Maximum daily and hourly precipitation in the Nagano Prefecture, Japan, Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano data from the Automated Meteorological Data Acquisition System (AMeDAS) provided by the Japan Meteorological Agency were used. The average annual precipitation, temperature, wind speed, and sunshine hours in Nagano Prefecture for 1991-2020 were also used. The precipitation, temperature, and wind speed data on October 12, 2019, in Nagano Prefecture were used. The latitude, longitude, and height above sea level at the AMeDAS observation points are listed in Table 1 and are shown in Figure 1 .

Principal component analysis (PCA) attempts to recombine many original indexes with certain correlations into a new set of linearly independent comprehensive indexes to replace the original indexes. We performed PCA to derive climate indices that describe the main spatial features of the climate in Nagano Prefecture, Japan, using the average annual precipitation, temperature, wind speed, and sunshine hours for 1991-2020. PCA was calculated using Python.

The k-means clustering method is widely used for weather classification because of its simplicity and speed. Here, k-means clustering was calculated using Python.

When data are taken to be the maxima (or minima) over certain blocks of time, as the annual maximum precipitation, it is appropriate to use the generalized extreme value (GEV) distribution as follows:

$G\left(z\right)=\{\begin{array}{l}\exp \left\{-{\left[1+\xi \left(\frac{z-\mu }{\sigma }\right)\right]}^{-1/\xi }\right\},\xi \ne 0\hfill \\ \exp \left\{-\exp \left[-\left(\frac{z-\mu }{\sigma }\right)\right]\right\},\xi =0\hfill \end{array}$(1)

where μ is the location parameter, σ is the scale parameter, and ξ is the shape parameter. G is defined for all z values such that (1 + ξ(z − μ)/σ) > 0 for ξ ≠ 0 and all z values for ξ = 0. Three families of GEV distributions were defined depending on the value of ξ. For ξ > 0, we obtain the Fréchet distribution with a heavy tail, ξ = 0, the Gumbel distribution with a lighter tail, and ξ < 0, the Weibull distribution with a finite tail.

A classical method for modeling the extremes of a stationary time series is the block maxima method, in which consecutive observations are grouped into non-overlapping blocks of length n, generating a series of m block maxima, M_n,1, ..., M_n,m, i.e., to which the GEV distribution can be fitted for a large value of n. The usual approach involves considering blocks of a given time length, thus yielding maxima at regular intervals [1] .

Once a GEV distribution has been fitted to empirical observations, it is possible to estimate the probability of an event that has not yet been observed. Estimates of the extreme quantiles of the annual maximum distribution were obtained by inverting Equation (1) as follows:

$z_p=\{\begin{array}{l}\mu -\frac{\sigma }{\xi }\left[1-{\left\{-\log \left(1-p\right)\right\}}^{-\xi }\right],\xi \ne 0\hfill \\ \mu -\sigma \log \left\{-\log \left(1-p\right)\right\},\xi =0\hfill \end{array}$(2)

Here, G(z_p) = 1 − p. The return level z_p is associated with the return period 1/p. Because of a reasonable degree of accuracy, the level z_p is expected to exceed on average once every 1/p year. More precisely, z_p is exceeded by the annual maximum in any particular year with probability p [1] .

Modeling was performed using the evd and is mev packages in R for GEV calculations.

The explained variances and loadings of the first two principal components calculated from annual average precipitation, temperature, wind speed, and sunshine hours data for 1991-2020 in Iiyama, Nagano, Omachi, Sugadaira, Ueda, Karuizawa, Matsumoto, Saku, Suwa, Kisofukushima, Ina, Iida, and Minamishinano are shown in Table 2(a) . The first and second principal components explained 74.7% of the variance. Figure 2 shows a principal component analysis (PCA) biplot diagram for height above sea level, average sunshine hours, temperature, annual precipitation, and wind speed. The relationship between each principal component and each feature is indicated by an arrow. The larger the dot product between the axis corresponding to the principal component and the arrow corresponding to the feature, the greater the correlation between the principal component and the feature. In the PCA results, the sunshine hours and wind speed exhibit the same trend because the red arrows are close together. The red arrows are almost in the opposite direction, so precipitation and sunshine hours (wind speed) have opposite trends. Temperature and height above sea level have opposite trends. The lower the height, the higher the temperature, and the more precipitation, the shorter the sunshine hours. Table 3(a) shows a strong inverse correlation between height and temperature (r = −0.89). There was an inverse correlation between annual precipitation and sunshine hours (r = −0.57). There was a correlation between wind speed and sunshine hours (r = 0.42).

The k-means clustering of the annual precipitation and sunshine hours is shown in Figure 3 . It can be classified into four types: high precipitation and low sunshine hours (Kisofukushima, Minamishinano, Iiyama), normal precipitation and low sunshine hours, low precipitation and high sunshine hours (Ueda, Matsumoto, Suwa, Iida), and low precipitation and normal sunshine hours.

The k-means clustering of the height above sea level and annual temperature is shown in Figure 4 . It is clear that the higher the height, the lower the temperature. It can be classified into three types: high height and low temperature (Sugadaira, Karuizawa), normal height and normal Temperature, and low height and high Temperature (Iiyama, Minamishinano, Nagano, Iida, Ueda).

The explained variances and loadings of the first two principal components calculated for average annual, maximum daily, and maximum hourly precipitation data for 1991-2020 in Iiyama, Nagano, Omachi, Sugadaira, Ueda, Karuizawa, Matsumoto, Saku, Suwa, Kisofukushima, Ina, Iida, and Minamishinano are shown in Table 2(b) . The first and second principal components explained 93.7% of the variance. Figure 5 shows a principal component analysis (PCA) biplot diagram for the studied variables. The PCA results showed that the annual and maximum daily precipitation tended to follow the same pattern, but the maximum hourly precipitation differed. Even if the annual precipitation is low, some places (Suwa, Sugadaira) have high maximum hourly precipitation, while others have low maximum one. In Minamishinano and Kisofukushima, both annual and maximum daily precipitation were large. Table 3(b) shows a strong correlation between annual and maximum daily precipitation (r = 0.80). There was also a correlation between annual precipitation and maximum hourly precipitation (r = 0.51). There was a correlation between maximum daily precipitation and maximum hourly precipitation (r = 0.60).

The k-means clustering of the annual and maximum daily precipitation data is shown in Figure 6 . The greater the annual precipitation, the greater the maximum daily precipitation. The locations were classified into three categories: locations with high annual and maximum daily precipitation (Minamishinano, Kisofukushima, Iida), locations with normal annual and normal maximum daily precipitation, and locations with low annual and low maximum daily precipitation (Nagano, Ueda, Matsumoto, Saku).

Maximum daily precipitation in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano are shown in Figure 7 . The changes in Karuizawa, Saku, and Minamishinano were larger than those in the other points. The peak in 2019 was on October 12, when Typhoon HAGIBIS (T1919) occurred. Figure 8 shows the daily precipitation versus average wind speed in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano on October 12, 2019. The weather at the seven locations can be roughly classified into four types based on the precipitation and wind speed. Saku and Sugadaira have high rainfall and low wind speed. Karuizawa have high precipitation and wind speeds. Minamishinano and Kisofukushima receive little precipitation and low wind speeds. Matsumoto and Nagano experienced little precipitation and high wind speeds.

Figure 9 shows the wavelet power spectra of the maximum daily precipitation in Karuizawa, Kisofukushima, and Minamishinano, where the characteristic periodicity was observed in some periods. In Karuizawa, for 1940-1960, a strong priority of eight years was observed. In Kisofukushima, for 1991-1992, two years was observed. In Minamishinshu, two years were observed between 1990 and 1991.

Table 4 shows the results of GEV modeling for the maximum daily precipitation in Matsumoto using the block maxima method. The GEV parameters were estimated using maximum likelihood estimation (MLE). The model has three parameters: the location parameter μ, scale parameter σ, and shape parameter ξ. As ξ was positive, the daily maximum rainfall did not have a finite upper limit. The various diagnostic plots for assessing the accuracy of the GEV model fitted to the daily maximum precipitation in Matsumoto are shown in Figure 10 . Neither the probability plot nor the quantile plot doubts the validity of the fitted model: each set of plotted points is near-linear. In the return level curve, the estimated curve is not linear because ξ is not close to zero. Finally, the corresponding density estimates appeared to be consistent with the data. Consequently, all four diagnostic plots support the fitted GEV model.

A greater accuracy of confidence intervals can usually be achieved using the profile likelihood. Figure 11 shows the profile log-likelihood for ξ in Matsumoto, where ξ is estimated to be 0.203 with a 95% confidence interval (CI) [0.0514, 0.367], which is only slightly different from the earlier calculation, as shown in Table 3 . Because ξ > 0, the maximum daily rainfall does not have a finite upper limit, and it is not useful to perform a detailed inference of the upper limit. Instead, we focus on the extreme return levels. Figure 11 shows the symmetry of the profile log-likelihood surface.

Table 5 shows the ξ for the maximum daily precipitation in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. The values of ξ in Matsumoto, Karuizawa, Sugadaira, and Saku were positive, therefore, it has no upper bound and tends to take large values, which is dangerous, and caution is required. The values of ξ in Nagano, Kisofukushima, and Minamishinano were close to zero and included zero as a confidence interval, therefore, we determined it to be zero. Hence, there is no upper limit; thus, the probability of obtaining a large value is low, and caution is required.

Maximum hourly precipitation in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano are shown in Figure 12 . Table 6 shows ξ for the maximum hourly precipitation in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. The values of ξ in Matsumoto, and Sugadaira were positive, therefore, it has no upper bound and tends to take large values, which is dangerous, and caution is required. The values of ξ in Nagano, Karuizawa, Saku, Kisofukushima, and Minamishinano were close to zero and included zero as a confidence interval, therefore, we determined it to be zero. Hence, there is no upper limit; thus, the probability of obtaining a large value is low, and caution is required.

Table 7 shows the predicted maximum return levels for the maximun daily precipitation for return periods of 10, 20, 50, and 100 years along with their respective 95% CI in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. In Matsumoto, the 10-year return level was estimated to be 106 mm, with 95% CI [93.1, 119]. The 100-year return level was 182 mm, with 95% CI [129, 236]. Another way to interpret the plot is to say that there is approximately a 1% chance (1/100) each year that the magnitude of the maximum daily precipitation will exceed 182 mm. There is an approximately 10% chance (1/10) each year that the magnitude of the maximum daily precipitation will exceed 106.

The better accuracy is due to the profile likelihood. Figure 13 and Figure 14 show the profile log-likelihood for the 10- and 100-year return levels for the maximum daily rainfall in Matsumoto, respectively. The surface is highly asymmetric, reflecting the greater uncertainty associated with large processes. From Figure 12 , Matsumoto obtained the 10-year return level as 105 mm, with a 95% CI—obtained at the intercept with the drawn horizontal line—[94.5, 121]. As shown in Figure 12 , the 100-year return level is 183 mm, with a 95% CI of [145, 266], which is similar to an earlier calculation. These results differ slightly from those of earlier calculations.

Figure 15 shows the return period versus return level for Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. It can be divided into Minamishinano, Karuizawa, and Saku with high return levels, and Sugadaira, Kisofukushima, Matsumoto, and Nagano with low return levels. The 100-year return levels were 285, 271, and 271 mm in Minamishinano, Saku, and Karuizawa, respectively, where the changes in maximum daily precipitation were large. The maximum daily precipitation in Japan was 922.5 mm in Hakone, Kanagawa Prefecture on October 12, 2019. Karuizawa’s maximum 10-min rainfall was 38.5 mm on August 2, 1960, ranking tenth in Japan.

Figure 16 shows the average annual precipitation versus the 100-year return level for the maximum daily and hourly precipitation in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. Regarding the average monthly precipitation, only in Minamishinano and Kisofukushima, precipitation in June and July (rainy season) is significantly higher than that in September and October (autumn rainy season). At the other five points, precipitation was almost the same in both seasons. These are some of the reasons why Minamishinano and Kisofukushima experienced high precipitation. Because Minamishinano, Karuizawa, and Saku have large 100-year return levels for the daily maximum rainfall, caution is required against heavy rainfall. Especially, in Saku and Karuizawa the annual average precipitation was low, but the 100-year return levels were high. The 100-year return levels for maximum hourly precipitation showed a smaller variance.

Table 8 shows the 100-year return levels for maximum hourly precipitation in Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. The 100-year return level was 107.2 mm in Sugadaira, where the changes in maximum hourly precipitation were large. Figure 17 shows the 100-year return level for the maximum daily precipitation versus the maximum hourly precipitation for Nagano, Matsumoto, Karuizawa, Sugadaira, Saku, Kisofukushima, and Minamishinano. Both were almost proportional, and the values were similar for Karuizawa and Saku. In Sugadaira, the 100-year return level for the maximum hourly precipitation (107.2 mm) was larger than that for the maximum daily precipitation. Hence, it is necessary to be very careful about short-term precipitation. The maximum hourly precipitation in Japan was 153 mm on October 27, 1999, in Katori, Chiba Prefecture, and 153 mm on July 23, 1982, in Nagauradake, Nagasaki Prefecture.