3.1 Statistical Description
The summary statistics for the variables under this study, including mean, maximum, and minimum temperature, as well as precipitation, are presented in Table 2. The minimum values observed for mean, maximum, and minimum temperature were 16.58°C, 22.76°C, and 10.72°C, respectively, while the maximum values were 30.69°C, 38.58°C, and 27.07°C, respectively. Precipitation ranged from a minimum of 0 mm to a maximum of 714.23 mm, with a mean and standard deviation of 191.36 mm and 175.37 mm, respectively. Notably, precipitation exhibited greater variability (CV = 91.64%) compared to the mean (CV = 14.63%), maximum (CV = 9.31%), and minimum (CV = 22.48%) temperatures. The coefficients of kurtosis and skewness for precipitation (-0.7 and 0.6), mean temperature (-0.82 and − 0.78), maximum temperature (-0.26 and − 0.48), and minimum temperature (-1.13 and − 0.61) indicate deviations from normality, rendering Pearson’s correlation coefficient unsuitable for measuring interdependence between the variables. Despite this deviation from normality, both correlation coefficients Kendall’s Tau (τ) and Pearson rho (ρ) revealed a strong positive and highly significant correlation between precipitation and mean temperature (τ = 0.524, ρ = 0.7), precipitation and maximum temperature (τ = 0.306, ρ = 0.456), and precipitation and minimum temperature (τ = 0.645, ρ = 0.795). However, it failed to capture the nonlinear correlation present among these variables. This nonlinear dependence can be effectively modeled using copulas, as suggested by previous studies (Eling & Toplek, 2009; Hussain et al., 2022; Junker et al., 2006).
Table 2
Summary statistics of Mean, Maximum and Minimum Temperature and Precipitation
Measures
|
Precipitation (mm)
|
Mean Temperature (0C)
|
Maximum Temperature (0C)
|
Minimum Temperature (0C)
|
Minimum
|
0
|
16.58
|
22.76
|
10.72
|
Maximum
|
714.23
|
30.69
|
38.58
|
27.07
|
Mean
|
191.36
|
25.09
|
29.74
|
21.22
|
Standard deviation
|
175.37
|
3.67
|
2.77
|
4.77
|
Coefficient of variation (CV%)
|
91.64
|
14.63
|
9.31
|
22.48
|
Kurtosis
|
-0.7
|
-0.82
|
-0.26
|
-1.13
|
Skewness
|
0.6
|
-0.78
|
-0.48
|
-0.61
|
Correlation with precipitation (Kendall Tau, τ)
|
-
|
0.524*
|
0.306*
|
0.645*
|
Correlation with precipitation (Pearson Rho, ρ)
|
-
|
0.700*
|
0.456*
|
0.795*
|
*Significant at the 1% level (2-tailed) |
Violin plots shown in Figs. 1a–d illustrate the month-wise variability in precipitation, mean temperature, maximum temperature, and minimum temperature, respectively. These plots indicate greater variability in precipitation during January (CV = 102.36%), February (CV = 81.83%), March (CV = 76.46%), November (CV = 111.02%), and December (CV = 233.28%), whereas less variability was observed during May (CV = 35.65%), June (CV = 25.59%), July (CV = 22.49%), August (CV = 22.26%), and September (CV = 23.88%) (Table 3). Similarly, variability in mean, maximum, and minimum temperature exhibited distinct patterns across different months. Overall, the results emphasize the importance of considering both linear and nonlinear correlations in understanding the complex interplay between temperature and precipitation variables, particularly in the context of climate change and its implications for various sectors.
The scatter plot (Fig. 2) revealed a highly dispersed distribution of points, indicating significant variability within the dataset. A few observations appear to align closely with the fitted regression line, while the majority deviate below and above this line. Moreover, the r2-values of regression were found to be 0.49, 0.208, and 0.612 for “precipitation-mean temperature," “precipitation-maximum temperature,” and “precipitation-minimum temperature,” respectively. Consequently, the regression line exhibits a poor fit with the observed data. Notably, large values of mean and minimum temperature correspond to large values of precipitation, suggesting a strong correlation, particularly in extreme events.
However, it is evident from the scatter plot that a linear fit is inadequate, as the points fail to exhibit a regular pattern between the variables. This nonlinear dependence emphasized the limitations of traditional approaches, such as correlation coefficients, in capturing complex dependence structures. Thus, the adoption of more robust tools, such as copulas, becomes imperative to effectively model the complex relationships between temperature and precipitation variables. Furthermore, regression analysis revealed that for every 1 ºC increase in mean, maximum, and minimum temperatures, there was an associated increase in precipitation of 33.48 mm, 28.91 mm, and 29.25 mm, respectively. Hussain et al. (2022) also conducted a linear regression fit, which revealed that for every 1 ºC increase in mean temperature, monthly precipitation increased by 0.694 mm. The study found a comparatively lower increase in precipitation because Pakistan has a much drier climate than Bangladesh.
Table 3
Summary Statistics of month-wise Precipitation, Mean Temperature, Maximum Temperature and Minimum Temperature
Precipitation
|
Measures
|
Jan
|
Feb
|
Mar
|
Apr
|
May
|
June
|
July
|
Aug
|
Sep
|
Oct
|
Nov
|
Dec
|
Minimum
|
0.12
|
0.08
|
5.95
|
17.33
|
83.91
|
221.14
|
242.89
|
224.27
|
164.43
|
8.56
|
0.21
|
0
|
Maximum
|
49.59
|
71.19
|
220.1
|
378.22
|
572.56
|
714.23
|
685.61
|
573.35
|
506.91
|
474.57
|
168.77
|
237.19
|
Mean
|
11.03
|
21.96
|
60.4
|
164.15
|
307.46
|
443.17
|
429.8
|
353.74
|
292.43
|
163.18
|
35.92
|
13.1
|
Standard Deviation
|
11.29
|
17.97
|
46.18
|
81.74
|
109.6
|
113.42
|
96.67
|
78.74
|
69.84
|
76.27
|
39.88
|
30.56
|
Coefficient of variation (CV%)
|
102.36
|
81.83
|
76.46
|
49.8
|
35.65
|
25.59
|
22.49
|
22.26
|
23.88
|
46.74
|
111.02
|
233.28
|
Mean Temperature
|
Measures
|
Jan
|
Feb
|
Mar
|
Apr
|
May
|
June
|
July
|
Aug
|
Sep
|
Oct
|
Nov
|
Dec
|
Minimum
|
16.58
|
18.99
|
22.89
|
24.8
|
26.04
|
26.83
|
26.6
|
26.41
|
26.29
|
24.91
|
21.54
|
17.34
|
Maximum
|
19.76
|
23.78
|
27.52
|
30.41
|
30.69
|
29.55
|
29.04
|
29.43
|
29.01
|
28.44
|
24.7
|
21.24
|
Mean
|
18.16
|
21.14
|
25.36
|
27.74
|
28.25
|
28.2
|
27.89
|
27.98
|
27.7
|
26.43
|
22.98
|
19.26
|
Standard Deviation
|
0.76
|
0.98
|
0.97
|
1.16
|
0.92
|
0.64
|
0.57
|
0.6
|
0.58
|
0.63
|
0.7
|
0.72
|
Coefficient of variation (CV%)
|
4.19
|
4.64
|
3.82
|
4.18
|
3.26
|
2.27
|
2.04
|
2.14
|
2.09
|
2.38
|
3.05
|
3.74
|
Maximum Temperature
|
Measures
|
Jan
|
Feb
|
Mar
|
Apr
|
May
|
June
|
July
|
Aug
|
Sep
|
Oct
|
Nov
|
Dec
|
Minimum
|
22.76
|
24.97
|
28.04
|
28.84
|
29.62
|
29.22
|
29.22
|
29.05
|
29.14
|
27.98
|
26.28
|
23.04
|
Maximum
|
26.7
|
30.95
|
35.3
|
38.58
|
37.4
|
33.03
|
31.98
|
32.62
|
32.55
|
32.16
|
29.59
|
27.1
|
Mean
|
24.66
|
27.63
|
31.6
|
33.04
|
32.39
|
31.33
|
30.69
|
30.87
|
30.82
|
30.23
|
28.22
|
25.45
|
Standard Deviation
|
0.92
|
1.3
|
1.45
|
1.74
|
1.35
|
0.85
|
0.63
|
0.69
|
0.67
|
0.79
|
0.84
|
0.77
|
Coefficient of variation (CV%)
|
3.73
|
4.71
|
4.59
|
5.27
|
4.17
|
2.71
|
2.05
|
2.24
|
2.17
|
2.61
|
2.98
|
3.03
|
Minimum Temperature
|
Measures
|
Jan
|
Feb
|
Mar
|
Apr
|
May
|
June
|
July
|
Aug
|
Sep
|
Oct
|
Nov
|
Dec
|
Minimum
|
10.72
|
13.32
|
17.86
|
20.93
|
23.4
|
24.8
|
24.88
|
24.67
|
24.3
|
21.51
|
16.18
|
11.78
|
Maximum
|
14.48
|
18.53
|
22.09
|
25.39
|
26.58
|
26.8
|
26.91
|
27.07
|
26.73
|
25.69
|
20.48
|
16.67
|
Mean
|
12.75
|
15.34
|
19.78
|
23.28
|
24.79
|
25.8
|
25.81
|
25.79
|
25.32
|
23.2
|
18.49
|
14.23
|
Standard Deviation
|
0.79
|
1.02
|
0.97
|
0.93
|
0.69
|
0.49
|
0.51
|
0.54
|
0.55
|
0.75
|
0.93
|
0.92
|
Coefficient of variation (CV%)
|
6.2
|
6.65
|
4.9
|
3.99
|
2.78
|
1.9
|
1.98
|
2.09
|
2.17
|
3.23
|
5.03
|
6.47
|
3.2 Copula Results
The best marginal fit among the many distribution models was analyzed and determined by a histogram and a Quantile-Quantile (Q-Q) plot (Fig. 3), including goodness-of-fit metrics like loglikelihood (LL), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) (Table 4). After a thorough analysis, it was found that different climate variables led to different distribution models being selected. The generalized Pareto distribution was found to provide the best marginal match for precipitation data, with the greatest loglikelihood value (-5316.3) and minimum AIC (10538.6) and BIC (10652.8) values (Table 4). This result was supported by Martins et al. (2020) and Singirankabo & Iyamuremye (2022), who also found generalized Pareto to model extreme rainfall events. On the other hand, the generalized extreme value distribution performed better as the best marginal fit for mean temperature data (Fig. 3b), as supported by higher loglikelihood (-2187.8) and lower AIC (4381.5) and BIC values (4395.8) (Table 4). In comparison to the extreme value and normal distributions, the Weibull distribution provided a more accurate representation of the maximum temperature data, as evidenced by the higher loglikelihood (-2058.9) and lower AIC (4121.9) and BIC values (4131.4) (Table 4). Finally, the Generalized Pareto distribution showed the best fit for minimum temperature data. These marginal distributions were also reported by other studies (Hussain et al., 2022; Laux et al., 2011; Schölzel & Friederichs, 2008).
Table 4
Selection of the marginal distribution
Margincal distribution for precipitation
|
Marginal distribution for mean temperature
|
Distribution
|
Parameters
|
LL
|
AIC
|
BIC
|
Distribution
|
Parameters
|
LL
|
AIC
|
BIC
|
GP
|
k = -0.278
𝜎̂ = 249.051
θ = 0
|
-5316.3
|
10638.6
|
10652.8
|
GEV
|
k = -0.616
𝜎̂ = 3.885
𝜇̂ = 24.394
|
-2187.8
|
4381.5
|
4395.8
|
Exponential
|
𝜇̂ = 191.360
|
-5328.5
|
10659.1
|
10663.8
|
EV
|
𝜇̂ = 26.749
𝜎̂ = 2.615
|
-2211.8
|
4427.5
|
4437.0
|
Normal
|
𝜇̂ = 191.360
𝜎̂ = 175.476
|
-5611.1
|
11226.3
|
11235.8
|
Weibull
|
A = 26.592
B = 9.307
|
-2241.3
|
4486.6
|
4496.1
|
Marginal distribution for maximum temperature
|
Marginal distribution for minimum temperature
|
Distribution
|
Parameters
|
LL
|
AIC
|
BIC
|
Distribution
|
Parameters
|
LL
|
AIC
|
BIC
|
Weibull
|
A = 30.957
B = 12.546
|
-2058.9
|
4121.9
|
4131.4
|
GP
|
k = -1.416
𝜎̂ = 23.1447
θ = 10.72
|
-2303.0
|
4612.1
|
4626.3
|
EV
|
𝜇̂ = 31.055
𝜎̂ = 2.479
|
-2075.9
|
4155.8
|
4165.3
|
GEV
|
k = -0.848
𝜎̂ = 5.256
𝜇̂ = 20.871
|
-2341.0
|
4688.0
|
4702.2
|
Normal
|
𝜇̂ = 29.745
𝜎̂ = 2.770
|
-2076.5
|
4157.0
|
4166.5
|
EV
|
𝜇̂ = 23.422
𝜎̂ = 3.528
|
-2458.7
|
4921.4
|
4930.9
|
After finalizing the marginal distributions, a subsequent step is required for fitting different copulas to the observed data. The initial plot in this context combined two previously presented plots: the scatter plot of the dataset and the kernel density estimation (KDE) of the marginal distributions. As depicted in Fig. 4, the scatter plot shows the joint behavior of temperature and precipitation variables, with temperature placed on the horizontal axis and precipitation on the vertical axis. Notably, the scatter plot illustrated a nonlinear dependence between temperature and precipitation, indicative of their interrelated dynamics. Moreover, the KDE of the marginal distributions revealed non-normal distributions. Specifically, the marginal distributions of mean, maximum, and minimum temperature were fitted with Generalized Extreme Value (GEV), Weibull, and Generalized Pareto (GP) distributions, respectively. Conversely, the Generalized Pareto (GP) distribution was most suitable for precipitation. These observations emphasized the diverse distributional characteristics exhibited by the climatic variables under investigation, highlighting the necessity of employing copula-based methodologies to capture their complex interdependencies accurately.
The initial step in copula analysis was to use the probability integral transformation to convert the marginal distribution of each variable into a standard uniform marginal distribution. The transformed marginal of precipitation with transformed marginals of mean, maximum, and minimum temperatures were illustrated using the scatter plot and kernel density estimate (KDE), as shown in Figs. 5a-c, respectively. Both the transformed variables on the horizontal and vertical axes, respectively, were found to belong within the interval [0, 1]. This finding indicates that the three temperature variables and precipitation were transformed into a standard uniform distribution, a crucial step in copula analysis for exploring the interdependence between variables.
After the initial step, bivariate frequency histograms (Figs. 6a-c) were constructed to gain a deeper understanding of the natural dependence structure between variables. As customary, the transformed marginals (u and v) were depicted on the x and y-axes, respectively, while the z-axis represented their bivariate frequency histogram. Notably, two connecting lines were delineated within the diagram: the main diagonal, corresponding to u = v, and the vice-diagonal, perpendicular to the main diagonal, representing the line when u + v = 1. Analysis of the bivariate frequency histogram revealed distinct distributions for minimum temperature and precipitation, which were densely concentrated along the main diagonal and less so along the vice-diagonal (Fig. 6c). Conversely, the individual bivariate frequency histogram of mean and maximum temperature with precipitation exhibited lower density along the main diagonal compared to the bivariate distribution of minimum temperature with precipitation (Fig. 6a-b). Nevertheless, the density remained higher than that along the vice-diagonal, suggesting asymmetrical frequencies in the upper and lower tails. Overall, variations in correlation were observed across different sections of the histogram, indicating a heterogeneous dependence structure between the variables (Hussain et al., 2022).
The above analysis proved that the dependence structure was not symmetrical, which led to the fitting of both elliptical (Gaussian) copulas and Archimedean (Gumbel, Clayton, and Frank) copulas. The parameters derived from this analysis, along with the in-sample evaluation criteria for the fitted copulas, are summarized in Table 5. Analyzing the table reveals that the Clayton copula emerged as the preferred choice for pairs involving "precipitation-mean temperature," "precipitation-maximum temperature," and "precipitation-minimum temperature." This selection was consistent across all evaluation criteria, including LL, AIC, BIC, and RMSE, underscoring the capability of the Clayton copula to capture the interdependence between temperature and precipitation variables in the context of Bangladesh. This result was corroborated by Pandey et al. (2018), who found the Clayton copula to be the best copula for precipitation and maximum temperature in Agartala (humid) regions.
The dependence parameters (θ) associated with the Clayton copula were determined to be 1.7086, 1.0440, and 3.0970 for the "precipitation-mean temperature copula," "precipitation-maximum temperature copula," and "precipitation-minimum temperature copula," respectively (Table 5). Notably, these parameters deviated significantly from the independence coefficient of this copula, which is conventionally set at 0 (Boateng et al., 2022). The positive dependence parameters seen in each case indicate the presence of significant interdependence between the respective pairs of variables.
Table 5
Goodness of fit of bivariate copulae
Precipitation- Mean temperature copula
|
Precipitation- Maximum temperature copula
|
Copula
|
θ
|
LL
|
AIC
|
BIC
|
RMSE
|
Copula
|
θ
|
LL
|
AIC
|
BIC
|
RMSE
|
Gaussian
|
0.578
|
2033.3
|
-4064.6
|
-4059.8
|
2.684
|
Gaussian
|
0.406
|
2183.9
|
-4365.7
|
-4361.0
|
2.249
|
Clayton
|
1.709
|
2154.4
|
-4306.8
|
-4302.0
|
2.328
|
Clayton
|
1.044
|
2276.3
|
-4550.5
|
-4545.8
|
2.018
|
Frank
|
4.022
|
2034.2
|
-4066.4
|
-4061.7
|
2.681
|
Frank
|
2.413
|
2179.5
|
-4357.1
|
-4352.3
|
2.260
|
Gumbel
|
1.501
|
1975.9
|
-3949.9
|
-3945.1
|
2.870
|
Gumbel
|
1.248
|
2142.6
|
-4283.2
|
-4278.4
|
2.360
|
Precipitation- Minimum temperature copula
|
|
Copula
|
θ
|
LL
|
AIC
|
BIC
|
RMSE
|
|
|
|
|
Gaussian
|
0.843
|
2145.2
|
-4288.4
|
-4283.6
|
2.363
|
|
|
|
|
Clayton
|
3.097
|
2277.4
|
-4552.8
|
-4548.1
|
2.015
|
|
|
|
|
Frank
|
9.024
|
2145.6
|
-4289.3
|
-4284.5
|
2.352
|
|
|
|
|
Gumbel
|
4.076
|
2096.5
|
-4190.9
|
-4186.2
|
2.492
|
|
|
|
|
Following the fitting of various copula models, the correlation values derived from the dependence parameters of the best-fitted copulas consistently exhibited positive values for Bangladesh across all scenarios. Notably, the correlation coefficient associated with the "precipitation-minimum temperature copula" attained the highest value (0.6076) among the three cases, indicating the strongest interdependence between the respective variables, followed by correlation coefficient (0.4607) of “precipitation-mean temperature copula”. Conversely, the correlation coefficient of the "precipitation-maximum temperature copula" showed the lowest value (0.3430), suggesting the weakest degree of interdependence within this context. These findings emphasized the varying strengths of interdependence observed across different pairs of variables, with precipitation and minimum temperature displaying the most pronounced correlation compared to other combinations.
Figs. 7-9 presented contour plots depicting the cumulative distribution functions (CDFs) of both empirical and four theoretical copula models for the "precipitation-mean temperature," "precipitation-maximum temperature," and "precipitation-minimum temperature" pairs, respectively. Within the figures, the black solid lines represented the empirical copula, while the color lines signified the fitted (theoretical) copulas, and the blue dots represented the observed data. Notably, the color of each line corresponded to the probability of values for temperature and precipitation, with the associated probabilities delineated in the color bar of the figures. These visual representations aided the comparison between empirical and theoretical copulas, providing insights into the probability distributions and interdependencies of temperature and precipitation variables within the context of the study.