If the relationship is nonlinear, we need to add center point to the predicting. Yun-Hsuan Yeh Level 1 Level 2 The real phenomenon Predicted by linear regression with 2 levels. ??? 2014/12/29 43 Level 1 Level 2 Narrow range Wide range
  8. 8. DOE (Full factorial designs) A class of designs that use k factors at 2 levels. t = nk where t = the number of tests n = number of levels k = number of factors Yun-Hsuan Yeh2014/12/29 44
  9. 9. DOE (Fractional factorial designs) We have a k factors and 2k-p fractional factorial designs, then the design follows the properties: 1.Sum of each column in the sign table is 0 1allfor,0 >=∑ js j j 2.Sum of the product of any two columns is 0 3.Sum of the square of elements in any column is 2k-p .,1,allfor,0 , kjjss kj kj ≠>=∑ 1allfor,22 >= − ∑ js pk j j A B C 1 -1 -1 1 2 1 -1 -1 3 -1 1 -1 4 1 1 1 23-1 Design Yun-Hsuan Yeh2014/12/29 45
  10. 10. DOE (Custom Design) D or I-optimal designs are the forms of design provided by a computer algorithm. The reasons for using D or I-optimal designs instead of standard classical designs generally fall into two categories: Standard factorial or fractional factorial designs require too many runs for the amount of resources or time allowed for the experiment. The design space is constrained (the process space contains factor settings that are not feasible or are impossible to run). Factors Full Fractional D-Optimal 5 32 16 16 6 64 32 28 7 128 64 35 8 256 128 43 9 512 256 52 Yun-Hsuan Yeh2014/12/29 46
  11. 11. D-Optimal (emphasize the corners) I-Optimal (emphasize the centers) DOE (Custom Design) 2014/12/29 Yun-Hsuan Yeh 47
  12. 12. Evaluation of DOE Prediction Variance Profile is used to determine the relative variability in the response depending on the DOE design. Add additional samples and the variance goes down. Modify the design by adding runs and see what happens to the predicted variance. We will consider DOEs for multiple linear regression using linear and polynomials and where errors are due to noise in the data. where is the nx1 vector, x is the nxp matrix and β is the px1 vector. because in using to estimate the responses, we effectively estimate parameters β . That is, we lose p degrees of freedom. βˆˆ xy = pn yy n j jj − ∑ − = = 1 2 )( ˆσ (MSE) yˆ yˆ Yun-Hsuan Yeh2014/12/29 48
  13. 13. Prediction Variance By definition we have, Then , so that different x has distinct variance. xxxx xx xy 1TT2 T ][ˆ )ˆVar( )ˆ(Var)ˆVar( − = = = σ β β xxxx y 1TT 2 ][ ˆ ]ˆVar[ − = σ The Prediction Variance Profile gives a profiler of the relative variance of prediction as a function of each factor at fixed values of the other factors. ˆσ The prediction variance is 0.2241 that means the variance of experimental error is 10, then the prediction variance of response at X1=0 is 10x0.2241=2.241. Yun-Hsuan Yeh2014/12/29 49
  14. 14. DOE (Correlation of Xs check) If Xs were correlated to any great degree (r > 0.5) then experiment was not well designed and should be redone, this will cause collinearity then There is no unique solution for regression coefficients. Partial coefficients are estimating something that does not occur in the data. This step is very important for uncontrolled variables, engineers always ignore uncontrolled variables usually. Yun-Hsuan Yeh2014/12/29 50
  15. 15. Regression Learn more about the relationship between several independent or predictor variables and a dependent or criterion variable. In the most general terms, least squares estimation is aimed at minimizing the sum of squared deviations of the observed values for the dependent variable from those predicted by the model. 2 )]([min∑ − fy θ where is a known function of , and are random variables, and usually assumed to have expectation of 0. )]([min∑ − i i f fy θ )(θf θ ii fy εθ += )( iε Yun-Hsuan Yeh2014/12/29 51 0H : Individual coefficient is equal to 0. 0H : All coefficients are equal to 0.
  16. 16. R2 A data set has values yi each of which has an associated modeled value f the yi are called the observed values and the modeled values f called the predicted value. y 2 ][∑ −= i ireg fySS 2 ][∑ −= i itot yySS tot reg SS SS R −=12 Yun-Hsuan Yeh2014/12/29 52 y
  17. 17. Adjusted R2 To correct the R2 for such situations, an adjusted R2 takes into account the degrees of freedom of an equation. When you suspect that an R2 is higher than it should be, calculate the R2 and adjusted R2. If the R2 and the adjusted R2 are close, then the R2 is probably accurate. If R2 is much higher than the adjusted R2, you probably do not have enough data points to calculate the regression accurately. 2 −− where n is the number of data points and m is the number of independent variables. 1 )1)(1( 1 2 2 −− −− −= mn nR adjR Yun-Hsuan Yeh2014/12/29 53
  18. 18. Regression (Parameter estimates) 2014/12/29 Yun-Hsuan Yeh 54 : the coefficient is zero.0H Therefore, the terms, Correlation, Leverage, Profit, Change of holder, Turnover(60), and Correlation*Profit have significant influence under α=0.05.
  19. 19. DOE (Replication) Replication improve the chance of detecting a statistically significant effect in the midst of natural process variation (noise). The variation can be determined from control charts and process capability studies. We can improve the power (1- β) of the DOE by adding actual replicates where conditions are duplicated.replicates where conditions are duplicated. Replication Yun-Hsuan Yeh2014/12/29 55
  20. 20. DOE (Residuals) Residuals can tell us whether our assumptions are reasonable and our choice of model is appropriate. Residuals can be thought of as elements of variation unexplained by the fitted model (normality, independence, and homoscedasticity). We expects them to be normal and independently distributed with a Residual = Observation - Predicted response We expects them to be normal and independently distributed with a mean of 0 and some constant variance. Yun-Hsuan Yeh2014/12/29 56
  21. 21. DOE (Response Surface Methods) High-resolution fractional or full factorial to understand interactions ( add center points at this stage to test for curvature). Response surface methods RSM) to optimize. Contour maps (2D) and 3D surfaces guide you to the optimal solutions. Set the limits of response To find the safe window Yun-Hsuan Yeh2014/12/29 57
  22. 22. Postscript Nowadays, the problem can be solved by the experiences that will not be a problem. Furthermore, those methods had been developed over 50 years. The rollback or missing anticipation always occur without evaluation and the cost is ignored usually. There is gap between prediction and practicality certainly, so that to evaluate the variances becomes the important objective in DOE. Sometimes, DOE predicts the effect is 80%, but the real effect is about 40% that may be caused by poor evaluation. Evaluation is that to reduce the risk and cost through the scientific methods. If we have too many features may fit the training set very well, but fail to generalize to new examples (predict the response on new examples). Yun-Hsuan Yeh2014/12/29 58
  23. 23. Statistical methods for EngineeringEngineering Yun-Hsuan Yeh Software JMP 2014/12/29 1Yun-Hsuan Yeh
  24. 24. Outline TYPE OF ANALYSIS APPLICATION Sample Plan Reduce excessive sampling Capability Study (SPC) Cpk analysis of Machine, Process or System Gr&R For all measuring equipment Comparative Analysis Compare of two or more groupsComparative Analysis Compare of two or more groups DOE For all new system & process (material, parameters) Correlation Correlation between two objectives Regression X & Y correlation (input and output relationship) Reliability Risk assessments of long term All the items linked with one another Yun-Hsuan Yeh2014/12/29 2
  25. 25. 6 sigma (DMAIC) •Xs/Ys SPC selection •Identify problem •Customer ‘s requirement •Select CTQ (Critical to Quality) •Goal setting •Time line •Sampling Plan •Measuring technology •Screening DOE •Optimization (Regression ) •Validation •Xs/Ys SPC selection •SPC control •Documentation •Reliability monitor Yun-Hsuan Yeh2014/12/29 3 •Hypothesis testing •ANOVA •Contingency & Cluster analysis
  26. 26. Population, Sample and Variable A population includes each element from the set of observations that can be made. A sample consists only of observations drawn from the population. A variable is an attribute that describes a person, place, thing, or idea. Variables can be classified as qualitative (aka, categorical) or quantitative (aka, numeric). 2014/12/29 Yun-Hsuan Yeh 4
  27. 27. Sample Plan Simple Random Sampling: Every member of the population has an equal chance of being selected for your sample. Stratified Sampling: The population is split into non-overlapping groups, then simple random sampling is done on each group to form a sample. Systematic Sampling: Every nth individual from the population is placed in the sample. That cannot obtain a frame of the population you wish to study. 2014/12/29 Yun-Hsuan Yeh 5 Cost Risk Size Frequency α (AQL) β (LTPD)
  28. 28. Sample Plan Alpha is the probability of mistakenly detecting a difference even when there is no difference. Beta is the probability of not being able to detect a significant difference. Power (1-Beta) is the probability of correctly detecting a shift in performance. Sample size in each group (assumes equal Represents the desired power Define the rational specifications via the pre-sampling. group (assumes equal sized groups) Represents the desired level of statistical significance (typically 1.96). Standard deviation of the outcome variable Effect Size (the difference in means) 2 2 /2 2 difference )Z(2 αβσ + = Z n Yun-Hsuan Yeh2014/12/29 6
  29. 29. Sample Plan (Difference) D The D means that we want to defect the difference under the power. D is the accuracy of the estimate. We can determine D by experiences and standard deviation form outcome, and power is an indicator to estimate the sample size. Yun-Hsuan Yeh2014/12/29 7
  30. 30. Sample Plan (K means) D1 D3 If we fix the Di, then the sample size increases as the total variances increase. When the Di is large, we detect the different easily with small sample size (refer to one-way ANOVA). 2014/12/29 Yun-Hsuan Yeh 8 D2 Total mena
  31. 31. Sample Size (Traditional industrial) Compute the UCI and LCI for the instant-in-time variation at 95% UCL=x+1.96xS, LCI=x-1.96xS, where S is the instant-in-time variation 2 timeofperiod 2 timeininstanttotal −−−− += SSS Yun-Hsuan Yeh2014/12/29 9
  32. 32. Sample Size (Traditional industrial) Determine the range of the Cl at various sample sizes and compare it as a percentage of the tolerance= (UCI-LCI)/(USL-LSL)x100 – Two side. a percentage of the margin= (Average-LCI)/(Average-LSL)x100 – one side. Percentage Sampling SizePercentage Sampling Size <7% Excessive 7~15% Efficient >15% Insufficient Yun-Hsuan Yeh2014/12/29 10 The percentages can be modify to fit industrial or process form experiences.
  33. 33. Sample Frequency (Traditional industrial) Use regression to correlate the run order to the data. Determine the change in the measurement based on one unit of time (drift rate). Make sure and examine the drift rate as the most dynamically changing segment of the data. Drift rate x Time/(USL-LSL) Sampling frequency 5~15% Efficient >15% Insufficient Yun-Hsuan Yeh2014/12/29 11 The percentages can be modify to fit industrial or process form experiences.
  34. 34. Attribute sampling (GO/NG) Using the Poisson to calculate probabilities, the probability of exactly x defects or defective parts in a sample n with the probability of defect occurrence p. ! )( )( x npe xP xnp− = If the lot has 3% defective, then it has a probability 70% of being α If the lot has 10% defective, then it has a probability 10% of being accepted with n=40. it has a probability 70% of being accepted with n=40. β AQL LTPD Yun-Hsuan Yeh2014/12/29 12
  35. 35. Attribute sampling (GO/NG) As an example, consider a wafer manufacturing process with a target of 4 defects per wafer. You want to verify that a new process meets that target within a difference of 1 defect per wafer with a significance level of 0.05. In the Counts per Unit window: 2014/12/29 Yun-Hsuan Yeh 13
  36. 36. Postscript Sample plan is a comparative calculation that depends on the detection and acceptable risk . How long should the period be decided? That must consider the operating period, such as period of job rotation, PM period, and period of material renewing.of material renewing. To reduce the sample size is always a easy way to cost down, simultaneously, we can detect the variances and risk under the anticipations. Yun-Hsuan Yeh2014/12/29 14
  37. 37. Gr&R True value True value Step 1: Set 3-10 parts and 3-5 operators for measuring 3-10 times . Step 2: Check the tool is ready and operators is clear for operating. Step 3: Ensure the parts will not be mixed up and operators do not know which part he/she are measuring (random). Step 4:Test, data collection, and analysis. Accuracy True value True value Reproducibility Operator 1 Operator 3 Operator 2 Repeatability Yun-Hsuan Yeh2014/12/29 15
  38. 38. Gr&R The calculation parameters, repeatability, reproducibility, and total measurement precision , 22 & RrrR SSS += where, , 15.5 1KR Sr × = nm R R m i n ij × = ∑∑= =1 1 15.5 r under the 99% confidence interval, ( ) 15.5 2 2 1 kn S KR S r x R × −× = nm R × = and where, m = number of operators, n = number of parts, k = number of repeated readings, Rij = rang of repeated readings for operator i and part j, and = the maximum operator average – the minimum operator average.x R Yun-Hsuan Yeh2014/12/29 16
  39. 39. Gr&R We have range and average to imply the K factors, since the range and estimated standard deviation have a relationship, 2 ˆ d R r =σ Where is called the relative range, σˆ R       = σˆ2 R Ed σˆ and then, 2 11 2 15.5 ityRepeatabil ˆ15.5ityRepeatabil d KRK d R =⇒= == σ let, Yun-Hsuan Yeh2014/12/29 17
  40. 40. Gr&R Trail K1 Operations K2 Parts K3 2 4.5656 2 3.6525 2 3.6524 3 3.0419 3 2.6963 3 2.6963 4 2.5012 4 2.2991 4 2.2991 5 2.2141 5 2.0766 5 2.0766 6 2.0324 6 1.9288 6 1.9288 7 1.9046 7 1.8198 7 1.8198 Similarly, we have K2 and K3. 7 1.9046 7 1.8198 7 1.8198 8 1.8089 8 1.7398 8 1.7398 9 1.734 9 1.6721 9 1.6721 10 1.6732 10 1.6195 10 1.6195 And the precision to variation ratio is 3KRS pp ×= Where is the range between the maximum and minimum part averages.pR Yun-Hsuan Yeh2014/12/29 18
  41. 41. Gr&R GRR Rating The precision-to-tolerance ratio, GRR, shows percent of the specification window is consumed by measurement uncertainty, and is defined as: %100 15.5 & × − × = UCLUSL S GRR rR GRR Rating >30% Needs improvement 10~30% Marginal <10% Acceptable Yun-Hsuan Yeh2014/12/29 19
  42. 42. Attribute Gauge (GO/NG) The result of the checks made by attribute gauges is not a numerical value, but a good/scrap status. Agreement rate of Part 6 is 5/9 (55.56%).6 is 5/9 (55.56%). These parts might have been more difficult to categorize. We usually set the criteria of false alarm to be <0.1. Yun-Hsuan Yeh2014/12/29 20
  43. 43. Gauge Linearity and Bias Study The gauge linearity and bias study is used extensively in quality control in manufacturing but also extends beyond this practical application. It is used to determine the accuracy of measurements through the expected measurement range. 2014/12/29 Yun-Hsuan Yeh 21 Criteria <5%
  44. 44. Degrees of Freedom The number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. In order to estimate sigma, we must first have estimated mu. Thus, mu is replaced by x-bar in the formula for sigma. Thus, degrees of freedom are n-1 in the equation for s below: )( 2 − = xx s 2014/12/29 Yun-Hsuan Yeh 22 For example, imagine you have four numbers (a, b, c and d) that must add up to a total of m; you are free to choose the first three numbers at random, but the fourth must be chosen so that it makes the total equal to m - thus your degree of freedom is three. 1 )( − − = n xx s
  45. 45. Central Limit Theorem The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal. Thus, the Central Limit theorem is the foundation for many statistical procedures, including Quality Control Charts, because the distribution of the phenomenon under study does not have to be Normal because its average will be. Furthermore, this normal distribution will have the same mean as the parent distribution, AND, variance equal to the variance of the parent 2014/12/29 Yun-Hsuan Yeh 23 parent distribution, AND, variance equal to the variance of the parent divided by the sample size. N=1 N=30
  46. 46. SPC (Cpk and Ppk) The Ca, Cp, and Cpk. where is estimated by range or sigma of subgroup. 2/)( LCLUSL x Ca − − = µ σˆ6 LCLUSL Cp − = CpCaCpk ×−= )1( σˆ The Ppk is calculated by the sigma from the raw data, that includes the variation within subgroup (the common cause) and the variation between subgroup (the special cause), so that Cpk ≥ Ppk. Cpk Ppk Yun-Hsuan Yeh2014/12/29 24
  47. 47. On the other hand, Cpk is we have SPC (Cpk and PPM of OOS)       −− = σσ ˆ3 , ˆ3 min LSLxxUSL Cpk 3 xUSL Cpkz − =×= LSLx z − =or By standard normal distribution, the defective parts per million is σˆ 3 CpkzU =×= σˆ zL = [ ] 1000000×> UzzP or [ ] 1000000×< LzzP Yun-Hsuan Yeh2014/12/29 25
  48. 48. SPC (Cpk Level) Level Cpk PPM A >1.67 <0.5443 Process capability measures the ability of the process to meet the design requirements. A >1.67 <0.5443 B 1.67~1.33 0.5443~66.07 C 1.33~1 66.07~2699.8 D <1 >2699.8 Yun-Hsuan Yeh2014/12/29 26
  49. 49. t-Test (σ is unknown) Populations are normally distributed. Samples are independent, The hypothesis is defined as: If judge by F-test, then we do the t-test with the equal variation (select the Means/Anova/Pooled t option). 210 : µµ =H 2 2 2 1 σσ = 21 XX t − = )1()1( 2 22 2 12 −+− = SnSn S Otherwise, we do the t-test with the unequal variation (select the t-Test option ). 2 2 2 1 2 1 21 n S n S XX t + − = 21 11 21 nnpS XX t + − = 2 )1()1( 21 2212 −+ −+− = nn SnSn Spwhere Yun-Hsuan Yeh2014/12/29 27
  50. 50. p-value p-value is the probability of getting an effect of the size or greater by random chance alone (tail probability), if p-value is small then t is large and the means of the two samples have significance (reject the hypothesis). Yun-Hsuan Yeh p-value t 2 α p-value -t 2 α Two tailed test 2014/12/29 28 The critical p-value is a subjective criterion that depends on your practical problem. Generally, we set the criterion as 0.05.
  51. 51. Test Standard Deviations An F-test is used to test if the variances of two populations are equal. The F hypothesis is defined as: F distribution is also used to test multiple means. Test statistic: 2 2 2 10 : σσ =H 222 == Test statistic: Where and are the sample variances. The more this ratio deviates from 1, the stronger the evidence for unequal population variances. 22 2 2 1 / tssF == 2 1s 2 2s Yun-Hsuan Yeh2014/12/29 29
  52. 52. Step 1: Unequal Variances Test p > .05 p < .05 t-Test Stdevs are unequal When the variation by group was not equal and therefore the t test assuming unequal variances should select the t-test. Step 2: Means/Anova/Pooled t t-Test p > .05 Variances are the same p < .05 Variances are different Yun-Hsuan Yeh2014/12/29 30
  53. 53. Chi-squared test There are basically two types of random variables and they yield two types of data: integral and categorical. A chi square (X2) statistic is used to investigate whether distributions of categorical variables differ from one another. the Chi Square statistic is calculated by the formula: ∑ − = ii E EO 2 2 )( χ 2014/12/29 Yun-Hsuan Yeh 31 where O = observed cell count and E = expected cell count. The hypothesis is defined as: Pearson Chi-Square results when one or more cells contain less than 5 observations cannot be trusted, an alternative to the Fisher’s Exact Test. ∑i iEall on.distributispecifiedafollowdataThe:0H
  54. 54. Chi-squared test (Lot-yield comparison) A test for independence Test statistic: t.independenaretionsclassificatwoThe:0H 2014/12/29 Yun-Hsuan Yeh 32 ∑∑ − = j i ij ijij E EO all all 2 2 )( χ
  55. 55. One way ANOVA (Means Test) The distribution of the response variable follows a normal distribution.. All populations have a common variance. All samples are drawn independently of each other. The hypothesis is defined as: An experiment with k groups and each group has n samples, j=1,…,k .2,...: 210 ≥=== nH nµµµ An experiment with k groups and each group has nj samples, j=1,…,k the test statistic: where wb MSMSF /= j j j ji jij w ni n xx MS ,...,1, )1( )( 2 , = − − = ∑ ∑ 1 )( 2 − − = ∑ k xxn MS j jj b Yun-Hsuan Yeh2014/12/29 33
  56. 56. Two way ANOVA The distribution of the response variable follows a normal distribution.. All populations have a common variance. All samples are drawn independently of each other. The groups must have the same sample size. The hypothesis is defined as: 0H :The means do not depend on factor A. 0H :The means do not depend on factor B. The statistics are 0H :The means do not depend on factor B. 0H :The means do not depend on factor AxB. wa MSMSF /= wb MSMSF /= wab MSMSF /= Yun-Hsuan Yeh2014/12/29 34
  57. 57. Two way ANOVA An experiment with a levels with factor A and b levels with factor B and sample size is r. 1 … a 1 1…r… 1…r b 1…r •• jx •ijx b 1…r )1( )( 2 ,, − − = ∑ • rab xx MS kji ijijk w 1 )( 2 − − = ∑ ••••• a xxrb MS i i a •••x••ix 1 )( 2 − − = ∑ ••••• b xxra MS j j b )1)(1( )( 2 −− +−− = ∑ •••••••• ba xxxxr MS ij jiij ab Yun-Hsuan Yeh2014/12/29 35
  58. 58. Two way ANOVA 0H :The pains do not depend on gender. 0H :The pains do not depend on drug. 0H :The pains do not depend on gender x drug. There is no interaction between the gender and drug. Yun-Hsuan Yeh2014/12/29 36
  59. 59. Hierarchical Clustering (Apply to tool mapping or product difference) The basic process of hierarchical clustering is: 1. Start by assigning each item to its own cluster, so that if you have N items, you now have N clusters, each containing just one item. Let the distances (similarities) between the clusters equal the distances 2014/12/29 Yun-Hsuan Yeh 37 (similarities) between the items they contain. 2. Find the closest (most similar) pair of clusters and merge them into a single cluster, so that now you have one less cluster. 3. Compute distances (similarities) between the new cluster and each of the old clusters. 4. Repeat steps 2 and 3 until all items are clustered into a single cluster of size N.
  60. 60. Hierarchical Clustering Single-link : is the distance between clusters and . Complete-link: ),(),( min, ba ba dCCd ji CC ji ∈∈ = ),( ji CCd iC jC ),(),( max badCCd = 2014/12/29 Yun-Hsuan Yeh 38 Average-link: where is the mean vector of . ),(),( max, ba ba dCCd ji CC ji ∈∈ = ∑∪∈ −= ji CC ji aCCd a µ),( µ ji CC ∪
  61. 61. K-means Clustering The process of K-means clustering is: 1. Define the initial groups' centroids. This step can be done using different strategies. A very common one is to assign random values for the centroids of all groups. Another approach is to use the values of K different entities as being the centroids. 2014/12/29 Yun-Hsuan Yeh 39 2. Assign each entity to the cluster that has the closest centroid. In order to find the cluster with the most similar centroid, the algorithm must calculate the distance between all the entities and each centroid. 3. Recalculate the values of the centroids. The values of the centroid's fields are updated, taken as the average of the values of the entities' attributes that are part of the cluster. 4. Repeat steps 2 and 3 iteratively until entities can no longer change groups.
  62. 62. DOE (WLB RDL Short Issue) Defect mode: RDL short (Width 15um / Spacing 10um) Factors: PR thickness, Exposure energy, Developer time, and PR ash time. 2014/12/29 Yun-Hsuan Yeh 40 Factors: PR thickness, Exposure energy, Developer time, and PR ash time. Faults: We should remove the trials that were anticipated. On the others side, we should not put the depending variables in the DoE. It is waste!!
  63. 63. Key factor: PR thickness Optimal solution: Actually, we just select one or two factors for this problem 2014/12/29 Yun-Hsuan Yeh 41 Actually, we just select one or two factors for this problem that will save many resources.
  64. 64. DOE Design of experiments are a structured development strategy for product/process engineering in order to characterize, optimize and control performance with minimal waste. This is accomplished by experimenting with many factors at the same time. Main Effects or Linear Interactions Powers or Quadratics Correlations between factor and respond. Yun-Hsuan Yeh2014/12/29 42
  65. 65. DOE (Levels) If unsure of levels, run a few pre-experiments to find an operating range, then begin the experiment. 3 levels are used if the factor range is wide. If the relationship is nonlinear, we need to add center point to the predicting. Yun-Hsuan Yeh Level 1 Level 2 The real phenomenon Predicted by linear regression with 2 levels. ??? 2014/12/29 43 Level 1 Level 2 Narrow range Wide range
  66. 66. DOE (Full factorial designs) A class of designs that use k factors at 2 levels. t = nk where t = the number of tests n = number of levels k = number of factors Yun-Hsuan Yeh2014/12/29 44
  67. 67. DOE (Fractional factorial designs) We have a k factors and 2k-p fractional factorial designs, then the design follows the properties: 1.Sum of each column in the sign table is 0 1allfor,0 >=∑ js j j 2.Sum of the product of any two columns is 0 3.Sum of the square of elements in any column is 2k-p .,1,allfor,0 , kjjss kj kj ≠>=∑ 1allfor,22 >= − ∑ js pk j j A B C 1 -1 -1 1 2 1 -1 -1 3 -1 1 -1 4 1 1 1 23-1 Design Yun-Hsuan Yeh2014/12/29 45
  68. 68. DOE (Custom Design) D or I-optimal designs are the forms of design provided by a computer algorithm. The reasons for using D or I-optimal designs instead of standard classical designs generally fall into two categories: Standard factorial or fractional factorial designs require too many runs for the amount of resources or time allowed for the experiment. The design space is constrained (the process space contains factor settings that are not feasible or are impossible to run). Factors Full Fractional D-Optimal 5 32 16 16 6 64 32 28 7 128 64 35 8 256 128 43 9 512 256 52 Yun-Hsuan Yeh2014/12/29 46
  69. 69. D-Optimal (emphasize the corners) I-Optimal (emphasize the centers) DOE (Custom Design) 2014/12/29 Yun-Hsuan Yeh 47
  70. 70. Evaluation of DOE Prediction Variance Profile is used to determine the relative variability in the response depending on the DOE design. Add additional samples and the variance goes down. Modify the design by adding runs and see what happens to the predicted variance. We will consider DOEs for multiple linear regression using linear and polynomials and where errors are due to noise in the data. where is the nx1 vector, x is the nxp matrix and β is the px1 vector. because in using to estimate the responses, we effectively estimate parameters β . That is, we lose p degrees of freedom. βˆˆ xy = pn yy n j jj − ∑ − = = 1 2 )( ˆσ (MSE) yˆ yˆ Yun-Hsuan Yeh2014/12/29 48
  71. 71. Prediction Variance By definition we have, Then , so that different x has distinct variance. xxxx xx xy 1TT2 T ][ˆ )ˆVar( )ˆ(Var)ˆVar( − = = = σ β β xxxx y 1TT 2 ][ ˆ ]ˆVar[ − = σ The Prediction Variance Profile gives a profiler of the relative variance of prediction as a function of each factor at fixed values of the other factors. ˆσ The prediction variance is 0.2241 that means the variance of experimental error is 10, then the prediction variance of response at X1=0 is 10x0.2241=2.241. Yun-Hsuan Yeh2014/12/29 49
  72. 72. DOE (Correlation of Xs check) If Xs were correlated to any great degree (r > 0.5) then experiment was not well designed and should be redone, this will cause collinearity then There is no unique solution for regression coefficients. Partial coefficients are estimating something that does not occur in the data. This step is very important for uncontrolled variables, engineers always ignore uncontrolled variables usually. Yun-Hsuan Yeh2014/12/29 50
  73. 73. Regression Learn more about the relationship between several independent or predictor variables and a dependent or criterion variable. In the most general terms, least squares estimation is aimed at minimizing the sum of squared deviations of the observed values for the dependent variable from those predicted by the model. 2 )]([min∑ − fy θ where is a known function of , and are random variables, and usually assumed to have expectation of 0. )]([min∑ − i i f fy θ )(θf θ ii fy εθ += )( iε Yun-Hsuan Yeh2014/12/29 51 0H : Individual coefficient is equal to 0. 0H : All coefficients are equal to 0.
  74. 74. R2 A data set has values yi each of which has an associated modeled value f the yi are called the observed values and the modeled values f called the predicted value. y 2 ][∑ −= i ireg fySS 2 ][∑ −= i itot yySS tot reg SS SS R −=12 Yun-Hsuan Yeh2014/12/29 52 y
  75. 75. Adjusted R2 To correct the R2 for such situations, an adjusted R2 takes into account the degrees of freedom of an equation. When you suspect that an R2 is higher than it should be, calculate the R2 and adjusted R2. If the R2 and the adjusted R2 are close, then the R2 is probably accurate. If R2 is much higher than the adjusted R2, you probably do not have enough data points to calculate the regression accurately. 2 −− where n is the number of data points and m is the number of independent variables. 1 )1)(1( 1 2 2 −− −− −= mn nR adjR Yun-Hsuan Yeh2014/12/29 53
  76. 76. Regression (Parameter estimates) 2014/12/29 Yun-Hsuan Yeh 54 : the coefficient is zero.0H Therefore, the terms, Correlation, Leverage, Profit, Change of holder, Turnover(60), and Correlation*Profit have significant influence under α=0.05.
  77. 77. DOE (Replication) Replication improve the chance of detecting a statistically significant effect in the midst of natural process variation (noise). The variation can be determined from control charts and process capability studies. We can improve the power (1- β) of the DOE by adding actual replicates where conditions are duplicated.replicates where conditions are duplicated. Replication Yun-Hsuan Yeh2014/12/29 55
  78. 78. DOE (Residuals) Residuals can tell us whether our assumptions are reasonable and our choice of model is appropriate. Residuals can be thought of as elements of variation unexplained by the fitted model (normality, independence, and homoscedasticity). We expects them to be normal and independently distributed with a Residual = Observation - Predicted response We expects them to be normal and independently distributed with a mean of 0 and some constant variance. Yun-Hsuan Yeh2014/12/29 56
  79. 79. DOE (Response Surface Methods) High-resolution fractional or full factorial to understand interactions ( add center points at this stage to test for curvature). Response surface methods RSM) to optimize. Contour maps (2D) and 3D surfaces guide you to the optimal solutions. Set the limits of response To find the safe window Yun-Hsuan Yeh2014/12/29 57
  80. 80. Postscript Nowadays, the problem can be solved by the experiences that will not be a problem. Furthermore, those methods had been developed over 50 years. The rollback or missing anticipation always occur without evaluation and the cost is ignored usually. There is gap between prediction and practicality certainly, so that to evaluate the variances becomes the important objective in DOE. Sometimes, DOE predicts the effect is 80%, but the real effect is about 40% that may be caused by poor evaluation. Evaluation is that to reduce the risk and cost through the scientific methods. If we have too many features may fit the training set very well, but fail to generalize to new examples (predict the response on new examples). Yun-Hsuan Yeh2014/12/29 58
  81. 81. Reliability (Censor) During the T hours of test we observe r failures (where r can be any number from 0 to n). The failure times are t1, t2 ,…, tr and there are (n-r) units that survived the entire T-hour test without failing. r1 fail r2 fail r3 fail ……………………… ∑= i irr 0 t1 t2 t3 n units start test tr i Yun-Hsuan Yeh2014/12/29 59 In the typical test scenario, we have a fixed time T to run the units to see if they survive or fail. The data obtained are called Censored Type I data.
  82. 82. Reliability (Weibull) The most general expression of the Weibull pdf is given by the two- parameter Weibull distribution expression, where is the shape parameter and is the scale parameter, F(T) is CDF. When , the Weibull reduces to the Exponential model. 0T,)( )( 1 ≥= − − β αβ β α β T eTTf β α 1=β Yun-Hsuan Yeh When , the Weibull reduces to the Exponential model. 1<β 0=β 1>β )(1 )( rateFailure TF Tf − = 1=β 2014/12/29 60 infant mortality wear out
  83. 83. Reliability (Life Distribution) The Life Distribution platform helps you discover distributional properties of time-to-event data. The failure rate is 0.188 at 5975 units of time. Yun-Hsuan Yeh2014/12/29 61
  84. 84. Reliability (Sample Size) From historical information an engineer knows that component's life follows a Weibull distribution with some and . Under the confidence level CL, the total number of units n can be determined by the following α β Yun-Hsuan Yeh2014/12/29 62 determined by the following equation with t cycle times. We want to know the required sample size to estimate the time till probability of units fail, with a two-sided absolute precision cycle times. )ln()( CL t n βα =
  85. 85. Reliability (Acceleration) Failure may be due to mechanical fatigue, corrosion, chemical reaction, diffusion, migration, etc. These are the same causes of failure under normal stress; the time scale is simply different. Then, an acceleration factor AF between stress and use means the following relationships hold: Time-to-Fail tu = AF × ts 2014/12/29 Yun-Hsuan Yeh 63 Time-to-Fail tu = AF × ts Failure Probability Fu(t) = Fs(t/AF) Reliability Ru(t) = Rs(t/AF) PDF or Density Function fu(t) = (1/AF)fs(t/AF) Failure Rate hu(t) = (1/AF) hs(t/AF) u: at use/ s: at stress
  86. 86. Reliability (Arrhenius) This empirically based model is known as the Arrhenius equation. The acceleration factor AF between a higher temperature T2 and a lower temperature T1 is given by where k is Boltzmann’s constant and is the activation energy depends on the failure mechanism and the materials involved, and typically ranges from 0.3 to 1.5,             − ∆ = 21 11 exp TTk H AF H∆ 2014/12/29 Yun-Hsuan Yeh 64 failure mechanism and the materials involved, and typically ranges from 0.3 to 1.5, or even higher. Fit life by Temperature The Hazard Profiler for failure probability of 0.00033 at 45 degrees is 2641.5 hours.
  87. 87. Reliability (Thermal Cycling Test) The AF is given by where f is temperature-cycling frequency m=1/3, n=1.9 (for SnPb solder), and =1414.                 − ∆       ∆ ∆       = su n u s m u s TTk H T T f f AF max,max, 11 exp kH /∆ 2014/12/29 Yun-Hsuan Yeh 65 =1414.kH /∆ Test conditions: 0°C ~ 100°C (36 cycles per day). Use conditions: 0°C ~ 35°C (2 cycles per day) Then AF=6.24. By fitting Weibull, the failure risk is 0.05136 at 400 cycles . So that the life time is 400x6.24/(365x2)=3.41 years.
  88. 88. Postscript AF (Arrhenius) can be estimated through the experiments and mode fitting. Acceleration experiment can reduce the testing period and to find the impact of factors. If there are different failure modes after reliability test, then we canIf there are different failure modes after reliability test, then we can analyze by modes to figure out which mode influences the life time critically. Yun-Hsuan Yeh2014/12/29 66

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