Dimensionality Reduction Method for Compact Sample Distribution Using Multi-Step Diffusion Mapping

Zhonghai He; Qiong Jia; Zhanbo Feng; Xiaofang Zhang

doi:10.3788/AOS240820

Acta Optica Sinica, Volume. 44, Issue 20, 2030001(2024)

Dimensionality Reduction Method for Compact Sample Distribution Using Multi-Step Diffusion Mapping

Zhonghai He^1,2、*, Qiong Jia¹, Zhanbo Feng¹, and Xiaofang Zhang³

Author Affiliations

¹School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, Hebei , China

²Hebei Key Laboratory of Micro-Nano Sensing, Qinhuangdao 066004, Hebei , China

³School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China

show less

Abstract Get PDF(in Chinese)

Figures & Tables(9)

Fig. 1. Typical bandwidth-aggregate similarity plot

Download full size

Fig. 2. Typical Shannon entropy variation curve with diffusion step

Download full size

Fig. 3. Spectrogram of soybean meal

Download full size

Fig. 4. Scatter plots of soybean meal samples with different values of diffusion step t. (a) t=1; (b) t=5; (c) t=150; (d) t=21

Download full size

Fig. 5. Scatter plots produced by the different dimensionality reduction methods show that the point cloud from the multi-step diffusion mapping method is more conducive to showing the specificity of the new samples. (a) Soybean meal (PCA); (b) soybean meal (multi-step diffusion mapping method); (c) corn gluten meal (PCA); (d) corn gluten meal (multi-step diffusion mapping method)

Download full size

Table 1. Automatic method for optimal bandwidth selection of kernel function based on the best linear fit degree

View table

Table 1. Automatic method for optimal bandwidth selection of kernel function based on the best linear fit degree

Algorithm 1: bandwidth selection algorithm
Input: N sample sets {X_N}
Output: the optimal bandwidth value of the kernel function $δ_{o p t}$
Initialization procedure: plot the range of bandwidth values on the X-axis using a logarithmic scale, $δ \in [10^{- 4}, 10^{4}]$ , select 81 bandwidth value points according to the ratio of $q = 10^{0.1}$ , $δ = \{δ_{1}, δ_{2}, . . ., δ_{81}\}$
% The similarity matrix is constructed and the total similarity is calculated
1 for a = 1: 81
2 $W (i, j) = e x p (- {‖x_{i} - x_{j}‖}^{2} / δ_{a})$
3 $L (δ) = \sum_{i}^{N} \sum_{j}^{N} W (i, j)$
4 end
%Calculate the range of the bandwidth value $δ \in [δ_{L}, δ_{U}]$ , where the subscripts L and U represent the sequence number of the lower upper bound element in the $δ$ set respectively
5 $\{\begin{matrix} δ_{L} = a r g [N + 0.1 (N^{2} - N)] \\ δ_{U} = a r g [N + 0.9 (N^{2} - N)] \end{matrix}$
%The two-point method is used to fit the line, and the fitting error value of each data point is calculated
6 for i=L+1: U-1
7 ${\hat{y}}_{i} = \frac{L (δ_{i + 1}) - L (δ_{i - 1})}{l g δ_{i + 1} - l g δ_{i - 1}} (l g δ_{i} - l g δ_{i - 1}) + L (δ_{i - 1})$
8 $C_{i} = \|L (δ_{i}) - {\hat{y}}_{i}\|$
9 end
%Select the point with the smallest sum of error values
10 $δ_{o p t} = a r g m i n (C_{i})$

Table 2. Selection algorithm of optimal diffusion step number $t_{o p t}$

View table

Table 2. Selection algorithm of optimal diffusion step number $t_{o p t}$

Algorithm 2: optimal diffusion step t selection algorithm
Input: Markov probability transition matrix P, the maximum $t_{m a x}$
Output: the most appropriate number of diffusion steps $t_{o p t}$
%The matrix P is decomposed by eigenvalues
1 $P x = λ x$ %N eigenvalues are calculated $\{λ_{1}, λ_{2}, \dots, λ_{N}\}$
%The eigenvalues of matrix P are normalized and the function is calculated $H (t)$
2 for t=1: t_max
3 $A (t) = \sum_{j = 1}^{N} λ_{j}^{t}$ %calculate the sum of N features worth t powers
4 for i=1:N
5 ${[η (t)]}_{i} = λ_{i}^{t} / A (t)$
6 end %calculate $\{{[η (t)]}_{1}, {[η (t)]}_{2}, . . ., {[η (t)]}_{N}\}$
7 $H (t) = - {\sum_{i = 1}^{N} [η (t)]}_{i} l o g {[η (t)]}_{i}$
8 end
%Divide $H (t)$ point by point, and the left and right sides of each equinoxes are fitted linearly, and the fitting error value is obtained
9 for j=2:t_max-1
10 for m=1:j
11 It is calculated by linear fitting with least square method $y_{j l} (t)$
12 $C_{j l} = \sum_{m = 1}^{j} \|H (m) - y_{j l} (m)\|$
13 end
14 for n=j:t_max-1
15 It is calculated by linear fitting with least square method $y_{j r} (t)$
16 $C_{j r} = \sum_{n = j}^{t_{m a x}} \|H (n) - y_{j r} (n)\|$
17 end
18 $C_{j} = C_{j l} + C_{j r}$
19 end
%Select the point with the smallest sum of fitting error values
20 $t_{o p t} = a r g m i n (C_{j})$

Table 3. Statistical parameters of the data set
View table
Table 3. Statistical parameters of the data set
Data set Component Minimum value /% Maximum value /% Average value /% Standard deviation /%
Soybean meal Moisture 11.96 15.21 13.15 0.91
Corn gluten meal Moisture 5.10 11.10 7.92 1.01

Table 4. RMSE of spectral prediction model
View table
Table 4. RMSE of spectral prediction model
Data set Old sample New sample 1 New sample 2 New sample 3
Soybean meal 0.2064 0.3261 0.4540 0.6391
Corn gluten meal 0.5935 0.6204 0.6878 0.8358

Tools

Get Citation

Copy Citation Text

Zhonghai He, Qiong Jia, Zhanbo Feng, Xiaofang Zhang. Dimensionality Reduction Method for Compact Sample Distribution Using Multi-Step Diffusion Mapping[J]. Acta Optica Sinica, 2024, 44(20): 2030001

Download Citation

EndNote(RIS)BibTex Plain Text

Set citation alerts for article

Save article for my favorites