# On the Regional Temperature Series Evolution in the South-Eastern Part of Romania
## 2 Methodology and Data Series  
### Methodology  
Figure 1 presents the flowchart of this study. Two statistical tests were performed on the data series. The first one was the Mann-Kendall trend test [45] that permits testing the null hypothesis of no monotonic trend, H\({}_{0}\), against the alternative hypothesis, H\({}_{1}\), that there is an increasing or decreasing monotonic trend of the temperature series. [PERSON]'s nonparametric procedure [46] was used to estimate the slope of such a trend (if it exists). The second test was that of [PERSON] [47], whose null hypothesis is that the distribution functions from which the samples are extracted are the same. In this case, the samples were the temperature series recorded at each site. Given \(k\) data series, each of them formed by \(m\) elements, let \(X=(x_{ij})\) (\(j=1,\ldots,\)\(m,\)\(i=1,\)\(\ldots,\)\(k\)) the matrix whose columns are the series elements (column 1--the elements of series 1, \(\ldots,\) column \(k\)--the elements of the series \(k\)). For determining the spatial trend, the series are recorded at the same time. MPPM has the following stages [41]:  
Figure 1: The study flowchart. Sim_no means the number of simulations. 1. For each row _j of X_ (_j_ = 1,..., _m_), compute the difference between the maximum and the minimum values, call it amplitude, and denote it as \(A_{j}\). 2. For each \(j\) (_j_ = 1,..., _m_), divide the interval between the minimum and maximum values on row \(j\) in \(n_{j}\) intervals of the same length that contain enough values. The number of intervals, \(n_{j}\), is chosen based on the user experience and/or a number of trials. 3. For each \(j\) (_j_ = 1,..., _m_), choose the interval with the highest number of elements (call this number frequency) among those formed in the previous step. Denote this interval by \(I_{j\max}\) and its number of elements \(f_{j\max}\). 4. Build the 'Regional series' whose values are the mean of the elements in all \(I_{j\max}\). 5. Represent the chart of the 'Regional series' in Cartesian coordinates, where the Ox is the time axis and Oy is the axis containing the values of the 'Regional series.' 6. Estimate the goodness of fit by computing MAE and MSE and build their charts. The MAE and MSE are computed using the errors obtained as the differences between the recorded values (at each station) and the 'Regional series' values. The following remarks on MPPM are necessary:
* At stage 2, \(n_{j}\) is not necessarily the same for each row in the matrix. For a small number of series, it is recommended to use the same \(n_{j}\). The number of values in an interval depends on the number of the series. At least ten series should be considered for building the regional series, and at least two values must be contained in each interval. Three values should be the minimum for a higher number of series to avoid intervals containing only extreme values or outliers. * If at stage 3 in MPPM there are many intervals with the maximum frequency, it is recommended to run the next steps of the algorithm for each case. The 'Regional series' will be that with the minimum MAE and MSE. This approach offers a better fit of the 'Regional series' than the original one, which only considered the interval whose average value was the closest to the average value of the data recorded at a specific moment. We name MPPM with this improvement, Method I. To run Method I, Climate Analyser, implemented in Matlab [44], was utilized. It involves the following stages:  
1. _Compute_Amplitude_ step: compute the extreme values for each year and then the amplitude. 2. _Amplitude Representation step_: draw the chart of amplitude. 3. _Compute the regional series:_ for each year, choose the number of intervals, compute the number of elements in each interval, select the interval with the highest number of elements, and compute the average of the elements in this interval. 4. _Collect the results_ in the Data Processing from Matlab destination tables (containing MAE and MSE). 5. _Draw the charts_ of 'Regional series', _MAE_, and _MSE_.