Co-authors: C-F Chien, Y-J Chen National Tsing Hua University ISMI - - PowerPoint PPT Presentation

Bayesian Inference Technique for Data mining for Yield Enhancement in Semiconductor Manufacturing Data Presenter: M. Khakifirooz Co-authors: C-F Chien, Y-J Chen National Tsing Hua University

ISMI 2015, 16th -18th Oct. KAIST, Daejeon, Korea

1

The Purpose

• f Bayesian

Inference Data Structure provided by Data Model Data Analysis Approach

• Bayesian Variable

Selection (BVS)

• Data Clearance
• Yield Classification

Conclusive Research Framework Final Decision Table

Conclusion & Path Forward

2

Outline

Bayesian Inference Naïve Bayesian Classifier Gaussian Bayesian Classifier … Bayesian Networks

3

Learning Curve

The Purpose of Bayesian Inference

4

Human Experience Human Experience + System Analysis

Yield Learning Curve of Semiconductor Manufacturing

Yield Learning Curve of Semiconductor Manufacturing: In addition to data analytics, Cumulative

Engineering Training and Experience

significantly enhanced yield improvement

Effron(1996), Tobin et al. (1999)

The Purpose of Bayesian Inference

5

Data Structure provided by Data Model

𝑗 = 1, … , 𝑁 𝑂 ⋕ of process stage sample size 1 ≤ 𝑙𝑗 ≤ 𝑂 ⋕ of specify tools at each stage 𝑜𝑗𝑘, 𝑘 = 1, … , 𝑙𝑗 1 ≤ 𝑄

𝑜𝑗𝑘 ≤ 𝑜𝑗𝑘

𝑞𝑚, 𝑚 = 1, … , 𝑄

𝑜𝑗𝑘

frequency of each specify tool ⋕ of exist chambers for each tool frequency of each exist chamber 𝑂 =

𝑘=1 𝑙𝑗 𝑚=1 𝑄𝑜𝑗𝑘

𝑞𝑚 𝑂 ∗ 𝑁 =

𝑗=1 𝑁 𝑘=1 𝑙𝑗 𝑚=1 𝑄𝑜𝑗𝑘

𝑞𝑚 Response Variable: %Yield (continues) Explanatory Variables: Stages (tools-chambers) (nominal) Stages (process time) (continues)

Obs. 𝐰𝐛𝐬𝟐 𝐰𝐛𝐬𝟑 𝑜1 𝑏1 𝑏2 𝑜2 𝑏1 𝑐2 𝑜3 𝑐1 Na Obs. 𝐰𝐛𝐬𝟐-𝒃𝟐 𝐰𝐛𝐬𝟐−𝒄1 𝐰𝐛𝐬𝟑-𝒃𝟑 𝐰𝐛𝐬𝟑-𝒄2 𝑜1 1 1 𝑜2 1 1 𝑜3 1

Nominal Variables Dummy Variables

6

Data Structure provided by Data Model

Yield 𝒕𝒖𝒃𝒉𝒇 𝟐 𝒕𝒖𝒃𝒉𝒇 𝟑

• bs. 1

𝑈𝑝𝑝𝑚 1 𝑈𝑝𝑝𝑚 2

• bs. 2

𝑈𝑝𝑝𝑚 1 𝑈𝑝𝑝𝑚 1

• bs. 3

𝑈𝑝𝑝𝑚 2 Tool 2 Yield 𝒕𝒖𝒃𝒉𝒇 𝟐 𝒕𝒖𝒃𝒉𝒇 𝟑

• bs. 1

Chamber 1 Chamber 2

• bs. 2

Chamber 2 Chamber 1

• bs. 3

Chamber 1 Chamber 2 Yield 𝒕𝒖𝒃𝒉𝒇 𝟐 𝒕𝒖𝒃𝒉𝒇 𝟑

• bs. 1

𝑈𝑝𝑝𝑚 1. Chamber 1 𝑈𝑝𝑝𝑚 2. Chamber 2

• bs. 2

𝑈𝑝𝑝𝑚 1. Chamber 2 𝑈𝑝𝑝𝑚 1. Chamber 1

• bs. 3

𝑈𝑝𝑝𝑚 2. Chamber 1 𝑈𝑝𝑝𝑚 2. Chamber 2 Yield 𝒕𝒖𝒃𝒉𝒇 𝟐 𝒕𝒖𝒃𝒉𝒇 𝟑

• bs. 1

𝐸𝑏𝑢𝑓 1.1 𝐸𝑏𝑢𝑓 1.2

• bs. 2

𝐸𝑏𝑢𝑓 2.1 𝐸𝑏𝑢𝑓 2.2

• bs. 3

𝐸𝑏𝑢𝑓 3.1 Date 3.2

Yield 𝒕 𝟐. 𝑼 𝟐. 𝑫𝒊 𝟐 𝒕 𝟐. 𝑼 𝟐. 𝑫𝒊 𝟑 𝒕 𝟐. 𝑼 𝟑. 𝑫𝒊 𝟐 𝒕 𝟑. 𝑼 𝟑. 𝑫𝒊 𝟑 𝒕 𝟑. 𝑼 𝟑. 𝑫𝒊 𝟑

• bs. 1

1 1

• bs. 2

1 1

• bs. 3

1 1

Yield 𝒕 𝟐. 𝑼 𝟐. 𝑫𝒊 𝟐 𝒕 𝟐. 𝑼 𝟐. 𝑫𝒊 𝟑 𝒕 𝟐. 𝑼 𝟑. 𝑫𝒊 𝟐 𝒕 𝟑. 𝑼 𝟑. 𝑫𝒊 𝟑 𝒕 𝟑. 𝑼 𝟑. 𝑫𝒊 𝟑

• bs. 1

𝐸𝑏𝑢𝑓 1.1 𝐸𝑏𝑢𝑓 1.2

• bs. 2

𝐸𝑏𝑢𝑓 2.1 𝐸𝑏𝑢𝑓 2.2

• bs. 3

𝐸𝑏𝑢𝑓 3.1 𝐸𝑏𝑢𝑓 2.3

7

Data Structure provided by Data Model

Obs. 𝐰𝐛𝐬𝟐-𝒃𝟐 𝐰𝐛𝐬𝟐−𝒄1 𝐰𝐛𝐬𝟐-𝒅𝟐 𝑜1 1 𝑜2 1 𝑜3 1 Pr(ith variable sellected) 1 3 1 3 1 3

var1−𝑏1, var1−𝑐1, var1−𝑑1

𝑒 Multinomial 1

3 , 1 3 , 1 3 1,0,0 0,0,1 0,1,0 𝐰𝐛𝐬𝟐-𝒃𝟐 𝐰𝐛𝐬𝟐-𝒅𝟐 𝐰𝐛𝐬𝟐−𝒄1 To randomly pick a point in this space, we need a continues distribution Distribution over Multinomial (posterior distribution): Dirichlet Distribution

selection probability based on engineer experience

 Critical Phenomena: i.

High dimensionality caused by transforming categorical variables to dummies

ii.

Multicollinearity caused by dummies nature

iii.

Complicated posterior distribution caused hardness for direct variable selection

• Remedy:

Approximate Inference with Sampling

Use random sampling (MCMC techniques: Gibbs sampler, Metropolis-Hastings,…) to approximate the distribution and selecting significant explanatories

8

Data Analysis Approach

9

Data Analysis Approach: Gibbs Sampler

Suppose 𝒚𝟐, 𝒚𝟑~𝐐𝐬 𝑦, 𝑦2 Beginning with initial value 𝒚𝟐

𝟏 , 𝒚𝟑 𝟏

Sampling at iteration t as follow:

Iteration Sample 𝐲𝟐 Sample 𝐲𝟑 k x𝟐

𝑢 ~𝐐𝐬 x𝟐|x𝟑 t−1

x𝟑

𝑢 ~𝐐𝐬 x𝟑|x𝟐 𝑢

Iterating the above step until the sample values have the same distribution as if they where sampled from the true posterior joint distribution

Based on frequency of visits, selecting the most probable variables

10

Data Analysis Approach: Data Clearance

When X is categorical (dummy var.) & Y is quantitative variable

• parametric or non-parametric?
• dependent or independent?
• unbalanced class?

Yield value Representative var. Bad Yield 53.12 < 1 Middle Yield 53.12 ≤ and ≤ 57.51 ignore Good Yield >57.51

11

Data Analysis Approach: Data Clearance

Level a Level b Level c fc𝑏 fc𝑐 Level d fd𝑏 fd𝑐

Variable I Variable II

If both 𝑤𝑏𝑠. 𝐽 & 𝑤𝑏𝑠. 𝐽𝐽 are explanatory:

• test the Interchangeability of measures
• measurement of the degree of Homogeneity

If 𝑤𝑏𝑠. 𝐽 is explanatory and 𝑤𝑏𝑠. 𝐽𝐽 is response:

• measurement of the Reliability of instrument (test/scale)
• measurement of the Objectivity or lack of bias

MEASURMENT of AGREEMENT

• W. S. Robinson(1957)

Cohen’s Kappa 𝓛

𝒧 < 0, "No agreement" 0 ≤ 𝒧 < 0.2, “Slight agreement“ 0.2 ≤ 𝒧 < 0.4, "Fair agreement" 0.4 ≤ 𝒧 < 0.6, "Moderate agreement" 0.6 ≤ 𝒧 < 0.8, "Substantial agreement" 0.8 ≤ 𝒧 ≤ 1, "Almost perfect agreement"

12

Research Framework (I)

Data Preparation Data Mining & Key Factor Screening Problem Definition Data Integration Dummy Variable Construction for Integrated Variables (1460 var.) Wrap the associate variables Cohen’s Kappa Statistics for each pairs of input variables

Agreement

Assign Cutting Point & Bad/Middle/Good Wafers

No Agreement

A Bayesian Framework for Semiconductor Manufacturing Data

Almost perfect agreement Substantial agreement Moderate agreement 3 109 1,764 Fair agreement Slight agreement No agreement 24,539 280,081 758,574 THE CLASS DISTRIBUTION FOR THE KAPPA TEST FOR EACH PAIR OF INPUT VARIABLES

13

Research Framework (II)

BVS via Gibbs Sampler Data Clearance 𝒧 ≤ 0.2

No Agreement Agreement

GLM Construction with Gaussian distribution & Repeated Random Sub-sampling Validation A Comparison to the Wrapped Variables Define Abnormal Devices & Time Model Construction, Evaluation & Interpretation Cohen’s Kappa Statistics for each pairs of X & Y Data Mining & Key Factor Screening

Model RMSE Adjusted R-squared Min Median Max Min Median Max Gibbs + GLM 1.842 2.653 2.841 0.046 0.371 0.711 GBM + GLM 2.534 3.051 3.332 0.000 0.053 0.337 RF + GLM 2.268 2.838 3.660 0.016 0.293 0.507 GLM 7.951 34.60 139.8 0.000 0.029 0.214

Number of resamples 20, Number of iterations 2

14

Decision Graph

High Yield Middle Yield Low Yield

Factors Date Bad Good Stage10 - Tool2 - Chamber3 before 8/29/2014 2:32 after 8/29/2014 12:50 Stage12 - Tool2 - Chamber1 between 8/30/2014 3:26 & 8/30/2014 3:43 before 8/29/2014 10:55 Stage12 - Tool2 - Chamber4 after 8/29/2014 7:36 till 8/30/2014 3:44 before 8/29/2014 7:36 Stage13 - Tool5 - Chamber2

• generally effected the high yield

Stage17 - Tool2 - Chamber2 after 8/30/2014 12:21 before 8/30/2014 10:37 Stage23-Tool3-Chamber2

• generally effected the high yield

Stage44 - Tool7.- Chamber2 and Chamber3 at 9/3/2014 at 9/1/2014 Stage49 - Tool1.- Chamber4 at 9/3/2014 at 9/2/2014 Stage57 - Tool1.- Chamber3

• generally effected the high yield

15

Decision Table



Based on the empirical results, we validate that the proposed approach has practical viability, which means adding the efficacy of domain knowledge and experience to the system could improve results.



Using the domain knowledge might be to restrict conjunctions in rules to tools, chambers and steps that are related to occurs within a reasonable time frame.



The data are not sampled from a stationary population, hence, over the time, the results may change significantly, or some empirical answer might be reject based on engineer domain knowledge, which doesn’t mean that the result is incorrect.



The result may be a proxy for one or more events that are occurring elsewhere or at the other periods of the time, hence, the simulation study is an essential tool for evaluation the accuracy of our proposed method.

16

Conclusion & Path Forward

17