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SCIENCE CHINA Information Sciences, Volume 62, Issue 7: 070204(2019) https://doi.org/10.1007/s11432-018-9714-5

Pigeon-inspired optimization and extreme learning machine via wavelet packet analysis for predicting bulk commodity futures prices

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  • ReceivedAug 15, 2018
  • AcceptedNov 30, 2018
  • PublishedMay 31, 2019

Abstract

In this paper, a hybrid approach consisting of pigeon-inspired optimization (PIO) and extreme learning machine (ELM) based on wavelet packet analysis (WPA) is presented for predicting bulk commodity futures prices. Firstly, WPA is applied to decompose the original futures prices into a set of lower-frequency subseries. Secondly, the PIO algorithm is used to optimize the parameters of ELM and then the optimized ELM is utilized to forecast the subseries. Finally, we adopt the hybrid method to calculate the final forecasting outcomes of futures prices. In order to further test the predictive ability of the hybrid forecasting model on bulk commodity futures prices, we use the prices of West Texas Intermediate crude oil futures and Chicago Board of Trade soybean futures to make one-step, two-step and four-step ahead predictions. In comparison with complete ensemble empirical mode decomposition with adaptive noise, empirical mode decomposition and singular spectrum analysis, WPA is the most suitable for decomposing bulk commodity futures prices. The experimental outcomes show that the hybrid WPA-PIO-ELM model has better performance on horizontal precision, directional precision and robustness.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61773401, 61304067, 11601524, 61761130081) and Foundation of Hubei Province of China (Grant Nos. 17G024, 2017132).


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  • Figure 1

    (Color online) Learning framework of the hybrid model.

  • Figure 2

    (Color online) Futures prices of WTI crude oil.

  • Figure 3

    (Color online) Futures prices of CBOT soybean.

  • Figure 4

    (Color online) Decomposition of WTI crude oil futures prices.

  • Figure 5

    (Color online) MAPE of WTI crude oil futures prediction.

  • Figure 10

    (Color online) Ds of WTI crude oil futures prediction.

  • Figure 11

    (Color online) Decomposition of CBOT soybean futures prices.

  • Figure 12

    (Color online) MAPE of CBOT soybean futures prediction.

  • Figure 17

    (Color online) Ds of CBOT soybean futures prediction.

  • Table 1   Calculation formulae of statistical indicators
    Statistical indicator Calculation method
    MAPE MAPE $=\frac{1}{T}~\sum_{t=1}^{T}~|~\frac{\hat{H}(t)-H(t)}{H(t)}|\times~100%$
    RMSE RMSE $=\sqrt{\frac{1}{T}~\sum_{t=1}^{T}(\hat{H}(t)-H(t))^2}$
    NRMSE NRMSE $=\frac{100}{\bar{H}}\sqrt{\frac{1}{T}~\sum_{t=1}^{T}(\hat{H}(t)-H(t))^2}\times~100%$
    TIC TIC $=\frac~{\sqrt{\frac{1}{T}~\sum_{t=1}^{T}(\hat{H}(t)-H(t))^2}}~{\sqrt{\frac{1}{T}~\sum_{t=1}^{T}~\hat{H}(t)~^2}~+~\sqrt{\frac{1}{T}~\sum_{t=1}^{T}~H(t)~^2}~}$
    IA IA $=~1-~\frac~{\sum_{t=1}^{T}~(\hat{H}(t)-H(t))^2}~{\sum_{t=1}^{T}~(~|~\hat~{H}(t)-\bar{H}~|~+~|~H(t)-\bar{H}~|~)^2}$
    Ds Ds $=~\frac~{1}~{T}~\sum_{t=2}^~{T}~\gamma(t)~\times~100~%~$
  •   

    Algorithm 1 Hybrid WPA-PIO-ELM

    Utilize WPA to decompose the original series into a set of subseries;

    Input the subseries, build an ELM model, set the initial weights and thresholds of ELM as the optimization objects ($X$), and set the prediction error as the fitness function of PIO $f(x)$;

    Initialize the parameters of the PIO algorithm, including the number of pigeons ($N$), the dimension ($D$) and the changing boundary $[m,n]$ of the searching room, the map and compasscoefficient ($R$), the maximum number of iterations affected by the two operators $(N_{O_1},N_{O_2})$ in PIO and a random velocity ($V_j$) and location ($X_j$) of every pigeon;

    while $l~\leq~N_{O_1}~$ do

    for each pigeon

    Update their velocities and locations by Eqs. 8 and 9 and calculate their fitness values;

    end for

    Update the location of the global best pigeon $X_{\rm~gbest}$;

    end while

    while $N_{O_1}+1~\leq~l~\leq~N_{O_2}$ do

    for each pigeon

    Update their velocities and locations by Eqs. 10, 11 and 12 and calculate their fitness values;

    end for

    Update the location of the global best pigeon $X_{\rm~gbest}$;

    end while

    return $X_{\rm~gbest}$;

    Set $X_{\rm~gbest}$ as the initial weights and thresholds of ELM and utilize the improved ELM to make predictions.

  • Table 2   DM test outcomes of one-step ahead prediction of WTI crude oil futures
    Benchmark model Tested model
    CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    WPA-PIO-ELM $-$2.6052 $-$1.9203 $-$3.1579 $-$2.9443 $-$3.3404 $-$4.9981
    mbox (0.0079) (0.0337) (0.0022) (0.0036) (0.0014) (2.34e$-$05)
    CEEMDAN-PIO-ELM mbox $-$1.8080 $-$3.0970 $-$2.9287 $-$3.3293 $-$4.9979
    mbox mbox (0.0419) (0.0025) (0.0038) (0.0015) (2.34e$-$05)
    PIO-ELM mbox mbox $-$2.0356 $-$2.6472 $-$3.1379 $-$4.9968
    mbox mbox mbox (0.0267) (0.0072) (0.0023) (2.35e$-$05)
    ELM mbox mbox mbox $-$1.7475 $-$2.5949 $-$4.9927
    mbox mbox mbox mbox (0.0470) (0.0081) (2.38e$-$05)
    LSSVM mbox mbox mbox mbox $-$3.8140 $-$4.9817
    mbox mbox mbox mbox mbox (0.0004) (2.44e$-$05)
    GRNN mbox mbox mbox mbox mbox $-$4.9730
    mbox mbox mbox mbox mbox mbox (2.49e$-$05)
  • Table 3   DM test outcomes of two-step ahead prediction of WTI crude oil futures
    Benchmark model Tested model
    CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    WPA-PIO-ELM $-$1.1835 $-$2.3021 $-$2.6263 $-$5.5253 $-$4.8258 $-$2.4828
    mbox (0.1243) (0.0154) (0.0075) (6.40e$-$06) (3.59e$-$05) (0.0104)
    CEEMDAN-PIO-ELM mbox $-$2.2151 $-$2.5941 $-$5.5220 $-$4.8200 $-$2.4798
    mbox mbox (0.0185) (0.0081) (6.45e$-$06) (3.65e$-$05) (0.0105)
    PIO-ELM mbox mbox $-$1.6824 $-$5.4803 $-$4.7432 $-$2.3821
    mbox mbox mbox (0.0530) (7.14e$-$06) (4.41e$-$05) (0.0129)
    ELM mbox mbox mbox $-$5.5512 $-$4.8177 $-$2.3426
    mbox mbox mbox mbox (6.01e$-$06) (3.67e$-$05) (0.0141)
    LSSVM mbox mbox mbox mbox 6.2658 4.5846
    mbox mbox mbox mbox mbox (1.0000) (0.9999)
    GRNN mbox mbox mbox mbox mbox 1.5720
    mbox mbox mbox mbox mbox mbox (0.9352)
  • Table 4   DM test outcomes of four-step ahead prediction of WTI crude oil futures
    Benchmark model Tested model
    CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    WPA-PIO-ELM $-$2.1870 $-$2.6353 $-$3.3719 $-$5.4831 $-$4.8650 $-$2.2768
    mbox (0.0196) (0.0074) (0.0013) (7.094e$-$06) (3.26e$-$05) (0.0162)
    CEEMDAN-PIO-ELM mbox $-$2.5505 $-$3.3183 $-$5.4777 $-$4.8556 $-$2.1432
    mbox mbox (0.0089) (0.0015) (7.189e$-$06) (3.337e$-$05) (0.0215)
    PIO-ELM mbox mbox $-$2.3883 $-$5.4729 $-$4.8288 0.4089
    mbox mbox mbox (0.0128) (7.272e$-$06) (3.568e$-$05) (0.6568)
    ELM mbox mbox mbox $-$5.5533 $-$4.8777 2.8062
    mbox mbox mbox mbox (5.976e$-$06) (3.159e$-$05) (0.9950)
    LSSVM mbox mbox mbox mbox 6.2270 5.4896
    mbox mbox mbox mbox mbox (1.0000) (1.0000)
    GRNN mbox mbox mbox mbox mbox 1.5720
    mbox mbox mbox mbox mbox mbox (1.0000)
  • Table 5   PT test outcomes of WTI crude oil futures prediction
    mbox WPA-PIO-ELM CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    One-step ahead 45.4319 25.6441 0.9511 $-$0.5960 $-$0.2667 0.7757 0.4879
    prediction (0.0000) (0.0000) (0.3416) (0.5512) (0.7897) (0.4379) (0.6256)
    Two-step ahead 37.7136 18.8974 0.6604 $-$1.4355 $-$0.7289 $-$0.7529 $-$0.7486
    prediction (0.0000) (0.0000) (0.5090) (0.1511) (0.4661) (0.4515) (0.4541)
    Four-step ahead 26.6502 15.1759 $-$0.7426 $-$0.7186 $-$1.2499 $-$0.3596 $-$0.9909
    prediction (0.0000) (0.0000) (0.4577) (0.4724) (0.2113) (0.7191) (0.3217)
  • Table 6   DM test outcomes of one-step ahead prediction of CBOT soybean futures
    Benchmark model Tested model
    CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    WPA-PIO-ELM $-$2.0316 $-$1.3856 $-$2.1278 $-$4.8871 $-$5.4659 $-$3.0946
    mbox (0.0270) (0.0896) (0.0222) (3.09e$-$05) (7.40e$-$06) (0.0026)
    CEEMDAN-PIO-ELM mbox $-$1.3097 $-$2.0564 $-$4.8841 $-$5.4644 $-$3.0944
    mbox mbox (0.1016) (0.0256) (3.11e$-$05) (7.43e$-$06) (0.0026)
    PIO-ELM mbox mbox $-$2.2538 $-$4.7482 $-$5.3783 $-$3.0891
    mbox mbox mbox (0.0170) (4.36e$-$05) (9.17e$-$06) (0.0026)
    ELM mbox mbox mbox $-$4.7634 $-$5.3927 $-$3.0884
    mbox mbox mbox mbox (4.20e$-$05) (8.85e$-$06) (0.0026)
    LSSVM mbox mbox mbox mbox $-$6.2995 $-$2.8916
    mbox mbox mbox mbox mbox (9.95e$-$07) (0.0041)
    GRNN mbox mbox mbox mbox mbox $-$2.6876
    mbox mbox mbox mbox mbox mbox (0.0066)
  • Table 7   DM test outcomes of two-step ahead prediction of CBOT soybean futures
    Benchmark model Tested model
    CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    WPA-PIO-ELM $-$2.4016 $-$2.3861 $-$1.9301 $-$5.0574 $-$5.6073 $-$1.9580
    mbox (0.0124) (0.0128) (0.0330) (2.02e$-$05) (5.24e$-$06) (0.0312)
    CEEMDAN-PIO-ELM mbox $-$2.3263 $-$1.9052 $-$5.0528 $-$5.6045 $-$1.9345
    mbox mbox (0.0146) (0.0347) (2.05e$-$05) (5.28e$-$06) (0.0327)
    PIO-ELM mbox mbox $-$0.4727 $-$4.8624 $-$5.4854 $-$0.9618
    mbox mbox mbox (0.3205) (3.28e$-$05) (7.05e$-$06) (0.1731)
    ELM mbox mbox mbox $-$5.1623 $-$5.7208 $-$0.6101
    mbox mbox mbox mbox (1.56e$-$05) (3.98e$-$06) (0.2739)
    LSSVM mbox mbox mbox mbox $-$6.4966 4.3509
    mbox mbox mbox mbox mbox (6.26e$-$07) (0.9999)
    GRNN mbox mbox mbox mbox mbox 5.1468
    mbox mbox mbox mbox mbox mbox (1.0000)
  • Table 8   DM test outcomes of four-step ahead prediction of CBOT soybean futures
    Benchmark model Tested model
    CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    WPA-PIO-ELM $-$2.3803 $-$3.3016 $-$1.8914 $-$5.7202 $-$6.2204 $-$4.7718
    mbox (0.0130) (0.0016) (0.0356) (3.98e$-$06) (1.20e$-$06) (4.11e$-$05)
    CEEMDAN-PIO-ELM mbox $-$3.1817 $-$1.8754 $-$5.7113 $-$6.2153 $-$4.7581
    mbox mbox (0.0021) (0.0368) (4.07e$-$06) (1.21e$-$06) (4.25e$-$05)
    PIO-ELM mbox mbox $-$1.5243 $-$5.5971 $-$6.1524 $-$4.6515
    mbox mbox mbox (0.0705) (5.37e$-$06) (1.41e$-$06) (5.55e$-$05)
    ELM mbox mbox mbox $-$5.8226 $-$6.6770 $-$4.2913
    mbox mbox mbox mbox (3.11e$-$06) (4.11e$-$07) (0.0001)
    LSSVM mbox mbox mbox mbox $-$7.1382 $-$1.2593
    mbox mbox mbox mbox mbox (1.43e$-$07) (0.1103)
    GRNN mbox mbox mbox mbox mbox 1.3198
    mbox mbox mbox mbox mbox mbox (0.9001)
  • Table 9   PT test outcomes of CBOT soybean futures prediction
    mbox WPA-PIO-ELM CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    One-step ahead 67.4621 25.4103 $-$2.8837 $-$0.6938 $-$4.0953 $-$3.8716 $-$2.7188
    prediction (0.0000) (0.0000) (0.0039) (0.4878) (4.2162e$-$05) (0.0001) (0.0066)
    Two-step ahead 42.0257 18.2267 1.1240 2.9933 $-$2.2243 $-$3.4929 $-$2.0437
    prediction (0.0000) (0.0000) (0.2610) (0.0028) (0.0261) (0.0005) (0.0410)
    Four-step ahead 29.9611 15.2254 1.0903 0.6843 $-$1.7956 $-$2.5590 0.0678
    prediction (0.0000) (0.0000) (0.2756) (0.4938) (0.0726) (0.0105) (0.9459)
  • Table 10   Robustness analysis of WTI crude oil futures prediction
    WTI forcasting Standard deviation WPA-PIO-ELM CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    mbox MAPE 0.0023 0.0033 0.0041 0.0274 0.4093 0.2818 1.1207
    One-step ahead RMSE 0.0007 0.0006 0.0005 0.0185 0.2896 0.1956 0.5645
    prediction IA 0.0005 0.0005 0.0005 0.0009 0.0025 0.0021 0.0165
    mbox Ds 0.6398 0.5829 0.5941 0.9787 1.1890 0.6727 0.9041
    mbox MAPE 0.0033 0.0060 0.0073 0.0062 0.3667 0.2453 1.8952
    Two-step ahead RMSE 0.0011 0.0009 0.0001 0.0040 0.2619 0.1745 1.1927
    prediction IA 0.0001 0.0002 0.0002 0.0001 0.0030 0.0022 0.0263
    mbox Ds 0.3200 0.3017 0.3200 0.3219 0.1882 0.8019 0.6411
    mbox MAPE 0.0064 0.0082 0.0146 0.0067 0.2408 0.7800 0.6390
    Four-step ahead RMSE 0.0142 0.0034 0.0029 0.0142 0.1709 0.1746 0.3951
    prediction IA 0.0006 0.0003 0.0003 0.0003 0.0034 0.0053 0.0017
    mbox Ds 0.0457 0.1371 0.1828 0.3206 0.4767 0.3535 1.2520
  • Table 11   Robustness analysis of CBOT soybean futures prediction
    WTI forcasting Standard deviation WPA-PIO-ELM CEEMDAN-PIO-ELM PIO-ELM ELM LSSVM GRNN BPNN
    mbox MAPE 0.0016 0.0027 0.0035 0.0425 0.1370 0.1668 0.5557
    One-step ahead RMSE 0.0043 0.0049 0.0043 0.5918 1.7537 2.1322 6.9070
    prediction IA 0.0003 0.0004 0.0003 0.0012 0.0186 0.0109 0.0331
    mbox Ds 0.3656 0.8311 0.3656 0.8799 2.2887 0.6377 2.3280
    mbox MAPE 0.0014 0.0048 0.0061 0.0841 0.1234 0.0882 0.8240
    Two-step ahead RMSE 0.0142 0.0379 0.0236 0.9338 1.6310 1.0437 7.9131
    prediction IA 0.0007 0.0008 0.0007 0.0023 0.0121 0.0107 0.0192
    mbox Ds 0.4570 0.5484 0.0914 0.8585 0.6717 0.7550 1.8164
    mbox MAPE 0.0011 0.0035 0.0030 0.0482 0.2231 0.0690 0.1893
    Four-step ahead RMSE 0.0147 0.0122 0.0092 0.5842 2.0020 0.9416 2.4118
    prediction IA 0.0004 0.0004 0.0001 0.0017 0.0357 0.0221 0.0151
    mbox Ds 0.2286 0.1975 0.0457 0.1293 0.6463 1.1003 1.7166
  • Table 12   Forecasting errors of natural gas futures and gold futures
    mbox Statistical indicator Nature gas futures prediction Gold futures prediction
    mbox MAPE (%) 0.1116 0.037
    mbox RMSE 0.0043 0.5987
    One-step ahead NRMSE (%) 0.1424 0.0482
    prediction TIC 6.95e$-$04 2.41e$-$04
    mbox IA 1.0000 0.9981
    mbox Ds (%) 97.532 97.8976
    mbox MAPE (%) 0.2781 0.1027
    mbox RMSE 0.0109 1.6174
    Two-step ahead NRMSE (%) 0.3638 0.1303
    prediction TIC 0.0018 6.51e$-$04
    mbox IA 0.9999 0.9975
    mbox Ds (%) 94.6069 94.4241
    mbox MAPE (%) 0.7844 0.3637
    mbox RMSE 0.0303 5.8779
    Four-step ahead NRMSE (%) 1.0091 0.4737
    prediction TIC 0.0049 0.0024
    mbox IA 1.0000 0.9816
    mbox Ds (%) 86.4717 81.8099
  • Table 13   Forecasting errors of SSA-K-means-ELM
    mbox Statistical indicator WTI crude oil futures prediction CBOT soybean futures prediction
    mbox MAPE (%) 1.1215 0.6760
    mbox RMSE 0.7707 10.5351
    One-step ahead NRMSE (%) 1.3259 1.0516
    prediction TIC 0.0063 0.0052
    mbox IA 0.9974 0.9846
    mbox Ds (%) 75.5027 75.3199
    mbox MAPE (%) 1.3259 0.6601
    mbox RMSE 0.7616 10.3798
    Two-step ahead NRMSE (%) 1.3104 1.0361
    prediction TIC 0.0062 0.0051
    mbox IA 0.9979 0.9830
    mbox Ds (%) 76.3254 75.5941
    mbox MAPE (%) 1.2198 0.7177
    mbox RMSE 0.8328 11.1091
    Four-step ahead NRMSE (%) 1.4329 1.1089
    prediction TIC 0.0068 0.0055
    mbox IA 0.9947 0.9661
    mbox Ds (%) 72.8519 73.5832
  • Table 14   Forecasting errors of EMD-RBF
    mbox Statistical indicator WTI crude oil futures prediction CBOT soybean futures prediction
    mbox MAPE (%) 1.3284 0.5209
    mbox RMSE 1.5300 10.1198
    One-step ahead NRMSE (%) 2.6246 1.0102
    prediction TIC 0.0125 0.0050
    mbox IA 0.9992 0.9994
    mbox Ds (%) 78.6106 83.9122
    mbox MAPE (%) 1.6458 0.6556
    mbox RMSE 2.1256 9.9750
    Two-step ahead NRMSE (%) 3.6478 0.9957
    prediction TIC 0.0174 0.0049
    mbox IA 0.9908 0.9927
    mbox Ds (%) 77.0567 78.9762
    mbox MAPE (%) 3.1846 0.7993
    mbox RMSE 2.6471 12.3883
    Four-step ahead NRMSE (%) 4.5502 1.2366
    prediction TIC 0.0216 0.0061
    mbox IA 0.9940 0.9944
    mbox Ds (%) 70.2393 73.2176

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