1 Introduction

The Earth’s climate has frequently changed over the past 4.5 billion years, with significant fluctuations in global average temperatures due to volcanic emissions, tectonic plate movements, changes in solar radiation, and several other factors (Caloiero et al. 2021; Lin and Qian 2022). However, since the last ice age, the Earth’s climate has been relatively stable, with global temperatures varying by less than 1 °C over a century in the last 10,000 years. Over the last 100 years, human activities have started to play a major role in increasing temperature levels, with alarming acceleration over the last twenty years (Schmidt and Hertzberg 2011). The sixth report (AR6) by the Intergovernmental Panel on Climate Change (IPCC 2013) evidenced an increase in the Earth’s surface temperature in the twentieth century that was around 1.1 °C above 1850–1900 in 2011–2020, with larger increases over land than over the ocean. An increase in temperature leads to greater evapotranspiration demand, accelerated hydrological cycle, an intensification, and an increase in extreme events, such as droughts, floods, cold spells and heat waves (Buttafuoco et al. 2016, 2018; Sirangelo et al. 2020). Several studies performed worldwide have shown an increase in the mean temperature. For example, in Eastern China, an increase of 1.52 °C over the past century has been detected (Zhao et al. 2014). Similarly, several increases in temperature have been detected in Switzerland (Ceppi et al. 2012), Pakistan (Nawaz et al. 2019), the United States, except for Pennsylvania and Maine (Martinez et al. 2012), and Senegal (Djaman et al. 2017). Several studies have also been carried out in the Mediterranean basin, a region highly susceptible to climate change since geographically, this region lies in a transition area between the hot, dry climate of Africa and the cold, humid air masses arriving from northern Europe (Goubanova and Li 2007; Vicente-Serrano et al. 2014). For instance, Giorgi (2002), at the annual scale, and Jacobeit (2000), in summer, demonstrated statistically significant warming trends over most parts of the Mediterranean area. Several authors have also detected a general increase in temperature in the Mediterranean basin over recent decades (Caloiero and Guagliardi 2020, 2021; Caloiero et al. 2016; Pellicone et al. 2019; Scorzini and Leopardi 2019; Gentilucci et al. 2020).

In these studies, climate change has been analysed considering variations in average temperature, but the frequency and severity of extreme events cause most of the social and economic costs linked with climate change (King et al. 2015). In fact, changes in climate extremes lead to more substantial impacts on human and natural systems than changes in the average climate (Infusino et al. 2022a; Ricca and Guagliardi 2015). For this reason, recently particular attention has been given worldwide to the analysis of extreme temperatures at different spatial scales, revealing differences in the characteristics of extreme temperatures, with marked variations in some areas and nonsignificant trends in others (Bonsal et al. 2001). Therefore, performing regional studies is crucial for evaluating the impacts of extreme events (Zarenistanak et al. 2014), even though this approach is not always possible due to the lack of complete and spatially coherent daily temperature series (Mishra and Singh 2011). Consequently, there is an absence of regional-scale studies on extreme changes in temperature (Murara et al. 2019).

Missing data are a common issue in practical data analysis, particularly for time series. In fact, it is well-known that for performing correct statistical analyses, working datasets must be complete. To address this issue effectively, ad hoc theoretical frameworks have been introduced to reconstruct incomplete datasets (Lo Presti et al. 2010). Many techniques for missing data reconstruction have been derived from these frameworks. With respect to temperature data, gap-filling methods can generally be separated into spatial and temporal groups. The methods of the first group are usually based on interpolation techniques such as inverse-distance weighting, kriging, multiple regressions, and thin-plate splines (Lompar et al. 2019). On the other hand, the methods used for the latter group depend on the autocorrelation of the meteorological time series (Lompar et al. 2019). For example, Claridge and Chen (2006) applied polynomial fitting and simple linear interpolation, while Liston and Elder (2006) reconstructed missing data using a linear combination of forecasts and hindcasts. Finally, there are several methods, such as empirical orthogonal functions that combine spatial and temporal interpolation (Von Storch and Zwiers 1999), even though the application of these methods is strictly related to the number of neighboring stations and the size of the data gap. Generally, older methods are single-valued, but more recently, multiple-valued methods have been introduced; consequently, multiple imputation methods have become available that are considered to be better than single-valued methods (Scheffer 2002). Another issue concerns how to perform an effective selection of the most appropriate method of missing data reconstruction. A relevant consequence of Wolpert’s theorem is that no method can outperform all others across all possible datasets (Wolpert 1996).

Within this context, the aim of this study is to propose a multiple-valued methodology aimed at providing the best missing data reconstruction with respect to the case study and the selected methods. The gap-filling database was subsequently used to perform a reliable trend analysis of extreme temperature data in the Calabria region in southern Italy.

2 Study area and data

Due to its position in the center of the Mediterranean basin and its climatic characteristics, Calabria is considered a highly susceptible area to climate change (Guagliardi et al. 2021), where even a small temperature increase could lead to various environmental problems. The Calabria region is located between 37° 54’ and 40° 09’ N and between 15° 37’ and 17° 13’ E, with an area of 15,080 km² (Caroletti et al. 2019; Iovine et al. 2018). Although the region has a long north-south orientation and does not have many high peaks, it is one of the most mountainous Italian regions (Gaglioti et al. 2019; Infusino et al. 2022b), with mountains occupying 42% of the region and hills covering 49% of the territory (Fig. 1).

Fig. 1
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Study area with selected temperature stations (the number close to the red dots corresponds to the code of the climatic series)

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The Köppen-Geiger classification (Köppen 1936) identifies the climate of the region as a hot-summer Mediterranean climate, with relatively mild winters (with rain) and very hot summers (often very dry). With this climate, the coldest month generally has an average temperature above 0 °C, at least one month’s average temperature reaches values higher than 22 °C, and at least four months present an average temperature above 10 °C. Moreover, due to its orography, the region exhibits sharp contrasts, with warm air currents coming from Africa affecting the Ionian side with short and heavy precipitation and western air currents affecting the Tyrrhenian side with high precipitation (Caroletti et al. 2019; Pellicone et al. 2018).

Given the findings of past studies showing that winter conditions are changing more rapidly than they are in any other season (Caloiero et al. 2017), in this study, particular attention was given to the temporal changes in frost days, i.e., the annual daily minimum temperature < 0 °C (Zhang et al. 2011). In fact, a decrease in frost days could have lasting impacts on ecosystems, especially in mountainous and forested areas such as the Calabria region. Indeed, frost days are linked to the forest health, by killing insect pests, and to the snowpack, which insulates soil providing subnivean refugia for some animals and preventing the freezing of roots and microbes (Contosta et al. 2019).

In Calabria, climatic data are managed by the Multi-Risk Functional Centre of the Regional Agency for Environmental Protection which publishes online daily temperature data.

The starting dataset was formed by daily minimum temperature related to 72 monitoring climatic gauges and covering an observation period ranging from 01/01/1990 to 31/12/2019, i.e. 30 years or 10957 days. The dataset was first analyzed to assess the missing data rate for each monitoring site, to check if there were time series with a missing data percentage too high to be reliably reconstructed. Percentage of missing data among sites ranged from 0.23 to 41.73% and, on average, the number of missing data rate was 8.4%. All the sites with a missing data rate larger than 10% have been dropped, consequently a reduced set of 28 sites have been retained for the analysis and, considering the elevation, 8 temperature series have been investigated with a remaining set of supporting sites of 20. These 8 series are well distributed in the region and fall within 4 of 5 of the main mountains of the region, the Sila Massif, the Coastal Chain, the Aspromonte Massif, the Serre Massif. In fact, in the Pollino Massif, no thermometric stations are available (Fig. 1; Table 1).

Table 1 Main characteristics of the target stations selected for this study
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Following Caloiero et al. (2017), the 8 selected temperature series were checked for errors and inhomogeneity by means of a multiple application of the Craddock test (Craddock 1979). The homogenization procedure was applied to minimum series and, due to the low number of temperature stations available for this analysis, each series was tested against other ones, in subgroups of five series.

3 Materials and methods

In this section, a new methodology aimed at providing the best missing data reconstruction with respect to the case study and the selected methods is presented (Fig. 2).

Fig. 2
figure 2

Flowchart of the methodology

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Following the linear regression approach, proposed by FAO (Fagandini et al. 2024), the series to be filled must have at least 70% of complete data; thus, the methodology assumes that the missing data cannot significantly change the shape or parameters of the complete data distribution if they are below a certain percentage (< 30%). Besides detecting the percentage of missing data, it is crucial to determine whether the missing data is isolated or chained together in a sequence of consecutive days without data. Therefore, the distribution shape and the characteristic parameters (mean and standard deviation) of the incomplete dataset can be considered within a certain approximation, the real distribution, and the real parameters. After applying several reconstruction methodologies, the distributions and parameters of the reconstructed datasets are compared with those of the original incomplete dataset. The reconstructed dataset that better overlapped the original dataset was considered the best. The core of the methodology is the application of several missing data recovery methods to an incomplete dataset, at least three and based on different assumptions, to account for all the specific features contained in the data. The need for using more than one method for data recovery derives from Wolpert’s theorem (Wolpert 1996), which states the inexistence of a universally optimal machine learning method in absolute terms but only regarding to a specific problem. After running the selected methods of data recovery, three (or more) different completed datasets are obtained. The analysis of the data loss mechanism is then carried out to check whether missing data can be classified as ignorable. In fact, the assumption of ignorability is essentially the belief that the available data are sufficient to estimate missing data; thus, this assumption is the basis of the missing data reconstruction concept.

After the missing data reconstruction, a trend analysis was performed to evaluate the possible trend in the gap-filled data.

3.1 Preliminary statistical analysis

First, descriptive statistics were computed to summarize the main features of the minimum temperature distribution. In addition, the hypothesis of normality was tested using the Anderson-Darling test (Barca et al. 2016, 2017). The descriptive statistics were generated by applying the R library summarytools and the related function descr (Comtois 2021).

3.2 Gap-filling procedure

The analysis and reconstruction of an incomplete dataset is a complex task, and to carry out this task correctly, a suitable methodology is needed. The methodology described hereafter is a multistep one.

The first step is preprocessing the incomplete dataset to obtain descriptive statistics useful for making correct decisions during the analysis. First, the quantification (in absolute and percentage terms) of the missing data (MD) for each incomplete station must be accomplished. Several numerical and graphical tools are currently available for performing such tasks. The aim of this analysis is manifold: (i) assess the percentages of MD to select the best reconstruction strategy (Table 2); (ii) assess wheter all the MDs are in principle recoverable; and (iii) recognize possible patterns among MDs by means of graphical tools, such as Margin Plots (Templ et al. 2012).

Table 2 Suggested methods considering the missing data rate (MD)
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The second step is the correlation analysis, which helps in collecting all the time series (TS) related to sites that are better correlated with the target site. The TSs are the basic information of all the missing data recovery methods. Another important methodological step concerns the identification of the mechanisms responsible for data value losses. Once the support TSs for the considered target site are gathered and the loss mechanisms are identified, the next step is to select a group of suitable missing data recovery methods. As mentioned before, more than one methodology is recommended because a method that outperforms all the others for any class of data does not exist.

After applying the selected missing data recovery methods, the performances of each method must be compared to safely choose the best data reconstruction for the analyzing case. At this step, the probability density function (pdf) associated with the incomplete target site TS is considered and compared with the pdfs of the completed TSs provided by the applied recovery data methods. The comparison is carried out by superimposing the pdf associated with the incomplete TS with each of the densities of the completed datasets and computing the percentage of overlap. For all the completed datasets, the dataset that overcomes 90% of the overlap, and has the least deviation from the characteristic parameters extracted from the incomplete original data is selected as the optimal solution.

3.2.1 Preprocessing and correlation analysis

In this study, space-time continuous data, i.e., time series data from different monitoring stations belonging to a permanent observational network, are analyzed. Generally, that kind of data is arranged as a bidimensional array endowed with the following structure:

  • each row is uniquely associated with the date on which the observation was made according to the considered time aggregation level (daily, monthly, or yearly),

  • each column is uniquely associated with a monitoring site.

Rows are often referred to as “items”, and columns are often referred to as “sites”.

As reported in Table 1, the missing data rate is the key factor for selecting the best method for rebuilding incomplete time series (Lo Presti et al. 2010). It is straightforward that not all incomplete datasets can be filled even if very powerful methods are currently available. In fact, some missing data percentage thresholds discourage any attempt to fill these gaps. Therefore, another important step is to drop all the columns of the matrix (i.e. the sites) affected by high missing data rates (> 30%) that could be severely distorted by reconstruction methods. Afterwards, the remaining columns are split into two groups: (i) the group containing the sites to be reconstructed (target sites) and (ii) the support site group, which provides the information needed to accomplish the reconstruction.

Once the target sites are recognized, a correlation analysis is performed to separate the columns (sites) from the support group that are not well correlated with the first group of sites.

3.2.2 Missing data method assessment

The subsequent methodological step concerns the identification of the mechanism responsible for data value losses. When the phenomenon of missing data is described in probabilistic terms, three different types of processes can be considered (Table 3): missing completely at random (MCAR), missing at random (MAR) and not missing at random (NMAR) (Little and Rubin 1987; Lo Presti et al. 2010; Leurent et al. 2013). The substantial difference between these types of loss mechanisms relies on the degree of dependence between the probability of the data loss and the data value.

Table 3 Formal definition of the data loss mechanisms (by Lo Presti et al. 2010)
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A comprehensive description of the theoretical framework can be found in Rubin (1976). If the loss mechanism is too difficult to assess due to the incompleteness of the supporting group sites, the usual assumption is MAR, and the consequent approach is multiple imputation. This approach is different from the others (e.g. uni-multivariate regression) in that it does not provide a single replacement for each missing entry in the dataset. In contrast, if a small distribution of realistic data is provided, afterwards, the user is burdened to select among the distributions the best suited replacement for the missing datum.

Once the features that the reconstruction approach should possess according to the percentage of missing data are identified, some algorithms for fixing the problem can be selected (see Table 1). The basic idea of the proposed methodology is to apply more than one algorithm, each endowed with different predictive characteristics (parametric, non-parametric, etc.).

3.2.3 Missing data reconstruction: predictive mean matching (PMM) outline

Predictive Mean Matching (PMM) is an algorithm used to perform multiple imputations for missing data; it is particularly useful for quantitative variables and has the advantage of being non-parametric (Rubin 1986; Little 1988). Such an approach to multiple imputations is known as multiple imputations by chained equations (MICE), sequential generalized regression, or the fully conditional specification (FCS) (Van Buuren 2007).

The algorithm follows this line: suppose there is a single variable x that has some cases with missing data and a set of variables \({z}_{i}\) that are used to impute x. The steps of the PMM algorithm are as follows:

  1. 1.

    for cases with no missing data, estimating a linear regression of x on\({z}_{i}\), producing a set of coefficients\({b}_{i}\);

  2. 2.

    to perform a random extraction from the “posterior predictive distribution” of b, producing a new set of coefficients b*. This step is necessary to produce sufficient variability in the imputed values and is common to all methods for multiple imputations;

  3. 3.

    using b*, generate predicted values for x for both cases with missing data and those with no missing data;

  4. 4.

    for each case with missing x, identifying a set of cases with observed x whose true values are close to the predicted value for the case with missing data;

  5. 5.

    among those close cases, randomly choose one and assign its observed value to substitute for the missing value;

  6. 6.

    repeating steps 2 through 5 for each completed dataset.

The R library mice and function mice (Van Buuren 2007) was used for missing data estimation.

3.2.4 Missing data reconstruction: the expectation maximization (EM) algorithm

The package norm implements missing data procedures via the Multivariate Expectation-Maximization (EM) method. The EM algorithm formalizes an elementary idea for dealing with missing data and consists of the following steps:

  1. 1.

    replacing missing values with estimated values (e.g. by averaging observations);

  2. 2.

    estimating the prediction model parameters;

  3. 3.

    re-estimating the missing data, assuming that the new estimated parameters are correct;

  4. 4.

    the parameters are re-estimated, and the procedure is repeated until convergence.

Therefore, the EM algorithm consists of two main stages, namely the Expectation and the Maximization, which are repeated until the missing data estimation changes significantly; otherwise, the process stops.

The EM algorithm is a parametric method and can be applied to any type of data, but it is particularly effective for distributions of the exponential family. The main advantage of using the EM algorithm is that it deterministically converges because it is simple to define pre-specified bounds that can be used to terminate the algorithm (King et al. 2001; Honaker and King 2010).

3.2.5 Missing data reconstruction: the missforest algorithm

MissForest is a random forest imputation algorithm for missing data, implemented in R in the missForest (Stekhoven and Buhlmann 2012) R package. Initially, all missing data were imputed using the mean/mode; subsequently, for each variable with missing values, MissForest was used to fit a random forest to the observed data and predict the missing data. This process of training and predicting repeats an iterative process until a stopping criterion is met, or a maximum number of user-specified iterations is reached. The reason for the multiple iterations is that, from iteration 2 onward, the random forests performing the imputation will be trained on better- and better-quality data that itself have been predictively imputed. The MissForest algorithm has an important drawback tied to the heavy computational cost that can be partially overcome by using the function in a parallel framework.

3.3 Time series serial correlation

To assess the significance of a time series trend, after data reconstruction, a test of serial correlation is needed. In fact, autocorrelation can adversely affect a trend test, lowering the significance threshold α and distorting the trend test outcomes. The serial correlation test is a graphical tool that is easily interpreted. The first quadrant of the Cartesian plane is shown, crossed by two dashed lines. From each lag, a vertical line is raised at lag 0, a line is generated that terminates at the 1.0 ordinate. This line is of little use because it indicates that the time series is correlated with itself. From a lag value of 1 on, if the vertical line overcomes the dashed line, the result can be considered significant. As an example, Fig. 3 shows that the line corresponding to lag 1 crosses the dashed line, which means that the time series is autoregressive of the first order (AR(1)) for the Nocelle-Arvo site. The functions acf and pacf of the R library stats were used to perform the time autocorrelation test.

Fig. 3
figure 3

Serial correlation test for the Nocelle-Arvo site (ID 1500). The time series appears to be AR(1)

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3.4 Trend tests

The Mann-Kendall (MK) test is a nonparametric method for detecting trends in a time series. This test is widely used in environmental science because it is simple and robust. After the first versions were proposed by Mann (1945) and Kendall (1975), the test has been improved in many ways by introducing covariance matrix analysis (Dietz and Kileen 1981), filtering seasonality (Hirsch et al. 1982), multiple monitoring sites (Lettenmaier 1988) and covariates representing natural fluctuations (Libiseller and Grimvall 2002). If a serial correlation in the data is detected, a trend test robust against its adverse effects should be applied. The Modified Mann-Kendall (MMK) test is an implementation of the MK test that is free from serial correlation effects (Kundzewicz and Robson 2000). The R library used was Kendall and the corresponding function was MannKendall (Hipel and McLeod 1994).

In addition to the MK test, to evaluate the trend magnitude, the Theil-Sen estimator (TSE) was applied. In fact, this test is considered more powerful than linear regression methods in trend magnitude evaluation, because it is not subject to the influence of extreme values (Sen 1968).

4 Results

4.1 Basic statistics

Table 4 shows the basic statistics related to the eight target sites described above. The considered variable is the daily minimum temperature (°C).

Table 4 Basic statistics for the 8 target sites (the analysed period is 1990–2019 and the variable is daily minimum temperature)
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Among the different statistics, the analysis of skewness, which yields values close to zero, is particularly important. This means that the empirical distributions of the target sites are all approximately Gaussian. This is a key feature because it guarantees the correct operation of parametric tests and applied methods (e.g. correlation). Unfortunately, this finding was not confirmed after the application of the Anderson-Darling Gaussianity test (Thode 2002; Barca et al. 2016), which rejects the Gaussianity hypothesis for all the empirical distributions related to the different TSs. This is probably due to the large size of the TS associated with each target size. Table 5 reports the values of the quantile imbalance (qi) that, if close to zero, assures that at least the central part of the distribution is approximately Gaussian (Brunsdon et al. 2002).

Table 5 Values of quantile imbalance related to the target stations
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The obtained values confirm that the hypothesis can be tested safely, and thus, parametric tests can be safely carried out (e.g. Pearson correlation).

4.2 Missing data and data reconstruction: target and supporting sets

The eight target sites selected for the analysis and their percentages of missing data are reported in Fig. 4.

Fig. 4
figure 4

Missing data percentages

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Figure 4 provides a specific visualization of the amount of missing data, with the location of missing values shown in black and providing information on the overall percentage of missing values. A careful analysis of the figure reveals that all the missing data are potentially recoverable. In fact, there is no black line crossing the entire gray area, and the missing data percentages are within the range (Table 1) to be reliably reconstructed by means of multiple imputation methods. It is evident that the missing data are evenly distributed throughout the observation period, without any significant sequence of consecutive days without data. The plot was drawn by means of the R function vis_miss from the naniar library (Tierney et al. 2015).

Unfortunately, no time series among those analyzed appears to be complete, which raises difficulty in identifying the underlying mechanism. The MAR mechanism was then assumed to be main driver of the data loss. The consequence of such an assumption is that the rebuilt time series should have the same distribution as the original data with the same parameters (mean and standard deviation). The following reconstruction methods were applied: mice, MissForestand imp.norm from R packages mice, missForest and norm, respectively (Van Buuren 2007; Feng et al. 2020; Anderson et al. 1994). As a result, the rebuilt method reporting the best performance was imp.norm with an overlap rate greater than 97% (Table 6).

Table 6 Percentages of overlap between the original data density and the reconstructed data density according to the applied methods
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As mentioned above, the validation is structured in two stages. During the first stage, an overall comparison between densities is carried out; in the second stage, the main parameters, mean and standard deviation, before and after the data reconstruction are computed and compared. To perform the first stage, the two produced densities (the original and the rebuilt density) are overlapped, and the percentage of overlap is computed. A percentage larger than 90% indicates very good reconstruction performance, and a percentage larger than 95% can be considered excellent (De Girolamo et al. 2022). As shown in Table 7, the overlap values were always greater than 97%; therefore, the performed reconstruction was judged as excellent. For the second stage, the average and the standard deviation of the original and reconstructed data were compared to assess the deviation of these parameters from one dataset to the other. Table 7 shows that such deviations can be considered substantially negligible; therefore, this further validation stage can be considered successful.

Table 7 Number of daily minimum temperature subzeros per year
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4.3 Number of subzeros data points after reconstruction

Table 7 reports the number of subzeros days after dataset reconstruction for each year.

The thirty years of data were analysed to determine possible trends underlying the presence of climate change effects on the minimum temperature distribution. Figure 5 shows the relationship between elevation and the number of subzero days; as expected, the number of subzeros increases with increasing elevation.

Fig. 5
figure 5

Relationships between subzero count (x) and elevation (y). The relationship was significant (p = 0.025)

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Another interesting analysis carried out on the sub-zero values was the cluster analysis, which involved crossing the subzero values and the elevation by means of a heatmap plot. By analysing the heatmap in Fig. 6, it can be concluded that there are two homogeneous clusters of monitoring sites formed by the 4 sites. Nocelle – Arvo (ID 1500), Cecita (ID 1100), Camigliatello (ID 1092) and Serra S. Bruno (ID 1980) belong to the first group, while Acri (ID 1120), Domanico (ID 1000), Antonimina - Canolo Nuovo (ID 2180) and Gambarie d’Aspromonte (ID 2470) belong to the second group. These two groups are consistent with the elevation.

Fig. 6
figure 6

Heatmap of the values reported in Table 8. The dendrogram on the left represents clusters between sites; the dendrogram on the top represents similarities between years

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4.4 Trend assessment

Due to the nonsignificant degree of temporal autocorrelation, the Mann-Kendall trend without correction for autocorrelation was applied. Table 8 shows the results of the trend tests.

Table 8 Parameters of the Mann-Kendall trend test
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The p-values in italics refer to the significant test outcomes. Only two out of eight series were nonsignificant: site Gambarie d’Aspromonte, (ID 2470) with a negative slope of -0.25, and site Antonimina - Canolo Nuovo (ID 2180), which had a slope of 0.14. This trend behavior can be influenced by the geographic position of the stations, that are the southernmost considered and are directly interested by African air currents, presenting high-temperature values with respect to the other stations. As regards the significant trends, the most marked negative values were detected in the Sila Massif and, specifically, in the Cecita, Acri and Nocelle - Arvo stations with slopes of -1.50, -1.21 and − 1.05, respectively. A relevant reduction of -1.14 has also been detected at the Serra San Bruno site, while less marked trends have been identified in Domanico (ID 1000, slope of -0.65) and Camigliatello (ID 1092, slope of -0.46).

The results of this study agree with the scientific findings related to projections on southern Europe’s frost regime (Torma and Kis 2022) and with previous investigations on maximum and minimum temperatures at a planetary scale that evidenced an increase in the minimum values rather than the maximum values (Caloiero and Guagliardi 2020).

5 Discussion

The results of this study proved that the outcomes of a data reconstruction process can be confidently analysed via statistical analysis. It is sufficient that those outcomes be rebuilt following the precautions needed; thus, the results can be effectively applied for climate change analysis. Therefore, tools for data reconstruction are invaluable for carrying out complex data analysis and obtaining important results that are impossible to obtain otherwise. The results of the present study, undoubtedly prove that the Calabria region is interested in a warming process. This conclusion is reached by showing that the eight- time series of the count subzero values in six out of the eight sites have a decreasing trend.

It is necessary to develop adaptation measures and to stimulate their implementation by decision- makers in politics, administration, the economy, and society considering that climate change affects different sectors, such as agriculture, forestry, urban and regional planning, nature conservation, and water management, and that its impacts will inevitably be felt more in the future.

By analysing the scientific literature, it is possible to find papers about old and new missing data reconstruction methodologies that are claimed by authors as “optimal”. Such papers are commonly structured as follows: an incomplete dataset, a method declared by the authors as the best, at least a couple of benchmark methods and a fine statistical analysis to confirm the claim. No one seems to realize about a paradoxical situation: often a benchmark in one paper, i.e., the weak method to be improved, becomes the best method in another paper. There is a straightforward explanation that solves the paradox: the optimality of a missing data reconstruction method must be considered not absolute but appropriately related only to the considered dataset. However, the crucial question to address is as follows: if the best absolute method does not exist, how can a suitable reconstruction method be selected at the start of the process of data reconstruction? The common approach is to select the most commonly used method, or one of the most recently introduced methods. However, the consequences of Wolpert theorem suggest a possible way to follow: by applying a suite of reconstruction methods and selecting the best method according to the rank of similarity between the incomplete dataset and the completed dataset. Obviously, the best-found method downstream of the proposed procedure cannot be considered the best absolute method but is the best with respect to the dataset used. In the present study, three methods were selected and compared, namely mice, missForest and norm; these methods have different technical characteristics, e.g. parametric vs. nonparametric. This approach was used to attempt to grasp, at most, the structure underlying the incomplete dataset. The goal of the proposed methodology is not to simply fill in the gaps in the incomplete dataset but accomplish this goal by maximizing confidence in the conclusive statistical analysis and in its results.

6 Conclusions

The present research attempts to show how assessments of high quality and consistency with temperature data are important for underpinning any analyses. This study aimed to explore the minimum temperature trend by means of a gap-filled database of 8 temperature series collected over a 30-year period (1990–2019) in the Calabria region in southern Italy and to provide gap filling of daily temperature data for a reliable trend analysis of frost days in the Calabria region (southern Italy).

The following conclusions can be drawn:

  • the performances of the gap-filling technique in estimating daily observed temperature data is composed of strict criteria (specific tests), although it allows adjustments by expert users, according to their knowledge of the local climate. Specifically, the percentages of missing data were assessed. If all of the missing data were reconstructed, the possible patterns among missing data, the correlation analysis between sites and target sites, the mechanisms responsible for data value losses, and the selection of a group of suitable missing data recovery methods that must be compared to safely choose the best data reconstructionwere assessed to determine the best reconstruction strategy;

  • after the missing data reconstruction, the Mann-Kendall nonparametric test showed a decreasing trend in the number of frost days for six out of the eight series, confirming an overall increase in the minimum temperature in the study area;

  • that outcome can be interpreted as an actual measurement of the climate change effect over the region of interest;

  • this approach includes novel and effective methodologies that, under suitable boundary conditions, are capable of reconstructing the missing data.