# A Study Concerning Soft Computing Approaches for Stock Price Forecasting

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Hidden Markov Model

#### 2.2. Support Vector Machine

^{2}) were determined via trial and error. The prediction results were much better than those of its counterparts BPNN and the case-based reasoning method (CBR). In addition to stock price prediction, Huang et al. [35] utilized SVM to forecast the weekly movement of the Nikkei 225 index by correlating with macroeconomic factors (viz. exchange rate and Standard and Poor’s 500 Index (S&P500)). The prediction of up-to-minute stock price was carried out by Henrique et al. [36]; they concluded that the incorporation of a moving window could improve the predictability of the SVR. Recently, the SVM algorithm has been combined with other feature selection techniques (e.g., filter method and wrapper method) to improve the accuracy of trend prediction [37,38].

#### 2.3. Artificial Neural Network

#### 2.4. Discrete Wavelet Transform-Based Models

#### 2.5. Empirical Mode Decomposition-Based Models

## 3. Data and Methods

#### 3.1. Data Sets

#### 3.2. Comparison Criteria

#### 3.3. HMM

#### 3.4. SVM

#### 3.5. ANN

**MSE**):

#### 3.6. DWT

#### 3.7. EMD

- (1)
- Identify all the local maxima and minima of $y(t)$.
- (2)
- Connect all the local maxima and local minima separately by the cubic spline to form the upper envelope line $h(t)$ and lower envelope line $l(t)$.
- (3)
- Calculate the mean value of the upper and lower envelope $m(t)=(h(t)+l(t))/2$.
- (4)
- Derive a new time-series $c(t)$ by subtracting the mean envelope, $c(t)=y(t)-m(t)$.
- (5)
- If $c(t)$ satisfies the properties of IMF [64]; then, $c(t)$ is regarded as an IMF, and $y(t)$ in step 1 is replaced with the new process $r(t)=y(t)-c(t).$ Otherwise, substitute $y(t)$ in step 1 by $c(t)$ and repeat all of the above process.

#### 3.8. Overall Process of Hybrid Machine Learning Models

## 4. Experimental Results

#### 4.1. MAPE Comparison

#### 4.2. Sample Size Effect

#### 4.3. Momentum and Mean Reversion Stock Pattern Effect

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIC | Akaike Information Criterion |

ANFIS | Adaptive Neuro Fuzzy Inference System |

ANN | Artificial Neural Network |

APE | Absolute Percentage Error |

AR | Autoregressive Model |

ARCH | Autoregressive Conditional Heteroskedasticity Model |

ARIMA | Autoregressive Integrated Moving Average |

BDI | Baltic Dry Index |

BP | Backpropagation |

BPNN | Backpropagation Neural Network |

CAPM | Capital Asset Pricing Model |

DJIA | Down Jones Industrial Average |

DWT | Discrete Wavelet Transform |

EM | Expectation-Maximization Algorithm |

EMD | Empirical Mode Decomposition |

EMH | Efficient Market Hypothesis |

GA | Genetic Algorithm |

GMM | Gaussian Mixture Model |

GRNN | General Regression Neural Network |

HMM | Hidden Markov Model |

HSCEI | Hong Seng China Enterprises Index |

HSI | Hang Seng Index |

IMF | Intrinsic Mode Function |

LDA | Linear Discriminant Analysis |

LS-SVR | Least Square Support Vector Regression |

MAPE | Mean Absolute Percentage Error |

MODWT | Maximum Overlap Discrete Wavelet Transform |

ML | Machine Learning |

NN | Neural Network |

RBF | Radial Basis Function |

RBFNN | Radial Basis Function Neural Network |

S.D. | Standard Deviation |

SVM | Support Vector Machine |

SVR | Support Vector Regression |

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**Table 1.**Summary of the hidden Makarov model (HMM) based prediction methods used by this study as references.

Article | Stock Market | Input Variables | Training Method | Comparative Studies | Best Model |
---|---|---|---|---|---|

[26] | - | Opening, high, low and closing price | HMM | ANN | Similar performance |

[28] | USA | Opening, high, low and closing price | ANN-HMM-GA | ARIMA | Similar performance |

[27] | - | Opening, high, low and closing price | HMM-Fuzzy | HMM, HMM-ANN-GA, ARIMA and ANN | HMM-Fuzzy |

[29] | USA | Fractional change, fractional high and low | Maximum a Posterior HMM | HMM-Fuzzy, ARIMA and ANN | HMM |

[30] | China | Open, high, low and closing price; news articles | Extended coupled HMM | SVM, TeSIA, CMT, ECHMM-NE, ECHMM-NC, ECHMM | ECHMM |

**Table 2.**Summary of support vector machine (SVM) based prediction methods used by this study’s references.

Article | Stock Market | Input Variables | Training Method | Comparative Studies | Best Model |
---|---|---|---|---|---|

[34] | USA | Daily closing price of five future contracts | SVM | BPNN | SVM |

[33] | Korea | 12 technical indicators | SVM | ANN and CBR | SVM |

[35] | Japan | Weekly price of S&P500 Index and USD/Yen exchange rate | SVM | LDA, QDA, EPNN | SVM |

[38] | USA | 29 technical indicators and lagged index price | SVM | BPNN | SVM |

[37] | Indonesia | 14 technical indicators | Particle swarm optimization-SVR | - | - |

[36] | Brazil, USA and China | 5 technical indicators for daily and 1-minute price | SVR | Random walk model | SVR |

**Table 3.**Summary of artificial neural network (ANN)-based prediction methods used by this study’s references.

Article | Stock Market | Input Variables | Training Method | Comparative Studies | Best Model |
---|---|---|---|---|---|

[40] | USA | Daily stock return | NN | - | Over-optimistic |

[41] | Japan | Technical indicators and economic indexes | NN | Multiple regression analysis | NN |

[44] | China | Daily closing price; quarterly book value; common share outstanding | ANN | CAPM, Fama and Frech’s 3 factor model | ANN |

[43] | Turkey | 10 technical indicators | ANN | SVM | ANN |

[47] | USA | Daily opening, high, low and closing price; trading volume | ANN | ARIMA | ANN |

[42] | Japan | Technical indicators | GA-ANN | - | - |

[45] | China | Weekly close price | BPNN | RBF, GRNN, SVR, LS-SVR | BPNN |

**Table 4.**Summary of discrete wavelet transform (DWT)-based prediction methods used by this study’s references.

Article | Stock Market | Input Variables | Training Method | Comparative Studies | Best Model |
---|---|---|---|---|---|

[62] | - | Weekly exchange rate GBP/USD | DWT-ANN/ARIMA | ARIMA, ANN and Zhang′s hybrid model | DWT-ANN/ARIMA |

[60] | India | Weekly closing price | MODWT-ANN/SVR | ANN, SVR | MODWT-ANN/SVR |

[56] | USA | Minute-in-day closing price | DWT-BPNN | ARMA, Random walk model | DWT-BPNN |

[59] | China | Monthly closing price | DWT-BPNN | BPNN | DWT-BPNN |

[58] | Taiwan, USA, UK, Japan | 10 technical indicators | DWT-ABC-RNN | DWT-BP-ANN, BNN | DWT-ABC-RNN |

[57] | USA | 9 technical indicators | DWT-FGP | - | - |

**Table 5.**Summary of empirical mode decomposition (EMD)-based prediction methods used by this study’s references.

Article | Stock Market | Input Variables | Training Method | Comparative Studies | Best Model |
---|---|---|---|---|---|

[22] | USA, UK | Daily crude oil spot price | EMD + ARIMA + ALNN; EMD-FNN-ALNN; EMD-ARIMA-Average | ARIMA, FNN | EMD - FNN - ALNN |

[71] | China | Closing price | EMD-SVM | SVM | EMD-SVM |

[69] | - | Daily exchange rate USD/NTD, JPY/NTD and RMB/NTD | EMD-LSSVR | EMD-ARIMA, LSSVR, ARIMA | EMD-SVR |

[68] | Taiwan | Closing price | EMD-SVR | AR, SVR | EMD-SVR |

[65] | Taiwan, HK | Closing price | EMD-ANFIS | SVR, AR, ANFIS | EMD-ANFIS |

[67] | USA | Intraday price | EMD-SVR | ARIMA, Directed SVR, Recursive SVR | EMD-SVR |

Statistics ^{1} | Hang Seng Index | Hang Seng China Enterprises Index | Tencent |
---|---|---|---|

Mean ($\mu $) | 0.05% | 0.04% | 0.19% |

Standard Deviation ($\sigma $) | 1.24% | 1.54% | 2.07% |

Skewness (${S}_{k}$) | 0.01 | 0.10 | 0.14 |

Kurtosis ($\beta $) | 2.51 | 1.98 | 2.04 |

^{1}$\text{}\mu =\frac{{{\displaystyle \sum}}_{i=1}^{N}{r}_{i}}{N}$, $\sigma =\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{N}{({r}_{i}-\mu )}^{2}}{N}}$, ${S}_{k}=\frac{1}{N}\frac{{{\displaystyle \sum}}_{i=1}^{N}{({r}_{i}-\mu )}^{3}}{{\sigma}^{3}}$, and $\beta =\frac{1}{N}\frac{{{\displaystyle \sum}}_{i=1}^{N}{({r}_{i}-\mu )}^{4}}{{\sigma}^{4}}$ where r and N stand for daily return and population number, respectively.

Methodology | Hang Seng Index | Hang Seng China Enterprises Index | Tencent | |||
---|---|---|---|---|---|---|

MAPE (%) | Rank | MAPE (%) | Rank | MAPE (%) | Rank | |

Benchmark S. D. | 1.24 | - | 1.54 | - | 2.07 | - |

Random walk | 0.97 | - | 1.13 | - | 1.68 | - |

HMM | 1.42 | 4 | 1.27 | 3 | 2.41 | 4 |

SVR | 1.69 | 6 | 1.86 | 6 | 2.70 | 5 |

DWT - SVR | 1.85 | 7 | 2.04 | 7 | 3.10 | 7 |

EMD - SVR | 1.45 | 5 | 1.57 | 5 | 2.60 | 6 |

ANN | 1.12 | 2 | 1.04 | 1 | 1.81 | 1 |

DWT - ANN | 1.11 | 1 | 1.20 | 2 | 1.83 | 2 |

EMD - ANN | 1.21 | 3 | 1.35 | 4 | 2.25 ^{1} | 3 |

^{1}MAPE for Tencent was obtained using a fixed training sample size of 200.

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**MDPI and ACS Style**

Shi, C.; Zhuang, X.
A Study Concerning Soft Computing Approaches for Stock Price Forecasting. *Axioms* **2019**, *8*, 116.
https://doi.org/10.3390/axioms8040116

**AMA Style**

Shi C, Zhuang X.
A Study Concerning Soft Computing Approaches for Stock Price Forecasting. *Axioms*. 2019; 8(4):116.
https://doi.org/10.3390/axioms8040116

**Chicago/Turabian Style**

Shi, Chao, and Xiaosheng Zhuang.
2019. "A Study Concerning Soft Computing Approaches for Stock Price Forecasting" *Axioms* 8, no. 4: 116.
https://doi.org/10.3390/axioms8040116