# Stock Index Return Volatility Forecast via Excitatory and Inhibitory Neuronal Synapse Unit with Modified MF-ADCCA

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multifractal Asymmetric Detrended Cross-Correlation Analysis (MF-ADCCA)

Algorithm 1. Algorithm to Moving-window MF-ADCCA Method. | ||

Input: Time Series: ${X}_{t}$; | < Size ($w\ast l$) > | |

Time Series: ${Y}_{t}$; | < Size ($w\ast l$) > | |

Days Scaling: $T$; | ||

Step: $t$; | ||

Output: Asymmetric generalized Hurst exponents; | ||

Function Moving-window MF-ADCCA (${X}_{t},{Y}_{t},T,t$) | ||

Initialize $\mathit{H}\mathit{u}\mathit{r}\mathit{s}\mathit{t}\leftarrow A$rray[0,…${N}_{w-T+1}$]$,\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{+}\leftarrow A$rray[0,…${N}_{w-T+1}$], $\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{-}\leftarrow A$rray[0,…${N}_{w-T+1}$] | ||

for $i$ in range (0, ${N}_{w-T+1}$, step) do | ||

$SeriesA\leftarrow {X}_{t}\left[i,\dots i+T\right]$ $SeriesB\leftarrow {Y}_{t}\left[i,\dots i+T\right]$ | ||

$\mathit{H}\mathit{u}\mathit{r}\mathit{s}\mathit{t},,\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{+},\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{-}\leftarrow MF-ADCCA\left(SeriesA,SeriesB\right)$ | ||

$\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}_{\left[\mathit{i}\right]}\leftarrow \mathit{H}\mathit{u}\mathit{r}\mathit{s}\mathit{t}$ | ||

$\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}_{\left[i\right]}^{+}\leftarrow \mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{+}$ | ||

$\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}_{\left[\mathit{i}\right]}^{-}\leftarrow \mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{-}$ | ||

return
$\mathit{H}\mathit{u}\mathit{r}\mathit{s}\mathit{t},,\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{+},\mathit{H}\mathit{u}\mathit{r}\mathit{s}{\mathit{t}}^{-}$ | ||

End function |

#### 2.2. Excitatory and Inhibitory Neuronal Synapse Unit (EINS)

Algorithm 2. Algorithm to Excitatory and Inhibitory Neural Synapse Model. | ||

Input: Time Series: ${T}_{t}$; | < Size ($batch,timestep,inputsize$) > | |

Input Size: $I$; Hidden Size: $H$; Output Size: $O$; | ||

Step: $t$; | ||

Output: Prediction Result: ${T}_{t+1}$; | ||

Procedure EINS ($n,p,i,j,{\theta}_{0}$) | ||

Initialize ${\mathit{E}}_{\mathit{t}}\leftarrow $0; ${\mathit{I}}_{\mathit{t}}\leftarrow 0$; ${\mathit{A}}_{\mathit{t}}\leftarrow 0$; $\theta \leftarrow {\theta}_{0}$; $i\leftarrow 0$; $j\leftarrow 0$. | ||

for ${X}_{i}$ in ${T}_{t}$ do | ||

${\mathit{D}}_{\mathit{t}}\leftarrow tanh\left({X}_{i}{W}_{d}^{I\ast H}-{b}_{d}^{H}\right)$ ${\mathit{E}}_{\mathit{t}}\leftarrow \delta \left({A}_{t-1}{W}_{ea}^{H\ast H}+{E}_{t-1}{W}_{ee}^{H\ast H}-{I}_{t-1}{W}_{ei}^{H\ast H}+{D}_{t}{W}_{ed}^{H\ast H}-{b}_{e}^{H}\right)$ | ||

${\mathit{I}}_{\mathit{t}}\leftarrow \delta \left({A}_{t-1}{W}_{ia}^{H\ast H}-{E}_{t-1}{W}_{ie}^{H\ast H}+{I}_{t-1}{W}_{ii}^{H\ast H}+{D}_{t}{W}_{id}^{H\ast H}-{b}_{i}^{H}\right)$ | ||

${\mathit{A}}_{\mathit{t}}\leftarrow \left({E}_{t}-{I}_{t}\right)\odot ex{p}^{-k\ast {D}_{t}^{2}}+{D}_{t}$ ${T}_{t+1}\leftarrow \left({A}_{t}{W}_{O}^{H\ast O}-{b}_{O}^{O}\right)$ return ${T}_{t+1}$ | ||

End for | ||

While $jp$ do | ||

Update $\theta $ by running training algorithm for $n$ steps | ||

$i\leftarrow i+n$ | ||

${T}_{t+1}\leftarrow ValidationSetError\left(\theta \right)$ | ||

if ${T}_{t+1}<{T}_{valid}$ then | ||

$j\leftarrow 0$ | ||

${\theta}^{\ast}\leftarrow \theta $ | ||

${i}^{\ast}\leftarrow i$ | ||

else | ||

$j\leftarrow j+1$ | ||

End if | ||

End while | ||

return ${\theta}^{\ast}$ and save the trained EINS model weights | ||

End Procedure |

## 3. Data and Experiments

#### 3.1. Data Description

#### 3.2. Experiments

## 4. Results and Discussion

^{2}of MF-ADCCA-based forecasting models outperformed MF-DFA under corresponding hyper-parameters. However, the multifractal elements obtained by MF-ADCCA method lacked stability when fitted into deep learning model due to proxy design diversities. Hence, it is recommended to use indicators related to financial technical analysis as proxies for MF-ADCCA method since prerequisite fintech knowledge is essential. It is noted that the improvement in MF-ADCCA performance to multiple proxies mechanism allowed to retain more time series’ past information indicating that asymmetric multifractal elements by MF-ADCCA method as input features are applicable to time series forecasting in deep learning.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Mean | Max | Min | Std | Skew | Kurt | J-Bera ^{1} | ADF ^{2} | KPSS ^{3} | |
---|---|---|---|---|---|---|---|---|---|

${r}_{t}$ | $4.6\times {10}^{-5}$ | $5.6\times {10}^{-3}$ | $-8.9\times {10}^{-2}$ | $1.3\times {10}^{-2}$ | −0.953 | 6.695 | 4372.8 * | −9.148 * | $8.27\times {10}^{-2}$ * |

${v}_{t}$ | $2.1\times {10}^{-4}$ | 3.7987 | −3.7436 | 0.8384 | $7.8\times {10}^{-3}$ | 1.0185 | 93.18 * | −15.769 * | $7.88\times {10}^{-2}$ * |

^{1}denotes Jarque-Bera statistic tests;

^{2}denotes Augmented Dickey–Fuller test;

^{3}denotes Kwiatkowski–Philips–Schmidt–Shin (KPSS) test; * represents 1% significance level. Note: (1) The ADF test uses null to express the existence of a unit root; (2) the KPSS test uses the null to express stationary.

Hyper-Parameters | Settings |
---|---|

Hidden neurons | 128,256 |

Time Horizons | 32,128 |

Learning rate | $1\times {10}^{-3}$ |

Dropout rate | 0.2 |

Epochs | 100 |

Optimizer | Adam |

Error function | Mean squared error |

**Table 3.**Forecasting performance with various hyper-parameters (note: best results highlighted in

**BOLD**).

Multifractal | Model | MSE | MAE | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

$\mathrm{Hidden}\mathrm{neurons}=128,\mathrm{Time}\mathrm{horizons}=32,\mathrm{Learning}\mathrm{rate}=1\times {10}^{-3}$ | ||||

MF-DFA | EINS | 0.02584 | 0.11069 | −0.01746 |

LSTM | 0.02704 | 0.11351 | −0.06485 | |

GRU | 0.02769 | 0.11494 | −0.09042 | |

MF-ADCCA | EINS | 0.02549 | 0.10999 | −0.00398 |

LSTM | 0.02577 | 0.11091 | −0.01469 | |

GRU | 0.02580 | 0.11130 | −0.01605 | |

$\mathrm{Hidden}\mathrm{neurons}=128,\mathrm{Time}\mathrm{Horizons}=128,\mathrm{Learning}\mathrm{rate}=1\times {10}^{-3}$ | ||||

MF-DFA | EINS | 0.02324 | 0.10689 | −0.00691 |

LSTM | 0.02337 | 0.10810 | −0.01262 | |

GRU | 0.02354 | 0.10885 | −0.02011 | |

MF-ADCCA | EINS | 0.02372 | 0.10770 | −0.02798 |

LSTM | 0.02507 | 0.11252 | −0.08636 | |

GRU | 0.02496 | 0.11118 | −0.08178 | |

$\mathrm{Hidden}\mathrm{neurons}=256,\mathrm{Time}\mathrm{Horizons}=32,\mathrm{Learning}\mathrm{rate}=1\times {10}^{-3}$ | ||||

MF-DFA | EINS | 0.02582 | 0.11069 | −0.01666 |

LSTM | 0.02727 | 0.11501 | −0.07390 | |

GRU | 0.02699 | 0.11419 | −0.06298 | |

MF-ADCCA | EINS | 0.02532 | 0.11003 | 0.00296 |

LSTM | 0.02704 | 0.11489 | −0.06500 | |

GRU | 0.02593 | 0.11145 | −0.02102 | |

$\mathrm{Hidden}\mathrm{neurons}=256,\mathrm{Time}\mathrm{Horizons}=128,\mathrm{Learning}\mathrm{rate}=1\times {10}^{-3}$ | ||||

MF-DFA | EINS | 0.02366 | 0.10756 | −0.02516 |

LSTM | 0.02467 | 0.11103 | −0.06923 | |

GRU | 0.02774 | 0.11885 | −0.20226 | |

MF-ADCCA | EINS | 0.02335 | 0.10705 | −0.01197 |

LSTM | 0.02419 | 0.10989 | −0.04810 | |

GRU | 0.02418 | 0.11025 | −0.04761 |

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**MDPI and ACS Style**

Wang, L.; Lee, R.S.T. Stock Index Return Volatility Forecast via Excitatory and Inhibitory Neuronal Synapse Unit with Modified MF-ADCCA. *Fractal Fract.* **2023**, *7*, 292.
https://doi.org/10.3390/fractalfract7040292

**AMA Style**

Wang L, Lee RST. Stock Index Return Volatility Forecast via Excitatory and Inhibitory Neuronal Synapse Unit with Modified MF-ADCCA. *Fractal and Fractional*. 2023; 7(4):292.
https://doi.org/10.3390/fractalfract7040292

**Chicago/Turabian Style**

Wang, Luochao, and Raymond S. T. Lee. 2023. "Stock Index Return Volatility Forecast via Excitatory and Inhibitory Neuronal Synapse Unit with Modified MF-ADCCA" *Fractal and Fractional* 7, no. 4: 292.
https://doi.org/10.3390/fractalfract7040292