Evolving a Bayesian network model with information flow for time series interpolation of multiple ocean variables
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Abstract: Based on Bayesian network (BN) and information flow (IF), a new machine learning-based model named IFBN is put forward to interpolate missing time series of multiple ocean variables. An improved BN structural learning algorithm with IF is designed to mine causal relationships among ocean variables to build network structure. Nondirectional inference mechanism of BN is applied to achieve the synchronous interpolation of multiple missing time series. With the IFBN, all ocean variables are placed in a causal network visually, making full use of information about related variables to fill missing data. More importantly, the synchronous interpolation of multiple variables can avoid model retraining when interpolative objects change. Interpolation experiments show that IFBN has even better interpolation accuracy, effectiveness and stability than existing methods.
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Key words:
- Bayesian network /
- information flow /
- time series interpolation /
- ocean variables
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4. Interpolation results with the data missing rate of 40%: WD (a), SST (b), SLP (c), SAL (d), RH (e), DEN (f).
A1. Interpolation results with the data missing rate of 50%: WD (a), SST (b), SLP (c), SAL (d), RH (e), DEN (f).
A2. Interpolation results with the data missing rate of 60%: WD (a), SST (b), SLP (c), SAL (d), RH (e), DEN (f).
Algorithm 1 GS algorithm Input $ V $ is variable set, $ D $ is complete data set of $ V $, $ {G}_{0} $ is an initial structure Output $ G $ is the optimal structure Step 1 Score the initial network structure ${G}_{0}\!\to\! oldscore$; Step 2 Perform arc addition, arc reduction and determine arc direction by IF, score the new network structure $ {G}'\!\to\! tempscore $; if $ tempscore\!>\!oldscore $ $ newscore\!\equiv\! tempscore $ and keep the corresponding arc operation; else $ newscore\!\equiv \!oldscore $ and discard the corresponding arc operation; end if Step 3 If $ newsore\!\to\! max $ return $ G\equiv {G}\!\to '\! $ 
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Table 1. Discretization standard of ocean variables
Ocean variable WD SST SLP SAL RH DEN Interval 1 m/s 0.1°C 1 Pa 0.1‰ 1% 0.1 kg/m3 State label 1–10 1–26 1–9 1–17 1–24 1–19
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Table 2. Discrete training time series
Variable Training data 1 d 2 d 3 d 4 d 5 d ··· 500 d 600 d WD/(m·s–1) 3 2 3 3 5 ··· 2 5 SST/°C 25 24 24 22 24 ··· 19 18 SLP/Pa 2 3 2 3 3 ··· 6 7 SAL/‰ 15 15 15 10 8 ··· 15 15 RH/% 8 12 14 17 10 ··· 9 7 DEN/(kg·m–3) 10 11 11 7 6 ··· 12 12
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Table 3. Standardized IF matrix
Variable WD SST SLP SAL RH DEN WD \ 0.133 6 –0.002 5 –0.012 6 0.023 8 0.087 0 SST –0.000 3 \ 0.005 2 0.194 7 0.109 9 0.239 1 SLP 0.011 9 –0.011 8 \ 0.027 2 0.006 0 –0.023 9 SAL 0.003 0 0.119 3 0.037 1 \ 0.017 1 0.088 6 RH 0.039 4 0.096 2 0.006 7 –0.020 2 \ –0.018 6 DEN –0.005 2 0.259 3 0.028 1 0.347 4 0.066 9 \ Note: \ means it cannot be calculated.
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Table 4. Conditional probability distribution of node
$ {\rm{SST}} $ Condition $P \left(\mathrm{S}\mathrm{S}\mathrm{T}\right|\mathrm{W}\mathrm{D})$ $ \mathrm{W}\mathrm{D}\!=\!1 $ $ \mathrm{W}\mathrm{D}\!=\!2 $ $ \mathrm{W}\mathrm{D}\!=\!3 $ $ \mathrm{W}\mathrm{D}\!=\!4 $ $ \mathrm{W}\mathrm{D}\!=\!5 $ $ \mathrm{W}\mathrm{D}\!=\!6 $ $ \mathrm{W}\mathrm{D}\!=\!7 $ $ \mathrm{W}\mathrm{D}\!=\!8 $ $ \mathrm{W}\mathrm{D}\!=\!9 $ $ \mathrm{W}\mathrm{D}\!=\!10 $ $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!1 $ 0 0 0 0 0 0 0.032 3 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!2 $ 0 0 0 0 0 0.029 0 0.064 5 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!3 $ 0 0 0 0 0.014 3 0.014 5 0 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!4 $ 0 0 0 0 0.028 6 0.058 0 0.032 3 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!5 $ 0 0 0 0.012 2 0.014 3 0.014 5 0.032 3 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!6 $ 0 0 0.021 3 0.024 4 0.028 6 0.029 0 0.032 3 0.153 8 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!7 $ 0 0.062 5 0 0.048 8 0.042 9 0.058 0 0.096 8 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!8$ 0 0.031 3 0.021 3 0.012 2 0.042 9 0.043 5 0.032 3 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!9 $ 0 0.031 3 0.042 6 0.109 8 0.014 3 0.014 5 0 0.153 8 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!10$ 0.1 0.031 3 0.042 6 0.024 4 0.028 6 0.014 5 0 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!11 $ 0 0.031 3 0.021 3 0.012 2 0.028 6 0.072 5 0.032 3 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!12 $ 0 0 0.063 8 0.048 8 0.071 4 0.087 0 0 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!13 $ 0.1 0 0.085 1 0.048 8 0.028 6 0.043 5 0.032 3 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!14 $ 0.1 0 0 0.036 6 0.057 1 0.058 0 0.096 8 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!15 $ 0 0.031 3 0.063 8 0.048 8 0.057 1 0.014 5 0.064 5 0.076 9 0.333 3 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!16 $ 0 0.031 3 0.042 6 0.048 8 0.014 3 0.014 5 0.064 5 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!17$ 0 0 0.042 6 0.012 2 0.028 6 0.043 5 0 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!18 $ 0.2 0.031 3 0.085 1 0.073 2 0.114 3 0.014 5 0.064 0 0 0.333 3 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!19 $ 0.1 0.156 3 0.021 3 0.122 0 0.128 6 0.072 5 0 0.076 9 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!20 $ 0 0.062 5 0.063 8 0.085 4 0.014 3 0.058 0 0.096 8 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!21 $ 0.2 0.187 5 0.148 9 0.134 1 0.071 4 0.058 0 0.032 3 0.076 9 0 1 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!22$ 0 0.031 3 0.042 6 0.036 6 0.028 6 0.058 0 0.064 5 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!23 $ 0.1 0.125 0 0.063 8 0.012 2 0.028 6 0.072 5 0.032 3 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!24 $ 0 0.093 8 0.021 3 0.048 8 0.085 7 0.029 0 0.096 8 0 0.333 3 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!25 $ 0.1 0.031 3 0.085 1 0 0.028 6 0.029 0 0 0 0 0 $ \mathrm{S}\mathrm{S}\mathrm{T}\!=\!26 $ 0 0.031 3 0.021 3 0 0 0 0 0 0 0 Note: WD=1, 2, ···, 10, and SST=1, 2, ···, 26 indicate WD and SST take different discrete states.
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Table 5. TIC of each model under different missing rate
Missing rate Interpolation model WD SST SLP SAL RH DEN 40% IFBN 0.091 5 0.001 1 0.001 4 0.000 9 0.011 5 0.001 7 CSI 0.182 9 0.005 3 0.002 7 0.002 2 0.022 1 0.004 3 BP 0.095 6 0.003 3 0.002 5 0.001 6 0.014 1 0.003 5 CBN 0.089 7 0.003 6 0.002 2 0.001 8 0.013 9 0.003 3 50% IFBN 0.117 1 0.001 4 0.001 5 0.001 1 0.017 5 0.001 3 CSI 0.177 6 0.013 2 0.006 1 0.005 6 0.039 5 0.009 8 BP 0.119 4 0.005 8 0.004 3 0.003 4 0.025 2 0.014 3 CBN 0.112 1 0.005 6 0.004 5 0.003 7 0.031 1 0.012 7 60% IFBN 0.106 5 0.003 7 0.002 4 0.002 2 0.018 9 0.001 8 CSI 0.215 8 0.015 4 0.010 9 0.009 3 0.062 9 0.013 1 BP 0.138 2 0.008 3 0.007 4 0.006 5 0.040 7 0.012 1 CBN 0.141 2 0.007 9 0.007 5 0.006 1 0.039 6 0.011 9 Note: The bold numbers are the results obtained by proposed model.
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Table 6. R of each model under different missing rate
Missing rate Interpolation model WD SST SLP SAL RH DEN 40% IFBN 0.811 5 0.989 5 0.987 2 0.987 1 0.788 1 0.992 7 CSI 0.731 5 0.926 9 0.941 2 0.975 9 0.719 9 0.968 1 BP 0.636 6 0.915 7 0.935 3 0.985 9 0.720 5 0.973 6 CBN 0.701 1 0.914 8 0.939 8 0.984 6 0.731 6 0.975 4 50% IFBN 0.678 6 0.988 8 0.972 3 0.993 5 0.690 8 0.990 4 CSI 0.651 0 0.807 2 0.838 1 0.867 0 0.578 4 0.839 9 BP 0.512 7 0.943 8 0.905 7 0.975 5 0.596 1 0.816 5 CBN 0.611 4 0.915 2 0.902 1 0.969 8 0.601 3 0.824 1 60% IFBN 0.667 8 0.934 2 0.923 1 0.929 1 0.654 7 0.982 7 CSI 0.585 9 0.716 4 0.793 6 0.860 2 0.517 0 0.768 0 BP 0.618 3 0.883 2 0.826 5 0.973 8 0.574 4 0.798 3 CBN 0.609 7 0.892 1 0.799 8 0.976 1 0.584 2 0.801 1 Note: The bold numbers are the results obtained by proposed model.
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Table 7. The Maximum and averaged length of the consecutive missing data for each variable at different missing rate
Missing rate Statistics length WD SST SLP SAL RH DEN 40% maximum/d 4.00 5.00 4.00 4.00 4.00 3.00 average/d 1.59 1.92 1.78 1.78 1.48 1.32 50% maximum/d 6.00 5.00 7.00 7.00 6.00 7.00 average/d 2.32 2.46 3.56 3.91 3.64 2.98 60% maximum/d 8.00 8.00 7.00 9.00 8.00 9.00 average/d 3.78 5.23 4.16 5.35 4.62 5.01
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Table 8. The imformation of new experiment data
Position name Peried Latitude-longitude coordinate A Jan. 1, 2013 to Jun. 30, 2015 5°N, 110°W B Apr. 1, 2009 to Jul. 31, 2012 12°N, 70°W C May 1, 2010 to Jan. 31, 2013 15°S, 90°E
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Table 9. Comparative analysis of interpolation results with different model in Position A
Evaluation indicator Interpolation model WD SST SLP SAL RH DEN MRE IFBN 0.167 7 0.036 5 0.037 9 0.027 1 0.134 4 0.028 7 CSI 0.245 8 0.065 4 0.093 9 0.042 3 0.229 1 0.062 1 BP 0.191 5 0.056 4 0.041 2 0.035 9 0.179 9 0.038 1 TIC IFBN 0.140 9 0.002 6 0.002 2 0.000 5 0.023 9 0.002 7 CSI 0.179 4 0.005 8 0.004 3 0.004 4 0.075 2 0.014 3 BP 0.147 6 0.003 2 0.003 1 0.002 6 0.049 5 0.009 8 R IFBN 0.618 6 0.968 8 0.960 2 0.950 8 0.633 3 0.910 3 CSI 0.582 7 0.923 8 0.835 7 0.865 5 0.514 4 0.736 5 BP 0.601 2 0.946 9 0.908 1 0.917 1 0.577 1 0.819 9 Note: The bold numbers are the results obtained by proposed model.
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Table 10. Comparative analysis of interpolation results with different model in Position B
Evaluation indicator Interpolation model WD SST SLP SAL RH DEN MRE IFBN 0.169 8 0.031 8 0.040 0 0.025 5 0.134 9 0.027 2 CSI 0.241 1 0.064 8 0.096 4 0.043 2 0.225 5 0.059 1 BP 0.189 3 0.055 2 0.039 0 0.033 1 0.176 4 0.035 6 TIC IFBN 0.136 4 0.005 3 0.002 1 0.000 3 0.021 5 0.003 9 CSI 0.175 4 0.008 8 0.005 9 0.002 0 0.078 6 0.014 0 BP 0.150 8 0.003 1 0.002 9 0.002 7 0.047 0 0.008 3 R IFBN 0.620 5 0.968 7 0.956 4 0.952 8 0.636 4 0.913 6 CSI 0.580 9 0.923 3 0.835 7 0.869 4 0.511 8 0.737 4 BP 0.605 7 0.948 4 0.912 7 0.921 7 0.581 4 0.820 4 Note: The bold numbers are the results obtained by proposed model.
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Table 11. Comparative analysis of interpolation results with different model in Position C
Evaluation indicator Interpolation model WD SST SLP SAL RH DEN MRE IFBN 0.171 9 0.039 3 0.036 0 0.022 9 0.130 5 0.031 7 CSI 0.243 7 0.069 7 0.094 2 0.039 6 0.233 7 0.061 4 BP 0.194 1 0.052 7 0.037 9 0.040 0 0.174 9 0.042 2 TIC IFBN 0.143 4 0.003 3 0.003 2 0.000 4 0.026 6 0.003 5 CSI 0.178 2 0.005 5 0.001 9 0.007 7 0.078 4 0.011 9 BP 0.148 3 0.003 7 0.004 6 0.003 1 0.053 2 0.006 3 R IFBN 0.614 4 0.967 2 0.962 1 0.955 8 0.629 1 0.906 7 CSI 0.578 2 0.920 4 0.838 2 0.861 3 0.513 4 0.740 2 BP 0.601 5 0.949 8 0.907 6 0.916 5 0.574 7 0.820 7 Note: The bold numbers are the results obtained by proposed model.
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