# Designing an Interactive Visual Analytics System for
## 5 Interactive Precipitation Data Analysis System
### Station-Specific Analysis Interface
#### 5.1.2 Precipitation Anomaly Detection  
A precipitation anomaly visualization (Figure 4) supports detecting precipitation anomalies by applying anomaly detection using three different distributions--normal, gamma, and Weibull. Red-colored glyphs indicate identified anomalies. It displays detailed measurements when highlighted. The visualization allows switching between distributions to detect potential anomalies under different statistical assumptions. The normal distribution uses z-scores to detect anomalies, with mean precipitation volume serving as the central reference point and standard deviation measuring variation from this average. The z-scores (\(z=\frac{x-\mu}{\sigma}\)) are calculated for each precipitation event to identify anomalous precipitation events. Possible anomalies are determined by evaluating how much they deviate from the mean. Specifically, it evaluates each precipitation event's z-score and identifies it as an \"anomalous\" event if it exceeds a predetermined threshold (typically 2 or 3 standard deviations). The gamma distribution is commonly used in rainfall data analysis because precipitation is nonnegative and often has skewed patterns (many small and fewer large events). It has been found that daily precipitation follows the gamma distribution [48]. It uses the gamma cumulative distribution function (CDF), \(F(x;k,\theta)=\frac{\gamma(k,\gamma/\theta)}{\Gamma(k)}\), to compute  
Figure 3: Visualizations of precipitation data analysis with (**A**) intensity–duration frequency (IDF) curve showing the relationship between rainfall intensity, duration, and return periods; and (**B**) hourly precipitation intensity distribution with exceedance probability analysis using Weibull distribution. the probability of a given rainfall value as \(P(X\geq x)=1-F(x)\), where \(k\) and \(\theta\) indicate shape and scale parameters, respectively. They are used to control the shape and scale of the distribution. The shape parameter is a useful metric and is broadly used to track changes in extremes in precipitation [48; 49]. It is computed as \(k=\frac{\mathbb{E}[X]^{2}}{\mathrm{Var}(X)}\), where \(\mathbb{E}[X]\) represents the mean of precipitation \(\mathbb{E}[X]=\frac{1}{n}\sum_{i=1}^{n}x_{i}\) and \(\mathrm{Var}(X)\) denotes the variance of the precipitation \(\mathrm{Var}(X)=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\mathbb{E}[X])^{2}\). Anomalies are determined by identifying values whose probability of occurrence is less than a specified threshold (0.05 or 0.01). Figure 4B presents an example of detecting possible precipitation anomalies (\(p<0.01\)) with the gamma distribution. It has detected numerous precipitation events as possible anomalies since 2013. This might be closely connected to climate change impacts observed in the U.S. [50]. We also found similar trends in precipitation when analyzing multiple stations' precipitation data with our designed system (see Section 6 for details). In the visualization, the Weibull distribution is also supported to detect precipitation anomalies [51]. The gamma distribution is well suited for evaluating long accumulation periods. However, the Weibull distribution is recommended for analyzing precipitation in short accumulation periods [52]. The Weibull distribution uses Weibull CDF, \(F(x;k,\lambda)=1-e^{-(x/\lambda)^{k}}\), where \(k\) and \(\lambda\) indicate shape and scale parameters, respectively. The shape parameter \(k\) has effects similar to those in the gamma distribution. It controls skewness and tail behavior. However, it uses a different scale parameter \(\lambda\) to determine the distribution of precipitation data. For estimating \(k\), the Newton-Raphson iteration method [53] is used. Once \(k\) is estimated, \(\lambda\) is computed using \(\lambda=\left(\frac{1}{n}\sum_{i=1}^{n}x_{i}^{k}\right)^{\frac{1}{k}}\).