# Designing an Interactive Visual Analytics System for
## 5 Interactive Precipitation Data Analysis System
### Station-Specific Analysis Interface
#### 5.1.4 Precipitation Trend Analysis  
Trend analysis visualization supports identifying precipitation patterns over time using statistical regression methods such as linear, polynomial, and exponential regression. Both long-term and seasonal trend analyses can be performed using these regression methods. Users can switch between different regression methods and long-term and seasonal trend analyses directly within the visualization. Long-term trend analysis evaluates all precipitation data that appeared within the user-selected period. For seasonal trend analysis, only selected seasonal data are utilized to identify seasonal variations in precipitation intensity. Linear regression is a commonly used method for identifying steady, consistent changes in various types of data [56]. It determines the relationship between time and precipitation intensity by fitting a straight line to the precipitation data. The measured R-squared value indicates how well the regression line fits the data. The Mann-Kendall test [57] is also applied to measure whether the determined regression trend is statistically significant or not. Figure 6A shows a linear trend in the precipitation data at the Reagan National Airport from 1950 to 2024. The linear model has a low explanatory power, as indicated by a low value of \(R^{2}=0.025\). Despite this low fit, the Mann-Kendall test indicates that the trend is statistically significant, representing a minor increasing trend (\(p<0.01\), \(\tau=0.043\)). This suggests a gradual yet observable increase in precipitation over seventy years, though the linear model explains only a small portion of the variation in the data.  
Polynomial regression generalizes linear regression by including higher-degree terms. In our visualization, a quadratic model (degree = 2) is used. It is effective for identifying more complex patterns that cannot be captured by linear regression with a simple straight line. Figure 6B shows a deceleration pattern around the 1950s and changes back to an acceleration pattern after 2000. Polynomial regression is useful for analyzing seasonal  
Figure 6: Visualizations of trend analysis with (**A**) linear regression, (**B**) polynomial regression, and (**C**) exponential regression using all available precipitation data at Reagan National Airport. (**D**) A trend analysis using only summer precipitation data with polynomial regression. Blue-colored glyphs represent individual precipitation events, while red-colored lines indicate the determined trend line for each regression method. The R-squared value of each regression method and statistical significance from the Mann–Kendall test are presented at the bottom left of each visualization.  
precipitation variations that may show a U-shaped pattern clearly (see Figure 6D). This polynomial regression of summer data shows a statistically significant trend representing a precipitation pattern shift (\(p<0.01\), \(\tau=0.064\)), characterized by higher summer rainfall after 2010. This visualization shows the pattern shift but does not explain its underlying causes. However, from the analysis of the pattern with the annual maximum precipitation visualization (Figure 5C), we found that the 1950s were characterized by prolonged periods of low-intensity rainfall, in contrast to the recent trend of having high-intensity burst rainfall events in that region [58]. Exponential regression is useful when data change exponentially over time. Thus, it is valuable for analyzing extreme precipitation events, where the rate of change increases or decreases exponentially. Exponential regression is often used in climate change studies where precipitation patterns might show accelerating changes rather than linear trends [59; 60]. Figure 6C shows a similar accelerated pattern compared to the linear regression result. It also represents a very low R-squared value (\(R^{2}=0.012\)) because precipitation data typically do not show continuous exponential growth.