# Prediction in Chaotic Environments Based on Weak Quadratic Classifiers

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Solving Method

#### 2.2. Formalization of the Problem Definition

#### 2.3. Special Solution Details

#### 2.4. Preliminary Smoothing

## 3. Results and Discussion

- P1. Precedent prediction (using the nearest-neighbor method) (3);
- P2. Average forecast based on three analogs (4);
- P3. Weighted average forecast using the three analogs (5);
- P4. Average forecast for analogs with similarity degree not lower than ${Y}_{0}$.

- Average absolute deviation ${K}_{1}:Err=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}abs\left({\tilde{Y}}_{t,t+L}-{Y}_{t+L}\right),$
- Average square of deviations ${K}_{2}:Err=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{\left({\tilde{Y}}_{t,t+L}-{Y}_{t+L}\right)}^{2}$.

## 4. Conclusions

- Building self-organizing forecasting algorithms which take into account changes in external factors affecting the controlled process;
- Application of multivariate analysis on the group of correlated processes using metrics of multivariate statistical analysis. In particular, the effectiveness of the precedent forecast based on the Mahalanobis distance, or other less common metrics such as the Hotelling’s trace or the Pillai’s trace is important.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Example of the current widow and analog windows in the general time schedule of a smoothed process.

**Figure 3.**Relative accuracy of the prediction for the technological process with nonstationary dynamics.

Observation Interval | Criterion | Prediction Alternative | |||
---|---|---|---|---|---|

P1 | P2 | P3 | P4 | ||

1 | ${K}_{1}$ | 34.09 | 23.89 | 24.57 | 27.70 |

${K}_{2}$ | 2152.81 | 1054.82 | 1203.23 | 1673.13 | |

2 | ${K}_{1}$ | 35.00 | 19.52 | 19.54 | 29.60 |

${K}_{2}$ | 2416.21 | 589.28 | 624.98 | 1778.80 | |

3 | ${K}_{1}$ | 34.88 | 23.23 | 22.77 | 25.09 |

${K}_{2}$ | 2085.85 | 980.67 | 1019.16 | 1286.75 | |

4 | ${K}_{1}$ | 43.94 | 35.17 | 35.85 | 39.52 |

${K}_{2}$ | 3051.40 | 2087.24 | 2153.68 | 2559.02 |

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**MDPI and ACS Style**

Musaev, A.; Borovinskaya, E.
Prediction in Chaotic Environments Based on Weak Quadratic Classifiers. *Symmetry* **2020**, *12*, 1630.
https://doi.org/10.3390/sym12101630

**AMA Style**

Musaev A, Borovinskaya E.
Prediction in Chaotic Environments Based on Weak Quadratic Classifiers. *Symmetry*. 2020; 12(10):1630.
https://doi.org/10.3390/sym12101630

**Chicago/Turabian Style**

Musaev, Alexander, and Ekaterina Borovinskaya.
2020. "Prediction in Chaotic Environments Based on Weak Quadratic Classifiers" *Symmetry* 12, no. 10: 1630.
https://doi.org/10.3390/sym12101630