# Influence of Regional Temperature Anomalies on Strawberry Yield: A Study Using Multivariate Copula Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

_{max}and Std. Y

_{min}are the standardised daily maximum and minimum yield over each month, Y

_{max}and Y

_{min}are the maximum and minimum daily yield over each month, ${\mu}_{max}$ and ${\mu}_{min}$ are the mean of the maximum and minimum yield time series data (for every month in the time period), and ${\sigma}_{max}$ and ${\sigma}_{min}$ are the standard deviation of the maximum and minimum yield time series data.

#### 2.2. Copula Analysis

_{1}and X

_{2}, with distribution functions F

_{1}(X

_{1}) and F

_{2}(X

_{2}), respectively. As per Sklar’s theorem there always exists a copula function (C) such that,

_{1}= x

_{1}, X

_{2}= x

_{2}) = C(F

_{1}(x

_{1}), F

_{2}(x

_{2}))

_{l},u

_{2}) is itself a distribution function where u

_{1}and u

_{2}are F

_{1}(x

_{1}), F

_{2}(x

_{2})), respectively.

#### 2.3. Construction of a 3-Dimensional Vine Copula

#### 2.4. Calculation of Joint Probability of Occurrence of Events Using Copula

#### 2.4.1. Univariate Probability

#### 2.4.2. Joint Probability for Tri-Variate Events

_{1}, X

_{2}, and X

_{3}, some of the formulae of tri-variate probability distributions is given as follows [49]:

#### 2.4.3. Conditional Probability of Tri-Variate Events

_{1}, X

_{2}, and X

_{3}, the conditional probability of occurrence of events can be estimated with the help of the underlying copulas. An example case is defined below [49]:

## 3. Preliminary Data Analysis

## 4. Results and Discussion

#### 4.1. Univariate Analysis

#### 4.2. Multivariate Copula Analysis Using Vine Copula

## 5. Limitations of This Study

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Maximum and minimum monthly strawberry yield at Santa Maria from 2011 to 2019; (

**b**) the observed temperature at Santa Maria from 2011 to 2019 and the temperature anomalies estimated; (

**c**) trend of monthly maximum temperature anomalies observed from 2011 to 2019.

**Figure 2.**Histogram of (

**a**) standardised maximum strawberry yield at Santa Maria; (

**b**) standardised minimum strawberry yield at Santa Maria; (

**c**) temperature anomaly at Santa Maria for each month.

**Figure 4.**The standardised maximum and minimum strawberry yield and temperature anomaly along with the corresponding simulated data using the fitted copula.

Statistics | Max Yield (Pounds/Acre) | Min Yield (Pounds/Acre) | Max Temp Anomaly (°F) |
---|---|---|---|

Min. | 57.94 | 1 | −11.9 |

1st Qu. | 128.5 | 17.72 | −2.6 |

Median | 185 | 73.74 | 4.5 |

Mean | 245.66 | 86.21 | 5.839 |

3rd Qu | 327.67 | 123.15 | 13.85 |

Max. | 686 | 356.32 | 25.9 |

**Table 2.**Probability of the yield loss events and temperature anomaly (T

_{A}) to be more than the given thresholds using the univariate probability distribution of the random variables.

Events | Probability of Occurrence | Remarks |
---|---|---|

Std. Y_{max} $\le -0.5$, Std. Y_{min} $\le -0.5$ | 0.004 | Yield loss |

Std. Y_{max} $\le -1.0$, Std. Y_{min} $\le -1.0$ | 0.0001 | Moderate or high yield loss |

Std. Y_{max} $\le -1.5$, Std. Y_{min} $\le -1.5$ | 0.25 × 10^{−6} | High yield loss |

T_{A}$>1,$ T_{A} $\le 2$ | 0.036 | Temperature anomaly is between the range of 1 °F and 2 °F |

T_{A}$>2$ T_{A} $\le 3$ | 0.042 | Temperature anomaly is between the range of 2 °F and 3 °F |

T_{A} $>3$ | 0.61 | Temperature anomaly greater than 3 °F |

Variables | Std. Max Yield | Std. Min Yield | Temperature Anomaly | Sum |
---|---|---|---|---|

Standardised maximum yield | 1 | 0.55 | −0.29 | 1.84 |

Standardised minimum yield | 0.55 | 1 | −0.417 | 1.967 |

Temperature anomaly | −0.29 | −0.417 | 1 | 1.707 |

**Table 4.**Multivariate probability of yield loss events for different temperature anomaly conditions.

No | Events | Conditional Probability of Occurrence of Events Given the Following T_{A} (°F) | Remarks | ||
---|---|---|---|---|---|

$1<$${\mathbf{T}}_{\mathbf{A}}\le 2$ | $2<$${\mathbf{T}}_{\mathbf{A}}\le 3$ | ${\mathbf{T}}_{\mathbf{A}}>3$ | |||

1 | $\mathrm{Std}.{Y}_{\mathit{max}}\le -0.5$$\mathrm{Std}.{Y}_{\mathit{min}}\le -0.5$|T_{A} | 0.584 | 0.56 | 0.8 | Yield loss |

2 | $\mathrm{Std}.{Y}_{\mathit{max}}\le -1$$\mathrm{Std}.{Y}_{\mathit{min}}\le -1$|T_{A} | 0.039 | 0.037 | 0.66 | Moderate or high yield loss |

3 | $\mathrm{Std}.{Y}_{\mathit{max}}\le -1.5$$\mathrm{Std}.{Y}_{\mathit{min}}\le -1.5$|T_{A} | 0.0006 | 0.0005 | 0.63 | High yield loss |

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**MDPI and ACS Style**

Unnikrishnan, P.; Ponnambalam, K.; Karray, F.
Influence of Regional Temperature Anomalies on Strawberry Yield: A Study Using Multivariate Copula Analysis. *Sustainability* **2024**, *16*, 3523.
https://doi.org/10.3390/su16093523

**AMA Style**

Unnikrishnan P, Ponnambalam K, Karray F.
Influence of Regional Temperature Anomalies on Strawberry Yield: A Study Using Multivariate Copula Analysis. *Sustainability*. 2024; 16(9):3523.
https://doi.org/10.3390/su16093523

**Chicago/Turabian Style**

Unnikrishnan, Poornima, Kumaraswamy Ponnambalam, and Fakhri Karray.
2024. "Influence of Regional Temperature Anomalies on Strawberry Yield: A Study Using Multivariate Copula Analysis" *Sustainability* 16, no. 9: 3523.
https://doi.org/10.3390/su16093523