本研究擬分析螺絲鍛造力訊號資料，作為螺絲品質分類之依據。此處之訊號為非負實數之壓電反應值。非負矩陣分解法將給定的高維度非負矩陣，分解成非負的「訊號基底矩陣」與非負的「權重矩陣」。再以權重矩陣作為降維後之數據，進行分類與分群，並與其他降維方法比較。本研究針對舊產線與新產線提出不同分類方法，建立螺絲品質分類之準則。期能將此準則應用在生產線上，建構一螺絲品質即時監控系統，進而改善工廠製程之良率，降低生產成本。

This thesis investigates the problem of identification of the quality of screws through the data of screws' forging force signals. For each sample, the signals are a sequence of non-negative voltage values generated by the piezoelectric effect. The non-negative matrix factorization decomposes a high dimensional non-negative matrix into a non-negative “signal bases matrix” and a “weights matrix”. The weights matrix is used as the dimensionally reduced data for further analysis such as classification and clustering. We compare its performance with other dimension reduction methodology. Moreover, in this work, we propose different classification methods for offline control with sufficient and insufficient prior data and establish the corresponding quality classification criterion. The offline control criterion with insufficient prior data can be applied to the production line to construct a screw quality real-time monitoring system, which can help to improve the yield rate of the production line and reduce the cost.

論文審定書 i

誌謝 iii

摘要 iv

Abstract v

1 Introduction p.1

2 Data Descriptions p.3
2.1 Data Generating System p.3
2.2 Experimental Data p.4
2.3 Real Data p.6

3 Methodology p.7
3.1 Non-negative Matrix Factorization (NMF) p.7
3.1.1 Notations and Assumption p.7
3.1.2 Measurement of Errors p.8
3.1.3 Gradient Descent p.8
3.1.4 Multiplicative Update (MU) Rule p.9
3.1.5 Pseudo Inverse Projection with Non-negative Constraint p.10
3.2 Regularizing Weights Matrix p.10
3.3 K-Nearest Neighbors Classification p.12
3.4 Density-based Spatial Clustering of Applications with Noise p.13
3.5 Other Dimension Reduction Methods p.16
3.5.1 Principal Component Analysis p.16
3.5.2 Functional Principal Component Analysis p.16

4 Analysis Procedure p.17
4.1 Analysis of Experimental Data p.17
4.2 Analysis of Real Data p.18
4.2.1 Offline Control with Sufficient Prior Data p.18
4.2.2 Offline Control with Insufficient Prior Data p.19

5 Empirical Study p.21
5.1 Experimental Analysis p.21
5.1.1 Dimension Reduction with the NMF p.21
5.1.2 Regularization of the Weight Matrix p.22
5.1.3 Pseudo Inverse Projection with Non-negative Constraint p.22
5.1.4 Regularization of the Weight Matrix with Testing Data p.23
5.1.5 K-Nearest Neighbors Classification p.23
5.1.6 Comparison with Other Methods with Cross Validation p.24
5.2 Offline Control with Sufficient Prior Data p.25
5.3 Offline Control with Insufficient Prior Data p.27

6 Conclusion and Future Work p.29
6.1 Conclusion p.29
6.2 FutureWork p.30

7 References p.32

A Appendix p.33

[1] Ding, C., He, X. and Simon, H.D. (2005). On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering. Proc. SIAM Int’l Conf. Data Mining, 606–610.

[2] Ester, M., Kriegel, H., Sander, J. and Xu, X. (1996). A density-based algorithm for discov- ering clusters in large spatial databases with noise. Proceedings of the Second International Conference on Knowledge Discovery and Data Mining. 226–231.

[3] James, G., Witten, D., Hastie, T. and Tibshirani, R. (2013). An Introduction to Statistical Learning with Applications in R. Springer, New York.

[4] Johnson, R.D. and Wichern, D.W. (2007). Applied Multivariate Statistical Analysis. Pear- son, Essex.

[5] Lee, D.D. and Seung, H.S. (1999). Learning the Parts of Objects by Nonnegative Matrix Factorization. Nature, 401, 788–791.

[6] Montgomery, D.C. (2012). Introduction to Statistical Quality Control, 7th Edition. John Wiley & Sons, New York.

[7] Ramsay,J.O.andSilverman,B.W.(2005).FunctionalDataAnalysis,2ndEdition.Springer, New York.