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Ph. D. ThesisPh. D. Thesis 6. Results – Multivariate Calibrations6. Results – Multivariate Calibrations 6.11. Conclusions6.11. Conclusions
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Ph. D. Thesis
  Abstract
  Table of Contents
  1. Introduction
  2. Theory – Fundamentals of the Multivariate Data Analysis
  3. Theory – Quantification of the Refrigerants R22 and R134a: Part I
  4. Experiments, Setups and Data Sets
  5. Results – Kinetic Measurements
  6. Results – Multivariate Calibrations
    6.1. PLS Calibration
    6.2. Box-Cox Transformation + PLS
    6.3. INLR
    6.4. QPLS
    6.5. CART
    6.6. Model Trees
    6.7. MARS
    6.8. Neural Networks
    6.9. PCA-NN
    6.10. Neural Networks and Pruning
    6.11. Conclusions
  7. Results – Genetic Algorithm Framework
  8. Results – Growing Neural Network Framework
  9. Results – All Data Sets
  10. Results – Various Aspects of the Frameworks and Measurements
  11. Summary and Outlook
  12. References
  13. Acknowledgements
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6.11.   Conclusions

In this chapter, several multivariate calibration methods were used for the quantification of the refrigerants R22 and R134a in mixtures measured by a single sensor SPR setup. It was demonstrated that the combination of the time-resolved measurement approach with a multi­variate data analysis allows the quantification of 2 analytes using only 1 single sensor.

Yet, the most common multivariate calibration method, PLS, was not capable of dealing with the nonlinear relationship between the sensor signals and the concentrations of the analytes resulting in unacceptably high prediction errors and systematical biases. Thus, different well-known methods for linearizing the data or for introducing nonlinearities into linear models were used. In principle, these methods successfully compensated the nonlinearities in the data structure. Yet, compared with the standard deviations of the sensor signals with 0.2% for R22 and 1.3% for R134a, which were calculated for 60 seconds exposure to analyte of different concentrations using reproduced measurements, the overall quality of predictions seems to be improvable.

The application of uniform fully connected neural networks showed the best results with respect to generalization ability. In contrast to the methods mentioned before, the neural networks make no assumption of the type of relationship between the input and the response variables (linear, quadratic...) and thus can approximate the relationship between the time-resolved sensor responses and the concentrations of the analytes quite well, for which no model exists at the moment. A significant drawback, which can also be seen in table 2, is the overfitting of the neural networks, which is observable as gap between the prediction errors of the calibration data and of the validation data. In contrast to all other methods mentioned before, the neural networks perform no variable selection or compression of the input information resulting in a high number of adjustable parameters and consequently running into the danger of overfitting. Therefore, neural networks combined with a principal component analysis were used for the compression of the input variables and two pruning algorithms were applied for thinning out the network structure. Whereas the pruning algorithms showed unstable and worse results, the combination of the PCA with neural networks demonstrated that in principle smaller neural network could calibrate the relationships at least as well as the non-optimized networks. As the PCA-NN is a very simple method to reduce the neural network size, a more sophisticated variable selection method is expected to show even better results. Thus, in the next chapter genetic algorithms are combined with neural networks to perform a variable selection and thus to reduce the network size.

 

Method

Calibration

Data Set

Validation

Data Set

Non-linearity

Over-fitting

R22

R134a

R22

R134a

PLS (Martens' Uncertainty)

11.89

11.40

10.27

9.94

High

No

PLS (Min. Crossvalidation)

10.47

8.51

8.69

7.63

High

No

Box-Cox Transformation

2.97

4.50

3.09

5.04

Low

Medium

INLR

2.25

2.81

3.47

4.02

No

Medium

QPLS

2.31

3.87

2.41

3.92

No

No

CART

3.81

4.85

8.79

11.20

No

High

Model Trees

7.19

7.59

10.29

11.20

No

Medium

MARS

1.46

2.27

2.96

3.71

No

Medium

Neural Networks

1.47

2.62

2.18

3.26

No

Medium

PCA - NN

1.98

3.08

2.16

3.24

No

No

MAG-Pruning - NN

2.34

3.16

2.48

3.34

No

No

OBS-Pruning - NN

2.10

3.22

2.12

3.32

No

No

table 2:      Comparison of the rel. RMSE of the calibration and validation data in %. Additionally the degrees of nonlinearity and overfitting of the predictions are listed.

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