Preprocessing…

In this part, we generate original training data set taken from the uniform distribution in (-1, 1). Then we use tangent sigmoid as a function to transfer the original training data set to a new one which is Gaussian-based. We make an assumption about whether the training data set is center distribution or more uniform will make better analyzed performance. They will give some illustration in following figures.

Figure 4.22 and 4.23 represent the original training data set X and the new training data set 'X which is transferred by tangent function respectively.

Figure 4.22 The original training data setX

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Figure 4.24 RMSE as a function of FWHM under SCSP

Figure 4.25 Correlation coefficient as a function of FWHM under SCSP

Chapter 5 Discussion

For regularization concept, almost all inverse problem methods involve a trade-off between two optimizations：agreement between data and solution, and smoothness of the solution. We define that the unconstrained minimum of agreement and the unconstrained minimum of smoothness is the best solution. Figure 5.1 will give you a brief thought about that. Here, we have a question for how to define or find out the location of the best solution between “Best smoothness” line and “Best agreement” line.

Figure 5.1 Where is the best solution

The estimated criterion RMSE and correlation coefficient would involve a trade-off relationship. In our data experiment results, we hope the RMSE is low and correlation coefficient is high to verify our proposed method. So, we need some

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verification to explain this problem. We make a assumption that our proposed method and PLS may have different curves as shown in Figure 5.2. In further study, we will have a fundamental proof for this issue.

Figure 5.2 Trade-off curves of Bayesian-based PLS and PLS

The preprocessing result, we transfer the original data set to Gaussian form to examine whether the performance is better or not. We make different widths for FWHM to verify our proposed method. But we could obviously find out the hypothesis for data preprocessing doesn’t accomplish to our expectation. The results after preprocessing might be influenced by the limitation of tangent function. The data after tangent function transferring may be divergent so that the analyzed results would be affected for this reason.

The local and global minimum problem is another issue we concern. We would like to find the best solution to approximate nearly global minimum.

Chapter 6

Conclusions and Future works

6.1 Conclusions

We have established a probability based analyzed method which combines the advantages of regularization and the properties of PLS for a novel calibration model.

The proposed method, Bayesian-based PLS, is able to reduce the noise signal hidden in the training data. And it has better analyzed results than original PLS method when training data accompanying noise signal during calibration phase. So we can apply our method to on-line analyzed system for further application.

6.2 Future works

In data preprocessing issue, we might to make tries for other kinds of transfer function (e.g., arcsine function) to make sure the data divergent problem and improve the limitation of transformation accuracy to obtain better performance for further study. The track of best solution between the agreement and smoothness is our next objective to achieve. Then, we also consider to make the results approximated to the global minimum so that we can apply the proposed method for weights initialization of backpropogation network. There still have another issue we have to take into account. The selection of appropriate prior would probably affect the analyzed result.

So we need to make a study about the prior probability to make sure that we don’t have a bad or wrong one.

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