# Climatic Variations in Macerata Province (Central Italy)

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## Abstract

**:**

## 1. Introduction

^{2}(Figure 1). Macerata is bordered by the province of Perugia (Umbria Region) to the west, by the Adriatic Sea, an arm of the Mediterranean Sea, to the east, and by three other provinces in the same Region, Ancona to the north, Fermo to the south and Ascoli Piceno to the southwest.

## 2. Materials and Methods

#### 2.1. Quality Control Procedures

#### 2.2. Basic Cartography and Correlation of Independent Variables

^{2}) [23], with an assessment of the goodness of correlation represented by the calculation of the standard error of the mean and the F-test. These 3 parameters can explain the relation between the topographic variables and temperature or precipitation; in fact the R

^{2}adjusted (R

^{2}

_{adj}) shows the amount of variation explained by the estimated regression line [28]:

^{2}. The variance is an index of variability and it is expressed by the formula:

#### 2.3. Geostatistical Analyses

- Geostatistics is the branch of statistics that is deals with the analysis and interpretation of geographical data. In this case the ordinary cokriging (with one or more independent variables) was chosen after a cross-validation assessment with simple cokriging, kriging (ordinary and simple), empirical Bayesian kriging and universal cokriging. Cokriging and in particular ordinary cokriging was the method that minimized the error (in terms of Mean Error, RMSE, Mean Standardized error, RMSSE, Mean standard error) more than all the others. The Ordinary co-kriging, [29] can be considered a particular case of the universal cokriging, in which the residuals mean is assumed constant and unknown.$${Z}_{OCK}\left(u\right)={\displaystyle \sum}_{{\alpha}_{1}=1}^{{n}_{1}\left(u\right)}{\lambda}_{{\alpha}_{1}}^{OCK}\left(u\right){Z}_{1}\left({u}_{{\alpha}_{1}}\right)+{\displaystyle \sum}_{{\alpha}_{2}=1}^{{n}_{2}\left(u\right)}{\lambda}_{{\alpha}_{2}}^{OCK}\left(u\right){Z}_{1}\left({u}_{{\alpha}_{2}}\right)$$${\lambda}_{{\alpha}_{1}}^{OCK}\left(u\right)and{\lambda}_{{\alpha}_{2}}^{OCK}\left(u\right)$ = weights of the data assigned to ${Z}_{1}\left({u}_{{\alpha}_{1}}\right)\text{}\mathrm{and}{Z}_{1}\left({u}_{{\alpha}_{2}}\right)$ and varies between 0 and 100%.${Z}_{1}\left({u}_{{\alpha}_{1}}\right)and{Z}_{1}\left({u}_{{\alpha}_{2}}\right)$ = regionalized data at a given location, primary and secondary data.
- Simple co-kriging, [30] is used when the mean is stationary and the residuals mean is considered a global constant and known in the whole study area, this method can be good only if there are a large number of sample points.$${Z}_{SCK}\left(u\right)-{\mu}_{1}={\displaystyle \sum}_{{\alpha}_{1}=1}^{{n}_{1}\left(u\right)}{\lambda}_{{\alpha}_{1}}^{SCK}\left(u\right)\left[{Z}_{1}\left({u}_{{\alpha}_{1}}\right)-{m}_{1}\right]+{\displaystyle \sum}_{{\alpha}_{2}=1}^{{n}_{2}\left(u\right)}{\lambda}_{{\alpha}_{2}}^{SCK}\left(u\right)\left[{Z}_{1}\left({u}_{{\alpha}_{2}}\right)+{m}_{2}\right]$$${m}_{1}\mathrm{and}{m}_{2}$ = mean of the primary and secondary data
- Universal Co-Kriging, [31] a generalization of the ordinary cokriging, is used when the mean isn’t stationary, i.e., if there is a trend, and the residual isn’t correlated to the trend (stationarity of the residuals).$${Z}_{UCK}\left(u\right)={\epsilon}_{1}+{\epsilon}_{2}+{\displaystyle \sum}_{{\alpha}_{1}=1}^{{n}_{1}\left(u\right)}{\lambda}_{{\alpha}_{1}}^{UCK}\left(u\right){Z}_{1}\left({u}_{{\alpha}_{1}}\right)+{\displaystyle \sum}_{{\alpha}_{2}=1}^{{n}_{2}\left(u\right)}{\lambda}_{{\alpha}_{2}}^{UCK}\left(u\right){Z}_{1}\left({u}_{{\alpha}_{2}}\right)$$${\epsilon}_{1}\mathrm{and}{\epsilon}_{2}$ = mean of the residuals in the primary and secondary variable.

## 3. Results

#### 3.1. Data Quality Control

#### 3.2. Assessment of Correlation between Topographical and Climatic Variables

^{2}

_{adj}; st. error and F-test).

#### 3.3. Interpolation and Climate Change Analysis

- Root Mean Square Error (RMSE)—the standard deviation between observed and predicted values: this parameter allows an assessment of the prediction errors for different weather stations. The value of RMSE should be the smallest possible and similar to the average standard error (SEM):$$\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{[\widehat{Z}\left({s}_{i}\right)-z\left({s}_{i}\right)]}^{2}}{n}}$$$\widehat{Z}\left({s}_{i}\right)$ = measured value at position ${s}_{i}$;$z\left({s}_{i}\right)$ = predicted value at position ${s}_{i}$;$n$= number of weather stations;$\widehat{\sigma}$= standard deviation of the population.
- Average Standard Error (SEM)—this statistical tool is known from the mean and it is used to estimate the standard deviation of a sampling distribution. A value close to zero and similar to RMSE represents a very low error in the estimation of the variability of the sampling distribution.$$\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\widehat{\sigma}}^{2}\left({s}_{i}\right)}{n}}$$
- Mean Standardized Error (MSE)—is similar to the mean error and calculates the difference between measured and predicted values; however, MSE values aren’t related to single variables, but it can be used to compare different variables.$$\frac{{{\displaystyle \sum}}_{i=1}^{n}[\widehat{\widehat{Z}}\left({s}_{i}\right)-z\left({s}_{i}\right)]/\widehat{\sigma}\left({s}_{i}\right)}{n}$$
- Root Mean Square Standardized Error (RMSSE)—allows assessment of the goodness of prediction models. It is desirable to have a value close to 1. If the value of RMSSE is lower than 1 the variability is overestimated, otherwise it is underestimated. This is a dimensionless statistical tool.$$\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{[\widehat{Z}\left({s}_{i}\right)-z\left({s}_{i}\right)/\widehat{\sigma}\left({s}_{i}\right)]}^{2}}{n}}$$

## 4. Discussion

## 5. Conclusions

- Homogenization and validation of about 80 years of climatic data relating to precipitations and temperatures, applying a new method [8].
- Identification of altitude as the geotopographic variable most closely related to temperature and precipitation.
- Interpolation of temperature and precipitation data for the province of Macerata with geostatistical techniques.
- Assessment of space and the average climate change between the standard periods 1931–1960 and 1991–2014.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Meanannual temperature (

**a**) and meanannual precipitation (

**b**) in the period 1991–2014 for Macerata province.

**Figure 6.**Interpolation annual mean of the 3 periods (1931–1960/1961–1990/1991–2014) for temperature (

**a**) and precipitation (

**b**).

**Figure 9.**Monthly temperature variations (maximum, mean and minimum) between the periods 1991–2014 and 1931–1960.

**Figure 10.**Trend of monthly variations in precipitation between the periods 1991–2014 and 1931–1960 for precipitations.

**Table 1.**Comparison between 3 interpolation methods (Inverse Distance Weighting (IDW), empirical Bayesian kriging (EBK) and Ordinary Co-Kriging).

Statistical Quality Parameters | IDW | EBK | Co-Kriging |
---|---|---|---|

Regression function | 0.7x + 3.2 | 0.7x + 2.7 | 0.9x + 1.3 |

Mean | −0.5278 | −0.2443 | −0.2371 |

Root-mean-square | 2.0862 | 1.7824 | 1.4917 |

Mean standardized | −0.0628 | −0.0850 | |

Root-mean-square-standardized | 0.8922 | 0.9547 | |

Mean standard error | 2.1955 | 2.9028 |

Regr. Stats for T | Alt.-yrs T 1931–2014 | Dist. Sea-yrs T 1931–2014 | Lat.-yrs T 1931–2014 | Dist. River-yrs T 1931–2014 | Aspect-yrs T1931–2014 | Dist. cre.-yrs T 1931–2014 |
---|---|---|---|---|---|---|

R^{2}_{adj} | 0.84 | 0.46 | 0.44 | 0.06 | −0.01 | −0.02 |

Std error | 0.76 | 1.40 | 1.43 | 1.86 | 1.92 | 1.93 |

Sign. F | 8.31 × 10^{−18} | 4.29 × 10^{−07} | 9.03 × 10^{−07} | 0.07 | 0.49 | 0.84 |

Regr. Stats for P | Alt.-yrs P 1931–2014 | Dist. Sea-yrs P 1931–2014 | Lat.-yrs P 1931–2014 | Dist. River-yrs P 1931–2014 | Aspect-yrs P 1931–2014 | Dist. cre.-yrs P 1931–2014 |
---|---|---|---|---|---|---|

R^{2}_{adj} | 0.70 | 0.69 | 0.26 | 0.07 | −0.02 | −0.02 |

Stderror | 102.32 | 103.50 | 159.80 | 178.40 | 187.33 | 187.39 |

Sign. F | 6.92 × 10^{−13} | 1.14 × 10^{−12} | 2.24 × 10^{−4} | 0.04 | 0.75 | 0.79 |

**Table 4.**Comparison between 3 interpolation methods ordinary cokriging (OCK), simple cokriging (SCK) and universal cokriging (UCK), for annual average precipitation 1991–2014.

Statistical Quality Parameters | OCK | SCK | UCK |
---|---|---|---|

Regression function | 0.7x + 291 | 0.7x + 278 | 0.7x + 298 |

Mean | 0.5272 | −37.2186 | −5.0803 |

Root-mean-square | 92.2624 | 96.5024 | 103.5851 |

Mean standardized | 0.0054 | −0.5402 | −0.0339 |

Root-mean-square-standardized | 0.9748 | 1.3718 | 0.9578 |

Mean standard error | 98.2177 | 75.9468 | 110.2958 |

**Table 5.**Period 1961–1990, statistical indicators for interpolations of maximum, mean and minimum temperatures.

P. 1961–1990 | RMSE | MSE | RMSSE | ASE | P. 1961–1990 | RMSE | MSE | RMSSE | ASE |
---|---|---|---|---|---|---|---|---|---|

Annual mean | 1.89 | −0.15 | 1.14 | 2.05 | Av. January | 1.36 | −0.13 | 1.01 | 1.62 |

Av. February | 1.71 | −0.14 | 1.13 | 1.89 | Av. March | 1.91 | −0.17 | 1.16 | 2.01 |

Av. April | 2.19 | −0.16 | 1.17 | 2.25 | Av. May | 2.39 | −0.16 | 1.21 | 2.58 |

Av. June | 2.21 | −0.17 | 1.16 | 2.31 | Av. July | 2.17 | −0.17 | 1.16 | 2.31 |

Av. August | 2.00 | −0.17 | 1.17 | 2.03 | Av. September | 2.38 | −0.14 | 1.13 | 2.62 |

Av. October | 1.83 | −0.15 | 1.08 | 1.95 | Av. November | 1.55 | −0.13 | 1.04 | 1.71 |

Av. December | 1.54 | −0.11 | 1.02 | 1.91 | |||||

Max ann. mean | 3.05 | −0.18 | 1.14 | 3.04 | Max January | 2.15 | −0.17 | 1.09 | 2.22 |

Max February | 2.70 | −0.19 | 1.12 | 2.70 | Max March | 3.05 | −0.20 | 1.23 | 2.95 |

Max April | 3.23 | −0.18 | 1.17 | 3.16 | Max May | 3.54 | −0.18 | 1.23 | 3.54 |

Max June | 3.44 | −0.20 | 1.22 | 3.30 | Max July | 3.56 | −0.20 | 1.20 | 3.43 |

Max August | 3.51 | −0.19 | 1.18 | 3.31 | Max September | 3.77 | −0.17 | 1.17 | 3.75 |

Max October | 2.97 | −0.19 | 1.12 | 2.83 | Max November | 2.42 | −0.17 | 1.09 | 2.38 |

Max December | 2.35 | −0.16 | 1.04 | 2.57 | |||||

Min ann. mean | 1.30 | −0.04 | 1.08 | 1.44 | Min January | 0.88 | −0.05 | 0.87 | 1.18 |

Min February | 0.97 | −0.03 | 0.93 | 1.33 | Min March | 1.15 | −0.07 | 0.99 | 1.32 |

Min April | 1.40 | −0.06 | 1.15 | 1.53 | Min May | 1.68 | −0.07 | 1.09 | 1.87 |

Min June | 1.60 | −0.05 | 1.14 | 1.61 | Min July | 1.86 | −0.05 | 1.18 | 1.69 |

Min August | 1.66 | 0.11 | 1.20 | 1.46 | Min September | 1.67 | −0.03 | 1.14 | 1.79 |

Min October | 1.36 | −0.01 | 1.17 | 1.41 | Min November | 1.20 | −0.01 | 1.12 | 1.28 |

Min December | 1.03 | −0.05 | 0.97 | 1.31 |

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**MDPI and ACS Style**

Gentilucci, M.; Barbieri, M.; Burt, P.
Climatic Variations in Macerata Province (Central Italy). *Water* **2018**, *10*, 1104.
https://doi.org/10.3390/w10081104

**AMA Style**

Gentilucci M, Barbieri M, Burt P.
Climatic Variations in Macerata Province (Central Italy). *Water*. 2018; 10(8):1104.
https://doi.org/10.3390/w10081104

**Chicago/Turabian Style**

Gentilucci, Matteo, Maurizio Barbieri, and Peter Burt.
2018. "Climatic Variations in Macerata Province (Central Italy)" *Water* 10, no. 8: 1104.
https://doi.org/10.3390/w10081104