Figure 13: Latent space mapping by UMAP when the network is trained only using only the Positive spectra without multiplying the integration value with the addition of batch normalization layers with different latent dimensions at (a) Latent dimension 3, (b) Latent dimension 4, (c) Latent dimension 5
Figure 14: Clustering accuracy depending on different multiplied Positive spectra ratio with different latent dimensions
p(yi|µi,Σi) = 1
2πn/2|Σ|1/2exp
(yi−µi)T|Σ|−1(yi−µi) 2
, (19)
log(p(yi|µi,Σi)) =log 1
2πn/2|Σ|1/2−(yi−µi)T|Σ|−1(yi−µi)
2 . (20)
Finally, the log-likelihood is changed as the form below,
−log(p(yi|µi,Σi)) ∝||yi−µi||2
2 = ||µi−fθ(xi)||2
2 . (21)
It can therefore be concluded that, when using MSE loss for neural network training, the neural network is trained based on the mean value of its dataset.
In Figure 14, the accuracy is compared with different ratios of the multiplied positive in the training
22
dataset. In the NMR spectra dataset, there are three different mean values,µND,µmul−PosandµPos. Since µmul−Pos is formed by multiplying 5.28 to µPos, µmul−Pos=5.28×µPos. If the ratio of the multiplied Positive spectra is 0.5, the autoencoder is trained on
0.5×(µPos+µmul−Pos) =0.5×(µPos+5.28×µmul−Pos) =3.14×µPos. (22) Then, the dimension of the multiplied Positive spectra is reduced between 5.28×µPosand 3.14×µPos
and the dimension of the Positive spectra are reduced between 3.14×µPos andµPos. GMM makes two multivariate normal distributions based onN (µr−ND|Σr−ND)andN (µr−mul−Pos|Σr−mul−Pos). In the test dataset, there are two multivariate normal distributionsN(µND|ΣND)andN (µPos|ΣPos). If the smaller value than 5.28 is multiplied to Positive spectra, two normal distribution from GMM is not capable to distinguish ND and Positive spectra. In Figure 15, if the bigger value than 5.28 is multiplied, the distance difference betweenµND−µPosandµPos−µmul−Posbecomes too big and the autoencoder considers ND and Positive as one distribution. The distance betweenµPos−µmul−Pos becomes about 25 times bigger than µND−µPos, so the autoencoder considers ND and Positive spectra as one distribution. It is not compatible with the purpose of differentiating ND and Positive spectra.
In Figures 16, 17 and 18, each figure is averaged in a latent plot with different ratios of multiplied Positive spectra in the training dataset. As shown in Figure 14, if the ratio is not 0.5, the ND, Positive, and multiplied Positive spectra are not aligned sequentially, and the accuracy is not as high as the ratio of 0.5.
23
Figure 15: Distance of mean vectors µmul−Pos,µPosandµND in case of multiplying different values in different latent dimension (a) Latent dimension 2, (b) Latent dimension 3, (c) Latent dimension 4, (d) Latent dimension 5
24
Figure 16: The averaged latent space mapping when the ratio of multiplied spectra is zero and different latent dimension (a) Latent dimension 3 (b) Latent dimension 4 (c) Latent dimension 5
Figure 17: The averaged latent space mapping when the ratio of multiplied spectra is 0.5 and different latent dimension (a) Latent dimension 3 (b) Latent dimension 4 (c) Latent dimension 5
Figure 18: The averaged latent space mapping when the ratio of multiplied spectra is zero and different latent dimension (a) Latent dimension 3 (b) Latent dimension 4 (c) Latent dimension 5
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VI Conclusion
In this study, we devise L-SNRNet to distinguish low SNR NMR spectra into impurity-included signal and just noisy signal. We supposed that even the determination made by NMR experts are not accurate, so the autoencoder was used for dimensionality reduction and clustering methods were chosen for semi- supervised learning. We also developed L-SNRNet using three steps of regularization methods. We have normalized the low SNR spectra in multiple aspects. Since each spectrum has different intensity, all spectra have been normalized into maximum value of each spectrum. Then, BN layers have been added to increase the representational capacity within latent space. Finally, to enhance the expressivity of each spectrum, the mean integration values are multiplied. The effect of the multiplied integration value is that it makes the ND and Positive spectra linearly farther apart, so GMM can make two normal distributions more in distinct way. L-SNRNet can distinguish low SNR NMR spectra well into two categories. L-SNRNet can be applied to analyze weak and low SNR spectra of any measurements.
26
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Acknowledgements
I would like to begin by expressing my genuine appreciation to my advisor, Prof. Hongsik Jeong, who gave me the precious opportunity to complete this master’s course in UNIST. I would also like to show my deepest gratitude to Dr. Dong-hyeok Lim. His essential insight in many fields and his thoughtful advice have been indispensable in helping me finish my master’s study.
I am sincerely thankful to my colleagues. With their assistance, I have learned countless things.
They have always encouraged me to keep up my research pace. I hope they achieve all of their goals in the future.
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