# Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

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## Abstract

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## 1. Fractional Calculus (FC) and Fractional-Order Thinking (FOT)

**fractional dynamics**has merged and gained followers in the statistical and chemical physics communities [6,7,8]. For example, optimal image processing has improved through the use of fractional-order differentiation and fractional-order partial differential equations as summarized in Chen et al. [9,10,11]. Anomalous diffusion was described using fractional-diffusion equations in [12,13], and Metzler et al. used fractional Langevin equations to model viscoelastic materials [14].

**complexity and inverse power laws (IPL)**.

#### 1.1. Complexity and Inverse Power Laws

#### 1.2. Heavy-Tailed Distributions

#### 1.2.1. Lévy Distribution

#### 1.2.2. Mittag–Leffler PDF

#### 1.2.3. Weibull Distribution

#### 1.2.4. Cauchy Distribution

#### 1.2.5. Pareto Distribution

#### 1.2.6. The $\alpha $-Stable Distribution

#### 1.3. Mixture Distributions

#### 1.3.1. Gaussian Distribution

#### 1.3.2. Laplace Distribution

#### 1.4. IPL Tail-Index Analysis

## 2. Big Data, Variability and FC

**variability**is the most important characteristic being discussed. Variability refers to several properties of big data. First, it refers to the number of inconsistencies in the data, which need to be understood by using anomaly- and outlier-detection methods for any meaningful analytics to be performed. Second, variability can also refer to diversity [89,90], resulting from disparate data types and sources, for example, healthy or unhealthy [91,92]. Finally, variability can refer to multiple research topics (Table 2).

**variability**, which refers to the behavior of the dynamic system. The ancient Greek philosopher Heraclitus (535 BC–475 BC) also realized the importance of variability, prompting him to say: “The only thing that is constant is change”; “It is in changing that we find purpose”; “Nothing endures but change”; “No man ever steps in the same river twice, for it is not the same river and he is not the same man”.

#### 2.1. Hurst Parameter, fGn, and fBm

#### 2.2. Fractional Lower-Order Moments (FLOMs)

#### 2.3. Fractional Autoregressive Integrated Moving Average (FARIMA) and Gegenbauer Autoregressive Moving Average (GARMA)

#### 2.4. Continuous Time Random Walk (CTRW)

#### 2.5. Unmanned Aerial Vehicles (UAVs) and Precision Agriculture

## 3. Optimal Machine Learning and Optimal Randomness

**Machine learning (ML)**is the science (and art) of programming computers so they can learn from data [139]. A more engineering-oriented definition was given by Tom Mitchell in 1997 [140], “A computer program is said to learn from experience E with respect to some task T and some performance measure P, if its performance on T, as measured by P, improves with experience E”.

**Reflection:**ML is, today, a hot research topic and will probably remain so into the near future. How a machine can learn efficiently (optimally) is always important. The key for the learning process is the optimization method. Thus, in designing an efficient optimization method, it is necessary to answer the following three questions:

- What is the optimal way to optimize?
- What is the
**more optimal**way to optimize? - Can we demand
**“more optimal machine learning”**, for example, deep learning with the minimum/smallest labeled data)?

**Optimal randomness:**In the section on the Lévy PDF, the Lévy flight is the search strategy for food the albatross has developed over millions of years of evolution. Admittedly, this is a slow optimization procedure [84]. From this perspective, we should call “Lévy distribution” an optimized or learned randomness used by albatrosses for searching for food. Therefore, we pose the question: “can the search strategy be more optimal than Lévy flight?” The answer is yes if one adopts the FC [145]! Optimization is a very complex area of study. However, a few studies have investigated using FC to obtain a better optimization strategy.

#### 3.1. Derivative-Free Methods

#### 3.2. The Gradient-Based Methods

#### Nesterov Accelerated Gradient Descent (NAGD)

#### 3.3. What Can the Control Community Offer to ML?

**stmcb( )**[162] to derive its discrete form. After the complex poles are included, one can have:

## 4. A Case Study of Machine Learning with Fractional Calculus: A Stochastic Configuration Network with Heavytailedness

#### 4.1. Stochastic Configuration Network (SCN)

#### 4.2. SCN with Heavy-Tailed PDFs

#### 4.3. A Regression Model and Parameter Tuning

#### Performance Comparison among SCNs with Heavy-Tailed PDFs

#### 4.4. MNIST Handwritten Digit Classification

#### Performance Comparison among SCNs on MNIST

## 5. Take-Home Messages and Looking into the Future: Fractional Calculus Is Physics Informed

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACF | Auto-Correlation Function |

AI | Artificial Intelligence |

ARMA | Autoregression and Moving Average |

CLT | Classical Central Limit Theorem |

CS | Cuckoo Search |

CTRW | Continuous Time Random Work |

EOM | Equation of Motion |

fBm | Fractional Brownian Motion |

fGn | Fractional Gaussian Noise |

FARIMA | Fractional Autoregressive Integrated Moving Average |

FC | Fractional Calculus |

FIGARCH | Fractional Integral Generalized Autoregressive Conditional Heteroscedasticity |

FLOM | Fractional Lower-Order Moments |

FOCV | Fractional-Order Calculus of Variation |

FODA | Fractional-Order Data Analytics |

FOEL | Fractional-Order Euler–Lagrange |

FOT | Fractional-Order Thinking |

GARMA | Gegenbauer Autoregressive Moving Average |

GD | Gradient Descent |

GDM | Gradient Descent Momentum |

GEV | Generalized Extreme Value |

IMP | Internal Model Principle |

IPL | Inverse Power Law |

ISE | Integral Squared Error |

LGD | Long Range Dependence |

LTI | Linear Time Invariant |

MAD | Modeling, Analysis and Design |

ML | Machine Learning |

MLL | Mittag–Leffler Law |

MNIST | Modified National Institute of Standards and Technology Database |

NAGD | Nesterov Accelerated Gradient Descent |

NDVI | Normalized Difference Vegetation Index |

NILT | Numerical Inverse Laplace Transform |

NN | Neural Networks |

PA | Precision Agriculture |

Probability Density Function | |

PID | Proportional, Integral, Derivative |

PSO | Particle Swarm Optimization |

RBF | Randomized Radial Basis Function (RBF) Networks |

RGB | Red, Green, Blue |

RMSE | Root Mean Squared Error |

RVFL | Random Vector Functional Link |

RW-FNN | Feed-Forward Networks with Random Weights |

SCN | Stochastic Configuration Network |

SGD | Stochastic Gradient Descent |

SLFNNs | Single-Layer Feed-Forward Neural Networks |

UAVs | Unmanned Aerial Vehicles |

USDA | United States Department of Agriculture |

wGn | White Gaussian Noise |

## Appendix A. SCN Codes

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**Figure 1.**Inverse power law (complexity “bow tie”): On the left are the systems of interest that are thought to be complex. In the center panel, an aspect of the empirical data is characterized by an inverse power law (IPL). The right panel lists the potential properties associated with systems with data that have been processed and yield an IPL property. See text for more details.

**Figure 2.**Complex signals (IPL): Here, the signal generated by a complex system is depicted. Exemplars of the systems are given as are the potential properties arising from the systems’ complexity.

**Figure 3.**Cauchy distributions are examples of fat-tailed distributions. The parameter a is the location parameter; the parameter b is the scale parameter.

**Figure 4.**Symmetric $\alpha $-stable distributions with unit scale factor. The most narrow PDF shown has the smallest IPL index and, consequently, the most weight in the tail regions.

**Figure 6.**(

**a**) The behavior of tail-index α during the iterations; (

**b**) The training and testing accuracy. At first, the α decreases very slowly; when α reaches its lowest level, which means longer tail distribution, there is a significant jump, which causes a sudden decrease in accuracy. Beyond this point, the process recovers again, and we see stationary behavior in α and an increasing behavior in the accuracy.

**Figure 9.**The 2-D Alpine function for derivative-free methods; there are (

**a**) single agent search and (

**b**) swarm-based search methods.

**Figure 10.**Sample paths. Wei et al. [148] investigated the optimal randomness in a swarm-based search. Four heavy-tailed PDFs were used for sample path analysis; there are (

**a**) Mittag-Leffler distribution, (

**b**) Weibull distribution, (

**c**) Pareto distribution, and (

**d**) Cauchy distribution. The Long steps, referring to the jump length, frequently happened for all distributions, which showed strong heavy-tailed performance. For more details, please refer to [148].

**Figure 12.**The integrator model (embedded in $G\left(z\right)$). The integrator in the forward loop eliminates the tracking steady-state error for a constant reference signal (internal model principle (IMP)).

**Figure 13.**Training loss (

**left**); test accuracy (

**right**). It is obvious that for different zeros and poles, the performance of the algorithms is different. One finds that both the $b=-0.25$ and $b=-0.5$ cases perform better than does the stochastic gradient descent (SGD) momentum. Additionally, both $b=0.25$ and $b=0.5$ perform worse. It is also shown that an additional zero can improve the performance, if adjusted carefully.

**Figure 15.**Performance of SCN, SCN-Lévy, SCN-Weibull, SCN-Cauchy and SCN-Mixture. The parameter L is the hidden node number.

Characteristics | Description |
---|---|

1. Volume | Best known characteristic of big data; more than 90 percent of the whole data were created in the past couple of years. |

2. Velocity | The speed at which data are being generated. |

3. Variety | Processing structured, unstructured and semistructured data. |

4. Variability | Inconsistent speed of data loading, multitude of data dimensions, and number of inconsistencies. |

5. Veracity | Confidence or trust in the data. |

6. Validity | Refers to how accurate and correct the data are. |

7. Vulnerability | Security concerns, data breaches. |

8. Volatility | Design policy for data currency, availability, and rapid retrieval of information when required. |

9. Visualization | Develop new tools considering the complex relationships between the above properties. |

10. Value | The most important of the 10 Vs; substantial value must be found. |

Topics | Description |
---|---|

1. Climate variability | Changes in the components of the climate system and their interactions. |

2. Genetic variability | Measurements of the tendencies of individual genotypes between regions. |

3. Heart rate variability | Physiological phenomenon where the time interval between heart beats varies. |

4. Human variability | Measurements of the characteristics, physical or mental, of human beings. |

5. Spatial variability | Measurements at different spatial points exhibit different values. |

6. Statistical variability | A measure of dispersion in statistics. |

**Table 3.**General second-order algorithm design. The parameter $\rho $ is the loop forward gain; see text for more details.

$\mathit{\rho}$ | 0.4 | 0.8 | 1.2 | 1.6 | 2.0 | 2.4 |
---|---|---|---|---|---|---|

a | −0.6 | −0.2 | 0.2 | 0.6 | 1 | 1.4 |

b | 1.5 | 0.25 | −0.1667 | −0.3750 | −0.5 | −0.5833 |

**Table 4.**General third-order algorithm design, with parameters defined by Equation (41).

$\mathit{\rho}$ | 0.4 | 0.8 | 1.2 | 1.6 | 2.0 | 2.4 |
---|---|---|---|---|---|---|

a | 0.6439 | 0.5247 | −0.4097 | −0.5955 | −1.0364 | −1.4629 |

b | 0.0263 | 0.0649 | 0.0419 | −0.0398 | 0.0364 | 0.0880 |

c | 1.5439 | 0.5747 | −0.3763 | −0.3705 | −0.5364 | −0.6462 |

d | 0.0658 | 0.0812 | 0.0350 | −0.0408 | 0.0182 | 0.0367 |

$\mathit{\rho}$ | 0.3 | 0.5 | 0.7 | 0.9 |
---|---|---|---|---|

$\alpha $ | 1.8494 | 1.6899 | 1.5319 | 1.2284 |

$\beta $ | 20 | 20 | 20 | 20 |

Properties | Values |
---|---|

Name: | “Stochastic Configuration Networks” |

Version: | “1.0 beta” |

L: | hidden node number |

W: | input weight matrix |

b: | hidden layer bias vector |

Beta: | output weight vector |

r: | regularization parameter |

tol: | tolerance |

Lambda: | random weight range |

L${}_{max}$: | maximum number of hidden neurons |

T${}_{max}$: | maximum times of random configurations |

nB: | number of node being added in one loop |

Models | Mean Hidden Node Number | RMSE |
---|---|---|

SCN | 75 ± 5 | 0.0025 |

SCN-Lévy | 70 ± 6 | 0.0010 |

SCN-Cauchy | 59 ± 3 | 0.0057 |

SCN-Weibull | 63 ± 4 | 0.0037 |

SCN-Mixture | 70 ± 5 | 0.0020 |

Models | Training Accuracy | Test Accuracy |
---|---|---|

SCN | 94.0 ± 1.9% | 91.2 ± 6.2% |

SCN-Lévy | 94.9 ± 0.8% | 91.7 ± 4.5% |

SCN-Cauchy | 95.4 ± 1.3% | 92.4 ± 5.5% |

SCN-Mixture | 94.7 ± 1.1% | 91.5 ± 5.3% |

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Niu, H.; Chen, Y.; West, B.J.
Why Do Big Data and Machine Learning Entail the Fractional Dynamics? *Entropy* **2021**, *23*, 297.
https://doi.org/10.3390/e23030297

**AMA Style**

Niu H, Chen Y, West BJ.
Why Do Big Data and Machine Learning Entail the Fractional Dynamics? *Entropy*. 2021; 23(3):297.
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**Chicago/Turabian Style**

Niu, Haoyu, YangQuan Chen, and Bruce J. West.
2021. "Why Do Big Data and Machine Learning Entail the Fractional Dynamics?" *Entropy* 23, no. 3: 297.
https://doi.org/10.3390/e23030297