# Estimating the Statistical Significance of Cross–Correlations between Hydroclimatic Processes in the Presence of Long–Range Dependence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{o})/not assumed (alternative hypothesis, H

_{1}) statistically significant, based on whether it is estimated within/outside of the confidence limits of the generalized Gaussian distribution.

^{b(H,q)}

_{1}H

^{2}+ p

_{2}H + p

_{3}

_{1}H

^{2}+ p

_{2}H + p

_{3}

_{1}, p

_{2}, p

_{3}are coefficients that can be selected from Table 1. It is noted that R

^{2}> 0.99 in all expressions. An important remark is that the above expressions correspond to Hurst values between 0.5 and 0.9, while values outside these limits could lead to erroneous extrapolations.

## 3. Applications

#### 3.1. Applications to Global-Scale Temperature, Wind Speed and Dew Point

#### 3.2. Application to Global-Scale Precipitation

#### 3.3. Application of the Statistical Tests

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A comparison of the probability density distribution functions of zero-lag cross–correlations estimated from 10,000 normally distributed series of n = 20 and 100 length.

**Figure 2.**Fitting of the generalized Gaussian (Gen Gaussian) probability distribution with the empirical one of the estimations of zero-lag cross–correlations of series with H = 0.9 (shown in Figure 2).

**Figure 3.**A comparison of the empirical probability density functions of cross–correlations from the LRD-generated series (through the SMA algorithm), with H = 0.5 and 0.9, and n = 20 (

**a**) and 100 (

**b**).

**Figure 5.**Quantile (c) of the estimator of the linear cross–correlation coefficient for q = 99% confidence level.

**Figure 6.**Distances (km) (

**a**) and differences in elevation (m) (

**b**) of stations measuring temperature, wind speed and dew point from their corresponding station measuring precipitation, for the 1058 out of 2090 stations where these variables are measured at different locations.

**Figure 7.**A comparison of quantiles (c) of the estimator of the linear cross–correlation coefficient for q = 99% confidence level for cross–correlation between processes with Hurst parameters H = 0.6 and H = 0.8 with the ones obtained from processes sharing the same H, depending on the sample’s length.

**Figure 8.**Zero-lag cross–correlations between mean temperature, wind speed, and dew point (annual scale, a sample of 20 stations).

**Figure 9.**Cross–correlations between global-scale temperature and wind-speed records of annual resolution. Results are color coded from red (strong positive) to blue (strong negative) cross–correlations.

**Figure 10.**Cross–correlations between global-scale temperature and dew-point records of annual resolution. Results are color coded from red (strong positive) to blue (strong negative) cross–correlations.

**Figure 11.**Cross–correlations between global-scale wind-speed and dew-point records of annual resolution. Results are color coded from red (strong positive) to blue (strong negative) cross–correlations.

**Figure 12.**Cross–correlations between global-scale precipitation and temperature records of annual resolution. Results are color coded from red (strong positive) to blue (strong negative) cross–correlations.

Confidence Interval | 70% | 80% | 95% | 99% | ||||
---|---|---|---|---|---|---|---|---|

H | a | b | a | b | a | b | a | b |

0.50 | 1.281 | −0.548 | 1.526 | −0.539 | 1.829 | −0.523 | 2.276 | −0.471 |

0.55 | 1.281 | −0.547 | 1.521 | −0.537 | 1.823 | −0.522 | 2.270 | −0.469 |

0.60 | 1.264 | −0.539 | 1.503 | −0.530 | 1.803 | −0.514 | 2.229 | −0.461 |

0.65 | 1.244 | −0.528 | 1.479 | −0.518 | 1.770 | −0.502 | 2.184 | −0.449 |

0.70 | 1.223 | −0.512 | 1.443 | −0.501 | 1.720 | −0.484 | 2.105 | −0.429 |

0.75 | 1.172 | −0.485 | 1.385 | −0.475 | 1.646 | −0.458 | 1.983 | −0.400 |

0.80 | 1.116 | −0.453 | 1.317 | −0.443 | 1.551 | −0.424 | 1.847 | −0.364 |

0.85 | 1.031 | −0.410 | 1.212 | −0.399 | 1.427 | −0.380 | 1.687 | −0.321 |

0.90 | 1.007 | −0.380 | 1.173 | −0.367 | 1.356 | −0.344 | 1.549 | −0.278 |

0.95 | 0.933 | −0.333 | 1.080 | −0.319 | 1.244 | −0.296 | 1.404 | −0.230 |

Model Parameters | ||||||||

p_{1} | −1.660 | 1.119 | −2.101 | 1.148 | −2.825 | 1.204 | −4.009 | 1.273 |

p_{2} | 1.601 | −1.138 | 2.023 | −1.169 | 2.746 | −1.234 | 3.785 | −1.301 |

p_{3} | 0.901 | −0.259 | 1.045 | −0.242 | 1.169 | −0.208 | 1.399 | −0.139 |

**Table 2.**Model parameters a and b for calculating cross–correlations between precipitation and temperature, for various confidence intervals, assuming H = 0.6 for precipitation and H = 0.8 for temperature.

Confidence Interval | 70% | 80% | 95% | 99% | ||||
---|---|---|---|---|---|---|---|---|

a | b | a | b | a | b | a | b | |

Precipitation-Temperature | 0.5797 | −0.4995 | 0.899 | −0.4919 | 1.56 | −0.4671 | 1.902 | −0.4374 |

**Table 3.**A list of stations selected for an example of cross–correlations between hydroclimatic processes and their locations.

Station | Latitude | Longitude | Approximate Location |
---|---|---|---|

1 | 77.00° | 15.50° | Svalbard |

2 | 78.25° | 15.47° | Adventfjorden, Svalbard |

3 | 69.68° | 18.92° | Tromsø, Norway |

4 | 70.25° | 19.50° | Karlsøy Municipality, Norway |

5 | 69.02° | 23.07° | Kautokeino, Norway |

6 | 69.98° | 23.37° | Alta, Norway |

7 | 70.07° | 24.98° | Lakselv, Norway |

8 | 71.02° | 25.98° | Valan, Norway |

9 | 71.03° | 27.83° | Gamvik Municipality, Norway |

10 | 71.10° | 28.22° | Gamvik Municipality, Norway |

11 | 70.87° | 29.03° | Berlevåg, Norway |

12 | 70.07° | 29.85° | Vadsø Municipality, Norway |

13 | 65.20° | 11.00° | Sklinna, Norway |

14 | 67.52° | 12.10° | Røst, Norway |

15 | 65.47° | 12.22° | Toft, Norway |

16 | 66.77° | 12.48° | Myken, Norway |

17 | 65.97° | 12.47° | Alstahaug Municipality, Norway |

18 | 66.37° | 12.62° | Lurøy Municipality, Norway |

19 | 65.78° | 13.22° | Kjærstad, Norway |

20 | 66.37° | 14.30° | Rana Municipality, Norway |

**Table 4.**A comparison between statistical significance methods, displaying the number of stations with a significant cross–correlation and their percentage out of the total number of stations reviewed in this study.

Classic Statistical Test | Stochastic Statistical Test | |||
---|---|---|---|---|

Cross–Correlations | No. of Stations | % of All Stations | No. of Stations | % of All Stations |

Temperature-Wind Speed | 683 | 37.57% | 526 | 28.93% |

Temperature-Dew Point | 1449 | 79.70% | 1362 | 74.92% |

Precipitation-Temperature | 295 | 24.83% | 278 | 23.40% |

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**MDPI and ACS Style**

Koskinas, A.; Zaharopoulou, E.; Pouliasis, G.; Deligiannis, I.; Dimitriadis, P.; Iliopoulou, T.; Mamassis, N.; Koutsoyiannis, D.
Estimating the Statistical Significance of Cross–Correlations between Hydroclimatic Processes in the Presence of Long–Range Dependence. *Earth* **2022**, *3*, 1027-1041.
https://doi.org/10.3390/earth3030059

**AMA Style**

Koskinas A, Zaharopoulou E, Pouliasis G, Deligiannis I, Dimitriadis P, Iliopoulou T, Mamassis N, Koutsoyiannis D.
Estimating the Statistical Significance of Cross–Correlations between Hydroclimatic Processes in the Presence of Long–Range Dependence. *Earth*. 2022; 3(3):1027-1041.
https://doi.org/10.3390/earth3030059

**Chicago/Turabian Style**

Koskinas, Aristotelis, Eleni Zaharopoulou, George Pouliasis, Ilias Deligiannis, Panayiotis Dimitriadis, Theano Iliopoulou, Nikos Mamassis, and Demetris Koutsoyiannis.
2022. "Estimating the Statistical Significance of Cross–Correlations between Hydroclimatic Processes in the Presence of Long–Range Dependence" *Earth* 3, no. 3: 1027-1041.
https://doi.org/10.3390/earth3030059