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A new technique for time series forecasting by using symbiotic organisms search

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Abstract

Symbiotic organisms search (SOS) is a new metaheuristic optimization algorithm proposed by Cheng and Prayogo (Comput Struct 139:98–112, 2014). In this paper, SOS has been applied to determine the functional forms of different time series which are used to predict the time series. There are some previous attempts by researchers where genetic algorithm has been used to find the functional form of a time series. Here, we explore this new algorithm in time series analysis. SOS mimics the symbiotic relationships among organisms in the ecosystem. Improvement in SOS in parasitism phase has been proposed here. Also, several types of time series have been tested to compare the performance of the original SOS with its improved version and with already well-established artificial neural network (ANN) in the field of time series forecasting.

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Acknowledgements

The first author would like to thank DST INSPIRE, India, for their help and supports to sustain the work.

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Authors and Affiliations

  1. Department of Mathematics, NIT Durgapur, Durgapur, West Bengal, 713209, India

    Shanoli Samui Pal & Samarjit Kar

  2. Department of Physics, Presidency University, Kolkata, West Bengal, 700073, India

    Saumyadip Samui

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  1. Shanoli Samui Pal
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Correspondence to Samarjit Kar.

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Appendix

Appendix

Five data sets on sunspots, domestic airline passengers, rainfall measurement, number of lynx and BSE sensex closing price have been given in following Tables 6, 7, 8, 9 and  10, respectively.

Table 6 Sunspots data collected yearly
Full size table
Table 7 Number of air passengers collected monthly
Full size table
Table 8 Rainfall data
Full size table
Table 9 Lynx data
Full size table
Table 10 Closing price of BSE sensex data
Full size table

Functional forms generated from the algorithmic steps for each time series are given in Eqs. (9)–(14).

  1. 1.

    Functional form for the sunspots time series with window size six is given below:

    $$\begin{aligned} x[t]&=(4.09\times 10^{-02})/x[t-5]/x[t-3]\nonumber \\&+\,(-1.62)+(-1.34)/x[t-1]*x[t-4]\nonumber \\&+\,(5.126\times 10^{-01})/x[t-1]*x[t-3]+(1.60)/\nonumber \\&x[t-3]+(3.79\times 10^{-01})*x[t-1]/x[t-1]\nonumber \\&+\,(9.92\times 10^{-02})/x[t-3]*x[t-3]\nonumber \\&+\,(1.15)*x[t-1]+(-9.21\times 10^{-01})/\nonumber \\&x[t-2]+(2.11\times 10^{-01})*x[t-3]\nonumber \\&+\,(-9.48\times 10^{-04})*x[t-3]*x[t-4]\nonumber \\&+\,(-4.58\times 10^{-02})*x[t-2]+(2.30\times 10^{-01})/\nonumber \\&x[t-4]+(-2.79\times 10^{-04})*x[t-3]/x[t-2]\nonumber \\&+\,(1.82\times 10^{-03})*x[t-4]*x[t-2]\nonumber \\&+\,(-2.65\times 10^{-04})*x[t-2]/x[t-5]+\nonumber \\&(-9.86\times 10^{-02})*x[t-5]+(2.19\times 10^{-01})*x[t-2]/x[t-3]\nonumber \\&+\,(-1.46\times 10^{-01})/x[t-4]/x[t-2]+(-9.86\nonumber \\&\times 10^{-01})/x[t-5]/x[t-2]+(1.24\times 10^{-01})*x[t-5]/x[t-3]\nonumber \\&+\,(1.04\times 10^{-01})*x[t-4]+(6.04\times 10^{-03})/\nonumber \\&x[t-5]+(-4.56\times 10^{-04})/x[t-4]/x[t-3]\nonumber \\&+\,(-5.73\times 10^{-04})*x[t-4]*x[t-5]\nonumber \\&+\,(-6.21\times 10^{-04})/x[t-4]*x[t-3]\nonumber \\&+\,(6.14\times 10^{-04})*x[t-4]/x[t-2]\nonumber \\&+\,(7.38\times 10^{-04})*x[t-1]*x[t-1]\nonumber \\&+\,(-3.64\times 10^{-03})/x[t-1]+(1.06\times 10^{-04})\nonumber \\&/x[t-1]*x[t-2]+(1.37\times 10^{-04})*x[t-2]*x[t-3]\nonumber \\&+\,(7.02\times 10^{-03})/x[t-1]/x[t-4]+(-1.99\times 10^{-04})/x[t-5]\nonumber \\&/x[t-5]+(1.73\times 10^{-01})/x[t-2]*x[t-2]\nonumber \\&+\,(6.19\times 10^{-01})/x[t-1]*x[t-5]\nonumber \\&+\,(5.05\times 10^{-01})*x[t-4]/x[t-1]+\nonumber \\&(1.97\times 10^{-03})*x[t-4]/x[t-5]\nonumber \\&+\,(5.37\times 10^{-03})/x[t-3]*x[t-4]\nonumber \\&+\,(-1.01\times 10^{-03})/x[t-2]/x[t-5]+\nonumber \\&(-3.06\times 10^{-04})/x[t-3]/x[t-2]\nonumber \\&+\,(-3.47\times 10^{-03})/x[t-5]*x[t-1]\nonumber \\&+\,(-4.89\times 10^{-03})*x[t-2]*x[t-2]+\nonumber \\&(2.80\times 10^{-03})/x[t-5]*x[t-5]\nonumber \\&+\,(-1.43\times 10^{-03})*x[t-5]*x[t-1]\nonumber \\&+\,(1.39\times 10^{-03})*x[t-5]/x[t-1]+\nonumber \\&(-2.90\times 10^{-03})*x[t-1]*x[t-4]\nonumber \\&+\,(1.37\times 10^{-02})/x[t-1]*x[t-1]\nonumber \\&+\,(8.12\times 10^{-04})/x[t-4]/x[t-4]+\nonumber \\&(3.00\times 10^{-04})*x[t-5]*x[t-4]\nonumber \\&+\,(1.38\times 10^{-03})*x[t-3]*x[t-2]\nonumber \\&+\,(7.09\times 10^{-04})*x[t-5]/x[t-5]+\nonumber \\&(1.24\times 10^{-03})*x[t-1]/x[t-3] \end{aligned}$$
    (9)
  2. 2.

    Functional form for the Mackey–Glass time series with window size six is given below:

    $$\begin{aligned} x[t]&=(2.02)*x[t-5]+(-9.70\times 10^{-01})/x[t-1]/x[t-4]\nonumber \\&+\,(3.58\times 10^{-01})*x[t-1]+(5.85\times 10^{-01})*x[t-4]+\nonumber \\&(-1.39)/x[t-5]*x[t-3]+(-1.06)*x[t-1]/x[t-4]\nonumber \\&+\,(-2.05\times 10^{-02})*x[t-4]/x[t-1]+(1.51)/x[t-1] \nonumber \\&+\,(-1.22\times 10^{-01})/x[t-2]+(1.45)/x[t-4]+(2.05\times 10^{-02})*x[t-4]\nonumber \\&+\,(1.49\times 10^{-01})*x[t-1]*x[t-3]+(-8.18\times 10^{-01})*\nonumber \\&x[t-3]+(-2.74\times 10^{-01})*x[t-5]/x[t-3]\nonumber \\&+\,(-4.08\times 10^{-01})/x[t-1]/x[t-1]\nonumber \\&+\,(4.61\times 10^{-01})/x[t-4]*x[t-3]+\nonumber \\&(1.67\times 10^{-02})*x[t-5]/x[t-3]+(1.02e+00)/x[t-3]\nonumber \\&+\,(1.07e+00)+(-1.85e-02)/x[t-3]+(-8.20e-01)/\nonumber \\&x[t-3]/x[t-2]+(-1.83\times 10^{-02})\nonumber \\&+\,(6.61\times 10^{-02})*x[t-3]/x[t-1]\nonumber \\&+\,(2.78\times 10^{-01})*x[t-2]*x[t-1]+(-7.98\times \nonumber \\&10^{-01})*x[t-5]*x[t-1]+(1.88)*x[t-3]*x[t-4]\nonumber \\&+\,(-1.02)/x[t-5]+(-1.90\times 10^{-01})*x[t-3]/x[t-4]+(3.91)/\nonumber \\&x[t-4]/x[t-2]+(-7.01\times 10^{-01})/x[t-3]/x[t-3]\nonumber \\&+\,(-2.43)/x[t-2]/x[t-1]\nonumber \\&+\,(-2.79)*x[t-4]/x[t-4]+(-1.73)/\nonumber \\&x[t-1]*x[t-1]+(2.18)*x[t-2]+(-4.70^{e-02})/x[t-4]/x[t-4]\nonumber \\&+\,(-5.30\times 10^{-01})*x[t-1]/x[t-3]+(-8.71\times \nonumber \\&10^{-01})*x[t-4]*x[t-2]+(-2.88\times 10^{-01})/x[t-4]/x[t-5]\nonumber \\&+\,(1.29\times 10^{-01})*x[t-1]/x[t-1]+(-2.31\times 10^{-01})/\nonumber \\&x[t-2]*x[t-4]+(-6.49\times 10^{-01})*x[t-1]*x[t-4]\nonumber \\&+\,(5.29\times 10^{-01})*x[t-1]/x[t-2]+(3.91\times 10^{-01})*x[t-5]/\nonumber \\&x[t-1]+(-8.03\times 10^{-01})/x[t-4]*x[t-2]\nonumber \\&+\,(7.90\times 10^{-01})/x[t-3]/x[t-1]+(6.31\times 10^{-01})/x[t-1]*x[t-5]+\nonumber \\&(-8.15\times 10^{-01})*x[t-4]*x[t-5]\nonumber \\&+\,(3.80\times 10^{-01})/x[t-2]/x[t-5]+(9.79\times 10^{-01})/x[t-2]*x[t-1]\nonumber \\ \end{aligned}$$
    (10)
  3. 3.

    Functional form for the air passengers time series with window size six is given below:

    $$\begin{aligned} x[t]&=(-8.16\times 10^{-01})/x[t-4]*x[t-2]\nonumber \\&+\,(-1.30)*x[t-1]/x[t-5]\nonumber \\&+\,(-2.33)/x[t-1]+(1.19\times 10^{-01})*x[t-3]+(9.25\times \nonumber \\&10^{-01})+(-1.66\times 10^{-01})/x[t-2]/x[t-1]+(1.00)*x[t-5]\nonumber \\&+\,(-1.30\times 10^{-02})/x[t-1]+(1.75\times 10^{-03})+\nonumber \\&(1.61\times 10^{-02})+(1.07)/x[t-2]\nonumber \\&+\,(9.11\times 10^{-01})/x[t-2]*x[t-1]\nonumber \\&+\,(4.79\times 10^{-03})*x[t-3]+(-1.03)/x[t-1]/\nonumber \\&x[t-3]+(1.39\times 10^{-03})*x[t-5]+(-3.28\times 10^{-03})\nonumber \\&+\,(-1.32)*x[t-2]/x[t-4]+(-5.35\times 10^{-01})/x[t-4]+\nonumber \\&(-1.07)/x[t-5]*x[t-2]+(1.72)*x[t-2]/x[t-2]\nonumber \\&+\,(-3.13\times 10^{-01})/x[t-4]*x[t-1]\nonumber \\&+\,(-1.66)*x[t-4]+(1.47)/\nonumber \\&x[t-1]*x[t-4]+(2.54\times 10^{-01})/x[t-5]*x[t-5]\nonumber \\&+\,(5.95\times 10^{-01})/x[t-3]/x[t-4]+(-2.32\times 10^{-03})*\nonumber \\&x[t-4]+(1.50)*x[t-2]+(8.36\times 10^{-03})*x[t-2]\nonumber \\&+\,(6.88\times 10^{-01})*x[t-4]/x[t-1]+(-8.11\times 10^{-01})/\nonumber \\&x[t-5]+(-1.74)/x[t-4]/x[t-4]+(9.06\times 10^{-01})/x[t-1]/x[t-1]\nonumber \\&+\,(-2.28\times 10^{-01})*x[t-1]/x[t-1]+\nonumber \\&(3.23\times 10^{-02})*x[t-1] \end{aligned}$$
    (11)
  4. 4.

    Functional form for the rainfall time series window size twelve is given below:

    $$\begin{aligned} x[t]&=(-1.43\times 10^{-01})*x[t-8]\nonumber \\&+\,(5.80\times 10^{-03})/x[t-11]\nonumber \\&+\,(-5.29\times 10^{-01})/x[t-4]\nonumber \\&+\,(3.13 \times 10^{-02})*x[t-4]+\nonumber \\&(-2.69\times 10^{-01})/x[t-2]\nonumber \\&+\,(7.30\times 10^{-03})/x[t-9]/x[t-11]\nonumber \\&+\,(4.70\times 10^{-04})*x[t-5]\nonumber \\&+\,(3.57\times 10^{-03})*x[t-1]/x[t-2]+\nonumber \\&(-4.89\times 10^{-03})/x[t-8]\nonumber \\&+\,(-2.26\times 10^{-03})/x[t-4]*x[t-9]\nonumber \\&+\,(1.30\times 10^{-03})*x[t-2]*x[t-9]\nonumber \\&+\,(-1.52\times 10^{-01})/\nonumber \\&x[t-3]+(2.06\times 10^{-04})*x[t-3]\nonumber \\&+\,(-2.94\times 10^{-02})*x[t-7]/x[t-8]\nonumber \\&+\,(2.63\times 10^{-02})+(-1.74\times 10^{-04})*x[t-9]*\nonumber \\&x[t-3]+(-1.22\times 10^{-01})*x[t-11]\nonumber \\&+\,(-1.78\times 10^{-01})/x[t-2]/x[t-8]\nonumber \\&+\,(8.25\times 10^{-02})*x[t-2]+(-4.82\times \nonumber \\&10^{-01})/x[t-11]*x[t-5]\nonumber \\&+\,(-2.60\times 10^{-01})*x[t-1]/x[t-9]\nonumber \\&+\,(3.08\times 10^{-02})*x[t-9]/x[t-1]+(-1.39\times 10^{-01})/\nonumber \\&x[t-7]+(6.88\times 10^{-02})/x[t-10]*x[t-5]\nonumber \\&+\,(-7.09\times 10^{-02})*x[t-6]\nonumber \\&+\,(-8.74\times 10^{-02})/x[t-9]+(7.63\times 10^{-02})*\nonumber \\&x[t-5]/x[t-7]+(1.53\times 10^{-01})*x[t-7]\nonumber \\&+\,(-2.11\times 10^{-02})/x[t-5]\nonumber \\&+\,(5.49\times 10^{-02})/x[t-9]/x[t-3]+\nonumber \\&(-1.07\times 10^{-02})/x[t-6]/x[t-6]\nonumber \\&+\,(-3.48\times 10^{-02})/x[t-5]/x[t-1]\nonumber \\&+\,(-2.18\times 10^{-02})/x[t-2]*x[t-1]+(3.40\times \nonumber \\&10^{-02})*x[t-9]+(-8.42\times 10^{-04})*x[t-2]*x[t-8]\nonumber \\&+\,(-1.10\times 10^{-02})*x[t-1]/x[t-5]+(-1.21\times 10^{-04})*\nonumber \\&x[t-11]/x[t-2]+(1.58\times 10^{-03})/x[t-9]*x[t-4]\nonumber \\&+\,(-9.85\times 10^{-04})/x[t-1]+(-1.04\times 10^{-03})*x[t-1]*x[t-6]+\nonumber \\&(-9.49\times 10^{-04})*x[t-10]/x[t-7]\nonumber \\&+\,(5.05\times 10^{-04})/x[t-1]/x[t-1]\nonumber \\&+\,(-1.25\times 10^{-03})/x[t-7]*x[t-5]+(7.06\times \nonumber \\&10^{-04})*x[t-9]/x[t-6]+(2.00\times 10^{-04})/x[t-6]/x[t-4]\nonumber \\&+\,(7.05\times 10^{-04})*x[t-6]/x[t-2]\nonumber \\&+\,(4.97\times 10^{-04})*x[t-5]*\nonumber \\&x[t-2]+(4.46\times 10^{-04})*x[t-11]*x[t-1]\nonumber \\&+\,(1.41\times 10^{-04})*x[t-9]*x[t-11]\nonumber \\&+\,(1.07\times 10^{-04})/x[t-11]*x[t-7] \end{aligned}$$
    (12)
  5. 5.

    Functional form for the Lynx time series with window size six is given below:

    $$\begin{aligned} \begin{aligned} x[t]&=(5.39\times 10^{-01})/x[t-4]/x[t-2]+(1.42)/x[t-3]/x[t-3]\\&+\,(-4.93\times 10^{-01})*x[t-3]+(-2.54)/x[t-2]/x[t-1]+\\&(-4.34)*x[t-4]/x[t-4]+(1.51)/x[t-1]*x[t-1]\\&+\,(-3.07)/x[t-2]+(-2.11\times 10^{-01})*x[t-2]\\&+\,(5.46\times 10^{-01})/\\&x[t-1]+(1.97\times 10^{-01})*x[t-5]\\&+\,(2.02)+(3.76\times 10^{-01})/x[t-5]*x[t-3]\\&+\,(3.19\times 10^{-01})*x[t-4]+(3.51)/\\&x[t-4]+(1.33)*x[t-3]/x[t-4]\\&+\,(1.83)/x[t-3]+(-3.24)/x[t-1]/x[t-4]\\&+\,(1.15)*x[t-1]+(-3.60)*x[t-1]/ \\&x[t-4]+(-9.90\times 10^{-01})*x[t-4]/x[t-1]\\&+\,(3.08\times 10^{-07})*x[t-4]/x[t-1]\\&+\,(-1.56\times 10^{-04})/x[t-4]/x[t-4]+ \\&(-1.46\times 10^{-04})/x[t-5]\\&+\,(-1.29\times 10^{-04})/x[t-4]/x[t-5]\\&+\,(-1.52\times 10^{-04})*x[t-1]*x[t-4]\\&+\,(-1.35\times 10^{-03})*x[t-1]/x[t-2]\\&+\,(1.94\times 10^{-04})/x[t-4]*x[t-2]\\&+\,(-8.19\times 10^{-04})/x[t-3]/x[t-1] \end{aligned} \end{aligned}$$
    (13)
  6. 6.

    Functional form for the BSE sensex closing price time series with window size six is given below:

    $$\begin{aligned} \begin{aligned} x[t]&=(4.60\times 10^{-01})/x[t-2]+(7.01\times 10^{-02})*x[t-1]\\&+\,(-3.45\times 10^{-01})/x[t-1]*x[t-1]+(4.95\times \\&10^{-01})/x[t-3]+(1.45\times 10^{-01})/x[t-1]\\&+\,(6.39\times 10^{-01})/x[t-5]*x[t-3]+(2.42\times 10^{-01})+\\&(2.23\times 10^{-01})*x[t-4]+(-3.19\times 10^{-01})*x[t-3]\\&+\,(-9.19\times 10^{-02})/x[t-4]+(-1.37\times 10^{-01})*\\&x[t-3]/x[t-4]+(6.85\times 10^{-01})*x[t-5]\\&+\,(1.92\times 10^{-01})/x[t-1]/x[t-4]\\&+\,(3.57\times 10^{-01})*x[t-1]/x[t-4]+\\&(2.61\times 10^{-02})*x[t-4]/x[t-1]\\&+\,(-3.23\times 10^{-01})*x[t-4]/x[t-4]\\&+\,(3.71\times 10^{-01})*x[t-2]+\\&(5.59\times 10^{-01})/x[t-4]/x[t-2]\\&+\,(-3.18\times 10^{-01})/x[t-3]/x[t-3]\\&+\,(-4.12\times 10^{-01})/x[t-2]/x[t-1]+\\&(1.30\times 10^{-04})*x[t-1]/x[t-2] \end{aligned} \end{aligned}$$
    (14)

We have neglected the terms with coefficient value less than \(10^{-04}\) for time series sunspots, air passengers, rainfall, lynx data, BSE sensex and coefficient value less than \(10^{-02}\) for Mackey–Glass in all the above expressions in our writing but still our computations remain unaffected.

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Pal, S.S., Samui, S. & Kar, S. A new technique for time series forecasting by using symbiotic organisms search. Neural Comput & Applic 32, 2365–2381 (2020). https://doi.org/10.1007/s00521-019-04134-8

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