Reliability doe the proper analysis approach for life data

Reliability DOE: The Proper Analysis Approach for Life Data可靠性DOE:寿命数据的正确分

析方法

Huairui Guo, Ph.D.郭怀瑞,博士

©2011 ASQ & Presentation GuoPresented live on May 11th, 2011

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Document Revision: 1.0.1

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Reliability DOE:The Proper Analysis Approach

for Life Data

可靠性DOE:寿命数据的正确分析方法

Huairui Guo, Ph.D.

郭怀瑞, 博士

2

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(DFR)

Introduction to Reliability Concepts,

Principles and Applications

DOE: Experiment Design and Analysis

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3

常用词中英文对照表常用词中英文对照表ANOVA: 方差分析

DOE: 实验设计

Factor: 因子

Level: 水平

2-Level Factorial Design: 两水平因子实验

2-Level Fractional Factorial

Design: 两水平部分因子实验

Response: 反应

Main Effect: 主效应

Interaction Effect: 交互效应

Coefficient: 系数

Critical Value: 关键值

Outlier: 离群值

Censored Data: 删失数据

MLE: 极大似然估计

Likelihood function: 似然函数

Life Characteristic: 寿命特征量

Life-Factor Relationship: 寿命-

因子关系

Life-Stress Relationship:寿命-

应力关系

Likelihood Ratio Test: 似然比检验

Probability density function

(pdf): 概率密度函数

Mean Squares (MS): 均方差

Mean Squares of Error: (MSE):

残方差

4

Introduction Example

引例

Consider an experiment to improve the reliability of fluorescent lights. Five factors A-E are investigated in the experiment. A 25-2 design with factor generators D=AC and E=BC was conducted*.

Objective: To identify significant factors and adjust them to improve life.

*Taguchi, 1987, p. 930.

5

Introduction Example (cont’d)

引例(继续)

Two replicates at each treatment.

Inspections were conducted every two days.

Results have interval data and suspensions.

A B C D E

-1 -1 -1 1 1 14~16 20+

-1 -1 1 -1 -1 18~20 20+

-1 1 -1 1 -1 8~10 10~12

-1 1 1 -1 1 18~20 20+

1 -1 -1 -1 1 20+ 20+

1 -1 1 1 -1 12~14 20+

1 1 -1 -1 -1 16~18 20+

1 1 1 1 1 12~14 14~16

Failure Time

6

Traditional DOE Approach

传统的DOE方法

Assumes that the response (life) is normally

distributed.

Treats suspensions as failures.

Uses the middle point of the interval data as

the failure time.

Problem: The above assumptions and

adjustments are incorrect and do not apply to

life data.

7

EDUCATION

7

Life Data AnalysisLife Data Analysis寿命数据分析简介寿命数据分析简介

8

Complete Data

Censored Data

Right Censored (Suspended)

Interval Censored

Life Data Types

寿命数据类型

9

Complete and Censored Data

完全数据和删失数据

Complete Data

Censored Data

Right Censored

Interval Censored

?

?

10

Complete Data: Example

完全数据例子

For example, if we tested five units and they all failed, we would

then have complete information as to the time of each failure in

the sample.

11

Right Censored (Suspended) Data: Example

右删失数据 (终止): 例子

Imagine we tested five units and three failed. In this scenario,

our data set is composed of the times-to-failure of the three units

that failed and the running time of the other two units without

failure.

This is the most common censoring scheme and is used

extensively in the analysis of field data.

12

Interval Censored Data: Example

区间删失数据: 例子

Imagine we are running a test on five units and inspecting them

every 100 hr. If a unit failed between inspections, we do not

know exactly when it failed, but rather that it failed between

inspections. This is also called “inspection data”.

13

Censored Data Analysis Example

删失数据计算例子100 pumps operated for three months.

One failed during the first month.

One failed during the second month.

Two failed during the third month.

What is the average time-to-failure?

You can’t answer this question without assuming a model for the data.

?25.24

)3(2)2(1)1(1

14

Common Distributions Used in Reliability

可靠性中常用的分布

Weibull distribution pdf:

Lognormal distribution pdf:

Exponential distribution pdf:

t

et

tf

1

)(

21 ln( )

21( )

2

t

f t et

m

t

em

tf1

)(

15Parameter Estimation:

Maximum Likelihood Estimation (MLE)

极大似然参数估计

Statistical (non-graphical) approach to parameter

estimation.

Given a data set, estimates the parameters that

maximize the probability that the data belong to that

distribution and that set of parameters.Constructs likelihood function as product of densities,

assuming independence.

Uses calculus to find the values that maximize the likelihood

function.

Has elegant statistical properties when the sample size is

large.

16

MLE Concept极大似然参数估计概念

Which model is more likely if two values are observed:

-3 and 3?

17

If T is a continuous random variable with pdf:

where 1, 2, … , k are k unknown parameters that need to be estimated,

and we conduct an experiment and obtain N independent observations,

T1, T2, … , TN, then the likelihood function is given by:

For a one-parameter distribution with a single parameter and data of

10, 20, 30, the likelihood of the function would be:

Likelihood Function: Complete Data

似然函数: 完全数据

1 2( ; , , , )kf T

1 2 1 2 1 2

1

( , , , , , , ) ( ; , , , )N

k N i k

i

L T T T f T

)30()20()10()30,20,10( fffL

18

The logarithmic likelihood function is:

The maximum likelihood estimators (MLE) of 1, 2, … , k are obtained

by maximizing either L or .

By maximizing , which is much easier to work with than L, the

maximum likelihood estimators (MLE) of 1, 2, … , k are the

simultaneous solutions of k equations such that:

Likelihood Function: Complete Data (cont‘d)似然函数: 完全数据 (继续)

1 2 1 2

1 2

1

ln ( , , , , , , )

ln( ( ; , , , ))

k N

N

i k

i

L T T T

f T

0, 1,2,... i

i k

19

Likelihood Function: Right Censored Data

似然函数: 右删失数据

The likelihood function for M suspension times,

S1,S2,…,SM, is given by:

1 2 1 2

1 2

1

1 2

1

( , ,..., | , ,..., )

1 ; , ,...,

; , ,...,

k M

M

j k

j

M

j k

j

L S S S

F S

R S

20

Likelihood Function: Interval Data

似然函数: 区间数据

The likelihood function for P intervals, IL1 , IU1; IL2 , IU2;…;

ILP , IUP, is given by:

1 2 1 1 2 2

1 2 1 2

1

( , ,..., | , , , ,..., , )

; , ,..., ; , ,...,

k L U L U LP UP

P

Ul k Ll k

l

L I I I I I I

F I F I

21

After completing the likelihood function for the different types of

data, the likelihood function (without the constant) can now be

expressed in its complete form:

The Complete Likelihood Function

完整的似然函数

1 2 1 2

1 1

1 2 1 2

1

; , ,..., ; , ,...,

; , ,..., ; , ,...,

N M

i k j k

i j

P

Ul k Ll k

l

L f T R S

F I F I

22

MLE Parameter Estimation

极大似然解

The logarithmic likelihood function is:

The maximum likelihood estimators (MLE) of 1, 2, … , k

are the simultaneous solutions of k equations such that:

1 2 1 1 1 1ln ( , , , , , , ,..., , , ,..., , )k N N U L UP LPL T T S S I I I I

0, 1,2,... i

i k

23

EDUCATION

23

Combining Reliability and DOECombining Reliability and DOE可靠性和可靠性和DOEDOE的结合的结合

24

Combining Reliability and DOE: Life-Factor Relationship

可靠性和可靠性和DOEDOE的结合的结合:: 寿命因子关系寿命因子关系

The graphic shows an example where life decreases when a factor is

changed from the low level to the high level.

It is seen that the pdf changes in scale only. The scale of the pdf is

compressed at the high level.

The failure mode remains the same. Only the time of occurrence

decreases at the high level.

25

Life-Factor Relationship Simplify: Life Characteristic

简化寿命-因子关系:寿命特征量

Instead of considering the entire scale of the pdf, the life characteristic

can be chosen to investigate the effect of potential factors on life.

The life characteristic for the 3 commonly used distributions are:

Weibull: Lognormal: Exponential: m

26

Life-Factor Relationship

寿命-因子关系

Using the life characteristic, the model to investigate the effect of factors on life can be expressed as:

Note that a logarithmic transformation is applied to the life characteristics of the Weibull and exponential distributions.

This is because and m can take only positive values.

......' 211222110 xxxx

where:

)ln(' or ' or )ln(' m

:jx jth factor value

27

MLE Based on Life-Factor Relationship

基于寿命-因子关系的极大似然解

Life-Factor Relationship

Failure Time Data

Suspension Data

Interval Data

MLE

...... 211222110

' iiiii xxxx

1

( ; , )N

f i i

i

L f T

1

( ; , )M

S j i

j

L R S

1

( ; , ) ( ; , )P

I Ul i Ll i

l

L F I F I

0 1 2, , ,... and for lognormal

28

Testing Effect Significance: Likelihood Ratio Test

检验效应的显著性: 似然比检验

( )( ) 2 ln

( )

L effect k removedLR effect k

L full Model

2

1,( )LR effect k

Life-factor relationship is

Likelihood ratio test

If

then effect k is significant or active.

...... 211222110

' iiiii xxxx

29

Fluorescent Lights R-DOE: Data and Design

荧光灯可靠性DOE: 数据和实验设计

The design is identical to traditional DOE.

Data entered includes suspensions and interval data.

30

Fluorescent Lights R-DOE: Results

荧光灯可靠性DOE: 结果

Life is assumed to follow the Weibull distribution.

31

Fluorescent Lights R-DOE: Analyzing Model Fit

荧光灯可靠性DOE: 模型拟合分析

Residual Probability Plot

When using the Weibull distribution for life, the residuals from the

life-factor relationship should follow the extreme value distribution

with a mean of zero.

32

Fluorescent Lights R-DOE: Analyzing Model Fit

(cont’d)

荧光灯的可靠性DOE: 模型拟合分析(继续)

Plot of residuals against run order

There should be no outliers or pattern.

33

Fluorescent Lights R-DOE: Interpreting the Results

荧光灯的可靠性DOE: 理解结果

From the results, factors A,B, D and E are significant at the risk Level of

0.10. Therefore, attention should be paid to these factors.

In order to improve the life, factor A and E should be set to the high

level; while factors B and D should be set to the low level.

Term Coefficient

A:A 0.1052

B:B -0.2256

C:C -0.0294

D:D -0.2477

E:E 0.1166

MLE Information

34

EDUCATION

34

Traditional DOE ApproachTraditional DOE Approach传统的传统的DOEDOE方法方法

35

Traditional DOE Approach: Model

传统的DOE方法: 模型

Traditional DOE uses ANOVA models.

…coefficients are estimated using least squares.

For the first observation:

…assuming that the interactions are absent.

0 1 1 2 2 12 1 2ˆ ...y x x x x

A B C D E

-1 -1 -1 1 1 14~16 20+

-1 -1 1 -1 -1 18~20 20+

-1 1 -1 1 -1 8~10 10~12

-1 1 1 -1 1 18~20 20+

1 -1 -1 -1 1 20+ 20+

1 -1 1 1 -1 12~14 20+

1 1 -1 -1 -1 16~18 20+

1 1 1 1 1 12~14 14~16

Failure Time

1 0 1 2 3 4 5ˆ ( 1) ( 1) ( 1) ( 1) ( 1)y

36

The ANOVA model is

F test

Traditional DOE Approach: Effect Significance

传统的DOE方法: 效应显著性检验

0 ( ) k

k

E

MSF

MS

0 ( )k criticalF f If

then effect k is significant or active.

0 1 1 2 2 12 1 2ˆ ... ...i i i k ik i iy x x x x x

37

Fluorescent Lights Example: Traditional DOE

Approach

荧光灯例子: 传统DOE分析方法Suspensions are treated as failures.

Mid-points are used as failure times for interval data.

Life is assumed to follow the normal distribution.

38

Fluorescent Lights Example: Traditional DOE

Approach Results

荧光灯例子:传统DOE分析结果

B and D come out to be significant using traditional DOE approach.

A, B, D and E were found to be significant using R-DOE.

Tradition DOE fails to identify A and E as an important factor at a

significance level of 0.1.

39

Where to Get More InformationWhere to Get More Information

哪里可以找到更多的信息哪里可以找到更多的信息1. http://www.itl.nist.gov/div898/handbook/

2. www.Weibull.com

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