Multivariate Analysis of Evaporation Drivers in Mbeya, Tanzania Using Principal Component Analysis

doi:10.21203/rs.3.rs-5336289/v1

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Multivariate Analysis of Evaporation Drivers in Mbeya, Tanzania Using Principal Component Analysis

https://doi.org/10.21203/rs.3.rs-5336289/v1

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Evaporation is a critical process in the hydrological cycle, contributing to approximately 70% of the water loss from Earth's surface. Understanding the drivers of evaporation, including meteorological factors like temperature, humidity, solar radiation, and wind speed, is essential for water resource management and agriculture. Traditional univariate models often oversimplify these interactions, but Principal Component Analysis (PCA) offers a powerful multivariate approach for analysing complex datasets. This study applies PCA to 10 years of meteorological data from Mbeya, Tanzania, including maximum and minimum temperature, wind speed, and solar radiation, to identify key factors influencing evaporation. The analysis highlights that solar radiation (mean = 17.60, SD = 6.01) and sunshine hours (mean = 6.96, SD = 2.87) are the most significant drivers, with strong positive loadings on Principal Component 1 (0.88 and 0.89, respectively). temperature also plays a crucial role, with maximum temperature (mean = 24.16, SD = 2.07) loading heavily on Principal Component 2 (0.75). Together, the first two components explain 60.32% of the total variance. The results demonstrate that PCA effectively reduces dimensionality, providing clearer insights into the dominant meteorological factors affecting evaporation. This dimensionality reduction not only simplifies complex relationships but also improves model predictions, making PCA a valuable tool in environmental studies.

Evaporation

Principal Component Analysis (PCA)

meteorological factors

multivariate analysis

dimensionality reduction

Evaporation, a fundamental process in the hydrological cycle, is vital for water resource management, agriculture, and ecosystem sustainability, with approximately 70% of water loss from the Earth’s surface attributed to evaporation (Moges et al., 2003). Meteorological factors such as temperature, humidity, solar radiation, wind speed, and atmospheric pressure are the primary drivers of evaporation (Monteith, 1965). Understanding the complex interplay of these variables is increasingly important, especially in the context of climate change and mounting water scarcity (IPCC, 2021). Recent advancements in data collection and analytical methods have enabled researchers to use multivariate analysis techniques, like Principal Component Analysis (PCA), to better identify the key factors influencing evaporation and streamline the interpretation of complex datasets (Xu & Singh, 2005; Tran et al., 2020). Traditional univariate models, which analyse variables in isolation, often fail to account for the interdependence of meteorological factors, leading to oversimplified predictions (Sharma & Panu, 2012). In contrast, multivariate approaches such as PCA offer significant advantages by transforming correlated variables into uncorrelated principal components, each representing a distinct portion of variance within the dataset (Wold et al., 1987; Jolliffe & Cadima, 2016). This dimensionality reduction allows for more accurate and insightful analysis of the drivers of evaporation while preserving the integrity of the original data (Kim et al., 2020), making it a powerful tool in environmental system studies.

The Principal Component Analysis (PCA) has been applied extensively in hydrological and meteorological research to identify key variables influencing evaporation (Wang et al., 2015). PCA has been particularly effective in analysing large, complex datasets where multiple meteorological variables, such as temperature, humidity, wind speed, and solar radiation, interact simultaneously (Xu et al., 2006). Rahman et al. (2022) applied PCA to assess the primary drivers of evaporation in South Asia, finding that solar radiation and air temperature were the most significant contributors to evaporation variability across the region. Similarly, Wang et al. (2018) used PCA to evaluate the relationship between climate variability and evaporation rates in Northern China, demonstrating that temperature and wind speed were the dominant factors in determining seasonal evaporation patterns. Their study highlighted that PCA simplifies the complexity of hydrological datasets, allowing for a more straightforward interpretation of the critical variables influencing evaporation. Jafari and Dinpashoh (2019) applied PCA in conjunction with multiple linear regression to model pan evaporation in Iran. Their results showed that the first three principal components accounted for more than 90% of the variance in the data, leading to significant improvements in model performance with R² values greater than 0.74.

Cao et al. (2011) used PCA alongside a radial basis function (RBF) neural network to predict water surface evaporation. Their study demonstrated a 95.3% prediction accuracy, a marked improvement over traditional models like the backpropagation network. In another study, Tianxiao et al. (2009) used PCA to analyse evaporation factors in northern China, revealing that sunshine was the most significant factor affecting evaporation when temperature was held constant. These studies highlight the effectiveness of PCA in identifying key meteorological drivers of evaporation, enabling more accurate and is the subject of this study for meteorological data observed in Mbeya, Tanzania.

With the fact that PCA is a widely used technique for dimensionality reduction in large datasets, the following steps outline the PCA methodology used and is summarized in Fig. 1:

Data Use

The meteorological data including maximum and minimum temperature, humidity, wind speed and solar radiation recorded over the period of 10 years was used. The dataset was structured as a matrix where each row represents an observation and each column represents a variable.

Data Standardization

The data was to ensure comparability between variables. Each variable is cantered by subtracting the mean and scaled by its standard deviation. For variable xi, the standardized value z_i is calculated as

$$\:{z}_{i}=\frac{{x}_{i}-{\mu\:}_{i}}{{\sigma\:}_{i}}$$

where $\:{\mu\:}_{i}$ is the mean and $\:{\sigma\:}_{i}$ is the standard deviation of $\:{x}_{i}$.

Calculation of the Covariance Matrix Calculation

The covariance matrix C is computed to measure the relationships between pairs of variables. For two variables x_i and xj, the covariance is calculated as

$$\:C\left({x}_{i},{x}_{j}\right)=\frac{1}{n-1}\sum\:_{k=1}^{n}\left({x}_{ki}-{\mu\:}_{i}\right)\left({x}_{kj}-{\mu\:}_{j}\right)$$

where n is the number of observations, and x_ki and x_kj are the values of variables i and j for the k-th observation.

Calculate the Eigenvalue and Eigenvector

Perform eigenvalue decomposition on the covariance matrix C. The eigenvalues λ_i lambda represent the variance explained by each principal component, and the eigenvectors (principal components) v_i represent the direction of maximum variance in the dataset. The relationship is defined as

$$\:C{v}_{i}={\lambda\:}_{i}{v}_{i}$$

Selection of the Principal Component

Select the principal components corresponding to the largest eigenvalues. The variance explained by each component is proportional to its eigenvalue. The proportion of total variance explained by the k-th principal component is given by

$$\:Explained\:Variance\:Ratio=\frac{{\lambda\:}_{k}}{\sum\:_{i=1}^{p}{\lambda\:}_{i}}$$

where p is the total number of variables.

Component Interpretation

The original data is projected onto the principal components by computing the score matrix Z as

$$\:Z=X\bullet\:V$$

where X is the standardized data matrix and V is the matrix of eigenvectors (principal components). The resulting principal component scores can be analysed to understand which variables contribute most to the variability in evaporation.

Results Interpretation and Analysis

The first few principal components were used, which explain the most variance, in regression models to identify the most influential meteorological factors that drive the evaporation process.

3.1. Descriptive statistics

Table 1 provides descriptive statistics and normality tests for key meteorological variables perceived to be divers of evaporation: radiation, wind speed, maximum temperature, sun shine hours, and minimum temperature, showing notable deviations from normality in most cases. Radiation has a mean of 17.60, with moderate variability (standard deviation 6.01), and is left-skewed (-1.48), indicating a concentration of values on the higher end. The kurtosis (1.94) points to a relatively normal distribution, but the Shapiro-Wilk test rejects normality (p < .001) (Hun Myoung Park, 2004). Wind Speed, with a mean of 7.15 and a large standard deviation of 10.20, exhibits significant positive skewness (5.23) and high kurtosis (31.03), suggesting extreme outliers. The Shapiro-Wilk test confirms non-normality (p < .001) (Ajay S. Singh, 2014). Maximum temperature is more stable with a mean of 24.16, minimal variability (standard deviation 2.07), and nearly symmetric distribution (skewness − 0.02, kurtosis 0.60), yet its Shapiro-Wilk test indicates non-normality (p < .001). Sun shine hours show moderate variability (standard deviation 2.87) and slight negative skewness (-0.64), with a flatter-than-normal distribution (kurtosis − 0.51). However, it also fails the Shapiro-Wilk normality test (p < .001). Finally, minimum temperature exhibits substantial variability (standard deviation 6.92) with strong positive skewness (3.97) and very high kurtosis (46.83), confirming the presence of extreme values. The Shapiro-Wilk test similarly confirms non-normality (p < .001) (Dago Dougba Noel, 2021).

These findings reveal that Wind Speed and Minimum Temperature show particularly significant departures from normality, with implications for any statistical analysis requiring normally distributed data.

Table 1

Descriptive statistics of the meteorological variables
	Radiation	Wind Speed	Max. Temp	Min. Temp	Sun Shine Hours
Mean	17.60	7.15	24.16	9.65	6.96
Standard deviation	6.01	10.20	2.07	6.92	2.87
Minimum	0.39	1.40	14.30	0.09	0.00
Maximum	31.40	109.00	36.50	99.00	11.70
Skewness	-1.48	5.23	-0.02	3.97	-0.64
Kurtosis	1.94	31.03	0.60	46.83	-0.51
Shapiro-Wilk W	0.85	0.36	0.99	0.67	0.94
Shapiro-Wilk p	< .001	< .001	< .001	< .001	< .001

3.2. Correlation Matrix of the meteorological variables

The correlation matrix of the minimum temperature, sun shine hours, maximum temperature, wind speed, and radiation is shown in Table 2. The strongest correlation is observed between sun shine hours and radiation (0.66), indicating a significant positive relationship. This means that as the number of sunshine hours increases, radiation levels also rise, which is consistent with established climatological principles where solar exposure is directly tied to radiation levels (Hun Myoung Park, 2004).

Minimum temperature shows moderate negative correlations with sun shine hours (-0.25) and radiation (-0.21), implying that cooler nights might be associated with fewer sunlight hours and lower radiation levels. This could reflect seasonal or diurnal patterns, where longer periods of darkness result in lower minimum temperatures (Ajay S. Singh, 2014). Additionally, minimum temperature shows a weak positive correlation with maximum temperature (0.16), suggesting that warmer daytime temperatures are somewhat linked to warmer nights, although the relationship is not very strong. Maximum temperature exhibits weak correlations with all other variables, with the strongest being its connection to minimum temperature (0.16). The weak correlations with sun shine hours (0.12) and wind speed (0.07) indicate that maximum temperature is not strongly dependent on these factors, which is in line with other studies where temperature variability is often influenced by more complex interactions (Kristoffer Ostrom et al., 2000). Wind speed shows the weakest correlations in the matrix, with negligible relationships to other variables like radiation (0.02) and sun shine hours (0.03). This suggests that wind operates largely independently of these other environmental factors, which might reflect localized meteorological phenomena that are not directly driven by solar radiation or temperature changes (Dago Dougba Noel, 2021).

The analysis shows that sun shine hours and radiation are the most strongly correlated, while other variables, particularly wind speed, show weak interrelationships. This highlights the complexity of systems, where certain factors may interact closely, while others are controlled by independent forces.

Table 2

Correlation matrix of the meteorological variables
	Min. Temp	Sun Shine Hours	Max. Temp	Wind Speed	Radiation
Min. Temp	1
Sun Shine Hours	-0.25	1
Max. Temp	0.16	0.12	1
Wind Speed	0.07	0.03	0.07	1
Radiation	-0.21	0.66	0.14	0.02	1

3.3. Eigenvalue, Eigenvector and Component Numbers

The analysis performed using PCA is visualized through a scree plot (Fig. 2), sum of squared (SS) loadings loadings of the component (Table 3), and a summary of component statistics (Table 4), providing a comprehensive overview of the data's dimensionality and variance. The scree plot shows the eigenvalues associated with each principal component, where the steep decline from the first to the second component followed by a more gradual tapering indicates that the first component explains the most variance in the dataset. This decline is characteristic of the "elbow method", which suggests retaining only the components that contribute significantly to the total variance. According to the Kaiser Criterion, components with eigenvalues greater than 1.0 should be retained, implying that the first two components are important for explaining the data’s variance (Jolliffe, 2002; Abdi & Williams, 2010).

The component loadings after varimax rotation further clarify the relationship between the variables and the extracted components. Principal Component 1 has high loadings for radiation (0.88) and Sun Shine Hours (0.89), suggesting that it represents factors related to solar exposure and energy. On the other hand, Principal Component 2 has high loadings for minimum temperature (0.62) and maximum temperature (0.75), indicating that this component is more related to temperature variations. The uniqueness values reveal how much variance is not explained by the components, with wind speed having the highest uniqueness (0.75), indicating that it is least explained by these components. The varimax rotation ensures that these loadings are more interpretable by maximizing the variance of squared loadings, making it clearer which variables contribute most to each component (Jolliffe, 2002; Abdi & Williams, 2010).

The summary of component statistics shows that the first two components explain a cumulative 60.32% of the total variance in the dataset, with Principal Component 1 accounting for 36.19% and Principal Component 2 contributing an additional 24.14%. This indicates that these two components together provide a good reduction in dimensionality while retaining a significant portion of the dataset’s variability. Typically, capturing 60–70% of the variance in the first few components is considered sufficient for reducing complexity while maintaining the integrity of the data. The retained components reflect the primary sources of variation in the environmental factors, with Principal Component 1 linked to solar-related factors and Principal Component 2 related to temperature variations (Jolliffe, 2002; Abdi & Williams, 2010).

Table 3

Sum of Square Loadings of the principal component
PC No.	SS Loadings	% of Variance	Cumulative %
1	1.81	36.19	36.19
2	1.21	24.14	60.32

Table 4

Summary of component statistics
	Principal Component		Uniqueness
	1	2	Uniqueness
Min. Temp	-0.46	0.62	0.41
Sun Shine Hours	0.89		0.21
Max. Temp		0.75	0.39
Wind Speed		0.5	0.75
Radiation	0.88		0.22

3.4. Factor score coefficients for the first two PCs

Figure 3 shows the factor score coefficients for first two principal and how different variables minimum temperature, sun shine hours, maximum temperature, wind speed, and radiation contribute to two principal components. Principal Component 1 is characterized by high positive loadings for sun shine hours and radiation (both around 0.9), indicating that these variables strongly influence this component. These high loadings suggest that Principal Component 1 primarily captures variability related to solar exposure and radiation, making it a dimension reflecting solar energy factors. Interestingly, minimum temperature has a negative loading (-0.46) on this component, implying an inverse relationship, where colder minimum temperatures correspond to lower radiation and sunlight hours, a relationship that can be observed in cooler, cloudier seasons.

In contrast, Principal Component 2 is largely driven by temperature-related variables, with high loadings for both maximum temperature (0.75) and minimum temperature (0.62), indicating that this component captures the thermal characteristics of the environment. Wind speed also has a moderate positive loading (0.50) on this component, suggesting that it contributes to the variance explained by temperature factors, possibly due to the role wind plays in heat distribution.

The negative loading of minimum temperature on Principal Component 1 combined with its positive loading on Principal Component 2 suggests that it plays a dual role: negatively associated with solar exposure while positively related to temperature variability. radiation and sun shine hours dominate Principal Component 1, indicating its focus on sunlight, while maximum and minimum temperatures drive Principal Component 2, highlighting its focus on thermal aspects. This division of factors into distinct components aligns with typical findings in environmental datasets where solar and thermal dimensions are key sources of variability (Jolliffe, 2002; Abdi & Williams, 2010).

The PCA results for meteorological data from Mbeya, Tanzania, show that solar radiation and sunshine hours are the dominant factors influencing evaporation. Together with temperature, these variables explain 60.32% of the total variance in the dataset. Solar radiation and sunshine hours both exhibited strong loadings on Principal Component 1 (0.88 and 0.89, respectively), which explained 36.19% of the variance. Maximum temperature was most strongly related to Principal Component 2, with a loading of 0.75, reflecting the role of temperature in driving evaporation. Wind speed, however, had a relatively low impact, with its highest loading being only 0.50 on Principal Component 2, and its uniqueness (0.75) suggests it is influenced by other unmeasured factors. These findings align with previous studies in other regions, reinforcing the importance of solar radiation and temperature in evaporation processes. The PCA approach allowed for better understanding and interpretation of complex datasets, enabling more accurate predictions of evaporation rates. By reducing the dataset to key components, PCA enhances the clarity of meteorological influences and supports improved decision-making in water resource management.

It is recommended that future studies continue using multivariate techniques, such as PCA, alongside other predictive models like neural networks, to refine evaporation forecasts under varying climatic conditions. Additionally, addressing non-normality in key variables, like wind speed (skewness = 5.23) and minimum temperature (skewness = 3.97), using transformations such as the Box-Cox method, would enhance the accuracy of predictions (Kristoffer Ostrom et al., 2000). Studies across diverse climates will further refine the understanding of evaporation drivers and improve resource management strategies.

Author Contribution

Z. K. did all the work it is a single authored manuscrip

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No competing interests reported.

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You are reading this latest preprint version

Multivariate Analysis of Evaporation Drivers in Mbeya, Tanzania Using Principal Component Analysis

Status:

Version 1

Abstract

Figures

1. Introduction

2. Application of the PCA and the data available

3. Results and discussion

3.1. Descriptive statistics

3.2. Correlation Matrix of the meteorological variables

3.3. Eigenvalue, Eigenvector and Component Numbers

3.4. Factor score coefficients for the first two PCs

4. Conclusion

5. Recommendation

Declarations

Author Contribution

References

Additional Declarations

Status:

Version 1