# Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Data

#### 2.2. Stationarity

#### 2.3. Fourier Analysis

#### 2.4. Lyapunov Exponents

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Period | coop1 | coop2 | coop3 | |||
---|---|---|---|---|---|---|

Mean | Error | Mean | Error | Mean | Error | |

2011 | 42,202 | 3598 | – | – | – | – |

2012 | 42,830 | 4382 | – | – | – | – |

2013 | 46,837 | 4448 | 34,364 | 3321 | – | – |

2014 | 47,946 | 4632 | 49,938 | 4088 | – | – |

2015 | 31,378 | 2427 | 44,235 | 2961 | 35,621 | 3124 |

2016 | 34,875 | 2959 | 41,278 | 2814 | 47,378 | 3692 |

2017 | 33,340 | 2421 | 28,908 | 2166 | 53,099 | 4029 |

2018 | 34,422 | 2765 | 24,656 | 2106 | 44,588 | 3744 |

2019 | 28,095 | 2159 | 29,036 | 2119 | 43,042 | 3724 |

2020 | 28,791 | 1845 | 22,207 | 1474 | 30,708 | 2353 |

$\mathit{\epsilon}$ | Range $\mathbf{\Delta}\mathit{n}$ | Exponent | Coefficient |
---|---|---|---|

2500 | $[36,\phantom{\rule{4pt}{0ex}}60]$ | $0.038733$ | $0.999612$ |

5000 | $[33,\phantom{\rule{4pt}{0ex}}50]$ | $0.042351$ | $0.999078$ |

7000 | $[34,\phantom{\rule{4pt}{0ex}}50]$ | $0.043439$ | $0.999411$ |

9000 | $[32,\phantom{\rule{4pt}{0ex}}47]$ | $0.043454$ | $0.999949$ |

14,000 | $[27,\phantom{\rule{4pt}{0ex}}37]$ | $0.049539$ | $0.999453$ |

$\mathit{\epsilon}$ | Range $\mathbf{\Delta}\mathit{n}$ | Exponent | Coefficient |
---|---|---|---|

2000 | $[38,\phantom{\rule{4pt}{0ex}}52]$ | $0.049485$ | $0.999835$ |

4000 | $[34,\phantom{\rule{4pt}{0ex}}44]$ | $0.049351$ | $0.999968$ |

8000 | $[32,\phantom{\rule{4pt}{0ex}}45]$ | $0.046593$ | $0.999930$ |

16,000 | $[30,\phantom{\rule{4pt}{0ex}}44]$ | $0.044334$ | $0.999872$ |

$\mathit{\epsilon}$ | Range $\mathbf{\Delta}\mathit{n}$ | Exponent | Coefficient |
---|---|---|---|

1000 | $[46,\phantom{\rule{4pt}{0ex}}74]$ | $0.040654$ | $0.999546$ |

2000 | $[44,\phantom{\rule{4pt}{0ex}}69]$ | $0.040315$ | $0.999373$ |

4000 | $[43,\phantom{\rule{4pt}{0ex}}71]$ | $0.038884$ | $0.999604$ |

8000 | $[40,\phantom{\rule{4pt}{0ex}}70]$ | $0.037983$ | $0.999479$ |

12,000 | $[36,\phantom{\rule{4pt}{0ex}}71]$ | $0.036804$ | $0.998665$ |

First 4 Seasons | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coefficient |

2500 | 2 | $[37,\phantom{\rule{4pt}{0ex}}60]$ | $0.033932$ | $0.999293$ |

5000 | 2 | $[37,\phantom{\rule{4pt}{0ex}}53]$ | $0.042953$ | $0.999331$ |

7000 | 2 | $[38,\phantom{\rule{4pt}{0ex}}53]$ | $0.042915$ | $0.998767$ |

9000 | 2 | $[40,\phantom{\rule{4pt}{0ex}}50]$ | $0.044363$ | $0.996814$ |

14,000 | 2 | $[39,\phantom{\rule{4pt}{0ex}}44]$ | $0.052637$ | $0.998350$ |

Last 6 Seasons | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coeficient |

2500 | 2 | $[34,\phantom{\rule{4pt}{0ex}}60]$ | $0.036119$ | $0.999693$ |

5000 | 2 | $[27,\phantom{\rule{4pt}{0ex}}52]$ | $0.039526$ | $0.999682$ |

7000 | 2 | $[24,\phantom{\rule{4pt}{0ex}}51]$ | $0.038135$ | $0.999653$ |

9000 | 2 | $[25,\phantom{\rule{4pt}{0ex}}48]$ | $0.036410$ | $0.999531$ |

14,000 | 2 | $[22,\phantom{\rule{4pt}{0ex}}41]$ | $0.036882$ | $0.999364$ |

First 4 Seasons | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coefficient |

2000 | 2 | $[38,\phantom{\rule{4pt}{0ex}}55]$ | $0.051889$ | $0.999930$ |

4000 | 2 | $[37,\phantom{\rule{4pt}{0ex}}48]$ | $0.049883$ | $0.999755$ |

8000 | 2 | $[35,\phantom{\rule{4pt}{0ex}}51]$ | $0.048558$ | $0.999732$ |

16,000 | 2 | $[31,\phantom{\rule{4pt}{0ex}}44]$ | $0.049202$ | $0.999739$ |

Last 6 Seasons | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coeficient |

2000 | 2 | $[45,\phantom{\rule{4pt}{0ex}}70]$ | $0.034716$ | $0.999751$ |

4000 | 2 | $[41,\phantom{\rule{4pt}{0ex}}61]$ | $0.035169$ | $0.999702$ |

8000 | 2 | $[38,\phantom{\rule{4pt}{0ex}}57]$ | $0.034492$ | $0.999950$ |

16,000 | 2 | $[36,\phantom{\rule{4pt}{0ex}}49]$ | $0.032439$ | $0.999622$ |

Seasons 1–3 | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coefficient |

1000 | 3 | $[44,\phantom{\rule{4pt}{0ex}}77]$ | $0.043092$ | $0.998316$ |

2000 | 3 | $[41,\phantom{\rule{4pt}{0ex}}73]$ | $0.043402$ | $0.998516$ |

4000 | 3 | $[38,\phantom{\rule{4pt}{0ex}}71]$ | $0.042802$ | $0.998661$ |

8000 | 3 | $[33,\phantom{\rule{4pt}{0ex}}63]$ | $0.042343$ | $0.998135$ |

12,000 | 3 | $[30,\phantom{\rule{4pt}{0ex}}61]$ | $0.040764$ | $0.998190$ |

Seasons 4–6 | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coefficient |

1000 | 2 | $[60,\phantom{\rule{4pt}{0ex}}97]$ | $0.029172$ | $0.999118$ |

2000 | 2 | $[57,\phantom{\rule{4pt}{0ex}}91]$ | $0.030104$ | $0.999720$ |

4000 | 2 | $[52,\phantom{\rule{4pt}{0ex}}85]$ | $0.029541$ | $0.999602$ |

8000 | 2 | $[50,\phantom{\rule{4pt}{0ex}}81]$ | $0.029783$ | $0.999707$ |

12,000 | 2 | $[45,\phantom{\rule{4pt}{0ex}}79]$ | $0.029731$ | $0.999593$ |

Seasons 1–4 | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coefficient |

1000 | 2 | $[42,\phantom{\rule{4pt}{0ex}}69]$ | $0.041033$ | $0.996666$ |

2000 | 2 | $[40,\phantom{\rule{4pt}{0ex}}69]$ | $0.041306$ | $0.997620$ |

4000 | 2 | $[38,\phantom{\rule{4pt}{0ex}}69]$ | $0.040852$ | $0.999009$ |

8000 | 2 | $[35,\phantom{\rule{4pt}{0ex}}69]$ | $0.039491$ | $0.998923$ |

12,000 | 2 | $[33,\phantom{\rule{4pt}{0ex}}69]$ | $0.038913$ | $0.998336$ |

Seasons 1–5 | ||||

$\mathbf{\epsilon}$ | Delay | Range $\Delta \mathit{n}$ | Exponent | Coefficient |

1000 | 1 | $[45,\phantom{\rule{4pt}{0ex}}69]$ | $0.039694$ | $0.999392$ |

2000 | 1 | $[43,\phantom{\rule{4pt}{0ex}}69]$ | $0.039060$ | $0.999476$ |

4000 | 1 | $[43,\phantom{\rule{4pt}{0ex}}69]$ | $0.037576$ | $0.999617$ |

8000 | 1 | $[41,\phantom{\rule{4pt}{0ex}}69]$ | $0.036650$ | $0.999072$ |

12,000 | 1 | $[39,\phantom{\rule{4pt}{0ex}}69]$ | $0.036414$ | $0.998659$ |

$\mathit{\epsilon}$ | Range $\mathbf{\Delta}\mathit{n}$ | Exponent | Coefficient |
---|---|---|---|

1000 | $[44,\phantom{\rule{4pt}{0ex}}77]$ | $0.025899$ | $0.998097$ |

2000 | $[42,\phantom{\rule{4pt}{0ex}}73]$ | $0.028804$ | $0.999045$ |

4000 | $[39,\phantom{\rule{4pt}{0ex}}65]$ | $0.030844$ | $0.999562$ |

8000 | $[36,\phantom{\rule{4pt}{0ex}}59]$ | $0.032047$ | $0.999483$ |

12,000 | $[33,\phantom{\rule{4pt}{0ex}}54]$ | $0.034171$ | $0.998611$ |

$\mathit{\epsilon}$ | Range $\mathbf{\Delta}\mathit{n}$ | Exponent | Coefficient |
---|---|---|---|

500 | $[66,\phantom{\rule{4pt}{0ex}}89]$ | $0.030747$ | $0.999539$ |

1000 | $[60,\phantom{\rule{4pt}{0ex}}87]$ | $0.028890$ | $0.998904$ |

2000 | $[54,\phantom{\rule{4pt}{0ex}}81]$ | $0.028803$ | $0.999281$ |

4000 | $[48,\phantom{\rule{4pt}{0ex}}73]$ | $0.028493$ | $0.999873$ |

8000 | $[47,\phantom{\rule{4pt}{0ex}}69]$ | $0.027050$ | $0.999313$ |

12,000 | $[45,\phantom{\rule{4pt}{0ex}}60]$ | $0.027428$ | $0.999044$ |

$\mathit{\epsilon}$ | Range $\mathbf{\Delta}\mathit{n}$ | Exponent | Coefficient |
---|---|---|---|

250 | $[57,\phantom{\rule{4pt}{0ex}}87]$ | $0.037408$ | $0.999269$ |

500 | $[55,\phantom{\rule{4pt}{0ex}}83]$ | $0.036858$ | $0.998921$ |

1000 | $[51,\phantom{\rule{4pt}{0ex}}79]$ | $0.037610$ | $0.999470$ |

2000 | $[48,\phantom{\rule{4pt}{0ex}}75]$ | $0.037513$ | $0.999424$ |

4000 | $[45,\phantom{\rule{4pt}{0ex}}71]$ | $0.035340$ | $0.998392$ |

Days | Dimension | Delay | Radius | Threshold |
---|---|---|---|---|

1–7 | 4 | 4 | 7000 | 16 |

8–14 | 10 | 10 | 7000 | 22 |

1–14 | 4 | 4 | 3250 | 16 |

Days | Dimension | Delay | Radius | Threshold |
---|---|---|---|---|

1–7 | 10 | 3 | 3825 | 22 |

8–14 | 5 | 13 | 1650 | 16 |

1–14 | 7 | 6 | 6000 | 22 |

Days | Dimension | Delay | Radius | Threshold |
---|---|---|---|---|

1–7 | 9 | 10 | 600 | 10 |

8–14 | 9 | 10 | 600 | 10 |

1–14 | 9 | 10 | 600 | 10 |

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**MDPI and ACS Style**

Borrero, J.D.; Mariscal, J.
Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series. *Mathematics* **2021**, *9*, 3034.
https://doi.org/10.3390/math9233034

**AMA Style**

Borrero JD, Mariscal J.
Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series. *Mathematics*. 2021; 9(23):3034.
https://doi.org/10.3390/math9233034

**Chicago/Turabian Style**

Borrero, Juan D., and Jesus Mariscal.
2021. "Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series" *Mathematics* 9, no. 23: 3034.
https://doi.org/10.3390/math9233034