## 2 Materials and Methods
### Trend Analysis  
The trend analysis was carried out using the Mann-Kendall test together with the Sen slope estimator. The non-parametric Mann-Kendall test, referred to as the Kendall tau test, is one of the most widely accepted non-parametric tests for detecting trends in time series [50; 54; 56; 57]. The Mann-Kendall test was applied, as in other research [58], as follows:  
\[S=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}sgn\big{(}x_{j}-x_{i}\big{)}\]  
Figure 1: California physical map and meteorological stations analysed (\(\bullet\)).  
\[sgn\big{(}x_{j}-x_{i}\big{)}=\left\{\begin{array}{l}1\text{ if }x_{j}-x_{i}>0\\ 0\text{ if }x_{j}-x_{i}=0\\ -1\text{ if }x_{j}-x_{i}<0\end{array}\right.\]  
where \(n\) is the length of the sample, \(x_{j}\) and \(x_{i}\) are from \(i=1,2,\,\ldots,\,n-1\) and \(j=i+1,\,\ldots,\,n\). 34. For \(n>8\), \(S\) is approximately normally distributed. The mean of \(S\) is 0 and the variance of S can be obtained as follows:  
\[V(S)=\frac{n(n-1)(2n+5)-\sum_{1=1}^{n}t_{i}\,i(i-1)(2i+5)}{18}\]  
The standardized test statistics (\(z\)) of the MK test and the corresponding \(p\)-value (\(p\)) for the one-tailed test are respectively given as follows:  
\[Z=\left\{\begin{array}{l}\frac{S-1}{\sqrt{Var(S)}}S>0\\ 0\,S=0\\ \frac{S+1}{\sqrt{Var(S)}}S<0\end{array}\right.\]  
If \(Z>0\), it indicates an increasing trend and vice versa. Given a confidence level \(\alpha\), the sequential data would be supposed to experience a statistically significant trend if \(|Z|>Z(1-\alpha/2)\), where \(Z(1-\alpha/2)\) is the corresponding value of \(P=\alpha/2\) following the standard normal distribution. In this study, a 0.05 confidence level was used.  
The slope of the linear trend is estimated with the nonparametric Sen slope estimator. It is a non-parametric procedure that estimates changes per unit of time in a series when there is a linear trend [59; 60; 61]. It allows to estimate the slope fitting a set of (time, \(x\) variable) elements in a non-parametric way, being much less sensitive to outliers. It is the median of all the slopes calculated between all the time instants whenever instant \(i\) is less than instant \(j\).  
_Sen's slope_ is defined as  
\[Set^{\prime}s\,\,slope=\,median\bigg{\{}\left(\frac{x_{j}-x_{i}}{j-i}\right) \,\forall\,\,i<j\bigg{\}}\]  
The Modified _Sen's slope_ aims to remove the autocorrelation of the \(x\) variable before computing the slopes. The trend of \(x\) time series is computed using an autoregressive model of order 1, an AR (1) model.  
With the autoregressive term \(\rho\) a new series \(x^{\prime}\) (with \(n-1\) elements) is constructed:  
\[x^{\prime}=x\{i=2,3,\ldots,n\}-x\{i=1,2,3,\ldots,n-1\}\rho\]  
_Sen's slope_ formula is applied considering the \(x^{\prime}\) transformed series and time values from 1 to \(n-1\).  
The modified Sen slope [62] was applied in this research. R package version 4.1.0 was used both to carry out the slope calculations, obtained with a modified Sen's slope method and the Mann-Kendall test [63]. All trend analysis was carried out at monthly, seasonal and annual temperature levels.  
Statistical interpolation of the values is necessary for a specific region. In order to carry out this interpolation, ArcGis 10.8 [64] software was used and, more specifically, an Empirical Bayesian Kriging geoprocessing tool (EBK). It is a method of interpolation of geographic statistics where the standard errors of the prediction are more precise than in other kriging methods [65; 66; 67]. In addition, 17 average temperature trend contour maps were designed with ArcGis 10.8 [6] and statistically significant areas, at a 95% confidence level, were also superimposed onto the contour maps.