# Forecasting Based on High-Order Fuzzy-Fluctuation Trends and Particle Swarm Optimization Machine Learning

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Step 1:
- Define the universe U and the number and length of the intervals;
- Step 2:
- Fuzzify the historical training time series into fuzzy time series;
- Step 3:
- Establish fuzzy logical relationships (FLR) according to the historical fuzzy time series and generate forecasting rules based on fuzzy logical groups (FLG);
- Step 4:
- Calculate the forecast values according to the FLG rules and the right-hand side (RHS) of the forecasted point.

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. PSO-Based Machine Learning Method

_{1}and c

_{2}are the self-confidence coefficient and social confidence coefficient, respectively, $Rand()\in [0,1]$ is a random number, and ${p}_{i,m}$ and ${p}_{g,m}$ are the personal best position found by particle i and the global best position found by all particles in the swarm up to time step m, respectively.

## 4. A Novel Forecasting Model Based on High-Order Fuzzy-Fluctuation Trends

## 5. EmpiricalAnalysis

#### 5.1. Forecasting TAIEX

_{1}= c

_{2}= 1.4962, and use the PSO algorithm listed in Figure 1 to determine the parameters $\varphi {}_{k}(k=1,2,...,n)$ and $\epsilon $. In the PSO process, each element in the generalizedEquation (8) is a particle and their personal best and global best positions are determined by the RMSE of the actual values and forecast values. The obtained global best parameters are shown in Table 1.

#### 5.2. Forecasting DAX30

#### 5.3. Forecasting SHSECI

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

PSO-Based Machine Learning Algorithm for the Training Process | |
---|---|

INPUT: | X: training time series, containing T cases, denoted as $X[1],X[2],...,X[i]...,X[T]$. |

S: a fuzzy-fluctuation time series of training data, containing T−1 cases, denoted as $S[2],S[3],...,S[i]...,S[T]$. | |

n: the number of nth-order. | |

itern: the number of iterations. | |

x_{min}, x_{max}: lower and upper bounds of space. | |

w, c_{1}, c_{1}: parameters described in Equations (3) and (4). | |

OUPUT: | Φ[k] and ε: parameters for the forecasting model, k = 1,2,…,n. |

1. | Initialize the position and velocity for each particle i: |

pn = T−1−n; | |

/* the number of particles. */ | |

For i = 1 to pn | |

For j = 1 to n | |

x[i,j] = rand(x_{min}, x_{max}); | |

v[i,j] = rand(x_{min}, x_{max}); | |

2. | Calculate the fitness value for each particle i according to Equation (6): |

Set x[pbest] to current x[i] for each particle. | |

Locate the global best fitness value x[gbest] and set Φ[k] and ε to the corresponding x[gbest]. | |

3. | for m=1 to itern loop |

For each particle i | |

Calculate particle velocity according to Equation (3). | |

Update particle position according to Equations (4) and (5) | |

If the fitness value is better than the best fitness value x[pbest] of particle i in history: Set current value as the new x[pbest] for particle i | |

Locate the current global best fitness value, if it is better than the x[gbest] in history: Set current global best fitness value as the new x[gbest], and set Φ[k] and ε to x[gbest]. | |

4. | Output Φ[k] and ε |

## Appendix B

Date (MM/DD/YYYY) | TAIEX | Fluctuation | Fuzzified | Date (MM/DD/YYYY) | TAIEX | Fluctuation | Fuzzified | Date (MM/DD/YYYY) | TAIEX | Fluctuation | Fuzzified |
---|---|---|---|---|---|---|---|---|---|---|---|

1/5/1999 | 6152.43 | - | - | 4/17/1999 | 7581.5 | 114.68 | 3 | 7/26/1999 | 7595.71 | −128.81 | 1 |

1/6/1999 | 6199.91 | 47.48 | 3 | 4/19/1999 | 7623.18 | 41.68 | 2 | 7/27/1999 | 7367.97 | −227.74 | 1 |

1/7/1999 | 6404.31 | 204.4 | 3 | 4/20/1999 | 7627.74 | 4.56 | 2 | 7/28/1999 | 7484.5 | 116.53 | 3 |

1/8/1999 | 6421.75 | 17.44 | 2 | 4/21/1999 | 7474.16 | −153.58 | 1 | 7/29/1999 | 7359.37 | −125.13 | 1 |

1/11/1999 | 6406.99 | −14.76 | 2 | 4/22/1999 | 7494.6 | 20.44 | 2 | 7/30/1999 | 7413.11 | 53.74 | 3 |

1/12/1999 | 6363.89 | −43.1 | 1 | 4/23/1999 | 7612.8 | 118.2 | 3 | 7/31/1999 | 7326.75 | −86.36 | 1 |

1/13/1999 | 6319.34 | −44.55 | 1 | 4/26/1999 | 7629.09 | 16.29 | 2 | 8/2/1999 | 7195.94 | −130.81 | 1 |

1/14/1999 | 6241.32 | −78.02 | 1 | 4/27/1999 | 7550.13 | −78.96 | 1 | 8/3/1999 | 7175.19 | −20.75 | 2 |

1/15/1999 | 6454.6 | 213.28 | 3 | 4/28/1999 | 7496.61 | −53.52 | 1 | 8/4/1999 | 7110.8 | −64.39 | 1 |

1/16/1999 | 6483.3 | 28.7 | 2 | 4/29/1999 | 7289.62 | −206.99 | 1 | 8/5/1999 | 6959.73 | −151.07 | 1 |

1/18/1999 | 6377.25 | −106.05 | 1 | 4/30/1999 | 7371.17 | 81.55 | 3 | 8/6/1999 | 6823.52 | −136.21 | 1 |

1/19/1999 | 6343.36 | −33.89 | 2 | 5/3/1999 | 7383.26 | 12.09 | 2 | 8/7/1999 | 7049.74 | 226.22 | 3 |

1/20/1999 | 6310.71 | −32.65 | 2 | 5/4/1999 | 7588.04 | 204.78 | 3 | 8/9/1999 | 7028.01 | −21.73 | 2 |

1/21/1999 | 6332.2 | 21.49 | 2 | 5/5/1999 | 7572.16 | −15.88 | 2 | 8/10/1999 | 7269.6 | 241.59 | 3 |

1/22/1999 | 6228.95 | −103.25 | 1 | 5/6/1999 | 7560.05 | −12.11 | 2 | 8/11/1999 | 7228.68 | −40.92 | 2 |

1/25/1999 | 6033.21 | −195.74 | 1 | 5/7/1999 | 7469.33 | −90.72 | 1 | 8/12/1999 | 7330.24 | 101.56 | 3 |

1/26/1999 | 6115.64 | 82.43 | 3 | 5/10/1999 | 7484.37 | 15.04 | 2 | 8/13/1999 | 7626.05 | 295.81 | 3 |

1/27/1999 | 6138.87 | 23.23 | 2 | 5/11/1999 | 7474.45 | −9.92 | 2 | 8/16/1999 | 8018.47 | 392.42 | 3 |

1/28/1999 | 6063.41 | −75.46 | 1 | 5/12/1999 | 7448.41 | −26.04 | 2 | 8/17/1999 | 8083.43 | 64.96 | 3 |

1/29/1999 | 5984 | −79.41 | 1 | 5/13/1999 | 7416.2 | −32.21 | 2 | 8/18/1999 | 7993.71 | −89.72 | 1 |

1/30/1999 | 5998.32 | 14.32 | 2 | 5/14/1999 | 7592.53 | 176.33 | 3 | 8/19/1999 | 7964.67 | −29.04 | 2 |

2/1/1999 | 5862.79 | −135.53 | 1 | 5/15/1999 | 7576.64 | −15.89 | 2 | 8/20/1999 | 8117.42 | 152.75 | 3 |

2/2/1999 | 5749.64 | −113.15 | 1 | 5/17/1999 | 7599.76 | 23.12 | 2 | 8/21/1999 | 8153.57 | 36.15 | 2 |

2/3/1999 | 5743.86 | −5.78 | 2 | 5/18/1999 | 7585.51 | −14.25 | 2 | 8/23/1999 | 8119.98 | −33.59 | 2 |

2/4/1999 | 5514.89 | −228.97 | 1 | 5/19/1999 | 7614.6 | 29.09 | 2 | 8/24/1999 | 7984.39 | −135.59 | 1 |

2/5/1999 | 5474.79 | −40.1 | 2 | 5/20/1999 | 7608.88 | −5.72 | 2 | 8/25/1999 | 8127.09 | 142.7 | 3 |

2/6/1999 | 5710.18 | 235.39 | 3 | 5/21/1999 | 7606.69 | −2.19 | 2 | 8/26/1999 | 8097.57 | −29.52 | 2 |

2/8/1999 | 5822.98 | 112.8 | 3 | 5/24/1999 | 7588.23 | −18.46 | 2 | 8/27/1999 | 8053.97 | −43.6 | 1 |

2/9/1999 | 5723.73 | −99.25 | 1 | 5/25/1999 | 7417.03 | −171.2 | 1 | 8/30/1999 | 8071.36 | 17.39 | 2 |

2/10/1999 | 5798 | 74.27 | 3 | 5/26/1999 | 7426.63 | 9.6 | 2 | 8/31/1999 | 8157.73 | 86.37 | 3 |

2/20/1999 | 6072.33 | 274.33 | 3 | 5/27/1999 | 7469.01 | 42.38 | 2 | 9/1/1999 | 8273.33 | 115.6 | 3 |

2/22/1999 | 6313.63 | 241.3 | 3 | 5/28/1999 | 7387.37 | −81.64 | 1 | 9/2/1999 | 8226.15 | −47.18 | 1 |

2/23/1999 | 6180.94 | −132.69 | 1 | 5/29/1999 | 7419.7 | 32.33 | 2 | 9/3/1999 | 8073.97 | −152.18 | 1 |

2/24/1999 | 6238.87 | 57.93 | 3 | 5/31/1999 | 7316.57 | −103.13 | 1 | 9/4/1999 | 8065.11 | −8.86 | 2 |

2/25/1999 | 6275.53 | 36.66 | 2 | 6/1/1999 | 7397.62 | 81.05 | 3 | 9/6/1999 | 8130.28 | 65.17 | 3 |

2/26/1999 | 6318.52 | 42.99 | 3 | 6/2/1999 | 7488.03 | 90.41 | 3 | 9/7/1999 | 7945.76 | −184.52 | 1 |

3/1/1999 | 6312.25 | −6.27 | 2 | 6/3/1999 | 7572.91 | 84.88 | 3 | 9/8/1999 | 7973.3 | 27.54 | 2 |

3/2/1999 | 6263.54 | −48.71 | 1 | 6/4/1999 | 7590.44 | 17.53 | 2 | 9/9/1999 | 8025.02 | 51.72 | 3 |

3/3/1999 | 6403.14 | 139.6 | 3 | 6/5/1999 | 7639.3 | 48.86 | 3 | 9/10/1999 | 8161.46 | 136.44 | 3 |

3/4/1999 | 6393.74 | −9.4 | 2 | 6/7/1999 | 7802.69 | 163.39 | 3 | 9/13/1999 | 8178.69 | 17.23 | 2 |

3/5/1999 | 6383.09 | −10.65 | 2 | 6/8/1999 | 7892.13 | 89.44 | 3 | 9/14/1999 | 8092.02 | −86.67 | 1 |

3/6/1999 | 6421.73 | 38.64 | 2 | 6/9/1999 | 7957.71 | 65.58 | 3 | 9/15/1999 | 7971.04 | −120.98 | 1 |

3/8/1999 | 6431.96 | 10.23 | 2 | 6/10/1999 | 7996.76 | 39.05 | 2 | 9/16/1999 | 7968.9 | −2.14 | 2 |

3/9/1999 | 6493.43 | 61.47 | 3 | 6/11/1999 | 7979.4 | −17.36 | 2 | 9/17/1999 | 7916.92 | −51.98 | 1 |

3/10/1999 | 6486.61 | −6.82 | 2 | 6/14/1999 | 7973.58 | −5.82 | 2 | 9/18/1999 | 8016.93 | 100.01 | 3 |

3/11/1999 | 6436.8 | −49.81 | 1 | 6/15/1999 | 7960 | −13.58 | 2 | 9/20/1999 | 7972.14 | −44.79 | 1 |

3/12/1999 | 6462.73 | 25.93 | 2 | 6/16/1999 | 8059.02 | 99.02 | 3 | 9/27/1999 | 7759.93 | −212.21 | 1 |

3/15/1999 | 6598.32 | 135.59 | 3 | 6/17/1999 | 8274.36 | 215.34 | 3 | 9/28/1999 | 7577.85 | −182.08 | 1 |

3/16/1999 | 6672.23 | 73.91 | 3 | 6/21/1999 | 8413.48 | 139.12 | 3 | 9/29/1999 | 7615.45 | 37.6 | 2 |

3/17/1999 | 6757.07 | 84.84 | 3 | 6/22/1999 | 8608.91 | 195.43 | 3 | 9/30/1999 | 7598.79 | −16.66 | 2 |

3/18/1999 | 6895.01 | 137.94 | 3 | 6/23/1999 | 8492.32 | −116.59 | 1 | 10/1/1999 | 7694.99 | 96.2 | 3 |

3/19/1999 | 6997.29 | 102.28 | 3 | 6/24/1999 | 8589.31 | 96.99 | 3 | 10/2/1999 | 7659.55 | −35.44 | 2 |

3/20/1999 | 6993.38 | −3.91 | 2 | 6/25/1999 | 8265.96 | −323.35 | 1 | 10/4/1999 | 7685.48 | 25.93 | 2 |

3/22/1999 | 7043.23 | 49.85 | 3 | 6/28/1999 | 8281.45 | 15.49 | 2 | 10/5/1999 | 7557.01 | −128.47 | 1 |

3/23/1999 | 6945.48 | −97.75 | 1 | 6/29/1999 | 8514.27 | 232.82 | 3 | 10/6/1999 | 7501.63 | −55.38 | 1 |

3/24/1999 | 6889.42 | −56.06 | 1 | 6/30/1999 | 8467.37 | −46.9 | 1 | 10/7/1999 | 7612 | 110.37 | 3 |

3/25/1999 | 6941.38 | 51.96 | 3 | 7/2/1999 | 8572.09 | 104.72 | 3 | 10/8/1999 | 7552.98 | −59.02 | 1 |

3/26/1999 | 7033.25 | 91.87 | 3 | 7/3/1999 | 8563.55 | −8.54 | 2 | 10/11/1999 | 7607.11 | 54.13 | 3 |

3/29/1999 | 6901.68 | −131.57 | 1 | 7/5/1999 | 8593.35 | 29.8 | 2 | 10/12/1999 | 7835.37 | 228.26 | 3 |

3/30/1999 | 6898.66 | −3.02 | 2 | 7/6/1999 | 8454.49 | −138.86 | 1 | 10/13/1999 | 7836.94 | 1.57 | 2 |

3/31/1999 | 6881.72 | −16.94 | 2 | 7/7/1999 | 8470.07 | 15.58 | 2 | 10/14/1999 | 7879.91 | 42.97 | 3 |

4/1/1999 | 7018.68 | 136.96 | 3 | 7/8/1999 | 8592.43 | 122.36 | 3 | 10/15/1999 | 7819.09 | −60.82 | 1 |

4/2/1999 | 7232.51 | 213.83 | 3 | 7/9/1999 | 8550.27 | −42.16 | 2 | 10/16/1999 | 7829.39 | 10.3 | 2 |

4/3/1999 | 7182.2 | −50.31 | 1 | 7/12/1999 | 8463.9 | −86.37 | 1 | 10/18/1999 | 7745.26 | −84.13 | 1 |

4/6/1999 | 7163.99 | −18.21 | 2 | 7/13/1999 | 8204.5 | −259.4 | 1 | 10/19/1999 | 7692.96 | −52.3 | 1 |

4/7/1999 | 7135.89 | −28.1 | 2 | 7/14/1999 | 7888.66 | −315.84 | 1 | 10/20/1999 | 7666.64 | −26.32 | 2 |

4/8/1999 | 7273.41 | 137.52 | 3 | 7/15/1999 | 7918.04 | 29.38 | 2 | 10/21/1999 | 7654.9 | −11.74 | 2 |

4/9/1999 | 7265.7 | −7.71 | 2 | 7/16/1999 | 7411.58 | −506.46 | 1 | 10/22/1999 | 7559.63 | −95.27 | 1 |

4/12/1999 | 7242.4 | −23.3 | 2 | 7/17/1999 | 7366.23 | −45.35 | 1 | 10/25/1999 | 7680.87 | 121.24 | 3 |

4/13/1999 | 7337.85 | 95.45 | 3 | 7/19/1999 | 7386.89 | 20.66 | 2 | 10/26/1999 | 7700.29 | 19.42 | 2 |

4/14/1999 | 7398.65 | 60.8 | 3 | 7/20/1999 | 7806.85 | 419.96 | 3 | 10/27/1999 | 7701.22 | 0.93 | 2 |

4/15/1999 | 7498.17 | 99.52 | 3 | 7/21/1999 | 7786.65 | −20.2 | 2 | 10/28/1999 | 7681.85 | −19.37 | 2 |

4/16/1999 | 7466.82 | −31.35 | 2 | 7/22/1999 | 7678.67 | −107.98 | 1 | 10/29/1999 | 7706.67 | 24.82 | 2 |

4/17/1999 | 7581.5 | 114.68 | 3 | 7/23/1999 | 7724.52 | 45.85 | 3 | 10/30/1999 | 7854.85 | 148.18 | 3 |

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ϕ_{1} | ϕ_{2} | ϕ_{3} | ϕ_{4} | ϕ_{5} | ϕ_{6} | E | RMSE |
---|---|---|---|---|---|---|---|

−0.1638 | 0.0803 | 0.1372 | −0.0321 | 0.0433 | 0.2546 | 1.4408 | 115.73 |

Date (MM/DD/YYYY) | Actual | Forecast | (Forecast–Actual)^{2} | Date (MM/DD/YYYY) | Actual | Forecast | (Forecast–Actual)^{2} |
---|---|---|---|---|---|---|---|

11/1/1999 | 7814.89 | 7869.35 | 2965.89 | 12/1/1999 | 7766.20 | 7705.59 | 3673.57 |

11/2/1999 | 7721.59 | 7825.35 | 10,766.14 | 12/2/1999 | 7806.26 | 7790.48 | 249.01 |

11/3/1999 | 7580.09 | 7704.00 | 15,353.69 | 12/3/1999 | 7933.17 | 7824.29 | 11,854.85 |

11/4/1999 | 7469.23 | 7573.21 | 10,811.84 | 12/4/1999 | 7964.49 | 7967.96 | 12.04 |

11/5/1999 | 7488.26 | 7460.24 | 785.12 | 12/6/1999 | 7894.46 | 7965.87 | 5099.39 |

11/6/1999 | 7376.56 | 7468.50 | 8452.96 | 12/7/1999 | 7827.05 | 7897.62 | 4980.12 |

11/8/1999 | 7401.49 | 7345.94 | 3085.80 | 12/8/1999 | 7811.02 | 7806.25 | 22.75 |

11/9/1999 | 7362.69 | 7400.03 | 1394.28 | 12/9/1999 | 7738.84 | 7823.68 | 7197.83 |

11/10/1999 | 7401.81 | 7379.30 | 506.70 | 12/10/1999 | 7733.77 | 7701.12 | 1066.02 |

11/11/1999 | 7532.22 | 7410.86 | 14,728.25 | 12/13/1999 | 7883.61 | 7718.38 | 27,300.95 |

11/15/1999 | 7545.03 | 7553.82 | 77.26 | 12/14/1999 | 7850.14 | 7921.86 | 5143.76 |

11/16/1999 | 7606.20 | 7569.42 | 1352.77 | 12/15/1999 | 7859.89 | 7862.87 | 8.88 |

11/17/1999 | 7645.78 | 7631.90 | 192.65 | 12/16/1999 | 7739.76 | 7857.12 | 13,773.37 |

11/18/1999 | 7718.06 | 7667.91 | 2515.02 | 12/17/1999 | 7723.22 | 7750.49 | 743.65 |

11/19/1999 | 7770.81 | 7750.58 | 409.25 | 12/18/1999 | 7797.87 | 7733.15 | 4188.68 |

11/20/1999 | 7900.34 | 7800.66 | 9936.10 | 12/20/1999 | 7782.94 | 7815.10 | 1034.27 |

11/22/1999 | 8052.31 | 7936.55 | 13,400.38 | 12/21/1999 | 7934.26 | 7781.74 | 23,262.35 |

11/23/1999 | 8046.19 | 8079.43 | 1104.90 | 12/22/1999 | 8002.76 | 7953.13 | 2463.14 |

11/24/1999 | 7921.85 | 8072.42 | 22,671.32 | 12/23/1999 | 8083.49 | 8060.46 | 530.38 |

11/25/1999 | 7904.53 | 7908.83 | 18.49 | 12/24/1999 | 8219.45 | 8119.70 | 9950.06 |

11/26/1999 | 7595.44 | 7912.20 | 100,336.90 | 12/27/1999 | 8415.07 | 8246.57 | 28,392.25 |

11/29/1999 | 7823.90 | 7576.21 | 61,350.34 | 12/28/1999 | 8448.84 | 8462.94 | 198.81 |

11/30/1999 | 7720.87 | 7823.06 | 10,442.80 | Root Mean Square Error(RMSE) | 99.31 |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

RMSE | 109.04 | 105.47 | 103.04 | 102.96 | 101.92 | 99.12 | 99.59 | 99.6 | 98.75 | 99 |

g | 3 | 5 | 7 | None |
---|---|---|---|---|

RMSE | 99.12 | 101.67 | 105.82 | 128.97 |

Year | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 |
---|---|---|---|---|---|---|---|---|---|

RMSE | 143.60 | 115.34 | 99.12 | 125.70 | 115.91 | 70.43 | 54.26 | 57.24 | 54.68 |

Methods | RMSE | S |
---|---|---|

Yu’s Method(2005) [36] | 145 | 1.62 ** |

Hsieh et al.’s Method(2011) [51] | 94 | −0.32 |

Chang et al.’s Method(2011) [48] | 100 | 0.11 |

Cheng et al.’s Method(2013) [50] | 103 | 0.34 |

Chen et al.’s Method(2013) [49] | 102.11 | 0.21 |

Chen and Chen’s Method(2015) [11] | 103.9 | 0.36 |

Chen and Chen’s Method(2015) [10] | 92 | −0.42 |

Zhao et al.’s Method(2016) [27] | 110.85 | 1.08 |

The Proposed Method | 99.12 | - |

Year | Yu (2005) [36] | Cheng et al. (2008) [37] | Wang et al. (2013) [25] | Rubio et al. (2017) [13] | Proposed Model |
---|---|---|---|---|---|

RMSE | 172.69 | 170.56 | 376.80 | 153.15 | 159.22 |

S | 1.31 | 1.23 | 3.68 ** | −0.26 | - |

Year | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 |
---|---|---|---|---|---|---|---|---|---|

RMSE | 113.11 | 55.28 | 49.59 | 45.73 | 28.45 | 25.05 | 19.86 | 41.44 | 59.5 |

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## Share and Cite

**MDPI and ACS Style**

Jia, J.; Zhao, A.; Guan, S.
Forecasting Based on High-Order Fuzzy-Fluctuation Trends and Particle Swarm Optimization Machine Learning. *Symmetry* **2017**, *9*, 124.
https://doi.org/10.3390/sym9070124

**AMA Style**

Jia J, Zhao A, Guan S.
Forecasting Based on High-Order Fuzzy-Fluctuation Trends and Particle Swarm Optimization Machine Learning. *Symmetry*. 2017; 9(7):124.
https://doi.org/10.3390/sym9070124

**Chicago/Turabian Style**

Jia, Jingyuan, Aiwu Zhao, and Shuang Guan.
2017. "Forecasting Based on High-Order Fuzzy-Fluctuation Trends and Particle Swarm Optimization Machine Learning" *Symmetry* 9, no. 7: 124.
https://doi.org/10.3390/sym9070124