Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States
Abstract
:1. Introduction
2. Materials and Methods
2.1. MFDFA Method
- (1)
- Determine the accumulated deviation of the time series signal X (x1, x2… xL) from its mean (so called ‘profile’) as:
- (2)
- Identify different independent sub-series of different lengths (called scale s) both in forward and backward directions.
- (3)
- Fit polynomial to each sub-series and find the difference between original sub-series and its fitted polynomial as:
- (4)
- Determine the fluctuation function (FF) for different combination of scale s and statistical moment order q, based on step 3:
- (5)
- Develop the log-log plot of fluctuation function FFq(s) vs. scale s. The slope of logarithmic plot of FF versus s is called generalized Hurst exponent (GHE) h(q) for different moment order q. The non-linear behavior of the plot between h(q) and q indicates the multifractality of the time series. For monofractal time series h(q) is independent of q. Multifractal behavior (a significant dependence of h(q) on q) will manifest itself only if small and large fluctuations scale differently in the time series. For positive values of q, h(q) describes the scaling behavior of the segments with large fluctuations, whereas for negative values of q, the scaling behavior of the segments with small fluctuations is described. While the precision of h(q) estimation depends mostly on the length of the time series, for large negative values of q (very small fluctuations) the precision of measurement is also starting to play a role. A detailed analysis on the precision of h(q) estimation for the MFDFA method was performed by López and Contreras [71]. The GHE value for q = 2 is considered to be equivalent to the classical Hurst exponent (H) [27,28]. From h(q) other useful exponents like mass exponent τ(q) or singularity exponent f(α) can be derived as follows:
2.2. MFCCA Method
- (1)
- For two time series xi and yi (I = 1, 2,…, N), determine the profiles as:
- (2)
- Divide each profile Px(j) and Py(j) into Ns non-overlapping segments both in progressive and retrograde directions, to avoid any omission of time series data from the head or tail end of the series.
- (3)
- For each 2Ns segments, compute local trend of both “profiles” Px(j) and Py(j) by fitting polynomials pX,υm(j) and pY,υm(j) of appropriate order m. The subtraction of the fitted polynomial from the original segment gives the covariance as given below:
- (4)
- Calculate detrended covariance by summation over all segments:
- (5)
- Check if FqXY(s) behaves as a power-law function of s (the scaling behavior), where s is the segmental sample size:
2.3. Study Area and Data
3. Results and Discussion
3.1. MFDFA of Agro-Meteorological Time Series
Origin of Multifractality
3.2. MFCCA of Meteorological Variables with ET0
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Station | Variable | Maximum | Minimum | Mean | Standard Deviation |
---|---|---|---|---|---|
Dagget | ET0 (mm) | 18 | 0.2 | 6.42 | 3.37 |
T (°C) | 37.2 | −5.3 | 19.72 | 8.47 | |
U (m/s) | 15.973 | 0.174 | 5.04 | 2.22 | |
SR (Wm−2) | 1576.2 | 102 | 993.9 | 359.5 | |
RH (%) | 100 | 7 | 30.20 | 14.29 | |
P (kPa) | 96.7 | 92.7 | 94.66 | 0.48 | |
Bakersfield | ET0 (mm) | 13 | 0.2 | 4.68 | 2.78 |
T (°C) | 37 | −2.8 | 18.58 | 8.04 | |
U (m/s) | 14.564 | 0 | 3.49 | 1.32 | |
SR (Wm−2) | 1521.3 | 129.6 | 895.3 | 400.7 | |
RH (%) | 100 | 12 | 48.10 | 18.56 | |
P (kPa) | 101.7 | 97.7 | 99.83 | 0.50 | |
Santa Maria | ET0 (mm) | 10.6 | 0.3 | 3.21 | 1.22 |
T (°C) | 33.3 | 5.7 | 16.72 | 3.36 | |
U (m/s) | 12.719 | 1.257 | 3.77 | 1.00 | |
SR (Wm−2) | 1492.7 | 49.6 | 850.7 | 327.4 | |
RH (%) | 100 | 7 | 65.22 | 16.07 | |
P (kPa) | 103 | 99.1 | 101.19 | 0.37 | |
Los Angeles | ET0 (mm) | 9 | 0.5 | 3.03 | 1.21 |
T (°C) | 26.7 | 0.5 | 13.35 | 2.80 | |
U (m/s) | 12.53 | 0.453 | 4.07 | 1.31 | |
SR (Wm−2) | 1527.7 | 157.2 | 889.6 | 346.7 | |
RH (%) | 98 | 16 | 65.34 | 13.78 | |
P (kPa) | 102.5 | 98.6 | 100.80 | 0.38 | |
San Diego | ET0 (mm) | 9.4 | 0.5 | 3.31 | 1.18 |
T (°C) | 32.2 | 6.7 | 17.55 | 3.43 | |
U (m/s) | 10.897 | 0.697 | 3.55 | 0.99 | |
SR (Wm−2) | 1488.5 | 156.8 | 864.5 | 314.1 | |
RH (%) | 100 | 10 | 63.99 | 14.02 | |
P (kPa) | 103.1 | 99.5 | 101.46 | 0.35 |
Variable | Multifractal Properties | |||||
---|---|---|---|---|---|---|
H | W | AI | ∆f(α) | ∆h(q) | α0 | |
Dagget | ||||||
ET0 | 0.758 | 0.458 | 0.362 | 0.370 | 0.240 | 0.804 |
T | 0.851 | 0.270 | 0.596 | 0.344 | 0.133 | 0.867 |
U | 0.616 | 0.383 | 0.454 | 0.371 | 0.212 | 0.655 |
SR | 0.687 | 0.551 | 0.260 | 0.232 | 0.313 | 0.747 |
RH | 0.814 | 0.472 | 0.308 | 0.271 | 0.264 | 0.864 |
P | 0.578 | 0.348 | 0.320 | 0.223 | 0.199 | 0.619 |
Bakersfield | ||||||
ET0 | 0.810 | 0.279 | 0.529 | 0.307 | 0.147 | 0.831 |
T | 0.748 | 0.325 | 0.403 | 0.275 | 0.173 | 0.778 |
U | 0.710 | 0.246 | 0.300 | 0.109 | 0.120 | 0.728 |
SR | 0.701 | 0.523 | 0.377 | 0.385 | 0.293 | 0.755 |
RH | 0.820 | 0.269 | 0.527 | 0.305 | 0.138 | 0.840 |
P | 0.634 | 0.309 | 0.210 | 0.132 | 0.172 | 0.672 |
Santa Maria | ||||||
ET0 | 0.742 | 0.207 | 0.411 | 0.176 | 0.105 | 0.759 |
T | 0.843 | 0.381 | 0.390 | 0.292 | 0.218 | 0.885 |
U | 0.626 | 0.464 | 0.522 | 0.460 | 0.260 | 0.665 |
SR | 0.652 | 0.421 | 0.405 | 0.350 | 0.224 | 0.690 |
RH | 0.706 | 0.477 | 0.373 | 0.336 | 0.286 | 0.765 |
P | 0.648 | 0.413 | 0.286 | 0.268 | 0.220 | 0.693 |
Los Angeles | ||||||
ET0 | 0.685 | 0.380 | 0.245 | 0.196 | 0.198 | 0.724 |
T | 0.812 | 0.501 | 0.185 | 0.162 | 0.289 | 0.876 |
U | 0.735 | 0.239 | 0.381 | 0.203 | 0.148 | 0.772 |
SR | 0.644 | 0.417 | 0.528 | 0.471 | 0.229 | 0.682 |
RH | 0.674 | 0.611 | 0.378 | 0.419 | 0.389 | 0.760 |
P | 0.636 | 0.337 | 0.209 | 0.138 | 0.182 | 0.673 |
San Diego | ||||||
ET0 | 0.678 | 0.336 | 0.128 | 0.070 | 0.172 | 0.712 |
T | 0.826 | 0.426 | 0.121 | 0.078 | 0.247 | 0.883 |
U | 0.685 | 0.205 | 0.111 | 0.047 | 0.109 | 0.709 |
SR | 0.616 | 0.441 | 0.510 | 0.498 | 0.235 | 0.655 |
RH | 0.684 | 0.497 | 0.247 | 0.212 | 0.321 | 0.763 |
P | 0.642 | 0.338 | 0.213 | 0.145 | 0.179 | 0.678 |
Link | MFCCA Parameters | Pearson Correlation Coefficient | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Scaling Exponent | Spectral Width | Asymmetry Index | Cross Correlation Coefficient | |||||||||
λXX | λYY | λXY | WXX | WYY | WXY | AIXX | AIYY | AIXY | ρXY(90) | ρXY(365) | ρ | |
Dagget | ||||||||||||
T-ET0 | 0.896 | 0.778 | 0.837 | 0.300 | 0.426 | 0.218 | −0.522 | −0.325 | −0.473 | 0.575 | 0.610 | 0.560 |
U-ET0 | 0.619 | 0.778 | 0.699 | 0.417 | 0.426 | 0.276 | −0.452 | −0.325 | −0.454 | 0.256 | 0.416 | 0.465 |
SR-ET0 | 0.708 | 0.778 | 0.743 | 0.548 | 0.426 | 0.365 | −0.348 | −0.325 | 0.071 | 0.615 | 0.610 | 0.447 |
RH-ET0 | 0.868 | 0.778 | 0.823 | 0.482 | 0.426 | 0.273 | −0.341 | −0.325 | −0.018 | −0.652 | −0.633 | −0.508 |
P-ET0 | 0.619 | 0.778 | 0.699 | 0.413 | 0.426 | 0.139 | −0.369 | −0.325 | −0.237 | 0.033 | 0.019 | −0.169 |
Bakersfield | ||||||||||||
T-ET0 | 0.824 | 0.804 | 0.814 | 0.374 | 0.440 | 0.276 | −0.379 | −0.468 | −0.138 | 0.707 | 0.585 | 0.617 |
U-ET0 | 0.690 | 0.805 | 0.748 | 0.369 | 0.515 | 0.351 | −0.786 | −0.468 | 0.068 | 0.136 | 0.310 | 0.402 |
SR-ET0 | 0.719 | 0.804 | 0.762 | 0.491 | 0.440 | 0.317 | −0.396 | −0.468 | −0.019 | 0.668 | 0.684 | 0.571 |
RH-ET0 | 0.855 | 0.804 | 0.830 | 0.309 | 0.440 | 0.210 | −0.445 | −0.468 | −0.287 | −0.784 | −0.706 | −0.700 |
P-ET0 | 0.675 | 0.804 | 0.739 | 0.345 | 0.440 | 0.075 | −0.220 | −0.468 | −0.745 | −0.145 | −0.113 | −0.118 |
Santa Maria | ||||||||||||
T-ET0 | 0.881 | 0.748 | 0.815 | 0.351 | 0.281 | 0.249 | −0.392 | −0.463 | −0.193 | 0.462 | 0.452 | 0.505 |
U-ET0 | 0.630 | 0.715 | 0.672 | 0.483 | 0.375 | 0.288 | −0.450 | −0.220 | −0.325 | 0.084 | 0.163 | 0.159 |
SR-ET0 | 0.672 | 0.748 | 0.710 | 0.469 | 0.281 | 0.272 | −0.432 | −0.463 | −0.354 | 0.742 | 0.627 | 0.679 |
RH-ET0 | 0.739 | 0.748 | 0.743 | 0.503 | 0.281 | 0.282 | −0.381 | −0.463 | −0.432 | −0.811 | −0.822 | −0.791 |
P-ET0 | 0.679 | 0.748 | 0.713 | 0.405 | 0.281 | 0.154 | −0.180 | −0.463 | −0.315 | 0.087 | 0.088 | 0.081 |
Los Angeles | ||||||||||||
T-ET0 | 0.853 | 0.708 | 0.781 | 0.514 | 0.425 | 0.400 | −0.116 | −0.177 | 0.102 | 0.428 | 0.454 | 0.522 |
U-ET0 | 0.708 | 0.708 | 0.708 | 0.326 | 0.425 | 0.193 | −0.158 | −0.177 | −0.219 | 0.057 | −0.011 | 0.172 |
SR-ET0 | 0.676 | 0.708 | 0.692 | 0.400 | 0.425 | 0.266 | −0.566 | −0.177 | −0.328 | 0.764 | 0.716 | 0.681 |
RH-ET0 | 0.712 | 0.708 | 0.710 | 0.604 | 0.425 | 0.456 | −0.286 | −0.177 | −0.358 | −0.863 | −0.853 | −0.848 |
P-ET0 | 0.659 | 0.708 | 0.684 | 0.407 | 0.425 | 0.255 | −0.176 | −0.177 | −0.233 | 0.031 | −0.003 | 0.060 |
San Diego | ||||||||||||
T-ET0 | 0.857 | 0.711 | 0.784 | 0.462 | 0.344 | 0.377 | −0.049 | 0.003 | 0.289 | 0.385 | 0.397 | 0.460 |
U-ET0 | 0.681 | 0.711 | 0.696 | 0.268 | 0.344 | 0.135 | −0.121 | 0.003 | −0.147 | −0.110 | −0.013 | −0.009 |
SR-ET0 | 0.662 | 0.711 | 0.686 | 0.437 | 0.344 | 0.263 | −0.564 | 0.003 | −0.326 | 0.807 | 0.745 | 0.738 |
RH-ET0 | 0.722 | 0.711 | 0.717 | 0.626 | 0.344 | 0.335 | −0.337 | 0.003 | −0.132 | −0.798 | −0.768 | −0.768 |
P-ET0 | 0.668 | 0.711 | 0.689 | 0.410 | 0.344 | 0.181 | −0.168 | 0.003 | −0.285 | 0.037 | −0.025 | 0.103 |
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Sankaran, A.; Krzyszczak, J.; Baranowski, P.; Devarajan Sindhu, A.; Kumar, N.P.; Lija Jayaprakash, N.; Thankamani, V.; Ali, M. Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States. Atmosphere 2020, 11, 1116. https://doi.org/10.3390/atmos11101116
Sankaran A, Krzyszczak J, Baranowski P, Devarajan Sindhu A, Kumar NP, Lija Jayaprakash N, Thankamani V, Ali M. Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States. Atmosphere. 2020; 11(10):1116. https://doi.org/10.3390/atmos11101116
Chicago/Turabian StyleSankaran, Adarsh, Jaromir Krzyszczak, Piotr Baranowski, Archana Devarajan Sindhu, Nandhineekrishna Pradeep Kumar, Nityanjali Lija Jayaprakash, Vandana Thankamani, and Mumtaz Ali. 2020. "Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States" Atmosphere 11, no. 10: 1116. https://doi.org/10.3390/atmos11101116