# Trends of Changes in Minimum Lake Water Temperature in Poland
## 2 Materials and Methods
### Statistical Analysis  
The determination of the trend of changes in minimum water temperatures in the lakes employed a modified Mann-Kendall test (MK) [28]. The approach is broadly applied in analyses regarding the trends of climatic and hydrological changes [29; 30; 31; 32; 33; 34]. A zero hypothesis was stated (H\({}_{0}\)) on the lack of a monotonic trend of minimum water temperatures, and an alternative hypothesis (H\({}_{1}\)) on the existence of a monotonic trend. The main assumption of the Mann-Kendall test is a lack of autocorrelation. In order to remove the autocorrelation from the series of data, the trend-free prewhitening procedure was used [35]. The determination of the magnitude of changes in minimum water temperatures in the lakes applied a non-parametric Sen test (S) [36]. The Sen test is insensitive to outliers. It can be much more accurate than a simple linear regression for skewed and heteroscedastic data. The analysis of trends and the magnitude of changes by means of MK and S tests employed the modified mk package developed by [PERSON] and [PERSON] [37]. The detection of change-points in a series of 1-day minimums of monthly annual water temperatures employed the Wild Binary Segmentation 2 method. Originally, the Wild Binary Segmentation method was developed by [PERSON] [38]. The WBS method permits the determination of several change-points. The Wild Binary Segmentation 2 method contains a recursive algorithm that provides a complete solution for the problem of change-point detection. Another improvement is a \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No**} & \multirow{2}{*}{**Lake**} & \multirow{2}{*}{**Latitude**} & \multirow{2}{*}{**Longitiude**} & **Area** & **Volume** & **Mean** & **Ice Cover** & \multirow{2}{*}{
\begin{tabular}{c} **Koppen’s** \\ **Climate** \\ **Classification** \\ \end{tabular} } \\  & & & **(ha)** & **(103 m3)** & **Depth** & **Max Thickness** & **Koppen’s** \\  & & & & **(ln)** & **(m)** & **(Mean)** & **Climate** \\ \hline
1 & Slawskie & 51.89 & 16.02 & 822.5 & 42,664.8 & 5.2 & 20.0 & Cfb \\
2 & Sepolefskie & 53.46 & 17.51 & 157.5 & 7501.6 & 4.8 & 22.6 & Cfb \\
3 & Charzykowskie & 53.77 & 17.50 & 1336.0 & 134,533.2 & 9.8 & 22.2 & Cfb \\
4 & Gardno & 54.71 & 17.39 & 2337.5 & 30,950.5 & 1.3 & 18.8 & Cfb \\
5 & Bachotek & 53.30 & 19.47 & 215.0 & 15,394.2 & 7.2 & 24.8 & Cfb \\
6 & Jeziorak & 53.72 & 19.62 & 3152.5 & 141,594.2 & 4.1 & 22.4 & Cfb \\
7 & Mikolajskie & 53.77 & 21.60 & 424.0 & 55,739.7 & 11.2 & 27.6 & Dfb \\
8 & Selmet Wielki & 53.83 & 22.48 & 1207.5 & 9963.9 & 7.8 & 30.2 & Dfb \\
9 & Studzieniczne & 53.87 & 23.12 & 244.0 & 22,073.6 & 8.7 & 29.9 & Dfb \\
10 & Harcza & 54.27 & 22.81 & 305.0 & 120,000.0 & 38.7 & 24.1 & Dfb \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lakes’ morphometric parameters [27] and environmental characteristics. new model selection procedure for change-point problems which is not based on penalties but uses thresholding. The developed method is fast and requires no selection of a window or span parameter [39]. The calculation of change-points employed the breakfast package developed by [40]. The analysis of changes in minimum lake water temperatures was performed in reference to changes in minimum air temperatures. For this purpose, data on air temperatures were analysed the same way, by applying a modified Mann-Kendall test, Sen test, and Wild Binary Segmentation 2 method. The statistical analysis of the results was performed by assuming standard levels of significance of 0.05 and 0.01. The generalisation of the results involved grouping months and lakes by means of cluster analysis. The clustering was based on the minimum water temperatures and the magnitude of changes in minimum water temperatures described by means of the Sen coefficient. The cluster analysis was performed by means of the Ward method. Square Euclidean distance was used as the similarity measure. The division of sets into clusters used the threshold value of 66%, and a threshold value of 25% for subclusters [41].